OPERATIONS RESEARCH. Dr. Mohd Vaseem Ismail. Assistant Professor. Faculty of Pharmacy Jamia Hamdard New Delhi

Size: px

Start display at page:

Download "OPERATIONS RESEARCH. Dr. Mohd Vaseem Ismail. Assistant Professor. Faculty of Pharmacy Jamia Hamdard New Delhi"

Joshua Singleton
5 years ago
Views:

1 OPERATIONS RESEARCH

3 OPERATIONS RESEARCH By Dr. Qazi Shoeb Ahmad Professor Department of Mathematics Integral University Lucknow Dr. Shakeel Javed Assistant Professor Department of Statistics & O.R. AMU, Aligarh Dr. Mohd Vaseem Ismail Assistant Professor Faculty of Pharmacy Jamia Hamdard New Delhi UNIVERSITY SCIENCE PRESS (An Imprint of Laxmi Publications Pvt. Ltd.) BANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD JALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI NEW DELHI BOSTON, USA

4 Copyright 2013 by Laxmi Publications Pvt. Ltd. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Published by : UNIVERSITY SCIENCE PRESS (An Imprint of Laxmi Publications Pvt. Ltd.) 113, Golden House, Daryaganj, New Delhi Phone : Fax : info@laxmipublications.com Price : ` Only. First Edition : 2013 OFFICES Bangalore Chennai Cochin , Guwahati , Hyderabad Jalandhar Kolkata Lucknow Mumbai , Ranchi UOR OPERATION RESEARCH-ISM Typeset at : ABRO Enterprises, Delhi. C 16931/09/01 Printed at : Ajit Printers, Delhi.

5 Contents CHAPTER 1: Graphical Solution and Formulation Linear Programming Structure of Linear Programming Model Advantages of Linear Programming General Mathematical Model of Linear Programming Problem Areas of Application of Linear Programming Basic Definitions Graphical Method of Solution LPP Some Special Cases in LPP Formulation of Problem as an LPP Examples on the Applications of Linear Programming 21 Objective Type Questions 30 Exercises 32 CHAPTER 2: Linear Programming Simplex Method Standard Form of Linear Programming Problem Some Basic Definitions Simplex Method Minimization Problems (All Constraints of Type ) (a) The Big-M Method (Method of Penalty) Solving LPP when Variables are Unrestricted Two Phase Method Degeneracy Resolving Degeneracy 140 Objective Type Questions 154 Unsolved Examples 156 CHAPTER 3: Duality and Sensitivity Analysis in Linear Programming Duality: Its Concept Formulation of Dual Linear Programming Problem Construction of Dual Problem from Primal Problem Main Points on Duality Advantages of Duality Comparison of Solutions to the Primal and its Dual 169 v

6 vi 3.7 Dual Simplex Method Sensitivity Analysis 185 Objective Type Questions 193 Exercises 195 Unsolved Problems 196 CHAPTER 4: Transportation Problem Transportation Problem Mathematical Model of Transportation Problem Some Basic Definitions Loops in Transportation Table and their Properties The Transportation Method North-West Corner Method Least Cost Method The Row-Minima Method The Column-Minima Method Vogel s Approximation Method (VAM) Test for Optimality The Transportation Simplex Method or Transportation Algorithm (MODI method) Stepping Stone Solution Degeneracy and its Resolution Prohibited Transportation Routes Maximization Transportation Problem Transshipment Problem 241 Objective Type Questions 250 Exercises 251 Unsolved Exercises 252 CHAPTER 5: Assignment Problems Assignment Problems Mathematical Model of Assignment Problem Solution of Assignment Problem Maximization Problem Alternate Optimal Solutions Unbalanced Assignment Problem Restrictions on Assignments in Assignment Problem Travelling Salesman Problem 281 Objective Type Questions 284 Exercises 286 Unsolved Problems 286

7 vii CHAPTER 6: Dynamic Programming Introduction Bellman s Principle of Optimality Characteristics of Dynamic Programming Forward and Backward Recursion Basic Steps of Dynamic Programming Applications of Dynamic Programming Model I: Single Additive Constraint, Multiplicative Separable Return Model II: Single Additive Constraint, Additive Separable Return Model III: Single Multiplicative Constraint, Additively Separable Return Model IV: Shortest Route Problem Solution of Some Other Problems by Using Dynamic Programming Solution of Linear Programming Problem by Dynamic Programming 306 Exercises 310 CHAPTER 7: Decision Theory Introduction Elements of a Decision Problem Types of Decision-Making Environment Decision-Making Under Uncertainty Decision-Making Under Risk Decision Tree 328 Exercises 334 CHAPTER 8: Theory of Games Introduction Assumptions of the Game Basic Terminology Two Person Zero-Sum Games Minimax Theorem Solution of Games with Saddle Point (Pure Strategy Games) Solution of Games Without Saddle Point (Mixed Strategies Games) 344 Exercises 366 CHAPTER 9: Stochastic Inventory Models Introduction Inventory Control Need of Inventory Inventory Classification Types of Inventories Inventory Costs Other Elements of Inventory Problem 373

8 viii 9.8 Economic Order Quantity (EOQ) Inventory Models Model (1) Model (2) Model (3) Model (4) EOQ Problems with Price Breaks EOQ Problem with One Price Break EOQ Problem with Two Price Breaks EOQ Problem with n Price Breaks 398 Exercises 400 CHAPTER 10: Simulation Introduction The Basic Steps of Simulation Process Advantages of Simulation Technique Disadvantages of Simulation Technique Applications of Simulation Monte Carlo Simulation Basic Characteristics Simulation Languages 407 Exercises 421 Chapter 11: Queueing Models Introduction Queueing System Elements of the Queueing System Distribution of Arrivals (Pure Brith Model ) Distribution of Inter-Arrival Times (Exponential Process) Distribution of Service Times Kendal s Notation for Representing Queueing Models Terminology and Notations for Queueing Models Definition of Transient and Steady State Model Model Multichannel Queueing Models Model Model Exercises 454 Index

9 Preface The authors feel great pleasure in presenting the first edition of the book Operations Research. This book is designed to meet the requirement of B.Tech, M.B.A., B.B.A. students of different Universities. The subject matter has been discussed in such a way that the students will find no difficulty to understand it. Each chapter of this book contains complete self-explanatory theory and a large number of solved examples, followed by a collection of good exercises. The language of the book is simple and easy to understand. The authors hope that the students, teachers and other readers will find the book interesting and to the point covering the whole course. We hope that the students will receive the book warmly. We have taken great care in eliminating the misprints, but if there are still any, we shall be highly obliged to those who will take trouble of pointing them out. Suggestions for the improvement of the book will be gratefully acknowledged. Authors ix

11 I UNIT NIT I LINEAR PROGRAMMING

13 CHAPTER 1 Graphical Solution and Formulation 1.1 LINEAR PROGRAMMING Linear programming deals with the maximization or minimization (or optimization ) of a function of variables known as objective function, subject to a set of linear equalities and/or inequalities known as constraints. Here, the objective function can be cost, profit or production capacity, which is to be obtained in the best possible manner. The constraints may be imposed by different sources such as market demand, storage capacity, raw material availability etc. The word linear here implies that the variables, considered here do not certain any powers greater than one. Thus, a given change in one variable will always cause a resulting proportional change in another variable. The word programming refers to modelling and solving a problem mathematically that involves the economic allocation of limited resources by choosing a particular course of action or a strategy among various alternative strategies to achieve the desired objective. G.B. Dantzig developed the technique primarily for solving military logistics problems. But nowadays, it is being used widely in all the areas of management, education, transportation planning, military operations, healthcare systems, education etc. 1.2 STRUCTURE OF LINEAR PROGRAMMING MODEL The general structure of LP model consists of mainly three basic components: (a) Decision variables. The Operation Researcher needs to evaluate various alternatives (or courses of action) for arriving at the optimal value of the objective function. If there is no alternative to be selection among various alternatives, then there is no need of linear programming. The evaluation of various alternatives can be obtained by the nature of objective function and availability of resources. Once the objective is decided, we pursue certain activities (also called as decision variables) usually denoted x i s (i = 1, 2,..., n). The value of these activities represents the extent to which each of these is performed. There are certain variables (decision), which are not always under the control of decision 3

14 4 OPERATIONS RESEARCH maker, and are called as uncontrollable variables. If the values are under the control of decision maker, then these variables are called as controllable variables. These decision variables, usually interrelated in terms of consumption of limited resources, require simultaneous solutions. One thing is to be kept in mind that all the decision variables in linear programming model are controlable, continuous and non-negative. i.e., x j 0, ( j = 1, 2,..., n) (b) The objective function. The main aim (or goal or objective) of each linear programming problem (LPP) is to either maximize the profit or minimize the costs involved. Thus, the objective function of each LPP can be expressed in terms of the decision variables to optimize (maximize or minimize) the criterion of optimality, for example, profit, cost, revenue, distance etc. Thus, the objective function of any LPP, by using the decision variables (discussed above) can be expressed as Maximize/Minimize Z = C 1 x 1 + C 2 x C n x n or Max/Min Z = Cx n j = 1 j j (j = 1, 2,..., n) where Z is the value of the objective function and C j : represents the contribution of a unit of the respective variables x j ( j = 1, 2,..., n), so that the value of z may be maximum or minimum. Now, the optimal value of Z (maximum or minimum) can be obtained graphically or by using the well known method known as simplex method (will be discussed later on). (c) The constrants. Constraints are imposed while formulating the linear programming model, as there is always a limitation of using the available resources. For example, man hour, labour, machine, raw-material, money, space etc. can be utilized upto a limit. These limited resources can be used upto some extent to which an objective of the model can be achieved. All the solutions obtained after simplifying the LP model, must satisfy the constraints imposed. (d) Non-negative restriction. All the decision variables must assume non-negative values as negative values of physical quantities is an impossible solution. 1.3 ADVANTAGES OF LINEAR PROGRAMMING There are various advantages in using the linear programming. Some of the advantages are discussed below: 1. It is very much helpful in attaining the optimum use of productive resources. If the resources are limited, then the linear programming model gives the optimum result after using the limited available resources. Linear programming shows how a decision maker can use his available limited resources effectively by selecting and distributing these resources. 2. Linear programming also helps in re-evaluating the basic plan for changing conditions. For example, if a particular plan is started and after sometime, certain conditions are changed that effects the model, then LP can determine other conditions that will adjust the remainder of the plan for best result. 3. With the use of linear programming, the executive builds into his planning a true reflection of the limitations and restrictions under which he must operate and whenever it becomes

15 GRAPHICAL SOLUTION AND FORMULATION 5 necessary to deviate from the best programme he can evaluate the cost or penalty involved. 4. In LPP, high lighting the bottlenecks in the process is the most significant advantage of this technique. For example, whenever bottleneck problem occurs, some of the machines remain idle for sometime while few cannot meet the respective demands. 1.4 GENERAL MATHEMATICAL MODEL OF LINEAR PROGRAMMING PROBLEM The general linear programming problem with n decision variables and m constraints can be stated in the following form: Find the values of decision variables x 1, x 2,..., x n so as to Maximize or Minimize Z = C 1 x 1 + C 2 x C n x n Subject to the linear constraints a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 : : : : : : : : : : : : a m1 x 1 + a m2 x a mn x n = b m The above formulation can also be written as follows: subject to linear constraints Max or Min Z = n j = 1 cijxj n j = 1 Cx j j...(1) = b i i = 1, 2,..., m...(2) j = 1, 2,..., n and x j 0...(3) The equation represented by (1) is the objective function; equation (2) is the set of constraints and equation (3) is the non-negativity condition. In the above LPP model, instead of equality sign (=), or (less than or equal to or greater than or equal to) sign can be used as per requirement in equation (2) for constraints. Here: C j : represents the coefficient of per unit contribution (profit/cost) of decision variables x j to the value of the objective function; the coefficients a ij (i = 1, 2,..., m and j = 1, 2,..., n) are referred to as the substitution or technological coefficients. These represents the amount of resources consumed per unit of variable a ij and b i (i = 1, 2,..., m) is the constant representing the requirement or availability of the ith constant. The above constrained optimization (maximization or minimization) problem may have (i) a unique optimal feasible solution

9789381159651 Author : Dr. Qazi Shoeb Ahmad, Dr. Shakeel Javed, Dr.Mohd.

16 Operations Research By Dr. Qazi Shoeb Ahmad, Dr. Shakeel Javed, Dr.Mohd. Vaseem Ismail 40% OFF Publisher : Laxmi Publications ISBN : Author : Dr. Qazi Shoeb Ahmad, Dr. Shakeel Javed, Dr.Mohd. Vaseem Ismail Type the URL : 60 Get this ebook

FUNDAMENTALS OF COMPUTER PROGRAMMING AND IT

FUNDAMENTALS OF COMPUTER PROGRAMMING AND IT SALIENT FEATURES OF THE PRESENT EDITION Motivates the unmotivated and provides the teachers an unequaled approach that allows them to teach students with a disparity