Book of Abstracts
Operational Research Practice
in Africa Conference 2015
April 20-22, Algiers
Conference Sponsors:
Preamble:
ORPA conferences are focused on the use of Operations Research (OR) in Africa both in
academic as well as application point of view. The conference is intended to bring academics to
discuss and present some of the most recent developments in the field of OR. It is also intended
as a vehicle to bring industry and the academic world together in order to produce
relationships that may result in beneficial implementations of OR in practice. We invite
submissions covering topics related to OR for development as well as OR applications in
government and industry. All of these applications are fundamentally based on theoretical OR
and Optimization topics also in the scope of the conference. Non-exhaustive lists of such topics
are: Combinatorial Optimization, Combinatorics, Integer Programming, Multicriteria
Optimization, Global Optimization, Stochastic Programming, Scheduling, Facility Location
Problems, etc.
History:
Operational Research Practice in Africa (ORPA) has emerged as an initiative aimed at promoting
the use of operational research approaches in Africa both in academic as well as application
point of view. The conference is intended to bring academics to discuss, and present some of
the most recent developments in our field.

The process began in 2005 with the first ORPA conference which was successfully
launched in Ouagadougou, Burkina Faso with the supports of EURO and IFORS.

ORPA/CAPPS 2006 (Improving Governance & Enhacing Policymaking in Africa) was held
in London. It aims to promote the application of OR in diverse areas of public
policymaking and organisational decision-making with a view to enhancing good
governance, sustainable development and socio-economic development in Africa.

ORPA'2007, was jointly organized with the Operations Research Society of South Africa
(ORSSA) and held September 10 – 14, 2007 in Cape Town, RSA. Jeff Camm (of the
University of Cincinnati College of Business Administration). James Cochran represents
INFORMS at this conference and document this experience in a forthcoming ORMS
Today article. This conference also featured the first ORPA/INFORMS/EURO/IFORS
Teaching Effectiveness Colloquium that was cosponsored by INFORMS. Partial support
for this conference was provided by INFORMS and EURO.

The ORPA'2008 Edition was held jointly with the INFORMS 2008 annual meeting in
Washington DC. The conference is focused on the use of Operations Research (OR)
in: Urban Transportation Management Issues in Africa and Water Resource
Management Issues in Africa.

ORPA'2010 was held in the Cheikh Anta Diop University for March 18 -20, 2010 with a
special focus in : OR in Transport and Logistic, OR in Water Resources Management, OR
in Energy and Environment Management.

In 2013, a workshop dealing with Transportation Network Design and Activity Location
had been organized in Dakar.
More information on the
link http://www.orpagroup.net/
ORPA
history
can
be
found
in
the
following
Program Committee:
Organizing Committee (USTHB):
 Serigne GUEYE (President), LIA, UAPV, France
 Mohamed El-Amine CHERGUI
(President)
 Hacène BELBACHIR, RECITS, USTHB, Algeria
 Jean-Charles BILLAUT, LI, Uni.of Tours, France
 Mourad BOUDHAR, RECITS, USTHB, Algeria
 Mohamed El-Amine CHERGUI, RECITS, USTHB, Algeria
 James COCHRAN, Louisiana Tech. Uni., USA
 Fayçal HAMDI, RECITS, USTHB, Algeria
 Hans W. ITTMANN, HWI Consulting, South Africa
 Hamamache KHEDDOUCI, LIRIS, Uni. of Lyon 1, France
 Ridha MAHJOUB, LAMSADE, Uni.of Paris, France
 Mozart MENEZES, Uni. of Calgary, Canada
 Philippe MICHELON, LIA, Uni. of Avignon, France
 Miloud MIHOUBI, RECITS, USTHB, Algeria
 Babacar M. NDIAYE, Uni. C.A.D. of Dakar, Senegal
 Ammar OULAMARA, Uni. of Lorraine, France
 Maurice POUZET, Uni.of Lyon 1, France
 Djamal REBAINE, UQAChicoutimi, Canada
 Celso RIBEIRO, Uni. Federal Fluminense, Brazil
 Ourida SADKI, RECITS, USTHB, Algeria
 Djenat SEDDIKI-MERAD, RECITS, USTHB, Algeria
 Theodor STEWART, Uni. of Capetown, South Africa
 Jacques TEGHEM, Mons Uni., Belgium
 Claude YUGMA, E.M. de St-Etienne, France
 Abdelhakim AÏTZAÏ
 Karim AMROUCHE
 Hacène BELBACHIR
 Mourad BOUDHAR
 Med Amine BOUTICHE
 Fayçal HAMDI
 Amina HANED
 Nawel KAHOUL
 Wafaa LABBI
 Abdelhak MEZGHICHE
 Mohamed YAGOUNI
Preface
The sixth edition of the international conference “Operations Research Practice in
Africa” (ORPA'2015) will take place at the University of Science and Technology Houari
Boumediene (USTHB) as part of the scientific events of 2015.
This conference has become an institution for the international academic community after
its success during the previous editions (ORPA'2005 Ouagadougou Burkina Faso, ORPA'2006
London (England), ORPA'2007 Cape Town (RSA), ORPA'2008 Washington DC (USA) and
ORPA'2010 Dakar (Senegal).
This scientific event aims to gather researchers in the field of Operations Research (OR)
and its applications, to exchange experiences on the practice of this discipline. It also links the
efforts of both academics and practitioners to ease the acquisition and the transmission of the OR
tools to the different user sectors and illuminates the progress in this field.
The ORPA'2015 conference will be held on April, 20th, 21st and 22nd coinciding with the 43rd
anniversary of USTHB University, expressing gratitude and recognition to those who have
contributed to the development of education and research at USTHB, in Mathematics in general
and especially in Operation Research.
The organization of the scientific event by the Laboratory of Operations Research,
Combinatorics, Theoretical Computer Science and Stochastic Methods (RECITS) of the Faculty
of Mathematics at USTHB is the result of the efforts as well as the commitment of young
researchers.
The project of organizing ORPA conference at USTHB began to germinate during Dakar
Edition in 2010. It is finally concretized with the contribution of eminent professors of the
scientific committee chaired by my friend Serigne Gueye, co -president of ORPA group, and the
efforts of the local organizing committee. I would like to thank all the members of both
committees for making the project work successfully. I would also like to thank all the
participants of the ORPA 2015 conference and especially our guest speakers who have agreed to
make their trip to Algiers and take part in the event, despite their tight schedules.
The project would not have been possible without the significant contributions of our
sponsors; I thank them for the support they have provided to carry out the various tasks in
organizing the conference.
President of organizing committee of ORPA'2015 Conference
Dr Mohamed El-Amine Chergui
Program ORPA'2015
Monday, April 20
8:00-9:00
Registration
9:00-9:30
Opening Ceremony
9:30-10:30
Combinatorics
Metaheuristic
Room 2
Room 3
Stochastic
Programming
Room 1
10:30-11:00
11:00-12:20
Plenary 1: Ridha Mahjoub, Network Survivability and Polyhedral Analysis
Chair: Serigne Gueye
Tea-Coffee Break
Contributed Paper Sessions
Chair: Abdelhakim AitZai
Mohamed Lakehal
A Tabu Search for a Bicriteria Scheduling Problem of Job-Shop with Blocking
Karima Bouibede
A Tabu Search and a Genetic Algorithm for solving a Bicriteria Parallel Machine Scheduling Problem
Wahiba Bouabsa
Some asymptotic normality result of k-Nearest Neighbour estimator
Résolution collective des problèmes de l'optimisation combinatoire ou la collaboration et la
coopération des métaheuristiques
Mohammed Yagouni
Chair: Abdelhafid Berrachedi
Mourad Rahmani
Oussama Igueroufa
Moussa Ahmia
Athmane Benmezai
Chair: Fayçal Hamdi
Wissam Bentarzi
Fares Ouzzani
Metaheuristic
Room 1
15:40-16:40
Periodic Integer-Valued GARCH
On the estimation of mixture periodic integer-valued ARCH models
Ines Brahimi
On the Inference of Periodic Mixture Generalized Autoregressive Conditional Heteroscedastic Model
Nadia Boussaha
Estimation des paramètres d'un modèle à volatilité stochastique périodique
Lunch Break
12:30-14:30
14:30-15:30
A three-term recurrence relation for computing Cauchy numbers
Generalized catalan numbers based on generalized pascal triangle
Unimodal rays in the q-ordinary bisnomial coefficient
The q-analogue of Fibonacci and Lucas sequences
Plenary 2: Hans W. Ittmann, Freight Transport Planning and Modelling - its application within a rail environment
Chair: Adnan Yassine
Contributed Paper Sessions
Chair: Mohammed Yagouni
Asma Boumesbah
An approximation of the Pareto frontier for the multi-objective spanning tree problem
Karim Amrouche
Complexity results of a chain reentrante shop with an exact time lag
Souad Larabi Marie-Sainte
Solving Real World University Examination Timetabling using Tribes Particle Swarm Optimization
Room 3
Transport
Combinatorial
Optimization
Room 2
Chair: Meriem Mechebbek
Oumar Sow
Data Envelopment Analysis with presolve as a decision making tool. Dakar Dem Dikk case study
Baldé Mouhamadou Amadou A new Heuristic method for Transportation Network and Land Use Problem
Ishak Khelassi
Chair: Karima Bouibede
Un modèle multi-objectif pour le choix de l'itinéraire d'un tramway
Mohamed El Amine Badjara
Contribution à la résolution du problème du stable multi-objectif
Amirouche Bourahla
Amélioration d'une méthode exacte pour la recherche des solutions entières efficaces
Bi-objective branch and bound algorithm to minimize total tardiness and system unavailability on
single machine problem
Tea-Coffee Break
Contributed Paper Sessions
Asmaa Khoudi
16:40:16:50
16:50-17:50
Applic. OR
Transport
Room 2
Room 3
Stochastic
Programming
Room 1
Chair: Babacar M. Ndiaye
Manel Boutouis
Nacim Nait Mohand
Lamia Meziani
Chair: Isma Bouchemakh
Omar Mosbah
Sécurité des aliments : quelle règle de responsabilité? Une approche par la théorie des jeux.
Production agro-alimentaire dans les pays en développement: qualité ou quantité? Un modèle
d'analyse
Logistique, concurrence oligopolistique et qualité des produits alimentaires
La dynamique des d'épidémies dans un réseau de petit monde
Mohammed Sbihi
A Mixed Integer Linear Programming Model for the North Atlantic Aircraft Trajectory Planning
Idres Lahna
Chair: Ourida Sadki
A game-based algorithm assigning selfish users on a road network
Nesrine Zidani
Approximation of multiserver retrial queues by truncation technique
Nadjet Stihi
Systèmes d'attente avec arrivées négatives et leurs applications dans les réseaux de neurones
Nabil Bacha
A condition based maintenance model using Bayesian control chart
Tuesday, April 21
8:30-9:30
9:30-10:30
Room 1
Graph Theory
Global Optimization
Room 3
12:30-14:30
14:30-18:00
Stochastic
Programming
Room 2
10:30-11:00
11:00-12:20
Plenary 3: Adnan Yassine, Optimisation des flux physiques sur un terminal roulier
Chair: Mourad Boudhar
Plenary 4: Sheetal Silal, Hitting a moving target: Analysing epidemiology using Operational Research
Chair: Imed Boudabbous
Tea-Coffee Break
Contributed Paper Sessions
Chair: Fatima Affif Chaouche
Hakim Harik
On bandwidth of a n-dimensional grid
Mohamed Amine Boutiche
Forwarding Indices of Some Graphs
Fatma Messaoudi
Some applications of the double and paired-dominating polynomials
Achemine Farida
New Concepts of Equilibrium in Extensive Finite Game with Uncertain Variables
Chair: Nawel Kahoul
Aicha Bareche
An optimal approximation of the GI/M/1 model with two-stage service policy
Gyongyi Bankuti
Animation of the Goal Function of Classical Parametric Problems with an Interactive Maple Tool
Adda Ali-Pacha
Approximation des Equations Différentielles des Attracteurs Chaotiques Appliquée à la Cryptographie
Belabbaci Amel
Une méthode de séparation pour l'optimisation exacte d'une fonction quadratique strictement
concave
Chair: Ali Berrichi
Salima Amrouche
Fatima Bellahcene
Baha Alzalg
Zahia Bouabbache
Multi-Objective Stochastic Integer Linear Programming
On the Solution of the Multiobjective Maximum Probability Problem
Stochastic second-order cone programming: Applications and algorithms
Optimal contol of dynamic systems with random input
Lunch Break
Social Program (visit to Casbah of Algiers)
Wednesday, April 22
8:30-9:30
9:30-10:30
Combinatorics
Global
Optimization
Room 2
Room 3
Stochastic
Programming
Room 1
10:30-11:00
11:00-12:20
12:30-14:30
Plenary 5: Imed Boudabbous, Indecomposable tournaments and critical vertices
Chair: Hacène Belbachir
Plenary 6: Hans W. Ittmann, City Logistics and Urban Freight Transport Challenges in Developing Countries
Chair: Sheetal Silal
Tea-Coffee Break
Contributed Paper Sessions
Chair: Sihem Bekkai
El-Sedik Lamini
Opitimisation de la largeur bit pour l'evaluation des polynômes
On complexity analysis of interior point method for semidenite programming based on a new kernel
Imene Touil
function
Mohamed Kecies
Application de la méthode du point fixe dans le cas p-adique
Mohammed Belloufi
Global convergence properties of the HS conjugate gradient method
Chair: Miloud Mihoubi
Zakaria Chemli
Shifted domino tableaux and the super shifted plactic monoid
Assia Medjerredine
Stirling and Bell Numbers of Some Join Graphs
Assia Fettouma Tebtoub
Successive associated stirling numbers
Amine Belkhir
Some identities for the bivariate hyperbonacci polynomials
Chair: Djenat Seddiki
Asymptotic behavior of a Local linear estimation of conditional hazard rate function for truncated
Ourida Sadki
data
Samra Dhiabi
Kernel estimator of conditional hazard rate function for associated censored data
Sara Stihi
Convergence de densité spectrale dans l'ensemble de Wishart
Selma Meradji
Les notations de base et les préliminaires de G-calcule stochastique
Lunch Break
Contributed Paper Sessions
Combinatorics Applic. RO
Room 2
Room 3
Combinatorial
Optimization
Room 1
14:30-15:50
Chair: Sadek Bouroubi
Amar Oukil
Inverse optimization: An efficient tool for resource reallocation in the farming sector
Abdelkader Mendas
MCDM and GIS to identify land suitability for agriculture
Chair: Mohamed Amine Boutiche
Ali Chouria
Bell polynomials in combinatorial Hopf algebras
Fariza Krim
Linear recurrence sequence associated to rays of negatively extended Pascal triangle
Imad Eddine Bousbaa
Associated r-Stirling numbers
Meriem Tiachachat
The r-Whitney numbers and the value of the high order Bernoulli polynomials
Chair: Amina Haned
Algorithmes exacts et approchés pour le problème d'ordonnancement préemptif sur machines
Ryma Zineb Badaoui
parallèles avec délais de transport
Ordonnancement sur deux machines avec contraintes de concordance : machines identiques et
Amine Mohabeddine
uniformes
Nour El Houda Tellache
Flow-shop scheduling problem with conflict graphs
Nadjet Meziani
Problème du flowshop à deux machines avec des opérations couplées sur la première machine
15:50-16:00
Tea-Coffee Break
16:00-16:30
Closing Ceremony
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Plenary sessions
Hans W. Ittmann
Freight Transport Planning and Modelling - its application within a rail environment
11
Adnan Yassine
Optimisation des flux physiques sur un terminal roulier …………………………….......
12
Ali Ridha Mahjoub
Network Survivability and Polyhedral Analysis ………………………………………….
13
Sheetal Silal
Hitting a moving target: Analysing epidemiology using Operational Research ………..
14
Imed Boudabbous
Indecomposable tournaments and critical vertices ………………………………………
15
Hans W. Ittmann
City Logistics and Urban Freight Transport Challenges in Developing Countries …....
16
Parallel sessions
Abdelkader Mendas
MCDM and GIS to identify land suitability for agriculture ………………………...…..
17
Nadjet Stihi & Natalia Djellab
Systèmes d'attente avec arrivées négatives et leurs applications dans les réseaux de
neurones ……………………………………………………………………………………..
18
M. Sbihi, O. Rodionova, D. Delahaye & M. Mongeau
A Mixed Integer Linear Programming Model for the North Atlantic Aircraft
Trajectory Planning ………………………………………………………………………...
19
S. Stihi, H. Boutabai & S. Meradji
Convergence de densité spectrale dans l'ensemble de Wishart ………………………….
21
S. Meradji, H. Boutabai & S. Stihi
Les notions de base et préliminaire de la G-calcule stochastique ………………………..
23
Nesrine Zidani & Natalia Djellab
Approximation of multiserver retrial queues by truncation technique …………………
25
Adda Ali-Pacha & Naima Hadj-Said
Approximation des Equations Différentielles des Attracteurs Chaotiques Appliquée à
la Cryptographie ……………………………………….…………………………………...
27
5
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Gy Bankuti
Animation of the Goal Function of Classical Parametric Problems with an Interactive
Maple Tool …………………………………………………………………………………..
28
Mohammed Belloufi & Rachid Benzine
Global convergence properties of the HS conjugate gradient method ………………….
30
Souad Larabi Marie-Sainte
Solving Real World University Examination Timetabling using Tribes Particle
Swarm Optimization ……………………………………...………………………………...
31
Nour El Houda Tellache & Mourad Boudhar
Flow-shop scheduling problem with conflict graphs ……………………………………..
33
Asma Boumesbah & Mohamed El-Amine Chergui
An approximation of the Pareto frontier for the multi-objective spanning tree
problem ……………………………………………………………………………………...
35
Meriem Tiachachat & Miloud Mihoubi
The r-Whitney numbers and the value of the high order Bernoulli polynomials ………
37
Ourida Sadki & Karima Zerfaoui
Asymptotic behavior of a Local linear estimation of conditional hazard rate function
for truncated data …………………………………………………………………………..
39
Omar Mosbah, khadidja Khalloufi, Yamina Baara, Abd el lah Dilem, Omar Harrouz &
Nourdinne Zekri
La dynamique des d'épidémies dans un réseau de petit monde …………………………
41
Samra Dhiabi & Ourida Sadki
Kernel estimator of conditional hazard rate function for associated censored data …..
42
Mohamed Kecies
Application de la méthode du point fixe dans le cas p-adique …………………………...
44
Karim Amrouche, Mourad Boudhar & Farouk Yalaoui
Complexity results of a chain reentrante shop with an exact time lag ………………….
46
Amar Oukil
Inverse optimization, an efficient tool for resource reallocation: application in the
farming sector ………………………...…………………………………………………….
48
Imene Touil & Djamel Benterki
On complexity analysis of interior point method for semidefinite programming based
on a new kernel function …………………………………………………………………...
50
Imad Eddine Bousbaa & Hacène Belbachir
Associated r-Stirling numbers ……………………………………………………………..
52
6
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Farida Achemine & Karima Fahem
New Concepts of Equilibrium in Extensive Finite Game with Uncertain Variables …..
54
Fariza Krim & Hacène Belbachir
Linear recurrence sequence associated to rays of negatively extended Pascal triangle ..
56
Ali Chouria
Bell polynomials in combinatorial Hopf algebras I ………………………………………
58
Lahna Idres & Mohammed Said Radjef
A game-based algorithm assigning selfish users on a road network ……………………
60
Nadjat Meziani, Ammar Oulamara & Mourad Boudhar
Problème du flowshop à deux machines avec des opérations couplées sur la première
machine ……………………………………………………………………………………...
62
Zahia Bouabbache & Mohamed Aidene
Optimal control of dynamic systems with random input ………………………………..
64
Amine Belkhir & Hacène Belbachir
Some identities for the bivariate hyperfibonacci polynomials …………………………...
65
Assia Fettouma Tebtoub & Hacène Belbachir
Successive associated Stirling numbers …………………………………………………...
67
A. Medjerredine, H. Belbachir & M.A Boutiche
Stirling and Bell Numbers of Some Join Graphs …………………………………………
69
Zakaria Chemli
Shifted domino tableaux and the super shifted plactic monoid …………………...…….
71
Lamia Meziani, A.Hakim Hammoudi & Mohammed Said Radjef
Logistique, concurrence oligopolistique et qualité des produits alimentaires ……….....
73
Mohamed El Amine Badjara & Mohamed El-Amine Chergui
Contribution à la résolution du problème du stable multi-objectif ……………………..
75
N. Nait Mohand, M.S Radjef & A. Hammoudi
Production agro-alimentaire dans les pays en développement : qualité ou quantité ?
Un modèle d’analyse ………………………………………………………………………..
77
Athmane Benmezai & Hacène Belbachir
The q-analogue of Fibonacci and Lucas sequences ………………………………………
79
Fatma Messaoudi & Miloud Mihoubi
Some applications of the double and paired-dominating polynomials ………………….
81
Moussa Ahmia & Hacène Belbachir
Unimodal rays in the q-bisnomial coefficient ……………………………………………..
83
7
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Oussama Igueroufa & Hacène Belbachir
Generalized Catalan Numbers Based On Generalized Pascal Triangle ………………..
85
Amel Belabbaci & Bachir Djebbar
Une méthode de séparation pour l'optimisation exacte d'une fonction quadratique
strictement concave …………………………………………………………………………
87
Baha Alzalg
Stochastic second-order cone programming: Applications and algorithms ……………
89
Fatima Bellahcene & Philippe Marthon
On the Solution of the Multiobjective Maximum Probability Problem ………………...
90
Mohammed Yagouni
Résolution collective des problèmes de l’optimisation combinatoire ou la collaboration
et la coopération des métaheuristiques ……………………………………………………
92
Aicha Bareche, Mouloud Cherfaoui & Djamil Aissani
An optimal approximation of the GI/M/1 model with two-stage service policy ………..
94
Manel Zahra Boutouis, Abdelhakim Hammoudi & Wassim Benhassine
Sécurité des aliments : quelle règle de responsabilité ? Une approche par la théorie
des jeux ………………………………………………………………………………………
96
Mourad Rahmani
A three-term recurrence relation for computing Cauchy numbers ……………………..
98
Nadia Boussaha & Fayçal Hamdi
Estimation des paramètres d.un modèle à volatilité stochastique périodique ………….
100
M.A Boutiche
Forwarding Indices of Some Graphs ……………………………………………………...
102
Oumar Sow, Babacar M. Ndiaye & Aboubacar Marcos
Data Envelopment Analysis with presolve as a decision making tool. Dakar Dem Dikk
case study ……………………………………………………………………………………
104
Amirouche Bourahla & Mohamed El-Amine Chergui
Amélioration d'une méthode exacte pour la recherche des solutions entières efficaces .
106
Hakim Harik & Hacène Belbachir
On bandwidth of a n-dimensional grid ……………………………………………………
108
Mouhamadou A.M.T. Baldé, Babacar M. Ndiaye & Serigne Gueye
A new Heuristic method for Transportation Network and Land Use Problem ………..
110
Wahiba Bouabsa, Mohammed Kadi Attouch
Some asymptotic normality result of k-Nearest Neighbour estimator ………………….
111
8
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Amine Mohabeddine, Mourad Boudhar & Ammar Oulamara
Ordonnancement sur deux machines avec contraintes de concordance : machines
identiques et uniformes …………………………………………………………………….
112
Ryma Zineb Badaoui, Mourad Boudhar & Mohammed Dahane
Algorithmes exacts et approchés pour le problème d’ordonnancement préemptif sur
machines parallèles avec délais de transport …………………………………………......
114
Salima Amrouche & Mustapha Moulai
Multi-Objective Stochastic Integer Linear Programming ………………………………. 116
Nabil Bacha, Isabel Lopes & Lino Costa
A condition based maintenance model using Bayesian control chart …………………... 118
Ines Brahimi & Mohamed Bentarzi
On The Inference Of Periodic Mixture Generalized Autoregressive Conditional
Heteroscedastic Model ……………………………………………………………………... 120
Fares Ouzzani & Mohamed Bentarzi
On the estimation of Mixture Periodic integer-valued ARCH models ………………….
122
Wissam Bentarzi & Mohamed Bentarzi
Periodic Integer-Valued GARCH (1,1) Model …………………………………………… 124
Karima Bouibede-Hocine, Drifa Hetak & Mourad Boudhar
A Tabu Search and a Genetic Algorithm for solving a Bicriteria Parallel Machine
Scheduling Problem ………………………………………………………………………... 126
Mohamed Lakehal & Karima Bouibede-Hocine
A Tabu Search for a Bicriteria Scheduling Problem of Job-Shop with Blocking ……...
128
El-sedik Lamini, Samir Tagzout, Hacène Belbachir & Adel Belouchrani
Optimisation de la largeur bit pour l'évaluation des polynômes ………………………... 130
Asmaa Khoudi, Ali Berrichi & Farouk Yalaoui
Bi-objective branch and bound algorithm to minimize total tardiness and system
unavailability on single machine problem ………………………………………………...
132
Ishak Khelassi
Un modèle multi-objectif pour le choix de l’itinéraire d’un tramway ………………….. 134
9
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
10
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Freight Transport Planning and Modellingits application within a rail environment
Hans W. Ittmann
LTS (Africa), University of Johannesburg, Johannesburg
South Africa 0002
[email protected]
20-22 April 2015
Abstract
The movement of freight is an integral part of an economy in a country. Sufficient, appropriate and reliable transport remains an essential element for sustained
economic growth. The production and consumption of goods and services are usually physically separated which requires the distance between the two needs to be
bridged by means of at least one mode of transportation. Providing the necessary
capacity for the different modes of transport requires careful and proper planning.
In this regard the forecasting of future demand is a critical component of the planning function. This paper will endeavour to present a short review of the planning
process, of freight transport modelling and then discuss the use and application of
different models within a rail environment. Various tools have been developed to
assist in the planning of future rail capacity and infrastructure. The paper will also
endeavour to illustrate the importance of an appropriate planning philosophy for
long-term planning.
11
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Optimisation des flux physiques sur un terminal roulier
Adnan Yassine
Laboratoire de Mathématiques Appliquées du Havre
Université du Havre, France
20-22 April 2015
Résumé
l’allocation d’un navire à un poste à quai dans un terminal roulier doit tenir compte de
plusieurs facteurs importants tels que la disponibilité des postes à quai, la compatibilité
entre le navire et le poste à quai (profondeur d’eau, longueur de quai, type de navire,
etc.), certaines règles de priorité concernant les problèmes de sécurité et de l’intérêt
génral, les distances entre les postes à quai et les zones dédiées au stockage de voitures et
les capacités de ces dernières, les règles de gestion adoptées par les différents opérateurs
de stockage, le nombre de voitures et le risque de croisements, etc. Notre objectif dans ce
travail est de développer un système d’aide à la décision pour optimiser les flux physiques
en minimisant à la fois le temps d’attente en mer et aux postes à quai, la distance globale
à parcourir par les voitures sur le terminal et les croisements entre les flux de voitures
sur le terminal.
12
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Network Survivability and Polyhedral Analysis
A. Ridha Mahjoub
LAMSADE, Université Paris-Dauphine, Paris, France
[email protected]
20-22 April 2015
Abstract
For the past few decades, combinatorial optimization techniques have shown
to be powerful tools for formulating, analysing and solving problems arising from
practical decision situations. In particular, the polyhedral approach has been successfully applied for many well known NP-hard problems. The equivalence between
separation and optimization has been behind a big development of this method.
The so-called Branch and Cut method, which is inspired from this equivalence, is
now widely used for obtaining optimal and near-optimal solutions for hard combinatorial optimization problems. In this talk we present these methods, and discuss
some applications to survivable network design models.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Hitting a moving target: Analysing epidemiology
using Operational Research
Sheetal Silal
Department of Statistical Sciences, Faculty of Science
at the University of Cape Town, South Africa
20-22 April 2015
Abstract
Infectious diseases pose the greatest challenge to the health of African people with the leading causes of disease burden being lower respiratory infections,
HIV/Aids, diarrheal diseases and malaria. It is necessary for countries to be able
to manage existing disease levels and to combat epidemics as and when they occur.
The Operational Research armamentarium has within it tools to assist with the
containment of diseases, predicting optimal resource allocation, strategies to minimize the evolution of drug resistance, assessment of new interventions to interrupt
disease transmission, and the assessment of operational feasibility. This talk will
provide an overview of current OR practice in epidemiology across the globe as well
as some of my research in mathematical modelling for the elimination of malaria in
South Africa.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Indecomposable tournaments and critical vertices
Imed Boudabbous
Faculté des Sciences de Sfax,
Département de Mathématiques, Tunisie
20-22 April 2015
Abstract
A tournament T = (V (T ), A(T )) or (V, A) consists of a finite vertex set V with
an arc set A of ordered pairs of distinct vertices satisfying: for x, y ∈ V , with x = y,
(x, y) ∈ A if and only if (y, x) ∈
/ A. For two distinct vertices x and y of a tournament
T, x → y means that (x, y) ∈ A(T ). For x ∈ V (T ) and Y ⊂ V (T ), x → Y (resp.
Y → x) signifies that for every y ∈ Y , x → y (resp. y → x).
Given a tournament T = (V, A), a subset I of V is an interval (or a clan or an
homogeneous subset ) of T provided that for every x ∈ V − I, x → I or I → x.
This definition generalizes the notion of interval of a total order. Given a tournament
T = (V, A), ∅, V and {x}, where x ∈ V , are clearly intervals of T , called trivial intervals.
A tournament is then said to be indecomposable, (or primitive) if all of its intervals
are trivial, and is said to be decomposable otherwise.
Let T be an indecomposable tournament T with V (T ) = ∅. A vertex x of T is
critical if the tournament T − x is decomposable. T is critical if each vertex x of T
is critical. If T admits a unique non critical vertex then T is (-1)-critical. The critical
tournaments are characterized by J.H. Schmerl and W.T. Trotter and the (-1)-critical
tournaments are characterized by H.Belkhechine, I. Boudabbous and J. Dammak.
This work describes the new characterization of the critical and (-1)-critical tournaments.
15
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
City Logistics and Urban Freight Transport Challenges
in Developing Countries
Hans W. Ittmann
LTS (Africa), University of Johannesburg, Johannesburg
South Africa 0002
[email protected]
20-22 April 2015
Abstract
Cities world-wide are growing larger and larger as the trend towards urbanisation
increases almost on a daily basis. Many of those attracted to cities are faced with
unemployment. In an eort to survive from day to day many small businesses are
established by those trying to just exist. This scenario is typical of developing
countries. South Africa is unique in the sense that a rst world economy co-exists
within a developing country with a third world economy. This paper will endeavour
to outline the concept of City Logistics and also address the challenges of urban
freight transport. In addition small business logistics challenges faced by small
businesses in urban areas will be highlighted. The use of Operations Research as a
scientic approach to problems solving to address these challenges, will be presented.
16
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
MCDM and GIS to identify land suitability for
agriculture
A. Mendas
Centre des Techniques Spatiales (Arzew, Algeria)
20-22 April 2015
Abstract
The integration of MultiCriteria Decision Making approaches (MCDM) in a Geographical Information System (GIS) provides a powerful spatial decision support
system which offers the opportunity to efficiently produce the land suitability maps
for agriculture. Indeed, GIS is a powerful tool for analyzing spatial data and establishing a process for decision support. Because of their spatial aggregation functions,
MCDM methods can facilitate decision making in situations where several solutions
are available, various criteria have to be taken into account and decision-makers are
in conflict. The parameters and the classification system used in this work are inspired from the FAO (Food and Agriculture Organization) approach dedicated to a
sustainable agriculture. A spatial decision support system has been developed for
establishing the land suitability map for agriculture. It incorporates the multicriteria analysis method ELECTRE Tri (ELimitation Et Choix Traduisant la REalit)
in a GIS within the GIS program package environment. The main purpose of this
research is to propose a conceptual and methodological framework for the combination of GIS and multicriteria methods in a single coherent system that takes
into account the whole process from the acquisition of spatially referenced data to
decision-making. In this context, a spatial decision support system for developing
land suitability maps for agriculture has been developed. The algorithm of ELECTRE Tri is incorporated into a GIS environment and added to the other analysis
functions of GIS. This approach has been tested on an area in Algeria. A land
suitability map for durum wheat has been produced. Through the obtained results,
it appears that ELECTRE Tri method, integrated into a GIS, is better suited to
the problem of land suitability for agriculture. The coherence of the obtained maps
confirms the system effectiveness.
Keywords: MultiCriteria Decision Analysis, Decision support system, Geographical Information System, Land suitability for agriculture.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Systèmes d’attente avec arrivées négatives et leurs
applications dans les réseaux de neurones.
Stihi Nadjet1 et Djellab Natalia2
1,2
Département de Mathématiques, Laboratoire LaPS, Université d’Annaba
20-22 April 2015
Abstract
Il est apparu ces dernières années dans la littérature des files d’attente, des
travaux portant sur les systèmes et les réseaux de files d’attente caractérisés
par la présence de deux types d’arrivées. D’un côté, les arrivées positives ou
régulières qui ont pour objectif l’occupation du service. D’autre côté, les arrivées négatives, dont la présence dans le système de files d’attente affecte ce
dernier de différentes manières: élimination individuelle, élimination par groupe,
le désastre (la catastrophe),... L’intért porté à cette nouvelle famille de réseaux
de files d’attente avec arrivées négatives, introduite par Gelenbe[2], était motivé
initialement par la modélisation des réseaux de neurones où les arrivées positives
représentent les signaux excitateurs, qui font crotre le potentiel du neurone et
sa tendance produire une impulsion, et inhibiteurs, qui diminuent le potentiel
du neurone et sa tendance produire une impulsion, respectevement. Puis, leurs
domaines d’applications se sont étendus pour toucher d’auters systmes plus complexes comme les réseaux informatiques avec infection par virus [1], élimination
des transactions dans les bases de données[3], les systèmes d’inventaires [4], les
systèmes de télécommunication, le systèmes de production, etc.
Dans ce travail, nous présentons les systèmes d’attente avec arrivées négatives
et leurs applications dans les réseaux de neurones.
References
[1] J.R. Artalejo and A. Gomez-Corall : Computation of the limiting distribution
in queueing systems with repeated attempts and disasters. RAIRO Operations
Research 33.(nro.), 371-382. (1999).
[2] E. Gelenbe : Random neural network with negative and positive signals and
product form solution. Neural Computation 1.(nro.), 502-510. (1989).
[3] E. Gelenbe : Producty form queueing network with negative and positive
customers. Journal of Applied Probability 28.(nro.), 656-663. (1991).
[4] G. Jain and K. Sigman : A Pollaczek-Khintchine formula for M/G/1 queue
with desasters. Journal of Applied Probability 33.(nro.), 1191-1200. (1996).
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
A Mixed Integer Linear Programming Model for the
North Atlantic Aircraft Trajectory Planning
M. Sbihi, O. Rodionova, D. Delahaye, M. Mongeau
ENAC, MAIAA, F-31055 Toulouse, France
Univ de Toulouse, IMT, F-31400 Toulouse, France
20-22 April 2015
Abstract
This paper discusses the trajectory planning problem for flights in the North
Atlantic oceanic airspace (NAT). We develop a mathematical optimization framework in view of better utilizing available capacity by re-routing aircraft. The
model is constructed by discretizing the problem parameters. A Mixed integer
linear program (MILP) is proposed. Based on the MILP a heuristic to solve
real-size instances is also introduced.
1
Context and problem description
The North Atlantic oceanic airspace accommodates air traffic between North
America and Europe. The major traffic flow takes place in two distinct traffic
flows during each 24-hour period due to passenger preference, time zone differences and the imposition of night-time noise curfews at the major airports. The
westbound flow departing from Europe in the morning, and the eastbound flow
departing from North America in the evening. Radar coverage is not, for most
parts, available in the NAT Region. Air Traffic Control (ATC) units monitor
the progress of flights on the basis of pilot position reports and uses procedural
methods. Such procedures are based on an Organized Track System (OTS) (see
[2]). The OTS consists of several, typically 4 to 7, quasi-parallel tracks each of
which represents a sequence of great circles joining successive significant waypoints (WPs). The variability of the wind patterns would make a fixed track
system unnecessarily penalizing in terms of flight time and consequent fuel usage. As a result, an OTS is set up on a diurnal basis for each of the Westbound
and Eastbound flows according to the prevailing winds. Adjacent tracks are separated by 60 Nautical Miles. In vertical direction each track consists of several
flight levels spaced by 1000 feet. The construction of OTS ensures vertical and
lateral separation between aircraft. The longitudinal (in-track) separation in
NAT is assessed in terms of differences in actual and estimated times of arrival
at common WPs; and expressed in clock minutes. Current regulations impose
large separation minima for aircraft on OTS. The time separation between subsequent aircraft following the same track is 10 minutes. When an aircraft wants
to shift from one track to an adjacent track, the separation must be at least 15
minutes with aircraft located on such adjacent track.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Due to the high density of traffic on tracks and these large separation standards, the re-routing maneuver inside OTS can rarely be applied. This leads to
increasing the length and the duration of the flights, as well as to increasing the
congestion in the continental airspace near the exit of the OTS.
The transition from present ATC tools to airborne-based systems will allow
to reduce significantly the present aircraft separation standards: consecutive
aircraft following the same track would be separated by only around 2 minutes,
and an aircraft performing a re-routing to the adjacent track would be separated by around 3 minutes from aircraft on this track. Obviously, with such
reduced separation standards aircraft will be able to change their tracks more
frequently [3]. To show the benefits that can be expected from such a reduction
of separation standards, the problem was investigated in [1] using genetic algorithms. The aim of this study is to give an approach based on mixed integer
linear programming.
2
Contributions
We developed a mathematical optimization framework to better utilize available
capacity by re-routing aircraft based on wind conditions and origin/destination
of flights. The model is constructed by discretizing the problem parameters.
The OTS is represented by a 3D-grid and an MILP path-based formulation is
proposed. The program takes (among others things) as input entry and exit
tracks and track-entry times of flights. The program then assigns a route to
each of these flights explicitly considering separation constraints to minimize
the total cruising time within the OTS. The approach can be used with a variety of other aviation-focused metrics.
When the number of aircraft increases, a scalability problem rises. The proposed
second methodology addresses the scalability problem issue through utilizing a
heuristic method. The heuristic recursively defines, and then solves short-term
(small-size) subproblems, and partially fixes routes. The heuristic was tested on
two test problems based on real air-traffic data and produces near-optimal solutions (typically within 2% of optimality) while standard optimization software
fails to solve such instances.
Acknowledgements
This work has been supported by French National Research Agency (ANR)
through JCJC program (project ATOMIC nANR 12-JS02-009-01).
References
[1] O. Rodionova, M. Sbihi, D. Delahaye, M. Mongeau, North Atlantic Aircraft
Trajectory Optimization. IEEE Transactions on Intelligent Transportation Systems DOI: 10.1109/TITS.2014.2312315 (2014).
[2] NAT Doc 007, Guidance concerning air navigation in and above the North
Atlantic MNPS airspace, 2012th ed., International Civil Aviation Organisation,European and North Atlantic Office of ICAO, (2012).
[3] A. Williams, Benefits assessment of reduced separations in North Atlantic
Organized Track System, CSSI Inc., Advanced Programs, 400 Virginia Ave.
SW, Washington, DC, Tech. Rep., (2005).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Convergence de densité spectrale dans l’ensemble de
Wishart
S. Stihi1 ; H.BOUTABAI
1,2,3
2
et S.MERADJI
3
Département de Mathématiques, Laboratoire LaPS, Université d’Annaba
20-22 April 2015
Abstract
In this work we define the matrices of wishart and their characteristics, we
study also the convergence of the spectral density, then we search to define a
model of a matrix process in the nonlinear part to then calculate the EDS of
the eigenvalues and the eigenvectors.
1
Introduction
The theory of matrices with random entries, originally devised as a tool to
understand and predict the spectra of heavy nuclei for which a detailed account
of the interactions between particles is too complicated, has seen a spectacular
resurgence of interest in recent years, with a number of unexpected and often
surprising applications. While Wigner and Dyson are usually regarded as the
pioneers in the field, John Wishart had already introduced random matrices in
1928 in his studies of multivariate populations. The Wigner-Dyson (Gaussian)
and Wishart ensembles (together with a few others) lie at the core of the classical
world of invariant matrices, characterized by the following main features:
• The joint probability distribution (jpd) of matrix entries, collectively denoted by P [X].
• The joint distribution of the N real eigenvalues P (λ1 , λ2 , ..., λN ) can be
generically written in the Gibbs-Boltzmann form,
P (λ1 , λ2 , ..., λN ) =
1
exp (−H (λ1 , λ2 , ..., λN ))
ZN
with the Hamiltonian H (λ1 , λ2 , ..., λN ) given by:
H (λ1 , λ2 , ..., λN ) =
N
V (λi ) − β
i=1
ln |λj − λk |
j<k
where V (X) a confining potential that depends on the precise form of the
joint distribution of matrix entries P [X]. For example, if the entries of X are
independent, the only allowed potential is quadratic V (X) = βx2 /2, which
correspond to the Gaussian ensembles. If correlations among the entries are
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
allowed, then different potentials (all corresponding to rotationally invariant
weights) are possible. For example, in the Wishart case, V (X) = ∞ for x < 0
(so that all the eigenvalues are non-negative) and V (X) = x/ 2σ 2 − α log x
for x ≥ 0.
ZN is a normalization constant.
2
wishart ensembles
Let X be a real (respectively complex) Gaussian random matrix of size M × N ,
i.e. a random matrix chosen in the space of M ×N real (resp. complex) matrices
according to the law:
1
P (X) dX ≡ exp − T r (X ∗ X) dX
2
where X ∗ is the Hermitian conjugate of X. In the following, we will denote
the real (resp. complex) Wishart ensemble by W β with β = 1 in the real case
(resp.β = 2 in the comple case).
The real (resp. complex) Wishart Ensemble is the ensemble of (N × N )
square matrices of the product form W := X ∗ X where X is a real (resp. complex) Gaussian random matrix of size N × M .They have appeared in many different applications such as communication technology, nuclear physics, quantum
chromodynamics, statistical physics of directed polymers in random media and
non intersecting Brownian motions, as well as Principal Component Analysis of
large datasets.
3 Continuous processes for real and complex
Wishart ensembles
We wish to define here a diffusive matrix process depending on a fictitious time
t > 0 that will converge to the Wishart Ensembles in the limit of large time.
The idea is simply to set
Wt := Xt∗ Xt
where Xt is a real (resp. complex) random matrix process (of size M × N )
following the Ornstein-Uhlenbeck law,
1
dXt = − Xt dt + σdBt
2
where Bt is a real Brownian (resp. complex) random matrix, i.e. a matrix whose entries are given by independent standard Brownian motions. By a
standard Brownian motion, one means a centered (zero-mean) Gaussian process
with covariance function E (Bt Bt ) = min (t, t ).
4
convergence of the spectral density
The spectral measure of the eigenvalues is given by:
ρβN =
N
1 δ (λ − λi )
N
i=1
References
[1] R. Allez, J. P. Bouchaud, S. N. Majumdar, and P. Vivo. , Invariant -Wishart
ensembles, crossover densities and asymptotic
2 corrections to the Mar cenkoPastur law, Arxiv:1209.6171v1 [cond-mat.stat-mech](2012)
[2] V.A. Marcenko and L.A. Pastur , Distribution of eigenvalues for some sets of
random matrices, Mathematics of the USSR-Sbornik, 1, 457 (1967)
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Les notions de base et prliminaire de la G-calcule
stochastique
S. MERADJI
1,2,3
1
; H. BOUTABAI
2
et S. Stihi
3
Département de Mathématiques, Laboratoire LaPS, Université d’Annaba
20-22 April 2015
Abstract
Dans ce travail nous donnons les notations de base et les préliminaires de
la G-espérance , nous introduions aussi la notion de G-mouvement brownien
liée à la distribution G-normale dans un espace d’espérance sous linéaire,nous
donnons également la définition de la G-intégrale stochastique et nous énonçons
la G-formule d’itô pour définir aprés la G-espérance sur l’ensemble des matrices
(les différents ensembles des matrices) et faire le lien avec les EDP et developper
plus le calcul stochastique dans le cadre non linéaire.
1
Introduction
la théorie d’espérance non linéaire a suscité un grand intérêt chez les chercheurs
pour ces applications potentielles dans les problèmes d’incertitude et ces nombreux outils riches, souples et élégants. C’est aussi le point de départ d’une
nouvelle théorie du calcul stochastique qui nous donne un nouvel aperçu pour
caractériser et pour calculer les différents types des risques financiers.
En particulier, Peng a étudié le théorème de représentation d’une espérance
sous-linéaire qui peut être exprimée comme un supremum des espérances linéaires
et a établit la théorie fondamentale de la G−espérance, où G est la fonction
génératrice d’une équation de la chaleur non linéaire. La notion de G−espérance
s’est développée très récemment et a ouvert la voix à l’introduction de variables
aléatoires G−normales, du G−mouvement Brownien et plus généralement des
G−intégrales stochastiques de type Itô, en vertu de laquelle Peng a introduit
également la G−normale distribution ainsi que le théorème de la limite centrale
et le concept du G−mouvement Brownien correspondant. Il a systématiquement
développé le calcul stochastique sous la G−espérance.
2
Généralités sur l’espérance sous linéaire
nous donnons dans cette section les notations de bases et les préliminaires de
l’espérance sous linéaire et les espaces d’espérance sous linéaire associés. On
donne aussi le théorème de représentation de l’espérance sous linéaire et certaines définitions qu’on utilise par la suite à savoir les notions de distribution et
d’indépendances en vertu de l’espérance sous linéaire.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
3 Notion de G−Mouvement Brownien et G−intégrale
d’Itô
On introduit le concept du G-mouvement Brownien et les G-intgrales stochastiques de type It passant par les dfinitions et les proprits ncessaires, ainsi que le
processus a variation quadratique associ.
References
[1]S. Peng , Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Processes and their Applications 118; 22232253 (2008).
[2] H. Boutabia et I. Grabsia : Chaos de Wiener par rapport au G-mouvement
Brownien. Université d’Annaba (2014).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Approximation of multiserver retrial queues by
truncation technique
Zidani Nesrine and Djellab Natalia
20-22 April 2015
Abstract
This paper discusses multiserver retrial queueing system with C servers. An
arriving customer finding a free server enters into service immediately, otherwise
the customer either enters into an orbit to try again after a random amount of
time or leave the system without service. As there are not closed-form solutions
to these systems, approximate methods are required. We use truncated system
that is obtained by modifying the orbit size for an approximation of stationary
of the queue length distribution. Also , we propose the necessary and sufficient
condition for stability of the system and show the convergence of approximation to the original model. An algorithmic solution for the stationary of the
queue length distribution in the truncated system and some numerical results
are presented.
1
INTRODUCTION
Systems with retrial and multiserveur appear in many application domains such
as computer networks and those of telecommunications. A study of this type of
system can be considered complete if the stationary distribution of the stochastic process describing the state of the system or at least an explicit analytical
dependence of the essential characteristics of the system and the initial parameters (arrival rate of primary customers, Service rate, rate of retrial and the
number of servers comprising the service area) is established. To date only a
few results can be considered valid, and this in the case of exponential service.
To overcome this difficulty, we used the approximation methods that allow for
quantitative estimates for certain performance characteristics. In this work, we
focus on the truncation method where the size of the orbit (i.e. the number
of sources of repeated calls is truncated by a constant M). For this purpose,
we consider the system retrial queue M/M/C, obtain the estimation of the stationary distribution of the system state and analyses the convergence of the
approximate solution to the exact solution.
References
[1] J.R Artalejo and A. Gomez-Corral : Retrial Queueing Systems: A Computational Approach. Springer, (2008).
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
[2] S.N. Stepanov, Markov models with retrials : The calculation of stationary
performance measures based on the concept of truncation. Mathematical and
computer Modelling 30, 207-228, (1999).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Approximation des Equations Différentielles des Attracteurs Chaotiques
Appliquée la Cryptographie
Adda ALI-PACHA – Naima HAD -SAID
Université des Sciences et de la Technologie d’Oran USTO,
BP1505 El M’Naouer Oran 31036 ALGERIE
*
Corresponding Author, Email:[email protected], a.alipacha @gmail.com
Résumé :
La résolution de la plupart des équations différentielles requiert donc l'utilisation de méthodes numériques.
Chacune de ces méthodes peut tre appliquée la résolution de la plupart des équations différentielles.
La question qui se pose comment choisir la méthode numérique qui approxime au mieux d’un point de vue
cryptographique l’attracteur qui nous délivre un flux chaotique de codon (streamkeys), qui va être additionné aux
données à sécuriser. Autrement, rechercher un critère de Choix ?
Les résultats obtenus montrent que malgré que les différentes méthodes d’analyses numériques approximent au
mieux les m mes équations différentielles globalement, elles ne donnent pas les m mes résultats localement pour
tous les points. C’est cette différence qu’on peut l’exploiter crryptographiquement. Autrement dit, la méthode
d’analyse numérique peut être un paramètre de la clé de chiffrement.
Mots clés : Chaos, Attracteurs Etrangers, EDP, Runge-Kutta, Euler, Cryptographie.
1. Introduction :
Beaucoup de phénomènes de physique, en mécanique et électricité par exemple, se ramènent l'étude de systèmes
dynamiques représentes par des équations différentielles EDP.
Par exemple si t o y(t) est la fonction qui à l’ instant t associe la position y d'un mobile, y’ sera la vitesse de ce
mobile et y l'accélération. Or dans un système dynamique, y est liée une force qui est souvent fonction de la
position et de la vitesse. Ceci conduit alors une équation entre y, y’ et y , équation différentielle dite du second
ordre 2.
Les équations différentielles ont été inventées par Newton 1, 5 . C'est le début de la physique moderne et
l'utilisation de l'analyse pour résoudre la loi de la gravitation universelle conduisant l'ellipsité des orbites des
planètes dans le système solaire. Leibniz érige l'analyse en discipline autonome mais il faut attendre les travaux
d'Euler et de Lagrange pour voir appara tre les méthodes permettant la résolution des équations linéaires. De nos
jours beaucoup de mathématiciens,
commencer par Poincaré, ont montré que les solutions d'équations
différentielles peuvent tre très instables; ces équations peuvent conduire des situations chaotiques cause d'une
grande sensibilité aux conditions initiales. Ces mathématiciens ont contribué détruire le mythe que l'on pouvait
décrire le monde uniquement partir d'équations différentielles qu'ils suffisaient alors de résoudre
Une équation différentielle est donc une relation d'égalité liant une fonction y et une voire plusieurs de ses dérivées
2 . Lorsque l'équation différentielle est complétée par une condition initiale, c'est- -dire par la connaissance de
l'image d'un réel particulier, on dit que l'on a résoudre un problème de Cauchy.
La majorité des attracteurs chaotiques qui sont définis par un système d'équations différentielles n'ont pas des
solutions analytiques sauf des approximations par des méthodes numériques. En effet, il existe plusieurs types
d'équations différentielles. Chaque type nécessite une méthode de résolution particulière. uelle est la méthode
numérique la plus appropriée, pour résoudre le système dynamique non linéaire de R ssler, ou bien de Lorenz
. Théorie du Chaos :
Le chaos est le terme utilisé pour décrire le comportement apparemment complexe de ce que nous considérons
tre simples 6]. Le chaos est défini généralement comme un comportement particulier d’un système dynamique
déterministe non linéaire. Du point de vue mathématique la notion générale de système dynamique est défini son
tour à partir d’un ensemble de variables qui forment le vecteur d’état
x
xi ‫ א‬R , i
1...n
O n représente la dimension du vecteur. Ce jeu de variables à la propriété de caractériser complètement l’état
instantané du système dynamique générique. En associant en plus un système de coordonnées on obtient l’espace
d’état qui est appelé également l’espace de phase 6 , il s’agit d’un espace de dimensions deux ou trois dans lequel
chaque coordonnée est une variable d’état du système considéré. Conjointement avec l’espace d’état un système
dynamique est défini aussi par une loi d’évolution, généralement désignée par dynamique, qui caractérise
l’évolution de l’état du système en temps.
27
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Animation of the Goal Function of Classical
Parametric Problems with an Interactive Maple Tool
Gy. Bankuti
20-22 April 2015
Abstract
In my presentation I want to show an Interactive Maple Tool for the Simplex Method based solution of parametric LP problems with visualization and
animation facility in 2d and 3d - an efficient Tool for teachers or students.
1
Introduction
Most of OR courses deals with Classical Parametric LP Problems (linear parameter only in the goal function) as it shows the role of a parameter. With the Tool
our aim was to avoid handy calculation and provide automatic visualization.
2
The Methodology
The Classical Solution of a Parametric Problem is the list of the Optimal Solutions in all Characteristic Ranges and the graph of the Goal Function (GF)
depending on parameter t. A 2d example solution is below.
Figure 1: Classical Solution of a 2d Parametric Problem: z = z (t) and x
Illustrating the new type visualization animation: (3 GF states above the
Feasibility Spaces (FS) with the optimal solution (red point)represents the
Figure 2: Feasibility Space with 3 states of Goal Function for every Optimality Range
28
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
motion of the goal function.). At the end of the first optimality range (t=1) the
GF reaches the horizontal axis. In the second range [1,3] it turns to the bottom
right boarder of the FS, and in the last range it turns to the top right boarder
- till it goes throughout (fixed to) the optimal points in every ranges.
Our Tool is able to calculate the Simplex Tables after each other and to move
back and further in them and to calculate the Optimality Ranges, to construct
graph of the FS and the GF and even animation of the GF (in 2d, 3d).
Figure 3: The interactive screens of the Maple Tool, left the Problems for Practice
right the New Problem Solver part window (with the execution buttons)
Visualization of the FS for inner parameter values (of the characteristic
ranges) are on the graphs below. In the characteristic points the goal function matches the border lines of the feasibility space - now specially axes x1 , x2 .
Figure 4: Visualization of a 3d problem for inner parameter values
The Tool needs of course Maple (at least 16) or Maple viewer, but the graphs,
animations can be exported to pdf or animated gif files and use for education.
Acknowledgements
Support of TAMOP-4.1.2.A/1-11/1-2011-0098 and Campus Hungary grants.
References
[1] Frederic S. Hillier Gerald J. Liebermann : Introduction to Operations Reserach,. McGraw-Hill Higher Education (2010).
[2] Gy. Bankuti: Visualization of Parametric Linear Programing with Maple.
4th International Conference of Economic Science, .Proceedings CD 104 112.p.
[3] Gy. Bankuti ; Gy. Kover: An interactive Maple Tool for Parametric Linear
Programming Problems, 26th European Conference on Operational Research
EURO-INFORMS Joint International Meeting (Rome 1-4 July 2013).
2
29
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Solving Real World University Examination
Timetabling using Tribes Particle Swarm
Optimization
Souad Larabi Marie-Sainte
College of Computer and Information Sciences, King Saud University
20-22 April 2015
Abstract
This paper is derived from an importance in creating an automated approach
to treat Examination-Timetabling Problems (ETP) in Information Technology
(IT) department at King Saud University (KSU). We propose a new hybrid
approach based on Particle Swarm Optimization algorithm. This approach is
tested and its efficiency is compared with that of Tabu search. The proposed
approach has considerable results and usually a good stability.
1
Introduction
ETP is a discrete and combinatorial optimization problem, usually handled with
heuristic and/or artificial intelligent approaches. ETP consists in allocating a
set exams in a finite period of time while satisfying a set of hard and soft
constraints. Hard constraints must be satisfied to get a feasible timetable while
soft constraints are those preferable to be satisfied. The hard and soft constraints
vary from one university to another. The best timetable must satisfy a maximum
number of soft constraints with respect to the hard constraints. The common
problem with ETP is the conflict of students having more than one exam at the
same timeslot, which always happens to the students who are taking courses
belonging to different levels, in addition to the courses that are not offered by the
department. In scheduling, it consists of avoiding overlapping of examinations
for every course with common enrolled students. Many universities tackle this
task manually, which is tough and time consuming. In addition, if the conflict
occurs and the preconditions change, the whole work must to be restart from
the beginning. Numerous approaches have been proposed to solve ETP ([1],
[4]), most of them were feasible but not considered as the best solution since the
constraints differ between universities. Our problem consists in handling exams
in IT department at KSU, there are 29 courses and more than 471 students
enrolled in different courses and/or different levels. The timeslot number (nbT)
must be less than 14 for each mid-term. The density of the conflict matrix is
equal to 0,9655. The hard constraints considered are: 1) Every exam must be
scheduled in exactly one timeslot; 2) One student cannot have two exams at the
same timeslot; 3) All exams must be scheduled. While the soft constraints are:
31
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
1) Exams for each student should be spread as far apart as possible; 2) nbT
should be minimized. In this paper, we consider the objective function defined
in [1] which represents the first soft constraint. The objectives of this paper are:
1) to produce an examination timetable suitable to the needs of IT Department;
2) to minimize nbT.
2
Tribes Particle Swarm Optimization (TPSO)
PSO is a population based optimization method motivated by social behavior
of fish schooling and bird flocking [2]. PSO has been successfully involved in
many research and application areas. This study utilizes a binary PSO inspired
from Tribes method which is a stochastic optimization and free parameter technique [3]. In TPSO, a particle P is defined as being a timetable. The algorithm
starts with one particle in a first tribe and then it consists in creating particles
and tribes. Along iterations, the position of the particles is updated according
to a new proposed strategy of displacement, which involves spreading as far as
possible two timeslots if their associated exams are conflicting. Creating particles and tribes relies on the number of iterations and the size of the swarm.
In fact, the creation of particles is done periodically so that it progresses in
order to have time to propagate the information between particles and explore
further the search space. The creation of a new tribe is based on the creation
of particles that compose this tribe. In each new tribe, the number of created
particles is equal to nbT/2 in order to enrich the tribe through the communication transferred between particles and at the same time this number is limited
to avoid overcrowding of particles. To ensure the diversification in the swarm,
we propose two types of particles generated using two proposed heuristics.
3
Experimental Results and Conclusion
TPSO is tested on different nbT starting with 14 and decreasing this number as
far as possible. Lower than 10, the algorithm cannot provide a feasible solution.
The fitness value decreases as the nbT increases. The best solution is found with
nbT equal to 10. Moreover, TPSO is compared with Tabu Search [4] based on
different nbT. TPSO performs better than Tabu Search. Tabu Search does not
provide solution when nbT is less than 11. To conclude, TPSO would assist the
department in scheduling the examinations of students without conflicts. TPSO
is an efficient examination timetabling technique designed for solving ETP.
4
References
[1] M. Alzaqebah, and S. Abdullah. An adaptive artificial bee colony and
late-acceptance hill climbing algorithm for examination timetabling. Journal
of Scheduling vol. (17), 249-262. (2014).
[2] J. Kennedy, and R. Eberhart. Swarm Intelligence. Yuhui Shi, (1995).
[3] M. Clerc. Particle swarm optimization. International Scientific and Technical
Encyclopaedia Wiley, Hoboken, (2206).
[4] S. Larabi Marie-Sainte. ITETT: An Automatic IT Examination Timetable
Tool. In : proceedings of International conference on Recent Trends in Power,
Control and Instrumentation Engineering. vol.(6), 530-535, (2013).
2
32
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Flow-shop scheduling problem with conflict graphs
Nour El Houda TELLACHE, Mourad BOUDHAR
RECITS Laboratory, Faculty of Mathematics, USTHB university, BP 32 El-Alia, Bab-Ezzouar, Algiers, Algeria
[email protected], [email protected]
20-22 April 2015
Abstract
This paper adresses the problem of flow-shop scheduling where conflicting jobs cannot be scheduled concurrently on diffrent machines because of certain technical constraints. These constraints are presented by a graph called conflict graph. The objective is to minimize the makespan. We prove the NP-hardness of several versions of
this problem and we exhibite some polynomial-time solvable cases. Lower bounds and
heuristic algorithms will also be presented along with an experimental study.
Keywords: Flow-shop scheduling, Conflict graph, Complexity, Heuristics.
1
Introduction
We consider the problem of flow-shop scheduling with conflict graph described as follows. Given a set of n jobs that have to be processed by m machines in the same order
and a conflict graph G = (V ; E) denoted by Conf lictG = (V ; E) over the jobs. Vertices
of V represent the jobs and each edge in E models a pair of conflicting jobs that cannot
be scheduled concurrently on different machines. Its complement is called agreement
graph and denoted by AgreeG. The processing time of the job Jj (j = 1 . . . , n) on the
machine Mi (i = 1, . . . , m) is denoted by pij . However, it is not required that each job
has an operation on each machine.
This combination of scheduling and conflict graph was studied on identical machines
and initially introduced by Baker and Coffman in [2]. As far as we know, there’s no
study dealing with the case of flow-shop scheduling problem and it seemed important
to us to go in depth in this matter taking that many possible applications in many fields
(hospitals, industry, sports, military, schedules, etc) may emerge from our findings.
2
Complexity
The problem described above is in general NP-hard. Consider the case of two machines,
we prove that the problem F 2 | AgreeG = (S1 , S2 ; E) | Cmax is NP-hard when the
bipartite agreement graph is complete. The problem F 2 | AgreeG = (V ; E), pij =
1 | Cmax is also NP-hard for planar graphs, bipartite graphs and chordal graphs. On
the other hand, there are many known special graph classes on which this problem is
solvable in polynomial time such as interval graphs, circular arc graphs, cocomparability
graphs, cographs, trees, block graphs and bipartite permutation graphs. This problem
remains NP-hard even when preemption is allowed.
3
Lower bound
Let an instance of the problem described above, and let C be the vector representing
the processing times of the jobs of this instance. If we regard the processing times
33
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
as weights in the agreement graph G, we obtain the weighted graph Gc = (V ; E; C).
Since non adjacent jobs must be scheduled in disjoint time intervals, the weight of
an independent set of maximum capacity represente a lower bound of this instance.
This lower bound has been proposed in [4] and we have adapted it to our case. The
maximum weighted independent set problem is NP-hard. Thus, a greedy algorithm
presented in [7] has been used.
4
Heuristic approaches
Two different heuristic approaches have been proposed. The first one is a constructive algorithm that searchs first an independent set of maximum cardinality in G and
schedule the corresponding jobs in the best possible way. Then, it picks the job with
the highest priority value, from the jobs that haven’t been processed yet, according to a
priority rule. Next, it finds the best sequence which minimizes the partial makespan by
placing the selected job at all possible positions in the partial sequence under construction without changing the relative positions of the already assigned jobs. The second
approach is also a constructive algorithm. In this approach, we partition the conflict
graph into a minimum number of independent sets and schedule the jobs of each set
in the best possible way. Then, we select the independent set with the highest priority
value, from the sets that haven’t been processed yet, according to a priority rule. Next,
we try to find the best sequence which minimizes the partial makespan by placing the
selected set at all possible positions in the partial sequence under construction without
changing the relative positions of the already assigned jobs.
5
Conclusion
In this paper, we showed that the flow shop problem with conflict graphs is NP-hard
in general even for two machines. Furthermore, we conducted a computational experiments where the above heuristic approaches and lower bounds were coded in C++
language and carried out on 7200 instances. The evaluation showed that the second
approach seems to be more efficient than the first one especially for large size instances
and for high densities.
References
[1] A. Oulamara, D. Rebaine, M. Serairi: Scheduling the two-machine open shop
problem under resource constraints for setting the jobs. Ann Oper Res 211., 333356. (2013).
[2] B.S. Baker, E.G. Coffman: Mutual Exclusion Scheduling. Theoretical Computer
Science 162., 225-243. (1996).
[3] E. Taillard: Some efficient heuristic methods for the flow shop sequencing problem.
OMEGA, The Internalional Journal of Management Science 47., 65-74. (1990).
[4] M. Bendraouche, M. Boudhar: Scheduling jobs on identical machines with agreement graph. Computers and Operations Research 39., 382-390. (2012).
[5] M. Bendraouche, M. Boudhar, A. Oulamara: Scheduling: Agreement graph vs
resource constraints. European Journal of Operational Research 240., 355-360.
(2014).
[6] M.R. Garey, D.S. Johnson: Computers and intractability: A Guide to the Theory
of NP-Completeness. New York: Freeman, (1979).
[7] S. Sakai, M. Togasaki, K. Yamazaki: A note on greedy algorithms for the maximum
weighted independent set problem. Discrete Applied Mathematics 126., 313-322.
(2003).
2
34
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
An approximation of the Pareto frontier for the
multi-objective spanning tree problem
Asma BOUMESBAH and Mohamed El-Amine CHERGUI
USTHB, Faculty of Mathematics, RECITS Laboratory
[email protected] [email protected]
20-22 April 2015
Abstract
The purpose of our study is an adaptation of the genetic algorithm proposed by
Deb [4], allowing the approximation of the Pareto frontier for the multi-objective
spanning tree problem (MOST) in a connected graph G having n vertices and m
edges, each edge provided with a weight vector of dimension r ≥ 2. This problem is
known to be NP-hard even for r = 2 [2] and exact methods are implemented only
for the bi-objective spanning tree problem [3,5].
The Proposed Algorithm
1. The edges of G are numbered. The encoding of a chromosome (corresponding
to a spanning tree) is a vector of dimension n − 1, wherein each coordinate
represents the associated number of an edge.
2. Generate an initial population of p size of spanning trees using to both optimal spanning trees corresponding to each criterion, random generation with
aggregation of criteria and applying Kruskal (or Prim) algorithm by randomly
selecting edges of graph G.
3. The single-point-crossover operator is used. The obtained offspring chromosomes don’t correspond necessarily to spanning trees. Indeed, either redundant edges, or cycles, or non connected sub-graphs can be obtained. To overcome this infeasibility and keep the spirit of genetic mixing of parents, we
have introduced procedures of rearrangement of chromosome edges for each
offspring who is not a spanning tree.
The edges of A 2 are chosen in the following sets: B2 \ A1 , B1 \ A1 and
E \ A1 respecting this order. In the same manner, those of B 2 are chosen in:
A2 \ B1, A1 \ B1 and E \ B1.
35
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
4. As genetically the mutation operator of a gene value in an offspring chromosome occurs rarely, and according to a definable mutation probability. This
probability should be set low. It consists to choose randomly a gene of offspring chromosome (an edge e = (x, y) of the spanning tree) and replace it
by another gene corresponding to an edge f whose weight vector dominates
that of e. f is not in the current spanning tree and belongs to the elementary
chain C in G connecting the vertices x and y, C ∪ e is then a cycle of G. This
allows to find another chromosome (i.e., a spanning tree) whose weight vector
dominates the offspring weight vector. If there is no edge in the chain C which
dominates the edge e, any edge is randomly chosen from the chain C and e is
replaced by this one.
5. The selection operator consists in proceed first with the partition of the population of 2p size, composed of parents and offsprings in classes, the first containing the non-dominated individuals, the second still contains non-dominated
individuals after removing elements of the first class, and so on. The new
population consists in choosing individuals respecting to the order of dominance classes. Individuals of the class k have to complete the p size of the new
population are selected according to a distance of ” crowding ” as proposed
in [4]. The first tests applied on graphs of medium size and the number of
criteria r > 2 seem to be conclusive. For the example in reference [1], 17.1
efficient solutions have been obtained in average among the 22 of the example,
by running our approach ten(10) times. At each run, the other solutions found
are close to the remaining efficient solutions of the cited example.
One of the obtained results.
References
[1] K. Andersen, K. Jörnsten and M. Lind: on bicriterion minimal spanning trees:
an approximation. Comput. Oper. Res vol.(23.), 11711182. (1996).
[2] P. Camerini, G. Galbiati and F. Maffioli: The complexity of weighted multiconstrained spanning tree problems. Colloquium on the Theory of Algorithms,
53101. (1984).
[3] C. G. da Silva and J. C. N. Clı́maco: A note on the computation of ordered supported non-dominated solutions in the bicriteria minimum spanning tree problems.
Journal of telecommunications and information technology, 11-15. (2007).
[4] K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan: A fast and elitist multiobjective genetic algorithm for multi-objective optimization. In Proceedings of the
Parallel Problem Solving from Nature VI (PPSN-VI), 849-85. (2000).
[5] H.W. Hamacher and G.Ruhe: On spanning tree problems with multiple objectives. Annals of Operations Research vol.(52.), 209-230. (1994).
2
36
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
The r-Whitney numbers and the value of the high
order Bernoulli polynomials
Meriem Tiachachat and Miloud Mihoubi
[email protected] [email protected]
RECITS Laboratory, Faculty of Mathematics, USTHB
20-22 April 2015
Abstract
The main object of this paper is to give an application of the r-Whitney
numbers on the values at rational arguments of the high order Bernoulli and
Euler polynomials. The obtained formulas generalize the known expressions of
the Bernoulli numbers of both kinds.
1
Introduction
The r-Whitney numbers wm,r (n, k) and Wm,r (n, k) of the first and the second
kinds,respectively, can be defined
when we write mn (x)n in the
as coefficients
k
n
basis (mx + r) ; k = 0, . . . n and (mx + r) in the basis (x)k ; k = 0, . . . n ,
i.e.
mn (x)n =
n
wm,r (n, k) (mx + r)k
k=0
and
(mx + r)n =
n
mk Wm,r (n, k) (x)k ,
k=0
where (α)n = α (α − 1) · · · (α − n + 1) if n ≥ 1 and (α)0 = 1.
As it is known, these numbers have exponential generating functions to be
n≥k
n≥k
tn
wm,r (n, k)
n!
Wm,r (n, k)
tn
n!
1
=
k!
=
1
k!
ln (1 + mt)
m
k
exp (mt) − 1
m
r
(1 + mt)− m ,
(1)
k
exp (rt) .
(2)
Merca [5] gaves links of these numbers to the symmetric functions and Mihoubi
et al. [6] gave some applications of the r-Stirling numbers on Bernoulli polynomials.
37
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Motivated by these works, we show in this paper some links of the Whitney numbers of both kinds to the values at rational numbers of the high order Bernoulli
and Euler polynomials.
As a results we have
For r, s, k, n, m be non-negative integers with m = 0, we have
Bn(k)
b(k)
n
r−s
m
r−s
m
=
n
1 j+k
k
mn
=
−1
n j+k
1
mn n!
(−k)
−
bn
r
m
r
m
k
j=0
and
wm,s (j + k, k) Wm,r (n, j) ,
(3)
j=0
Bn(−k)
−1
Wm,r (j + k, k) wm,s (n, j)
−1
=
1
mn
=
1
k!
wm,r (n + k, k) .
mn (n + k)!
n+k
k
(4)
Wm,r (n + k, k) ,
(5)
(6)
References
[1] H. Belbachir, I. E. Bousbaa, Translated Whitney and r-Whitney numbers:
A combinatorial approach, J. Integer Seq. 16 (2013) Article 13.8.6.
[2] M. Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math.,
159 (1996) 13–33.
[3] A. Z. Broder, The r-Stirling numbers, Discrete Math., 49 (1984) 241–259.
[4] G.-S. Cheon, J.-H. Jung, The r-Whitney numbers of Dowling lattices, Discrete Math., 312 (15) (2012) 2337–2348.
[5] M. Merca, A note on the r-Whitney numbers of Dowling lattices, C. R. Acad.
Sci. Paris, Ser. I, 351 (2013) 649–655.
[6] M. Mihoubi and M. Tiachachat, The values of the high order Bernoulli polynomials at integers and the r-Stirling numbers, available at http://arxiv.org/abs/1401.5958.
2
38
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Asymptotic behavior of a Local linear estimation of
conditional hazard rate function for truncated data
Ourida Sadki
Karima Zerfaoui
Laboratoire Recits,Faculte de Mathematiques
USTHB, bp 32 El Alia, 16111, Alger, Algeria
Univ. Oum El Bouaghi 04000, Algeria
[email protected]
Faculte de Mathematiques
Univ. des Sci. et Tech. Houari Boumédiène
bp 32, El Alia, 16111, Algeria
[email protected]
20-22 April 2015
Abstract
In this paper we propose a new nonparametric estimator for the conditional
hazard rate function, based on a local linear estimation techniques when data
are truncated. The asymptotic behavior of the proposed estimator is studied.
Somme Simulations with R are given to show the performance of the estimator
for finite samples sizes.
1
Introduction
Hazard function plays an important role in reliability and survival analysis since
it quantifies the instantaneous risk of failure or death at a given time point.
Parametric and nonparametric methods have been proposed to estimate hazard
rates. Surveys on nonparametric kernel rate estimation are provided by Singpurwala and Wong(1983). There is a vast literature on nonparametric estimation
of the hazard function for incomplete data, censoring or truncated data. A local
linear estimator of the conditional hazard rate for censored data has been studied by Kim et al (2010).In the present paper, we consider the estimation of the
conditional hazard rate function when the observations are subject to random
left truncation in the i.i.d. case, using a local polynomial regression techniques.
2
Definitions and main results
Let Y and T be independent random variables with distribution functions F and
G respectively, both assumed to be continuous, and let (Y1 , T1 ), (Y2 , T2 ), , (YN , TN )
be N independent and identically distributed (iid) copies of (Y, T ), where the
size N is deterministic, but unknown.
In the random left-truncation model Y is interfered by the truncation random
variable T , such that both quantities Y and T are observable only if Y ≥ T
, whereas nothing is observed if Y < T . Without possible confusion, we still
denote (Yi , Ti )1≤i≤n , as a consequence of the truncation, the size of the actually
39
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
observed sample, n , with n ≤ N . let γ = P (Y ≥ T ) be the probability that we
observe the rv of interest Y . It is clear that if γ = 0 no data can be observed.
Therefore, we suppose throughout the paper that γ > 0. Then the nonparametric maximum likelihood estimates of F and G are the product-limit estimators
obtained by Lynden-Bell (1971).We denote by f (.|x) and S(.|x) the conditional
density function and survival function respectively of Y given X = x. The conf (.|x)
,
ditional hazard rate function of Y given X = x is defined by λ(.|x) =
S(.|x)
for x such that S(.|x) > 0.
Local linear estimation is a special case of local polynomial regression and
the resulting estimators have superior bias properties compared to NadarayaWatson estimators . Let Let K and L be two kernel functions and let h1 and
h2 the bandwidths associated with the kernels L and K respectively. Define
T
minimizes:
fˆ(y|x) = β̂0 (y|x) , where β̂(y|x) = β̂0 (y|x), β̂1 (y|x)
n 2
γn Kh2 (Yi − y) − β0 + β1 (Xi − x) Gˆ−1 (Yi )Lh1 (Xi − x).
i=1
Our proposed estimator is given by λ̂(y|x) =
fˆn (y|x)
,
Sˆn (y|x)
where Sˆn , the estimator
of the survival function S is obtained similarly as fˆn . Under some classical
regularity asumptions on K, L,, h1 and h2 , we study the asymptotic behavior
of λ̂. Our first result is the uniform almost sure convergence with rate of the
conditional hazard rate function estimator. The second result is the asymptotic
normality of our proposed estimator. Some Simulations are given to show the
performance of the estimator for finite samples sizes.
References
[1] Kim, C., Oh, M. ,Yang,S.J., and Choi,H. (2010). A local linear estimation of conditional hazard function in censored data , J. Korean Stat Soc
39(3),347-355.
[2] Lynden-Bell, D. (1971). A method of allowing for known observational
selection in small samples applied to 3CR quasars, Monthly Notices Royal
Astronomy Society 155, 95-118.
[3] Sinpurwalla, N. D. and Wong, M. Y. (1983). Estimation of the failure rate- A survey of nonparametric models. PartI: Non-Bayesian methods
Commun. Statis. Theory Methods 12, 559-588.
2
40
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
La dynamique des d’épidémies dans un réseau de petit monde
O.Mosbah , A.Dilem, k.khelloufi, Y.Baara, O.Harrouz, N.Zekri
Université des sciences et de technologie (Mohamed Boudiaf)
LEPM, B.P 1505 El Mnaouer Oran, Algérie
E-mail mosbah31omar yahoo.fr
LEPM, B.P 1505 El Mnaouer Oran, Algérie
E-mail nzekri yahoo.com,
Résumé En utilisant le modèle SIR (Susceptible, infecté, récupéré), nous étudions
numériquement dans ce travail les propriétés dynamiques de propagation d’un virus dans
un réseau dynamique 1D de Petit Monde. Les paramètres choisis dans ce modèle
correspondent
un réseau réaliste dans une situation épidémique o les voisins
représentent les familles alors que les courts-circuits correspondent un marché ou un
établissement scolaire(les amis à l’école). Le modèle inclue également les périodes
latentes et d’infection. L’augmentation exponentielle du nombre d’infectés est retrouvée
pour différentes temps de latence et d’infection. La période de l’épidémie semble aussi
augmenter exponentiellement avec le temps de latence.
Mots clés :
Réseau petit monde, Dynamique de propagation, temps d’infection, temps de latence.
1/Introduction :
La propagation des maladies a été mise en équation depuis le 17émé siècle par Bernoulli (le modèle
SIR).Mais pour reproduire les épidémies dans différentes population, il est nécessaire d’effectue des
simulations introduisant les propriétés statistique de la population. Parmi ces propriété ‘’la
caractérisation du réseau’’. La recherche de ce type de réseau a débuté avec l’expérience de Milgram
1 qui a montré que la connexion entre deux individus pour une transmission par courrier nécessite en
moyenne six étapes (appelée six degrés de séparation). Ceci a conduit ensuite la mise au point du
Réseau du Petit Monde (RPM) par Watts et Strogatz [2], à partir d’un réseaux régulier
(unidimensionnel par exemple) de sites occupés (exposés à l’infection) repartis aléatoirement avec
une probabilité d’occupation (P). Plusieurs études numériques ont tenté de déterminer le seuil de
percolation 3 (épidémie- endémie). D’autres points caractérisant la propagation par simulation M.C
(Monte Carlo) 4 .
Ce travail introduit des effets réalistes comme les temps de latences (tlatt) et d’infections (tinf), ainsi
qu’une procédure dynamique de l’infection. Nous reproduisons l’augmentation exponentielle du
nombre de cas infectés
ܰ௜௡௙ ሺ‫ݐ‬ሻ ‫ ݁ ן‬௧Ȁఛ
(1)
Nous examinons également l’effet de tlatt sur la période d’épidémie.
Référence:
1
2
3
4
S. Milgram, Psychology Today 2 (1967) 60.
D. . Watts, Small Worlds, Princeton University Press, Princeton, 1999.
M. .Newman D. . Watts, Phys. Lett. A 263 (1999) 341.
N.Zekri, and .P.Clerc, Phys.Rev.E
(2002) 046108
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Kernel estimator of conditional hazard rate function
for associated censored data
Samra DHIABI
Ourida SADKI
département de Mathematiques
Univ. Mohamed Khider, Biskra, Algeria
Laboratoire RECITS-USTHB
bp 32 El Alia, 16111, Alger, Algeria
& Univ. Oum El Bouaghi 04000, Algeria
[email protected]
[email protected]
20-22 April 2015
Abstract
In this paper we study a smooth estimator of the conditional hazard rate
function in the censorship model, when the data exhibit some dependence structure. We show that, under some regularity conditions, that the kernel estimator
of the conditional hazard rate function suitably normalized is asymptotically
normally distributed
1
Introduction and model
Let T1 , ..., Tn be a sequence of the survival times of n individuals in a life table.
These random variables (r.v.s) are not assumed to be mutually independent but
are positively-associated and strictly stationary with common unknown absolutely distribution fonction (df) F . In many situations, we observe only censored
lifetimes of items under study. That is, assuming that {Ci ; i ≥ 1} is a sequence
of independent censoring times with common unknown df G, we observe only
the n pairs {(Yi , δi ) , i = 1, 2, ..., n} , with Yi = Ti ∧ Ci and δi = I{Ti ≤Ci } where
∧ denotes minimum and I{.} is the indicator r.v. of the specified event.
We will suppose that T and C are independent to ensures the identifiability of
the model.
Let X1 , ..., Xn be a stationary sequence of real-valued r.v.s,
and F (. | .) be the
conditional df of T given X = x, that is, F (t | x) = E 1{T ≤t} | X = x wich
can be writen as F (t | x) =:
F1 (t,x)
,
l(x)
where l is the marginal density of X with
F (t, x)
where F1 (., .) is the joint
respect to Lebesgue measure and f (t | x) =: 1
l(x)
probabilty density function of (T, X) .
The conditional hazard rate function, also known as the force of mortality or
f (.|x)
the failure rate, of T given X = x is defined by h(.|x) =
, for x such
1 − F (.|x)
that F (.|x) < 1.
It follows that, h(t|x)dt can be interpreted in reliability, as the conditional probability of a failure of the component in the interval (t, t+dt) given that the failure
42
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
had occurred in ]0, t] or as the instantaneous risk of death at time t, conditioned
by the fact that the subject is still alive at time t in medical trials .
The definition of the underlying dependence considered here is as follows
The random variables (X1 , X2 , ..., Xn ) is said to be positively-associated if for
every g1 , g2 : Rn → R, which are cordinatewise nondecreasing, and for which
Eg12 (Xj , 1 ≤ j ≤ n) < ∞, Eg22 (Xj , 1 ≤ j ≤ n) < ∞, it hold that
cov(g1 (X) , g2 (X)) ≥ 0.
Association has found extensive applications in systems reliability and various
problems in statistical mechanics (see, for example, Esary et al. (1967)). Properties of associated random sequences are given in Bulinski et al.(2007).
2
Main result
Several methods exist in the literature for estimating the hazard rate function,
nonparametric estimation using kernel smoothing method has received considerable attention in the statistical literature. Many authors continu to develop
differents aspects of nonparametric estimation of the hazard rate function under
right censoring ; González-Manteiga, W., Cao, R. and Marron, J.S. (1996) employed the bootstrap for bandwidth selection, Spierdijk, L.(2008) consider the
local linear method, Bakgavos, D. and Patill, P. (2008) estimated the hazard
rate function via a local polynomail fitting, Diallo, A. and Louani, D. (2013)
applied the moderate and large deviation principles for the hasard rate function.
In this paper, we propose a smooth estimator of h(.|x) using a double kernel
method. We study his asymptotic properties for the case in which the underlying failure times and the covariate are assumed to be positively associated and
under the condition that C and (X, T ) are independent.
References
[1] Bagkavos, D. and Patil, P. (2008). Local polynomial fitting in failure rate
estimation. IEEE Transactions on Reliability, 57, 41–52.
[2] Bulinski, A. and Shashkin, A. (2007). Limit theorems for associated random
fields and related systems, Vol. 10. Advanced series on statistical science &
applied probabilty.
[3] Diallo, A. and Louani, D. (2013). Moderate and large deviation principles
for the hazard rate function kernel estimator under censoring. Statistics
and Probability Letters, 83, 735–743.
[4] Esary, J., Proschan, F., Walkup, D. (1967). Association of random variables with applications.Annals of Mathematical Statistics, 38, 1466–1476.
[5] González-Manteiga, W., Cao, R. and Marron, J.S. 1996. Bootstrap selection of the smoothing parameter in nonparametric hazard rate estimation.
Journal of the American Statistical Association, 91, 1130–1140.
[6] Spierdijk, L.(2008). Nonparametric conditional hazard rate estimation:
a local linear approach. Computational Statistics and data analysis, 52,
2419–2434.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Application de la méthode du point fixe dans le cas
p-adique
Kecies Mohamed, Centre universitaire de Mila
20-22 April 2015
Résumé
Les nombres p-adiques sont une extension des nombres rationnels qui sont
utilisés en théorie des nombres pour calculer modulo une puissance d’un nombre
premier p. Ils sont inventés au début du vingtième siècle par le mathématicien
Allemand Kurt Hensel et jouent un rôle dominant dans la théorie des fonctions.
Dans ce travail nous avons appliqué la méthode du point fixe classique dans
le cas p-adique pour calculer l’inverse d’un nombre p-adique a ∈ Qp et ceci à
travers le calcul de la solution approchée du zéro d’une équation de la forme
f (x) = x1 − a = 0. Nous avons également déterminé la vitesse de convergence,
le nombre d’itérations en utilisant la technique de la norme p-adique. On a
basé dans ce travail sur les études faites par Michael P. Knapp et Christos
Xenophontos (voir [5]).
1
Introduction
La connaissance des propriétés arithmétiques et algébriques des nombres padiques est utile à l’étude de leurs propriétés diophantiennes et des problèmes
d’approximation. Il s’agit, dans ce travail, d’une application intéressante des
outils de l’analyse numérique à la théorie des nombres. On propose d’étudier le
problème suivant :
⎧
⎨ f (x) = x1 − a = 0
(1)
⎩
a ∈ Q∗p , p-premier
Le but est de calculer les développements finis p-adiques de l’inverse de a ∈ Q∗p .
La solution de (1) est approchée par une suite des nombres p-adiques (xn )n ∈ Q∗p
construite par la méthode du point fixe.
La méthode du point fixe consiste à remplacer la recherche de zéro de
l’équation f (x) = 0 par la recherche du point fixe de l’équation x = g(x),
sous réserve que ces deux formulations soient mathématiquement équivalentes.
Les conditions qui permettent la détermination de la fonction g sont :
- La fonction polynôme g ne doit pas avoir l’inverse de a dans ses coefficients
et g( a1 ) = a1 , g (k) ( a1 ) = 0, k = 1, s − 1 , g (s) ( a1 ) = 0 où s est la vitesse de
convergence de cette méthode. Pour cela, on pose
g(x) = x(1 + (1 − ax) + (1 − ax)2 + (1 − ax)3 + ... + (1 − ax)s−1 )
44
(2)
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
La suite des itérées associée à g(x) est définie par
∀n ∈ N : xn+1 = xn 1 + (1 − axn ) + (1 − axn )2 + ... + (1 − axn )s−1
−vp (a)
(3)
−m
= p , m ∈ Z. Il
Soit a un nombre p-adique non nul tel que |a|p = p
−1 ∗
est clair que si b ∈ Qp est l’inverse de a, alors |b|p = a p = pm , m ∈ Z. Donc
la suite des nombres p-adiques (xn )n devrait tendre vers b ∈ Q∗p . Ainsi à partir
d’un certain rang on a |xn |p = |b|p = pm , m ∈ Z.
Théorème 1.1
1) Si xn0 est l’inverse de a d’ordre r, alors xn+n0 est l’inverse de a d’ordre σn .
Où
(4)
∀n ∈ N : σn = sn r − m
2) La suite des écarts est définie par
∀n ∈ N : xn+n0 +1 − xn+n0 ≡ 0
Telle que
∀n ∈ N : σn = sn r − m
mod pσn
(5)
(6)
Corollaire1.2
Soient S = a ∈ Qp : |a|p = 1 , D = a ∈ Qp : |a|p < 1 et D = a ∈ Qp : |a|p > 1 .
Alors
ln(| M +m
|)
r
.
1. Le nombre nécessaire d’itérations n pour obtenir M chiffres est n =
ln s
2. Si m = 0, alors la vitesse de convergence est d’ordre s pour tout nombre
p-adique de S.
3. Si m > 0, alors la vitesse de convergence est moins rapide pour tout nombre
p-adique appartient à l’ensemble D.
4. Si m < 0, alors la vitesse de convergence est plus rapide pour tout nombre
p-adique appartient à l’ensemble D .
Références
[1] A.J. Baker :An Introduction to p-adic Numbers and p-adic Analysis, Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland,
(2004).
[2] A. Vimawala : P-adic Arithmetic Methods for Exact Computation of Rational
Numbers. School of Electrical Engineering and Computer Science, Oregon State
University. June (2003).
[3] C. k. Koc : A Tutorial on P-adic Arithmetic. Electrical and Computer Engineering. Oregon State University, Corvallis, Oregon 97331. Technical Report,
April (2002).
[4] F. Bajers. Vej : P-adic Numbers. Aalborg University. Departement Of Mathematical Sciences. 7E 9222 Aalborg ∅st. Groupe E3-104, 18-12-2000.
[5] M. Knapp, C. Xenophotos : Numerical analysis meets number theory : using
rootfinding methods to calculate inverses mod pn . Appl. Anal. Discrete Math.
4.23-31.(2010).
[6] S. Katok : Real and p-adic analysis. Course notes for Math 497C, Mass
Program, Fall 2000 (2001).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Complexity results of a chain reentrante shop with an exact
time lag
Karim Amrouche1 ,
Mourad Boudhar2 ,
Farouk Yalaoui3
1
University of Algiers 3, Faculty of Economics and Management sciences,
2 street Ahmed Waked, Dely Brahim, Algiers, Algeria. [email protected]
2
RECITS laboratory, Faculty of Mathematics, USTHB University,
BP 32 Bab-Ezzouar, El-Alia 16111, Algiers, Algeria. [email protected]
3
University of Technology of Troyes, LOSI laboratory, ICD UAR CNRS 6281
12 street Marie Curie, BP 2060, Troyes 10010, France. [email protected]
Mots-clés : Reentrant flow shop, time lags, makespan, complexity, heuristics.
1
Introduction
In this paper we consider the flowshop scheduling problem where n independent tasks T =
{T1 , T2 , ..., Tn } should be scheduled on a set of m machines M = {M1 , M2 , . . . , Mm }. Each task
must be processed according to the following order (Figure 1) :
M1 -
-
M2
-
M3
- ···
-
Mm
FIG. 1 – Order of excuting tasks on machines.
with a time lag li separating the completion time of the first operation on the first machine
and the start time of its second operation on the same machine. We consider the case where
for all tasks li = L. The problem is noted F m|chain − reentrant, li = L|Cmax .
The considered notation are given below :
– ai : the processing time of the first operation of the task Ti on the first machine.
– bij : the processing time of the task Ti on machine Mj , 2 ≤ j ≤ m and
m
j=2
bij ≤ L.
– ci : the processing time of the second operation of the task Ti on the first machine.
To illustrate the problem, we consider the following instance where n = 5, m = 3 and L = 8.
TAB. 1 – processing times of the 5 tasks
Ti
ai
bi2
bi3
ci
T1
2
2
6
5
T2
3
2
4
2
A schedule is given in Figure 1.
46
T3
2
3
3
3
T4
4
4
2
5
T5
2
6
2
6
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
machines
L
L -L
6 c[1] c[2] c[3]
a[4] @ a[5] HH
c[4]
c[5]
M1 a[1] @ a[2] a[3]
H
@
H
@
h
X
(
h
XXX
hhhh
((
M2 @ b[12]Q
(((
hhh
b
b
b
b
(
X
Q
(
[22]
[32]
[42]
[52]
XX
(
h
Q
@
(
P
hhhh
(
HH
(((
P
(
h
(
H
h
P
M3 H
(
hhhh b[43] H b[53] P
b[13]
b[23] b[33] ((((
P
P H
H
h
H
0 2
4
7
10
14
17
20
25
29 31
37
43 time
FIG. 2 – 5 tasks schedule instance on the machines.
2
Polynomial subproblems
We dedicated this section to identify a discuss on some new polynomial cases. Some of
them reduces to the maximum weight matching problem ; such as F m|chain − reentrant, aj >
L/2, cj > L/2, lj = L|Cmax and F 2|chain − reentrant, bj = L, aj + cj ≥ L, lj = L|Cmax . And
the others have been proved using their own particularities.
Theorem 1 The following problems are polynomially solved :
-F m|chain − reentrant, aj > L/2, cj > L/2, lj = L|Cmax
-F 2|chain − reentrant, aj = bj = L, lj = L|Cmax .
-F 2|chain − reentrant, bj = cj = L, lj = L|Cmax .
-F 2|chain − reentrant, aj = L/2, bj ≤ L/2, cj = c, lj = L|Cmax .
-F 2|chain − reentrant, cj = L/2, bj ≤ L/2, aj = a, lj = L|Cmax .
-F 2|chain − reentrant, aj = L/2, bj ≤ L/2, cj ≤ L/2, lj = L|Cmax .
-F 2|chain − reentrant, bj = L, aj + cj ≥ L, lj = L|Cmax .
3
Constant seconde stage
We are dealing with the case where m = 2 and bj = L, the NP-hardness of the problem
remains open. This problem was inspired in the operating blocks, where a patient should undergo a surgical operation, in the following steps :
1- The surgeon examines the patient and prepare him for the operation ; once the surgical
operation completed (aj the duration of the first step).
2- A nurse will take care of the patient for a while (bj = L the duration of the seconde).
3- After that ; the surgeon revisits his patient (cj the duration of the third step).
For the resolution of this problem we have proposed some heuristics based on random search
with numerical experimants.
4
Conclusion
We studied the chain reentrant problem with an exact time lag denoted F m|chain−reentrant, lj =
L|Cmax . The general problem is NP-Hard. We give several sub problems which are polynomially
resolved. And some heuristics based on random search for the resolution of a particular case.
References
[1] Wang MY, Sethi SP, Van De Velde SL. Minimizing makespan in a class of reentrant shops.
Oper. Res, 1997 ; 45 :702-712.
[2] V. Lev and I. Adiri, V-shop scheduling. European J. Opnl. Res. 18 (1984) 51-56.
47
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Inverse optimization, an efficient tool for resource
reallocation: application in the farming sector
Amar Oukil
∗
20-22 April 2015
Abstract
A mathematical model involving inverse optimization theory and data envelopment analysis (DEA) as an efficiency measurement tool has recently been
developed to handle the reallocation of resources in the context of Mergers &
Acquisitions (Gattoufi et al.[8]). This paper highlights the practical importance
of these concepts through an application in the farming sector, using a sample
of 45 farms from the Batinah region, in the sultanate of Oman. Through a
random selection of merging farms, the proposed InvDEA method is shown to
enable not only reallocating resources but also reversing the merger’s decision if
the contribution of one of the merging entities is found to be poor. In order to
gauge the practical importance of the InvDEA model, its application scope can
be expanded to other sectors, like banking, insurance, education, healthcare,
etc.
Keywords: Inverse optimization, data envelopment analysis, resource allocation.
1
Introduction
As defined in Ahuja & Orlin [1] ”An inverse optimization problem consists of
inferring the values of the model parameters (cost coefficients, right-hand side
vector, and constraint matrix) given the values of observable parameters (optimal decision variables)”.
Inverse problems were first studied by geophysical scientists (e.g. Tarantola [12]).
Within the Operations Research community, the work of Burton & Toint ([4],
[5]) on inverse shortest path problems is presented as the instigator of research
interest in the field of inverse optimization [1]. In this paper, we use Inverse
linear programming (LP) (see, e.g., [7], [10], [13], [14], [16]) to assist decision
makers (DM) in reallocating resources to decision making units (DMUs) prior
to a merger.
Mergers & acquisitions (M&A) are strategic decisions involving resource consolidation of a group of companies in order to expand market share, gain access to
new markets or enhance production capabilities [9]. As such, managing the most
vital resources optimally is a crucial pre-merger step for better performance of
∗ Department of Operations Management & Business Statistics, Sultan Qaboos University, Oman.
E-mail: [email protected]
48
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
the emerging corporation.
Performance and resource management being perfectly handled within frontier
analysis frameworks, Data envelopment Analysis (DEA) is used to estimate expected cost and profit efficiency gains, employing the existing levels of inputs
and outputs of the merging entities ([3], [11], [15]).
At a subsequent stage, the Inverse DEA (InvDEA) model proposed in Gattoufi
et al. [8] sets proper decision strategies for these resources. InvDEA assumes an
efficiency target of the new corporation and looks for the optimal levels of inputs
and outputs to achieve this efficiency [8, 14]. The proposed InvDEA method is
applied on a dataset of 45 farming units, and it is shown through a randomly
selected merger of two farms that the model can not only assist in reallocating
resources but it also enables reversing the merger’s decision if the contribution
of one of the merging entities is found to be poor. In order to gauge the practical importance of the InvDEA model, its application scope can be expanded to
other sectors, like banking, insurance, education, healthcare, etc.
2
Contextual setting
The Batinah agricultural area represents over 53% of the total cropped area
of Oman, where most of the vegetables supplying the capital and coastal city
markets are produced. Approximately 80% of the farms cover an area of less
than 2.1 hectares, and only 1% of the farms have an average size above 21.6
hectares. Date palms represent the most important agricultural product, followed by vegetables and field crops, including, mango, banana, lime, melon, watermelon, tomato, onion, pepper, potato, cucumber, okra, tobacco, and grass.
These specifications comply with the DEA assumption that all farms should
operate in a relatively homogeneous region, hence preventing undesirable effects
of climatic and bio-physical constraints on the technical efficiency of farms.
3
Farm efficiency evaluation
The evaluation of each farms efficiency is a necessary step to identify inefficient
farms that may be candidates for merger in a view of efficiency improvement.
Data envelopment analysis (DEA) is a non-parametric approach for the evaluation of relative efficiency on the grounds of an efficient production frontier.
DEA enables not only the identification of efficiency ratios but also estimation
of the allowable reduction of the inputs without altering any of its outputs. The
DEA models that are most frequently applied in agriculture are the CCR model
(Charnes et al. [6]), which assumes constant returns to scale (CRS), and BCC
model (Banker et al. [2]), which allows variable returns to scale (VRS). These
models are formulated as linear programmes (LPs).
4
Merging farms using InvDEA
Although tactical decisions, like input reduction, could contribute to improve
farms technical efficiencies, the effect of farms mergers are more likely to be perceived on a long range horizon. Resource reallocation among potential merging
farms is conducted via the InvDEA model developped by Gattoufi et al. [8]. An
illustrative example is used to emphasize the practical scope of this model as a
support for decision making.
2
49
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
On complexity analysis of interior point method for
semidefinite programming based on a new kernel
function
TOUIL Imene, Jijel University and BENTERKI Djamel, Sétif University
20-22 April 2015
Abstract
The aim of this work is to improve the complexity analysis of large-update
primal-dual interior point method (IPM) for semidefinite programming problem (SDP). We define a proximity function for SDP based on a new efficient
kernel function and we prove that the worst-caseiteration bound for the new
pq+1
√
n (ln n) pq ln n with p, q ≥ 1 , which imcorrespondent algorithm is O
prove
know iteration bound given by B. K. Choi and al [2] namely
√ the best
q+1
q
n (ln n)
ln n where q ≥ 1 .
O
1
Position of the problem
We consider the primal semidefinite programming problem
(SDP ) min {tr (CX) : tr (Ai X) = bi , i = 1, ..., m, X 0}
And it dual
(SDD) max
t
by:
m
(1)
yi Ai + S = C, S 0.
(2)
i=1
where Ai , i = 1, ..., m and C are symmetric (n × n) matrices, and b, y ∈ Rm .
0, ( or 0, ) means that the matrix is symmetric positive semidefinite
(or positive definite). We assume that the matrices Ai , i = 1, ..., m are linearly
independent. Finding an optimal solution of SDP and SDD is equivalent to
solving⎧
the following linear system which obtained by using the Newton’s method
⎪ tr(Ai ΔX) = 0, i = 1, ..., m,
(b)
⎨ m
Δyi Ai + ΔS = 0,
⎪
⎩ i=1
XΔS + ΔXS = μ+ I − XS; X, S 0.
W hereμ > 0, and I denotes the (n × n) identity matrix. A decisive observation
for SDP is that (b) might have nonsymmetric solution ΔX. For this, many
researchers have proposed different ways of symmetrizing the third equation.
50
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
In our work, we consider the symmetrization scheme that yields N T directions,
then (b), can be written as follows
⎧
tr(Ai DX ) = 0, i = 1, ..., m,
⎪
⎨ m
Δyi Ai + DS = 0,
⎪
⎩ i=1
(3)
DX + DS = V −1 − V.
D
−1
=P = X
1
2
1
2
X SX
1
2
− 12
X
1
2
− 12
= S
−1
2
1
2
S XS
1
2
12
S
−1
2
− 12
. (4)
1
1
1 1
V = √ D−1 XD−1 = √ DSD = √ D−1 XSD 2 .
μ
μ
μ
1
1
1
Ai = √ DAi D, i = 1, ..., m, DX = √ D−1 ΔXD−1 , DS = √ DΔSD.
μ
μ
μ
(5)
(6)
2
(V )
Define the classical kernel function ψC (t) = t 2−1 − log t, t > 0. Then −ψC
is the same as the right-hand side of last equation in (3) with
ψ(V ) = Q−1
V diag(ψ (λ1 (V )) , ψ (λ2 (V )) , ..., ψ (λn (V )))QV .
(7)
For our IP M , in place of ψC (t), we use our new kernel function ψ.
The proximity function (measure) for SDP is
Φ (X, S, μ) = Ψ (V ) = tr (ψ (V )) =
n
ψ (λi (V )) ,
(8)
i=1
We introduce the norm-based proximity measure σ as follows
σ = DX + DS = ψ (V ) =
2
tr ψ (V )2 =
DX 2 + DS 2 .
(9)
Main results
Let θ be such that 0 < θ < 1, then, we have
1. A decreasing of proximity function with our default step size α.
2. The total number of iterations is not more than
√
1
√ 1+ pq
1
6 2
1 n
2 2+
Ψ02
.
(2pq + 1) 1 + ln 3 2p Ψ0
ln
p
θ
ε
References
[1] G.M. Cho, An interior-point algorithm for linear optimization based on a
new barrier function. Appl. Mat. Comput. 218, pp. 386-395. (2011).
[2] B.K. Choi, G.M. Lee, On complexity analysis of the primal-dual interiorpoint method for semidefinite optimization problem based on a new proximity
function . Nonlinear Anal. 71, pp. 2628-2640. (2009).
[3] J. Peng, C. Roos, T. Terlaky, Self-regular functions and new search directions
for linear and semidefinite optimization. Math. Program. 93, pp. 129-171.
(2002).
[4] K. Roos, M. El Ghami, A comparative study of kernel functions for primaldual interior-point algorithms. regional centre of the Hungarian academy of sciences, university of Pannonya, Hungary, (2006). Available at http://www.isa.ewi.tudelft.nl/ roos.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Associated r-Stirling numbers
Imad Eddine Bousbaa and Hacène Belbachir
USTHB, Faculty of Mathematics
RECITS Laboratory, DG-RSDT
BP 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria
[email protected] & [email protected]
20-22 April 2015
Abstract
By the combinatorial interpretation of the associate Stirling numbers and
the r-Stirling numbers, We introduce the associated r-Stirling numbers and we
establish some recurrence relations and convolution identities. We give also
generating functions and nested sums related to binomial coefficient.
Broder [4] gives a generalization of the Stirling numbers of the first and
second kind, see [1, 3], theso-called
numbers of the first and second
r-Stirling
kind, denoted respectively nk r and nk r , by adding restriction on the elements
of Zn = 1, . . . , n : the nk r is the number of permutations of the set Zn with
k cycles such that the r first elements are in distinct cycles and the nk r is the
number of partitions of the set Zn into k subsets such that the r first elements
are in distinct subsets. They satisfy the following recurrence relations
n−1
n−1
n
=
+ (n − 1)
,
(1)
k−1
k
k
r
r
r
n
n−1
n−1
=
+k
,
(2)
k
k−1
k
r
r
r
with nk r = nk r = δn,k for k = r , where δ is the Kronecker delta, and
n
n
= k r = 0 for n < r.
k r
Comtet [5] define an other generalization of the Stirling numbers of both
kinds by adding a restriction on the number of elements by cycle or subset and
(s)
call them, for s 1, the s-associated Stirling numbers of the first kind nk
n(s)
n(s)
and of the second kind k
. The k
is the number of permutations of the
(s)
is
set Zn with k cycles such that, each cycle has at least s elements. The nk
the number of partitions of the set Zn into k subsets such that, each subset has
at least s elements. They have, respectively, the following generating functions
(s)
k
s−1 i
n
1
xn
x
,
(3)
=
− ln (1 − x) −
k
n!
k!
i
i=1
n≥sk
(s)
k
s−1 i
n
xn
x
1
.
(4)
=
exp (x) −
k
n!
k!
i!
i=0
n≥sk
52
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
We introduce the s-associated r-Stirling numbers of the both kinds, one can
see [2], as follows.
Definition 1 The s-associated r-Stirling numbers of the first kind count the
number of permutations of the set Zn with k cycles such that the r first elements
are in distinct cycles and each cycle contains at least s elements.
The s-associated r-Stirling numbers of the second kind count the number of
partitions of the set Zn into k subsets such that the r first elements are in distinct
subsets and each subset contains at least s elements.
They satisfy the following recurrence relations.
Theorem 2 Let r, k, s, and n be nonnegative integers such that n sk and
k r, we have
(s)
(s)
(s)
(s)
(n − r − 1)! n − s
n−r−1
n−s
n−1
n
+r
(s − 1)!
=
+ (n − 1)
,
s−2
k−1
k
k
(n − r − s)! k − 1
r
r
r−1
r
(s)
(s)
(s)
(s)
n−r−1
n−s
n−r−1
n−s
n−1
n
=
+r
+k
.
s−1
k−1
s−2
k−1
k
k
r
r
r−1
r
The s-associated r-Stirling numbers satisfy the following convolution relations.
Theorem 3 Let p, r, k and n be nonnegative integers such that p r k and
n sk, we have
(s)
(s)
n−p−s(k−p)
(n − r)!
i − p (s − 2) − 1 n − p − i
n
=
,
(n − r − i)!
p−1
k−p
k
i=(s−1)p
r
(s)
n
k
n−r−(s−1)p
=
r
i=p−r+s(k−p)
r−p
(n − r)!
i+r−p
i!((s − 1)!)p (n − p(s − 1) − r − i)!
k−p
(s)
pn−p(s−1)−r−i .
r−p
Their exponential generating functions are
Theorem 4 We have
(s)
n+r
xn
k+r
n!
n≥sk+(s−1)r
n≥sk+(s−1)r
(s)
r
xn
n!
s−1 i
x
i
i=1
k r
(−1)k
k!
=
xi k
xi r
1
(exp(x) −
) (exp(x) −
) .
k!
i!
i!
i=0
i=0
r
n+r
k+r
=
ln (1 − x) +
s−1
xs−1
1−x
,
s−2
References
[1] H. Belbachir and I. E. Bousbaa. Convolution identities for the r-Stirling
numbers. Submitted.
[2] H. Belbachir and I. E. Bousbaa. Associated Lah numbers and r-Stirling
numbers. ArXiv, arXiv:1404.5573v2(10), 2014.
[3] H. Belbachir and I. E. Bousbaa. Combinatorial identities for the r-Lah
numbers. Ars Combinatoria, 115:453–458, 2014.
[4] A. Z. Broder. The r-Stirling numbers. Discrete Math., 49(3):241–259, 1984.
[5] T. Comtet. Advanced Combinatorics. D. Reidel, Boston, DC, 1974.
2
53
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
New Concepts of Equilibrium in Extensive Finite
Game with Uncertain Variables
Achemine Farida and Fahem Karima
University of Mouloud Mammeri, Tizi-ouzou,Algeria
email : [email protected],[email protected]
20-22 April 2015
Abstract
In this paper, we consider a finite extensive game in which payoffs functions
are uncertain variables. We introduce new concepts of equilibrium and provide
sufficient conditions to the existence of our concepts. Finally, we present a
numerical example which illustrate these new concepts.
1
Problem Description and its Solution
We consider a finite extensive game with perfect information and chance moves.
We assume that the payoff function of each player is an uncertain variable.
Γ = N, H, P, fc , (ui )
(1)
Where
N = {1, ...., n} is the set of players.
A set H of finite consequences that satisfies the following two properties.
- The empty sequence ∅ is a member of H. Empty history means that nothing
has happened yet.
- If (ak )k=1,...,K ∈ H and L < K then (ak )k=1,...,L ∈ H.
(Each member of H is a history; each component of a history is an action taken
by a player.)
A history (ak )k=1,...,K ∈ H is terminal if there is no aK+1 such that (ak )k=1,...,K+1 ∈
H. The set of terminal histories is denoted T .
A function P from the nonterminal histories in H to N ∪{c} : P : H\T −→ ∪{c}.
P (h) = c means that chance determines the action taken after the history h.
For each h ∈ H with P (h) = c, fc (.|h) is a probability measure on the action
set of history h; each such probability measure is assumed to be independent
of every other such measure. (fc (a|h) is the probability that a occurs after the
history h.)
- Let h be a history of length k; we denote by (h; a) the history of length
k + 1 consisting of h followed by a.
We interpret such a game as follows. After any nonterminal history h player
54
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
P (h) ∈ N chooses an action from the set A(h) = {a / (h, a) ∈ H}.
If P (h) = c, chance determines the action taken after the history h.
The empty history is the starting point of the game.
For each possible choice a0 from this set player (or chance) P (a0 ) subsequently
chooses a member of the set A(a0 ); this choice determines the next player to
move, and so on.
- Each player, when making any decision, is perfectly informed of all the events
that have previously occurred (perfect information).
At the end of the game, each player i receives a gain ui . We assume that ∀i ∈ N ,
ui is an uncertain variable of an uncertainty space (Γ, L, M) [].
Using the ranking criterions given by Liu [1], we propose the generalization of
the Nash equilibrium [2] to the game (1).
References
[1] B. Liu, Uncertainty theory (2nd ed.) (Springer-Verlag, Berlin 2007)
[2] Nash, Jr., John F. Noncooperative games. Annals of Mathematics (1951)
289-295.
[3] M.J. Osborne and A. Rubinstein. A Course In Game Theory. The MIT
Press, Cambridge, Massachusetts and London, England, 1994.
2
55
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Linear recurrence sequence associated to rays of
negatively extended Pascal triangle
Hacène Belbachir1 and Fariza Krim2
1
USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT
2
USTHB, Faculty of Mathematics, LAROMADE Laboratory, DG-RSDT
P.B. 32 El Alia, 16111, Algiers, Algeria.
[email protected] & [email protected]
20-22 April 2015
Abstract
We consider the extension of Pascal’s triangle to negatively rows and describe
the recurrence relation associated to the sum of diagonal elements laying along
a finite ray. We also give the corresponding generating function. We conclude
by an application to negatively subscripted Fibonacci numbers.
Our aim is to extend the results of Belbachir, Komatsu, Szalay [1] to the
negative subscripted Pascal’s triangle. We use Sprugnoli’s approach to do the
work, see[2].
gives the number of k combinations of n
The positive integer (−1)k −n
k
with repetition,
−n
= (−1)k n+k−1
(1)
k
k
−n
The entry k in the extension of Pascal’s triangle to negatively rows is
determined for n ≥ 1, as
⎧ −n+1 −n − k−1
for k > 0,
k
−n ⎨
=
1
for
k = 0,
k
⎩
0
for k < 0.
(r,q,p)
Our aim is to determine V−n
in the negative Pascal’s triangle.
(r,q,p)
V−n
=
n−1
−q
k=0
, the sum of elements lying along a finite ray
−n − qk −n−p−(r+q)k p+1+rk
y
x
p + rk
with V0 = 0, and for r ∈ N+ , q ∈ N− , p ∈ N, p < r.
(r,q,p)
Theorem 1 The terms of sequence V−n
satisfy for n > −r −q the linear
n
recurrence relation
r
r
2 r
r
V−n−2 + · · · + (−x)
V−n−r = y r V−n−q−r . (2)
V−n−1 + x
V−n − x
2
r
1
56
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
We also determine the generating function of sum of elements lying along a
finite ray
(r,q,p)
Theorem 2 For q < 0, r+q < 0, the generating function associated to (V−n )n∈N
is given by
y p+1 z r−p−1
(−1)p
z 1−
x
yxr
G (z) = .
z r
+ (−1)r
z −q
1−
x
x
We conclude by the following nice application to the negative subscripted
Fibonacci numbers.
Proposition 3 We have the following identity
n−1
−n + k
−n
k
(−i − 2)
=
F−n+1 = Re i
2k
k=0
(−1)
0≤2j<k<n−2
k+j j
4
−n + k + 1
2k
References
[1] H. Belbachir, T. Komatsu and L. Szalay, Linear recurrences associated to
rays in Pascal’s triangle and combinatorial identities, Math. Slovaca 64
(2014), no. 2, 287–300.
[2] D. E. Knuth, The Art of Computer Programming, Vol. 1., Second edition,
Addison-Wesley Publishing Company, 1981.
[3] R. Sprugnoli, Negation of binomial cefficients, Discrete Math. 308 (2008),
5070-5077.
2
57
k
.
2j
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Bell polynomials in combinatorial Hopf algebras I
Ali Chouria
Laboratoire LITIS - EA 4108, Université de Rouen,
Avenue de l’Université BP 876801 Saint-tienne-du-Rouvray Cedex
[email protected]
20-22 April 2015
Abstract
We show that most of the results on Bell polynomials can be written in terms
of symmetric functions and transformations of alphabets. Using specializations,
sum and product of alphabets we establish more easily many equalities on Bell
polynomials by manipulation of the generating functions and prove existent ones.
1
Bell polynomial as symmetric functions
Partial multivariate Bell polynomials were defined by E.T. Bell [1] in 1934. Their
applications in Combinatorics, Analysis, Algebra, Probabilities etc. are numerous
(see e.g [4]). They are usually defined on an infinite set of commuting variables
{a1 , a2 , . . .} by the following generating function:
tn
tm k
1 Bn,k (a1 , . . . , ap , . . . ) = (
am
(1)
) .
n!
k! m1
m!
n0
Without loss of generality, we will suppose a1 = 1in the sequel.Indeed, if a1 = 0,
a
then one obtains Bn,k (a1 , . . . , ap , . . . ) = ak1 Bn,k 1, aa21 , · · · , ap1 and when a1 =
n!
0, Bn,k (0, a2 , . . . , ap , . . . ) = (n−k)!
Bn,k (a2 , . . . , ap , . . . ) if n ≥ k and 0 otherwise.
These polynomials are related to several combinatorial sequences which involve set
partitions. For instance, Bn,k (1, 1, . . . ) is the Stirling number of the second kind
Sn,k , which counts the ways to partition a set of n elements into k non empty
subsets.
The sum of all monomials of
total degree n in the variables x1 , x2 , . . . , is the complete
symmetric function hn = |λ|=n mλ . It is well known that hn and the power sum
symmetric functions, pn (X), defined by pn (X) = i1 xn
i generate the algebra of
symmetric fuctions Sym [3].
Given two alphabets X and Y, we also define (as in [2]) the alphabet X + Y
by pn (X + Y) = pn (X) + pn (Y), and for α ∈ C, the alphabet αX (resp XY) by:
pn (αX) = αpn (X)( resp pn (XY) = pn (X)pn (Y)). For our purpose, and without loss
of generality, we consider an alphabet X satisfying a1 = 1 and ai = i!hi for all i > 1.
k
k
ai i
tn
1
= tk! σt (kX), where
One obtains:
i1 i! t
n0 Bn,k (a1 , a2 , . . . ) n! = k!
σt (X) =
n0
hn (X)tn =
(1 − xi t)−1 = exp{
i1
n1
58
pn (X)
tn
}.
n
(2)
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Theorem 1 Hence, for any alphabet X (for hn (X) = n!hn (X)), we have
n!
n Bn,k (1, 2!h1 , . . . , (m + 1)!hm (X), . . . ) = hn−k (kX) =
hn−k (kX).
k
k!
2
(3)
Binomial functions and sums of alphabets
In this section we are interested in some equalities of [4]. Let us first recall that a
binomial
is a family of functions (fn )n∈N satisfying f0 (x) = 1 and fn (a +
n sequence
b) = n
k=0 k fk (a)fn−k (b) for all a, b ∈ C and n ∈ N. This last identity is nothing
but the sum of two alphabets stated in terms of modified complete functions hn .
Theorem 2 As a direct consequence of (3), we obtain
n n
Bn,k (1, . . . , ifi−1 (a), . . . ) = Bn,k (A) =
hn−k (kA) =
fn−k (ka).
(4)
k
k
Using properties of symmetric functions, we establish the following result,
n
n
k 1 + k2
Bi,k1 Bn−i,k2 .
Bn,k1 +k2 =
i
k1
i=0
(5)
The results involving binomial functions can be seen as a generalization of the
(so-called) convolution formula for Bell polynomials (see e.g. [4]):
n−k
i=k
n n
Bi,k (a1 , a2 , . . . )Bi,k (b1 , b2 , . . . ) =
Bn−k,k
i
k
m
1 m + 1
ai bm+1−i , . . .
a1 b 1 , . . . ,
i
m + 1 i=1
Again, without loss of generality, we can consider a1 = b1 = 1, ai = i!hi−1 (X)
and bi = i!hi−1 (Y). This equality can be proved directly using the standard rules
involving the sum of two alphabets. Let (an )n and (bn )n be two sequences of
numbers such that a1 = b1 = 1 and a−n = b−n = 0 for each n ∈ N. Consider
also three integers k, k1 and k2 such that k = k1 k2 . The following identity seems
laborious to prove:
b
a
λ
−i+j+1
λ
−i+j+1
i
i
det det Bn,k . . . , n!
(λi − i + j + 1)! , . . . =
(λi − i + j + 1)! λn−1
Bλi −i+j+k2 ,k2 (b1 , b2 , . . . ) Bλi −i+j+k1 ,k1 (a1 , a2 , . . . ) n! (λ)
.
(k1 !k2 !)
det det k!
(λi − i + j + k1 )!
(λi − i + j + k2 )!
λn−k
But it looks rather simpler when we recognize Bn,k (XY) =
hn (kXY) = λn sλ (k1 X)sλ (k2 Y).
n!
h
(kXY)
k! n−k
and apply
References
[1] E. T. Bell. Exponential polynomials. Annals of Mathematics, pages 258–277,
1934.
[2] J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet. Word symmetric functions
and the Redfield-Pólya theorem. DMTCS Proceedings, (01):563–574, 2013.
[3] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford University
Press, 1998.
[4] M. Mihoubi. Polynômes multivariés de Bell et polynômes de type binomial.
Thèse de Doctorat, L’Université des Sciences et de la Technologie Houari
Boumediene, 2008.
2
59
.
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
A game-based algorithm assigning selfish users on a
road network
1
Lahna IDRES1
Mohammed Said RADJEF1
Laboratory LaMOS, Department of Operational Research ,
Bejaia University, Algeria.
20-22 April 2015
Abstract
We investigate the problem of road traffic assignment, with N selfish users,
sharing the same Origin-Destination pair. We use a network congestion game
model to describe the interaction among the network’s users. We focus on
Rosenthal’s results to guarantee the existence of Pure Nash Equilibrium (PNE).
Then, we study the applicability of an algorithm basing on GBR principle in
finding a PNE, for a general network.
keywords: Traffic assignment, Congestion Game, Nash Equilibrium, Algorithm.
1
Introduction
The classical mathematical model of traffic assignment was formulated by Beckmann [1]. The solution of this model is a Wardrop equilibrium [8]. Nevertheless,
the variables of the model are assumed to be non-negative. Thus, we cannot
guarantee that the solution is an integer one. As the flow is indivisible, we may
have some difficulties to interpret the result. This relevant remark was raised
by Rosenthal [6]. He also proposed to use non-cooperative game theory to deal
with this disagreement. In this case the adequate solution concept is Nash equilibrium, which -under some conditions– is an equivalent of Wardrop equilibrium
[4].
The problem:
In our work, we consider N selfish road users and we look for an integer assignment. This assignment must verify Nash conditions. That is to say, once the
assignment done, none of the users would like to change unilaterally his path.
2
The methodology used
We model the interaction among a road network’s users (wishing to travel from
a common origin to a common destination) as a network congestion game[5].
Where the users are the players, their resources are the network’s arcs (roads),
60
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
the strategies are the paths (itineraries) and their costs are given by a BPR
function[7].
We focus on Rosenthal’s results[5] to guarantee the existence of Pure Nash
Equilibrium (PNE). Then, we use an algorithm basing on GBR principle[2] and
establish sufficient conditions that make it succeeds in finding PNE for a general
network. To ensure obtaining a PNE, we complete the algorithm by a procedure
that checks if the result returned is a PNE. Otherwise the solution is iteratively
corrected till it becomes a PNE.
3
Our contribution
The advantage of this approach over the classical approach (Beckmann’s model)
is that provides an integer assignment. The main contribution of our work is
the study of an algorithm finding a PNE to the network congestion game under
consideration. The following table summarizes the essential differences between
the previous works and ours.
[3]
[2]
Our work
Algorithm
GBR
GBR
GBR principle
GBR principle +
correcting procedure
applicability
Serries-parallel graphs
Parallel-graphs
General graph
under conditions
General graph
complexity
Polynomial
Polynomial
Polynomial
Polynomial
under conditions
We have tested the algorithm on seven subnetworks of Bejaia city. The results
are encouraging and the extension of the work to several origin-destination pairs
is promising.
References
[1] M. Beckmann, C.B. McGuire, C.B. Winsten: Studies in the economics of
transportation. Yale University Press.(1956).
[2] D. Fotakis, S. Kontogiannis, P. Spirakis: Symmetry in network congestion
games: Pure equilibria and anarchy cost. 3 rd International wokshop WAOA.
161-175. (2006).
[3] D. Fotakis, S. Kontogiannis, E. Koutsoupias, M. Mavronicolas, P. Spirakis:
The structure and complexity of Nash equilibria for a selfish routing game.
Theoretical computer science, 3305-3326. (2009)
[4] A. Haurie, P. Marcotte: On the relationship between Nash-Cournot and
Wardrop Equilibria. Networks vol.(15), 295-308. (1985).
[5] R.W. Rosenthal: A class of game possessing pure-strategy Nash equilibria.
Int. J. Game Theor vol.(2), 65-67. (1973).
[6] R.W. Rosenthal: The network Equilibrium Problem in Integers. Networks
vol.(3), 53-59. (1973).
[7] Sétra. Approche de la congestion routière: Méthode de caclcul du temps
gêné. technical report. Service d’étude sur les transports, les routes et leurs
aménagement (2009).
[8] J.G. Wardrop: Some theoritical aspect of road traffic research. Proceedings
of the Institute of civil engineers. Part II vol.(1), 325-378. (1952)
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Problème du flowshop à deux machines avec des
opérations couplées sur la première machine
Meziani Nadjat
Université Abderrahmane Mira Béjaia, Algérie
[email protected]
Oulamara Ammar
LORIA - UMR 7503, Université de Lorraine, Campus Scientifique - BP 239,
54506 Vandoeuvre-les-Nancy Cedex, France
[email protected]
Boudhar Mourad
Laboratoire RECITS, Faculté de Mathématiques,USTHB,
BP 32, Bab-Ezzouar, El-Alia 16111, Alger, Algérie
[email protected]
20-22 April 2015
Résumé
Dans ce travail, nous étudions le problème du flowshop à deux machines avec
des tâches couplées telle que chaque tâche est composée de deux opérations séparées par un délai exact sur la première machine et d’une seule opération sur la
deuxième machine. Le but est de minimiser le makespan. Nous abordons la complexité d’un sous problème et nous montrons qu’il est NP-difficile comme nous
présentons quelques sous problèmes polynomiaux. Nous proposons également
des heuristiques pour la résolution du problème général avec des expérimentations numériques.
Mots clès : flowshop, tâches couplées, time lag, makespan
1
Introduction
Le problème d’ordonnancement de tâches couplées a été introduit pour la
première fois par Shapiro [3]. Le problème consiste à ordonnancer un ensemble
de n tâches sur une seule machine. Une tâche est constituée de deux opérations,
séparées par un délai exact, qui s’exécutent dans l’ordre. Chaque tâche est notée
par le triplet (aj , Lj , bj ) où aj et bj représentent les temps d’exécution de la
première et de la deuxième opération respectivement et Lj le délai qui s’écoule
entre la date de fin de traitement de la première opération et de la date de début
d’exécution de la deuxième opération. Pendant ce délai, la machine est inactive
et une autre tâche ou opération peut être exécutée.
La motivation de ce problème vient d’un problème d’ordonnancement des
tâches d’un radar qui consiste dans l’émission des impulsions et la réception
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
des réponses après un temps d’attente. Ce problème apparaît également dans
des ateliers de production chimique où une machine doit exécuter plusieurs opérations d’une même tâche et un délai exact est imposé entre l’exécution de
chaque deux opérations consécutives. Orman et Potts [2] ont étudié la complexité du problème de tâches couplées sur une seule machine pour minimiser le makespan. Dans [1], les auteurs ont abordé la complexité du problème
F 2/Coup − Oper(1), ai , bi = Li = p, ci /Cmax et ils ont exposé des sous problèmes polynomiaux. Dans ce travail, nous considérons le problème du flowshop
à deux machines. Chaque tâche contient un couple d’opérations O1,j sur la première machine et une seule opération O2,j sur la deuxième machine. Chaque
couple d’opérations de tâche O1,j est notée par le triplet (aj , Lj , bj ) telle que aj
et bj sont les durées de traitement de la première et de la deuxième opération
du couple d’opération O1,j respectivement. cj est le temps d’exécution de l’opération O2,j qui ne peut être traitée que si le traitement du couple d’opérations
O1,j est terminé. Notre objectif est de minimiser le makespan.
2
Résultats
Nous avons prouvé que le problème F 2/Coup − Oper(1), ai = Li = p, bi , ci /
Cmax est NP-difficile et que les problèmes F 2/Coup−Oper(1), ai = Li = p, bi =
b, ci /Cmax et F 2/Coup − Oper(1), ai = bi = p, Li = L, ci /Cmax sont polynomiaux. Pour la résolution du problème général, nous proposons des heuristiques.
L’heuristique J_P se base sur la règle de jonhson pour déterminer la séquence
de tâches à ordonnancer et pour le calcul du makespan, nous introduisons la méthode en parallèle en considérant les tâches qui s’entrelacent. Les heuristiques
Lpt_ai _P et Lpt_bi _P se basent sur la règle LP T des ai et bi respectivement
pour déterminer la séquence de tâches à ordonnancer. L’heuristique Spt_Li _P
se base sur la régle SP T des Li . Pour le calcul du makespan de ces trois heuristiques, nous introduisons la méthode en parallèle en considérant les tâches à
entrelacer. La dernière heuristique H ∗ se base sur l’ordonnancement de tâches
suivant la règle LP T des ai et bi .
3
Conclusion
Nous avons traité le problème du flowshop à deux machine avec des opérations couplées sur la première machine séparées par un délai exact et d’une seule
opération sur la deuxième machine. Notre objectif est de minimiser le makespan.
Nous avons étudié la complexité de ses sous problèmes et nous avons proposé
des heuristiques avec des expérimentations numériques pour la résolution.
Références
[1] N. Meziani, A. Oulamara, M. Boudhar : Minimizing the makespan on twomachines flowshop scheduling problem with coupled-tasks. Proceedings du
Neuvième Colloque sur l’Optimisation et les Systèmes d’Information (COSI’2012, Tlemcen (Algérie)) : 192-203. (Mai 2012).
[2] A.J. Orman, C.N. Potts : On the complexity of coupled-Task Scheduling.
Discrete Applied Mathematics vol.(72.), 141-154. (1997).
[3] R.D. Shapiro : Scheduling Coupled Tasks. Naval Research logistics quartely
vol.(20.), 489-498. (1980).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Optimal control of dynamic systems
with random input
ZahiaBouabbache1 , M ohamedAidene2
L2CSP,Universitie Mouloud Mammeri de Tizi Ouzou, Algerie
1
bouabbache− [email protected] r
2
[email protected]
Abstract
Our work consists to solve a terminal problem of a linear dynamic system
with perturbed initial condition.The aim of the problem is to reach a target
with a given probability
Keywords
optimal control, support control, random variable, stochastic dynamic problem.
1.
Introduction
Bibliographie
When a real problem is modeled and solved by means of mathematical
may happen that some of the parameters which gure in
[1] R.problem
Gabasov.it Adaptive
method of linear programming. Preprint of the
the problem are unknown, whether it be in the objective function or
University
of Karlsruhe. Institue of Statistics and Mathematics. Karsruhe,
in the constraints. If these parameters of unknown value can be taken
Germany
(1993).
random variables the resulting problem is a stochastic problem one.
[2] B.asOksendal.
Stochastic Differential Equations. An introduction with
In this study, we assume that the components of the initial state are
Applications. Fifth Edition, corrected printing (2000).
random variables dened on space of probability (Ω, Ξ, P )
[3] E.continous
Trelat. contrôle
optimal : Theorie et application . Vuibert, collection
given, of know distribution with a probability to reach the target, so
Mathematiques
concretes (2005), 246 pages.
the stochastic dynamic system which we are considering is as follows :
⎧
J(u(t)) = E(ćx(tf )) → maxu (1)
⎪
⎪
⎪
⎪
⎨ ẋ = Ax + bu, x(t0 ) = xt0 (ω) (2)
(3)
P (Hi x(tf ) ≤ gi ) = αi
⎪
⎪
(4)
d1 ≤ u(t) ≤ d2
⎪
⎪
⎩
t ∈ T = [t0 , tf ]
(Ip )
x(t) ∈ n , x(t0 ) the initial state,A ∈ n × n , the rank of
H = m < n,b ∈ m , g ∈ m ,ć ∈ n ,u(t) ∈ ,t ∈ T the control,J(u(t))
where,
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Some identities for the bivariate hyperfibonacci
polynomials
Amine Belkhir and Hacène Belbachir
USTHB, Faculty of Mathematics
RECITS Laboratory, DG-RSDT
BP 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria
[email protected] & [email protected]
20-22 April 2015
Abstract
We define the bivariate hyperfibonacci polynomials and give a combinatorial
interpretation using tilings approach, which allowed us to obtain some properties
such as recurrences relations and explicit formulas. Relations with bivariate
incomplete Fibonacci polynomials are obtained. Furthermore, we establish their
generating function.
The hyperfibonacci numbers were introduced by Dil and Mező [5], there are
defined by the relation.
(r)
(r)
Fn(r) = Fn−1 + Fn(r−1) , with Fn(0) = Fn and F0
(r)
= 0, F1
= 1,
where r is a positive integer and Fn is the n-th Fibonacci number defined recursively by
Fn = Fn−1 + Fn−2 , for n ≥ 2, and F0 = 0, F1 = 1.
Definition 1 The bivariate hyperfibonacci polynomials are defined by the recurrence relation
(r)
(1)
Fn(r) (x, y) = xFn−1 (x, y) + yFn(r−1) (x, y),
(0)
(r)
with the initials Fn (x, y) = Fn (x, y), F0 (x, y) = 0 and Fn (x, y) is the bivariate Fibonacci polynomials defined by F0 (x, y) = 0, F1 (x, y) = 1, and for
n ≥ 2,
(2)
Fn (x, y) = xFn−1 (x, y) + yFn (x, y),
In [3], the authors gave tilings proofs of the hyperfibonacci numbers and some
others proprieties. Our aim is to define the bivariate hyperfibonacci polynomials
and provide a combinatorial interpretation. Further, relation with bivariate
incomplete Fibonacci polynomials are deduced.
The relation (1) can be generalized as follows.
Theorem 2 [2] For any n, r ≥ 0, we have
r
r k r−k (r−k)
(r)
Fn+k (x, y).
y x
Fn+r (x, y) =
k
k=0
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(3)
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Remark 3 If we take x = y = 1 the following identity holds,
r
r
(r)
(r−k)
Fn+r =
.
F
k n+k
(4)
k=0
In the following identity, we express the hyperfibonacci polynomials in terms
of Fibonacci polynomials.
Theorem 4 [2] For any n ≥ 0, we have
n
n + r − k − 1 r n−k
(r)
Fn (x, y) =
Fk (x, y).
y x
r−1
(5)
k=1
References
[1] T. Amdeberhan, Xi Chen, V. Moll, and B. E. Sagan. Generalized Fibonacci
polynomials and Fibonomial coefficients, arXiv:1306.6511, 2013.
[2] H. Belbachir, A. Belkhir. The combinatorics of bivariate hyperfibonacci and
hyperlucas polynomilas, Submitted.
[3] H. Belbachir, A. Belkhir. Combinatorial expressions involving Fibonacci,
Hyperfibonacci, and incomplete Fibonacci numbers, Journal of Integer Sequences, Article 14.4.3, vol 17, (2014).
[4] A. T. Benjamin, J. J. Quinn. Proofs that really count : The Art of Combinatorial Proof, The Mathematical Association of America,2003.
[5] A. Dil, I. Mező. A symmetric algorithm for hyperharmonic and Fibonacci
numbers, Appl. Math. Comput. 206 (2008), 942–951.
[6] A. Pinter, H. M. Strivastava. Generating Functions of the Incomplete Fibonacci and Lucas Numbers, Rend. Circ. Mat. Palermo 48.2 (1999): 591–
596.
[7] M. Shattuck. Combinatorial identities for incomplete tribonacci polynomials,
arXiv:1406.2755v1, 2014.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Successive associated Stirling numbers
Assia Fettouma Tebtoub and Hacène Belbachir
USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT
Po. Box 32, El Alia,16111, Algiers, Algeria
[email protected] & [email protected]
20-22 April 2015
Abstract
Using a combinatorial approach, we introduce successive associated Stirling
numbers with constraints. We give the recurrence relation and the generating
function.
Keys words: Successive associated Stirling numbers; Recurrence relations;
Generating function.
The Stirling numbers of second kind, denoted by, S(n, k) are defined as:
xn =
n
x(x − 1) · · · (x − k + 1)S(n, k).
k=0
It is well known that S(n, k) is the number of partitions of the set {1, 2, · · · , n}
into k non empty subsets. It satisfies the following recurrence relation:
S(n, k) = k S(n − 1, k) + S(n − 1, k − 1).
For other properties one can see [1, ch. 5],[2] and [3, ch. 4].
The associated Stirling numbers of second kind denoted by S2 (n, k) (see [3]),
are the number of partitions of the set {1, 2, ..., n} into k parts which contains
at least 2 elements. They satisfy the recurrence relation
S2 (n, k) = k S2 (n − 1, k) + (n − 1)S2 (n − 1, k − 1).
It is important in combinatorics to know if the sequences lying over rays
of arithmetical triangles are unimodal or not; specially in theorical computer
science, the mode constitute the maximal value to consider in programming.
The examples knowing in the literature are closed to Pascal’s triangle. It is ,
for us the first example of sequences lying over diagonal of Stirling triangle (as
arithmetical triangle) which are log-concave and then unimodal.
Our aim is to introduce successive associated Stirling numbers with some
contraints.
We denote these numbers by T2 (n, k) where n ≥ 2k. Based on the combinatorial
interpretation , we derive recurrence relations and we establish combinatorial
identity and also compute the generating functions.
We start by introducing the successive associated Stirling numbers,
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Definition 1 The successive associated Stirling numbers, denoted by T2 (n, k),
count the number of partitions of the set {1, 2, ..., n} into k parts, so that, each
part contains at least two consecutive numbers and such that the last element n:
• even it forms only a part with it’s predecessor, or
• it belongs to another part which satisfy already the previous property.
They satisfy the following recurrence relation.
Theorem 1
T2 (n, k) = kT2 (n − 1, k) + T2 (n − 2, k − 1), (n ≥ 2),
(1)
where T2 (0, 0) = 1, T2 (n, n − 1) = 0 and T2 (n, 0) = 0 (∀n ≥ 1).
The successive associated Stirling numbers the following vertical recurrence
relation.
Theorem 2 Let n, k ∈ N
T2 (n, k) =
n−2k
ki T (n − i − 2, k − 1).
(2)
i=0
Now, we give the generating function of successive associated Stirling numbers and an explicit expression.
Theorem 3 For m ≥ 1, the generating function of the successive associated
Stirling numbers is given, by:
Ak (x) :=
n≥2k
T2 (n, k)xn =
x2k
.
(1 − x)(1 − 2x) · · · (1 − kx)
(3)
Theorem 4 The successive associated Stirling number T2 (n, k) is given by
T2 (n, k) =
k
1 k n−k
(−1)(k−p)
for n ≥ 2k.
p
p
k! p=1
(4)
with T2 (0, 0) = 1 and T2 (n, k) = 0 for n < 2k.
References
[1] L. Comtet. Advanced Combinatorics. D. Reidel, Boston, DC, 1974.
[2] R. L. Graham, D. E. Knuth, O. Patashnik. Concrete Mathematics. AddisonWesley, Reading, MA, second edition, 1994.
[3] J. Riordan. An introduction to combinatorial analysis. Dover Publications Inc, Mineola, NY, 2002. Reprint of 1958 original [Wiley, New York;
MR0096594 (203077)].
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Stirling and Bell Numbers of Some Join Graphs
A. Medjerredine1 , H. Belbachir1 and M.A Boutiche2
1
Laboratory LRECITS,
2
Laboratory LaROMaD,
Faculty of Mathematics, USTHB,
BP 32 El Alia, 16111 Bab Ezzouar, Algeria.
[email protected], {hbelbachir,mboutiche}@usthb.dz
20-22 April 2015
Abstract
for a graph G is the number of
The Stirling number of the second kind G
k
independent partitions of V (G) into k subsets. The Bell number BG for a graph
G is the number of independent partitions of V (G). In this paper, we determine
the Stirling and the Bell numbers of join of two special graphs.
Key words: Stirling numbers, Bell numbers, Join graphs.
1
Introduction
Let G be a simple (finite) graph. A partition of V (G) is called an independent
partition if each block is an independent vertex set (i.e. adjacent vertices belong
to distinct blocks). Then for
a positive integer k ≤ |V (G)|, let the Stirling
for graph G be the number of independent
number of the second kind G
k
= 0, and define the Bell
partitions of V (G) into k subsets, moreover let G
0
number BG for graph G as the number of independent partitions of V (G). Then
G
|V (G)|
BG =
k=0
k
.
It’s known that for any simple graph G, G
= 0 if 0 ≤ k ≤ χ(G) − 1, where
k G = 1 and |V (G)|−1
=
χ(G) is the chromatic number of G. Moreover |V G
(G)|
|V (G)|(|V (G)| − 1)/2 − |E(G)|.
Stirling and Bell numbers can be computed using a recurrence relation on
graphs (see Theorem 1 by Duncan and Peele [2]).
Theorem
1 If G
G G−e
is a simple graph, e ∈ E(G) and 0 ≤ k ≤ |V (G)| − 1, then
=
− G/e
and BG = BG−e − BG/e , where G − e and G/e are the
k
k
k
simplified graphs obtained by deleting and contracting edge e from G, respectively.
Stirling and Bell numbers of graphs were defined by Duncan and Peele [1],
Kereskényi-Balogh and Nyul [3] summarized known properties related to these
numbers. Duncan [2] described these numbers for graphs having two components.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
2
Main Results
For vertex-disjoint graphs G and H, their join G ∨ H is the supergraph of G ∪ H
in which each vertex of G is adjacent to every vertex of H and both G and H
are induced subgraphs. Duncan [3] showed that BG∨H = BG .BH . Our main
contribution in this paper is to show the Stirling number of join graphs.
Theorem 2 Let G and H be simple graphs of order n ≥ 1, p ≥ 1 respectively,
with χ(G) + χ(H) ≤ k ≤ n + p. Then
G∨H
k
=
k G
H
j=0
j
k−j
.
Knowing the Stirling and the Bell numbers of some fewer graphs; En , Kn ,
Sn , Pn , Pn , Cn , Cn , Km,n that denote the empty graph, the complete graph,
the star graph, the path graph, the complementary of the path graph, the cycle
graph, the complementary of the cycle graph and the complete bipartite graph,
respectively, we compute the Stirling and the Bell numbers of some join of two
special graphs listed above.
References
[1] B. Duncan and R. Peele, Bell and Stirling numbers for graphs, J. Integer
Seq. 12 (2009), Article 09.7.1.
[2] B. Duncan, Bell and Stirling numbers for disjoint unions of graphs, Congr.
Numer. 206 (2010), 215-17217.
[3] Z. Kereskényi-Balogh and G. Nyul, Stirling numbers of the second kind and
Bell numbers for graphs, Australasian Journal of Combinatorics 58(2) (2014),
264-17274.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Shifted domino tableaux
and the super shifted plactic monoid
Zakaria Chemli
University of Paris-Est Marne-la-Vallée, LIGM
5 Boulevard Descartes, 77420 Champs-sur-Marne, France
[email protected]
20-22 April 2015
Abstract
We introduce new combinatorial objects called the shifted domino tableaux (ShDT). These objects
are in bijection with the pairs of shifted Young tableaux. This bijection shows that these objects can
be seen as elements of the super shifted plactic monoid (SShPl), which is the shifted analog of the
super plactic monoid.
Young tableaux were introduced by Young [You00] more than a century ago as a tool for invariant
theory. Later, he showed that they can give informations about representations of symmetric groups.
Since, Young tableaux play an important role in many fields of mathematics, from enumerative combinatorics to algebraic geometry. They allow to define certain symmetric functions: Schur functions,
which encode the characters of the irreductible representations of symmetric groups. Lascoux and
Schützenberger in [SL81] endowed the set of Young tableaux with the structure of a monoid (the
plactic monoid), which is of great importance for applications in representation theory and the theory of symmetric functions. The first significant application of the plactic monoid was to provide a
complete proof of the Littlewood-Richardson rule, a combinatorial algorithm for multiplying Schur
functions, which has been used for almost 50 years before being fully understood.
By extending Young tableaux to shifted Young tableaux, Sagan [Sag87] and Worley [Wor84]
developed independently a combinatorial theory of shifted Young Tableaux. The shifted Young
tableaux allow to define the P-Schur and the Q-Schur functions. These functions represent the
characters of irreductible projective representations of symmetric groups. A first version of the
shifted Littlewood-Richardson rule was proved by Stembridge in [Ste89]. A few years ago, Serrano
[Ser10] introduced the shifted plactic monoid, a shifted analog of the plactic monoid with similar
properties and applications. The author used the shifted plactic monoid to give a new proof and a
new version of the shifted Littlewood-Richardson rule.
The domino tableaux are another extension of Young tableaux. Carré and Leclerc in [CL95]
studied a bijection between domino tableaux and pairs of Young tableaux. The authors gave an
easier description of this bijection which highlights the role of the diagonals of tableaux. This allows
them to extend the plactic monoid of Lascoux and Schützenberger to dominoes, which define the
super plactic monoid. They also introduced a new expression of the Littlewood-Richardson rule in
terms of particular domino tableaux, and presented a new family of symmetric functions defined
by domino tableaux: The H functions (Hλ (q)) which depend on a parameter q, specialize to Schur
functions for q = 0, to a product of two Schur functions for q = 1, and to a plethysm of Schur
functions for q = −1 and for certain partitions. This specifically entailing a combinatorial description
of some plethysms of Schur functions.
In this paper, we extend naturally shifted Young tableaux to new combinatorial objects: The
shifted domino tableaux, a shifted analog of domino tableaux. The purpose of this extension is to
develop a theory of shifted domino tableaux parallel to the theory of domino tableaux, to investigate
whether as in the case of domino tableaux, the shifted domino tableaux will lead to the definition of
a new family of symmetric functions that is the shifted analog of the H functions [CL95].
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Definition 1. Given a certain partition λ paved with dominoes, a ShDT is a filling of the dominoes
which are cut by D2k (see [CL95] for more details) for any positive integer k by X, and the remaining
dominoes by letters in {1 < 1 < 2 < 2 < · · · } such that
- rows and columns are non-decreasing from left to right and from bottom to top;
- a letter ∈ {1, 2, 3, . . . } appears at most once in each column;
- a letter ∈ {1 , 2 , 3 , . . . } appears at most once in each row;
- there is no letter ∈ {1 , 2 , 3 , . . . } on the diagonal D0 .
We give below an example of ShDT of shape (8, 5, 5, 5, 5):
T =
X
X
X
4
X
2
1
2’
1
2’
5
5’
2
3
.
One of our most important result is a genaralisation of the correspondance between the set of
domino tableaux and the set of pairs of Young tableaux given in [CL95] to the shifted case.
Theorem 1. Let λ be a valid shape of ShDT of 2-quotient (μ, ν) [Mac95]. The set of ShDT of shape
λ and the set of pairs (t1 , t2 ) of ShYT of shape (μ, ν) are in bijection.
For example, the image of the ShDT T is the following pair of ShYT:
⎞
⎛
⎝
X X 5
X 4
1 2’
⎠
X 2 5’
,
1 2’ 2
3
Theorem 1 allows us to extend the shifted plactic monoid (ShPl) to dominoes, which define the
super shifted plactic monoid.
Definition 2. Let A1 and A2 be two totally ordered infinite alphabets. The super shifted plactic
monoid on A1 ∪ A2 , denoted by SShPl(A1 , A2 ) is the quotient of the free monoid (A1 ∪ A2 )∗ generated
by the shifted plactic relations of ShPl(A1 ) and ShPl(A2 ) such that the letters of A1 commute with
the letters of A2 .
Theorem 2. Each super shifted plactic class is represented by a unique ShDT.
The shifted domino tableaux will lead as in the case of domino tableaux, to the definition of a new
family of symmetric functions, the shifted analog of the H functions of Carré and Leclerc [CL95].
This analog functions can allow us to shed lights on the combinatorial properties of Q-Schur and
P-Schur functions. We expect also to find a new expression of the shifted Littlewood-Richardson rule
with coefficients in terms of shifted domino tableaux. Another direction, in which we can investigate,
is to find properties and applications of the super shifted plactic monoid.
References
[CL95]
C. Carré and B. Leclerc. Splitting the square of a Schur function into its symmetric and
antisymmetric parts. Journal of algebraic combinatorics, 4(3):201–231, 1995.
[Mac95] I. G. Macdonald. Symmetric functions and Hall polynomials. New York, 1995.
[Sag87] B. E. Sagan. Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley. Journal
of Combinatorial Theory, Series A, 45(1):62–103, 1987.
[Ser10]
L. Serrano. The shifted plactic monoid. Mathematische Zeitschrift, 266(2):363–392, 2010.
[SL81]
M. P. Schützenberger and A. Lascoux. Le monoı̈de plaxique. Ricerca Scient, (109), 1981.
[Ste89]
J. R. Stembridge. Shifted tableaux and the projective representations of symmetric groups.
Advances in Mathematics, 74(1):87–134, 1989.
[Wor84] D. R. Worley. A theory of shifted Young tableaux. PhD thesis, Massachusetts Institute of
Technology, 1984.
[You00] A. Young. On quantitative substitutional analysis. Proceedings of the London Mathematical
Society, 1(1):97–145, 1900.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Logistique, concurrence oligopolistique et qualité
des produits alimentaires
Lamia Meziani, A.Hakim Hammoudi et Mohammed Said Radjef
20-22 Avril 2015
Résumé
Ce travail s’inscrit dans le contexte des rapports Nord/Sud et de la régulation
internationale de la sécurité sanitaire des aliments. Il s’agit d’évaluer l’effet de la
logistique ainsi que des contrôles effectués aux frontières des pays développés sur
la concurrence et le comportement stratégique des Producteurs/Exportateurs
(Ps/Es) des pays en développement et des pays développés, notamment sur
l’incitation à investir dans la qualité des pratiques de production afin de se
conformer à la norme publique exigée par les pays développés pour protéger la
santé des consommateurs. La possibilité d’un soutien des PD à l’amélioration de
la logistique des PED est étudiée. Le modèle présenté est un modèle d’économie
industrielle. L’amélioration des infrastructures logistiques dans les pays en développement est analysée avec ses effets sur le comportement stratégique de ces
derniers, ainsi que sur leurs concurrents, et sur la réduction du risque sanitaire
dans le pays importateur.
1
Description du modèle
On considère une chaîne d’approvisionnement en un certain produit d’un
pays développé (PD) importé à partir de deux catégories de pays indexées par
i = 1, 2 : la première comprend N1 Pays En Développement (PED) et la seconde N2 Pays Développés (PD). Chaque pays est représenté par un Producteur/Exportateur (P/E). Les Ps/Es des PED sont de taille q1 et les Ps/Es des
PD sont de taille q2 .
Ces deux catégories de pays se font concurrence sur leurs niveaux d’investissement pour que leurs produits soient conformes à la norme s, (s ∈ [0, 1]) exigée
par le (PD) importateur.
La logistique permet de faciliter le respecter des réglementations, c’est-à-dire
diminuer le niveau d’investissement pour se conformer à cette dernière, et répondre aux demandes du marché. Comme les PED sont caractérisés par un
manque dans les infrastructures logistiques, on suppose que les PD ont une logistique parfaite, alors que dans les PED la logistique est imparfaite (μ1 ∈]0, 1[
et μ2 = 1)
1.1
Déroulement du jeu
Étape 1 : On suppose que l’état du pays importateur impose initialement
le seuil maximal de contamination admis s et fixe le niveau de perfection du
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
système de contrôle frontalier β.
Étape 2 : le PD importateur (dans le cadre bilatéral d’aide au développement) choisit de subventionner l’amélioration de la logistique des PED. Pour
cela, le pays choisit un niveau de subvention v qui lui permet de maximiser son
bien-être W (v) donné comme suit :
W (v) = N2 π2 (v, k2 , β) + 12 [QI (v, k1 , k2 , β)]2 − σγ
QC (v,k1 ,k2 ,β)
QI (v,k1 ,k2 ,β)
− N1 λ(v − μ1 )2 ,
Étape 3 : Les deux catégories de Ps/Es choisissent simultanément leurs niveaux
d’investissement pour se conformer à la norme publique s et maximiser leurs profits
individuels définis par :
πi (β, ki ) = ωqiI (β, ki ) − rqiR (β, ki ) − Ci (ki ), i = 1, 2,
(1)
où
– ω est le prix unitaire d’une unité de produit exportée ;
– Ci > 0 est le coût de la mise en conformité de la catégorie i = 1, 2 de pays
exportateurs ;
– r > 0 est le coût associé à chaque unité du produit rejetée ;
– qiI ≥ 0, qui est en fonction de β, (respectivement qir ≥ 0) représente la quantité
du produit qui passe l’inspection (respectivement la quantité du produit rejetée)
aux frontières du pays importateur (PD).La quantité qiI décroît en β ;
– QI est l’offre totale, QC est la quantité totale contaminée qui passe l’inspection ;
– σ et γ sont respectivement : le coût marginal lié à des intoxications et la probabilité qu’il y ait intoxication ;
– λ(v − μ1 ) : le coût de l’aide logistique.
1.2
Résolution du jeu
Premièrement, on calcule l’investissement optimal ki∗ (ω) de chaque producteur i,
i = 1, 2 en maximisant l’expression du profit (1). Le prix 1 ω est obtenu en égalisant
l’offre QI et la demande D = a − ω. A ce stade, plusieurs résultats ont été obtenus
concernant l’effet de l’amélioration de la logistique sur le comportement stratégique des
producteurs.
Par la suite, on détermine quel est le niveau de logistique que le PD doit subventionner pour maximiser son bien-être. Dans cette étape, on s’intéresse à l’équilibre parfait
du jeu.
References
[1] C. Grazia, A. Hammoudi and O. Hamza : Sanitary and Phytosanitary standards :
Does consumers’ health protection justify developing countries’ producers’ exclusion ?
The French Development Agency (AFD) Research Department. (2011).
[2] A. Hammoudi, R. Hoffmann and Y. Surry : Food safety standards and agri-food
supply chains : an introductory overview. European Review of Agricultural Economics.36.(4.),(2009).
[3] E. Giraud-Heraud, A. Hammoudi, R. Hoffmann and L-G Soler :Joint Private Safety
Standards and Vertical Relationships in Food Retailing. Journal of Economics and
Management Strategy.21.(1.). (2012).
1. la détermination d’un tel prix de marché se fait conformément à une procédure classique d’équilibre dans un marché concurrentiel (voir Pierre Picard, "Éléments de microéconomie : Tome 1, Théorie et
applications, Chapitre 7, Editions Montchrestien, 2011).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Contribution à la résolution du problème du stable
multi-objectif
BADJARA Mohamed El Amine∗, CHERGUI Mohamed El-Amine†
20-22 April 2015
Abstract
L’objet de notre travail est la génération de l’ensemble des solutions efficaces du
problème du stable multi-objectif dans un graphe valué. Pour ce faire, une méthode
exacte basée sur le principe ”séparation et évaluation” est proposée. Une procédure
de tri combiné est décrite pour l’étape ”séparation” sur les sommets du graphe. Pour
l’étape ”évaluation”, une approximation I du point idéal est calculée en chaque nœud
de l’arborescence de recherche, permettant la stérilisation du nœud correspondant
si I est dominé par une des solutions déjà trouvée.
Mot de passe: Problème du stable de poids maximum, optimisation multi-objectif,
séparation et évaluation.
1
Introduction
Étant donné un graphe G = (V, E), dont chaque sommet est valué par un vecteur
poids. Le problème du stable multi-objectif (M OISP ) consiste à générer l’ensemble
des stables efficaces de G. C’est un problème connu pour être difficile appartenant
à la classe N P -dur. Le modèle mathématique associé est (P ) :
{M in(Cx), Ax ≤ 1, x ∈ {0, 1}n }
où n = |V |, C est la matrice des poids sur les sommets et A la transposée de la
matrice d’incidence du graphe G. Pour sa résolution, nous proposons une méthode
par séparation et évaluation qui exploite la structure particulière du système des
contraintes.
2
Principe de la méthode
Sachant qu’on a fixé au préalable le nombre de niveaux mesurant la profondeur de
l’arborescence de recherche h, la séparation est faite sur la base de la règle ”retenir”
ou ”ne pas retenir” un sommet i de G dans les solutions ”stables efficaces” (xi = 1
ou xi = 0). Ceci se fait selon un tri préétabli des sommets. L’effet domino (si on
fixe un sommet, ses adjacents doivent être supprimés) a une incidence directe sur la
réduction de la taille du problème en chaque itération. Dans le pire des cas, on aura
∗ [email protected]
† [email protected]
; Laboratoire RECITS - USTHB
; Laboratoire RECITS - USTHB
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
2n feuilles dans l’arborescence et la fixation de la profondeur de l’arborescence à
h = n/3, a pour effet d’éviter d’explorer tous les cas possibles. En chaque feuille de
l’arborescence, on utilise la méthode de Chergui et al. décrite dans [2] et [3] pour la
résolution du sous-problème induit par la branche correspondante de l’arborescence.
C’est une méthode dédiée à la résolution des problèmes d’optimisation linéaire multiobjectif en nombres entiers (M OILP ).
L’évaluation est faite en approximant le point idéal local I =
(Z1 , · · · , Zn ) de
chaque sous-problème sur chaque nœud par cette formule: Zj = i∈{1,...,n}:cj ≥0 cji .
i
Lors de la séparation, si un sommet i est retenu dans une branche de l’arborescence,
la mise à jour de Zj est faite par l’ajout des poids négatifs associés au sommet i à Zj .
Si le sommet i n’est pas retenu, la mise à jour de Zj est faite par la soustraction des
poids positifs associés au sommet i à Zj . Un nœud de l’arborescence peut être sondé
si son évaluation est dominée au sens de Pareto par une solution potentiellement
efficace déjà trouvée.
3
Contribution de la méthode
Dans l’expérimentation numérique, plusieurs procédures préalablement établies
ont été testées. Il en ressort deux résultats majeurs:
• Si on n’utilise plus la méthode de Chergui et al. et on termine le processus de
séparation et évaluation jusqu’à l’obtention d’un stable (épuisement de tous
les sommets), cela va diminuer considérablement le temps d’exécution jusqu’à
la moitié sur beaucoup d’instances.
• Si on utilise le tri combiné entre le tri décroissant et le tri par domination et
adjacence (voir [1] et [2]), cela va réduire le nombre de nœuds de l’arborescence
et le temps d’exécution d’une façon remarquable.
4
Conclusion et perspectives
L’exploitation des particularités du programme mathématique du problème du
stable, a permis d’adapter une méthode par séparation et évaluation pour sa résolution.
Le choix des procédures de séparation et celles d’évaluation influe de façon notable
sur le temps d’exécution de la méthode proposée.
À partir de ce point de vue, d’autres paramètres peuvent être envisager pour améliorer
encore le temps d’exécution de la méthode proposée, tels que le choix de traitement
en priorité de la ”meilleure” branche dans l’arborescence selon un critère à fixer.
References
[1] BADJARA M.E.A. : Approches exacte et approhée pour le problème du stable
multi-objectif. Mémoire de Magister - USTHB (2013)
[2] BADJARA M.E.A. et CHERGUI M.E-A. : The multiobjective independent set
problem. Proceedings EURO26. Rome - Italy (2013).
[3] CHERGUI M.E-A., MOULAÏ M. and OUAÏL F.Z.: Solving the multiple objective
integer linear pogramming problem. Modelling, Computation and Optimization in
Information Systems and Management Sciences Communications in Computer and
Information Science vol.(14), 69-76 (2008).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Production agro-alimentaire dans les pays en développement: qualité ou
quantité? Un modèle d’analyse.
N. NAIT MOHAND, M.S RADJEF, A. HAMMOUDI
20-22 April 2015
Abstract
Les pays en développement (PED) souffrent pour la plupart d’une persistante insécurité
alimentaire. Les pays d’Afrique subsaharienne et d’Asie du Sud marquent les GHI (Global
Hunger Index) les plus élevés en 2013, et aux pays du Sahel 20 millions de personnes soit
26% de la population souffrent de faim en 2014 [2]. Par ailleurs, les crises sanitaires (Aflatoxines en Afrique en 2004, Cachir en Algérie en 1988, Lait infantile en Chine en 2009, etc.)
montrent que les populations des PED sont les plus exposées à des produits alimentaires
contaminés [4]. Dans ce travail, nous proposons un modèle d’économie industrielle qui
analyse les conditions qui rendent compatibles l’objectif de disponibilité de l’offre sur les
marchés domestiques (quantité de produits), la sécurité sanitaire de cette offre (qualité des
produits) et participation des producteurs (non exclusion de ceux-ci de l’activité agricole).
On considère une filière agricole composée de grands et petits producteurs. Les autorités
publiques doivent définir les réglementations sanitaires et veiller à ce que cela ne freine
pas trop l’activité (quantité de produits et nombre de producteurs participants). On considère un jeu séquentiel à deux niveaux [5], que l’on résout via la méthode d’induction en
amont [1]. Les résultats montrent des effets contre intuitifs à l’amélioration des contrôles
sur les marchés : le renforcement des contrôles alimentaires peut augmenter l’entrée de
producteurs dans l’activité. Nous montrons qu’à l’équilibre du jeu, plusieurs structures de
production (nombre de producteurs et tailles de ceux-ci) peuvent émerger et ces structures
ont des impacts différents sur la quantité totale de produits offerts sur le marché domestique sur leur niveau de qualité sanitaire.
Mots clés: Backward induction, hétérogénéité des producteurs, sécurité sanitaire, pays en
développement.
1
Représentation de modèle
Considérons une filière domestique composée de N producteurs de taille 1, et M producteurs de taille q, q ∈ [0, 1[, qui produisent un produit agro-alimentaire destiné au marché
domestique. La demande D sur le marché domestique est donnée par :
D
=
a−ω
(1)
Afin de protéger la santé des consommateurs, les autorités publiques exigent un seuil
maximal s, s ∈ [0, 1], autorisé en contamination dans le produit final, elles mettent en
place un système de contrôle de conformité à la norme s. Le contrôle peut être imparfait
illustré par une probabilité β, β ∈ [0, 1] pour q’un échantillon contaminé soit détecté lors
du contrôle [3].
En réponse à la réglementation sanitaire, chaque producteur i de taille qi , qi ∈ {1, q}
investit un niveau ki , ki ∈ [0, 1] en bonnes pratiques agricoles sur son site de production.
L’investissement ki engendre un coût C(F, ki ):
C(F, ki ) = F ki2
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(2)
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Le producteur i qui investit ki anticipe une proportion f (s, ki ) de sa production qi conforme à la norme sanitaire s, et la proportion g(s, β, ki ) de sa production qi qui passent
l’inspection. Ces fonctions sont données respectivement par:
f (s, ki ) = 1 − (1 − s)(1 − ki ),
g(s, β, ki ) = f (s, ki ) + (1 − β)(1 − f (s, ki )).
(3)
Le profit d’un producteur i s’écrit:
πi (s, β, ki , qi , ω, F ) = ω qi g(s, β, ki ) − C(F, ki ).
2
(4)
Le jeu
On considère le jeu séquentiel suivant:
Étape 1: A s donné1 . Les autorités publiques décident des moyens à mettre en inspection
sur le marché (niveau β).
Étape 2: Les N et M producteurs de taille 1 et q respectivement, observent (s, β, ω) et
décident simultanément d’entrer ou ne pas entrer sur le marché spot.
Etape 3: Les n (n ≤ N ) et m, (m ≤ M ) producteurs de taille 1 et qrespectivement, qui
sont entrés sur le marché écoulent la totalité de leurs quantités produites sur le marché,
les quantités admises par le contrôle détermineront le prix ω selon la loi de l’offre et la
demande, et chaque producteur i de taille qi cherche à maximiser son profit (4).
3
Équilibre de jeu
La résolution du jeu à deux niveaux ainsi construit s’est basée sur la technique d’induction
en amont [1]. Dans un premier temps, chaque producteur i ajuste son niveau d’investissement
en bonnes pratiques agricoles au niveau ki∗ (ω) qui maximise son profit π(ki ); les quantités
admises par le contrôle détermineront le prix d’équilibre ω ∗ par l’égalisation de l’offre à
la demande [6]. En deuxième étape, étant donnés le prix à l’équilibre du marché ω ∗ et
l’investissement optimal ki∗ (ω ∗ ), on détermine les nombres n∗ , m∗ de producteurs qui feront des profits πi (ki∗ (ω ∗ ), ω ∗ ) positifs et auront intérêt à entrer sur le marché domestique.
References
[1] R. Aumann: Backward induction and common knowledge of rationality. Games and
Economic Behavior. 8.(1), 9-19. (1995).
[2] Commission Européenne : Sahel: crise alimentaire et nutritionnelle. Commission Européenne, (2014).
[3] A. Hammoudi, C. Grazia, and Y. Surry: Sécurité sanitaire des aliments : régulation,
analyses économiques et retours d’expérience. Paris (France): Tec & Doc - Lavoisier. 325
p, (2014).
[4] F. Keck: L’affaire du lait contaminé. Perspectives chinoises. 1.(1), 96-101. (2009).
[5] J.F Nash: Non-cooperative Games. Journal of Annals of Mathematics. 54. 286-295.
(1951).
[6] P. Picard: Eléments de microéconomie : Tome 1 : Théorie et applications. Chapitre 7,
Editions Montchrestien, 2011.
1 On
suppose qu’il est issu de considérations toxicologiques.
2
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The q-analogue of Fibonacci and Lucas sequences
Athmane Benmezai and Hacène Belbachir
University Algiers 3, Faculty of Economic Sciences
RECITS Laboratory, DG-RSDT
BP 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria
[email protected] & [email protected]
20-22 April 2015
Abstract
We give a new approach of the q-analogue of Lucas polynomials and we use
it with the q-analogue of Fibonacci polynomials to generalize some identities
related to Fibonacci and Lucas polynomials.
The Fibonacci and Lucas polynomials, denoted (Fn ) and (Ln ) respectively,
are defined by
L0 = 2, L1 = 1,
F0 = 0, F1 = 1,
and
Fn = Fn−1 + zFn−2 (n ≥ 2) ,
Ln = Ln−1 + zLn−2 (n ≥ 2) .
It is established that
n−k k
z ,
k
k=0
n/2
n
n−k k
z (n ≥ 1) .
n−k
k
n/2
Fn+1 (z)
=
Ln (z)
=
k=0
One of important relations satisfied by the Fibonacci numbers is
Gn+k = Gk Fn−1 + Gk+1 Fn ,
(1)
which is verified for all sequences Gn satisfying Gn+2 = Gn+1 + Gn
As q-analogues of Fibonacci and Lucas polynomials, J. Cigler [2] considers,
for n ≥ 0, the following expressions
n/2
k+1 +m k n − k
(
)
(
)
2
q 2
zk ,
Fn+1 (z, m) =
k
k=0
q
n/2
(1+m) k n − k
[n]q
(2)
q
zk ,
Lucn (z, m) =
[n − k]q
k
k=0
q
with the q-notations
[n]q = 1 + q + · · · + q
n−1
n−k
, [n]q ! = [1]q [2]q · · · [n]q ,
k
79
=
q
[n]q !
.
[k]q ![n − k]q !
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
He showed that the q-Fibonacci polynomials satisfy the following relations
(2)
Fn+1 (z, m) = Fn (z, m) + q n−1 zFn−1 q m−1 z, m ,
(3)
Fn+1 (z, m) = Fn (qz, m) + qzFn−1 q m+1 z ,
and the shortest recurrence relation
of linght 4. Our first aim is to give
polynomials such that the first one
satisfy the recursion (3). Our second
satisfied by the sequence Lucn (z, m) , is
a pair of new expressions of the q-Lucas
satisfy the recursion (2) and the second
aim is to give a q-analogue of relation (1)
Main results
Theorem 1 The explicit formulae of the new pair of q-Lucas polynomials is
given by
n/2
k +m k n − k [k]q
)
(
)
(
2
2
Ln (z, m) : =
q
1+
zk ,
k
[n − k]q
k=0
q
n/2
k+1 +m k n − k [k]q
(
)
(
)
2
q 2
1 + q n−2k
zk .
Ln (z, m) : =
[n − k]q
k
k=0
q
Theorem 2 The polynomials Ln (z, m) satisfy the recursion (2) and the polynomials Ln (z, m) satisfy the recursion (3)
Ln+1 (z, m) = Ln (z, m) + q n−1 zLn−1 q m−1 z, m ,
Ln+1 (z, m) = Ln (qz, m) + qzLn−1 q m+1 z, m .
Theorem 3 for n ≥ 0 and z = 0, we have
L−n (z, m)
=
L−n (z, m)
=
Ln q −mn z, m
,
(q −mn z)n
−mn
n Ln q
z, m
(−1)n q −m( 2 )
.
−mn
(q
z)n
n
(−1)n q −m( 2 )
Now we
consider the following sets:
Ωm = U = (Un (z))n∈Z : Un+2 (z) = Un+1 (z) + q n zUn q m−1 z with U0 (z) , U1 (z) ∈ R [z] .
Θm = U = (Un (z))n∈Z : Un+2 (z) = Un+1 (qz) + qzUn q m+1 z with U0 (z) , U1 (z) ∈ R [z] .
The generalization of relation (1) is given as follows
Theorem 4 Pour G ∈ Ω1 (resp. G ∈ Ω2 ), on a pour n ≥ 0;
Gk+n z/q k
= Gk z/q k ∗ zFn−1 (q m z, m) + Gk+1 z/q k ∗ Fn (z, m) ,
z
Gk+n (z) = Gk (z) Δq n−1 zFn−1 q m−1 z, m + Gk+1
ΔFn (z, m) .
q
d
d
i
i
where
∗Un (z) = di=c αi z i Un q (m−1)i z and
ΔUn (z) =
i=c αi z
i=c αi z
d
i ni
(m−1)i
z whith c, d ∈ Z
i=c αi z q Un q
References
[1] H. Belbachir, A. Benmezai, On the expansion of Fibonacci and Lucas polynomials, revisited, submitted.
[2] J. Cigler, Some beautiful q-analogues of Fibonacci and Lucas polynomials,
arXiv:11042699.
3
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Some applications of the double and
paired-dominating polynomials
Fatma Messaoudi and Miloud Mihoubi
20-22 April 2015
Abstract
The double (respectively paired )domination polynomial of graph G is the
i
n
polynomial P (G, x) = Σn
i=γ×2 d(G, i)×x (respectively Q(G, x) = Σi=γpr q(G, i)×
i
x ) where d(G, i) ( respectively q(G, i) ) is the double (respectively paired)
dominating sets of G of size i and γ×2 (G) (respectively γpr (G) )is the double
(respectively paired) domination number of G. The double and paired domination was introduced by Haynes and Slater[2]. In this paper,we obtain some
properties of the coefficients both of the double domination polynomials and
the paired domination polynomials of specific graphs than we give the recursive
formula of the double and paired domination polynomials for some graphs.
1
Introduction
Let G = (V, E) be a simple graph of order n. A set S of G is a dominating
set of G if every vertex in V \ S is adjacent to at least one vertex in S. A
double domination set is the set S of G which S dominates every vertex of G
at least twice. A matching in graph G is a set of independent edges in G, a
perfect matching in G is a matching in G such that every vertex of G is incident to an edge of M . A paired dominating set of G is a set S of V which S
is a dominating set and the induced subgraphs < S > has a perfect matching.
The double (respectively paired) domination is the minimum cardinality of double(respectively paired) dominating set in G. For a detailed treatment of this
parameter, the reader is referred to[2]. Let P (G, i) (respectively Q(G, i))be the
family of dominating sets of G with cardinality i and let d(G, i) = |P (G, i))|
(respectively q(G, i) = |Q(G, i)|) then the double (respectively paired ) dominai
tion polynomial of G is defined as P (G, x) = Σn
i=γ×2 d(G, i) × x (respectively
n
i
Q(G, x) = Σi=γpr d(G, i) × x [1,3].
2
Mens results
we give some proprieties and some recurrence relations for the coefficient of
the double and paired dominating polynomials for cycles , paths,trees, graph
product, complete graph, and the joint of two graphs.... We show for i = γ×2
and for every path Pn , n ≥ 2, if n ≡ 2[3] then d(G, i) = 1 also for every Cn ,
n ≥ 3, if n ≡ 0[3] than d(G, i) = 3. For the corona of any graphs G of order
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
n or for any graph G if γ×2 (G) = n then d(G, n) = 1, also for the joint of two
graphs we have P (G1 ∪ G2 , x) = P (G1 , x) × P (G2 , x).
Acknowledgements
our work is inspired by the mean results of the domination polynomials of graphs,
specially by some papers of Saeid Alikhani [3,4,5]
References
[1] S.Alikhani: Dominating Sets and Dominating Polynomials of Graphs Ph.D
Thesis. University Putra Malaysia (2009)
[2] T.W.Haynes, S.T.Hedetniemi et P.J.Slate : Fundamentals of domination in
graphs.. Marcel Decker,Inc.New york, (1998).
[3] S.Alikhani,Y.H.Peng: Dominating Sets and Dominating polynomial of cycles
Global journal of Pure and Applied Mathematics vol 4 ( nro2.)151-162 .(2008).
[4] S.Alikhani,Y.H.Peng: Dominating Sets and Dominating polynomial of certain graphs II Opuscula Mathematica vol 30.( nro1.),37-51. (2012).
[5] S.Alikhani,Y.H.Peng: Dominating Sets and Dominating polynomial of paths
International journal of Mathematics and Mathematics Sciences vol 09. (nro2.),10
(2009).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Generalized Catalan Numbers Based On Generalized
Pascal Triangle
Oussama Igueroufa & Hacne Belbachir
USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT
Po. Box 32, El Alia, 16111, Algiersi Algeria
[email protected] & [email protected]
20-22 April 2015
Abstract
In[1], Thomas Koshy found three formulas of Catalan numbers through the
subtraction of some special collumns of Pascal triangle. Our aim is to establish
that there are also three equal formulas deducing by the same method if we use
the generalized Pascal triangle, this similarity alow us to define a new sequence
of integers: the generalized Catalan numbers. We work on the special case of
generalized Pascal triangle for s an odd integer and give the first elements of
the generalized Catalan numbers for s = 3.
Let us start by the following definition.
Definition 1. Let s ≥ 1, L ≥ 0 and k ∈ {0, 1, . . . , sL}. The bis nomial coefficient
is the kth element of the developpment:
L
2
s L
(1 + x + x + · · · + x ) =
xk .
k
k≥0
s
For the properties of these numbers we refer to [2].As the central binomial
, n ∈ N, we define the
coefficient have the well known expression Bn = 2n
n
central bis nomial coefficients as follows.
Definition 2. Let n, s ∈ N, we define the central bis nomial coefficient relatively
to the parity of the number s as follows:
2n
s
Bn =
, If s is an odd integer,
sn
s
n
s
Bn = s
, if s is an even integer.
n
2
s
As well known, the Pascal triangle (for s = 1) is a triangular array of binomial
coefficients such that, each number in the triangle is the sum of the two directly
above it. The generalized Pascal triangle is a triangular array of bis nomial
coefficients such that each number in the triangle is the sum of s + 1 directly
above it.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
1
1
3
1
1
4
...
3
12
40
44
...
546
580
7728
3
31
155
135
20
101
546
2128
1
6
40
155
...
2
10
12
65
456
2128
1918
8092
7728
.
.
.
C0
.
.
.
C1
336
1554
216
5328
4
···
120
···
2472
···
35
1128
6728
728
3823
.
.
.
C4
.
.
.
C3
.
.
.
C2
1
10
Figure 1: Generalized Pascal triangle, for s = 3
Remark 3. The generalized Pascal triangle is symmetric relatively to the central
collumn which contains all the central bis nomial coefficients.
Now, we introduce a new sequence of integers called s-Catalan numbers
denoted as {Cn,s }n∈N , for a fixed value of s ∈ 2N + 1. The basic idea is to use
the generalized Pascal triangle, reader can refer to ([1] chap.12).
Definition 4. For n ≥ 0, and for all odd number s, we define for n ≥ 0 the
s-Catalan numbers as follows:
2n
2n
Cn,s =
−
.
sn
sn + 1
s
s
Theorem 5. For s an odd number, we have
2n − 1
2n − 1
−
, n ≥ 1.
Cn,s =
sn
sn + 1
s
and
Cn+1,s =
s
2n
2n
−
, n ≥ 0.
sn
sn + (s + 1)
s
s
If s = 3, we have the following sequence of 3-Catalan numbers:
{Cn,3 } = C0 − C1 = C2 − C3 = C0 − C4 = {1, 1, 4, 34, 364, . . .}.
2n Remark 6. C0 = { 2n
}, C1 = { 3n+1
}, C2 = { 2n−1
}, C3 = { 2n−1
},
3n 3
3n 3
3n+1 3
3
2n C4 = { 3n+4 3 }.
References
[1] Thomas. Koshy : Catalan Numbers with Applications. OXFORD, (2009).
[2] H. Belbachir, S. Bouboubi, A. Khelladi : Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution.
Annales Mathematicae et informaticae 35. 21-30. (2008).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Une méthode de séparation pour l'optimisation
exacte d'une fonction quadratique strictement
concave.
Belabbaci Amel: doctorante, Djebbar Bachir :professeur
20-22 April 2015
Abstract
Le but de ce travail est de présenter un algorithme permettant de chercher
le sommet le plus proche d'un point externe à un convexe , et ce pour localiser
la zone où se trouve la solution optimale d'un programme quadratique strictement concave. L'algorithme proposé peut être utilisé pour des applications à
grande taille telles que les SVM (support vector machines traduit en français
par "séparateur à vastes marges"), techniques qui permettent, en particulier de
résoudre des problèmes de classement et de les reformuler comme des problèmes
d'optimisation quadratique.
1 Introduction
P
Considérons le problème quadratique suivant: Maxf(x) où f (x) = i αi xi +
βi x2i , sous des contraintes de la forme: Ax b, où A est une matrice réelle,
et b un vecteur de Rm
+ . Les coecients αi sont des nombres réels de signes
quelconques et βi < 0, ∀i. La fonction f (x) est alors strictement concave. Soit
Ω le convexe formé par les inégalités de contraintes. Ω est alors un convexe
fermé et borné de Rn .
Il est bien connu que le maximum de f est unique et que si f (x) atteint
son maximum local en un point intérieur de Ω, alors c'est le maximum global.
Sinon, le maximum est atteint sur la frontière de Ω.
Plusieurs méthodes peuvent être utilisées pour trouver l'optimum de f : la
méthode de Frank-Wolfe [2]; transformer le problème en un programme linéaire
et appliquer la méthode du simplex [3]; de nombreuses autres méthodes existent.
Cependant toutes ces méthodes ne donnent que des solutions approximatives.
Un nouvel algorithme a été proposé pour trouver la solution optimale exacte
sans aucune introduction de nouvelles variables. L'algorithme est basé sur la
localisation du point critique x∗ qui est l'optimum local de la fonction f . Si
x∗ ∈ Ω, c'est alors l'optimum global. Sinon, l'optimum global est atteint en la
projection de x∗ sur les hyperplans de séparation passant par le sommet le plus
proche du point critique [1]. Pour cette raison, nous devons d'abord chercher
le sommet le plus proche de x∗ pour projeter ensuite x∗ sur les hyperplans de
séparation passant par ce sommet.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
2 Méthode utilisée
La méthode proposée pour chercher le sommet le plus proche de x∗ est une
méthode de séparation et d'élimination basée sur le principe des boules concentriques.
Pour une boule centrée en x∗ et de rayon x − x∗ , où x est un point extrême
de Ω , nous cherchons une autre boule centrée en x∗ et de rayon x́ − x∗ , où
x́ est un autre point extrême de Ω plus près de x∗ que x. Nous continuons la
recherche de ces boules de centre x∗ et de rayon xk − x∗ jusqu'à ce qu'il n'y
ait plus de point extrême xk qui satisfait l'inégalité xk − x∗ < xk−1 − x∗ (ici xk−1 est le dernier point extrême considéré avant xk ). La dernière boule de
rayon xp − x∗ dénit exactement la zone où se trouve la solution optimale de
f (x).
3 Domaine d'application
Il est bien connu que les SVM sont considérés comme des problèmes d'optimisation convexe [4]. Plusieurs algorithmes ont été proposés à cet eet. Cependant
les SVM sont souvent appliquées à des problèmes de grande taille. Ces nombreux
algorithmes et méthodes proposés n'ont pu être appliqués ecacement [5].
La méthode proposée ici est facile pour être appliquée dans la pratique. Une
étude expérimentale a montré qu'elle peut être utilisée ecacement pour des
problèmes de grande taille.
References
[1] A.Belabbaci,B.Djebbar, and A.Mokhtari: Optimisation of a strictly concave
quadratic form. Lecture Notes in Management Science vol 4, 16-20. (2012).
[2] M.Frank, and Ph.Wolfe: An algorithm for quadratic programming. Naval
Res vol 3, 95-110. (1956).
[3] P.Wolfe: The simplex method for quadratic programming. Econometrica 27,
3 . (1959).
[4] L. Hamel :Knowledge discovery with support vector machines. John Wiley&
Sons, (2009).
[5] S.SRA, S.Nowozin ans S.J. Wright :Optimization for machine learning. The
MIT Press, (2012).
2
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Stochastic second-order cone programming:
Applications and algorithms
Baha Alzalg
Department of Mathematics, The University of Jordan
20-22 April 2015
Abstract
Second-order cone programming is a class of nonlinear optimization problems that enable mathematical modelers to model a large variety of real-world
applications. Two-stage stochastic second-order cone programs (SSOCPs) with
recourse have been introduced to handle uncertainty in data defining deterministic second-order cone programs. A diverse set of real-world applications can be
modeled as SSOCP problems (see [1, 2]). In this talk, we describe an application
of SSOCP, we also address some important algorithms for solving this problem
(see [3, 4]).
Keywords: Second-order cone programming, Stochastic second-order cone
programming, Optimization algorithms.
References
[1] B. Alzalg: Stochastic second-order cone programming: Application models,
Appl. Math. Model. 36, 5122–5134 (2012).
[2] F. Maggioni, E. Allevi, M.I. Bertocchi, F. Potra: Stochastic second-order
cone programming in mobile ad hoc networks, J. Optim. Theory Appl. 143,
309–322 (2009).
[3] B. Alzalg: Decomposition-based interior point methods for stochastic quadratic
second-order cone programming. Applied Mathematics and Computation. 249,
1–18 (2014).
[4] B. Alzalg: Homogeneous self-dual algorithms for stochastic second-order cone
programming. Journal of Optimization Theory and Applications 163(1), 148–
164 (2014).
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
On the Solution of the Multiobjective Maximum
Probability Problem
Fatima Bellahcene and Philippe Marthon
LAROMAD, Université Mouloud Mammeri, Tizi-Ouzou, Algérie.
[email protected]
ENSEEIHT, 2 Rue Camichel, BP 7122, Toulouse Cedex 7, France.
[email protected]
20-22 April 2015
Abstract
This study considers a multiobjective maximum probability problem. Since
the model contains random variables coefficients, it is almost impossible to solve
it directly. The usual route used in stochastic programming is followed here by
replacing the multiple stochastic model by a nonlinear multiobjective deterministic equivalent problem. The special structure of this last problem encouraged
us to design a new methodology to solve it rather simply than accurately. Then,
an analytical approach based on the bisection method is developed.
1
Problem statement
We focus on the multiple objective maximum probability problem with levels
u1 , u2 , ..., up formulated as:
max Pr[Ckt x ≥ uk ] , k = 1, · · · , p
subject to
Ax ≤ b, x ≥ 0
(1)
where x is an n-dimensional decision variable column vector, A is an m × n
coefficient matrix and b an m-dimensional column vector. We assume that the
feasible set S = {x ∈ Rn | Ax ≤ b, x ≥ 0} is nonempty and compact in Rn . We
assume also that each vector Ck has a multivariate normal distribution with
mean C k and covariance matrix Vk . For instance, this is the situation when
expected value and minimum variance of the profit are considered not to be a
good measure of criteria [1]. The difficulty of this model lies in the need to know
the distribution functions of the stochastic objectives. Consequently, the present
literature on this problem is surprisingly thin. To our knowledge there exist only
one popular method for solving the multiple minimum risk problem. This can
be found in [3] or [4]. Since the model contains random variables coefficients,
it is almost impossible to solve it directly. The usual route used in stochastic
programming is followed here by replacing the multiple stochastic model by the
following nonlinear multiobjective deterministic equivalent problem:
t
C x−u
max fk (x) = √k t k , k = 1, · · · , p
x Vk x
subject to x ∈ S
90
(2)
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
2
Solving the deterministic problem
Various ways of finding Pareto optimal solutions for multiobjective nonlinear
problems are discussed in the literature (see, for example [2]). Our choice is
focused on the use of a synthesis function mink=1,...,p {fk (x)} and maximize
it. We obtain the so-called Tchebycheff problem maxx∈S mink=1,...,p {fk (x)} or
equivalently the problem
max h
√
t
subject to uk − C k x + h xt Vk x ≤ 0, k = 1, · · · , p
x∈S
h≥0
(3)
where both x and h are variables. Problem (3) is nonlinear and nonconvex, it
is in general, difficult to find a solution directly and analytically. However, if
the value of h is fixed to h, solving problem (3) is equivalent to determining a
feasible solution xh in the convex set
D=
u k − C t x + h√ x t V k x ≤ 0
k
n x ∈ R+
Ax ≤ b
In order to cope with problem (3), we intend to construct a relaxed set Dl of
closed half spaces containing D. Then, as soon as an optimal solution of the
relax problem is obtained for a fixed value of h, it is tested for its optimality to
the problem (3) by computing the corresponding probabilities of achieving the
goals. The suggested approximation is based on the eigenvalues of the covariance
matrices Vk , k = 1, · · · , p and the developed solution algorithm uses the bisection
method.
Acknowledgements
This research was supported by the Operational Research and Mathematics
Decision Aid Laboratory (LAROMAD) of the High Education Algerian Ministry.
References
[1] R. Caballero, M. Del Mar, L. Rey : Efficient Solution Concepts and their
Relations in Stochastic Multiobjective programming. Journal of Optimization
Theory and Applications vol.110(1), 53–74. (2001).
[2] K.M. Miettinen: Nonlinear Multiobjective Optimization. Kluwer’s International Series, (1999).
[3] Slowinski, R., Teghem, J.: Stochastic Versus Fuzzy Approaches to Multiobjective, Mathematical Programming Under Uncertainty. Dordrecht: Kluwer
Academic Publishers, (1990).
[4] Stancu-Minasian, I., Tigan, S.: The Vectorial Minimum Risk Problem.
In Proceedings of the Colloquium on Approximation and Optimization, ClujNapoca, 321–328, (1984).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Résolution collective des problèmes de l’optimisation
combinatoire ou la collaboration et la coopération
des métaheuristiques
Mohammed Yagouni
Laromad, Faculté de Mathématiques - USTHB
BP 32 El-Alia, 16111 Alger Algérie
20-22 April 2015
Résumé
Ce papier présente la conception d’un système basé sur la collaboration et
la coopération des métaheuristiques pour la résolution collective des problèmes
difficiles de l’optimisation combinatoire (OC). Ce système réalisé vise à fédérer la complémentarité des propriétés désirables, telles que l’intensification et la
diversification, dans une approche résolvant une même instance d’un problème
donné. Le paradigme de la programmation parallèle distribuée via l’utilisation
de MPI (Message Passing Interface) pour assurer les échanges de solutions, au cours du traitement -, entre les différents processus. Chacun de ces derniers,
implémente une méthode, et participe, par l’échange de solutions, à la résolution collective de l’instance. L’approche proposée a été testée et validée sur une
batterie d’instances du célèbre problème du voyageur de commerce, les premiers
résultats obtenus sont probants et augurent de l’efficacité et de la pertinence de
ce mode de résolution.
Keywords : Métaheuristiques, Collaboration, Coopération, Hybridation
1
Introduction
Les métaheuristiques sont des méthodes approchées de résolution de problèmes difficiles de l’optimisation. La recherche foisonnante dont elles ont bénéficiée ces dernières années, témoigne de leur utilité et de leur importance
notamment quand il s’agit d’attaquer des instances de tailles réalistes de problèmes jouissant d’un intérêt pratique dans divers domaines où, les décisions à
prendre sont souvent stratégiques. Ces outils se caractérisent dans leur conception par des propriétés importantes telles que l’intensification et la diversification
(I&D) ; [1] L’intensification, signifiant l’exploitation d’une zone de recherche, est
utilisée dès que le processus d’optimisation se trouve dans une zone favorable
à l’existence d’une solution de bonne qualité ; la diversification, quant à elle,
désigne l’exploration de l’espace de recherche, elle est souhaitée dès lors que
le processus se trouve dans une zone stérile. L’analyse de ces méthodes fait
ressortir que ces deux propriétés désirables, en l’occurrence, l’I&D, se trouvent
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
inégalement présentes et souvent présentent d’une manière exclusive dans les différentes métaheuristiques. Ce constat souvent signalé dans la littérature abondante traitant des métaheuristiques et de leurs applications, a été à l’origine
de la recherche, sans cesse, de nouvelles stratégies visant à renforcer l’une ou
l’autre de ces propriétés, à la faveur d’un bon équilibre et d’un meilleur compromis de ces caractéristiques, en combinant divers schémas métaheuristiques
dans des configurations hybrides faisant intervenir des fragments de différents
algorithmes, voire des algorithmes entiers différents dans des approches tant
inventives qu’originales [2][4]. La limite principale, de ces différentes stratégies
d’hybridation, réside à trois niveaux, d’abord, une augmentation substantielle
de leurs temps d’exécution, ensuite une complexification de l’instanciation des
différents paramètres de leurs composantes, et en fin, une absence de garantie
tant dans l’amélioration de la qualité des solutions produites que dans la robustesse de ces méthodes. L’approche proposée tire son originalité dans le mode mis
en oeuvre pour la collaboration des métaheuristiques dans un mode collectif et
participatif de résolution efficace des POCs [3].
2 Le schéma collaboratif et coopératif pour
les métaheuristiques
La collaboration et la coopération sont deux termes souvent indifféremment
utilisés et, à tort, confondus dans leur signification générale, car en dépit du fait
qu’ils expriment que le travail est réalisé collectivement, la nuance réside dans
la manière de partager ce travail entre les membres (participants) du groupe.
Il s’agira d’un travail coopératif, quand deux ou plusieurs entités d’un groupe
travaillent conjointement dans un même objectif, chacun ayant à sa charge une
part bien déterminée et limitée du travail à réaliser :"chacun sa partie du tout"
. En revanche, dans un travail collaboratif, il n’existe pas de répartition a priori
des rôles, les membres du groupe travaillent pour un but commun, mais chacun,
œuvre et cherche individuellement à atteindre l’objectif en entier, "le tout pour
tous". C’est principalement, ces notions qui sont mises oeuvre dans l’approche
que nous proposons pour résoudre des POCs. Cette nouvelle approche a été
appliquée pour la résolution du PVC. Les méthodes participantes implémentées
sont le Recuit Simulé, la Recherche Taboue, les Algorithmes Génétiques et la
Recherche à Voisinages Variables. Les résultats des premiers tests réalisés sur des
benchmarks de la TSPLIB, et d’autres en cours, nous laissent escompter que ce
schéma générique puisse être adapté à d’autres domaines telles que l’optimisation
non convexe et l’optimisation multiobjectif.
Références
[1] C. Blum and A. Roli : Metaheuristics in combinatorial optimization : Overview and conceptual comparison. ACM Computing Surveys : 268–308, 2003.
[2] A. Le Bouthillier and T. G. Crainic : A cooperative parallel meta-heuristic
for the VRPTW. Computers & OR, (32) :1685–1708, 2005.
[3] M. Yagouni, A. H. Le Thi : A Collaborative Metaheuristic Optimization
Scheme : Methodological Issues. Advances in Intelligent Systems and Computing, Springer International Publishing Switzerland, 282, 3-14. (2014).
[4] G. Zäpfel, R. Braune, M. Bögl : Metaheuristic Search Concepts. Springer,
2010.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
An optimal approximation of the GI/M/1 model
with two-stage service policy
Aicha Bareche, Mouloud Cherfaoui and Djamil Aı̈ssani
Research Unit LaMOS (Modeling and Optimization of Systems),
University of Bejaia, 06000 (Algeria).
Tel : (213) 34 21 08 00. Fax : (213) 34 21 51 88
E-mail : aicha [email protected], [email protected],
lamos [email protected]
20-22 April 2015
Abstract
In this paper, we consider an GI/M/1 system with two-stage service policy, for which we determine its global transition operator. After that, with
using the strong stability method we establish the approximation conditions for
the stationary characteristics of this system by those of the standard GI/M/1
system, and we estimate the deviation (stability inequalities) between the stationary distribution of the quoted systems. To do so, the situation is modeled
by a mathematical optimization problem which belongs to the minimization of
a constrained nonlinear multi-variable function.
A queueing system with two stage-policy (sometimes called hysteretic policy)
is such a system in which the server starts to serve with rate of μ1 customers
per unit time until the number of customers in the system reaches λ. At this
moment, the service rate is changed to that of μ2 customers per unit time and
this rate continues until the system is empty. The well known N -policy is a
hysteretic policy, in which the server is turned off when the system is empty and
turned on again when the queue size reaches the number N .
The optimal control of queues under N -policy, launched by Yadin and Naor
(1963), has been the subject of numerous research papers. This is because it is a
very important problem that finds applications in various systems, such as production, inventory, transportation, telecommunication, and computer systems.
The literature on the optimal control of N -policy queueing systems is rich and
varied. Various scenarios have been considered by the researchers. However,
the hysteretic service policy has not been often applied to the GI/M/1 queueing system. For this system with two available choices, Dudin and Klimenok
(1991) obtained the stationary distribution of the queue length at the customer
arrivals. Zhang and Tian (2004) derived the stationary queue length and waiting time in a GI/M/1 queueing system with the N -policy. Kim et al. [2], in
the aim to obtain the stationary distribution of the number of customers in the
GI/M/1 queue with two-stage service policy, propose to decompose the process
representing the number of customers into two processes according to the service rate, to obtain the stationary distribution of each process by making use
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
of an embedded Markov chain and the technique of level-crossing. Then, they
combined the two stationary distributions to obtain the stationary distribution
of the original process.
Moreover, note that for some practical situations modeled by queueing systems with N -stage service policy, the performance evaluation is complex or
impossible. Therefore, we seek sometimes to evaluate the performances of the
threshold policies or/and the optimal policy. On the other hand, if the cost due
to switching service rate, in the hysteretic queues, is not negligible compared
to the costs of processing and queues, it is known that the hysteretic policy is
preferable to the threshold policy. For this, it is very important to define the
domain within the switching cost is non-negligible (or define the domain within
the switching cost is negligible).
To illustrate how this domain is determined, and to enrich studies on the
GI/M/1 system with two-stage service policy, we propose, in this paper, to
use the strong stability method [1] which is an approximation approach of own
interest. In addition of studying a performance measure in a quantitative fashion, this method attempts to reveal the relationship between the performance
measures and the parameters of the system. It allows us to realize both a
qualitative and a quantitative analysis of some complex systems described by
a homogeneous Markov chain [1], in order to substitute them by more simpler
ones. Indeed, in addition to the qualitative assertion of continuity, this method,
which supposes that the perturbations of the transition kernel are small with
respect to some norm in the operator space, gives better stability estimates and
enables us to find precise asymptotic expansions of the characteristics of the
”perturbed” chain with an exact computation of the constants. For the applicability of the strong stability method to several queueing models, see Bareche
and Aı̈ssani (2011) and references therein.
We study the quality of the approximation of the stationary characteristics
of a GI/M/1 system with two-stage service policy by those of the system with
threshold policies and by those of the system with the optimal policy. To do
so, we elaborate the conditions which allow us to say that the two considered
systems are sufficiently close. After that, we measure the associate deviation
between their stationary characteristics when the conditions are hold. In other
words, we will answer the following questions: Under what conditions can we
consider a GI/M/1 system with two-stage service policy to behave approximately in the same manner as a standard GI/M/1 system? If these conditions
are hold, what is the deviation between their stationary characteristics?
Finding an optimal approximation, of the characteristics of the GI/M/1
queue with two-stage service policy, consists to determine the parameters of the
system with a single policy that minimize the deviation between the characteristics of the two systems. Determining the parameters of the new system in
our case amounts to solve a problem that belongs to the constrained nonlinear
minimization problem of a nonlinear multi-variable function.
References
[1] D. Aı̈ssani and N.V. Kartashov : Ergodicity and stability of Markov chains
with respect to operator topology in the space of transition kernels. Doklady
Akademii Nauk Ukrainskoi S.S.R. vol. 11, pages 3-5. (1983).
[2] S. Kim, J. Kim and E.Y. Lee : Stationary distribution of queue length in
G/M/1 queue with two-stage service policy. Math. Meth. Oper. Res. vol. 64,
pages 467-480. (2006).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Sécurité des aliments: quelle règle de responsabilité?
Une approche par la théorie des jeux
Manel Zahra Boutouis, Abdelhakim Hammoudi et Wassim Benhassine
20-22 April 2015
Abstract
Ce travail s’inscrit dans le contexte actuel de régulation de la sécurité sanitaire des
aliments. Il s’intègre dans le cadre des préoccupations liées à la sécurité sanitaire dans
les filières domestiques des pays en développement (PED). En effet, les aliments sont
soumis à de nombreux risques tout au long du processus de mise à la consommation. Ces risques peuvent avoir plusieurs origines (microbiologiques, chimiques, etc.).
Certains peuvent autre du fait d’une mauvaise pratique de production ou de transformation et d’autres du fait de mauvaises pratiques de distribution (stockage, respect de
la chaîne du froid, etc.). Dans un tel contexte et notamment en absence d’ un système
de traçabilité, l’identification et l’imputation de la contamination à un acteur précis
de la chaîne alimentaire peut s’ avérer difficile. A cet effet, nous analysons les effets
des instruments de régulation publique, notamment la règle de responsabilité juridique,
sur les décisions stratégiques des producteurs / distributeurs de produits alimentaires,
exprimées en termes d’ investissement en bonnes pratiques, et sur la manière dont
cela va contribuer à protéger la santé des consommateurs. Le modèle que nous proposons, s’ appuie sur une approche d’ économie industrielle. Cette approche nous a
permis d’ identifier les conditions (caractéristiques du système de contrôle, niveau de la
norme, montant de l’ amende liée aux rejets des quantités non-conformes écoulées sur le
marché), selon lesquelles les autorités publiques choisissent une règle de responsabilité
partagée entre le producteur et le distributeur (cette responsabilité est liée au payement
des coûts du rejet, des quantités non-conformes, dans ce cas, elle incombe aux deux
opérateurs) ou une règle de responsabilité exclusivement basée sur le distributeur (le
distributeur seul subit ces coûts).
1
Présentation du modèle et déroulement du jeu
Le modèle que nous proposions comporte quatre intervenants de la chaîne alimentaire:
producteur, distributeur, consommateurs et autorités publiques. A ce titre, nous supposons qu’une filière du marché domestique d’un PED, est constituée d’un producteur
de taille q, correspondant à sa capacité de production vendue intégralement à un distributeur qui paye un prix unitaire ω. Le distributeur à son tour, revend ce produit au
consommateur à un prix P . Nous supposons que l’aval de la filière est soumis à une
procédure de contrôle imparfaite. Cette imperfection est mesurée par des erreurs dans
les tests de contrôle A ce titre nous considérons le jeu suivant:
Étape 1: Les autorités publiques décident du type de règle de responsabilité à mettre
en place: une règle partagée ou une règle de responsabilité exclusive au distributeur.
Étape 2: Après avoir observé le type de la règle imposée ainsi que d’autres paramètres
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
liés à la réglementation (seuil de la norme s, degré de fiabilité des contrôles et montant
de la pénalité), le producteur décide de son niveau d’investissement en qualité.
Étape 3 : Le distributeur observe la décision du producteur et décide de son côté de
son niveau d’investissement en qualité.
2
Résolution du jeu
Le jeu est résolu en utilisant la méthode de Backward Induction. Nous déterminons
d’abord le niveau optimal d’investissement et puis, les conditions pour que l’équilibre
parfait du jeu soit tel que l’état choisit une règle partagé ou une règle exclusivement
basée sur la responsabilité du distributeur. Selon la règle de responsabilité fixée par
les autorités publiques, le producteur (respectivement distributeur) choisit son niveau
d’investissement optimal afin de se conformer à la norme en vigueur (s).Il maximise
alors son profit par rapport à k0 , (resp, k1 ). Les profits sont données par:
max πPRP (r, s, β, q, w, F, k0 , k1 )
=
RP
max πD
(r, s, β, q, w, F, k0 , k1 )
=
RED
max πD
(r, s, β, q, w, F, k0 , k1 )
=
max πPRED (r, s, β, q, w, F, k0 , k1 )
=
k0
k1
k1
k0
1
1
βq(1 − s)(1 − (k0 + k1 ))(w + r(1 − k0 )) − F k02
2
2
1
1
1
( p − w)q − βq(1 − s)(1 − (k0 + k1 ))( (p − w + r(1 − k1 ))) − F k12
2
2
2
1
1
1
( p − w)q − βq(1 − s)(1 − (k0 + k1 ))( (p − w + r(1 − k1 ))) − F k12
2
2
2
1
1
wq − βqw(1 − s)(1 − (k0 + k1 )) − F k02
2
2
wq −
πPRP , πPRED : profit du producteur en règle partagée (resp, en règle exclusive au distributeur).
RP
RED
,πD
: profit du distributeur en règle partagée (resp, en règle exclusive au
πD
distributeur).
q I : quantité inspectée (celle qui passe l’inspection de contrôle).
q C : quantité du produit contaminée qui passe l’inspection de contrôle.
r : coût lié à chaque unité de produit rejetée.
k0 : niveau de qualité associé aux pratiques du producteur. Il représente les moyens
investis sur les sites de production (infrastructures, équipements, etc.) pour maximiser
la probabilité que le produit soit conforme à la norme en vigueur.
k1 : niveau de qualité associé aux pratiques du distributeur. Il représente également
les moyens investis sur les sites de distribution.
La décision des autorités publiques relative à la détermination de la règle de responsabilité appliquée, s’appuie sur le critère
sanitaire, mesuré par le taux de contamination
c
à l’équilibre parfait du jeu (tc = qqI ). Il s’agit d’identifier la règle qui permettra
d’enregistrer le taux de contamination le plus faible.
Références
[1] C. Grazia, A. Hammoudi, O. Hamza :Sanitary and phytosanitary standards: Does
consumers health protection justify developing countries producers exclusion?. Review
of Agricultural and Environmental Studies 2.(93.),145-170. (2012).
[2] Pouliot, S., & Sumner, D. A :Traceability, liability and incentives for food safety
and quality. American Journal of Agricultural Economics 1.(90.),15-27. (2008).
[3] Oualid Hamza : Sécurité sanitaire des aliments, commerce et développement: approche par l’économie industrielle. Thèse de Doctorat, l’Université Panthéon-Assas,
Paris (2012)
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
A three-term recurrence relation for computing
Cauchy numbers
Mourad RAHMANI
USTHB, Faculty of Mathematics
P.O. Box 32, El Alia, 16111, Algiers, Algeria
20-22 April 2015
Abstract
In this talk we define a new family of p-Cauchy numbers by means of the
confluent hypergeometric function. We establish some basic properties. As
consequence, a number of algorithms based on three-term recurrence relation
for computing Cauchy numbers of both kinds, Bernoulli numbers of the second
kind are derived.
1
Introduction
The Cauchy numbers of both kinds are introduced by Comtet [1] as integral of
the falling and rising factorials
c(1)
n :=
1
{x}n dx, and c(2)
n :=
0
1
(x)n dx.
0
These numbers appears in diverse contexts, like number theory, special functions and approximation theory. Recently, many research articles have been devoted to Cauchy numbers of both kinds and many generalization are introduced
(poly-Cauchy numbers, hypergeometric Cauchy numbers, generalized Cauchy
numbers). As an example of a recent application of the Cauchy numbers of
both kinds see [2], where Masjed-Jamei et al. have derived the explicit forms of
the weighted Adams-Bashforth rules and Adams-Moulton rules in terms of the
Cauchy numbers of the first and second kind.
In this talk, a number of algorithms based on three-term recurrence relation
for calculating Cauchy numbers of both kinds are derived. In particular, we
prove the following recurrence relation for computing Bernoulli numbers of the
second kind. First, we construct an infinite matrix (bn,p )n,p≥0 as follow: the
first row of the matrix is b0,p := 1 and each entry is given by
bn+1,p =
1−n
p+1
bn,p −
bn,p+1 .
1+n
(p + 2) (1 + n)
Finally, the first column of the matrix are Bernoulli numbers of the second kind
bn,0 := bn .
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
References
[1] L. Comtet, Advanced combinatorics, D. Reidel Publishing Co. (1974).
[2] M. Masjed-Jamei, M. A. Jafari and H. M. Srivastava, Some applications of
the Stirling numbers of the first and second kind, J. Appl. Math. Comput. 2014
1-22 (2014).
[3] M. Rahmani, On p-Cauchy numbers. To appear in Filomat (2014).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
!"$ "#" !"$ " "! ;<15)<176 ,-; 8):)5F<:-; ,B=6 57,F4- D >74)<141<G
;<7+0);<19=- 8G:17,19=),1) %## -< )@E)4 )*7:)<71:- "$# %#$ 4/-: 4/G:1
8:14 )6; 4- 8:7*4F5- ,- 4B-;<15)<176 ,-; 8):)5F<:-; ,B=6 57,F4- D >74)<141<G
;<7+0);<19=- #& 4) 84=8):< ,-; <:)>)=? <:)6;.7:5-6< 4- 57,F4- #& -6 =6 57
,F4- -;8)+- ,BG<)< 416G)1:- )>-+ ,-; -::-=:; 676 /)=;;1-66-; -< 4) .76+<176 ,9=);1>:)1;-5*4)6+- ,= 57,F4- 8-=< H<:- G>)4=G- )= 57@-6 ,= A4<:- ,- )4
5)6 7= ,- 0)6,:);-30): B)=<:-; 5G<07,-; ,- 76<- ):47 ;76< +76;1,G:G-;
+755- ,-; )4<-:6)<1>-; D 4B-;<15)<176 8): 4) 5G<07,- 9=);15)?15=5 ,- >:)1
;-5*4)6+- )6; +-<<- +755=61+)<176 67=; =<141;76; 4-; 5G<07,-; ,- 76<):47 ;G9=-6<1-44-; 9=1 7C:-6< =6 +),:- ,- 57,G41;)<176 16.7:5)<19=-5-6< -.
A+)+- -6 8):<1+=41-: 87=: 4B-;<15)<176 ,-; 8):)5F<:-; ,B=6- 4):/- /)55- ,57,F4-; 7=; ),)8<76; 84=; 8:G+1;G5-6< 4-; )4/7:1<05-; ,- A4<:)/- -< ,- 41;
;)/- 8):<1+=4)1:- )88419=G; )=? 57,F4-; -;8)+- ,BG<)<; 8:787;G; 8): 15 87=: 4B-;<15)<176 ,-; 8):)5F<:-; ,B=6 57,F4- 8G:17,19=- A6 ,BG<=,1-: 4) 8-:
.7:5)6+- ,-; ,-=? 8:7+G,=:-; 67=; 8:G;-6<76; =6- G<=,- ,- ;15=4)<176 -< 67=;
)88419=76; 4-; 8:7+G,=:-; G<-6,=-; D 4) 57,G41;)<176 ,- 4) ;G:1- 27=:6)41F:- ,4B16,1+- 76;1,G:76; 4- 57,F4
7I -< ;76< ,-=? ;=1<-; 16,G8-6,)6<-; ,- >):1)*4-; +-6<:G-; -<
,- >):1)6+- G/)4- D -< 4-; 8):)5F<:-; -< ;76< 8G:17,19=-; -6 ,- 8G:17,- - 57,F4- )88-4G 57,F4- D >74)<141<G ;<7+0);<19=- 8G:17,19= ) G<G 8:787;G -< G<=,1G 8): 367=+0- -< )4 7<76; 9=- -;< ,1CF:-6< ,= 57,F4- 8:787;G 8): $;1)3); "GG+:1>76; 4- 57,F4- ,GA61 ,)6; -6 .)1;)6< :-;;7:<1: 4) 8G:17
,1+1<G +755- ;=1<
-< ) .7:5- -;8)+- ,BG<)< 416G)1:- ,= 57,F4- -;< 7*<-6=- 8): 8);;)/- )=
47/):1<05
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
7I #1 -;<
=6- >):1)*4- /)=;;1-66- 4) 57@-66- -< 4) >):1)6+- ,- 8-=>-6< H<:)88:7?15G-; 8): -< :-;8-+<1>-5-6<
B-;<15)<176 ,-; 8):)5F<:-; ,-; 57,F4-; D 8):<1: ,- 4-=:; :-8:G
;-6<)<176; -;8)+- ,BG<)<; -;< 4B7*2-< ,- +-<<- +755=61+)<176 6 8:-51-: 41-=
67=; :)88-476; 4- 8:7*4F5- ,- 4BG>)4=)<176 ,- 4) .76+<176 9=);1>:)1;-5*4)6+,B=6 57,F4- >1) 4) 5G<07,- ,- 4B1667>)<176 7I 4-; 1667>)<176; -581
:19=-; ;76< 7*<-6=-; 8): 4- A4<:- ,- )45)6 7= 8): 4-; G9=)<176; ,- 0)6,:);-
30): 8G:17,19=-; 367=+0- -< )5,1 6 ;-+76, 41-= 67=; =<141;76; 4-;
5G<07,-; ,- 76<- ):47 ;G9=-6<1-44-; 9=1 .)+141<-6< 4) 57,G41;)<176 -6 8):
<1+=41-: 87=: 4B-;<15)<176 ,-; 8):)5F<:-; ,B=6- 4):/- /)55- ,- 57,F4-; 15
-< *1*417/:)801- 16<:) 7=; ),)8<76; 84=; 8:G+1;G5-6< 4-; )4/7:1<05-;
,- A4<:)/- -< ,- 41;;)/- 8):<1+=4)1:- )88419=G; )=? 57,F4-; -;8)+- ,BG<)<; 8:7
87;G; 8): 15 -< 15 -< #<7C-: 87=: 4B-;<15)<176 ,-; 8):)5F<:-;
,B=6 57,F4- 8G:17,19=- 7=; 8:G;-6<76; G/)4-5-6< =6- G<=,- ,- ;15=4)<176
87=: +758):-: 4) 8-:.7:5)6+- ,-; ,-=? 8:7+G,=:-; -< 67=; )88419=76; -6A6
4-; 8:7+G,=:-; G<-6,=-; D 4) 57,G41;)<176 ,- 4) ;G:1- 27=:6)41F:- ,- 4B16,1+- '( 367=+0- 1*1 )6, )5,1 !-:17,1+ )=<7:-/:-;;1>- ;<7+0);<1+
>74)<141<@ 7=:6G-; ,- #<)<1;<19=- 1;3:) >:14 '
( 367=+0- )6, )5,1 ?<-6;176 ,= A4<:- ,- 0)6,:);-30): )= +); ,-;
57,F4-; -;8)+- ,BG<)<; 8G:17,19=-; ! $% % '( 15 # " !0 <0-;1; %61>-:;1<@ 7.
!1<<;*=:/0 !1<<;*=:/0 ! '( 15 )6, # #<7C-: 1<<16/ ;<7+0);<1+ >74)<141<@ 57,-4; 16 <0- 8:-;-6+7. 1::-/=4): ;)58416/ >1) 8):<1+4- 5-<07,; )6, <0- -5 )4/7:1<05 ! # '( $;1)3); !-:17,1+ #<7+0);<1+ &74)<141<@ )6, )< $)14; ! 101
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Forwarding Indices of Some Graphs
M.A Boutiche
Laboratory ROMaD,
Faculty of Mathematics, USTHB,
BP 32 El Alia, 16111 Bab Ezzouar, Algeria.
[email protected]
20-22 April 2015
Abstract
In this paper, we determine lower bounds of vertex and edge forwarding
indices of the subdivision graph of the complete graph, tadpole graphs and the
wheel graphs.
Key words: Forwarding indices, Subdivision graphs, Tadpole graphs, Wheel
graphs.
1
Introduction
A routing R of a connected graph G = (V, E) of order n is a set of n(n − 1)
elementary paths R(u, v) specified for all (ordered) pairs u, v of vertices of G. A
routing R is said to be minimal if all the paths R(u, v) of R are shortest paths
from u to v, denoted by Rm .
To measure the efficiency of a routing deterministically, Chung et Al. [1]
introduced the concept of forwarding index of a routing.
The load of a vertex v (resp. an edge e) in a given routing R of G = (V, E),
denoted by ξ(G, R, v) (resp. π(G, R, e)), is the number of paths of R going
through v (resp. e), where v is not an end vertex.
The parameters
ξ(G) = min ξ(G, R) and π(G) = min π(G, R)
R
R
are defined as the vertex forwarding index and the edge forwarding index of
G, respectively. For more complete results on forwarding indices, we can refer
to the survey of Xu et al. [4].
The Tn,k Tadpole graph [2] is the graph obtained by joining a cycle graph
Cn to a path of length k. The wheel graph Wn+1 is defined as the graph
K1 + Cn , where K1 is the center vertex and Cn is the cycle graph such that
each vertex in Cn is adjacent to the center vertex. The subdivision graph S(G)
is the graph obtained from G by replacing each of its edge by a path of length
2, or equivalently, by inserting an additional vertex into each edge of G.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
2
Some Lemmas
For a given connected graph G = (V, E) of order n and number of edges m, set
⎛
⎞
1 ⎝ (dG (u, v) − 1)⎠ ,
A(G) =
n
B(G) =
v∈V \{u}
u∈V
1
m
dG (u, v),
(u,v)∈V ×V
where dG (u, v) denotes the distance from the vertex u to the vertex v in G.
The following bounds of ξ(G) and π(G) were first established by Chung et
al. [1] and Heydemann et al. [3], respectively.
Lemma 1 (Chung et al. [1])
Let G be a simple connected graph of order n. Then
A(G) ≤ ξ(G) ≤ ξm (G) ≤ (n − 1)(n − 2).
Lemma 2 (Heydemann et al. [3])
Let G = (V, E) be a simple connected graph of order n. Then
B(G) ≤ π(G) ≤ πm (G) ≤ 12 n2 .
3
Main results
In this section, we derived an expression for A(G) and B(G) of the subdivision
graphs of the complete, tadpole graphs and the wheel graphs. Moreover, we
give a lower and an upper bound for the vertex and edge forwarding indices of
these graphs.
Theorem 3 Let S(Kn ) be the subdivision graph of the complete graph Kn .
Then
A(S(Kn )) =
n2 (n−1)
n+1
− 1 and B(S(Kn )) = n2 /2.
Theorem 4 Let S(Wn+1 ) be the subdivision graph of the Wheel graph Wn+1 .
Then
⎧
⎨
A(S(Wn+1 )) =
2n(9n−13)
3n+1
−1
⎩
175
13
43
5
if
if
if
n≥5
n=4 ,
n=3
B(S(Wn+1 )) =
9n−13
2
47
4
8
Theorem 5 Let S(Tn,k ) be the subdivision graph of the tadpole graph Tn,k .
2
−1)k
kn
n3
+ n+k
+ n(n+1)
+ kn(2k+n)
+ 2(n+k)
− 1,
Then A(S(Tn,k )) = (4k
6(n+k)
4(n+k)
n+k
and B(S(Tn,k )) =
(4k2 −1)k
6(n+k)
+
kn
n+k
+
n(n+1)
4(n+k)
+
kn(2k+n)
n+k
+
n3
.
2(n+k)
References
[1] F. K. Chung, E. G. Coffman, M. I. Reiman and B. Simon, The forwarding
index of communication networks, IEEE Transactions on Information Theory.
33(2) (1987), 224–232.
[2] D.M. Cvetkocić, M. Doob, H. Sachs, Spectra of Graphs - Theory and Application. Academic Press, New York, 1980.
[3] M. C. Heydemann, J. C. Meyer and D. Sotteau, On forwarding indices of
networks, Discrete Applied Mathematics. 23 (1989), 103–123.
[4] J. Xu and M. Xu, The forwarding indices of graphs - A survey, Opuscula
Mathematica. 33(2) (2013), 345-372.
2
103
if
if
if
n≥5
n=4
n=3
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Data Envelopment Analysis with presolve as a
decision making tool. Dakar Dem Dikk case study
Oumar SOW† , Babacar M. NDIAYE‡ and Aboubacar MARCOS†
†
‡
Institute of Mathematics and Physical Sciences - IMSP
University of Abomey Calavi. BP 613, Porto-Novo, Benin
{oumar.sow, abmarcos}@imsp-uac.org
Laboratory of Mathematics of Decision and Numerical Analysis
LMDAN-FASEG, University of Cheikh Anta Diop
BP 45087 Dakar-Fann, Dakar, Senegal
[email protected]
Abstract
This paper deals with presolve techniques applied on Data Envelopment
Analysis (DEA) model in order to find a reduced DEA model which is more
efficiently solved. The DEA is a tool for measuring the relative efficiency of
organizations (called Decision Making Units (DMUs)) via weights associated to
input and output measures. It is based on linear programming techniques, and
can be used to measure technical efficiency, allocation effectiveness of inputs and
outputs, and economics performance of production means. The linear presolving techniques are techniques which aim to reduce the problem size by deleting
rows and columns and tightening bounds on variables, and to discover whether
the problem is unbounded or infeasible. In our study, we propose a combination
with DEA and presolve techniques which lead to solve the problem efficiently.
An application on Dakar Dem Dikk, the main public transportation company in
Dakar, is presented. A comparative analysis between the original DEA model
and the reduced one will be discussed. Finally, numerical simulations on real
datasets of Dakar Dem Dikk are carried out.
keywords: Data Envelopment Analysis, presolve, network transportation, optimization
References
[1] D.A. Erling and D.A. Knud, “Presolving in linear programming”, Mathematical Programming, 71, 221-245, 1995.
[2] A.L. Brearley, G. Mitra and H.P. Williams, “Analysis of mathematical programming problems prior to applying the simplex algorithm”, Mathematical
Programming, 8 (1), 54-83, 1975.
[3] S.E. Chang and S.T. McCormick, “A hierachical algorithm for making
sparse matrices sparser”, Mathematical Programming, 56 (1), 1-30, 1992.
[4] J. Gondzio, “Presolve Analysis of Linear Programs Prior to Applying An
Interior Point Method”, INFORMS Journal on computing, 9 (1), 73-91,
1997.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
[5] A. Charnes, W.W. Cooper and E. Rhodes, “Measuring the efficiency of
decision making units”, European Journal of Operational Research, 2, 429444, 1978.
[6] J.E. Beasley, “Allocating fixed costs and ressources via data envelopment
analysis”, European Journal of Operational Research, 147, 198-216, 2003.
[7] D.T. Barnum, J.M. Gleason and B. Hemily, “Using Panel Data Analysis to
Estimate Confidence Intervals for the DEA Efficiency of Individual Urban
Paratransit Agencies”, Great Cities Institute Publication Number: GCP07-10, A Great Cities Institute Working Paper, December 2007.
[8] J. Zhu, “Quantitative Models for Performance Evaluation and Benchmarking: Data Envelopment Analysis with Spreadsheets and DEA Excel Solver”,
Kluwer Academic Publishers, Boston, 2002.
[9] W.W. Cooper et al. (eds.), “Handbook on Data Envelopment Analysis”,
International Series of Operations Research & Management Science 164,
c Springer Science+Business
DOI 10.1007/978 − 1 − 4419 − 6151 − 8 11, Media LLC 2011.
[10] W.W. Coopers, L.M. Seiford and K. Tone, “Data envelopment analysis:
a comprehensive text with models, applications, references and DEA-solver
software”, Springer-Verlag New York Inc., 2nd ed. 2007.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Amélioration d'une méthode exacte pour la recherche
des solutions entières ecaces
Amirouche BOURAHLA 1 & Mohamed El-Amine CHERGUI 2
1,2
USTHB,Faculté de Mathématiques, Laboratoire RECITS
20-22 April 2015
Abstract
Dans ce travail, nous apportons une amélioration à une méthode exacte pour la
recherche des solutions entières ecaces d'un problème multiobjectif. Un nouveau test
d'arrêt est rajouté à la méthode exacte permettant de sonder un n÷ud de l'arborescence
de recherche si l'estimation par excès du point idéal correspondant au n÷ud est dominée
par une solution préalablement trouvée.
Mots clés: Programmation linéaire discréte, Optimisation multiobjectif, Solution
ecace, Point idéal.
1 Principe de la méthode
Dans le cadre des travaux de doctorat, notre travail se focalise sur la recherche de
l'ensemble des solutions ecaces d'un programme multiobjectif linéaire discret (M OILP ),
considérant p critères:
⎧
max(Z j )j=1...p
⎪
⎨
(M OILP )
A×x≤b
⎪
⎩ x≥0
x vecteur entier
A la lumière des travaux récents sur le problème (M OILP ), [1], [2], [3], [4], nous avons
mis au point une amélioration de la méthode décrite dans [3],dont le principe est le
suivant :
La méthode du branch & bound est utilisée pour résoudre le problème (P0) et le tableau
utilisé est augmenté de p nouvelles lignes pour permettre aux critères d'être évalués en
même temps que le critère Z(1) de (P0).
⎧
maxZ 1 = C 1 × x
⎪
⎨
(P 0)
A×x≤b
⎪
⎩ x≥0
x vecteur entier
La méthode est basée sur le principe du branch & bound pour la recherche de solutions entières, à chaque n÷ud l de l'arborescence, un programme (Pl) est résolu.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Avant la résolution d'un programme (M OILPl ) relatif à un n÷ud l de l'arborescence
$ du point
de recherche de la méthode décrite dans [3],on calcule l'approximation Id
$ l ) domine z(Idl ). Ce point est calculé en utilisant
idéal Id correspondant tel que z(Id
les bornes des variables sur la branche se terminant au n÷ud l dans l'arborescence
$ l ) est dominée par une solution
de recherche. Si l'approximation du point idéal z(Id
potentiellement ecace déjà trouvée, le n÷ud correspondant l est stérilisé, sans avoir
à explorer le domaine associé, sinon le programme (M OILPl ) est résolu.Ce procédé se
répète tant qu'il existe des n÷uds non stérilisés.
2 Expérimentation
L'expérimentation, en cours de réalisation, montre que la nouvelle méthode converge
plus rapidement que celles cités en références.
(n,m,p)
(10,10,2)
(10,10,3)
(10,10,5)
(17,10,2)
(17,10,4)
(20,10,2)
(20,10,3)
sol.EFF
Moy
4.9
14.2
120.5
7.8
139.75
8.1
27.6
CPU(s)
Min Moy
Max
0.179 0.37
1.15
0.759 1.11 1.391.25
1.14
2.13
3.7
21.539 89.31 256.883
157.44 3.14 395.69
20.104 48.21 75.725
64 162.62 269.32
Min[3]
0.55
1.24
1.64
82.981
232.10
63.287
95.87
Moy[3]
1.33
1.87
3.14
228.31
519.77
108.03
264.41
Max[3]
2.17
2.19
4.2
404.48
612.21
148.57
366.73
3 Conclusion
La structure de recherche arborescente de la méthode décrite dans [3], nous a permis
de rajouter un nouveau test d'arrêt basé sur une évaluation approchée du point idéal
en chaque n÷ud de l'arborescence. L'amélioration du temps de calcul laisse présager
une possibilité d'aborder la résolution d'instances de plus grandes dimensions.
References
[1] Abbas M. and Chaabane D : An algorithm for solving multiple objective integer
linear programming problem. RAIRO Operations Research 36,pp.351-364, 2002.
[2] Abbas M., Moulaï M. : Solving multiple objective integer linear programming.
Ricerca Operativa 29/89,15.38,1999.
[3] Chergui M.E-A., Moulaï M. and Ouaïl F.Z. : Solving the Multiple Objective Integer Linear Programming Problem. Modelling, Computation and Optimization in
Information Systems and Management Sciences, Communications in Computer and
Information Science,Volume 14, Part 1, Part 1, 69-76, DOI: 10.1007/978-3-540-874775-8, Springer, 2008.
[4] Özlen M. and Azizoglu, M. :Multiobjective integer programming : A general approach for generating all non-dominated solutions. European Journal of Operational
Research,Vol. 199, pp.25-35, 2009.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
On bandwidth of a n-dimensional grid
Hakim Harika,b and Hacène Belbachira
a
USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT
BP 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria
[email protected] or [email protected]
b
5 Rue des frères Aissou, Ben Aknoun, Algiers, Algeria
[email protected]
20-22 April 2015
Abstract
We denote by G = (V, E) a simple undirected graph of order |V (G)| and size
|E(G)|. N (v) is a set of vertices adjacent to vertex v in G and deg(v) = |N (v)|
denote its degree. A path Pn , from a vertex v1 to a vertex vn , n ≥ 2, is a
sequence of vertices v1 , . . . , vn and edges vi vi+1 , for i = 1, . . . , n − 1. A grid,
(q)
denoted by Qn , is a graph with elements of the set {0, 1, . . . , q}n , and two
vertices u and v are connected by an edge if and only if u and v differ only at
one component in which they have absolute difference one. A grid can be seen
(1)
(q)
as product of n paths Pq ; i.e Qn = Pq × Pq × · · · × Pq . For q = 1, Qn is the
classical n−dimensional cube Qn .
A numbering of a vertex set V is any function f : V → {1, 2, . . . , |V |}, which
is one-to-one (and therefore onto). A numbering uniquely determines a total
order, ≤, on V as follows u ≤ v if f (u) < f (v) or f (u) = f (v) . Conversely, a
total order defined on V uniquely determines a numbering of the graph.
The bandwidth of a numbering of a graph G = (V, E) is the maximum
difference
bw (f ) =
max |f (u) − f (v)|
uv∈E(G)
The bandwidth of a graph G is the minimum bandwidth over all numberings,
f , of G, i.e.
bw (G) = min bw (f )
f
The bandwidth problem appears in a lot of areas of Computer Science such
as VLSI layouts and parallel computing, see surveys [7,9].
Harper shows that the labeling given by the Hales order solves the bandwidth
problem for Qn and gives the following explicit formula [4] [6].
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Theorem 1 We have,
bw (Qn ) =
n−1
m=0
m
m/2
(1)
Our aim is to give an explicit expression for the bandwidth of the d−product
(q)
of a path with n edges, Qn using the notion of bis nomial coefficients [1], [2],
(q)
[3], [5]. The vertices of Qn are labeled with elements of the set {0, 1, . . . , q}n ,
and two vertices u and v are connected by an edge if and only if u and v differ
only at one component in which they have absolute difference one.
References
[1] G. E. Andrews, J. Baxter, Lattice gas generalization of the hard hexagon
model III q-trinomials coefficients, J. Stat Phys., 47, 297–330 (1987).
[2] H. Belbachir, A. Benmezai, A q−analogue for bis nomial coefficients and
generalized Fibonacci sequence, Cahier de Recherche Academie Scientifique
de Paris Serie 1, 352, 167-171 (2014).
[3] R. C. Bollinger, A note on Pascal T -triangles, Multinomial coefficients and
Pascal Pyramids, The Fibonacci Quaterly, 24, 140–144 (1986).
[4] L.H. Harper, Optimal numbering and isoperimetric problems on graphs,
Journal of Combinatorial Theory 1, 385-393 (1966).
[5] C. Smith, V. E. Hogatt, Generating functions of central values of generalized
Pascal triangles, The Fibonacci Quaterly, 17, 58–67 (1979).
[6] X. Wang, X. Wu, S. Dumitrescu, On explicit formulas for bandwidth and
antibandwidth of hypercubes, Discrete Applied Mathematics, 157 (8), 1947–
1952 (2009).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
A new Heuristic method for Transportation Network
and Land Use Problem
Mouhamadou A.M.T. Baldé† , Babacar M. Ndiaye† and Serigne Gueye‡
†
Laboratory of Mathematics of Decision and Numerical Analysis
LMDAN-FASEG, University of Cheikh Anta Diop
BP 45087 Dakar-Fann, Dakar, Senegal
{mouhamadouamt.balde,babacarm.ndiaye}@ucad.edu.sn
‡
LIA, Université d’Avignon et des Pays de Vaucluse,
339 chemin des Meinajaries,
BP 1228, 84911 Avignon Cedex 9, France
[email protected]
Abstract
Our paper deals with the Transportation Network and Land Use (TNLU)
problem. The problem consists in finding, simultaneously, the best location of
the activities of an urban area, as well as of the road network design in such
a way to minimize as much as possible the moving cost in the network, and
the network costs. NTLU has been introduced by Marc Los (1978) who gave
an exact and heuristic methods, assuming that the Origin-Destination flows are
routed via the shortest paths on the road network. Under the same assumption,
we propose a new mixed integer programming formulation of the problem, and
a new heuristic method for the resolution of TNLU. Numerical examples including the comparison with Los’s method, and simulations based on the scenario
of Dakar city are given.
keywords: transportation network, land use plan, quadratic assignment,
heuristic, simulations.
References
[1] M. Los. Simultaneous optimization of land use and transportation. A Synthesis of the Quadratic Assignment Problem and the Optimal Network
Problem,Regional Science and Urban Economics 8, 21-42, 1978.
[2] B.M. Ndiaye, M. Baldé, S. Gueye. Technical Report, Project ORTRANS Activity 1, 2012.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Some asymptotic normality result of k-Nearest
Neighbour estimator.
Wahiba Bouabsa, Mohammed Kadi Attouch
20-22 April 2015
Abstract
In this paper, we study the nonparametric estimator of the conditional mode
using the k-Nearest Neighbors (k-NN) estimation method for a scalar response
variable given a random variable taking values in a semi-metric space. We establish the asymptotic normality for independent functional data, and propose
confidence bands for the conditional mode function. Some simulations for real
data application have been driven to show how our methodology can be implemented.
1
Introduction
The first result of the asymptotic normality on the kernel estimator comes from
Masry (2005), he considers the case of α-mixing data but he did not give the
explicit expression of the dominant asymptotic terms of bias and variance. After Ferraty et al.(2007) gives the explicit expression of the asymptotic law (that
is the dominant terms of bias and variance) In the case of set of independent
data . The results exposed in the papers of Delsol (2007a, 2008b) make the link
between these articles, they generalize the results of Ferraty et al.(2007) in the
case of data α-mixing, Attouch and Benchikh (2012) established the asymptotic
normality of robust nonparametric regression function.
For the k-NN conditional mode estimator Attouch and Bouabça (2013) obtained
the almost complete convergence with rates in independent and identically distributed (i.i.d.) functional data case.
Let (Xi , Yi ), i = 1, .., n be n copies of random vector identically distributed as
(X, Y ) where X is valued in infinite dimensional semimetric vector space (F ,d)
and Y’s are valued in IR, In most practical applications, S is a normed space
which can be of infinite dimension (e.g. Hilbert or Banach space) with norm .
so that d(x, x ) = x − x .
We will denote the conditional distribution function of Y by:
∀y ∈ IR
F x (y) = IP(Y ≤ y|X = x).
Then we will denote by f x (respf x(j) ) the conditional density (resp its j th order
derivative) of Y .
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Ordonnancement sur deux machines avec
contraintes de concordance : machines identiques et
uniformes
Amine Mohabeddine ∗
Mourad Boudhar ∗
[email protected]
[email protected]
Ammar Oulamara †
[email protected]
20-22 April 2015
1
Introduction
La gestion de projets est une démarche visant à structurer et à optimiser le bon
déroulement d’un projet ayant plusieurs tâches à organiser. Toutefois, on se retrouve
très rapidement face à des limitations qui ne permettent pas une gestion libre du projet. Une des plus importantes contraintes rencontrées est la contrainte de ressources.
En effet l’exécution d’une tâche peut nécessiter une ou plusieurs ressources. Et dans
certains cas ces ressources ne peuvent pas être partagées entre les tâches. Ce qui empêche certaines tâches de s’exécuter simultanément e.g. deux tâches qui doivent être
supervisées par une même personne ne pourront pas être exécutées simultanément.
Ce problème de gestion de projet avec des contraintes de ressources non partageables
entre les tâches représente l’application la plus répondue du problème d’ordonnancement avec graphe de concordance, problème auquel nous allons nous intéresser.
2
Définition du problème
Le problème d’ordonnancement avec graphe de concordance consiste à ordonnancer un ensemble de n tâches sur m machines identiques, dans le but de minimiser la
date de fin de traitement. Chaque tâche a un temps de traitement et un temps de
disponibilité. Les tâches sont ordonnancées selon des contraintes de concordance représentées par un graphe appelé graphe de concordance, où chaque pair de tâches peut
s’ordonnancer simultanément si les tâches sont reliées par une arête dans le graphe
de concordance.
∗ Laboratoire
† Laboratoire
RECITS, Université de USTHB, Alger, Algerie.
LCOMS, Université de Lorrain, Metz, France.
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3
Résultats connus
Even et al. ont prouvé dans [1] que le problème était NP-difficile sur deux machines
lorsque le graphe de concordance est un graphe biparti avec des temps de traitement
des tâches égaux à 1, 2, 3, ou 4. Les auteurs ont également proposé un algorithme
polynomial qui résout le problème sur deux machines avec un graphe de concordance
arbitraire, lorsque les temps de traitement des tâches valent 1 ou 2. Bendraouche et
Boudhar dans [2] ont amélioré le résultat présenté dans [1], et ont prouvé la NPdifficulté du problème sur deux machines avec un graphe de concordance bipartie et
des temps de traitement des tâches égaux à 1, 2, ou 3. Bendraouche et al. dans [3] ont
prouvé l’NP-difficulté du problème sur deux machines et un graphe de concordance
bipartie avec deux temps de traitement des tâches différents égaux à a et 2a + b avec
b = 0.
4
Contribution
Notre travail consiste à étudier la complexité du problème lorsque le graphe de
concordance appartient à d’autres classes de graphe. Les classes de graphe étudiées
sont : les chemins, les cycles, les étoiles, les forêts, les graphes threshold, et les graphes
d’intervalle propre. Nous avons montré que le problème est polynomial lorsque le
graphe de concordance est un chemin, un cycle, et une étoile, en proposant un algorithme polynomial pour résoudre chacun de ces cas. Nous avons également prouvé la
NP-difficulté du problème lorsque le graphe de concordance est une forêt, un graphe
threshold, et un graphe d’intervalle propre. Dans la deuxième partie du travail, nous
nous intéressons au même problème lorsque les machines sont uniformes. Les résultats
NP-difficile présentés dans la première partie restent valides pour le problème lorsque
les machines sont uniformes. Dans cette partie nous allons proposer une preuve pour
la NP-difficulté du problème sur deux machines lorsque le graphe de concordance
est une étoile. Nous prouvons aussi que le problème sur deux machines avec temps
de traitement unitaire est NP-difficile. Nous terminons, en proposant un algorithme
d’approximation pour résoudre le dernier cas cité.
Références
[1] G. Even, M. M. Halldarsson, L. Kaplan, D. Ron. Scheduling with conflicts : online
and offline algorithms. Journal of Scheduling, 12 :199–224, 2009.
[2] M. Bendraouche, M. Boudhar. Scheduling jobs on identical machines with agreement graph. Computers & Operations Research, 39 :382–390, 2012.
[3] M. Bendraouche, M. Boudhar, A. Oulamara. Scheduling : Agreement graph vs
resource constraints. European Journal of Operational Research, 240 :355–360, 2015.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Algorithmes exacts et approchés pour le problème
d’ordonnancement préemptif sur machines
parallèles avec délais de transport
Ryma Zineb Badaoui ∗
Mourad Boudhar ∗
[email protected]
[email protected]
Mohammed Dahane †
[email protected]
20-22 April 2015
1
Introduction
Nous étudions le problème d’ordonnancement d’un ensemble de tâches indépendantes,
sur des machines parallèles identiques. Lorsque le traitement d’une tâche est interrompu sur une machine pour être poursuivi sur une autre, la contrainte considérée
consiste à prendre en charge les temps de transport de la tâche entre ces deux machines. Dans ce travail, nous supposons que ces temps de transport entre les machines
dépendent uniquement de la distance entre celles-ci.
En 2005, Fishkin et al.[1] ont étudié le problème avec des délais de transport fixes. Les
auteurs ont montré que pour certaines valeurs du paramètre de transport (noté d), le
problème est NP-difficile au sens fort. Ils ont également proposé une caractérisation
d’un ordonnancement optimal pour les cas de deux et de trois machines.
Haned et al.[2] ont étudié le cas de deux machines en se basant sur la caractérisation
de fishkin et al.[1]. Un algorithme de résolution pseudo-polynomial ainsi qu’un schéma
d’approximation complètement polynomial (FPTAS) pour le cas de deux machines
ont été proposés. Les auteurs[3] ont développé plusieurs heuristiques, cependant il
n’existe pas, à notre connaissance, d’autres méthodes approchées notamment de type
métaheuristique ayant été adaptées au problème.
2
Description du problème
Il s’agit d’exécuter un ensemble de n tâches indépendantes et morcelables sur un
ensemble de m machines parallèles identiques. Chaque tâche est disponible à l’instant
zéro et peut être traitée par au plus une machine à la fois, sachant qu’une machine
∗ RECITS,
† LGIPM,
Université U.S.T.H.B, Alger, Algérie.
École National d’ingénieurs de Metz (ENIM), Metz, France.
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ne peut traiter qu’une tâche à la fois. L’exécution d’une tâche Ti requiert pi unités de
temps. L’ordonnancement est préemptif, ce qui signifie que le traitement d’une tâche
sur une machine Mj peut être interrompu à tout instant pour être poursuivi sur une
autre machine Mj . La reprise de la tâche interrompue ne peut se faire qu’après une
période minimale djj unités de temps. Cette période est appelée délai de transport
entre les deux machines Mj et Mj . L’objectif est de minimiser le makespan. Le
problème se note P m|pmtn(delayjj )|Cmax.
3
Contribution
Afin d’exploiter la caractérisation d’un ordonnancement optimal pour le problème
P m|pmtn(delay = d)|Cmax avec m = 2 et m = 3, cette présente étude propose
une extension du travail effectué dans [2] en étudiant le même problème dans le cas
de trois machines. Nous présentons un programme dynamique pour résoudre le problème classique P 3||Cmax, puis un algorithme pseudo-polynomial pour résoudre le
problème initial. Enfin, nous montrons que le problème à trois machines admet un
5
FPTAS s’exécutant en O( n ).
Nous tentons ensuite de contribuer à la résolution du problème à m machines en
proposant une stratégie de résolution qui se compose de plusieurs méthodes à savoir
les métaheuristiques Variable Neighborhood Search (VNS) et le Simulated annealing
(SA). L’idée principale est de déterminer une séquence de machines, selon laquelle
tous les éventuels transports des tâches préemptées vont être effectués. Cette première phase de la stratégie de résolution est effectuée en utilisant la méthode VNS.
La deuxième phase consiste à déterminer l’ordre des tâches par la méthode SA, cet
ordre va servir à construire l’ordonnancement en utilisant la règle de Mc Naughton.
Les tâches sélectionnées pour être préemptées vont être transportées vers d’autres
machines suivant la séquence obtenue lors de la première phase.
Pour comparer les résultats obtenues par le SA, nous avons mis en place une heuristique de résolution qui s’inspire de la conjecture établie par Fishkin[1] et qui construit
une solution avec au plus m − 1 tâches préemptées et transportées suivant la séquence
obtenue par VNS.
En l’absence d’instances connues, les approches développées ont été appliquées sur
plusieurs instances générées aléatoirement. Une étude expérimentale et comparative
est en cours
References
[1] A.V Fishkin, Klaus Jansen, S.V Sevastyanov, and R.Sitters. Preemptive scheduling of independent jobs on identical parallel machines subject to migration delays.
In Algorithms-ESA 2005, pages 580-591. Springer, 2005.
[2] A.Haned, A.Soukhal, M.Boudhar, and Nguyen Huynh Tuong. Scheduling on parallel machines with preemption and transportation delays. Computers & Operations
Research, 39(2) : 374-381, 2012.
[3] Mourad Boudhar and Amina Haned. Preemptive Scheduling in the presence of
transportation times. Computers & Operations Research, 36(8) : 2387-2393, 2009.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Multi-Objective Stochastic Integer Linear
Programming
Salima Amrouche and Mustapha Moulaı̈
USTHB, Faculty of Mathematics, LaROMaD Laboratory
[email protected] [email protected]
20-22 April 2015
Abstract
Multi-objective stochastic integer programming is an optimization technique in
which the objective functions and some constraints of an optimization problem
contains integer variables and random data which follow discrete probability distribution [2]. Once a problem requires a stochastic formulation, a first step consist
in transforming the problem into an equivalent deterministic formulation [3]. In
the second case, it is necessary to transforming the multi-objective problem into
a mono-objective problem. An algorithm combined the integer L-shaped method
and branch and bound method with efficient cuts concept for the search of integer
efficient solutions [1].This approach has the advantage to give the decision maker
the efficient solution and their corresponding optimal cost values of the random
constraint violations.
Problem Formulation
We consider multi-objective integer linear programming problems involving random
variable coefficients in both objective functions and some constraints of the following
model
(P )
min Ci (ξ) x, i = 1, ..., k
T (ξ) x = h (ξ)
x ∈ S, x integer
where k ≥ 2; Ci , T and h are random matrices defined on some probability space
(Ω, E, P ) ; S = {x ∈ Rn |Ax = b, x ≥ 0}. The vector b ∈ Rm and A the m × n real
matrix are given and x is to be determined.
The transformed deterministic model
Assume that we have a joint finite discrete probability distribution of the random
data : {(ξ r , pr ) , r = 1, ..., R} . R is the number of realizations (scenarios).
For each realization ξ r we associate a criterion Ci (ξ r ) x, a matrix T (ξ r ) , a
vector h (ξ r ) and a recourse matrix W (ξ r ) = W. In this article the recourse matrix
W does not change, this is called fixed recourse [3]. We assume that the decision
maker is able to satisfactorily specify the penalties q r of the constraint violations
zr . The above problem is equivalent to the so-called Deterministic Equivalent (DE)
Multi-Objective Integer Linear Programs M OILP.
(DE)
min Zi = Źi + Q (x) , i = 1, ..., k
x ∈ S, x integer
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(1)
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
where Źi = Eξ [Ci (ξ) x] , Q (x) = Eξ [Q (x, ξ)] and
Q (x, ξ r ) = min (q r )t z | T (ξ r ) x + W (ξ r ) z = h (ξ r ) , z ≥ 0 .
(2)
The transformed mono-objective model
We transform
problems in the following way
the DE
min λT Ź + Q (x)
(DEλ )
x ∈ S, x integer
where λT = (λ1 , ..., λk ) ≥ 0, with at least one component strict inequality.
Solution method
⎧ T T
λ E Ci (ξ) x + θ
⎪
⎨
(Pl )
Dl x ≥ dl ,
⎪
⎩ El x + θ ≥ e l ,
l = 1, ..., N
l = 1, ..., M
(3)
x ∈ Sl
(Pl ) obtained at node l in a structured tree.
of efficient solution xl ;
Nl is the
indices iset of non-basic variables
i
i
cjl < 0 , where %
cjl is the component j of the reduced cost vector
Hl = j ∈ N l | %
of the objective
function
Ź
.
i
&
'
Sl+1 =
x ∈ Sl \
j∈Hl
xj ≥ 1 .
N and M indicate the number of feasibility cuts and optimality cuts, respectively,
added until step l.
Starting with θ = −∞ and without
feasibility
cuts, optimality cuts and effi
ciency cuts. The objective λT E C T (ξ) x is minimized under the deterministic
constraints. If for some realizations the second stage problems yielded by the solution x are not feasible, a feasibility cut is introduced. Then, the problem is
optimized again to obtain another feasible point x. If x is not integer, create two
new branches at fractional component of x; append the new nodes to the list of
pendant nodes; the branching process is started to search an integer feasible point
r
recours
xl . Using xl , we
solve
the
l problem for all realizations ξ , r ∈ {1, ..., R},
l
l
and compute Q x . If θ < Q x then a new optimality cut El x ≥ el is added to
the
current problem(Pl ) . In presence of integer optimal solution, An efficient cut
xj ≥ 1 is then added for deleting integer solutions that are not efficient and
j∈Hl
the new program is solved. The method terminates when all the created nodes are
saturated.
References
[1] M. Abbas; M.E. Chergui; M. Ait Mehdi ”Efficient cuts for generating the
non-dominated vectors for Multiple Objective Integer Linear Programming”, International Journal of Mathematics in Operational Research. Volume 2008, DOI:
10.1504/IJMOR.2012.046690, pages:302-316.
[2] S. Amrouche, M. Moulaı̈ ”Multi-objective stochastic integer linear programming
with fixed recourse”, International Journal of Multicriteria Decision Making (IJMCDM), 2012 Vol. 2 No. 4, pages:355-378.
[3] J.R Birge, F. Louveaux, Introduction to Stochastic Programming. Springer,
1997.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
A condition based maintenance model using
Bayesian control chart
Nabil Bacha, Isabel Lopes, and Lino Costa
Algoritimi Research Centre, University of Minho, Portugal
20-22 April 2015
Abstract
This paper introduces a model for Condition based maintenance (CBM) incorporating simultanuously two types of data: key quality control measurement
and equipment condition parameters.
1
Introduction
With the growth of systems complexity and the advance in technology in modern
industry and the intensive international competition, manufacturing companies
have to adopt new management maintenance practices. Since equipment in a
production system has become more advanced, more expensive, and may have
a significant impact on production function and products quality, the cost spent
in preventive maintenance has become higher and higher.
Maintenance strategies have progressed from breakdown maintenance, to
preventive maintenance, and then to CBM. Breakdown maintenance is reactive
in nature while the action of repair or replacement is done only when equipment
has already failed. Preventive maintenance is proactive in nature and consists of
a set of tasks (replacements, adjustments, inspections and lubrications, etc.) in
order to prevent catastrophic failures or to eliminate any degradation in equipment. If the deterioration level of items is correlated strongly with a control
parameter, the decision about the realization of preventive maintenance operations can be based on system’s condition, this is condition-based maintenance [4].
CBM is a set of maintenance actions performed based on real-time or near realtime assessment of equipment condition obtained from embedded sensors, external tests or measurements using portable equipment. The data obtained from
condition monitoring helps maintenance managers to decide if maintenance is
necessary or not by analyzing the actual condition of equipment [3]. Therefore,
CBM is considered as an intelligent preventive maintenance and a suitable strategy for forecasting items failures. CBM models provide lower inventory costs
for spare parts, reduces unplanned outage and minimize the risk of catastrophic
failure, avoiding high penalties associated with losses of production or delays.
The main goal of CBM models is to avoid unnecessary maintenance tasks by
taking maintenance actions only when there is evidence of abnormal behaviors
of an item [3]. CBM policies can be more cost effective than preventive maintenance policies and the least expensive and most effective strategies for proactive
maintenance [2][5].
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Using statistical control charts for maintenance planning leads to a significant
benefit in term of reducing maintenance costs [1][3]. Several maintenance models
using control charts for maintenance optimization have been extensively developed, and they can be classified into two different groups. The first one regards
the use of control chart to monitor a process through some quality characteristic for maintenance planning; the second one regards the use of control chart
to monitor the health state of equipment while in operation through a health
parameter. Control charts might be considered an effective tool to indicate and
detect the early signs of health state of a deterioration or malfunction of equipment. Control charts use parameters values obtained from periodic condition
monitoring of equipment in order to decide whether maintenance action should
be performed or not. However, Bayesian control charts have been proved to be
superior tools to control the process compared with the non-Bayesian charts [6].
In fact, there is a direct relationship between equipment maintenance and product quality. Products quality is often affected by the equipment health state,
improving equipment performance would also enhance product quality [1].
Nowadays, in maintenance area, decision-maker faces a challenge in terms
of choosing an appropriate and accurate decision. Proper and well-performed
CBM models are beneficial for maintenance decision making. However, CBM
models need to integrate new type of information in maintenance modeling in
order to improve the results. The main aim of this paper is to present a new
conceptual model based on a Bayesian control chart for predicting failures of
item incorporating simultaneously two types of data: key quality caracteristics
control measurement and equipment condition parameters.
Acknowledgements
This work has been supported by FCT Fundação para a Ciência e Tecnologia
in the scope of the project: PEst-OE/EEI/UI0319/2014.
References
[1] C.R. Cassady, R.O. Bowden, L. Liew, E.A. Pohl: Combining preventive
maintenance and statistical process control: a preliminary investigation. IIE
Transactions vol. 32, n. 6, 471–478, 2000.
[2] M. Ilangkumaran, S. Kumanan: Selection of maintenance policy for textile industry using hybrid multi-criteria decision making approach. Journal of
Manufacturing Technology Management vol. 20, n. 7, 1009–1022, 2009.
[3] A.K.S., Jardine, D. Lin, D. Banjevic. A review on machinery diagnostics
and prognostics implementing condition-based maintenance. Mechanical systems and signal processing vol. 20, n. 7, 1483–1510, 2006.
[4] M. Rabbani, N. Manavizadeh, S. Balali: S. A stochastic model for indirect
condition monitoring using proportional covariate model. International Journal
of Engineering Transactions a Basics, vol. 21, n. 1, p. 45, 2008.
[5] S.K. Yang: A condition-based failure-prediction and processing-scheme for
preventive maintenance. IEEE Transactions on Reliability vol. 52, n. 3, 373–
383, 2003.
[6] Z. Yin: Multivariate bayesian process control. PhD thesis, University of
Toronto, 2008.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
On The Inference Of Periodic Mixture Generalized
Autoregressive Conditional Heteroscedastic Model
Ines BRAHIMI1 & Mohamed BENTARZI1
20-22 April 2015
Abstract
This communication deals, essentially, with the estimation of parameters of
the first order Periodic Mixture Generalized Autoregressive Conditional Heteroscedastic Model M P GARCHS (K, 1, 1). The estimation is done by the Maximum Likelihood method via the Expectation-Maximisation algorithm (EM ).
An intensive simulation study is done.
1
Introduction
The standard tools modeling time series which exhibit clustering volatility and
kurtosis, heavy tails, excess of kurtosis, skewness (asymmetry) and multimodality in their innovation density are the autoregressive conditionally heteroscedastic (ARCH) model, introduced by Engel (1982), and the generalized autoregressive conditionally heteroscedastic (GARCH) models, which are a generalization
of ARCH models, introduced by Bollerslev (1986). However, these models are
not adequate neither when the series’s innovation density is affected by a high
kurtosis and heavy tails nor when the autocovariance structure of the series is
affected by periodicity. To compensate the first fact, Wong and Lee (2000,2001),
introduced Mixture ARCH when to compensate the second, Shao (2006) introduced the Mixture of Periodic Autoregressive models. Many years after, the
Periodic Mixture GARCH model, which is a larger class of models and which
takes in consideration the two precedents facts is introduces by Bentarzi and
Ouzzani (to appear).
We are mainly interested by the estimation of the parameters of the first order M P GARCHS (K, 1, 1) while using the EM algorithm. The EM algorithm
use the maximum likelihood to complete the missing data to finely estimate
the parameters of the generator model. In the rest of the article, we present
the underlined model. In the second section we present the EM algorithm and
estimate the parameters of our model while in the last section, we propose a
simulation study.
2
Definition and main assumption
A periodically correlated (in the sense of Gladishev (1961)) process,{Xt , t ∈ Z} ,
is said to satisfy a Mixture Periodic Generalized Autoregressive Conditionally
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Heteroscedastic model, with period S, denoted M P GARCHS (K, q1 , ..., qK , p1 , ..., pK ),
if it s a solution of the following non linear stochastic difference equation:
⎧
K
⎪
Xs+τ S
⎪
⎪
√
F
(X
/F
)
=
λ
Φ
, t, τ ∈ Z, s = 1..S
t
t−1
k
⎪
⎨
hk,s
k=1
qk
pk
⎪
⎪
(k)
(k) 2
(k)
⎪
=
α
+
α
X
+
βj,s hk,(s−j)+τ S .
h
⎪
(s−i)+τ S
0,s
i,s
⎩ k,s
i=1
(k)
αi,s
where:
=
odic in time).
3
(k)
αi,s+τ S
and
j=1
(k)
βi,s
=
(k)
βi,s+τ S
(k)
(k)
and σi,s = σi,s+τ S (functions peri-
Estimation by the EM algorithm
Briefly, if we put: X = (X1 , X2, ..., XN S , Z1,t , Z2,t , , ...ZK,t ) the vector of complete data, Z = (Z1,t , Z2,t , , ..., ZK,t ) the vector of missing data where Zk,t = 1
if the t− eme observation comes from the
model k, 0 else,
(k)
(k)
(k)
(k)
(k)
θk,s = α0,s , α1,s , ..., αqk,s ; β1,s , ..., βpk,s and λ = (λ1 , ..., λK−1 ) then suppos(i)
(i)
ing that λk and θk,s are known and equals λk and θk,s respectively, we can then
replace the missing data by their estimation obtained by their conditional expectations on parameters and observations, to calculate the maximum likelihood,
K
under constraints: λk = 1, which is the Expectation step of the algorithm.
k=1
Now,we can obtain the estimation of of parameters calculating the maximum
likelihood function and this is the maximization step of the algorithm. We replace the parameters obtained to estimated again the missing data and replace
the missing data thus obtained to estimate the parameters. We repeat this
operations until obtain the wished precision.
References
[3] M. Bentarzi and F. Ouzzani : On Mixture Periodic GARCH (1,1) model. To
appear.
[6] T. Bollerslev : Generalized Autoregressive Conditional Heteroscedasticity.
Journal of Econometrics vol.(31), 307-327. (1986).
[5] R. Engle : Autoregressive Conditional Heteroscedasticity with Estimates of
United Kingdom Inflation. Econometrica vol.(50), 987-1008. (1982).
[4] Q. Shao : Mixture Periodic Autoregressive Time Series Model. Statistics and
Probability Letters vol.(76), 609-618. (2006).
[1] C. S.Wong and W. K. Li : On a mixture autoregressive model. Journal of
the Royal Statistical Society vol.(62.), 95-115. (2000).
[2] C. S.Wong and W. K. Li : On a mixture autoregressive conditional heteroscedastic model. Journal of the American Statistical Association vol.(96),
982-995. (2001).
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
On the estimation of Mixture Periodic integer-valued
ARCH models
Ouzzani Fares1 and Bentarzi Mohammed1
20-22 April 2015
Abstract
This work deals with the mixture of periodic integer-valued ARCH models,
which is useful to describe the heteroscedasticity feature, the multimodality of
the generating process, and the hidden periodicity in the autocovariance structure. The main goal of this work is to estimate the model’s parameters using
the Expectation-Maximization algorithm, so we present in first time some probabilistic and statistical properties, then we study the EM estimation method,
and finally we give some simulations of the periodic MINARCH model.
1. Introduction
We encounter In life several phenomena where the counting reign, therefore the
modeling of these phenomena by mathematical models designed for real-valued
problems is not adequate. That is why a class of linear and non linear time
series models was introduced, namely the class of integer-valued models.
This new class of models has been very useful for modeling the counting process
depending on discrete time, including epidemiology (number of infected persons
by a certain type of disease), criminology (number of crimes in a state), and finance (discrete financial transactions). Therefore, the analysis of integer-valued
time series encountered nowadays incited many mathematicians to investigate
a variety of models to capture and describe the discreteness nature of the phenomena encountered in several areas. One can cite for example the INGARCH
models which captures the heteroskedasticity feature (see Ferland et al - 2006),
and mixtures of non-linear integer-valued models, which deal with the multimodality in law of the generating process (see Zhu et al - 2010). However, these
models do not deal with the hidden periodicity in the autocovariance structure.
Therefore, the investigation of the periodic integer-valued models is a primordial
goal.
This work aims to treat the mixture of periodic integer-valued autoregressive
conditionally Heteroscedastic (periodic MINARCH). First, we present the structure of the model and take the necessary notations and main assumptions like
the stationary conditions, then we present a method for estimating the model
parameters by adopting the Expectation-Maximization algorithm concept, introduced for the first time by Dempster (1977), and we finish with a simulation
study to support the results.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
2. Model definition and main assumptions
A periodically correlated (in the Gladyshev sense - 1963) integer-valued process
{yt ; t ∈ Z} is said to satisfy a Periodic Mixture Integer-Valued of Autoregressive
Conditionally Heteroskedastic (P M IN ARCH) model, with period S and orders
p1 , p2 , ..., pK , denoted P M IN ARCHS (K; p1 , p2 , ..., pK ), if it is given by :
Xt =
K
1 (ηt = k) Yk,t , such that Yk,t /Ft−1 → P (λk,t ) ,
k=1
with λk,t = β0,k +
pk
(2.1)
βi Xt−i ,
i=1
where 1 (.) and Ft−1 denote, as usual, the indicator function and the σ−algebra
based on the information available up to time t − 1 and where ηt is a sequence
of independent identically distributed
variables with P (ηt = k) = αk and where
αk > 0, k = 1, 2, ..., K such that K
k=1 αk = 1.
(k)
The parameters βi,t , i = 0, 1, ..., pk , are periodic in t, with period S, i.e.,
(k)
(k)
βi,t+rS = βi,t , i = 0, ..., pk , k = 1, ..., K, r ∈ Z and t ∈ Z. To avoid the
possibility of zero or negative conditional variances, the following conditions
(k)
(k)
(k)
for βi,t ’s must be imposed: β0,t > 0, k = 1, ..., K and t ∈ Z, and βi,t ≥ 0,
i = 1, ..., pk , k = 1, ..., K and t ∈ Z. Moreover, the process ηt is assumed, to be
independent of the past of the underlying process yt , that is ηt is independent
of yt−i , ∀i > 0 and ∀t ∈ Z. For the purpose of manipulation simplicity, we con(k)
sider pk , k = 1, ..., K, constant and equals to p = maxpk while taking αi,t = 0
k
for i > pk . It is worth noting that in the time-invariant case, i.e., S = 1, the
model (2.1) reduces to the mixture integer-valued ARCH model introduced and
investigated by Zhu et al (2010) :
Xt =
K
1 (ηt = k) Yk,t , such that Yk,t /Ft−1 → P (λk,t ) ,
k=1
with λk,t = β0,k +
pk
(2.2)
βi Xt−i ,
i=1
References
[1] Dempster, A. P., N. M. Laird, and D. B. Rubin : Maximum likelihood
from incomplete data via the em algorithm. Journal of the Royal Statistical
Society.(39), 1-38. (1977).
[2] Ferland, R., A. Latour, and D. Oraichi : Integer-valued garch process. Journal of Time Series Analysis. (27), 923-42. (2006).
[3] Gladyshev, E. G. (1963). Periodically and almost PC Random processes
with continuous time parameter. Theory Probab. Appl., 8, 173 − 177.
[4] Zhu F., Q. Li, and D. Wang : A mixture integer-valued arch model. Journal
of Statistical Planning and Inference. (140), 2025-2036. (2010).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Periodic Integer-Valued GARCH (1, 1) Model
Wissam Bentarzi∗ and Mohamed Bentarzi*
20-22 April 2015
Abstract
This paper deals with some probabilistic and statistical properties of a periodic integer-valued GARCH (1, 1) model. Necessary and sufficient conditions
for the periodically stationary, both in mean and second order, are established.
The closed-forms of the mean and the second moment are, under these conditions, obtained. The condition of the existence of higher moment orders and
their explicit formula in terms of the parameters are established. The autocovariance structure is studied, while providing the closed-form of the periodic
autocorrelation function. The Yule-Walker and the likelihood estimations of the
underlying parameters are obtained. A simulation study and an application on
real data set are provided.
1. Introduction
Much attention has been given in the last two decades to the modelling and
studying the probabilistic and statistical properties of linear and nonlinear nonnegative integer-valued time series models (see, Alzaid and Al-Osh (1990), Latour (1998), Silva and Oliveira (2000), Doukhan et al (2006), Ferland et al
(2006), Barczy et al (2011) and many others). However, despite the fact that
many nonnegative integer-valued time series encountered in the precedent different domains and others reveal the periodicity feature in their autocovariance
structures (as examples Number of cases of campylobacterosis infections time
series studied by Ferland et al (2006), Monthly number of short-term unemployed people in Penamacor County Portugal, studied by Monteiro et al 2010),
it seems that the study of periodic integer-valued time series (linear and non linear) models has not received much attention in the literature of count time series.
Indeed, in our knowledge, the first paper dealing with the modeling of the periodically correlated process, in the Gladyshev’s sense (1963), is this of Monteiro
et al (2010). It is recognized that many economic, financial and environmental
integer-valued time series, encountered in practice, exhibit a periodical autocorrelation structure, a feature that cannot be adequately accounted and described
by time invariant parameter integer-valued time series models. This fact gave a
good reason and motivation to extend this class of time-invariant models to the
periodic ones. In this communication, we provide the some probabilistic and
∗ Faculty of Mathematics, University of Science and Technology Houari Boumediene, Algiers,
Algeria. Corresponding author : Wissam Bentarzi Email. [email protected].
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
statistical properties a Periodic Integer-Valued Generalized Autoregressive Conditional Heteroskedastic, P IN GARCH (q, p). The estimation of the parameters
is done via the Yule-Walker and Conditional Maximum Likelihood estimation.
A simulation study and an application on real data set are provided.
2. Notations, Definitions and main Assumptions
A Periodically correlated Integer-Valued process, in the sence of Gladyshev
(1963), {yt ; t ∈ Z} is said to satisfy a periodic, with period S, Integer-Valued
Generalized Autoregressive Conditional Heteroskedastic model, with orders 1
and 1, noted P IN GARCHS (1, 1) , if it is given by :
yt /Ft−1 P (λt ) ,
λt = α0,t + α1,t yt−1 + βt λt−1 , ∀t ∈ Z,
(2.1)
where the parameters αi,t , i = 0, 1 and βt , are periodic in t, with period S, i.e.,
αi,t+rS = αi,t , i = 0, 1 and βt+rS = βt , t, r ∈ Z. Moreover, these parameters are
such that : α0,t > 0, α1,t ≥ 0 and βt ≥ 0, ∀t ∈ Z.
This model extends the following time-invariant IN GARCH (1, 1) studied
by Ferland et al (2006) to the time periodic case:
yt /Ft−1 P (λt ) where λt = α0 + α1 yt−1 + β λt−1 , ∀t ∈ Z.
(2.2)
References
[1] Alzaid, A. A. and Al-Osh, M. A.(1990). Integer-Valued p th-Order Autoregressive Structure (IN AR (p)) Process. J. App. Prob. 27, 314 − 324.
[2] Barczy, M., Ispany, M. and Pap, G. (2011). Asymptotic behavior of unstable
IN AR (p) processes. Stochastic Processes and their Applications. Time. Series
Anal. 121, 583 − 608.
[3] Doukhan, P. Latour, A. and Oraichi, D. (2006). A Simple Integer-Valued
Bilinear Time Series Model. Adv. Appl. Prob. 559 − 578.
[4] Ferland, R., Latour, A. and Oraichi, D. (2006). Integer-Valued GARCH
Process. J. Time Ser. Anal., Vol. 27. No. 6, 923-942.
[5] Gladyshev, E. G. (1963). Periodically and almost PC Random processes
with continuous time parameter. Theory Probab. Appl., 8, 173 − 177.
[6] Latour, A. (1998). Existence and Stochastic Structure of a Non-Negative
Integer-Valued Autoregressive Process Process. J. Time. Series Anal. Vol. 19,
No. 4, 439 − 455.
[7] Monteiro, M., Scotto, M. G. and Pereira, I. (2010). Integer-valued autoregressive processes with periodic structure. Journal of Statistical Planning and
Inference. pp. 1529 − 1541.
[8] Silva, M. D. and Oliveira, V. L. (2000). Difference equations for the higherorder moments and cumulants of the IN AR (1) model. J. Time. Series Anal.
Vol. 25, No. 3, 317 − 333.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
A Tabu Search and a Genetic Algorithm for solving a
Bicriteria Parallel Machine Scheduling Problem
Karima Bouibede-Hocine
[email protected]
Drifa Hetak
[email protected]
Mourad Boudhar
[email protected]
20-22 April 2015
Abstract
This paper deal with a uniform parallel machines scheduling problem. The
objective is the minimization of two citeria, the maximum completion time and
the maximum lateness, and we are interested to solve it by means of a fast
and ecient multiple-objective evolutionary algorithm based on NSGA-II. The
initial population is randomly generated by using a tabu serach algorithm which
is based on the minimisation of a linear combinaition of criteria.
Tabu Search, Bi-criteria, NSGA II, Scheduling
Keywords:
1 Introduction
The scheduling problem that we consider can be stated as follows. Consider n
jobs to be scheduled without preemption on m uniform parallel machines, such
that each machine has its own speed denoted by Vj , 8j = 1; :::; m. Each job Ji
has a release time ri and a due date di . Job Ji can be performed by any machine Mj , thus requiring a processing time pij with pij = Vpji , 8i = 1; :::; n; 8j =
1; :::; m, where pi is the absolute processing time of the job Ji . The bicriteria
problem involves the makespan and the maximum lateness minimisation. For
a given schedule, each job Ji completing at time Ci , the two objectives can be
computed as follows:
Cmax = maxi=1::n fCi g and Lmax = maxi=1::n fCi di g.
2 Problimatic
The aim of our work is to nd the so-called Pareto optimal solutions or nondominated solutions which can be formally dened as follows. Let S be the
set of feasible schedules. A schedule s 2 S is a strict Pareto optimum i there
does not exist another schedule s0 2 S such that (Lmax (s0 ) Lmax (s) and
Cmax (s0 ) < Cmax (s)) nor (Lmax (s0 ) < Lmax (s) and Cmax (s0 ) Cmax (s)).
The set of strict Pareto optimum is denoted by E 2 S . The criteria vector
associated to a strict Pareto optimum is called a non dominated criteria vector
and the set of such vectors is denoted by ZE . Notice that jZE j jE j. To solve
the bicriteria problem we enumerate set ZE by calculating one strict Pareto
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
optimum per non dominated criteria vector. The calculation of the strict Pareto
optima for this problem is NP-hard.
Few related works have been found in the literature. When only criterion Cmax
is minimised, Dell'Amico and Martello ([3]) propose for the P jjCmax a branchand-bound exact algorithm which is capable of solving, in the most favorable
cases, instances with up to 1000 jobs in size. The problem with unrelated parallel
machines, referred to as RjjCmax , has been solved by Van de Velde ([8]) and
Martello, Soumis and Toth ([7]) who propose an exact algorithms. Fanjul-Peyro
and R. Ruiz[4]propose also a new metaheuristics based on the Iterated Greedy
methodologie for the same problem. When only criterion Lmax is minimised,
Carlier ([1]) proposes a branch-and-bound algorithm that can be used to solve
the P jri ; di jLmax problem. This algorithm has been next extended by Carlier
and Pinson ([2]). Haouari and Gharbi ([5]) focus on the design of lower bounds
based on network ow formulations that outperform the one proposed by Carlier
and Pinson.
3 Resolution method
We propose to heuristically solve this problem by using a tabu search and a
genetic algorithm. The last one is a population-based algorithm which means
that it make evolving a set of solutions towards the set of strict Pareto optima.
The initial population of NSGA-II is generated by using a tabu search algorithm, that minimizes a linear combination of the two criteria considered for
our problem.
References
[1] Carlier, J.: Scheduling jobs with release dates and tails on identical machines
to minimize the makespan. Eur. J. of Oper. Res. 29 (1987) 298{306.
[2] Carlier, J., Pinson, E.: Jackson's pseudo preemptive schedule for the
Pmjrj ; qj jCmax scheduling problem. Ann. of Oper. Res. 83 (1998) 41{58.
[3] Dell'Amico, M., Martello, S.: Optimal scheduling of tasks on identical parallel processors. ORSA J. on Comp. 7 (1995) 191{200.
[4] L. Fanjul-Peyro and R. Ruiz Iterated greedy local search methods for unrelated parallel machine scheduling European Journal of Operational Research
205 2010 55{69.
[5] Haouari, M., Gharbi, A.: An improved max-ow-based lower bound for
minimizing maximum lateness on identical parallel machines. Oper. Res.
Let. 31 (2003) 49{52.
[6] Lancia, G.: Scheduling jobs with release dates and tails on two unrelated
parallel machines to minimize makespan. Eur. J. of Oper. Res. 120 (2000)
277{288.
[7] Martello, S., Soumis, F., Toth, P.: Exact and approximation algorithms for
makespan minimization on unrelated parallel machines. Disc. App. Math.
75 (1997) 169{188.
[8] Van de Velde, S.L.: Duality-based algorithms for scheduling unrelated parallel machines. ORSA J. on Computing 3(1) (2005) 1{21.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
A Tabu Search for a Bicriteria Scheduling Problem of
Job-Shop with Blocking
Mohamed LAKEHAL
USTHB
Algiers, Algeria,
[email protected]
Karima Bouibede-Hocine
USTHB
Algiers, Algeria,
[email protected]
20-22 April 2015
Abstract
This paper deal with a Bicriteria Scheduling Problem of Job-Shop with
Blocking. The objective is the minimization of two citeria, the maximum completion time and the maximum lateness, and we are interested to solve it by
means of a tabu search algorithm and the two phases method.
Keywords: Tabu Search, Bi-criteria, two phases, Scheduling, jobshop
1
Introduction
The problem considered in this paper is an extension of the classical job shop.
This scheduling problem is called job shop with blocking constraints or Blocking
Job Shop. We consider J as a set of n jobs, denoted by J = J1 , ..., Jn . It must
be scheduled on a set M of m machines, denoted by M = M1 , ..., Mm . Each
job Jj is a sequence of sj operations, Jj = Oj1 , Oj2 , , OjSj which have to be
processed on m different machines with no storage space and according to a
given order. An operation Oij requires Pij time units and must be executed
without interruption. To each job Jj a due date dj is associated. The aim is
to determine a performing machine and a starting time for each operation to
calculate the completion time Cj of job Jj . This scheduling is done in order to
minimize two criteria. The first one, referred to as the makespan, defined by
Cmax = max{Cj } . The second criterion is the maximum lateness of jobs and
is defined by Lmax = max{Cj − dj }
2
Problematic
The aim of our work is to find the so-called Pareto optimal solutions or nondominated solutions which can be formally defined as follows. Let S be the set
of feasible schedules. A schedule s ∈ S is a strict Pareto optimum if and only
if there does not exist another schedule s ∈ S such that (Lmax (s ) ≤ Lmax (s)
and Cmax (s ) < Cmax (s)) nor (Lmax (s ) < Lmax (s) and Cmax (s ) ≤ Cmax (s)).
The set of strict Pareto optimum is denoted by E ⊆∈ S. The criteria vector
associated to a strict Pareto optimum is called a non dominated criteria vector
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
and the set of such vectors is denoted by ZE. Notice that |ZE| ≤ |E|. To solve
the bi-criteria problem we enumerate set ZE by calculating one strict Pareto
optimum per non dominated criteria vector. The calculation of the strict Pareto
optima for this problem is NP-hard.
The literature contains a great number of papers dealing with job shop problem with blocking constraints. But to the best of our knowledge no work related
to the J/ri, di, blck/Cmax , Lmax problem exists. When only criterion Cmax is
minimised, Mascis and Pacciarelli ([3]), propose a branch-and-bound method
and three specialized dispatching heuristics for several types of job shop problems (classical, with blocking and no-wait ).In Mati et al. ([4]), a tabu search
based on a geometric approach, is proposed for a multi-resource job shop problem with blocking constraints. They also, propose a tabu search which applies
permutation moves on the critical path. Groflin and Klinkert ([2]) introduce a
generalized disjunctive graph framework for modelling various types of scheduling problems. They propose a local search approach for the generalized Blocking
Job Shop problem. In AitZai et al ([1]), authors propose a new parallel branch
and bound method and develop a sequential and parallel heuristics based on
Particle Swam Optimization (PSO).
3
Resolution method
We propose to heuristically solve this problem by using a tabu search algorithm
based on the two phases method. The combination of these methods solves the
job shop problem by exploring only the relevant search space, and it is able to
calculate an approximation of two types of solutions: Supported solutions and
Non-supported solutions. The first phase consists in finding supported solutions
by solving linear aggregations, of the form λCmax + (1 − λ)Lmax with λ = 0..1,
of the two criteria Lmax and Cmax . Once the approximation of the supported
solutions(Pareto solutions which are on the convex hull) has been found, the
second phase consists in finding an approximation of the Pareto solutions which
are not on the convex hull(Non-supported solutions).
References
[1] H. Ait Zai M. Boudhar: Parallel branch and bound and parallel PSO algorithms for the scheduling problem with blocking, International Journal of
Operations Research. 16 (1) (2013) 14-37.
[2] H.Groflin, A.Klinkert: A new neighbourhood and tabu search for the Blocking, Discrete Applied Mathematics, 157, (2009) 3643-3655.
[3] A. Mascis, D. Parcciarelli : Job Shop scheduling with blocking and no-wait
constraints, European Journal of Operation Research 143 (2002) 498-517.
[4] Y. Mati, N. Rezg, X. Xie : scheduling problems of Job-Shop with blocking:
A taboo search approach, Extended Abstracts, MIC 2001-4th Metaheuristics
International Conference, Portugal,, 2001, PP. 643-648.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Opitimisation de la largeur bit pour l’évaluation des
polynômes
El-sedik Lamini1 2 ,Samir Tagzout3 , Hacène Belbachir1 and Adel Belouchrani4
1
USTHB, Faculty of Mathematics RECITS Laboratory,DG-RSDT
BP 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria
2
Département Procédés Technologiques Avancés et Gestion de Projets
3
Division de Microélectronique et Nanotechnologie
CDTA, BP17 Baba Hassen, Algiers, Algeria.
4
Electrical Engineering Department Ecole Nationale Polytechnique
BP 16028,El Harrach, Algiers, Algeria.
[email protected] & [email protected] & [email protected]
& [email protected]
20-22 April 2015
Abstract
Optimisation de la largeur de bit (Bit width optimization, BWO) est la
détermination du nombre exact de bits pour les signaux et les constantes dans
le chemin de données. Pour les circuits intégrés (CI) aux virgules fixes, Le BWO
consiste à minimiser le nombre de bits pour la partie entière (Integer bit, IB)
afin de prévenir tout débordement et minimiser la partie fractionnaire (FB) pour
répondre aux exigences de précision.
Plusieurs stratégies existent dans la litérature pour l’évalution des polynômes.
chaque stratégie à ses avantages et ses inconvénients. La stratégie de Horner[3],
pour un polynôme de degré n, on utilise n multiplicateurs et n additionneurs
et on calcule les largeurs des 2(n − 1) segments intermédiaires, des constantes
(les coefficients ci de polynôme), du signale d’entrée (x) et de sortie (y). Nous
proposons l’utilisation d’une stratégie de planification avec un accumulateur,
un additionneur, un multiplexeur, un retardateur et deux multiplicateurs pour
l’évaluation d’un polynôme de degré n quelconque, voir la (Figure 1).
Des méthodes arithmétiques (intervalle ou affine) sont utilisées pour léstimation
des largeurs des données, voir [1, 5, 4], et qui donnent parfois des sur-ésitmations.
D. Lee and J. D. Villasenor [2] utilisent le théorème des accroissements fini
dans l’analyse de rang pour estimer l’IB. Nous utilisons des formules démontré
mathématiquement pour les analyses de rang et de précision.
L’analyse de rang : Soit le signale d’entrée x ∈ [xmin , xmax ] et ci les
coefficients de polynôme avec i = 0..n. Il est nécessaire d’estimer la valeur
maximale de chaque segment pour lui attribuer son propre IB.
maxSC = max1≤i≤n {|ci |}
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Figure 1: Circuit adopté pour l’évaluation d’un polynôme de degré 3
M axSX = max{|xmin |, |xmax |, (|xmin |)n , (|xmax |)n }
(2)
M axSXC = max1≤i≤n {|ci (xmin )i |, |ci ximax |}
(3)
M axSACM = max1≤i≤n {rangpoly (
i
(ck xk )}
(4)
k=1
Où rangpoly ( ik=1 (ck xk )) est une fonction qui calcule la valeur maximale de
la sortie du polynôme ik=1 (ck xk ) par la méthode d’analyse de rang proposée
dans [2].
L’analyse de précision : Après développement mathématique on obtient
la formule de la propagation de l’erreur dans le circuit proposé, notée ey , qui
s’exprime comme suit
e y = e c0 +
n
(ci )f
i=1
i−1
k=0
i−1
i
xk exi−k +
(x+ex )j etr (xi−j )+eci xi +etr (ci xi ) (5)
k
j=0
Où (x)f est la valeur de la représentation en virgule fixe du réel x, ex est
l’erreur produite lors du passage à la représentation, et on a x = (x)f − ex et
etr (x) est l’erreur de troncature après chaque multiplication.
A l’aide de ces formules, on peut facilement donner des bornes supérieures pour
le rang et l’erreur et ainsi l’IB et le FB nécessaire. Le choix du circuit de la
Figure 1 a permit de diminuer le nombre d’opérateurs et de variables (segments)
ainsi un gaine en terme d’area.
References
[1] L. Dong.U, A. Gaffar, O. Mencer, and W. Luk. Minibit: Bit-width optimization via affine arithmetic. DAC., (2):837–840, 2005.
[2] D. Lee and J. D. Villasenor. A bit-width optimization methodology for
polynomial-based function evaluation. IEEE ., 56(4):567–571, 2007.
[3] J. Muller.
Elementary Functions:
Birkhauser, Verlag AG, 1997.
Algorithms and Implementation.
[4] W. Osborne, R. Cheung, J. Coutinho, W. Luk, and O. Mencer. Automatic
accuracy-guaranteed bit-width optimization for fixed and floating-point systems, proc. Intl Conf., 32:617–620, 2007.
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Bi-objective branch and bound algorithm to
minimize total tardiness and system unavailability on
single machine problem
Asmaa Khoudi*, Ali Berrichi* and Farouk Yalaoui
*LIMOSE Laboratory, Computer Science Department,Faculty of Sciences,University
M´Hamed Bougara of Boumerdes, Independency Avenue, 35000 Algeria.
Charles Delaunay Institute, UMR CNRS 6281, LOSI,University of Technology of
Troyes, Troyes, France.
[email protected], [email protected], [email protected]
20-22 April 2015
Abstract
In this paper we propose a bi-objective branch and bound algorithm to deal
with production scheduling and preventive maintenance (PM) planning on single
machine problem. The aim is to find an appropriate assignment of production
jobs and a PM planning to minimize total tardiness and system unavailability
simultaneously. The start-times of PM actions and their number are not fixed in
advance but considered, with the execution dates of production jobs, as decisions
variables of the problem. The proposed algorithm allows finding the exact tradeoff solution set (Pareto front) between the two objective functions. Experimental
results showed the effectiveness of the upper and lower bounds and the proposed
dominance rules.
keywords: Scheduling, Preventive Maintenance, Bi-objective optimization,
Branch and Bound
1 Production Scheduling and Maintenance
planning
Production scheduling and Maintenance are two key functions which can potentially determine the degree of success of any manufacturing company. Their
optimization can improve profit margin and company´s effectiveness.
Production scheduling deals with the allocation of resources to tasks over
given time periods and its goal is to optimize one or more objectives [1]. One
objective may be the minimization of the completion time of the last task and
another may be the minimization of the total tardiness of tasks completed after
their respective due dates. Maintenance service role is to determine maintenance
policies in order to keep production equipments in good conditions, generally
through preventive maintenance actions. Indeed, effective preventive maintenance (PM) strategy can avoid machine failure then enhance productivity of
manufacturing systems.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
In research literature, production scheduling and maintenance planning are
generally studied and executed separately. However, dealing with these two activities separately may results in conflicts. So, the two services must collaborate
to reach a high level of productivity. To avoid such conflicts, some works have
been proposed in the literature to tackle the two problems simultaneously [2].
The majority of these works propose metaheuristics as solution methodology.
2
Resolution Methods
In this paper, we develop a bi-objective exact method based on branch and
bound algorithm to deal with single machine scheduling to minimize total tardiness and maintenance planning to minimize system unavailability simultaneously. In addition, we assume preemption of jobs is not allowed and PM
activities are performed to restore the machines to ”as good as new” conditions.
As we know, the Single machine problem to minimize total tardiness is proved
NP-hard. Considering PM activities will make the problem harder. To our
knowledge, there is no published works minimizing total tardiness and system
availability on single machine problem using exact algorithms. In the proposed
bi-objective branch and bound algorithm (BOBB); we propose a branching strategy, upper and lower bounds as well as dominance rules to accelerate the BOBB
algorithm.
In the branching strategy scheme, we consider that after each job we have
to decide if a PM action should be performed or no.
The decision is expressed by the following binary variable:
1 if a PM action is performed after a job
z=
0 if a PM action is not performed
It is worth considering a bi-objective metaheuristic approach to compute
an initial approximate Pareto front. For the proposed BOBB algorithm, the
well-known multi-objective NSGA-II algorithm [3] is used.
To test the proposed algorithm, a series of experiments was conducted. We
first analyze the effectiveness the proposed dominance rules. We also evaluate
the importance of considering an initial Pareto set. Indeed, our propositions
influence significantly the computational time.
References
[1] Michael Pinedo. Scheduling: Theory, and Systems, (2008).
[2] Ali Berrichi, Lionel Amodeo, Farouk Yalaoui, Eric Chatelet and Mohamed
Mezeghiche; Bi-objective optimization algorithms for joint production and maintenance scheduling: application to the parallel machine problem : journal of
intelligent manufacturing, 20: 389-400 (2009).
[3] Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal and T Meyarivan. A fast
and elitist multiobjective genetic algorithm: NSGA-II: IEEETransactions on
Evolutionary Computation, 6(2):18297, (2002).
2
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
OPERATIONAL RESEARCH PRACTICE IN AFRICA (ORPA) 2015
Un modèle multi-objectif pour le choix de litinéraire
d’un tramway
Khelassi Ishak
Bureau dEtude des Transports BETUR
20-22 April 2015
Résumé
La forte demande de déplacements dans les villes Algériennes a rendu le recours à
l’établissement des moyens de transport en commun capacitaires nécessaire pour faire
face aux problèmes d’insuffisances et dysfonctionnement de l’offre du transport urbain.
L’insertion d’un moyen de transport de masse nécessite une étude approfondie sur le comportement quotidien des voyageurs et aussi sur l’impact de ce nouveau type de transport
sur la ville, en Algérie, des études pareilles qui ont l’intitulé courant ”Etude de faisabilité d’une ligne de Tramway” ont été déjà réalisées dans plusieurs villes en citant
Alger, Annaba, Constantine, Oran, Tlemcen, Béchar, Ouargla etc., dont l’objectif est de
chercher un équilibre entre la demande et l’offre qui enregistre une forte croissance ces
dernières annèes.
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OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Author Index
Achemine Farida …………………..
Ahmia Moussa ……………………..
Aidene Mohamed ………………….
Aissani Djamil ……………………..
Ali-Pacha Adda ……………………
Alzalg Baha ………………………..
Amrouche Karim ………………….
Amrouche Salima ………………….
Attouch Mohammed Kadi …………
54
83
64
94
27
89
46
116
111
Baara Yamina ……………………...
Bacha Nabil ………………………..
Badaoui Ryma Zineb ………………
Badjara Mohamed El Amine ………
Baldé Mouhamadou A.M.T. ……….
Bankuti Gy …………………………
Bareche Aicha ……………………..
Belabbaci Amel ……………………
Belbachir Hacène…………………..
Belkhir Amine …………………….
Bellahcene Fatima …………………
Belloufi Mohammed ………………
Belouchrani Adel …………………..
Benhassine Wassim ……………….
Benmezai Athmane ……………….
Bentarzi Mohamed ………………..
Bentarzi Wissam …………………..
Benterki Djamel ……………………
Benzine Rachid …………………….
Berrichi Ali ………………………...
Bouabbache Zahia …………………
Bouabsa Wahiba …………………..
Boudabbous Imed …………………
Boudhar Mourad …………………..
Bouibede-Hocine Karima ………….
Boumesbah Asma ………………….
Bourahla Amirouche ..……………..
Bousbaa Imad Eddine …..…………
Boussaha Nadia ……………………
Boutabai H. ………………………..
Boutiche M.A ……………………...
Boutouis Manel Zahra ……………..
Brahimi Ines ……………………….
41
118
114
75
110
28
94
87
52, 56, 65, 67, 69, 79, 83, 85, 108, 130
65
90
30
130
96
79
120, 122, 124
124
50
30
132
64
111
15
33, 46, 62, 112, 114, 126
126, 128
35
106
52
100
21, 23
69, 102
96
120
Chemli Zakaria …………………….
Cherfaoui Mouloud ………………..
Chergui Mohamed El-Amine ……...
Chouria Ali ………………………...
71
94
35, 75, 106
58
135
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Costa Lino …………………………
118
Dahane Mohammed ……………….
Delahaye D. ………………………..
Dhiabi Samra ………………………
Dilem Abd el lah …………………..
Djebbar Bachir …………………….
Djellab Natalia ……………………..
114
19
42
41
87
18, 25
Fahem Karima ……………………..
54
Gueye Serigne ……………………..
110
Hadj-Said Naima …………………..
Hamdi Fayçal ………………………
Hammoudi Abdelhakim …………...
Harik Hakim ……………………….
Harrouz Omar ……………………..
Hetak Drifa ………………………..
27
100
73, 77, 96
108
41
126
Idres Lahna ……………………….. 60
Igueroufa Oussama ………………... 85
Ittmann Hans W …………………… 11, 16
Kecies Mohamed ………………….
Khalloufi khadidja …………………
Khelassi Ishak ……………………..
Khoudi Asmaa …………………….
Krim Fariza ………………………..
44
41
134
132
56
Lakehal Mohamed …………………
Lamini El-sedik ……………………
Larabi Marie-Sainte Souad ………..
Lopes Isabel ……………………….
128
130
31
118
Mahjoub Ali Ridha ………………...
Marcos Aboubacar …………………
Marthon Philippe …………………..
Medjerredine A. ……………………
Mendas Abdelkader ……………….
Meradji S. ………………………….
Messaoudi Fatma ………………….
Meziani Lamia ……………………..
Meziani Nadjat …………………….
Mihoubi Miloud ……………………
Mohabeddine Amine ………………
Mongeau M. ……………………….
Mosbah Omar ……………………...
Moulai Mustapha ………………….
13
104
90
69
17
21, 23
81
73
62
37, 81
112
19
41
116
136
OPERATIONAL RESEARCH PRACTICE IN AFRICA CONFERENCE (ORPA'2015)
Nait Mohand N. ……………………
Ndiaye Babacar M. ………………..
77
104, 110
Oukil Amar ………………………... 48
Oulamara Ammar …………………. 62, 112
Ouzzani Fares …………………….. 122
Radjef Mohammed Said ………….. 60, 73, 77
Rahmani Mourad …………………. 98
Rodionova O. ……………………… 19
Sadki Ourida ……………………….
Sbihi M. ……………………………
Silal Sheetal ……………………….
Sow Oumar ………………………..
Stihi Nadjet ………………………..
Stihi S. ……………………………..
39, 42
19
14
104
18
21, 23
Tagzout Samir ……………………..
Tebtoub Assia Fettouma …………..
Tellache Nour El Houda …………..
Tiachachat Meriem ………………..
Touil Imene ………………………..
130
67
33
37
50
Yagouni Mohammed ……………… 92
Yalaoui Farouk ……………………. 46, 132
Yassine Adnan …………………….. 12
Zekri Nourdinne …………………...
Zerfaoui Karima …………………..
Zidani Nesrine ……………………..
41
39
25
137