A BI-OBJECTIVE STOCHASTIC SINGLE FACILITY LOCATION

MODEL FOR A SUPERMARKET

AROUSHA HAGHIGHIAN ROUDSARI

A project report submitted in partial fulfillment of the

requirements for the award of the degree of

Master of Engineering (Industrial Engineering)

Faculty of Mechanical Engineering

Universiti Teknologi Malaysia

JANUARY 2013

iii

This thesis is dedicated to my parents

without their love and support it could not have

been produced.

ACKNOWLEDGEMENT

I wish to acknowledge and thank the following persons for their help and support

given throughout the period of preparation of this thesis.

My supervisor, Dr. Wong Kuan Yew for his enthusiasm, guidance, constructive

discussion and support throughout all stages of this research.

To the faculty of mechanical Engineering, University technology of Malaysia, for

their support and help towards my postgraduate affairs.

To Fereshteh Talesh Ensandost for all her help in gathering essential data for the aim

of this study.

Finally I wish to thank my parents Razieh, Mamoud and my brother Ali for all the

love, kindness, prayers and help shown to me during the period of studies and

preparation of this thesis.

Arousha Haghighian Roudsari, January 2013

v

ABSTRACT

One of the important strategic challenges in today’s competitive

environment is to select the best location for new facilities. Selecting a new location

is vital for retail stores such as chain supermarkets and can be considered as a huge

competitive advantage which may result in their failure or success. Location models

are hard to implement in the real world problems, firstly because of the uncertainty

of the input parameters and secondly, due to the intensive computations involved

when the solution space is large such as a city. In this study, a stochastic bi-

objective model is developed for points and area destinations with the purpose of

finding a single new location for a chain supermarket that aims to be close to more

customers and also have the minimum number of competitors near to the new

location. Customer locations are considered to be regional with uniform probability

distribution. A reduced gradient solution procedure is used as an algorithm for

solving the model. The problem is solved with the help of Matlab software due to

the high computations involved. The result of this study shows that more firms and

companies can be drawn to actually implementing these models that are closer to

reality to increase their profit and minimize their cost which is the aim of all

business firms.

vi

ABSTRAK

Salah satu cabaran strategik yang penting dalam persekitaran yang

kompetitif pada hari ini adalah untuk memilih lokasi terbaik untuk kemudahan baru.

Memilih lokasi baru adalah penting untuk kedai runcit seperti pasar raya rantaian

dan boleh dianggap sebagai kelebihan persaingan yang besar yang boleh

mengakibatkan kegagalan atau kejayaan mereka. Model lokasi adalah sukar untuk

dilaksanakan dalam masalah dunia sebenar, pertamanya kerana ketidaktentuan

parameter input dan kedua, disebabkan pengiraan intensif yang terlibat apabila

ruang penyelesaian adalah besar seperti sebuah bandaraya. Dalam kajian ini, model

stokastik bi-objektif dibangunkan untuk titik dan destinasi kawasan dengan tujuan

mencari lokasi tunggal baru untuk rantaian pasar raya yang bertujuan untuk menjadi

dekat kepada lebih ramai pelanggan dan juga mempunyai bilangan minimum

pesaing berhampiran lokasi baru. Lokasi pelanggan dianggap sebagai daerah dengan

taburan kebarangkalian seragam. Prosedur pengurangan kecerunan digunakan

sebagai algoritma untuk menyelesaikan model. Masalah itu diselesaikan dengan

bantuan perisian Matlab kerana pengiraan yang tinggi yang terlibat. Hasil kajian ini

menunjukkan bahawa lebih banyak firma-firma dan syarikat-syarikat boleh diambil

untuk benar-benar melaksanakan model-model yang lebih dekat kepada realiti untuk

meningkatkan keuntungan mereka dan mengurangkan kos mereka yang merupakan

matlamat semua firma-firma perniagaan.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURES xii

1 INTRODUCTION 1

1.1 Introduction 1

1.2 Background of the Study 2

1.3 Problem Statement 3

1.4 Objective of the Study 4

1.5 Scope of the Study 4

1.6 Significance of the Study 5

1.7 Organization of the Thesis 5

2 LITERATURE REVIEW 7

2.1 Introduction 7

2.2 Terms and Concepts 7

2.2.1 Facility Location Problems 7

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4 PROBLEM FORMULATION 26

4.1 Introduction 26

4.2 Model Development 26

4.2.1 Model Assumptions 26

4.2.2 Mathematical Formulation 28

4.2.2.1 Deterministic Model 29

4.2.2.2 Stochastic Model 30

4.3 A brief Introduction of the Case Study 32

4.4 Conclusion 35

5 DATA COLLECTION 36

5.1 Introduction 36

5.2 Data Related to Customer Regions 36

5.2.1 Location of Customer Regions 37

5.2.2 Weights of Customer Regions 39

2.2.1.1 Covering Objectives 8

2.2.1.2 Minimax Objective 8

2.2.1.3 Minisum Objective 9

2.2.2 Solution Method 9

2.2.3 Multi Objective 10

2.3 Stochastic Facility Location Models 10

2.3.1 Single Facility Stochastic Location Models 11

2.3.2 Multi Facility Stochastic Location Models 13

2.4 Competitive Facility Location Models 15

2.4.1 Deterministic Competitive Location Problems 17

2.4.2 Competitive Location Problems with Uncertainty 18

2.5 Conclusion 21

3 RESEARCH METHODOLOGY 22

3.1 Introduction 22

3.2 Research Design 22

3.3 Conclusion 25

ix

5.3 Data Related to Competitors 44

5.3.1 Location of Competitors 45

5.3.2 Factors for Assigning Weights to Competitors 46

5.3.3 Competitors Weights 50

5.4 Conclusion 53

6

RESULTS AND DISCUSSION 54

6.1 Introduction 54

6.2 The Procedure of Solving the Model 54

6.2.1 Combining the Objective Functions 55

6.2.2 The Method for Solving the Model 57

6.3 Solving the Model Using Matlab 59

6.3.1 Phase 1- Importing Data and Defining the

Parameters

61

6.3.2 Phase 2- Removing the Overlaps and Determining

the New Ranges

61

6.3.3 Phase 3- Calculating Sk 62

6.3.4 Phase 4- Calculating derivative for Determining

the Optimum 62

6.4 Validating the Program 63

6.5 Results of the Model Using Matlab 69

6.5.1 Depicting the Optimum Solution on the Map and

Discussion

75

6.6 Sensitivity Analysis 78

7 CONCLUSION AND FUTURE WORKS 82

7.1 Introduction 82

7.2 Limitations of the Study 82

7.3 Future Research Directions 83

7.4 Conclusion 84

x

REFERENCES

85

Appendices 91-95

xi

LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Previous works in stochastic location problems 20

5.1 Location of customer regions 38

5.2 Population and weights of customer regions 40

5.3 Marginal probability densities of the regions 42

5.4 Location of competitors 45

5.5 The size of competitors and related weights 46

5.6 Product variety scores 48

5.7 Customer services scores 49

5.8 Final scores and weights for product variety 50

5.9 Final scores and weights for customer service 50

5.10 Weights of factors 51

5.11 Final weights of competitors 52

6.1 New weights for customer regions along the x-axis 55

6.2 New weights for competitors along the x-axis 57

6.3 Data related to regions of problem B 63

6.4 Data related to competitors of problem B 64

6.5 The results of problem B using Excel 65

6.6 The result of phases of the algorithm using Matlab 69

6.7 Comparing the alternative locations 78

6.8 Altered data of the area weights in problem B 79

6.9 Altered data of competitors weights in problem B 80

xii

LIST OF FIGURES

FIGURE NO. TITLE PAGE

3.1 Research methodology diagram 23

4.1 Picture of four branches of the supermarket 32

4.2 Inside view of the supermarkets 33

4.3 Location of the main branch 34

4.4 The city map of Rasht and location of the four branches 34

5.1 Customer regions 37

5.2 Close up view of some regions 41

5.3 Location of competitors on the city map 44

6.1 Overview of Matlab R2012a V 7.14 60

6.2 Depicting the overlaps along the x-axis for problem B 64

6.3 Results of problem B using Matlab 67

6.4 Location of the optimum solution of problem B 68

6.5 Optimum x value of the model using Matlab 74

6.6 Location of the optimum solution on the map 76

6.7 Alternative available locations near optimum 77

6.8 Optimum solution of problem B with altered weights of

the regions 80

6.9 Optimum solution of problem B with altered weights of

competitors 81

xiii

LIST OF APPENDICES

APPENDIX TITLE PAGE

A Matlab Script File 91

CHAPTER 1

INTRODUCTION

1.1 Introduction

Facility location problems have a large history and have received

considerable research interest for decades. The location of a new facility is a long-

term investment due to the high costs and difficulties associated with locating or

altering the location of a facility after it is placed. Therefore, one of the important

strategic challenges in today‟s competitive environment is to select the best location

for new facilities (Owen & Daskin, 1998).

Generally facility location models and their application can be categorized

differently on the basis of the number of facilities to be located, their size, the

objective, the solution space, and the distance measure which are used in the models

to make them comparable with the real world problems. Single-facility minisum

location problem is a category of location problems that intends to find a single new

location for the facility in a way that the total weighted sum of the distances between

the new facility and all the existing facilities or demand points is minimized. The

space can be considered as to be planar or network and also continuous or discrete.

Continuous location problems means that the new facility location can be placed

anywhere on the space whereas discrete location problems include predefined sets of

locations for the new facility. Different types of distance measures can be chosen

based on the problem definition such as rectilinear distance for instances like urban

2

settings due to the street layout or Euclidean distances in situations where the

distance can be considered as a straight-line (Tompkins et al., 2010)

However, in classical facility location models the environment in which the

decision is being made may change and considered parameters such as cost,

demands, and distances may face uncertainties. Therefore this raises the need and

importance of facility location models that consider uncertainty. Different

approaches exist for overcoming this uncertainty in facility location problems which

helps the costly decision of finding a new location for a facility that will result in its

long-term profitability (Snyder, 2006).

1.2 Background of the Study

Changes in environmental factors, population and market trend make it very

hard to find a robust facility location and for doing so, uncertainties of future

occurrences should be taken into consideration by the decision makers. Most of the

researches of location problems have been limited by this difficulty and considered

as static and deterministic models (Owen & Daskin, 1998). For modeling more

realistic situations the uncertainty of inputs such as demands, cost, travel time can be

taken into account by stochastic (probabilistic) facility location models which

consider these uncertainties in risk situations that uncertain parameters are defined by

a known probability distribution in the facility location model (Arabani & Farahani,

2012).

Considering only a single objective in facility location problems has an old

background. Moreover, since management science has been acquainted with the

concept of multi-objective decision making, it has paved the ground for its

application in location problems. In the past decade this concept has grown

considerably and a new window has been opened to facility location science and its

application in various businesses. This growth is because of the realization that for

3

modeling location problems that are closer to reality and more practical, sometimes it

is necessary to consider more than one objective. There is no bound for the

application of location science and as it can be seen in the literature it has been

applied to a variety of scopes such as national and international, public and private

facilities, business areas and military environments (Farahani et al., 2010).

Most of the models that include more than one objective are considering

deterministic parameters. As mentioned before, uncertainties should be taken into

account in the location models in order to be closer to reality. Therefore, using

stochastic optimization and robustness concept containing uncertainty, random

parameters or probability distributions and combining it with multi-objective location

problems can help to depict more realistic and practical problems (Farahani et al.,

2010). Also, there are only few stochastic location problems that have been actually

applied to real world problems (Snyder, 2006).

1.3 Problem Statement

As most of the facility location problems have focused on deterministic

models, there has been a significant growth in the need for considering more

stochastic models that can cope with the uncertainties firms are facing in the real

world when locating a new facility. In problems such as locating a new retail store or

a supermarket because of the competitive environment that exists, besides the

uncertainties, more than one objective should be considered to find more practical

locations. However, to the best of my knowledge only few stochastic facility location

models have been proposed in a competitive environment. In addition, the stochastic

location models that have been actually applied to real world problems are limited.

This study deals with finding a new location for the expansion of a chain

supermarket that faces demand uncertainty which wants to be closer to more

customers and also to have the minimum number of competitors around the new

location in order to attract more market share. Customer locations or demand points

4

are considered to be regional with a probability distribution instead of just being

points in a plane.

1.4 Objective of the Study

i. To develop a mathematical model for finding a practical single new

location for a chain supermarket that aims to be located closer to more

customers and to have minimum number of competitors around the new

location with regional probability distributed customer locations.

ii. To determine the optimum location for the supermarket in the real

solution space.

1.5 Scope of the Study

The scope of the study is:

i. Case study is a chain supermarket (S) that exists in one of the cities in

Iran, Rasht.

ii. The solution space of the location problem will be the city map of Rasht

considering only the central part of the city that shows the population

distribution.

iii. The study will only consider rectilinear distance measure.

iv. Customer locations are not considered as point but instead will be

considered as rectangular regions with probability distribution.

v. For solving the model, Matlab R2012a version 7.14 will be used.

5

vi. The cost of the land will not be considered for this study.

vii. Google Earth is used for the measurements of the coordinates of the

regions and points.

1.6 Significance of the Study

The selection of location is a vital decision for retail stores such as chain

supermarkets and can be considered as a huge competitive advantage which may

result in their failure or success. This decision becomes even more important because

once a location is selected it will be very costly to change the location and therefore

is a long term investment. Hence, modeling more realistic problems in a competitive

environment by considering the uncertainties of the real world and more than one

objective will have a considerable effect on location decisions. The significance of

this study is to use the stochastic facility location models in a competitive

environment where the demand points or customer locations are uncertain and to

consider more than one objective. This would help to make the model more practical

and closer to reality and then applying the model to a case study. By doing so and

achieving a good result, more firms and companies can be drawn to actually

implementing these models to increase their profit and minimize their cost which is

the aim of all business firms.

1.7 Organization of the Thesis

In this thesis chapter one contains a brief introduction of the study, its

background, problem statement, objective and scope. A categorized literature review

relevant to location problems is provided in chapter two. In chapter three, research

methodology is explained and its structure and research design is shown. Chapter

6

four will contain the formulation of the model and also a brief introduction of the

case study.

The data collection for the case study is provided in chapter five.

Furthermore, chapter six includes the solution procedure for solving the model and

the relevant results and discussion. Finally, chapter seven will contain the conclusion

and the recommended future research directions.

85

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