Introduction to Management Science

Transcription

1 GLOBAL EDITION Introduction to Management Science Eleventh Edition Bernard W. Taylor III

2 Editorial Director: Sally Yagan Editor in Chief: Donna Battista Senior Acquisitions Editor: Chuck Synovec Senior Acquisitions Editor, Global Edition: Steven Jackson Editorial Project Manager: Mary Kate Murray Editorial Assistant: Ashlee Bradbury Director of Marketing: Maggie Moylan Executive Marketing Manager: Anne Fahlgren Marketing Manager, International: Dean Erasmus Senior Managing Editor: Judy Leale Production Project Manager: Jane Bonnell Senior Operations Supervisor: Arnold Vila Operations Specialist: Cathleen Petersen Creative Director: Blair Brown Senior Art Director/Supervisor: Janet Slowik Art Director: Steve Frim Interior Design: Mike Fruhbeis Cover Design: Jodi Notowitz Cover Image: (c) Scanrail - Fotolia.com Associate Media Project Manager, Editorial: Sarah Peterson Media Project Manager, Production: John Cassar Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: Pearson Education Limited 2013 The right of Bernard W. Taylor III to be identified as the author of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act Authorised adaptation from the United States edition, entitled Introduction to Management Science, 11 th Edition, ISBN by Bernard W. Taylor III, published by Pearson Education 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 either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. Microsoft and/or its respective suppliers make no representations about the suitability of the information contained in the documents and related graphics published as part of the services for any purpose. All such documents and related graphics are provided as is without warranty of any kind. Microsoft and/or its respective suppliers hereby disclaim all warranties and conditions with regard to this information, including all warranties and conditions of merchantability, whether express, implied or statutory, fitness for a particular purpose, title and non-infringement. In no event shall Microsoft and/or its respective suppliers be liable for any special, indirect or consequential damages or any damages whatsoever resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of information available from the services. The documents and related graphics contained herein could include technical inaccuracies or typographical errors. Changes are periodically added to the information herein. Microsoft and/or its respective suppliers may make improvements and/or changes in the product(s) and/or the program(s) described herein at any time. Partial screen shots may be viewed in full within the software version specified. Microsoft and Windows are registered trademarks of the Microsoft Corporation in the U.S.A. and other countries. This book is not sponsored or endorsed by or affiliated with the Microsoft Corporation. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN-13: ISBN-10: Typeset in 10/12, Times New Roman by PreMediaGlobal Printed and bound by Courier Kendallville in the United States of America The publisher's policy is to use paper manufactured from sustainable forests.

3 M06_TAYL6407_11_SE_C06.qxd 12/22/11 9:08 PM Page 251 CHAPTER 6 Linear Programming Transportation Models 251

4 252 CHAPTER 6 LINEAR PROGRAMMING TRANSPORTATION MODELS See Web site Module B for a chapter on the Transportation and Assignment Solution Methods. In this chapter, we examine three special types of linear programming model formulations transportation, transshipment, and assignment problems. They are part of a larger class of linear programming problems known as network flow problems. We are considering these problems in a separate chapter because they represent a popular group of linear programming applications. These problems have special mathematical characteristics that have enabled management scientists to develop very efficient, unique mathematical solution approaches to them. These solution approaches are variations of the traditional simplex solution procedure. Like the simplex method, we have placed these detailed manual, mathematical solution procedures called the transportation method and assignment method on the companion Web site that accompanies this text. As in previous chapters, we will focus on model formulation and solution by using the computer, specifically by using Excel and QM for Windows. In a transportation problem, items are allocated from sources to destinations at a minimum cost. The Transportation Model The transportation model is formulated for a class of problems with the following unique characteristics: (1) A product is transported from a number of sources to a number of destinations at the minimum possible cost; and (2) each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product. Although the general transportation model can be applied to a wide variety of problems, it is this particular application to the transportation of goods that is most familiar and from which the problem draws its name. The following example demonstrates the formulation of the transportation model. Wheat is harvested in the Midwest and stored in grain elevators in three different cities Kansas City, Omaha, and Des Moines. These grain elevators supply three flour mills, located in Chicago, St. Louis, and Cincinnati. Grain is shipped to the mills in railroad cars, each car capable of holding 1 ton of wheat. Each grain elevator is able to supply the following number of tons (i.e., railroad cars) of wheat to the mills on a monthly basis: Grain Elevator Supply 1. Kansas City Omaha Des Moines 275 Total 600 tons Each mill demands the following number of tons of wheat per month: Mill Demand A. Chicago 200 B. St. Louis 100 C. Cincinnati 300 Total 600 tons The cost of transporting 1 ton of wheat from each grain elevator (source) to each mill (destination) differs, according to the distance and rail system. (For example, the cost of

5 THE TRANSPORTATION MODEL 253 shipping 1 ton of wheat from the grain elevator at Omaha to the mill at Chicago is $7.) These costs are shown in the following table: Mill Grain Elevator A. Chicago B. St. Louis C. Cincinnati The linear programming model for a transportation problem has constraints for supply at each source and demand at each destination. 1. Kansas City $6 $ 8 $10 2. Omaha Des Moines The problem is to determine how many tons of wheat to transport from each grain elevator to each mill on a monthly basis to minimize the total cost of transportation. A diagram of the different transportation routes, with supply and demand, is given in Figure 6.1. FIGURE 6.1 Network of transportation routes for wheat shipments Des Moines 200 Chicago Omaha Kansas City 300 Cincinnati 150 St. Louis 100 The linear programming model for this problem is formulated as follows: minimize Z = $6x 1A + 8x 1B + 10x 1C + 7x 2A + 11x 2B + 11x 2C + 4x 3A + 5x 3B + 12x 3C subject to x 1A + x 1B + x 1C = 150 x 2A + x 2B + x 2C = 175 x 3A + x 3B + x 3C = 275 x 1A + x 2A + x 3A = 200 x 1B + x 2B + x 3B = 100 x 1C + x 2C + x 3C = 300 x ij Ú 0 In this model the decision variables, x ij, represent the number of tons of wheat transported from each grain elevator, i (where i = 1, 2, 3), to each mill, j (where j = A, B, C). The objective function represents the total transportation cost for each route. Each term in the objective function reflects the cost of the tonnage transported for one route. For example,

6 254 CHAPTER 6 LINEAR PROGRAMMING TRANSPORTATION MODELS In a balanced transportation model in which supply equals demand, all constraints are equalities. In an unbalanced transportation model, supply is greater than demand or demand is greater than supply. if 20 tons are transported from elevator 1 to mill A, the cost ($6) is multiplied by x 1A (= 20), which equals $120. The first three constraints in the linear programming model represent the supply at each elevator; the last three constraints represent the demand at each mill. As an example, consider the first supply constraint, x 1A + x 1B + x 1C = 150. This constraint represents the tons of wheat transported from Kansas City to all three mills: Chicago (x 1A ), St. Louis (x 1B ), and Cincinnati (x 1C ). The amount transported from Kansas City is limited to the 150 tons available. Note that this constraint (as well as all others) is an equation (=) rather than a inequality because all the tons of wheat available will be needed to meet the total demand of 600 tons. In other words, the three mills demand 600 total tons, which is the exact amount that can be supplied by the three grain elevators. Thus, all that can be supplied will be, in order to meet demand. This type of model, in which supply exactly equals demand, is referred to as a balanced transportation model. Realistically, however, an unbalanced problem, in which supply exceeds demand or demand exceeds supply, is a more likely occurrence. In our wheat transportation example, if the demand at Cincinnati is increased from 300 tons to 350 tons, a situation is created in which total demand is 650 tons and total supply is 600 tons. This will result in the following change in our linear programming model of this problem: minimize Z = 6x 1A + 8x 1B + 10x 1C + 7x 2A + 11x 2B + 11x 2C + 4x 3A + 5x 3B + 12x 3C subject to x 1A + x 1B + x 1C = 150 x 2A + x 2B + x 2C = 175 x 3A + x 3B + x 3C = 275 x 1A + x 2A + x 3A 200 x 1B + x 2B + x 3B 100 x 1C + x 2C + x 3C 350 x ij Ú 0 One of the demand constraints will not be met because there is not enough total supply to meet total demand. If, instead, supply exceeds demand, then the supply constraints will be. Sometimes one or more of the routes in the transportation model may be prohibited. That is, units cannot be transported from a particular source to a particular destination. When this situation occurs, we must make sure that the variable representing that route does not have a value in the optimal solution. This can be accomplished by assigning a very large relative cost as the coefficient of this prohibited variable in the objective function. For example, in our wheat-shipping example, if the route from Kansas City to Chicago is prohibited (perhaps because of a rail strike), the variable x 1A is given a coefficient of 100 instead of 6 in the objective function, so x 1A will equal zero in the optimal solution because of its high relative cost. Alternatively, the prohibited variable can be deleted from the model formulation. Time Out for Frank L. Hitchcock and Tjalling C. Koopmans Several years before George Dantzig formalized the linear programming technique, in 1941 Frank L. Hitchcock formulated the transportation problem as a method for supplying a product from several factories to a number of cities, given varying freight rates. In 1947 T. C. Koopmans independently formulated the same type of problem. Koopmans, an American economist originally from the Netherlands and a professor at Chicago and Yale, was awarded the Nobel Prize in 1975.

7 COMPUTER SOLUTION OF A TRANSPORTATION PROBLEM 255 Management Science Application Commercial Transportation Taking New Roads with GPS GPS (global positioning system) has become a useful tool for drivers who might be unfamiliar with an area they are driving through. Many of the GPS programs you can buy for your automobiles include important updates on traffic information, as well as the locations of restaurants, businesses, and local residences. Commercial drivers have also used GPS as a way to find the quickest route to their next pick-up or drop-off location. However, the routes will sometimes send drivers with large trucks down narrow roads, which could jam traffic as well as delay the trucker s schedule. Local drivers and delivery drivers alike get frustrated when streets are clogged because a truck is stuck on a narrow road. One U.K. company, Navevo, decided to create a product that would make it easier for these commercial drivers to get where they need to go. The company created ProNav, a unique GPS system that allows commercial drivers to see information that will help them choose the best route for their delivery. Drivers can enter the specific information about the truck they re driving (such as height, weight, destination), and ProNav will map out the most efficient route, avoiding areas where the truck might encounter difficulties. ProNav allows the drivers to see which roads are truck-free zones, which roads have obstacles like steep hills or sharp turns that a truck might have difficulty navigating, the weight restrictions on different bridges, as well as the locations of loading zones and other locations of transport-specific interest. The software also includes more standard features, such as the locations of restaurants, businesses, and up-to-date traffic reports. As a leslie sanders - Fotolia.com bonus feature, the software can also switch between two different modes: commercial and standard. This capability eliminates the drivers need to have separate systems for their trucks and their own personal vehicles. In 2008, Navevo was given the Frost & Sullivan Award for Product Innovation for their work on ProNav. The company received the award because it used innovation and refined previously existing technology to better suit the needs of commercial industry. Sources: ProNav Product Information, _information.htm; Navevo ProNav Wins Frost and Sulllivan 2008 U.K. Telematics Product Innovation Award, Navevo (April 4, 2008): Navevo ProNav Recognized for U.K. Heavy Goods Software, GPS World (April 5, 2008): Navevo-ProNav-Recognized-for-UK-Heavy-Goods-Softwa/Article- Standard/Article/detail/ Computer Solution of a Transportation Problem Because a transportation problem is formulated as a linear programming model, it can be solved with Excel and using the linear programming module of QM for Windows that we introduced in previous chapters. However, management science packages such as QM for Windows also have transportation modules that enable problems to be input in a special transportation tableau format. We will first demonstrate the solution of the wheat-shipping problem solved earlier in this chapter, this time by using Excel spreadsheets. Computer Solution with Excel We will first demonstrate how to solve a transportation problem by using Excel. A transportation problem must be solved in Excel as a linear programming model, using Solver, as demonstrated in Chapters 3 and 4. Exhibit 6.1 shows a spreadsheet setup to solve our wheat-shipping example. Notice that the objective function formula for total cost is contained in cell C10 and is shown on the formula bar at the top of the spreadsheet. It is constructed as a SUMPRODUCT of the decision variables in cells C5:E7 and a cost array in cells K5:M7. Cells G5:G7 in column G and cells C9:E9 in row 9, labeled Grain Shipped, contain the constraint formulas for supply