KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY, KUMASI INSTITUTE OF DISTANCE LEARNING SITE SELECTION FOR FIRE STATION IN BIRIM NORTH DISTRICT BY: MANU MICHAEL YAW (B.ED MATHEMATICS) A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS IN PARTIAL FULFILLMENT OF THE REQUIEMENTS FOR THE AWARD OF DEGREE OF MASTER OF SCIENCE IN INDUSTRIAL MATHEMATICS JULY, 2012 DECLARATION I hereby declare that this submission is my own work toward the M Sc. 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 due acknowledgement as been made in the text. MANU MICHAEL YAW …………………… …………………. (Student number: PG 4067610) Signature Date Certified by: MR. CHARLES SEBIL …………………. …………………. Supervisor Signature Date Certified by: PROF. S. K AMPONSAH ……..…………… ………………….. Head of Department Signature Date ii DEDICATION To the Glory of God I dedicate this project to my wife, Mrs. Janet Manu, my lovely daughters Josephine, Anastasia and Wilhelmina Manu and my colleagues for their moral and mutual support. ii i ACKNOWLEDGEMENT It is with great pleasure that I take this opportunity to recognize those who have played a major role in bringing this significant work to its full realization. It has been satisfying to see all the pieces come together, often in ways much better than I expected. It is therefore, my pleasure to express my greatest indebtedness to my supervisor, Mr. Charles Sebil who was always ready to assist me in one way or the other in the compilation of this work, may God richly bless him. I also wise to express my profound gratitude to all the lecturers in the Mathematics Department especially Prof. S. K Amponsah, Head of Department and Mr F. K Darkwa who in one way or the other contributed immensely to my success. I owe a considerable measure of appreciation to all staff of Nkawkaw fires service and Birim North District Assembly especially Mr. Bright Tetrafe, the district engineer for helping me with information about the map of the district. Simple words of acknowledge and appreciation seem inadequate to my wife Mrs Janet Manu for her love, support and encouragement. iv ABSTRACT This thesis seeks to identify the optimal location for sitting of a fire station in Birim North District using absolute centre heuristic method to ensure quick response to fire incidents and help fight fire outbreaks within the district thereby ensuring the future safety of people and property within the service coverage area. We focused on markets, hospitals, schools, residential facilities etc. and these were considered as the nodes or demand points. The Karis and Hakimi algorithm was used to compute the absolute centre from the distances obtained from the Floyd-Warshall’s all pairs shortest path algorithm. The absolute centre was identified as (0) zero meters from node C. In other words the absolute centre was at node C and the facility has to be sited at C. The maximum distance to be travelled from the facility to a farthest node, node O (Akokoase) shall be 4745metres. The fire station should be located at New Abirem to ensure minimum response time and travel distance in the service coverage area. v TABLE OF CONTENTS Contents Page DECLARATION ------------------------------------------------------------------------------------- ii DEDICATION --------------------------------------------------------------------------------------- iii ACKNOWLEDGEMENT ------------------------------------------------------------------------- iv ABSTRACT ------------------------------------------------------------------------------------------ v LIST OF FIGURES -------------------------------------------------------------------------------- viii CHAPTER 1 ----------------------------------------------------------------------------------------- 1 INTRODUCTION ---------------------------------------------------------------------------------- 1 1.1 Background ------------------------------------------------------------------------------------- 1 1.1.1 Causes of fire ----------------------------------------------------------------------------------- 2 1.1.2 Firefighting History --------------------------------------------------------------------------- 4 1.1.3 Fire Hydrants ----------------------------------------------------------------------------------- 8 1.1.4 Fire Management ----------------------------------------------------------------------------- 10 1.1.5 Structural fire suppression in the Birim North District --------------------------------- 10 1.2 Problem Statement ---------------------------------------------------------------------------- 12 1.3 Objectives of the Study ----------------------------------------------------------------------- 13 1.4 Methodology ----------------------------------------------------------------------------------- 13 1.5 Justification ------------------------------------------------------------------------------------ 13 1.6 Limitations of Study -------------------------------------------------------------------------- 14 1.7 Organization of the Study -------------------------------------------------------------------- 15 1.8 Summary --------------------------------------------------------------------------------------- 15 CHAPTER 2 ---------------------------------------------------------------------------------------- 16 LITERATURE REVIEW ------------------------------------------------------------------------ 16 2.1 Introduction ------------------------------------------------------------------------------------ 16 2.2 Review of Location Problem Models ------------------------------------------------------ 16 2.3 Summary --------------------------------------------------------------------------------------- 25 v i CHAPTER 3 ---------------------------------------------------------------------------------------- 26 METHODOLOGY -------------------------------------------------------------------------------- 26 3.0 Introduction ------------------------------------------------------------------------------------ 26 3.1 Shortest Path Problem ------------------------------------------------------------------------ 26 3.1.1 Single- Source Shortest Path Problem --------------------------------------------------- 27 3.1.2 All Pairs Shortest Path Problem ---------------------------------------------------------- 28 3.2 The Centre Problem Model ------------------------------------------------------------------ 28 3.3 Basic Assumptions of the Centre Problem ------------------------------------------------ 29 3.4 The Centre Problem --------------------------------------------------------------------------- 30 3.5 Finding the Absolute Centre ------------------------------------------------------------------ 32 3.6 Construction of Upper Envelope ------------------------------------------------------------ 37 3.7 Invariance of local with respect to choice of any node as the origin. ------------------ 44 3.8 Summary --------------------------------------------------------------------------------------- 48 CHAPTER 4 ---------------------------------------------------------------------------------------- 49 DATA COLLECTION AND ANALYSES --------------------------------------------------- 49 4.1 Introduction ------------------------------------------------------------------------------------ 49 4.1 All pairs shortest path for the data collected. --------------------------------------------- 53 4.1.1 Locating the Vertex/Node Centre -------------------------------------------------------- 53 4.1.2 Locating the Absolute Centre ------------------------------------------------------------- 53 4.2 Results ------------------------------------------------------------------------------------------ 55 4.3 Summary --------------------------------------------------------------------------------------- 56 CHAPTER 5 ---------------------------------------------------------------------------------------- 57 CONCLUSION AND RECOMMENDATION ---------------------------------------------- 57 5.0 Introduction ------------------------------------------------------------------------------------ 57 5.1 Summary of Findings ------------------------------------------------------------------------- 57 5.2 Recommendations ----------------------------------------------------------------------------- 58 REFERENCES -------------------------------------------------------------------------------------- 60 APPENDIX I --------------------------------------------------------------------------------------- 67 APPENDIX II -------------------------------------------------------------------------------------- 71 APPENDIX III ------------------------------------------------------------------------------------- 72 vi i LIST OF TABLES Tables Pages Table 3.1: --------------------------------------------------------------------- 33 Table 4.1: --------------------------------------------------------------------- 52 Table 4.2: --------------------------------------------------------------------- 54 Table 4.3: --------------------------------------------------------------------- 81 Table 4.4: --------------------------------------------------------------------- 81 LIST OF FIGURES Figures Pages Figure 3.1: --------------------------------------------------------------------- 33 Figure 3.2: --------------------------------------------------------------------- 34 Figure 3.3: --------------------------------------------------------------------- 35 Figure 3.4: --------------------------------------------------------------------- 39 Figure 3.5: --------------------------------------------------------------------- 41 Figure 3.6: --------------------------------------------------------------------- 41 Figure 3.7: --------------------------------------------------------------------- 42 Figure 3.8: --------------------------------------------------------------------- 42 Figure 3.9: --------------------------------------------------------------------- 43 Figure 3.10: --------------------------------------------------------------------- 49 Figure 3.11: --------------------------------------------------------------------- 50 Figure 4.1: --------------------------------------------------------------------- 53 vi ii CHAPTER 1 INTRODUCTION 1.1 Background Fires start when a flammable and or a combustible material with adequate supply of oxygen or another oxidizer is subjected to enough heat and is able to sustain a chain reaction. This is commonly called the ‘Fire tetrahedron’ Fire cannot exist without all of those elements being in place at the right conditions. Once ignited, a chain reaction must take place whereby the fire can sustain its own heat by further release of heat energy in the process of the combustion. It may propagate provided there is a continuous supply of an oxidizer and fuel. Propagation may be achieved through convection, conduction and radiation. Fire may be broadly grouped into two; structural fire and bush/wild/forest fire. Since creation, fire has been and continues to be a useful tool in the very existence of human lives on this planet. Fire may be a blessing or a scourge depending on how it is managed. People have benefited from fire in the areas of transportation, energy generation, vegetation management, farming, heating cooking etc. However, others have had their properties and their livelihood destroyed by fire. Whole communities had been wiped out by fire. Offices such as the ministry of foreign affairs, markets such as Kumasi central market, Kantamanto market in Accra and many others have been engulfed by fire. Thus fire is indispensable on this planet but if it is not managed or controlled it will become a scourge and would threaten the very survival of human race. The destruction and damages left in the trail of fire outbreaks may translate into poverty, joblessness, homelessness and famine. The emotional stress victims of fire go through cannot be quantified. 1 It must be stated that all fires start very small but if not detected and acted upon quickly they escalate and become very destructive. The purpose of this study is to use mathematical method to identify a fire station in the Birim North District so that fires that break out can be quickly responded by fire fighters and suppress them before they escalate. 1.1.1 Causes of fire As mentioned earlier, fires may be broadly grouped into two; Structural fires and wild/bush/forest fires reflecting the type of vegetation or fuel. Wild fire is an uncontrolled fire that occurs in the countryside or wilderness area. It differs from structural fire by its extensive size, the speed at which it can spread out from the original source and its ability to change direction unexpectedly and to jump gaps such as roads, rivers, streams and firebreaks. Wild fire may be caused either naturally or artificially through the activities of human beings. Four major natural causes are: a. Ignition by lightning; b. Ignition by volcanic eruption; c. Ignition by sparks from rock falls d. Spontaneous combustion. Some of the man-made that causes fire outbreak are i. Farming ii. Honey extraction iii. Group hunting 2