NATIONAL OPEN UNIVERSITY OF NIGERIA FACULTY OF SCIENCES COURSE CODE: CIT 907 COURSE TITLE: ADVANCED OPERATIONS RESEARCH (3 UNITS) 1 COURSE GUIDE CIT 907: ADVANCED OPERATIONS RESEARCH UNITS: 3 Course Team Course Developer/Writer Dr. Ogonnaya Michael Romanus African University of Science and Technology, Abuja Dr. Gregory Onwodi – NOUN Dr. Maduabuchi Ejike Okpala – Baze University, Abuja. Content Editor Prof. Benjamin Aribisala. LASU Course Material Coordination Dr. Vivian Nwaocha, Dr. Greg Onwodi & Dr. Francis B.Osang Department of Computer Science National Open University of Nigeria NATIONAL OPEN UNIVERSITY OF NIGERIA Headquarters Plot 91, Cadastral Zone, Nnamdi Azikiwe Expressway, Jabi – Abuja, Nigeria Lagos Office 14/16 Ahmadu Bello Way Victoria Island Lagos e-mail: [email protected] URL: www.nou.edu.ng Published by National Open University of Nigeria Printed 2020 ISBN: All Rights Reserved 2 Contents COURSE GUIDE ................................................................................................................. 2 INTRODUCTION ................................................................................................................ 4 THE COURSE ..................................................................................................................... 4 COURSE AIMS AND OBJECTIVES ........................................................................................ 4 WORKING THROUGH THIS COURSE ................................................................................... 5 COURSE MATERIALS ......................................................................................................... 5 STUDY UNITS .................................................................................................................... 5 TEXTBOOKS/REFERENCES ................................................................................................. 6 ASSESSMENT .................................................................................................................... 6 SELF-ASSESSMENT EXERCISES ........................................................................................... 6 END OF COURSE EXAMINATION ........................................................................................ 6 SUMMARY ....................................................................................................................... 7 MODULE 1 : LINEAR PROGRAMMING IN OPERATION RESEARCH .................................... 8 UNIT 1 CONCEPTS OF OPERATIONS RESEARCH ................................................................. 8 UNIT 2 MODELLING IN OPERATIONS RESEARCH .............................................................. 17 UNIT 3 LINEAR PROGRAMMING (1) ................................................................................. 23 UNIT 4 LINEAR PROGRAMMING (2) ................................................................................. 45 UNIT 5 LINEAR PROGRAMMING (3) ................................................................................. 73 MODULE 2: OPERATION RESEARCH TECHNIQUES ............................................................ 82 UNIT 1 CONCEPT OF INTEGER PROGRAMMING ............................................................ 82 UNIT 2 THE CONCEPT OF DYNAMIC PROGRAMMING ...................................................... 90 UNIT 3 CONCEPT OF GOAL PROGRAMMING ................................................................... 95 UNIT 4 TRANSPORTATION MODEL ................................................................................ 106 UNIT 5 ASSIGNMENT MODEL ......................................................................................... 135 MODULE 3: OPERATION RESEARCH MODELLING ........................................................... 147 UNIT 1 INVENTORY MODEL............................................................................................ 147 UNIT 2 NETWORK ANALYSIS .......................................................................................... 166 UNIT 3 THE PROBLEMS IN SEQUENCING ........................................................................ 190 UNIT 4 WAITING LINE IN QUEUING THEORY ................................................................. 209 UNIT 5 REPLACEMENT THEORY ..................................................................................... 242 MODULE 4: ADVANCES IN OPERATION RESEARCH......................................................... 253 UNIT 1 THEORY OF GAMES OR COMPETITIVE STRATEGIES ............................................ 253 UNIT 2 SOLVING GAMES ................................................................................................ 259 3 INTRODUCTION You should have become familiar with the Basic Operations Research concepts introduced to you in your undergraduate course. It is the objective of this course to build upon the previous lessons learnt, and formally introduce to you the more advanced concepts with the view to reinforcing your understanding and placing you in a pivotal position to intellectually understand developmental work and research in the rapidly advancing field of operations research. THE COURSE CIT 907: Advanced Operations Research consists of 17 units distributed across three modules as follows: Module 1 is composed of 5 units Module 2 is composed of 5 units Module 3 is composed of 5 units Module 4 is composed of 2 units Module 1 guides you into the realm of operations research by explaining the different approaches to operations research problem solving. It also introduces you into the concepts of operations research which is the primary concern of unit 1. Unit 2 treats modeling, while unit 3, 4 and 5 take you through progressively, through advanced levels of linear programming. In Module 2, units 1, 2 and 3 treat integer programming, dynamic programming and goal programming respectively, while unit 4 revisits transportation model which you certainly must have encountered. This module concludes with the assignment model and discusses amongst others, the two methods of enumeration and reduced matrix which are very crucial in solving assignment problems. Furthermore, Module 3 exposes you to inventory, network analysis, sequencing, line waiting and replacement. In Module 4, unit 1 introduces the basis of game theory as well as its important principles. While in unit 2 several methods of solving games were elaborately discussed with numerous examples. COURSE AIMS AND OBJECTIVES The aim of CIT 907 is to further intimate you with operations research, acquaint you with the mathematical calculations and the practical approximation of the idealized theorems which allows you establish their practicable applications and indispensability in the real world. You should also bear in mind the practical limitations of the concepts idealized in the real world. You are required to conscientiously and diligently work through this course and upon completion of this course, you should be able to: understand the meaning of operations research highlight the historical development of operations research describe the scientific nature of operations research identify the importance and uses of operations research with respect to the various topics to be treated in the study state the limitations of operations research state the meaning of model in operations research describe the various types of model describe how to construct a model 4 list some standard operations research model state the usefulness of linear programming in operations research state the properties of a linear programming model identify some areas of application of linear programming formulate a linear programming model state the usual assumptions of a linear programming model solve a two-variable linear programming model graphically prepare LPD for use of simplex explain the use of simplex list the steps involved in using a simplex method prepare a simplex table and understand its various components demonstrate the use of simplex method for solving an LLP solve LPP using maximization problem describe optimal – dual concept describe dual formulation procedure interpret dual programming model explain the concept of game theory understand a good number of solution methods for games. WORKING THROUGH THIS COURSE This course requires you to spend quality time to read. Whereas the content of this course is quite comprehensive, it is presented in clear language with lots of illustrations that you can easily relate to. The presentation style is qualitative and descriptive. This is deliberate and it is to ensure that your attention in the course content is sustained. You should take full advantage of the tutorial sessions because this is a veritable forum for you to “rub minds” with your peers, which provides you with valuable feedback as you have the opportunity of comparing knowledge with your course mates. COURSE MATERIALS You will be provided with course materials prior to commencement of this course. The course materials will comprise your course guide as well as your study units. You will also receive a list of recommended textbooks. These textbooks are however not compulsory. STUDY UNITS You will find listed below the study units which are contained in this course and you will observe that there are three modules. Each module comprises five units each. Module 1 Linear Programming in Operation Research Unit 1 Concepts of Operations Research Unit 2 Modeling Unit 3 Linear Programming (1) Unit 4 Linear Programming (2 Unit 5 Linear Programming (3) Module