OPTIMAL LOCATION OF AN ADDITIONAL HOSPITAL IN EJURA – SEKYEDUMASE DISTRICT (A CONDITIONAL P – MEDIAN PROBLEM) By GYAMERA MICHAEL (BSc. MATHS) A Thesis Submitted To the Department of Mathematics, Kwame Nkrumah University of Science and Technology In partial fulfillment of the requirement for the degree of MASTER OF SCIENCE Industrial Mathematics Institute of Distance Learning NOVEMBER 2013 i DECLARATION I hereby declare that this submission is my own work towards the MSc. and that, to the best of my knowledge, it contains no material previously published by another person nor material which has been accepted for the award of any other degree of the University, except where due acknowledgment has been made in the text. GYAMERA MICHAEL PG3013209 ………………….…… ………………………….. Student Name & ID Signature Date Certified by: Mr K. F. Darkwah ………………….…… ………………………….. Supervisor Name Signature Date Certified by: Prof. S. K. Amponsah. ………………….…… ……………………….. Head of Dept. Name Signature Date ii ABSTRACT This dissertation focuses mainly on conditional facility location problems on a network. In this thesis we discuss the conditional p – median problem on a network. Demand nodes are served by the closest facility whether existing or new. The thesis considers the problem of locating a hospital facility (semi – obnoxious facility) as a conditional p – median problem, thus some existing facilities are already located in the district. This thesis uses a new a new formulation algorithm for for the conditional p- median problem on a network which was developed by Oded Berman and Zvi Drezner (2008) to locate an additional hospital in Ejura – Sekyedumase district. A 25 – node network which had four existing hospital was used. The result indicated that additional hospital should be located at Frante (node 7) with an optimal objective function value of 113252. The additional facility at Frante will largely help reduce the pressure on the existing hospitals and improved the quality of service. iii TABLE OF CONTENTS CONTENT PAGE NUMBER DECLARATION i ABSTRACT ii TABLE OF CONTENT iii LIST OF TABLES vii LIST OF FIGURES viii DEDICATION ix ACKNOWLEDGEMENT x CHAPTER 1 INTRODUCTION 1.0 Introduction 1 1.1 The Hospital 3 1.1.1 Types of Hospital 4 1.1.2 Historical Background of Hospitals in Ghana 5 1.2 Background of the Study 7 1.3 Statement of the Problem 10 1.4 Objective of Study 11 1.5 Methodology 11 1.6 Justification of the Study 12 1.7 Organization of the Thesis 13 iv CHAPTER 2 REVIEW OF LITERATURE 2.0 Introduction 14 2.1 Approaches to Facility Location Problems 15 2.2 Set Covering 18 2.3 The p – center Problems 19 2.4 The p – median Problem 22 2.4.1 Conditional Location Problem 26 CHAPTER 3 METHODOLOGY OF THE STUDY 3.0 Facility Location Problems 27 3.1 Total or Average Distance models 28 3.2 Maximum Distance Models 29 3.3 P – Center Problem 34 3.4 The P – Median Problem 36 3.5 The Conditional P – Median Problem 37 3.5.1 The Algorithm of Berman and Simchi – Levi 39 3.5.2 Berman and Drezner‟s Algorithm 46 3.6 Factor Rating Method 54 v CHAPTER 4 DATA COLLECTION, ANALYSIS AND DISCUSSION OF RESULTS 4.0 Introduction 57 4.1 Data Collection 58 4.2 Data Analysis 62 4.3 Model Formulation 64 4.4 Algorithm used to solved the problem 65 4.5 Computation and Results 65 4.6 Discussion of Results 72 CHAPTER FIVE CONCLUSION AND RECOMMENDATIONS 5.1 Conclusion 73 5.2 Recommendation 76 REFERENCES 77 APPENDICES Appendix 1.0 86 Appendix 2.0 87 Appendix 3.0 88 Appendix 4.0 89 Appendix 5.0 91 vi LIST OF TABLES TABLE TITLE PAGE NO. NO. 1.0 Summary of statement of out-patients in Ejura-Sekyedumase district 9 hospital from 2008 – 2010 3.0 All pairs shortest path distance matrix D 41 3.1 Modified shortest Distance matrix . 42 3.2 The modified shortest distance matrix, with the existing facility 43 node removed 3.3 Optimal new location matrix using the modified shortest distance 45 matrix , 3.4 All pairs shortest path distance matrix, D 48 3.5 Modified shortest path distance matrix, , 50 3.6 Modified shortest path distance matrix, , with existing facility node 50 removed 3.7 Optimal new location matrix, using , 53 3.8 Rating weight of relevant factors and their respective location rates 55 On a 1 to 100 basis 3.9 Relative scores of factors for the health center 56 4.0 Major communities in Ejura – Sekyedumase district and their 60 respective nodes 4.1 Major communities in Ejura – Sekyedumase district and their 60 respective population vii 4.2 Matrix of network in figure 4.0 indicating Towns and their pair of 61 Distances 4.3 Summary of Shortest Distance Matrix between pair of nodes 63 4.4 Summary of Modified Shortest Path Distance matrix, , 68 4.5 Modified Shortest Path Distance Matrix, , with existing facility 69 nodes removed A2.0 All pairs shortest path distance matrix, D 87 A3.0 Modified Shortest Path Distance Matrix, , 88 viii LIST OF FIGURES FIGURE TITLE PAGE NO. NO. 3.0 A 6 - node sample network for p – median problem 40 3.1 Example of a 6 – node network for p – median problem 47 4.0 Network of 25 towns in Ejura – Sekyedumase district 59 5.0 Network of the 25 towns in Ejura – Sekyedumase district 75 indicating the location of the additional hospital A1.0 District map of Ejura – Sekyedumase district 86 ix DEDICATION This work is dedicated to the Almighty Lord who is my source of strength and knowledge and to my beloved parents, Mr. and Mrs. Benefo who have always stood by me, inspired and supported me. x ACKNOWLEDGEMENT This would not have been possible without the strength, knowledge and wisdom from the Almighty God. I owe deep debt of gratitude to my supervisor, Mr. K.F. Darkwah, for his continuous support, skillful guidance, enthusiasm, and patience over the past 4years. Mr Darkwah gave me not only invaluable remarks on my thesis, but also insightful advices for my career development. Without his help this work would have not been possible. I would like to express my sincere to all workers of Ejura-Sekyedumase District assembly. Thanks so much to my dear friends and colleagues in and out of KNUST especially Agyei Gyamfi David and Adjei Yeboah. They really encouraged me. Thanks full of love to my dearest one Afriyie Linda Dornic for her understanding and constant support. Finally, I thank all those who I have not mentioned, but have encouraged me in so many ways. Gyamera Michael. xi CHAPTER 1 1.0 INTRODUCTION Almost every public and private sector enterprise that we can think of has been faced with the problem of locating facilities. Government agencies need to determine locations of offices and other public services such as schools, hospitals, fire stations, ambulance bases, and so on. Industrial firms must determine locations for fabrication and assembly plants as well as warehouses. In these cases, the success or failure of facilities depends in part on the locations chosen for those facilities. Such problems are known as facility location problems. In other words, facility location problems investigate where to physically locate a set of facilities (i.e. resources, servers) to satisfy some set of demand (i.e., customers, clients). The goal is to place these facilities such that the quality of service provided is optimized. This optimization may vary depending on the particular objective function chosen. The function could be either: minimize average travel time or cost, minimize average response time, minimize maximum travel time or cost and or maximize net income (Amponsah, 2007). A facility is considered as a physical entity that provides services. Facility location problems arise in a wide range of practical applications in different fields of study: economic, management, planning, production and many others. Welch et al (1997) also classified facilities into three categories: non – obnoxious (desirable), semi – obnoxious and obnoxious (non- desirable) 1 In most location problems we are interested in locating facilities that are desirable. Schools, hospitals, production plants post offices, ambulances and fire stations are all considered as facilities that are desirable. Facilities at times can produce an undesirable effect, which may be present even though a high degree of accessibility is required to the facility. If the undesirable effect outweighs the accessibility requirement, then the facility can be classified as obnoxious. Some examples of obnoxious facilities are the nuclear power stations, military installations, pollution produced by industrial plants and recycling centers. Although necessary for society, these facilities are undesirable and often dangerous to the surrounding inhabitants so lowering local house prices and quality of life. Sometimes though a facility produces a negative or undesirable effect, this effect may be present even though a high degree of accessibility is required by the facility. For example waste disposal sites and football stadium. These facilities are referred to as semi – obnoxious.(Brimberg and Juel, 1998) This thesis aims to locate a hospital as an example of semi – obnoxious facility. Hospitals are useful and necessary for the community, but they are a source of negative effects, such as noise from the hospital‟s ambulance and also the solid waste materials from the hospitals that emits unpleasant smell which make it undesirable. The combination of the two makes the facility semi – obnoxious, (Gordillo et al, 2007). In real life, we always encounter health issues. At times these occurrences need emergency attention within the shortest possible time; otherwise the result may turn out to be disaster with attendant morbidity and mortality. The hospital is always the best option in these cases and hence its location is always of interest 2 1.1 THE HOSPITAL The word “hospital” comes from the latin word “hospes” which refers to either a visitor or the host who receives the visitor.