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I OPTIMAL RESOURCES ALLOCATION; a CASE STUDY OF OPTIMAL RESOURCES ALLOCATION; A CASE STUDY OF SEKYERE CENTRAL DISTRICT ASSEMBLY, NSUTA-ASHANTI By OWUSU-BEMPAH EMMANUEL (B.Ed Mathematics) A Thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology Kumasi, in partial fulfillment of the requirements for the award of MASTER OF PHILOSOPHY (APPLIED MATHEMATICS) June, 2013 i DECLARATION I hereby declare that this submission is my own work towards the award of the Master of Philosophy (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. OWUSU-BEMPAH EMMANUEL . Student (PG6202411) Signature Date Certified by: PROF S. K. AMPONSAH . Supervisor Signature Date Certified by: PROF. S. K AMPONSAH . Head of Department Signature Date i ACKNOWLEDGEMENTS At a distance from the efforts of me, the success of any project depends largely on the support and guiding principle of many others. I seize this opportunity to express my gratitude to the people who have been influential in the triumphant completion of this project. I would like to show my greatest appreciation to my supervisor, Prof. S. K Amponsah. I cannot say thank you enough for his remarkable support and help. I feel motivated and encouraged every time I meet him. Devoid of his back-up and direction this project would not have materialized. The guidance and support received from all loved ones, especially my wife Christiana Owusu-Bempah and my family who contributed to this project, was vital for the success of the project. I am grateful for their constant support and help. ii DEDICATION This thesis is dedicated to my parents, Mr. and Mrs. Owusu-Bempah who have supported me all the way since the beginning of my studies. Also, this thesis is dedicated to my family who has been a great source of motivation and inspiration. Finally, this thesis is dedicated to Katakyie Kwame Dwumah of the Ghana Revenue Authority, Accra and all those who believe in the richness of learning. iii ABSTRACT Education is input to eradicate poverty in the recent civilization and this cannot be overemphasized and however, the financial outlay required to provide this very important service is far from being enough as a lot of schools are under trees. In this thesis, we consider the problem of allocating resource at the Sekyere central District Assembly, Nsuta-Ashanti with the aim of minimizing unnecessary lapses during budget allocation for resources by the assembly. The problem was formulated as an Integer Linear Programming (ILP) problem using the available data from the District Assembly. This problem was solved with the Branch and- Bound of Method of solving Integer Linear Programming (ILP). It was found that the out of the ten different locations considered and budget of Three Hundred and Sixty Thousand Ghana Cedis, the optimal number of classroom to be built was fifteen (15) representing a 3-unit classroom and two 6-unit classroom buildings at three different locations within the District at a minimum budget of Three Hundred and Seventeen Thousand Ghana Cedis (GH¢ 317,000) respectively. We concluded that the Knapsack problem for selecting required sites in critical situations such as construction of school buildings was useful and it can be applied to any situation where allocation of funds in the sector of educational development becomes a serious setback. Scientific modeling of allocating resources was recommended as it can be used to reduce financial loss in budget allocations. iv TABLE OF CONTENTS DECLARATION …………………………………………………………. I ACKNOWLEDGEMENTS ……………………………………………… II DEDICATION ……………………………………………………………. III ABSTRACT ……………………………………………………………… IV TABLE OF CONTENTS ………………………………………………… V LIST OF TABLES………………………………………………………. IX CHAPTER 1 1 1.0 Introduction …………………………………………………. 1 1.1 Background of the Study …………………………………… 2 1.2 Problem Statement ………………………………………….. 4 1.3 Objectives …………………………………………………… 4 1.4 Justification …………………………………………………. 5 1.5 Methodology ……………………………………………….. 5 1.6 Scope of the Study …………………………………………. 6 1.7 Limitations of the study …………………………………….. 6 1.8 Organization of the Study …………………………………... 6 1.9 Summary …………………………………………………….. 7 CHAPTER 2 LITERATURE REVIEW ………………………………… 8 2.0 Introduction …………………………………………………………. 8 2.1 Relevant Literatures …………………………………………………… 8 2.2 Summary ………………………………………………………. 34 v CHAPTER 3 METHODOLOGY……………………………………………. 35 3.0 Introduction ……………………………………………………. 35 3.1 profile of Sekyere Central District Assembly ………………….. 35 3.2 Linear Programming ……………………………………………. 36 3.3 Terminologies in Linear Programming ………………………… 37 3.3.1 Decision Variables ……………………………………………… 38 3.3.2 Objective Function ……………………………………………. 38 3.3.3 Constraints ……………………………………………………. 38 3.3.4 Simple Upper Bound …………………………………………… 39 3.3.5 Non-Negativity Restrictions …………………………………. 39 3.3.6 Complete Linear Programming ……………………………… 40 3.3.7 Parameters …………………………………………………… 40 3.4 Basic Assumptions of Linear Programming ………………… 41 3.5 Fundamental Theorem of Linear Programming …………….. 42 3.6 Importance of Linear Programming …………………………. 42 3.7 Simplex Method ……………………………………………… 42 3.8 Summary of the Simplex Method …………………………………. 43 3.9 Simplex Algorithm ……………………………………………. 44 3.10 The Initial Tableau …………………………………………….. 44 3.11 Data Modeling …………………………………………………. 45 3.11.1 Dynamic Programming ………………………………………… 46 3.11.2 Knapsack Problem ……………………………………………… 46 3.12 Types of Knapsack Problem …………………………………… 48 3.12.1 Multi-Objective Knapsack Problem …………………………… 48 vi 3.12.2 Subset-Sum Knapsack Problem ……………………………….. 49 3.12.3 The Change-Making Problem ………………………………….. 49 3.12.4 Cutting-Stock Problem ………………………………………… 50 3.12.5 Multiple Knapsack Problem …………………………………… 51 3.12.6 The Quadratic Knapsack Problem …………………………….. 51 3.12.7 Unbounded Knapsack Problem ……………………………… 52 3.12.8 Bounded Knapsack Problem ………………………………… 53 3.12.9 0-1 Knapsack Problem ………………………………………. 54 3.13 Genetic Algorithm ……………………………………………. 55 3.14 Duality and Integer Programming …………………………… 57 3.14.1 Duality …………………………………………………………. 57 3.14.2 Integer Programming ………………………………………….. 58 3.15 Methods for Solving Linear Integer Programming Optimization (Lip) Problems ………………………………… 59 3.16 Branch and Bound …………………………………………… 64 3.17 General Description of Branch and Bound ………………….. 65 3.18 Sensitivity Analysis ………………………………………….. 67 CHAPTER 4 DATA COLLECTION AND ANALYSIS ……………….. 69 4.0 Introduction ………………………………………………………….. 69 4.1 Data Collection ………………………………………………………. 70 4.2 Results of the Analysis ………………………………………………. 73 4.3 Sensitivity Analysis on the whole solution ………………………….. 80 vii CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS ………. 81 5.0 Introduction …………………………………………………………. 81 5.1 Conclusions ………………………………………………………….. 81 5.2 Recommendations …………………………………………………… 82 REFERENCE ……………………………………………………………… 82 APPENDICES ……………………………………………………………… 89 viii LIST OF TABLES 4.1 Respective towns with their Budget allocations ……………………… 73 4.2 Breakdown of Budget allocation for 3-unit classroom at Bonkuro………. 76 4.3 Breakdown of Budget allocation for 6-unit classroom at Appiahkurom … 78 4.4 Breakdown of Budget allocation for 6-unit classroom at Asasebonso …… 80 A.1 Breakdown of Budget Allocations for Various Sites …………………… 90 A.2 Description of 3-unit classrooms ………………………………………… 92 A.3 Description of 6-unit classrooms ………………………………………… 92 A.4 Description of 9-unit classrooms ………………………………………… 92 A.5 Optimal Solutions for the various iterative stages (from QM software) … 93 ix CHAPTER ONE 1.0 INTRODUCTION It is a universal truth that education is key to eradicating poverty in the modern society and this cannot be overemphasized. In a developing country like Ghana education will help citizens to acquire the needed skills and knowledge. The skill and knowledge acquired will make the citizens functionally literate and productive to facilitate poverty alleviation and promote the rapid socio-economic growth. The government of Ghana embarked on the Basic Education Sector Improvement Program and more popular the Free Compulsory and Universal Basic Education Program (BESIP/FCUBE), which was aimed at providing every child of school-going age with good basic education. However, the financial outlay required to provide this very important service is far from being enough as a lot of schools are under trees. Ghana Education Trust fund was established in September 2000 by the government in providing educational finance supplementation particularly at the basic and secondary level where there exists an urgent need to provide adequate classroom infrastructure for our rapidly growing in- school population. (http://www.getfund.gov.gh). The limited funds allocated to the various Metropolitan, Municipal and District Assemblies should be efficiently used. The problem that frequently arises in resource allocation where there are financial constraints can be considered as a knapsack problem. This study seeks to provide a scientific way of allocating funds in the building of unit classroom that will ensure it optimality using linear integer programming with Sekyere Central District as a case study. 1 1.1 BACKGROUND OF STUDY Education is a fundamental human right for all children and this right may not be realized in Ghana if strategic measures are not put in place to ensure adequate infrastructure provision to schools, especially in rural communities. School infrastructure
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