KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY INSTITUTE OF DISTANCE LEARNING DEPARTMENT OF MATHEMATICS OPTIMAL RESOURCE ALLOCATION OF A COCOA PROCESSING COMPANY: APPLICATION OF INTEGER PROGRAMMING WITH IMPLICIT ENUMERATION BY TIMOTHY JOBSON MITCHUAL APRIL 2012 i DECLARATION I hereby declare that this submission is my own work towards the Master of Science degree and that, to the best of my knowledge it contains no material previously published by another person nor material which has been accepted for award of any other degree of the university except where due acknowledgement has been made in the text. Timothy Jobson Mitchual, PG3015409 ….……..………. ………… Student‘s Name & ID Signature Date Certified By Dr. S. K. Amponsah …………………. ……….. Supervisor‘s Name Signature Date Certified By Mr. S. K. Darkwah ………………….. .……….. Head of Department‘s Name Signature Date ii ABSTRACT Most cocoa producing countries are moving from the traditional export of raw cocoa beans to cocoa processing countries due to the hike in profit at the finished end of the production chain. For example; cocoa processing Company Ghana Limited has been increasing in production tonnage consistently from two thousand (2,000) tonnes in 1983 to sixty two thousand five hundred (62,500) tonnes, currently. This thesis describes the resource allocation problem of Cocoa Processing Company Ltd. as an integer programming problem. We applied integer programming algorithm to solve the company‘s resource allocation problem. Our research focused on the use of the integer programming problem for resource allocation given limited available funds for Cocoa Processing Company Ltd. in Ghana. The objective of this study is to mathematically model resource allocation problem of the Ghana cocoa processing company in the processing of the cocoa beans as a means of adding value to Ghana‘s cocoa to earn higher foreign exchange. In our methodology, we used the implicit enumeration algorithms in solving our problem. First, the algorithm was presented. A real life computational study was performed to evaluate the algorithms. We used the Quantitative Method (QM 32) to analyze our data. Cocoa Processing Company Ltd. resource allocation problem for the processing of the cocoa beans into various consumable products was modelled as integer programming problem. The Quantitative Method (QM 32) software was used to analyze the data collected from the Company Processing Company Ltd. The optimal value was GH¢770,000.00. It was observed that the solution that gave this achievable value was the utilization of the available resources to produced 3,000kg Royale, 7,000kg Altime, and 1,000kg Vitaco iii ACKNOWLEDGEMENT I would like to give thanks to the Jehovah God for granting me the strength and knowledge for understanding this course and the completion of this write-up. I am very grateful to my supervisor, Dr. S. K. Amponsah of the Department of mathematics, who painstakingly read through every line of the text and offered through his rich experience all the necessary guidance, direction, encouragement, advice and correction for the timely completion of this thesis. I will also give thanks to my wife Mrs. Sarah Jobson-Mitchual, my children Comfort Ekua Jobson-Mitchual and Jacob Kobi Jobson-Mitchual for their support during the entire course. Finally my sincere thanks go to all who in diverse ways helped in bringing this project to a successful end. God richly bless you all. iv TABLE OF CONTENTS DECLARATION i ABSTRACT ii ACKNOWLEDGEMENT iii TABLE OF CONTENTS iv LIST OF TABLES vi CHAPTER ONE 1 1.0 INTRODUCTION 1 1.1 Background of Study 1 1.2 Problem Statement 14 1.3 Objectives 14 1.4 Methodology 14 1.5 Justification 14 1.6 Limitations 15 1.7 Organization of the Thesis 15 v CHAPTER TWO 16 LITERATURE REVIEW 16 CHAPTER THREE 41 METHODOLOGY 41 3.0 Introduction 41 3.1 Polyhedral Theory 42 3.1.1 Implicit Enumeration 47 3.1.2 Branch and Bound 49 3.1.3 Branch and Bound for 0-1 Integer Program 50 CHAPTER FOUR 51 DATA COLLECTION AND ANALYSIS 51 4.0 Introduction 51 4.1 Data Collection and Analysis 51 CHAPTER FIVE 63 CONCLUSIONS AND RECOMMENDATIONS 63 5.0 Introduction 63 5.1 Conclusions 63 5.2 Recommendations 64 REFERENCES 65 vi LIST OF TABLE 4.1 Required Resources for the Processing of the Seven Products per 1,000kg 53 4.2 Summary of Results of Data Analyses 55 vii CHAPTER ONE 1.0 INTRODUCTION Established in 1965, the Cocoa Processing Company Limited (CPC) based in Tema, near Accra in Ghana, was incorporated in November 1981 as a Limited Liability Company. CPC, whose main objective is to process cocoa beans by adding value to them, comprises two factories, namely the Cocoa Factory and the Confectionery Factory. The former processes raw cocoa beans in to semi-finished products - cocoa liquor, butter, natural/alkalized cake or powder whilst the Confectionery Factory manufactures the Golden Tree chocolate bars, couverture, pebbles(chocolate coated peanut), VITACO and ALLTIME drinking chocolate Powder, Chocó Delight(Chocolate spread), Chocó Bake and Royale natural cocoa powder. The CPC factories process only the choicest premium Ghana cocoa beans without any blending, probably the only factory in the world, which can make such a claim. Through intensive research and product development, CPC turns out products, which meet international quality standards and also consumer satisfaction (Shaski and Vigneri, 2003). In the year 2002 CPC's position as one of the world's best chocolate producers was re- confirmed at the Monde Selection Competition held in Paris, France. At this Competition all the seven brands of chocolate and ALLTIME Drinking chocolate powder presented won gold medals on account of their distinctive quality. As a good corporate citizen, Cocoa Processing Company supports educational programmes in Ghana. The Company has established a GH₵500million-Endowment Fund to assist in equipping the distressed laboratories of the Science Faculties of the universities in Ghana. The Company is currently engaged in an expansion programme, which is aimed at increasing its cocoa throughout capacity from the viii original twenty five thousand (25,000) metric tonnes per annum to sixty five thousand (65,000) metric tonnes per annum. The expansion programme is in two phases. Phase one, which involved the construction of a new state-of-the-art factory with a capacity to process thirty thousand (30,000) metric tonnes of cocoa beans into cocoa liquor. Phase Two of the programme entails the rehabilitation of the existing factory into a modern factory with an increased capacity capable of processing thirty five thousand (35,000) metric tonnes of cocoa beans into cocoa butter cake and powder. The Ghana Cocoa Board has overtaken the Government of Ghana as the largest shareholder of Cocoa Processing Company Limited. This follows a conversion of an outstanding debt of US$14,087,120.50 owed to Ghana Cocoa Board into equity. As a result of the conversion, COCOBOD‘s total holding in CPC has increased from 21.74% to 57.73%. In March 2011, shareholders of the confectionery maker at an Annual General Meeting ratified the decision to convert the US$14,087,120.50 owed to Ghana Cocoa Board. Subsequent to the approval, the corporate action was executed at an agreed exchange rate of US$1.00 to GH₵1.65 which represents GH₵23, 243,748.83 for 937,247,936 ordinary shares. A price of GH₵0.0248 per share was used in the transaction representing a 26-week average of CPC share price between June 6, 2011 and December 5, 2011. By this action, COCOBOD has exceeded the 30% mandatory takeover threshold under the Ghana Stock Exchange‘s Listing Rules. However, there are indications that COCOBOD has no such plan to takeover CPC since that will go contrary to the principle behind the divestiture of CPC in 2002 from COCOBOD to the public via the Government of Ghana. Government of Ghana‘s stake in CPC was reduced from 48.38% to 26.13% while SSNIT maintains its spot as the third largest shareholder though its holding was diluted from 18.78% to 10.15%. The stake of the investing public has also been diluted from ix 11.10% to 5.99% as a result of the conversion. In a related development, CPC has listed an additional 937,247,936 ordinary shares on the GSE, following the conversion of the US$14,087,120.50 owed to Ghana Cocoa Board. In relation to this, the total issued shares of CPC have been increased from 1,100,826,240 to 2,038,074,176. In this chapter of the thesis, an overview of Ghana cocoa would be given; a brief description of the problem statement of the thesis is also presented together with the objectives, the methodology, the justification and the organization of the thesis. 1.1.0 COCOA IN GHANA No other country comes to mind more than Ghana when one speaks of cocoa. Likewise, one cannot think of Ghana without thinking of its cocoa sector, which offers livelihoods for over seven hundred (700,000) farmers in the southern tropical belt of the country. From long time immemorial, Ghana‘s main exports, cocoa has been central to the country‘s debates on development, reforms, and poverty alleviation strategies since independence in 1957. The cocoa sector in Ghana has not been an unmitigated success, however. After emerging as one of the world‘s leading producers of cocoa, Ghana experienced a major decline in production in the 1960s and 1970s, and the sector nearly collapsed in the early 1980s. Production steadily recovered in the mid-1980s after the introduction of economy wide reforms, and the 1990s marked the beginning of a revival, with production nearly doubling between 2001 and 2003.