KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY
OBNOXIOUS FACILITY PROBLEM: OPTIMAL SITE FOR REFUSE DUMP IN EJISU
TOWNSHIP
BY
EMMANUEL ADARKWA ANKOMAH
(B. A. COMPUTER SCIENCE AND MANAGEMENT STUDIES)
A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS, KWAME
NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF PHILOSOPHY
COLLEGE OF SCIENCE
JUNE, 2013
DECLARATION
I hereby declare that this submission is my own work towards the award of the M.Phil. degree
and that, to the best of my knowledge, it contains no material previously published by another
person nor material which had been accepted for the award of any other degree of the university,
except where due, acknowledgement had been made in the text..
Emmanuel Adarkwa. Ankomah
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SIGNATURE
DATE
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SIGNATURE
DATE
(PG6186711)
STUDENT
Certified by:
Mr. F. K. Darkwa
SUPERVISOR
Certified by:
Prof. S. K. Amponsah
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HEAD OF DEPARTMENT
SIGNATURE
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DATE
ABSTRACT
The study seeks to find the optimal sites to locate a refuse dump at Ejisu, small town and the
capital of Ejisu-Juaben Municipal, a district in the Ashanti Region of Ghana, using the Maxisum
location model. We seek to locate facilities so that the average distance covered by the
inhabitants of Ejisu will be minimized. Various location models and methods were studied and
we proposed the appropriate model for the location problem. Greedy Add heuristic was used to
estimate the demand-weighted distance. Data on distances between various suburbs and
population were collected and the Floyd’s algorithm was used to find the shortest distance matrix
between suburbs. The results show that the first facility has to be located at Krapa No. 2 and the
second facility, at Serwaa Akura. Overall solution is 23465km (Total demand weighted distance)
and in this case the average maximum of distance (4.13km) is covered by the inhabitants. The
researcher recommended that corporate bodies including Zoomlion Ghana, Ejisu Municipal
Assembly and individuals, who want to develop a refuse dump at Ejisu Township must be sited
at the two stated towns accordingly.
ii
DEDICATION
This thesis is dedicated to the Almighty God for giving me life and protection. It is also
dedicated to my dad who taught me that the best kind of knowledge to have is that which is
learned for its own sake. It is also dedicated to my mum, who taught me that even the largest task
can be accomplished if it is done one step at a time.
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ACKNOWLEDGEMENT
The Grace of God That Bringeth Salvation Hath Appeared to All Men. Shalom
I have taken efforts in this project. However, it would not have been possible without the kind
support and help of many individuals and organizations. I would like to extend my sincere thanks
to all of them.
I am highly indebted to Mr. Stephen Gyekye-Darko, Mr. David Ofori-Atta (Zoomlion Ghana
Limited), Mr. F. Nsoah (Ejisu Municipal Assembly - Statistical Service Department) and Mr.
Robert Boakye (Town and Country Planning), for their guidance and constant supervision as
well as for providing necessary information regarding the project and also for their support in
completing the project.
I also take this opportunity to express my profound gratitude and deep regards to my supervisor
Mr. F. Kwaku Darkwah (Mathematics Department, KNUST)
for his exemplary guidance,
monitoring and constant supervision throughout the course of this thesis.
I would like to express my gratitude towards my parents, brothers and sisters for their kind cooperation and encouragement which help me in completion of this project.
My thanks and appreciations also go to my colleagues (Samuel Okyere, Gabriel Obed Peters,
Gogovi Gideon and John Obeng Berko ) in developing the project and people who have
willingly helped me out with their abilities.
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LIST OF TABLES
Table 3.1: Road networks of the four locations
31
Table 3.2: Shortest path distance matrix with demand
32
Table 3.3: Demand time distance
32
Table 3.4 An illustrative example of the factor rating method
36
Table 3.5 Map coordinates and shipping loads
38
Table 3.6: Fixed and variable cost (in cedis) for a manufacturing plant site
40
Table 4.1: The nine suburbs of Ejisu with their respective population
42
Table 4.2: Edge distance in matrix form
44
Table 4.3: Shortest path distance matrix (
)
45
Table 4.4: Road distance matrix and the population
46
Table 4.5: The first greedy solution to the maxisum problem
48
Table 4.6: The second greedy solution to the maxisum problem
49
Table 4.7: Summary of solution by Greedy Add on the maxisum model
50
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LIST OF FIGURES
Figure 3.1: Road network with population
31
Figure 4.1: Road distances and the population at the nodes
43
vi
Table of Contents
DECLARATION ............................................................................................................................. i
ABSTRACT.................................................................................................................................... ii
DEDICATION ............................................................................................................................... iii
ACKNOWLEDGEMENT ............................................................................................................. iv
LIST OF TABLES .......................................................................................................................... v
LIST OF FIGURES ....................................................................................................................... vi
CHAPTER ONE ............................................................................................................................. 1
GENERAL INTRODUCTION ....................................................................................................... 1
1.0
INTRODUCTION ............................................................................................................ 1
1.1
BACKGROUND OF STUDY ......................................................................................... 2
1.2
PROFILE OF STUDY AREA ......................................................................................... 5
1.3
PROBLEM STATEMENT .............................................................................................. 5
1.4
OBJECTIVES .................................................................................................................. 6
1.5
METHODOLOGY ........................................................................................................... 6
1.6
THESIS JUSTIFICATION .............................................................................................. 7
1.7
THESIS ORGANISATION ............................................................................................. 7
CHAPTER TWO ............................................................................................................................ 8
LITERATURE REVIEW ............................................................................................................... 8
2.0
INRODUCTION .............................................................................................................. 8
2.1
REVIEW OF LOCATION PROBLEMS ......................................................................... 8
2.2
REVIEWS ON SHORTEST PATH PROBLEM ........................................................... 17
CHAPTER THREE ...................................................................................................................... 23
METHODOLOGY ....................................................................................................................... 23
3.0
INTRODUCTION .......................................................................................................... 23
3.1
FACILITY LOCATION PROBLEMS .......................................................................... 23
3.1.1
MAXIMUM DISTANCE MODELS ......................................................................... 24
3.1.2
TOTAL OR AVERAGE DISTANCE MODELS ...................................................... 26
3.2
MAXISUM LOCATION MODEL ................................................................................ 28
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3.3
SOLUTION APPROACHES FOR LOCATION MODELS ......................................... 29
3.3.1
HEURISTIC ALGORITHMS .................................................................................... 30
3.3.1.1
Greedy Heuristic ................................................................................................. 30
3.3.1.2
Stingy Heuristic ................................................................................................... 33
3.3.1.3
Improvement heuristics: ...................................................................................... 33
3.4
FACILITY LOCATION METHOD .............................................................................. 35
3.4.1
FACTOR RATING METHOD............................................................................... 35
3.4.2
THE CENTRE OF GRAVITY METHOD ............................................................. 36
3.4.3
THE LOCATION BREAK EVEN ANALYSIS .................................................... 39
CHAPTER FOUR......................................................................................................................... 42
DATA ANALYSIS AND RESULTS ........................................................................................... 42
4.0
DATA COLLECTION ................................................................................................... 42
4.1
RESULTS BY GREEDY ADD ALGORITHM FOR THE MAXISUM PROBLEM .. 47
4.1.1
FORMULATION OF PROBLEM INSTANCE ..................................................... 47
4.1.2
SITING SECOND FACILITY ............................................................................... 49
4.2 DISCUSSION OF RESULTS ............................................................................................ 50
CHAPTER FIVE .......................................................................................................................... 51
CONCLUSION AND RECCOMENDATION............................................................................. 51
5.1
INTRODUCTION ......................................................................................................... 51
5.2
CONCLUSION .............................................................................................................. 51
5.3
RECOMMENDATION ................................................................................................. 52
REFERENCE................................................................................................................................ 53
APPENDIX ................................................................................................................................. 588
MATLAB PROGRAM FOR THE FLOYD-WARSHAL'S ALGORITHM .............................. 588
viii
CHAPTER 1
GENERAL INTRODUCTION
1.0
INTRODUCTION
Facility location problems form an important class of industrial optimization problems.
These problems typically involve the optimal location of facilities. A facility is just a
physical entity that assists with the provision of a service or the production of a product.
Examples include: ambulance depot, emergency care centers, fire stations, workstations,
schools, libraries etc. Facility is classified into three categories: non – obnoxious
(desirable), semi-obnoxious and obnoxious (non-desirable), (Welch et al., 1997).
In most location problems, we are interested in locating desirable facilities. Ambulances,
fire stations, schools, hospitals, post offices, warehouses, and production plants etc., in
each case the facilities should be closer to the users or undesirable when they should be
far away. Many facilities provide benefits or services to their users while having an
adverse effect on the people nearby. Such facilities include chemical plants, nuclear
power stations, plants for treatment of residual waters, airports, etc. The growing interest
in location modeling for undesirable facilities may be attributed to our growing concerns
over the environment. In fact, any type of modern facility will have some detrimental
effects on the quality life, as a result of human perceptions evidenced by the physical
effects of different forms of pollution such as air, water and noise pollution. In general,
the objective is to locate the facility as far away as possible from an identified set of
population. The obnoxious facility location problem has drawn much recent attention
1
among researchers. Erkut and Neuman (1989) defined an obnoxious facility as one that
generates a disservice to the people nearby while producing an intended product or
service.
Facilities which cannot be classified as being purely desirable or purely obnoxious,
Brimberg and Juel introduced the term semi – desirable facility in 1998 to cater for such
facilities. For instance Airports and power plants are typical semi – desirable facilities.
These facilities may produce a negative or undesirable effect such as noise and pollution,
but a high degree of accessibility is required by these facilities.
This thesis aims to locate a refuse dump as an example of obnoxious facility. Refuse
dumps are useful and necessary for the communities to dispose of the waste produce by
the population but the disposal site may be offensive to look at, and also it emits
offensive odour which pollutes the air and the environment. Since the undesirable
features (perceived or real) of these facilities dominate the desirable ones, the facility is
classified as obnoxious.
1.1
BACKGROUND OF STUDY
Refuse dump is a site for the disposal of waste materials (Wikipedia, 2008). Waste(s)
(also known as rubbish, trash, refuse, or garbage) is unwanted or useless materials.
According to the Basel convention organized by the United Nations Environment
problem "Wastes are materials that are not prime products (that is products produced for
the market) for which the initial user has no further use in terms of his/her own purposes
of production, transformation or consumption, and of which he/she wants to dispose.
2
Wastes may be generated during the extraction of raw materials, the processing of raw
materials into intermediate and final products, the consumption of final products, and
other human activities. Residuals recycled or reused at the place of generation are
excluded." Organization for Economic Cooperation and Development defined waste as
materials that are not prime products (that is, products produced for the market) for which
the generator has no further use in terms of his/her own purposes of production,
transformation or consumption, and of which he/she wants to dispose. Under the Waste
Framework Directive, the European Union defines waste as "an object the holder
discards, intends to discard or is required to discard." There are many waste types defined
by modern systems of waste management, notably including: Municipal waste (includes;
household waste, commercial waste, and demolition waste), hazardous waste (includes
industrial waste), Bio-medical waste (includes clinical waste) and special hazardous
waste (includes radioactive waste, explosive waste and electronic waste).
Waste attracts rodents and insects which harbour gastrointestinal parasites, yellow fever,
worms, the plague and other conditions for humans. Exposure to hazardous wastes,
particularly when they are burned, can cause various other diseases including cancers.
Waste can contaminate surface water, groundwater, soil, and air which causes more
problems for humans, other species, and ecosystems. Waste treatment and disposal
produces significant green house gas (GHG) emissions, notably methane, which are
contributing significantly to global climate change.
3
Waste management is a significant environmental justice issue. Many of the
environmental burdens cited above are more often borne by marginalized groups, such as
racial minorities, women, and residents of developing nations. NIMBY (not-in-my-backyard) is a popular term used to describe the opposition of residents to a proposal for a
new development close to them. However, the need for expansion and siting of waste
treatment and disposal facilities is increasing worldwide. There is now a growing market
in the trans boundary movement of waste, and although most waste that flows between
countries goes between developed nations, a significant amount of waste is moved from
developed to developing nations
The economic costs of managing waste are high, and are often paid for by municipal
governments. Money can often be saved with more efficiently designed collection routes,
modifying vehicles, and with public education. Environmental policies such as pay as
you throw can reduce the cost of management and reduce waste quantities. Waste
recovery (that is, recycling, reuse) can curb economic costs because it avoids extracting
raw materials and often cuts transportation costs. The location of waste treatment and
disposal facilities often has an impact on property values due to noise, dust, pollution,
unsightliness, and negative stigma. The informal waste sector consists mostly of waste
pickers who scavenge for metals, glass, plastic, textiles, and other materials and then
trade them for a profit. This sector can significantly alter or reduce waste in a particular
system, but other negative economic effects come with the disease, poverty, exploitation,
and abuse of its workers
4
1.2
PROFILE OF STUDY AREA
Ejisu is a small town and is the capital of Ejisu-Juaben Municipal, a district in the
Ashanti Region of Ghana. Ejisu-Juaben Municipal is one of the 27 administrative and
political Districts in the Ashanti Region of Ghana. The Municipality is known globally
for its rich cultural heritage and tourists attractions notably the booming kente weaving
industry.
The Municipality stretches over an area of 637.2 km constituting about 10% of the entire
Ashanti Region. Currently it has four urban settlements namely, Ejisu, Juaben, Besease
and Bonwire.
The Municipality is located in the central part of the Ashanti Region and provides
enormous opportunity for creating an inland port for Ghana to serve northern section of
the country. It lies within Latitude 1° 15’ N and 1 ° 45’ N and Longitude 6° 15’W and 7°
00’W. Ejisu-Juaben Municipality shares boundaries with six (6) other Districts in the
Region.To the North East and North West of the Municipal are Sekyere East and Kwabre
East Districts respectively, to the South are Bosomtwe and Asante-Akim South Districts,
to the East is the Asante-Akim North Municipal and to the West is the Kumasi
Metropolitan.
1.3
PROBLEM STATEMENT
Waste management is a significant environmental issue. Waste attracts rodents and
insects which harbour gastrointestinal parasites, yellow fever, worms, the plague and
other conditions for humans. Exposure to hazardous wastes, particularly when they are
burned, can cause various other diseases including cancers. Waste can contaminate
5
surface water, groundwater, soil, and air which cause more problems for humans, other
species, and ecosystems. Waste treatment and disposal produces significant green house
gas (GHG) emissions, notably methane, which is contributing significantly to global
climate change.
Waste is linked to people development. Litter refers to waste disposed of improperly.
People in Ghana and Ejisu to be precise, litter everywhere due to improper sitting of
refuse dump. This work therefore seeks to find the optimal sites to locate a refuse dump
at Ejisu using the Maxisum location model.
1.4
OBJECTIVES
The objectives are;

To locate a refuse dump at suitable sites at Ejisu using Maxisum location
model

To locate the facilities so that the average distance covered by the
inhabitants will be minimized.
1.5
METHODOLOGY
We locate a refuse dump site using Maxisum location model. Various location models
and methods were studied and we proposed the appropriate model for the location
problem under study. Greedy Add heuristic was used to estimate the demand-weighted
distance. Data on distances between various suburbs and population were collected and
6
used. Floyd’s algorithm was used to find the shortest distance matrix between suburbs.
We use matlab to analyse the distance matrix.
1.6
THESIS JUSTIFICATION
Properly disposing of waste is not just a personal responsibility; some kinds of waste,
usually hazardous, must be properly disposed of according to law set forth by the
Environmental Protection Agency. Toxic waste can seep into the ground and contaminate
our water supplies, and sometimes cause widespread disease. Even non-toxic waste
causes pollution that contributes to global warming and a general negative impact on the
public health. Properly sitting refuse dump in the communities help overcome these
problems cause by improper disposal of waste.
1.7
THESIS ORGANISATION
This thesis is organized into five main chapters. Chapter one presents the introduction of
the thesis. This consists of the background of the study, the research problem statement,
objectives of the research, methodology, and organization of the thesis. Chapter two is
the literature review, which looks at briefly work done by other researchers on the topic.
Chapter three is the formulation of the mathematical model. Chapter four contains the
Data Analysis and Results. Chapter five looks at Conclusions and Recommendation of
the analyzed data.
7
CHAPTER 2
LITERATURE REVIEW
2.0
INRODUCTION
This chapter is concerned with the review of related works done on facility location. We
locate either non – obnoxious, semi obnoxious or obnoxious facility. In this thesis we aim
to locate obnoxious facility (refuse dump site) using the factor rating method, hence we
review related works on facility location problems. We also use Floyd’s algorithm to find
shortest path distance matrix.
2.1
REVIEW OF LOCATION PROBLEMS
Obnoxious facilities are those facilities which cause exposure to people as well as to the
environment i.e dump sites, chemical industrial plants, electric power supplier networks,
nuclear reactors and so on. Ballard and Kuhn (1996) developed and tested a facility
location model for the siting of a nuclear fuel waste disposal facility in Canada. The
model is based on successful Canadian siting processes related to hazardous waste and
low level radioactive waste facilities, as well as attributes of facility siting found in the
literature. The proposed model was presented to a sample of participants in the federal
environmental assessment review of the technical feasibility of the Canadian Nuclear
Fuel Waste Disposal Concept (CNFWDC) held throughout Canada in 1990. Their results
demonstrate that despite the fact that over half of the survey respondents did not support
the CNFWDC during the public hearings; the majority favorably rated the proposed
facility location model. Components of the model that were tested included siting criteria
and goals, decision-making groups, and siting safeguards. On the basis of these results,
they concluded that the siting of a nuclear fuel waste disposal facility must make the
8
decentralization of decision – making authority to local communities and governments a
priority.
Cappanera et al. (2002) defined a discrete combined location – routing model, which we
refer to as Obnoxious Facility Location and Routing model (OFLR). OFLR is a NP-hard
problem for which a Lagrangean heuristic approach is presented. The Lagrangean
relaxation proposed allows to decompose OFLR into a Location subproblem and a
Routing subproblem; such subproblems are then strengthened by adding suitable
inequalities. Based on this Lagrangean relaxation two simple Lagrangean heuristics are
provided. An effective Branch and Bound algorithm is then presented, which aims at
reducing the gap between the above mentioned lower and upper bounds. Their Branch
and Bound exploits the information gathered while going down in the enumeration tree in
order to solve efficiently the subproblems related to other nodes. This is accomplished by
using a bundle method to solve at each node the Lagrangean dual. Some variants of the
proposed Branch and Bound method are defined in order to identify the best strategy for
different classes of instances. A comparison of computational results relative to these
variants is presented.
Hazardous waste management involves the collection, transportation, treatment and
disposal of hazardous wastes. Alumur and Kara (2005) proposed a new multiobjective
location – routing model. Their model also includes some constraints, which were
observed in the literature but were not incorporated into previous models. The aim of
their proposed model was to answer the following questions: where to open treatment
9
centers and with which technologies, where to open disposal centers, how to route
different types of hazardous waste to which of the compatible treatment technologies, and
how to route waste residues to disposal centers. The model has the objective of
minimizing the total cost and the transportation risk. A large – scale implementation of
the model in the Central Anatolian region of Turkey was presented.
Snyder et al., (2004) presented a stochastic version of the Location Model with Risk
Pooling.
(LMRP) that optimizes location, inventory, and allocation decisions under random
parameters described by discrete scenarios. The goal of their model (called the stochastic
LMRP, or SLMRP) is to find solutions that minimize the expected total cost (including
location, transportation, and inventory costs) of the system across all scenarios. The
location model explicitly handles the economies of scale and risk – pooling effects that
result from consolidating inventory sites. The SLMRP framework can also be used to
solve multi-commodity and multi-period problems. They presented a Lagrangian –
relaxation {based exact algorithm for the SLMRP. The Lagrangian subproblem is a non –
linear integer program, but it can be solved by a low - order polynomial algorithm. They
discuss simple variable - fixing routines that can drastically reduce the size of the
problem. They presented quantitative and qualitative computational results on problems
with up to 150 nodes and 9 scenarios, describing both algorithm performance and
solution behavior as key parameters change.
10
The problem of optimally locating obnoxious facilities such as hazardous waste
repositories, dump sites, or chemical incinerators has been the source of much
controversy. Often, the final decision on where to locate the facility has been based on
criteria that are difficult to quantify. Previous efforts have focused on where to locate
such a facility to minimize the total population which will be exposed to the facility.
Similarly, efforts have been made to route the waste as to minimize the exposed
population. Attempts at combining the two elements of risk to determine locations have
been restricted to selecting from a given set of candidate locations. Stowers and Palekar
(1993) developed a combined model that quantifies the total exposure of the population
during transportation as well as long term storage. Some properties of the exposure
function are developed that help in the solution of special cases of the problem. They
showed that the problem admits a nodal optimality result when there are no location
risks. The result extends even for bi-objective problems involving exposure and cost.
When location risks are considered and population is concentrated at nodes, the problem
exhibits a finite dominating set. With uniform population distribution on arcs and node
populations, it is possible to partition the arcs of the network and search for the global
optimum solution.
Cokelez and Peacock (2008) developed a mixed integer linear programming model for
locating health care facilities. The parameters of the objective function of this model were
based on factor rating analysis and grid method. Subjective and objective factors
representative of the real life situations were incorporated into the model in a unique way
permitting a trade – off analysis of certain factors pertinent to the location of hospitals.
11
This results in a unified approach and a single model whose credibility is further
enhanced by inclusion of geographical and demographical factors.
Caruso et al (1993) presented a model for planning an urban solid waste management
system. Incineration, composition and recycling were considered for the processing phase
and sanitary landfills were considered for the disposal phase. Heuristic techniques
(embedded in the reference point approximation) were used to solve the model and, as a
consequence, “approximate Pareto solutions” were obtained. By varying the reference
point, different solutions can be obtained. The results for a case study (Lombardy region
in Italy) were presented and discussed.
Fonseca and Captivo (1996; 2006; 2007) study the location of semi obnoxious facilities
as a discrete location problem on a network. Several bi-criteria models were presented
considering two conflicting objectives, the minimization of obnoxious effect and the
maximization of the accessibility of the community to the closest open facility. Each of
these objectives was considered in two different ways, trying to optimize its average
value over all the communities or trying to optimize its worst value. The Euclidean
distance was used to evaluate the obnoxious effect and the shortest path distance was
used to evaluate the accessibility. The obnoxious effect was considered inversely
proportional to the weighted Euclidean distance between demand points and open
facilities, and demand directly proportional to the population in each community. All the
models were solved using Chalmet et al (1986), non- interactive algorithm for Bi-criteria
Integer Linear Programming modified to an interactive procedure by Ferreira et al
12
(1994). Several equity measures were computed for each non – denominated solution
presented to the decision – maker, in order to increase the information available to the
decision – maker about the set of possible solutions.
Giannikos (1998) presents a discrete model for the location of disposal or treatment
facilities and transporting hazardous waste through a network linking the population
centers that produce the waste and the candidate locations for the treatment facilities
method to choose the location for a waste treatment facility in a region of Finland.
Location problems are, in general, multidimensional in nature, particularly if sustainable
development planning is required. So, multicriteria approaches seem adequate in many
situations. Nevertheless, only a very small percentage of the publications in this area
concern multicriteria models or tools. Generally, the different criteria are formulated as
constraints imposing some minimum or maximum value, or are addressed by a surrogate
criterion (like distance) on a single objective structure. Captivo and Climaco (2008)
outline the more relevant multicriteria mixed - integer location models and approaches
taking into account several issues. The adequacy of the available models to reality was
discussed. They also put in evidence the importance of interactive approaches, namely,
discussing a decision support tool in which they are co-authors.
Ballou (1998) states that exact centre of gravity approach is simple and appropriate for
locating one depot in a region, since the transportation rate and the point volume are the
only location factors. Given a set of points that represent source points and demand
points, along with the volumes needed to be moved and the associated transportation
13
rates, an optimal facility location could be found through minimizing total transportation
cost. In principle, the total transportation cost is equal to the volume at a point multiplied
by the transportation rate to ship to that point multiplied by the distance to that point.
Furthermore, Ballou outlines the steps involved in the solution process in order to
implement the exact centre of gravity approach properly.
Pierce and Davidson (1982) applied linear programming to investigate the relative costs
of regional and statewide hazardous waste management schemes. The focus was the
identification of a cost effective configuration of transportation routes, transfer stations,
processing facilities and secure long – term storage impoundments. Wastes generated in
North Carolina were studied as a useful example of linear programming applications in
general and options available within a given state in particular. The value of the
techniques were highlighted, as were their limitations. The usefulness in developing
relative costs of alternatives was stressed, particularly in the ability of the techniques to
conduct sensitivity analyses in a topic area where data may not be generally available.
Suggestions were made for overcoming data shortcomings. In the case study, the options
were seen to revolve around the state of North Carolina’s expressed desire to locate one
large centralized storage landfill. From a pure cost standpoint, other management
facilities like transfer stations and incinerators appear to be precluded even with optimal
routing to and from the facilities. From other viewpoints, including risk aversion to spills
while the waste is in transit, the inclusion of these facilities in the state’s program can be
supported.
Cappanera (1999) presents a survey of mathematical models for undesirable location
problems in the plane and particularly on networks; solution procedures are briefly
14
described. A brief review of extensive (obnoxious) facility location problems in networks
is also given. Finally critical aspects of existing models are identified and some directions
for future search are suggested.
Ben – Moshe and Segal (2000) present efficient algorithms for several instances of the
following facility location problem. Place k obnoxious facilities, with respect to n given
demand sites and m given regions, where the goal is to maximize the minimal distance
between a demand site and a facility, under the constraint that each of the regions must
contain at least one facility. They also presented an efficient solution to the following
planar problem that arises as a subproblem. Given n transmitters, each of range r,
construct a compact data structure that supports coverage queries, i.e., determine whether
a query polygonal / rectangular region is fully covered by the transmitters.
Geoffrion (1978) solved location problems employing a wide range of objective criterion
and methodology used in the decision analysis, for instance Geoffrion, included
decomposition, mixed integer linear programming, simulation and heuristics that may be
used in analyzing location problems. He notes that a suitable methodology for supporting
managerial decisions should be computationally efficient, lead to an optimal solution, and
be capable of further testing.
Goldengorin et al, (1999) considered the simple plant location problem. This problem
often appears as a sub-problem in other combinatorial problems. Several branch and
bound techniques have been developed to solve these problems. Their work considered
new approaches called branch and peg algorithms, where pegging refers to assigning
values to variables outside the branching process. An exhaustive computational
15
experiment shows that the new algorithms generate less than 60% of the number of subproblems generated by branch and bound algorithms, and in certain cases requires less
than 10% of the execution times required by branch and bound algorithms. Firstly, for
each sub-problem generated in the branch and bound tree, a powerful pegging procedure
was applied to reduce the size of the sub-problem. Secondly, the branching function was
based on predictions made using the Beresnev function of the sub-problem at hand. They
saw that branch and peg algorithms comprehensively out perform branch and bound
algorithms using the same bound, taking on the average, less than 10% of the execution
time of branch and bound algorithms when the transportation cost matrix is dense. The
main recommendation from the results of the experiment was that branch and peg
algorithms should be used to solve SPLP instances.
Lai et al. (2010) presented a new hybrid algorithm for a classical capacitated plant
location problem. Benders' decomposition algorithm has been successfully applied in
many areas. A major difficulty with this decomposition lies in the solution of master
problem, which is a ''hard'' problem, costly to compute. Their proposed algorithm, instead
of using a costly branch-and-bound method, incorporates a genetic algorithm to obtain
''good'' suboptimal solutions to the master problem at a tremendous saving in the
computational effort. The performance of the proposed algorithm is tested on randomly
generated data and also well-known existing data. The computational results indicated
that the proposed algorithm was effective and efficient for the capacitated plant location
problem and competitive with the Benders' decomposition algorithm.
16
2.2
REVIEWS ON SHORTEST PATH PROBLEM
Computing shortest paths in graphs is one of the most fundamental and well-studied
problems in combinatorial optimization. Numerous real-world applications have
stimulated research investigations for more than 50 years. Finding routes in road and
public transportation networks is a classical application motivating the study of the
shortest path problem. Shortest paths are also sought by routing schemes for computer
networks: the transmission time of messages is less when they are sent through a short
sequence of routers. The problem is also relevant for social networks: one may more
likely obtain a favor from a stranger by establishing contact through personal
connections.
Sommer (2010) investigates the problem of efficiently computing exact and approximate
shortest paths in graphs, with the main focus being on shortest path query processing.
Strategies for computing answers to shortest path queries may involve the use of precomputed data structures (often called distance oracles) in order to improve the query
time. Designing a shortest path query processing method raises questions such as: How
can these data structures be computed efficiently? What amount of storage is necessary?
How much improvement of the query time is possible? What are the tradeoffs between
pre-computation time, storage, query time, and approximation quality? Sommer (2010)
prove a space lower bound implying that distance oracles with good precision and very
low query costs require large amounts of space. A second contribution consists of spaceand time-efficient data structures for a large family of complex networks. They prove that
exploiting well-connected nodes yields efficient distance oracles for scale-free graphs. A
third contribution is a practical method to compute approximate shortest paths. By means
17
of random sampling and graph Voronoi duals, their methods successfully accommodates
both highly structured graphs stemming from transportation networks and less structured
graphs stemming from complex networks such as social networks.
The shortest path problem often appears as a subproblem when solving difficult
combinatorial problems like multicommodity network flow (MCNF) problems. Most
shortest path algorithms in the literature are either to compute the 1-ALL single source
shortest path (SSSP) tree, or to compute the ALL-ALL all pairs shortest paths (APSP).
However, most real world applications require only multiple pairs shortest paths (MPSP),
where the shortest paths and distances between only some specific pairs of origindestination nodes in a network are desired. Wang, (2003) survey and summarize many
shortest path algorithms, and discuss their pros and cons. They also investigated the Least
Squares Primal - Dual method, a new LP algorithm that avoids degenerate pivots in each
primal - dual iteration, for solving 1-1 and 1-ALL shortest path problems with
nonnegative arc lengths, show its equivalence to the classic Dijkstra's algorithm, and
compare it with the original primal-dual method. They proposed two new shortest path
algorithms to save computational work when solving the MPSP problem. Their MPSP
algorithms are especially suitable for applications with fixed network topology but
changeable arc lengths. They discussed the theoretical details and complexity analyses.
They test several implementations of their new MPSP algorithms extensively and
compare them with many state-of-the-art SSSP algorithms for solving many families of
artificially generated networks and a real Asia-Pacific flight network. Their MPSP
computational experiments show that there exists no "killer" shortest path algorithm.
18
Ohshima (2008) considered a generalization of the shortest path problem in which the
edge length is time – variable, which we call the time - dependent shortest path problem.
This kind of problems has many applications in the fields of navigation systems and
others. Since the first algorithm was proposed by Cooke and Halsey in 1966, many
studies have been done for this problem. Currently the fastest algorithm is due to Dreyfus
and others (1969–1990) who proposed a straightforward generalization of the famous
Dijkstra algorithm that is originally developed for the classical shortest path problem. In
their, work they gave an even faster algorithm at a small amount of extra preprocessing
cost. Their proposed algorithm was based on the ALT algorithm proposed by Goldberg
and Harrelson (2005) for the shortest path problem, in which the main idea was to use
pre-calculated landmarks in determining the interim distance labels for vertices.
Experimental results show their algorithms are several times faster than the generalized
Dijkstra algorithm.
Bertsekas and Tsitsiklis (1991) consider a stochastic version of the classical shortest path
problem whereby for each node of a graph, we must choose a probability distribution
over the set of successor nodes so as to reach a certain destination node with minimum
expected cost. The costs of transition between successive nodes can be positive as well as
negative. They prove natural generalizations of the standard results for the deterministic
shortest path problem, and they extend the corresponding theory for undiscounted finite
state Markovian decision problems by removing the usual restriction that costs are either
all nonnegative or all nonpositive.
19
Chester (2007) presents the Geodesic Path – Switching algorithm, a method for solving
an all-to-all shortest path problem in a continuous region. Given a domain and a metric,
the goal was to compute the shortest geodesic path between a collection of points given
in the domain. For every pair of starting and end points, the path corresponds to the
solution of the Eikonal equation for a wave starting at the source and ending at the end
point. Their algorithm presented is a dynamic programming method. At its core, it
exploits methods borrowed from discrete graph theory to compute the solution to their
continuous problem. They find shortest paths in subproblems, then use those paths as
segments of a different shortest path. They applied concepts of methods used to solve the
all-to-all shortest path problem on a graph to their continuous setting.
Awerbuch (1989) paper, is concerned with distributed algorithm for finding shortest
paths in an asynchronous communication network. For the problem of Breadth First
Search, the best previously known algorithms required either
time, or
communication. They presented new algorithm, which requires
O
messages, for any
time, and
. (Here, V is number of nodes, E is number of edges
and D is the diameter.) This constitutes a major step towards achieving the lower bounds,
which are (E) communication and (D) time. For the general (weighted) shortest paths
problem, previously known shortest – paths algorithms required
messages and
time. Their algorithm requires
messages and
time. Their results enable to improve
significantly solutions for other basic network problems (e.g. leader election).
20
Chabini (1997) solves what appears to be a 30 years old problem dealing with the
discovery of most efficient algorithms possible to compute all-to-one shortest paths in
discrete dynamic networks. This problem lies at the heart of efficient solution approaches
to dynamic network models that arise in dynamic transportation systems, such as
Intelligent Transportation Systems (ITS), applications. While the main objective of their
paper was the study of the all – to – one dynamic shortest paths problem, one-to-all
fastest paths problems were studied as well. Early results were revisited and new
properties are established. They establish the exact complexity of these problems and
develop optimal, in the run time sense, solution algorithms. A new and simple solution
algorithm was proposed for all-to-one all departure time intervals shortest path problems.
It is proved, theoretically, that the new solution algorithm has an optimal run time
complexity that equals the complexity of the problem. Computer implementations and
experimental evaluations of various solution algorithms support the theoretical findings
and demonstrate the efficiency of the proposed solution algorithm.
Dean (2004) studied the problem of computing minimum – cost paths through a time –
varying network, in which the travel time and travel cost of each arc are known functions
of one’s departure time along the arc. For some problem instances, the ability to wait at
nodes may allow for less costly paths through the network. When waiting is allowed, it is
constrained by a (potentially time-varying) waiting policy that describes the length of
time one may wait and the cost of waiting at every node. In discrete time, time-dependent
shortest path problems with waiting constraints can be optimally solved by
21
straightforward dynamic programming algorithms; however, for some waiting policies
these algorithms can be computationally impractical. In their paper they survey several
broad classes of waiting policies and show how techniques for speeding up dynamic
programming can be effectively applied to obtain practical algorithms for speeding up
dynamic programming can be effectively applied to obtain practical algorithms for these
different problem variants
22
CHAPTER 3
METHODOLOGY
3.0
INTRODUCTION
In this chapter we employ the methods used in locating facilities. First of all we discuss
various location methods and models and also various shortest path problems and hence
propose the appropriate model for the study.
3.1
FACILITY LOCATION PROBLEMS
Facility location problems have occupied an important place in operations research since
the early 1960's. They investigate where to physically locate a set of facilities so as to
optimize a given function subject to a set of constraints. Facility location models are used
in a wide variety of applications. Examples include locating warehouses within a supply
chain to minimize the average travel time to the markets, locating hazardous material
sites to minimize exposure to the public, locating railroad stations to minimize the
variability of delivery schedules (Hale and Moberg, 2003).
There are different types of facility location problems. Some basic classes of facility
location problems are listed below (Berman and Krass, 2002).

Discrete facility location problem: location problem where the sets of demand
point and potential facility locations are finite.
23

Continuous facility location problem: location problem in a general space
endowed with some metric, e.g.,
norm. Facilities can be located anywhere in
the given space.

Network facility location problem: location problem which is confined to the
links and nodes of an underlying network.

Stochastic facility location problem: location problem where some parameters,
e.g., demand or travel time, are uncertain.
We can furthermore classify a model as capacitated as opposed to uncapacitated where
the former term refers to the upper bound on the number of clients (or demand) that a
facility can serve. Models are called dynamic (as opposed to static) if the time element is
explicitly represented (Wesolowsky, 1973). Current et al. (2002) listed several basic
discrete network location models: covering (including set-covering and maximal
covering), p-center, p-dispersion, p-median, fixed charge, hub, and maxisum. Distances
or some related measures (e.g., travel time or cost) are fundamental to such problems.
Consequently, we classify them according to their consideration of distance. The first
four are based on maximum distance and the last four are based on total (or average)
distance.
3.1.1 MAXIMUM DISTANCE MODELS
In some locations problems, an acceptable distance is set a priori. In the facility location
literature, a priori acceptable distances such as these are known as “covering” distances.
Demand within the covering distance of its closest facility is considered “covered.” An
24
underlying assumption of this measure of covering distance is that demand is fully
satisfied if the nearest facility is within the coverage distance and is not satisfied if the
closest facility is beyond that distance. That is, being closer to a facility more than the
covering distance does not improve satisfaction.

Set covering location model: The first location covering location problem was
the set covering problem (Toregas et al., 1971). Here the objective is to locate the
minimum number of facilities required to “cover” all of the demand nodes.

Maximal covering location problem: An underlying assumption of the set
covering location problem is that all of the demand nodes must be covered. In
essence, there is no budget constraint. However, in many facility planning
situations, a budget does exist. For example, many school districts would like to
have an elementary school within walking distance of all of its elementary age
students. However, satisfying such a requirement may require more schools than
the district is prepared to build. The maximal covering location problem (MCLP,
Church and ReVelle, 1974) was formulated to address planning situations which
have an upper limit on the number of facilities to be sited. The objective of the
MCLP is to locate a predetermined number of facilities, p, in such a way as to
maximize the demand that is covered. Thus, the MCLP assumes that there may
not be enough facilities to cover all of the demand nodes. If not all nodes can be
covered, the model seeks the siting scheme that covers the most demand.

P-Center Problem: The p-center problem (Hakimi, 1964;1965) addresses the
problem of minimizing the maximum distance that demand is from its closet facility
given that we are siting a pre-determined number of facilities. There are several
possible variations of the basic model. The “vertex” p-center problem restricts the set
25
of candidate facility sites to the nodes of the network while the “absolute” p-center
problem permits the facilities to be anywhere along the arcs or the network. Both
versions can be either weighted or unweighted. In the unweighted problem, all
demand nodes are treated equally. In the weighted model, the distances between
demand nodes and facilities are multiplied by a weight associated with the demand
node. For example, this weight might represent a node‟s importance or, more
commonly, the level of its demand.

The p-dispersion problem: For all of the models discussed above the concern is
with the distance between demand and new facilities. Also, an unspoken
assumption is that being close to a facility is desirable. The p-dispersion problem
(PDP) differs from those problems in two ways (Kuby, 1987). First, it is
concerned only with the distance between new facilities. Second, the objective is
to maximize the minimum distance between any pair of facilities. Potential
applications of the PDP include the siting of military installations where
separation makes them more difficult to attack or locating franchise outlets where
separation reduces cannibalization among stores.
3.1.2 TOTAL OR AVERAGE DISTANCE MODELS
Many facility location planning situations in the public and private sections are concerned
with the total travel distance between facilities and demand nodes. An example in the
private sector might be the location of production facilities that receive their inputs from
established sources by truckload deliveries. In the public sector, one might want to locate
a network of service providers such as licensing bureaus in such a way as to minimize the
total distance that customers must traverse to reach their closest facility. This approach
26
may be viewed as an “efficiency” objective as opposed to the “equity” objective of
minimizing the maximum distance, which is mentioned in other models.

The P-median problem: One classic model in this area is the p-median model
(Hakimi, 1964, 1965) which finds the locations of p facilities to minimize the
demand-weighted total distance between demand nodes and the facilities to which
they are assigned.

Fixed Charge Location Problem: The p-median problem makes three important
assumptions that may not be appropriate for certain siting scenarios. First, it
assumes that each potential site has the same fixed costs for locating a facility at
it. Secondly, it assumes that the facilities being sited do not have capacities on the
demand that they can serve. In the parlance of the literature, it is an
“uncapacitated” problem. Finally, it also assumes that one knows, a priori, how
many facilities should be opened (i.e., p). The fixed charge location problem
(FCLP) relaxes all three of these assumptions. The objective of the FCLP is to
minimize total facility and transportation costs. In so doing, it determines the
optimal number and locations of facilities, as well as the assignments of demand
to a facility. Given the fact that the facilities have capacities, demand may not be
assigned to its closest facility, as was the case in the previous models presented in
this chapter.

Hub location problems: Many logistics systems such as less-than-truckload
carrier networks, airline networks, and inter-modal carriers, employ hub and
spoke systems. These systems are designed to utilize larger capacity or faster
vehicles or modes over the long-haul portion of an origin to destination delivery.
27
Consequently, these systems reduce average per mile transportation cost or total
delivery time. Numerous models (e.g. O’Kelly, 1986a, 1986b; and Campbell,
1990, 1994) have been formulated to locate the hubs and delivery routes of hub
and spoke systems. Most of these models attempt to minimize total cost (as a
function of distance).

The Maxisum Location Problem: The average distance models discussed above
assume that locating facilities as close as possible to demands is desirable. For
many facilities this is the case. However, for undesirable facilities (e.g., prisons,
power plants, and solid waste repositories) at least one objective involves locating
facilities far from demand nodes. The maxisum location problem seeks the
locations of p facilities such that the total demand-weighted distance between
demand nodes and the facilities to which they are assigned is maximized. Hence
we propose this model for the study.
3.2
MAXISUM LOCATION MODEL
The Maxisum location model may be formulated as follows:
 h d
iI jJ
i
ij
yij
Subject to
x
jJ
j
y
jJ
ij
p
1
i  I
28
m
 y [h]
k 1
i
i
 x[m]i  0
{
}
{
}
This formulation is identical to that of the p-median problem with two notable
exceptions. First, the objective (3.10) is to maximize the demand weighted total distance
and not to minimize it. The unfortunate impact of this objective is that it forces demands
to be assigned to the most remote facility. Thus, the formulation has been extended with
constraint (3.14), which ensures that demands are assigned to the nearest facility. In this
constraint, [k]i is the index of the kth farthest candidate location from demand node i.
Constraint (3.14) then states that if the mth closest facility to demand node i is opened
then demand node i must be assigned to that facility or to a closer facility. The other
constraints; constraint (3.11) stipulates that p facilities are to be located. Constraint set
(3.12) requires that each demand node be assigned to exactly one facility. Constraint set
(3.13) restricts demand node assignments only to open facilities. Constraint set (3.15)
established the siting decision variable as binary. Constraint set (3.16) requires the
demand at a node to be assigned to one facility only.
3.3
SOLUTION APPROACHES FOR LOCATION MODELS
Discrete location models are generally constructed as mixed-integer linear programs.
However, formulating an appropriate model is only one step in analysing a location
problem. Another (and often larger) challenge is identifying the optimal solution.
Typically, the first approach to finding the optimal solution to such problems is to apply
29
one of the well-known algorithms such as branch and bound or cutting planes. While
these methods work on at least some instances of most location models, they are typically
only useful on small problems. Realistically scaled location models can easily have
thousands even hundreds of thousands of constraints and variables. Attempting a solution
with these standard optimization methods will quite often consume unacceptable
computational resources in terms of both computer memory and time and with no
guarantee of success. The reason is that even the most basic location models are
classified as NP-Hard (Garey and Johnson, 1979). As a result, the location analyst must
devise other methods to identify optimal solutions and, failing that, at least find very
good solutions. A method of the latter type is known as a heuristic, which is an algorithm
that can find good solutions to a decision problem, but will not guarantee finding the
optimal solution. Various heuristics we will discuss are applicable to most location
models.
3.3.1 HEURISTIC ALGORITHMS
In this section, we will explore several of the most common solution approaches used by
location analysts for the maxisum location model.
3.3.1.1 Greedy Heuristic
Greedy heuristic or Greedy add (Kuehn and Hamburger, 1963). When faced with
selecting a subset of things that will optimize some objective, there are numerous tactics
or “rules-of-thumb” that quickly suggest themselves. The most common is a sequential
approach that begins by evaluating each site individually and selecting the one facility
site that yields the greatest impact on the objective. That facility site is then fixed open.
The location of the next facility is then identified by enumerating all remaining possible
30
locations and choosing the site that provides the greatest improvement in the objective.
Each subsequent facility is located in an identical manner. The method stops when the
required number of facilities, p, have been sited. For obvious reasons, this approach is
known as a greedy heuristic.
To illustrate the approach for the Maxisum model, we consider the network of figure 3.1.
Numbers in boxes next to the nodes are demands,
10
5
1
A
B
3
4
2
7
C
10
D
2
Figure 3.1: Road network with population
The road network is shown in Table 3.1
Table 3.1: Road networks of the four locations
A
B
C
A
0
1
3
B
1
0
2
4
C
3
2
0
2
4
2
0
D
31
D
By using the Floyd’s algorithm, we obtain the shortest path distance matrix for the above
network. This is shown in Table 3.2
Table 3.2: Shortest path distance matrix with demand
A
B
C
D
10
A
0
1
3
5
B
1
0
2
4
10
C
3
2
0
2
7
D
4
2
0
We find
and sum the entries in the various columns. The column with the
largest value gives solution to the Maxisum problem. This is shown in Table 3.3
Table 3.3: Demand time distance
A
B
C
D
A
0
10
30
50
B
5
0
10
20
C
30
20
0
20
D
35
28
14
0
Total
70
58
54
90
From Table 3.3 we locate the facility at node D since it has the largest optimal value of
90.
32
3.3.1.2 Stingy Heuristic
The Stingy heuristic (Feldman, et al., 1966), also known as Drop or Greedy-Drop starts
with facilities located at all potential sites, and then removes (drops) the facility that has
the least impact on the objective function. We continue to drop facilities until p facilities
remain.
3.3.1.3 Improvement heuristics:
While both the Greedy-Add and the Greedy-Drop heuristics are effective at identifying a
feasible solution with modest computational effort, neither can be relied upon to
consistently produce good solutions. Therefore, several different approaches have been
developed that begin with a good (or at least feasible) solution and seek to improve upon
it. Not surprisingly, these are known as improvement or search heuristics.
Neighborhood search algorithm:
One of the earliest improvement heuristics is the neighborhood search algorithm
(Maranzana, 1964). In this method, we begin with any feasible solution or specifically a
set of p facility sites. Demand nodes are then assigned to their nearest facility. The set of
nodes assigned to a facility constitutes a “neighborhood” around that facility. Within each
neighborhood, the 1-median problem can be solved optimally by simply evaluating each
potential site in the neighborhood and selecting the best. The facilities are then relocated
to the optimal 1-median locations within each neighborhood. Then, if any facility sites
are relocated, new neighborhoods can be defined and the algorithm is repeated. This
cycle continues until there are no further changes in the facility sites or neighborhoods.
33
Exchange or Interchange Heuristic:
The most widely known improvement method was introduced by Teitz and Bart (1968).
The basic idea is to move a facility from the location it occupies in the current solution to
an unused site. Each unused location is tried in turn and when a move produces a better
objective function value, then that relocation is accepted and we have a new (improved)
solution. When an improved solution is obtained, the search process is repeated on the
new solution. The procedure stops when no better solution can be found via this method.
Although commonly used as a p-median problem, this approach has been found useful in
innumerable facility location models. While seemingly straightforward in concept, the
exchange heuristic has a number of alternative approaches that can be used in
implementing it. One, of course, is the process described above, where every time an
exchange is found that yields a better solution, the search process is restarted and applied
to improve this new solution. Alternatively, we could select the best solution after
considering all possible moves for a given facility site, or even choose the best after all
possible exchanges for all sites are examined. There are many other variations possible,
and these often influence the computational speed of the heuristic. The most efficient
implementation of the exchange algorithm was presented by Whitaker (1983). His “Fast
Interchange” method is described in detail in Hansen and Mladenovic (1997). One issue
in using improvement heuristics is to decide how the initial solution is generated. An
obvious choice is to use the result of another heuristic, such as one of the greedy
heuristics mentioned earlier. However, since the interchange heuristic is relatively fast,
many analysts have applied it to a series of randomly generated solutions, selecting the
best solution among all of the local optima found as the one to be implemented.
34
3.4
FACILITY LOCATION METHOD
There are three (3) main location methods. These include the centre of gravity method,
location break – even analysis and factor rating method.
3.4.1
FACTOR RATING METHOD
The factor rating method is a method used to find a suitable location for a facility
considering a number of factors. The factors include: labour cost (wages, unionization,
and productivity), labour availability, proximity to raw materials and supplier, proximity
to markets, state and local government fiscal policies, environmental regulations, utilities,
site cost, transportation, and quality of life issues within the comm. In using the factor
rating method, the following six steps must be followed.

Develop a list of relevant factors.

Assign a weight to each factor to reflect its relative importance in
management’s objective.

Develop a scale for each factor (for example, 1 to 10 or 1 to 100)

Have management or related people score each relevant factor, using the
scale developed above.

Multiply the score by the weight assigned to each factor and total the score
for each location.

Make a recommendation based on the maximum point score; considering
the result of quantitative approaches as well.
35
Example 1: Juaben Oil Mills wants to set up one of its factory at Esikuma. Factors
considered are shown in Table 3.1 below.
Table 3.4 An illustrative example of the factor rating method.
Factor
Factor name
1
Availability
of labour
Workforce
attitude
Community
desire
Cost of
transportation
Equipment
supplies
2
3
4
5
Rating
Weight
5
Location
A
B
90
80
Ratio of
Rating
0.3125
Location
A
B
28.13
25
3
60
85
0.1875
11.25
15.94
3
75
70
0.1875
14.06
13.13
2
60
50
0.125
7.5
6.25
3
80
55
0.1875
15
10.31
75.94
70.63
From Table 3.4 Location A would be considered since it has the highest aggregate score
of 75.94.
3.4.2
THE CENTRE OF GRAVITY METHOD
The centre of gravity method is a mathematical technique used for finding the location of
a distribution center that will minimize distribution cost. For instance in the location of a
market, the method takes into account the volume of goods shipped to those markets and
shipping cost in finding the best location for the distribution centre. The first step in the
centre of gravity method is to place the locations on a coordinate system. The coordinates
of each location must be carefully noted. The origin of the coordinate system is arbitrary,
just as long as the relative distances are correctly represented. This can be done easily by
placing a grid over an ordinary map of the location in question. The centre of gravity is
determined by equations (3.17) and (3.18) below;
36
∑
∑
∑
∑
Where
Coordinate of the centre of gravity
y – Coordinate of the centre of gravity
- Coordinate of location
y – Coordinate of location
Volume of goods to or from location
The centre of gravity
is determined by equation
the centre of gravity on the nearest map coordinate if
. We locate
does not fall directly on a
city. In the case where there is more than one city that can be used as possible location,
the factor rating method can be used to select one. Table 3.5 below illustrates an example
of the centre of gravity model.
Example 2: Table 3.5 gives the map coordinates and shipping loads for a set of cities that
that we wish to connect through a central “hub”. Near which map coordinates should the
hub be located.
Table 3.5 Map coordinates and shipping loads
37
Site
Map Coordinates
(
Shipping load
)
A
(4,2)
14
B
(3,4)
11
C
(7,5)
12
D
(4,10)
22
E
(5,11)
15
F
(3,2)
17
G
(4,4)
21
H
(6,8)
22
I
(3,7)
27
J
(9,5)
24
38
The centre of gravity
does not fall into any of the city coordinates;
therefore we locate it at the nearest city. Hence we find the distance between the centre of
gravity and the city coordinates. The nearest city is H with coordinates (6, 8), hence we
locate the Hub at city H.
3.4.3
THE LOCATION BREAK EVEN ANALYSIS
The Location break-even analysis is the use of cost-volume analysis to make an
economic comparison of location alternative. By identifying fixed and variable cost and
graphing them for each location, we can determine which one provides the lowest cost.
Location break-even analysis can be done mathematically or graphically. The graphic
approach has advantage of providing a range of volumes over which each location is
preferable.
The location break-even analysis steps are as follows:

Determine the fixed and variable cost for each location.

Plot the cost for each location, with cost on the X – axis of the graph and the
annual volume on the Y – axis, and finally,

Select the location that has the lowest total cost for the expected production.
39
The location break – even analysis is determined by the equation;
Where;
Variable cost
Fixed cost
Volume of business
Cost of business
Example 3: For the Break-Even analysis in Table 3.6, over what range of production is
each location optimal?
Table 3.6: Fixed and variable cost (in cedis) for a manufacturing plant site
Site
Fixed cost (b)
Variable cost (a)
A
800
9
B
900
6
C
1000
5
We relate the given Table 3.6 as;
40
For a volume of
volume,
units, site A is the best since it gives a cost of
, site B is the best, this gives a minimum cost in the range
. For a volume,
of
. For a
, site C is the best, this gives a minimum cost
.
41
CHAPTER 4
DATA ANALYSIS AND RESULTS
4.0
DATA COLLECTION
Nine suburbs in the Ejisu Township were selected. We needed a population and road
distances data of the nine suburbs. Secondary data was obtain which were 2010
population and housing census data from the Municipal statistical service department and
road distances data from town and country planning. The suburbs and their respective
population are shown in Table 4.1 and the road distances connecting the various suburbs
are shown in Figure 4.1.
Table 4.1: The nine suburbs of Ejisu with their respective population
Name of Suburb
Node
Population
New Krapa
A
743
Zongo
B
921
Serwaa Akura
C
758
New Town
D
520
Asomaya
E
822
Railway
F
435
Low cost
G
523
Krapa no. 1
H
620
Krapa no. 2
I
335
5677
Total Population
42
520
2
758
D
C
3
1
1
82
2
1
921
743
523
2
E
1
B
A
G
1
1
1
1
2
I
1
2
F
1
435
1
H
620
Figure 4.1: Road distances and the population at the nodes
43
335
Table 4.2: Edge distance in matrix form
A
B
C
D
E
F
G
H
I
A
B
C
D
E
F
G
H
I
Floyd’s Algorithm was used to find the shortest path distance matrix
, this is shown in
Table 4.3.
Matlab code written by Floyd Warshall was adopted and run on Dell Computer (Dell
XPS L502x Laptop - Intel Core i5 2450M 2.5GHz, 6GB DDR3 RAM Memory, 750GB
44
Hard Disk Drive, 15.6" HD LED, NVIDIA GT 540M 2GB Graphics, DVD-RW, HDMI,
Windows 7 Home Premium 64, Wi-Fi N, Bluetooth, 2MP Webcam) on two trials.
The code is displayed in the appendix.
Table 4.3: Shortest path distance matrix (
A
B
C
)
D
E
A
B
C
D
E
F
G
H
I
45
F
G
H
I
We now introduce the Population ( ) in the first column and shortest path distance
matrix
taken the others columns of Table 4.4
Table 4.4: Road distance matrix and the population
A
743
A
921
B
758
C
520
D
822
E
435
F
523
G
620
H
335
I
B
C
D
46
E
F
G
H
I
4.1
RESULTS BY GREEDY ADD ALGORITHM FOR THE MAXISUM
PROBLEM
4.1.1
FORMULATION OF PROBLEM INSTANCE
From Table 4.4 we find the results of the first Greedy add algorithm
Max hi dij
Where
hi  Population of each suburb in the first column
dij  The shortest path distance matrix (i, j )
Algorithm Steps
1. Compute the total demand weighted-distance (hi  dij ) for each row
2. Compute the sum
h d
i ij
for each column
3. We locate the facility at node or column with Max hi dij .
The results of the above steps are shown in Table 4.5.
47
Table 4.5: The first greedy solution to the maxisum problem
A
B
C
D
E
F
G
H
I
11900
12668
8169
9277
12618
13041
17926
A
B
C
D
E
F
G
H
I
Total
We calculate the column total and choose the one with the maximum value. From Table
4.5 the maximum value is 17926 and it occurs at node I. So we locate the refuse dump at
node I which is Krapa No. 2. To access the refuse dump, an individual in the nine suburbs
has covered an average distance of
.
48
4.1.2
SITING SECOND FACILITY
{
To locate a second facility, we compute
pair
} for each node location
. Hence we adjust the distance matrix and the results is shown in Table 4.6
Table 4.6: The second greedy solution to the maxisum problem
A
B
C
D
E
F
G
H
I
23465
22172
20074
18596
18261
18261
17926
A
B
C
D
E
F
G
H
I
Total
49
From Table 4.6 node C has the maximum value, which is
. Hence we locate the
second facility at node C which is Serwaa Akura. Citizens have to cover an average
distance of
in order to access the two facilities.
4.2 DISCUSSION OF RESULTS
We summarized the results, and this is shown in Table 4.7 below.
Table 4.7: Summary of solution by Greedy Add on the maxisum model
Solution by Greedy
Location
Add
1st Facility
Krapa No. 2
2nd Facility
Serwaa Akura
Total demand
Average demand
weighted distance
weighted distance
17926
3.16
Results as summarized in Table 4.7 shows, we need to locate first facility at Krapa No. 2,
in this case the population of 5677 has to cover a total demand weighted-distance of
17926 and an average distance of 3.16 in other to access the facility. For undesirable
facilities at least one objective involves locating facilities far from demand nodes. The
second refuse dump is to be located at Serwaa Akura. In this case the people have to
cover an average distance of 4.13 in order to access any of these two refuse dumps.
50
CHAPTER 5
CONCLUSION AND RECCOMENDATION
5.1 INTRODUCTION
In this chapter we conclude based on the discussed results obtained from greedy add
algorithm for the maxisum problem in chapter four. Recommendations are made based
on the solution by Greedy Add on the maxisum model.
5.2
CONCLUSION
The main objective of the thesis was to locate an undesirable facility (refuse dump) at
suitable sites in Ejisu Township and also to locate the facility so that the average distance
covered by the inhabitants will be maximized. Nine suburb of the Ejisu Township is
selected. Various models were studied and the maxisum location model was proposed for
the study of the refuse dump since it is more appropriate for the study of the undesirable
facility.
The Greedy add heuristic was used to solve the maxisum location problem for locating
undesirable facility in Ejisu Township. Because of it effectiveness and efficiency at
identifying a feasible solution with modest computational effort.
Results from Table 4.5 shows that, maximum of the total columns is 17926 and it occurs
at node I. So we locate the refuse dump at node I which is Krapa No. 2. To access the
refuse dump, an individual in the nine suburbs has covered an average distance
of
. While as the maximum value from Table 4.6 is at node
C. The maximum values is given as
. Hence we locate the second facility at node
51
C which is Serwaa Akura. Individuals located at the entire nine suburbs have to cover an
average distance of
in order to access the two facilities.
The results show that the first facility has to be located at Krapa No. 2 and the second
facility, at Serwaa Akura. Overall solution is 23465 (Total demand weighted distance)
and in this case the average maximum of distance (4.13km) is covered by the inhabitants.
5.3
RECOMMENDATION
In view of the result obtained in this study, the following recommendations are made:

Corporate bodies such as the Zoomlion Ghana, Ejisu Municipal Assembly as well
as the individuals, who wants to develop a refuse dump at Ejisu Township must
site it at Krapa No. 2 and in case of second facility it must be sited at Serwaa
Akura.

In this thesis we proposed greedy heuristic to solve maxisum location problem,
other heuristics algorithms such as stingy heuristics and improvement heuristics
can be consider by other researches as a future study.

Corporate bodies who would like to apply this method on the field need to
training people due to mathematical model employed in locating the facility
52
REFERENCE

Welch S. B., Salhi S., Drezner Z., (2005). The multifacility maximin planar
location problem with facility interaction. IMA journal of management
mathematics page 1 of 1

Erkut E., and Neuman S., (1989). Analytical models for locating undesirable
facilities. European Journal of Operational Research.Volume 40, No. 3, 1989, pp.
275-291. Doi: 10.1016/0377 -2217(89)90420-7.

Brimberg J., Juel H., Schobel A.,(1988) Linear facility location in three
dimensions-models and solution methods.
http://goedoc.uni-goettingen.de/bitstream/handle/1/5711/02-Brimberg-Juel-SchOR.pdf?sequence=1 [Accessed: 20th August, 2012]

Captivo E. M. and Climaco N. J., (August, 2008). On Multicriteria Mixed Integer
Linear Programming Based Tools for Location Problems-An Updated Critical
Overview Illustrated with a Bicriteria DSS.

Cooke and Halsey (1966). Optimization Methods and Applications

Alumur S. and Kara B. Y., (2005) A new model for the hazardous waste locationrouting
problem.
http://prolog.univie.ac.at/teaching/LVAs/KFK-
Seminar/SS06/sdarticle1.pdf [Accessed: 17th September, 2012]

Fonseca M.C. and Captivo M. E., (1996) Location of semi obnoxious facilities
with capacity constraints", Studies in Locational Analysis, 9: 51–52 (1996).
53

Dean B. C.,(2004) Algorithms for Minimum – Cost Paths in Time - Dependent
Networks with Waiting Policies.

Gabor Melli’s Research Knowledge Base, 2012. Dijkstra’s shortest path
algorithm.
http://www.gabormelli.com/RKB/Dijkstra%27s_Shortest_Path_Algorithm

Giannikos I., (1998). A multiobjective programming model for locating treatment
sites and routing hazardous wastes. European Journal of Operational Research,
104:
333–342
(1998).
http://www.sciencedirect.com/science/article/pii/S0377221797001884
[Accessed: 11th August, 2012]

Awerbuch
B.,
(1989).
Distributed
shortest
paths
algorithms.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.124.3299&rep=rep1&t
ype=pdf

[Accessed: 3rd August, 2012]
Ballard K. R. and Kuhn R. G., (2006). Developing and Testing a Facility Location
Model for Canadian Nuclear Fuel Waste.
http://onlinelibrary.wiley.com/doi/10.1111/j.15396924.1996.tb00833.x/abstract
[Accessed: 10th October, 2012]

Ben – Moshe B., Katz M. J., Segal M., (July, 2000) Obnoxious facility location:
Complete service with minimal harm. International Journal of computational
geometry, vol 10, issue 06, December 2000. http://www.worldscientific.com
[Accessed: 1st October, 2012]
54

Bertsekas P. D. and Tsitsiklis J. N. (1991). An analysis of stochastic shortest path
problems. http://mathor.highwire.org/content/16/3/580.abstract
[Accessed: 3rd October, 2012]

Cappanera P., (1999). A survey on obnoxious facility location problems.
http://dl.acm.org/citation.cfm?id=898115 [Accessed: 1st October, 2012]

Cappanera P., Gallo G., and Maffoli F., (2002) Discrete facility location and
routing
of
obnoxious
activities.
http://www.di.unipi.it/~gallo/Papers/ObnoxiousFacilityLocation.pdf
[Accessed:
27th September, 2012]

http://www.cic.ipn.mx/sitioCIC/images/revista/vol12-02/v12no2_Art06.pdf [19th
September, 2012]

Caruso C., Colorni A. and Paruccni M., (1986) “The regional urban solid waste
management system: A modeling approach”, European Journal of Operation
Research, 70:16-30

Chabini I., (2010). Discrete dynamic shortest path problems in transportation
applications: Complexity and algorithms with optimal run time.

Goldberg A. V. and Harrelson C., (2005). Computing the shortest path: A search
meets graph theory. SODA '05 Proceedings of the sixteenth annual ACM-SIAM
symposium
on
Discrete
algorithms
Pages
156
-
http://dl.acm.org/citation.cfm?id=1070455 [Accessed: 12th September, 2012]
55
165

Lai M. C., Sohn H. S., Tseng T. L., Chiang C. (2010). A hybrid algorithm for
capacitated plant location problem. Journal Expert Systems with Applications: An
International Journal archive Volume 37 Issue 12, December, 2010 Pages 85998605. http://dl.acm.org/citation.cfm? id=1842306 [Accessed: 15th September,
2012]

Ohshima T. (2006). A landmark algorithm for the time – dependent shortest path
problem
http://www-or.amp.i.kyoto-
u.ac.jp/members/ohshima/Paper/MThesis/MThesis.pdf [Accessed: 11th August,
2012]

Ohshima T., (2008). A landmark algorithm for the time – dependent shortest path
problem
http://www-or.amp.i.kyotou.ac.jp/members/ohshima/Paper/MThesis/MThesis.pdf
[Accessed: 3rd October, 2012]

Peirce J.J. and Davidson G. M., (1982). Linear programming in hazardous waste
management. Journal of the Environmental Engineering Division, Vol. 108, No.
5, September/October 1982, pp. 1014-1026.
http://cedb.asce.org/cgi/WWWdisplay.cgi?34608
[Accessed: 13th September,
2012]

Snyder L. V., Daskin S. M., and Teo C., (2004). The Stochastic Location Model
with Risk Pooling. http://www.lehigh.edu/~lvs2/Papers/slmrp.pdf
[Accessed: 27th September, 2012]
56

Sommer C. (2010) Approximate shortest path and distance queries in networks
http://www.sommer.jp/thesis.htm [21st September, 2012]

Stowers L. C. and Palekar S. U., (1993) Location Models with Routing
Considerations
for
a
Single
Obnoxious
http://transci.journal.informs.org/content/27/4/350.abstract
Facility.
[Accessed:
12th
August, 2012]

Wang
I.,
(2003)
shortest
path
and
multicommodity
network
flows
http://ilin.iim.ncku.edu.tw/ilin/phdthesis/vita.html [accessed: 20th October, 2012]

Wikipedia, (2008). Naples waste management issue. http://en.wikipedia.org/wiki
/Naples_waste_management_issue [Accessed: 13th October, 2012]

Wikipedia,
2011.
Floyd–Warshall
algorithm.
Retrieve
http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm
on
[Accessed:
20th September, 2012]

Zhou, B., and Eskandarian, A. A Non-Deterministic Path Generation Algorithm
for Traffic Networks
http://www.trforum.org/forum/downloads/2006_8B_PathGeneration_paper.pdf
[Accessed: 27th Sept ember, 2012]
57
APPENDIX
MATLAB PROGRAM FOR THE FLOYD-WARSHAL'S ALGORITHM
This is the actual algorithm:
# dist(i,j) is "best" distance so far from vertex i to vertex j
# Start with all single edge paths.
For i = 1 to n do
For j = 1 to n do
dist(i,j) = weight(i,j)
For k = 1 to n do # k is the `intermediate' vertex
For i = 1 to n do
For j = 1 to n do
if (dist(i,k) + dist(k,j) < dist(i,j)) then # shorter path?
dist(i,j) = dist(i,k) + dist(k,j)
The Programme
Here’s the code in matlab(floyd_warshall.c):
function floyd_warshall()
% Floyd - Warshall Algorithm to find the shortest path
clear all;
58
close all;
clc
global n;
global q;
global adj;
global dist;
global edge;
global dist_given;
global dist_new;
global dist_modify;
prompt = {'Enter the number of Vertices : ','Enter the number of Edges :'};
dlg_title = 'Vertices/Edges';
num_lines = 1;
def = {'4','5'};
answer = inputdlg(prompt,dlg_title,num_lines,def);
n = str2num(answer{1});
q = str2num(answer{2});
temp3 = n;
temp2 = q;
[temp3 temp2] = check1(temp3,temp2);
59
n = temp3;
q = temp2;
columnname = {};
columnformat = {};
rowname = {};
for i=1:q
for j=1:3
dist{i,j} = '';
end
columnformat{i} = 'numeric';
end
columnname{1} = 'Node 1' ;
columnname{2} = 'Node 2' ;
columnname{3} = 'Distance' ;
for i=1:q
rowname{i} = strcat('Edge ',num2str(i));
end
60
P = figure('Name','Enter the Edge Information','NumberTitle','off','Position',[50 50 320
400] );
t = uitable('Parent',P,...
'Position',[0 0 320 320],'Data', dist,...
'ColumnName', columnname,...
'ColumnEditable', true,...
'FontSize',12,'ForegroundColor','k', ...
'FontName','Comic Sans MS', ...
'RowName',rowname);
b = uicontrol('Parent',P,...
'Style','Pushbutton',...
'Units','points', ...
'Callback',@Pushbutton1_Callback,...
'Position',[120 250 83.1724 30.4138], ...
'String','Click To Solve', ...
'Tag','checkbox1' );
str1 = strcat('Vertices = ',num2str(n));
y1 = uicontrol('Parent',P,...
'Style','text',...
61
'Position', [30 340 110 30.4138], ...
'FontSize',10,'ForegroundColor','k', ...
'FontWeight','bold', ...
'FontName','Arial Black', ...
'String',str1);
P;
function Pushbutton1_Callback(hObject,eventdata)
if (get(hObject,'Value') == get(hObject,'Max'))
g = get(t,'Data');
for i=1:q
for j=1:2
temp1 = g{i,j};
temp1 = check(temp1,i,j,g);
g{i,j} = temp1;
end
end
dist_modify = g;
new = {};
62
for i=1:q
for j=1:3
new{i,j} = str2num(g{i,j});
end
end
A = {};
adj = {};
for i=1:n
for j=1:n
A{i,j} = 0;
adj{i,j}=0;
end
end
for i=1:q
A{new{i,1},new{i,2}} = new{i,3};
adj{new{i,1},new{i,2}} = 1;
A{new{i,2},new{i,1}} = new{i,3};
adj{new{i,2},new{i,1}} = 1;
end
dist = A;
63
dist_given = A;
for k=1:n
for i=1:n
for j=1:n
if (dist{i,k}*dist{k,j} ~= 0) && (i ~= j)
if (dist{i,k} + dist{k,j} < dist{i,j}) || (dist{i,j} == 0)
dist{i,j} = dist{i,k} + dist{k,j};
end
end
end
end
end
plot_graph(adj,n,dist_given,dist)
end
end
64
%% Final Plot Function
function plot_graph(dist,n,weight,dist_new)
close all;
Q = figure('Name','Floyd Warshall Shortest Path Matrix','NumberTitle','off','Position',[50
50 700 300],...
'DeleteFcn',@Figure_Close_Callback);
v = uitable('Parent',Q,'Position',[0 0 700 300],'Data', dist_new,...
'ColumnEditable', false);
v;
theta = 2*pi/n;
vertices = [];
r = 450;
c = 1;
for x=0:theta:(2*pi)
xcord = r*cos(x);
ycord = r*sin(x);
vertices(c) = (xcord) + 1i*(ycord);
c = c+1;
end
for i=1:n
65
X(i) = real(vertices(i));
Y(i) = imag(vertices(i));
end
T = figure('Name','Floyd Warshall : Given Graph','NumberTitle','off','Position',[50 50
1200 600],...
'DeleteFcn',@Figure_Close_Callback);
plot(real(vertices),imag(vertices),'ob','LineWidth',4,...
'MarkerSize',10)
for i=1:n
if X(i)>=0 && Y(i)>=0
text(X(i)+20,Y(i)+20,num2str(i),'FontSize',18,'Color','r');
elseif X(i)<=0 && Y(i)>=0
text(X(i)-30,Y(i)+30,num2str(i),'FontSize',18,'Color','r');
elseif X(i)<=0 && Y(i)<=0
text(X(i)-30,Y(i)-30,num2str(i),'FontSize',18,'Color','r');
elseif X(i)>=0 && Y(i)<=0
text(X(i)+20,Y(i)-20,num2str(i),'FontSize',18,'Color','r');
end
end
% Displaying weights
66
for i=1:n
for j=1:n
g = [X(i) X(j)];
h = [Y(i) Y(j)];
if X(i) <= X(j)
Wx(i,j) = X(i) + abs(X(j)-X(i))/2;
if Y(i) <= Y(j)
Wy(i,j) = Y(i) + abs(Y(j)-Y(i))/2;
else
Wy(i,j) = Y(j) + abs(Y(i)-Y(j))/2;
end
else
Wx(i,j) = X(j) + abs(X(i)-X(j))/2;
if Y(i) <= Y(j)
Wy(i,j) = Y(i) + abs(Y(j)-Y(i))/2;
else
Wy(i,j) = Y(j) + abs(Y(i)-Y(j))/2;
end
end
if dist{i,j} == 1
line(g,h,'LineWidth',1.5)
end
end
67
end
for i=1:n
for j=1:n
if X(i)~=X(j) && Y(i)~=Y(j)
if weight{i,j} ~= 0
text(Wx(i,j),Wy(i,j),num2str(weight{i,j}),'FontSize',18,'Color','m');
end
end
end
end
end
%% Check Function For Bounds
function temp1 = check(temp1,i,j,g)
if str2num(temp1) > n
str = strcat('Edge ',num2str(i)','--Node',num2str(j));
waitfor(msgbox('Vertex Number exceeds Total number of vertices ! ReEnter the Value.',str,'warn'));
prompt = {'Enter the Vertex Number :'};
dlg_title = str;
num_lines = 1;
68
def = {'1'};
answer = inputdlg(prompt,dlg_title,num_lines,def);
temp1 = answer{1};
temp1 = check(temp1,i,j,g);
end
end
%% Check Function for Max Edges
function [temp3 temp2] = check1(temp3,temp2)
temp_new = temp3*(temp3-1)/2;
if temp2 > temp_new
str = 'ERROR !';
str1 = strcat('Number of Edges cannot exceed --> ',num2str(temp_new),'
<-- for -->',num2str(temp3),' <-- Vertices');
waitfor(msgbox(str1,str,'warn'));
prompt = {'Enter the number of Vertices : ','Enter the number of Edges :
'};
dlg_title = str;
num_lines = 1;
def = {'4','5'};
answer = inputdlg(prompt,dlg_title,num_lines,def);
temp3 = str2num(answer{1});
temp2 = str2num(answer{2});
69
[temp3 temp2] = check1(temp3,temp2);
end
end
function Figure_Close_Callback(hObject,eventdata)
% Construct a questdlg with two options
choice = questdlg('Would you like to solve a new problem ?', ...
'Solution Achieved. Try another ?', ...
'Yes','No','View/Modify Input Data','No');
% Handle response
switch choice
case 'Yes'
close all; clear all;
start();
case 'No'
return
case 'View/Modify Input Data'
floyd_warshall_modify(dist_modify,n,q);
end
end
end % floyd
70
An illustrative example of Floyd-Warshal's Algorithm
A
B
C
D
E
A
B
C
D
E
F
G
H
I
71
F
G
H
I