Multicriteria Scheduling: Theory
and Models
Vincent T’KINDT
Laboratoire d’Informatique (EA 2101)
Dépt. Informatique - Polytech’Tours
Université François-Rabelais de Tours – France
[email protected]
Multicriteria Scheduling: Theory and Models
Structure
• Theory of Multicriteria Scheduling,
–
–
–
–
Optimality definition,
How to solve a multicriteria scheduling problem,
Application to a bicriteria scheduling problem,
Considerations about the enumeration of optimal solutions.
• Some models and algorithms,
– Scheduling with intefering job sets,
– Scheduling with rejection cost.
• Solution of bicriteria single machine problem by mathematical
programming
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Multicriteria Scheduling: Theory and Models
What is Multicriteria Scheduling?
• Multicriteria Optimization: How to optimize several
conflicting criteria?
• Scheduling: How to determine the « optimal » allocation of
tasks (jobs) to resources (machines) over time?
Multicriteria Scheduling =
Scheduling + Multicriteria Optimization.
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• What about multicriteria optimization?
Z2
– K criteria Zi to minimize,
– The notion of optimality is defined
by means of Pareto optimality,
– We distinguish between:
• Strict Pareto optimality,
• Weak Pareto optimality.
A solution x is a strict Pareto optimum
iff there does not exist another solution
y such that Zi(y) ≤Zi(x), i=1,…,K, with
at least one strict inequality.
E
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A solution x is a weak Pareto optimum
iff there does not exist another solution
y such that Zi(y) < Zi(x), i=1,…,K.

Z1
WE
4
Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Multicriteria scheduling (straigth extension),
– Determine one or more Pareto optimal (preferrably strict)
allocations of tasks (jobs) to resources (machines) over time.
• General fundamental considerations,
– How to calculate a strict Pareto optimum ?
– How to calculate the “best” strict Pareto optimum ?
 This depends on Decision Maker’s preferences.
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• How can be expressed decision maker’s preferences?
–
–
–
–
By means of weights (wi for criterion Zi),
By means of goals (fex: Zi  [LB;UB]),
By means of bounds (Zi  ei),
By means of an absolute order.
• Numerous studies can be found in the literature,
–
–
–
–
–
Convex combination of criteria (Geoffrion’s theorem),
e-constraint approach,
Lexicographic approach,
Parametric approach,
…
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• How to calculate the “best” strict Pareto optimum ?
Algorithms based on
an a priori method
Algorithms based on
an interactive method
Algorithms based on
an a posteriori method
Module which takes into
account the criteria
Module which takes into
account the criteria
Module which takes into
account the criteria
value of the parameters
value of the
parameters
a Pareto
optimum
Module which solves
the scheduling problem
Module which solves
the scheduling problem
a Pareto optimum
a Pareto optimum
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value of the
parameters
a Pareto
optimum
Module which solves
the scheduling problem
a set of Pareto optima
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Convex combination of criteria,
Min Si aiZi(x)
st
xS
ai  [0;1], Si ai = 1
• e-constraint approach,
Min Z1(x)
st
xS
Zi  ei,  i=2,…,K
 Strong convex hypothesis
(Geoffrion’s theorem).
 Discrete case: supported vs non
supported Pareto optima.
 Weak Pareto optima,
 Often used in a posteriori algorithms.
• Lexicographic approach: Z1  Z2  …  ZK
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Illustration on an example problem: 1|di|Lmax, C
• A single machine is available,
• n jobs have to be processed,
• pi : processing time,
• di : due date,
• Minimize Lmax=maxi(Ci-di) and C=Si Ci,
p1
C1 C2
Machine
1
2
1C3
2
3
d1
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3
d2
d3
time
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Illustration on an example problem: 1|di|Lmax, C
Design of an a posteriori algorithm1
A strict Pareto optimum is calculated by means of the e-contraint
approach
Known results :
– The 1||C problem is solved to optimality by Shortest Processing
Times first rule (SPT),
– The 1|di|Lmax problem is solved to optimality by Earliest Due Date
first rule (EDD),
1
L. van Wassenhove and L.F. Gelders (1980). Solving a bicriterion scheduling problem, EJOR, 4:42-48.
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• To calculate a Pareto optimum, solve the 1|di|e(C/Lmax)
problem:
Lmax  e
maxi(Ci-di)  e
Ci-di  e,  i=1,…,n
Ci  Di =di + e,  i=1,…,n
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Decision Aid module,
i.
ii.
iii.
iv.
Solve the 1|di|Lmax problem => Lmax* value.
Solve the 1||C problem => s0, C(s0), Lmax(s0).
E={s0}, e=Lmax(s0)-1.
Lmax
While e > Lmax* Do
L (s0)
i. Solve the 1|Di=di+ e | C problem => s,
ii. E=E//{s}, e=Lmax(s)-1.
v. End While.
vi. Return E;
s
Lmax*
e
e
C(s0)
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e
max
C
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Scheduling module (how to solve the 1|Di|C problem),
1
2
Machine
1
2
3
3
7
0
time
D1
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D2 D3
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
•
•
•
•
•
Candidate list based algorithm,
This a posteriori algorithm is optimal,
The scheduling module works in O(nlog(n)),
There are at most n(n+1)/2 non dominated criteria vectors,
This enumeration problem is easy,
• A polynomial time algorithm for calculating a strict
Pareto optimum,
• A polynomial number of non dominated criteria vectors.
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• The enumeration of Pareto optima is a challenging issue,
• How hard is it to perform the enumeration?
 Complexity theory.
• How conflicting are the criteria?
 A priori evaluation,
 Algorithmic evaluation,
 A posteriori evaluation (experimental evaluation).
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• From a theoretical viewpoint… complexity theory,
– Originally dedicated to decision problems,
• Scheduling problems are often optimisation
problems,
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• But now what happen for multicriteria
optimisation?
– We minimise K criteria Zi,
– Enumeration of strict Pareto optima,
Counting problem C
Input data, or instance, denoted by
I (set DO).
Question: how many optimal
solutions are there regarding the
objective of problem O?
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Enumeration problem E
Input data, or instance, denoted
by I (set DO).
Goal: find the set SI the optimal
solutions regarding the objective
of problem O.
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
Problems which can be
solved in polynomial time
in the input size and
number of solutions
Spatial complexity
vs
Temporal complexity,
V. T’kindt, K. Bouibede-Hocine, C. Esswein
(2007). Counting and Enumeration Complexity
with application to Multicriteria Scheduling,
Annals of Operations Research, 153:215-234.
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• There are some links between classes,
– If E  P then O  PO and C  FP,
– If O  NPOC and C  #PC then E  ENPC.
• .. in practice…
…if O  NPOC then E  ENPC
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• A priori conflicting measure: analysis on the
potential number of strict Pareto optima,
• Cone dominance,
Consider the following bicriteria / bivariable MIP problem:
x2
1
Min Si ci xi
Min Si ci2 xi
st
c1
Ax  b
c2
c1 and c2 are the
x  N2
generators of cone C
x1
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
x2
x1
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Consider the following problem: 1||Si uiCi, Si viCi
• The criteria can be formulated as:
Si uiCi = Si Sk ui pk xki
and
Si viCi = Si Sk vi pk xki
with xki = 1 if Jk precedes Ji
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• The generators are:
c1 = [u1p1,…,u1pn,u2p1,…,u2pn,…,unpn]
and
c2 = [v1p1,…,v1pn,v2p1,…,v2pn,…,vnpn]
• The cone C is defined by:
C={y  Rn2 / c1.y≥ 0 and c2.y ≥ 0}
If C is tight, then the number of Pareto optima
c1
C
is possibly high.
c2
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• The maximum angle between c1 and c2 is obtained
for :
ui=0, i=1,…,l, and ui≥0, i=l+1,…,n
and
vi ≥0, i=1,…,l, and vi=0, i=l+1,…,n
as the weights are non negative.
This can be helpful to identify/generate instances
with a potentially high number of strict Pareto
optima.
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Multicriteria Scheduling: Theory and Models
Theory of Multicriteria Scheduling
• Drawback: the number of strict Pareto optima also
depends on the spreading of solutions
(constraints),
• Drawback: not easy to generalize to max criteria.
• Generally, the number of strict Pareto optima is
evaluated by means of an algorithmic analysis,
• See for instance the 1|di|Lmax, wCsum problem,
• But we have a bound on the number of non
dominated criteria vectors.
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Multicriteria Scheduling: Theory and Models
Structure
• Theory of Multicriteria Scheduling,
–
–
–
–
Optimality definition,
How to solve a multicriteria scheduling problem,
Application to a bicriteria scheduling problem,
Considerations about the enumeration of optimal solutions.
• Some models and algorithms,
– Scheduling with interfering job sets,
– Scheduling with rejection cost.
• Solution of bicriteria single machine problem by mathematical
programming
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Multicriteria Scheduling: Theory and Models
Some models and algorithms
• A classification based on model features and not
simply on machine configurations,
•
•
•
•
•
•
•
•
•
Scheduling with controllable data,
Scheduling with setup times,
Just-in-Time scheduling,
Robust and flexible scheduling,
Scheduling with interfering job sets,
Scheduling with rejection costs,
Scheduling with completion times,
Scheduling with only due date based criteria,
….
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Multicriteria Scheduling: Theory and Models
Scheduling with interfering job sets
• 2 sets of jobs to schedule,
• Set A: nA, evaluated by criterion ZA,
• Set B: nB, evaluated by criterion ZB,
• Potentially large number of Pareto optima
(remember the cone dominance approach).
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Multicriteria Scheduling: Theory and Models
Scheduling with interfering job sets
• Consider the 1||Fl(Cmax, wCsum) problem,
Fl(Cmax, wCsum) = CAmax + awCBsum
wi‘=awi
p1‘=p1+p2+p3 / w1’=1
1
1’
2
4
3
5
wCBsum
CAmax
4
Machine
0
6
1’
5
6
time
WSPT on the fictitious A job and B jobs with weights wi’
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Multicriteria Scheduling: Theory and Models
Scheduling with interfering job sets
Problem
Reference
Note
1|di|Fl(Cmax,Lmax)
Baker and Smith (2003)
Yuan et al. (2005)
Polynomial in O(nB log(nB)).
1|di|Fl(Cmax,wCsum)
Baker and Smith (2003)
Polynomial in O(nB log(nB)).
1|di|Fl(Lmax,wCsum)
Baker and Smith (2003)
Yuan et al. (2005)
NP-hard. Polynomial for wi=1.
1||e(fAmax/fBmax)
Agnetis et al. (2004)
O(n2A+nB log(nB)). At most nAnB Pareto.
1||e(wCsumA/fBmax)
Agnetis et al. (2004)
NP-hard. Polynomial for wi=1 (at most
nAnB Pareto).
1|di|e(UA/fBmax)
Agnetis et al. (2004)
O(nA log(nA)+nB log(nB)).
1|di|e(UA/UB)
Agnetis et al. (2004)
O(n2A nB+n2B nA).
1|di|SjwUj
Cheng and Juan (2006)
m job sets. Strongly NP-hard.
1|di|e(wCsumA/UB)
Agnetis et al. (2004)
NP-hard.
1||e(CsumA/CsumB)
Agnetis et al. (2004)
NP-hard (at most 2n Pareto).
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Multicriteria Scheduling: Theory and Models
Scheduling with interfering job sets
• Multiple machines problems,
Problem
Reference
Note
J|di|ZA,ZB
Agnetis et al. (2000)
ZA and ZB are quasi-convexe functions of
the due dates. Enumerate the Pareto.
F2||e(CmaxA/CmaxB)
Agnetis et al. (2004)
NP-hard.
O2||e(CmaxA/CmaxB)
Agnetis et al. (2004)
NP-hard.
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Multicriteria Scheduling: Theory and Models
Scheduling with rejection costs
• A set of n jobs to be scheduled,
• A job can be scheduled or rejected,
• Minimize a « classic » criterion Z,
• Minimize the rejection cost RC=Si rci,
 Often Fl(Z,RC)=Z+RC is minimized.
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Multicriteria Scheduling: Theory and Models
Scheduling with rejection cost
• Consider the 1||Fl(Csum, RC) problem,
Fl(Csum, RC) = Csum + RC
1
2
3
1
Machine
2
rci
4
5
time
Fl=23
2
4
1
5
4
0
Compute the variations in the objective function Di:
Di =[ -2;-3;-10;-7]
SPT to get the initial sequencing
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pi
Job i:
1
1
pi: processing
2 time,
2
rci: rejection cost.
3
4
4
3
i
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Multicriteria Scheduling: Theory and Models
Scheduling with rejection cost
• Consider the 1||Fl(Csum, RC) problem,
Fl(Csum, RC) = Csum + RC
3
1
Machine
2
pi
rci
1
1
2
2
2
4
3
4
1
4
5
5
time
Fl=13
4
0
Compute the variations in the objective function Di:
Di =[ -1;-1;--;-3]
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i
34
Multicriteria Scheduling: Theory and Models
Scheduling with rejection cost
• Consider the 1||Fl(Csum, RC) problem,
Fl(Csum, RC) = Csum + RC
3
1
Machine
4
pi
rci
1
1
2
2
2
4
3
4
1
4
5
5
time
Fl=10
2
0
Compute the variations in the objective function Di:
Di =[ 0;1;--;--]
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i
35
Multicriteria Scheduling: Theory and Models
Scheduling with rejection costs
• Single machine problems,
Problem
Reference
Note
1||Fl(wCsum,RC)
Engels et al. (1998)
Weakly NP-hard. Dyn. Prog and
approx. scheme. Polynomial if wi=w or
pi=p.
1|ri,prec|Fl(wCsum,RC)
Engels et al. (1998)
Approximation scheme.
1|di|Fl(Lmax,RC)
Sengupta (1999)
Weakly NP-hard. Dyn. Prog and
approx. scheme.
1|ri,di,pi contr|Fl(Si
(Ri-wiTi-cixi),RC)
Yang and Geunes (2007)
Ri: profit, xi: compression amount, ci:
compression cost. NP-hard. Heuristic.
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Multicriteria Scheduling: Theory and Models
Scheduling with rejection costs
• Multiple machines problems,
Problem
Reference
Note
P||Fl(Cmax, RC)
Bartal et al. (2000)
Approximation algo for the off-line
case and competitive algo for the online case.
P|pmtn|Fl(Cmax, RC)
Seiden (2001)
Competitive algorithm for the on-line
case.
P,Q|pmtn|Fl(Cmax,RC)
Hoogeveen et al. (2000)
Weakly NP-hard. Approx. scheme.
R|pmtn|Fl(Cmax,RC)
Hoogeveen et al. (2000)
Strongly NP-hard. Approx. scheme.
O|pmtn|Fl(Cmax,RC)
Hoogeveen et al. (2000)
Strongly NP-hard. Approx. scheme.
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Multicriteria Scheduling: Theory and Models
Structure
• Theory of Multicriteria Scheduling,
–
–
–
–
Optimality definition,
How to solve a multicriteria scheduling problem,
Application to a bicriteria scheduling problem,
Considerations about the enumeration of optimal solutions.
• Some models and algorithms,
– Scheduling with interfering job sets,
– Scheduling with rejection cost.
• Solution of bicriteria single machine problem by mathematical
programming
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Nous considérons le problème d’ordonnancement suivant,
• Le
•
•
•
•
problème est noté 1|di | Lmax, Uw,
n travaux,
pi : durée de traitement,
di : date de fin souhaitée,
wi : un poids associé au retard.
On souhaite calculer un optimum de Pareto pour les critères Lmax et Uw.
• Lmax=maxi(Ci-di), le plus grand retard algébrique,
• Uw =Si wiUi, avec Ui=1 si Ci>di et 0 sinon, nombre pondéré de travaux
en retard.
 NP-difficile (quel sens ?)
Baptiste, Della Croce, Grosso, T’kindt (2007). Sequencing a single machine with due dates and
deadlines: an ILP-Based Approach to Solve Very Large Instances, à paraître dans Journal of
Scheduling.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Utilisation de l’approche e-contrainte,
Minimiser Uw
sc
Lmax  e
(A)
• La contrainte (A) est équivalente à :
Ci  Di=di+ e, i=1,…,n
• Pour calculer un optimum de Pareto on résout le problème
noté 1|di , Di| Uw
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Qu’avons-nous fait pour résoudre le problème 1|di , Di| Uw ?
• Partant d’un modèle mathématique…
• … proposition d’une heuristique (borne inférieure)
• …mise en place de techniques de réduction de problème
Tous ces éléments ont été intégrés dans une PSE.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Modélisation linéaire en variables bivalentes,
• xi = 1 si Ji est en avance,
• Bt = {i/Dit} et At = {i/di>t},
• Formulation indexée sur le temps (|T|2n),
T={di,Di}i
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Calcule d’une borne inférieure (heuristique),
Propriété : Soit pipj, dj di, Di Dj, wj wi, avec au moins une inégalité
stricte. On a (i >> j) :
1. Si i est en retard, j l’est aussi,
2. Si j est en avance, i l’est aussi.
• Algorithme basé sur le LP et la notion de « core
problem »,
• Mettre dans le « core problem » les variables
fractionnaires,
• Mettre les variables entières non dominées,
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Résoudre le « core problem » à l’aide du MIP (5% des
var),
• La solution du MIP donne la LB,
• Recherche locale en O(n3) par swap de travaux en
avance et en retard.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Preprocessing : traitement visant à réduire l’espace de
recherche (parfois en réduisant la taille du problème),
• Différents types de preprocessing,
Contraintes :
- ajout de contraintes redondantes,
- élimination de contraintes
redondantes,
-…
Variables :
- réduction des bornes,
- fixation de variables,
-…
• On s’est intéressé à des techniques de preprocessing sur les
variables.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Une technique générale de fixation de variables,
• Basée sur la résolution de la relaxation linéaire,
• Soit LB une borne inférieure et UBlp la borne relachée,
• On sait que pour toute solution x du problème mixte :
cx=UBlp+ SjHB rj xj
avec HB l’ensemble des variables hors base dans une solution donnant
UBlp.
avec rj le coût réduit (négatif ou nul) associé à xj
 UBlp+ SjHB rj xj ≤ LB
 SjHB rj xj ≤ LB-UBlp
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• On en déduit la condition de fixation suivante :
Si rj≥ LB-UBlp alors xj=0
• De même on peut fixer des variables à 1 en introduisant
des variables d’écart sj :
xj+sj=1
… et en tenant le même raisonnement si sj est fixé à 0
alors xj doit être fixé à 1.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• On utilise également une technique de fixation basée sur
les pseudocosts uj et lj
Soit xj une variable réelle de base du LP et on pose :
lj : une binf sur la diminution unitaire du coût si xj=0
uj : une binf sur la diminution unitaire du coût si xj=1
Si (1-xj)*uj ≤ UBlp-LB alors xj=0
Si xj*lj ≤ UBlp-LB alors xj=1
• Pour calculer lj et uj on peut utiliser les pénalités de
Dantzig1
1
Dantzig (1963). Linear Programming and Extensions, Princeton University Press, Princeton.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Algorithme de preprocessing,
(1) Résoudre le LP,
(2) Fixer des variables par les coûts réduits,
(3) Fixer des variables par les pseudocosts,
(4) Si l’étape 3 a permis de fixer des variables, aller en
(1).
 Permet de fixer environ 95% des variables.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Algorithme de la PSE proposée :
• Preprocessing,
• Branchement sur une variable binaire,
• Choix de la variable :
• La variable avec le max des pseudo-costs.
• Profondeur d’abord,
• UB: LP + procédure de réduction,
• Si à un nœud il y a moins de 1.4 107 coefficients non
nuls on résout le sous problème directement par le MIP.
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Quelques résultats,
• Cplex seul résout jusqu’à n=4000 en moins de 290s en
moyenne,
G=100*(UB-Opt)/Opt
G=100*(LB-Opt)/Opt
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Multicriteria Scheduling: Theory and Models
Bicriteria scheduling and Math. Prog.
• Pas de résultat sur l’énumération des optima de Pareto,
• Approche testée sur un autre problème d’ordonnancement1,
• Le problème F2|di=d, d unknown | d, U,
• Le calcul d’un optimum de Pareto se fait jusqu’à n=3000
(Cplex limité à n=2000 et la litérature à n=900),
• On fixe environ 85% des variables.
• L’énumération des (n+1) optima de Pareto strict se fait
jusqu’à n=500 en moins de 800s.
1
T’kindt, Della Croce, Bouquard (2007). Enumeration of Pareto Optima for a Flowshop Scheduling Problem
with Two Criteria, Informs JOC, 19(1):64-72.
Vincent T’kindt
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Multicriteria Scheduling: Theory and Models
Now what’s going on?
• Investigation of structural properties of the Pareto
set for scheduling problems,
• How to quickly calculate a Pareto optimum
starting with a known one?
• Generalized dominance conditions,
• Measuring the conflictness of criteria: from
cone dominance to the complexity of counting
problems,
• Complexity of exponential algorithms,
• …
Vincent T’kindt
53
Multicriteria Scheduling: Theory and Models
Now what’s going on?
• Investigation of emerging models,
• Scheduling with interfering job sets,
• Scheduling with rejection costs,
• Scheduling for new orders,
• Combined models: scheduling with rejection
costs and new orders, …
Vincent T’kindt
54
Multicriteria Scheduling: Theory and Models
Now what’s going on?
• Industrial applications,
• Are often multicriteria by nature,
• Practical application of theoretical models.
Vincent T’kindt
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Multicriteria Scheduling: Theory and Models
You want to know more?
V. T’kindt, JC. Billaut (2006). Multicriteria
Scheduling: Theory, Models and
Algorithms. Springer.
Vincent T’kindt
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