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Quantitative Analysis for Management For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition Global edition Global edition Quantitative Analysis for Management twelfth edition Barry Render • Ralph M Stair, Jr • Michael E Hanna • Trevor S Hale twelfth edition Render • Stair • Hanna • Hale This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada you should be aware that it has been imported without the approval of the Publisher or Author Pearson Global Edition RENDER_129205932X_mech.indd 03/02/14 5:01 PM Quantitative Analysis for Management Twelfth Edition Global Edition Barry Render Charles Harwood Professor of Management Science Crummer Graduate School of Business, Rollins College Ralph M Stair, Jr Professor of Information and Management Sciences, Florida State University Michael E Hanna Professor of Decision Sciences, University of Houston–Clear Lake Trevor S Hale Associate Professor of Management Sciences, University of Houston–Downtown Boston Columbus Indianapolis New York San Francisco Upper Saddle River  Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo A01_REND9327_12_SE_FM.indd 11/02/14 8:19 PM To my wife and sons—BR To Lila and Leslie—RMS To Zoe and Gigi—MEH To Valerie and Lauren—TSH Editor in Chief: Donna Battista Head of Learning Asset Acquisition, Global Edition: Laura Dent Acquisitions Editor: Katie Rowland Senior Acquisitions Editor, Global Edition: Steven Jackson Editorial Project Manager: Mary Kate Murray Editorial Assistant: Elissa Senra-Sargent Director of Marketing: Maggie Moylan Senior Marketing Manager: Anne Fahlgren Marketing Assistant: Gianna Sandri Managing Editor: Jeff Holcomb Senior Production Project Manager: Kathryn Dinovo Manufacturing Buyer/Procurement Specialist: Carol Melville Associate Project Editor, Global Edition: Uttaran Das Gupta Senior Manufacturing Controller, Production, Global Edition: Trudy Kimber Art Director, Cover: Blair Brown Cover Designer: PreMediaGlobal Cover Image: © Evlakhov Valeriy/Shutterstock Media Project Manager: Lisa Rinaldi Full-Service Project Management: PreMediaGlobal Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within text Pearson Education Limited Edinburgh Gate Harlow Essex CM 20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2015 The rights of Barry Render, Ralph M Stair, Jr., Michael E Hanna, and Trevor S Hale to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Quantitative Analysis for Management, 12th edition, ISBN 978-0-13-350733-1, by Barry Render, Ralph M Stair, Jr., Michael E Hanna, and Trevor S Hale, published by Pearson Education © 2015 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 license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, 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is available from the British Library 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 Typeset in 10/12 Times Roman by PreMediaGlobal Printed and bound by Courier Kendallville in The United States of America A01_REND9327_12_SE_FM.indd 11/02/14 8:19 PM About the Authors Barry Render is Professor Emeritus, the Charles Harwood Distinguished Professor of Operations Management, Crummer Graduate School of Business, Rollins College, Winter Park, Florida He received his B.S in Mathematics and Physics at Roosevelt University and his M.S in Operations Research and his Ph.D in Quantitative Analysis at the University of Cincinnati He previously taught at George Washington University, the University of New Orleans, Boston University, and George Mason University, where he held the Mason Foundation Professorship in Decision Sciences and was Chair of the Decision Science Department Dr Render has also worked in the aerospace industry for General Electric, McDonnell Douglas, and NASA Dr Render has coauthored 10 textbooks published by Pearson, including Managerial Decision Modeling with Spreadsheets, Operations Management, Principles of Operations Management, Service Management, Introduction to Management Science, and Cases and Readings in Management Science More than 100 articles of Dr Render on a variety of management topics have appeared in Decision Sciences, Production and Operations Management, Interfaces, Information and Management, Journal of Management Information Systems, Socio-Economic Planning Sciences, IIE Solutions, and Operations Management Review, among others Dr Render has been honored as an AACSB Fellow and was named twice as a Senior Fulbright Scholar He was Vice President of the Decision Science Institute Southeast Region and served as software review editor for Decision Line for six years and as Editor of the New York Times Operations Management special issues for five years From 1984 to 1993, Dr Render was President of Management Service Associates of Virginia, Inc., whose technology clients included the FBI, the U.S Navy, Fairfax County, Virginia, and C&P Telephone He is currently Consulting Editor to Financial Times Press Dr Render has taught operations management courses at Rollins College for MBA and Executive MBA programs He has received that school’s Welsh Award as leading professor and was selected by Roosevelt University as the 1996 recipient of the St Claire Drake Award for Outstanding Scholarship In 2005, Dr Render received the Rollins College MBA Student Award for Best Overall Course, and in 2009 was named Professor of the Year by full-time MBA students Ralph Stair is Professor Emeritus at Florida State University He earned a B.S in chemical engineering from Purdue University and an M.B.A from Tulane University Under the guidance of Ken Ramsing and Alan Eliason, he received a Ph.D in operations management from the University of Oregon He has taught at the University of Oregon, the University of Washington, the University of New Orleans, and Florida State University He has taught twice in Florida State University’s Study Abroad Program in London Over the years, his teaching has been concentrated in the areas of information systems, operations research, and operations management Dr Stair is a member of several academic organizations, including the Decision Sciences Institute and INFORMS, and he regularly participates in national meetings He has published numerous articles and books, including Managerial Decision Modeling with Spreadsheets, Introduction to Management Science, Cases and Readings in Management Science, Production and Operations Management: A Self-Correction Approach, Fundamentals of Information Systems, Principles of Information Systems, Introduction to Information Systems, Computers in Today’s World, Principles A01_REND9327_12_SE_FM.indd 11/02/14 8:19 PM 4   About the Authors of Data Processing, Learning to Live with Computers, Programming in BASIC, Essentials of BASIC Programming, Essentials of FORTRAN Programming, and Essentials of COBOL Programming Dr Stair divides his time between Florida and Colorado He enjoys skiing, biking, kayaking, and other outdoor activities Michael E Hanna is Professor of Decision Sciences at the University of Houston–Clear Lake (UHCL) He holds a B.A in Economics, an M.S in Mathematics, and a Ph.D in Operations Research from Texas Tech University For more than 25 years, he has been teaching courses in statistics, management science, forecasting, and other quantitative methods His dedication to teaching has been recognized with the Beta Alpha Psi teaching award in 1995 and the Outstanding Educator Award in 2006 from the Southwest Decision Sciences Institute (SWDSI) Dr Hanna has authored textbooks in management science and quantitative methods, has published numerous articles and professional papers, and has served on the Editorial Advisory Board of Computers and Operations Research In 1996, the UHCL Chapter of Beta Gamma Sigma presented him with the Outstanding Scholar Award Dr Hanna is very active in the Decision Sciences Institute, having served on the Innovative Education Committee, the Regional Advisory Committee, and the Nominating Committee He has served on the board of directors of the Decision Sciences Institute (DSI) for two terms and also as regionally elected vice president of DSI For SWDSI, he has held several positions, including ­president, and he received the SWDSI Distinguished Service Award in 1997 For overall service to the profession and to the university, he received the UHCL President’s Distinguished Service Award in 2001 Trevor S Hale is Associate Professor of Management Science at the University of Houston– Downtown (UHD) He received a B.S in Industrial Engineering from Penn State University, an M.S in Engineering Management from Northeastern University, and a Ph.D in Operations Research from Texas A&M University He was previously on the faculty of both Ohio University–Athens, and Colorado State University–Pueblo Dr Hale was honored three times as an Office of Naval Research Senior Faculty Fellow He spent the summers of 2009, 2011, and 2013 performing energy security/cyber security research for the U.S Navy at Naval Base Ventura County in Port Hueneme, California Dr Hale has published dozens of articles in the areas of operations research and quantitative analysis in journals such as the International Journal of Production Research, the European Journal of Operational Research, Annals of Operations Research, the Journal of the Operational Research Society, and the International Journal of Physical Distribution and Logistics Management among several others He teaches quantitative analysis courses in the University of Houston–Downtown MBA program and Masters of Security Management for Executives program He is a senior member of both the Decision Sciences Institute and INFORMS A01_REND9327_12_SE_FM.indd 11/02/14 8:19 PM Brief Contents Chapter Introduction to Quantitative Analysis 19 Chapter 13 Simulation Modeling 487 Chapter Probability Concepts and Applications 41 Chapter 14 Markov Analysis 527 Decision Analysis 83 Chapter 15 Chapter Statistical Quality Control 555 Chapter Regression Models 131 Appendices 575 Chapter Forecasting 167 Chapter Inventory Control Models 205 Chapter Linear Programming Models: Graphical and Computer Methods 257 Chapter Linear Programming Applications 309 Chapter Transportation, Assignment, and Network Models 341 Chapter 10 Integer Programming, Goal Programming, and Nonlinear Programming 381 Chapter 11 Project Management 413 Chapter 12 Waiting Lines and Queuing Theory Models 453 Online Modules Analytic Hierarchy Process  M1-1 Dynamic Programming  M2-1 Decision Theory and the Normal Distribution M3-1 Game Theory  M4-1 Mathematical Tools: Determinants and Matrices  M5-1 Calculus-Based Optimization  M6-1 Linear Programming: The Simplex Method M7-1 Transportation, Assignment, and Network Algorithms  M8-1 A01_REND9327_12_SE_FM.indd 11/02/14 8:19 PM Contents Preface  13 Introduction to Quantitative Analysis 19 1.1 Introduction 20 1.2 What Is Quantitative Analysis? 20 1.3 Business Analytics 21 1.4 The Quantitative Analysis Approach 22 Chapter 1.5 1.6 1.7 1.8 Defining the Problem 22 Developing a Model 22 Acquiring Input Data 23 Developing a Solution 23 Testing the Solution 24 Analyzing the Results and Sensitivity Analysis 24 Implementing the Results 24 The Quantitative Analysis Approach and Modeling in the Real World 26 Chapter Probability Concepts and Applications 41 2.1 Introduction 42 2.2 Fundamental Concepts 42 2.3 2.4 2.5 2.6 How to Develop a Quantitative Analysis Model 26 The Advantages of Mathematical Modeling 27 Mathematical Models Categorized by Risk 27 The Role of Computers and Spreadsheet Models in the Quantitative Analysis Approach 28 Possible Problems in the Quantitative Analysis Approach 31 Defining the Problem 31 Developing a Model 32 Acquiring Input Data 33 Developing a Solution 33 Testing the Solution 34 Analyzing the Results 34 Implementation—Not Just the Final Step 35 Lack of Commitment and Resistance to Change 35 Lack of Commitment by Quantitative Analysts 35 Summary 35  Glossary 36  Key Equations 36  Self-Test 36  Discussion Questions and Problems 37  Case Study: Food and Beverages at Southwestern University Football Games 39 Bibliography 39 2.7 2.8 Two Basic Rules of Probability 42 Types of Probability 43 Mutually Exclusive and Collectively Exhaustive Events 44 Unions and Intersections of Events 45 Probability Rules for Unions, Intersections, and Conditional Probabilities 46 Revising Probabilities with Bayes’ Theorem 47 General Form of Bayes’ Theorem 49 Further Probability Revisions 49 Random Variables 50 Probability Distributions 52 Probability Distribution of a Discrete Random Variable 52 Expected Value of a Discrete Probability Distribution 52 Variance of a Discrete Probability Distribution 53 Probability Distribution of a Continuous Random Variable 54 The Binomial Distribution 55 Solving Problems with the Binomial Formula 56 Solving Problems with Binomial Tables 57 The Normal Distribution 58 Area Under the Normal Curve 60 Using the Standard Normal Table 60 Haynes Construction Company Example 61 The Empirical Rule 64 2.9 The F Distribution 64 2.10 The Exponential Distribution 66 2.11 Appendix 2.1: Arnold’s Muffler Example 67 The Poisson Distribution 68 Summary 70  Glossary 70 Key Equations 71  Solved Problems 72 Self-Test 74 Discussion Questions and Problems 75 Case Study: WTVX 81 Bibliography 81 Derivation of Bayes’ Theorem  81 A01_REND9327_12_SE_FM.indd 11/02/14 8:19 PM Contents    7  Chapter Decision Analysis 83 3.1 Introduction 84 3.2 The Six Steps in Decision Making 84 3.3 Types of Decision-Making Environments 85 3.4 Decision Making Under Uncertainty 86 3.5 3.6 3.7 3.8 3.9 3.10 Optimistic 86 Pessimistic 87 Criterion of Realism (Hurwicz Criterion) 87 Equally Likely (Laplace) 88 Minimax Regret 88 4.6 4.7 4.11 4.12 Evaluating the Multiple Regression Model 147 Jenny Wilson Realty Example 148 Binary or Dummy Variables 149 Model Building 150 Stepwise Regression 151 Multicollinearity 151 Nonlinear Regression 151 Cautions and Pitfalls in Regression Analysis 154 Summary 155  Glossary 155  Key Equations 156  Solved Problems 157  Self-Test 159  Discussion Questions and Problems 159  Case Study: North–South Airline 164 Bibliography 165 Formulas for Regression Calculations  165 Appendix 4.1: QM for Windows 95 Excel QM 96 Chapter 5 Forecasting 167 5.1 Introduction 168 5.2 Types of Forecasting Models 168 A Minimization Example 93 Using Software for Payoff Table Problems 95 Decision Trees 97 Efficiency of Sample Information 102 Sensitivity Analysis 102 How Probability Values Are Estimated by Bayesian Analysis 103 Calculating Revised Probabilities 103 Potential Problem in Using Survey Results 105 5.3 5.4 5.5 Utility Theory 106 5.6 5.7 5.8 Coefficient of Determination 136 Correlation Coefficient 136 Assumptions of the Regression Model 138 Estimating the Variance 139 Testing the Model for Significance 139 Triple A Construction Example 141 The Analysis of Variance (ANOVA) Table 141 Triple A Construction ANOVA Example 142 Using Computer Software for Regression 142 Excel 2013 142 Excel QM 143 QM for Windows 145 A01_REND9327_12_SE_FM.indd 4.9 4.10 Multiple Regression Analysis 146 Expected Monetary Value 89 Expected Value of Perfect Information 90 Expected Opportunity Loss 92 Sensitivity Analysis 92 Chapter Regression Models 131 4.1 Introduction 132 4.2 Scatter Diagrams 132 4.3 Simple Linear Regression 133 4.4 Measuring the Fit of the Regression Model 135 4.5 4.8 Decision Making Under Risk 89 Measuring Utility and Constructing a Utility Curve 107 Utility as a Decision-Making Criterion 110 Summary 112  Glossary 112  Key Equations 113  Solved Problems 113  Self-Test 118  Discussion Questions and Problems 119  Case Study: Starting Right Corporation 127  Case Study: Blake Electronics 128 Bibliography 130 5.9 Qualitative Models 168 Causal Models 169 Time-Series Models 169 Components of a Time-Series 169 Measures of Forecast Accuracy 171 Forecasting Models—Random Variations Only 174 Moving Averages 174 Weighted Moving Averages 174 Exponential Smoothing 176 Using Software for Forecasting Time Series 178 Forecasting Models—Trend and Random Variations 181 Exponential Smoothing with Trend 181 Trend Projections 183 Adjusting for Seasonal Variations 185 Seasonal Indices 186 Calculating Seasonal Indices with No Trend 186 Calculating Seasonal Indices with Trend 187 Forecasting Models—Trend, Seasonal, and Random Variations 188 The Decomposition Method 188 Software for Decomposition 191 Using Regression with Trend and Seasonal Components 192 Monitoring and Controlling Forecasts 193 Adaptive Smoothing 195 Summary 195  Glossary 196  Key Equations 196  Solved Problems 197  Self-Test 198  Discussion Questions and Problems 199  Case Study: Forecasting Attendance at SWU Football Games 202  Case Study: Forecasting Monthly Sales 203 Bibliography 204 11/02/14 8:19 PM 8   Contents Chapter Inventory Control Models 205 6.1 Introduction 206 6.2 Importance of Inventory Control 207 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 Decoupling Function 207 Storing Resources 207 Irregular Supply and Demand 207 Quantity Discounts 207 Avoiding Stockouts and Shortages 207 7.4 Inventory Decisions 208 Economic Order Quantity: Determining How Much to Order 209 Inventory Costs in the EOQ Situation 210 Finding the EOQ 212 Sumco Pump Company Example 212 Purchase Cost of Inventory Items 213 Sensitivity Analysis with the EOQ Model 214 Reorder Point: Determining When to Order 215 EOQ Without the Instantaneous Receipt Assumption 216 Annual Carrying Cost for Production Run Model 217 Annual Setup Cost or Annual Ordering Cost 217 Determining the Optimal Production Quantity 218 Brown Manufacturing Example 218 Quantity Discount Models 220 7.5 7.6 7.7 7.8 Use of Safety Stock 224 Single-Period Inventory Models 229 Marginal Analysis with Discrete Distributions 230 Café du Donut Example 231 Marginal Analysis with the Normal Distribution 232 Newspaper Example 232 ABC Analysis 234 Dependent Demand: The Case for Material Requirements Planning 234 Material Structure Tree 235 Gross and Net Material Requirements Plan 236 Two or More End Products 237 Just-In-Time Inventory Control 239 Enterprise Resource Planning 240 Inventory Control with QM for Windows  255 Flair Furniture Company 259 Graphical Solution to an LP Problem 261 Graphical Representation of Constraints 261 Isoprofit Line Solution Method 265 Corner Point Solution Method 268 Slack and Surplus 270 Solving Flair Furniture’s LP Problem Using QM for Windows, Excel 2013, and Excel QM 271 Using QM for Windows 271 Using Excel’s Solver Command to Solve LP Problems 272 Using Excel QM 275 Solving Minimization Problems 277 Holiday Meal Turkey Ranch 277 Four Special Cases in LP 281 No Feasible Solution 281 Unboundedness 281 Redundancy 282 Alternate Optimal Solutions 283 Sensitivity Analysis 284 High Note Sound Company 285 Changes in the Objective Function Coefficient 286 QM for Windows and Changes in Objective Function Coefficients 286 Excel Solver and Changes in Objective Function Coefficients 287 Changes in the Technological Coefficients 288 Changes in the Resources or Right-Hand-Side Values 289 QM for Windows and Changes in Right-HandSide Values 290 Excel Solver and Changes in Right-Hand-Side Values 290 Summary 292  Glossary 292  Solved Problems 293 Self-Test 297  Discussion Questions and Problems 298  Case Study: Mexicana Wire Works 306  Bibliography 308 Brass Department Store Example 222 Summary 241  Glossary 241  Key Equations 242  Solved Problems 243  Self-Test 245  Discussion Questions and Problems 246  Case Study: Martin-Pullin Bicycle Corporation 253 Bibliography 254 Appendix 6.1: Linear Programming Models: Graphical and Computer Methods 257 7.1 Introduction 258 7.2 Requirements of a Linear Programming Problem 258 7.3 Formulating LP Problems 259 Chapter Chapter Linear Programming Applications 309 8.1 Introduction 310 8.2 Marketing Applications 310 8.3 Media Selection 310 Marketing Research 311 Manufacturing Applications 314 Production Mix 314 Production Scheduling 315 A01_REND9327_12_SE_FM.indd 11/02/14 8:19 PM Contents    9  8.4 8.5 8.6 8.7 Employee Scheduling Applications 319 Labor Planning 319 Financial Applications 321 Portfolio Selection 321 Truck Loading Problem 324 Ingredient Blending Applications 326 Other Linear Programming Applications 329 Transportation, Assignment, and Network Models 341 9.1 Introduction 342 9.2 The Transportation Problem 343 Chapter 9.3 9.4 9.5 9.6 9.7 The Assignment Problem 348 Linear Program for Assignment Example 348 The Transshipment Problem 350 Linear Program for Transshipment Example 350 Capital Budgeting Example 388 A01_REND9327_12_SE_FM.indd Nonlinear Programming 397 General Foundry Example of PERT/CPM 415 Drawing the PERT/CPM Network 417 Activity Times 417 How to Find the Critical Path 418 Probability of Project Completion 423 What PERT Was Able to Provide 424 Using Excel QM for the General Foundry Example 424 Sensitivity Analysis and Project Management 425 11.3 PERT/Cost 427 Using QM for Windows  378 Modeling with 0–1 (Binary) Variables 388 Example of Goal Programming: Harrison Electric Company Revisited 394 Extension to Equally Important Multiple Goals 395 Ranking Goals with Priority Levels 395 Goal Programming with Weighted Goals 396 Chapter 11 Project Management 413 11.1 Introduction 414 11.2 PERT/CPM 415 Shortest-Route Problem 355 Minimal-Spanning Tree Problem 356 Harrison Electric Company Example of Integer Programming 382 Using Software to Solve the Harrison Integer Programming Problem 384 Mixed-Integer Programming Problem Example 386 Goal Programming 392 Nonlinear Objective Function and Linear Constraints 398 Both Nonlinear Objective Function and Nonlinear Constraints 398 Linear Objective Function with Nonlinear Constraints 400 Summary 400  Glossary 401  Solved Problems 401 Self-Test 404  Discussion Questions and Problems 405  Case Study: Schank Marketing Research 410  Case Study: Oakton River Bridge 411 Bibliography 412 Example 353 Integer Programming, Goal Programming, and Nonlinear Programming 381 10.1 Introduction 382 10.2 Integer Programming 382 10.3 10.5 Maximal-Flow Problem 353 Chapter 10 Linear Program for the Transportation Example 343 Solving Transportation Problems Using Computer Software 343 A General LP Model for Transportation Problems 344 Facility Location Analysis 345 Summary 360  Glossary 361  Solved Problems 361 Self-Test 363  Discussion Questions and Problems 364  Case Study: Andrew–Carter, Inc. 375  Case Study: Northeastern Airlines 376  Case Study: Southwestern University Traffic Problems 377 Bibliography 378 Appendix 9.1: 10.4 Diet Problems 326 Ingredient Mix and Blending Problems 327 Summary 331  Self-Test 331  Problems 332  Case Study: Cable & Moore 339 Bibliography 340 Limiting the Number of Alternatives Selected 390 Dependent Selections 390 Fixed-Charge Problem Example 390 Financial Investment Example 392 11.4 11.5 Planning and Scheduling Project Costs: Budgeting Process 427 Monitoring and Controlling Project Costs 430 Project Crashing 432 General Foundary Example 433 Project Crashing with Linear Programming 434 Other Topics in Project Management 437 Subprojects 437 Milestones 437 Resource Leveling 437 Software 437 Summary 437  Glossary 438  Key Equations 438  Solved Problems 439  11/02/14 8:19 PM www.downloadslide.net M8-28   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms From To Path: plant to project A = +9 - +6 + +8 - +5 + +4 - +10 = +0 (closed path: 3A to 3C to 2C to 2B to 1B to 1A) Project A Project B 10 PLANT 40 PLANT Project C  70 PLANT + 30 From To 40 - 30  30 60 150 50 Path: plant to project B = +7 - +6 + +8 - +5 = + +4 (closed path: 3B to 3C to 2C to 2B) Project A Project B 10 PLANT 40 PLANT Project C 11 30 12 PLANT 40 Plant ­Capacities  70 20 - PROJECT ­REQUIREMENTS  50 + Path: plant to project A 20 11 30 + 12 PROJECT ­REQUIREMENTS Plant ­Capacities + 30  50 Path: plant to project B + - 30  30 50 60 150 Since all indices are greater than or equal to zero (all are positive or zero), this initial solution provides the optimal transportation schedule, namely, 40 units from to A, 30 units from to B, 20 units from to B, 30 units from to C, and 30 units from to C Had we found a path that allowed improvement, we would move all units possible to that cell and then check every empty cell again Because the plant to project A improvement index was equal to zero, we note that multiple optimal solutions exist Solved Problem M8-2  The initial solution found in Solved Problem M8-1 was optimal, but the improvement index for one of the empty cells was zero, indicating another optimal solution Use a stepping-stone path to develop this other optimal solution Z03_REND9327_12_SE_M08.indd 28 01/03/14 5:33 PM www.downloadslide.net Solved Problems   M8-29 Solution Using the Plant to Project A stepping-stone path, we see that the lowest number of units in a cell where a subtraction is to be made is 20 units from plant to project B Therefore, 20 units will be subtracted from each cell with a minus sign and added to each cell with a plus sign The result is shown here: From To Project A Project B 10 PLANT Project C 20 Plant ­Capacities 11 50 PLANT  70 12 8 50 PLANT 20 PROJECT ­REQUIREMENTS  50 40 50 10  30 60 150 Solved Problem M8-3  Prentice Hall, Inc., a publisher headquartered in New Jersey, wants to assign three recently hired college graduates, Jones, Smith, and Wilson to regional sales districts in Omaha, Dallas, and Miami But the firm also has an opening in New York and would send one of the three there if it were more economical than a move to Omaha, Dallas, or Miami It will cost $1,000 to relocate Jones to New York, $800 to relocate Smith there, and $1,500 to move Wilson What is the optimal assignment of personnel to offices? Hiree Office Omaha Miami Dallas JONES $800 $1,100 $1,200 SMITH $500 $1,600 $1,300 WILSON $500 $1,000 $2,300 Solution a The cost table has a fourth column to represent New York To balance the problem, we add a dummy row (person) with a zero relocation cost to each city Hiree Office Omaha Miami Dallas New York JONES $800 $1,100 $1,200 $1,000 SMITH $500 $1,600 $1,300 $800 WILSON $500 $1,000 $2,300 $1,500 DUMMY 0 0 b Subtract smallest number in each row and cover zeros (column subtraction will give the same ­numbers and therefore is not necessary) Hiree Z03_REND9327_12_SE_M08.indd 29 Office Omaha Miami Dallas New York JONES 300 SMITH 1,100 800 300 WILSON 500 1,800 1,000 DUMMY 0 0 400 200 01/03/14 5:33 PM www.downloadslide.net M8-30   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms c Subtract smallest uncovered number (200) from each uncovered number, add it to each square where two lines intersect, and cover all zeros Office Hiree Omaha Miami JONES   0 100 SMITH   0 WILSON   0 DUMMY 200 Dallas New York 200 900 600 100 300 1,600 800   0 0 d Subtract smallest uncovered number (100) from each uncovered number, add it to each square where two lines intersect, and cover all zeros Office Hiree Omaha Miami Dallas New York JONES   0   0 100   0 SMITH   0 800 500 100 WILSON   0 200 1,500 800 DUMMY 300   0 100 e Subtract smallest uncovered number (100) from each uncovered number, add it to squares where two lines intersect, and cover all zeros Office Hiree Omaha Miami Dallas New York JONES 100   0 100   0 SMITH   0 700 400   0 WILSON   0 100 1,400 700 DUMMY 400   0 100 f Since it takes four lines to cover all zeros, an optimal assignment can be made at zero squares We assign Dummy (no one) to Dallas Wilson to Omaha Smith to New York Jones to Miami Cost = +0 + +500 + +800 + +1,100 = +2,400 Solved Problem M8-4  PetroChem, an oil refinery located on the Mississippi River south of Baton Rouge, Louisiana, is designing a new plant to produce diesel fuel Figure M8.11 shows the network of the main processing centers along with the existing rate of flow (in thousands of gallons of fuel) The management at PetroChem would like to determine the maximum amount of fuel that can flow through the plant, from node to node Z03_REND9327_12_SE_M08.indd 30 01/03/14 5:33 PM www.downloadslide.net Solved Problems   M8-31 Figure M8.11 4 0 1 1 3 6 3 Solution 00 4,0 3,00 1,0 00 3, 00 5,000 5,000 1,0 00 1,000 1,00 00 Figure M8.12 Using the maximal-flow technique, we arbitrarily choose path 1–5–7, which has a maximum flow of The capacities are then adjusted, leaving the capacities from to at 0, and the capacity from to also at The next path arbitrarily selected is 1–2–4–7, and the maximum flow is When capacities are adjusted, the capacity from to and the capacity from to are 1, and the capacity from to is The next path selected is 1–3–6–7, which has a maximum flow of 1, and the capacity from to is adjusted to The next path selected is 1–2–5–6–7, which has a maximum flow of After this, there are no more paths with any capacity Arc 5–7 has capacity of While arc 4–7 has a capacity of 1, both arc 2–4 and arc 5–4 have a capacity of 0, so no more flow is available through node Similarly, while arc 6–7 has a capacity of remaining, the capacity for arc 3–6 and the capacity for arc 5–6 are Thus, the maximum flow is 10 15 + + + 12 The flows are shown in Figure M8.12 2, Solved Problem M8-5  The network of Figure M8.13 shows the highways and cities surrounding Leadville, Colorado ­Leadville Tom, a bicycle helmet manufacturer, must transport his helmets to a distributor based in Dillon,­ Colorado To this, he must go through several cities Tom would like to find the shortest way to get from Leadville to Dillon What you recommend? Figure M8.13 Leadville Z03_REND9327_12_SE_M08.indd 31 12 16 18 14 10 12 16 Dillon 01/03/14 5:33 PM www.downloadslide.net M8-32   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms Solution The problem can be solved using the shortest-route technique The nearest node to the origin (node 1) is node Give this a distance of and put this in a box next to node Next, consider nodes 3, 4, and since there is an arc to each of these from either node or node 2, and both of these have their d­ istances established The nearest node to the origin is (coming through node 2), so the distance to put in the box next to node is 14 18 + 62 Then consider nodes 4, 5, and The node nearest the origin is node 4, which has a d­ istance of 18 (directly from node 1) Then consider nodes and The node with the least distance from the origin is node (coming through node 2), and this distance is 22 Next, consider nodes and 7, and node is selected since the distance is 26 (coming through node 3) Finally, node is considered, and the shortest distance from the origin is 32 (coming through node 6) The route that gives the shortest distance is 1–2–3–6–7, and the distance is 32 See Figure M8.14 for the solution Figure M8.14 8 16 18 Leadville 22 14 12 14 10 32 12 16 18 Dillon 26 Self-Test Before taking the self-test, refer to the learning objectives at the beginning of the module, the notes in the margins, and the glossary at the end of the module ● Use the key at the back of the book to correct your answers ● Restudy pages that correspond to any questions that you answered incorrectly or material you feel uncertain about ● If the total demand equals the total supply in a transportation problem, the problem is a degenerate b unbalanced c balanced d infeasible In a transportation problem, what indicates that the minimum cost solution has been found? a all improvement indices are negative or zero b all improvement indices are positive or zero c all improvement indices are equal to zero d all cells in the dummy row are empty An assignment problem may be viewed as a transportation problem with a a cost of $1 for all shipping routes b all supplies and demands equal to c only demand constraints d only supply constraints If the number of filled cells in a transportation table does not equal the number of rows plus the number of columns minus 1, then the problem is said to be a degenerate b unbalanced Z03_REND9327_12_SE_M08.indd 32 c optimal d maximization problem If a solution to a transportation problem is degenerate, then a a dummy row or column must be added b it will be impossible to evaluate all empty cells without removing the degeneracy c there will be more than one optimal solution d the problem has no feasible solution If the total demand is greater than the total capacity in a transportation problem, then a the optimal solution will be degenerate b a dummy source must be added c a dummy destination must be added d both a dummy source and a dummy destination must be added The Hungarian method is a used to solve assignment problems b a way to develop an initial solution to a transportation problem c also called Vogel’s approximation method d only used for problems in which the objective is to maximize profit 01/03/14 5:33 PM www.downloadslide.net  In an assignment problem, it may be necessary to add more than one row to the table a False b True When using the Hungarian method, an optimal assignment can always be made when every row and every column has at least one zero a False b True 10 The first step of the maximal-flow technique is to a select any node b pick any path from the start to the finish with some flow c pick the path with the maximum flow d pick the path with the minimal flow e pick a path where the flow going into each node is greater than the flow coming out of each node Discussion Questions and Problems   M8-33 11 In which technique you connect the nearest node to the existing solution that is not currently connected? a maximal tree b shortest route c maximal flow d minimal flow 12 In the shortest-route technique, the objective is to determine the route from an origin to a destination that passes through the fewest number of other nodes a True b False 13 Adjusting the flow capacity numbers on a path is an important step in which technique? a minimal flow b maximal flow c shortest route Discussion Questions and Problems Discussion Questions M8-1 What is a balanced transportation problem? Describe the approach you would use to solve an unbalanced problem M8-2 The stepping-stone method is being used to solve a transportation problem The smallest quantity in a cell with a minus sign is 35, but two different cells with minus signs have 35 units in them What problem will this cause, and how should this difficulty be resolved? M8-3 The stepping-stone method is being used to solve a transportation problem There is only one empty cell having a negative improvement index, and this index is -2 The stepping-stone path for this cell indicates that the smallest quantity for the cells with minus signs is 80 units If the total cost for the current solution is $900, what will the total cost be for the improved solution? What can you conclude about how much the total cost will decrease when developing each new solution for any transportation problem? M8-4 Explain what happens when the solution to a transportation problem does not have m + n - occupied squares (where m = number of rows in the table and n = number of columns in the table) M8-5 What is the enumeration approach to solving assignment problems? Is it a practical way to solve a row * column problem? a * problem? Why? M8-6 How could an assignment problem be solved using the transportation approach? What condition will make the solution of this problem difficult? M8-7 You are the plant supervisor and are responsible for scheduling workers to jobs on hand After estimating Z03_REND9327_12_SE_M08.indd 33 the cost of assigning each of five available workers in your plant to five projects that must be completed immediately, you solve the problem using the ­Hungarian method The following solution is reached and you post these job assignments: Jones to project A Smith to project B Thomas to project C Gibbs to project D Heldman to project E The optimal cost was found to be $492 for these assignments The plant general manager inspects your original cost estimates and informs you that increased employee benefits mean that each of the 25 numbers in your cost table is too low by $5 He ­suggests that you immediately rework the problem and post the new assignments Is this necessary? Why? What will the new optimal cost be? M8-8 Sue Simmons’s marketing research firm has local representatives in all but five states She decides to expand to cover the whole United States by transferring five experienced volunteers from their current locations to new offices in each of the five states Simmons’s goal is to relocate the five representatives at the least total cost Consequently, she sets up a * relocation cost table and prepares to solve it for the best assignments by use of the Hungarian method At the last moment, Simmons recalls that although the first four volunteers did not pose any objections to being placed in any of the five new cities, the fifth volunteer did make one restriction 01/03/14 5:33 PM www.downloadslide.net M8-34   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms M8-9 M8-10 M8-11 M8-12 That person absolutely refused to be assigned to the new office in Tallahassee, Florida—fear of southern roaches, the representative claimed! How should Sue alter the cost matrix to ensure that this assignment is not included in the optimal solution? Describe the steps of the maximal-flow technique What are the steps of the shortest-route technique? Describe a problem that can be solved by the ­shortest–route technique Is it possible to get alternate optimal solutions with the shortest-route technique? Is there an automatic way of knowing if you have an alternate optional solution? (a) Use the northwest corner rule to establish an initial feasible shipping schedule and calculate its cost (b) Use the stepping-stone method to test whether an improved solution is possible (c) Explain the meaning and implications of an improvement index that is equal to What decisions might management make with this information? Exactly how is the final solution affected? M8-14 The Hardrock Concrete Company has plants in three locations and is currently working on three major construction projects, each located at a different site The shipping cost per truckload of concrete, daily plant capacities, and daily project requirements are provided in the table below (a) Formulate an initial feasible solution to Hardrock’s transportation problem using the northwest corner rule Then evaluate each unused shipping route by computing all improvement indices Is this solution optimal? Why? (b) Is there more than one optimal solution to this problem? Why? Problems* M8-13 The management of the Executive Furniture Corporation decided to expand the production capacity at its Des Moines factory and to cut back production at its other factories It also recognizes a shifting market for its desks and revises the requirements at its three warehouses Data for Problem M8-13 New Warehouse Requirements New Factory Capacities Albuquerque (A) 200 desks Des Moines (D) 300 desks Boston (B) 200 desks Evansville (E) 150 desks Cleveland (C) 300 desks Fort Lauderdale (F) 250 desks Table for Problem M8-13 To From Albuquerque Boston Cleveland DES MOINES EVANSVILLE FORT LAUDERDALE Data for Problem M8-14 To From Project A Project B Project C Plant Capacities PLANT $10 $4 $11  70 PLANT 12  5  50 PLANT  7  30 40 50 60 150 PROJECT REQUIREMENTS *Note:  means the problem may be solved with QM for Windows; solved with QM for Windows and/or Excel QM Z03_REND9327_12_SE_M08.indd 34 means the problem may be solved with Excel QM; and means the problem may be 01/03/14 5:33 PM www.downloadslide.net Discussion Questions and Problems   M8-35  Table for Problem M8-16 To From Supply House Supply House Supply House Mill Capacity (Tons) Pineville $3 $3 $2 25 OAK RIDGE 40 MAPLETOWN 3 30 SUPPLY HOUSE ­DEMAND (TONS) 30 30 35 95 M8-15 Hardrock Concrete’s owner has decided to increase the capacity at his smallest plant (see Problem M8-14) Instead of producing 30 loads of concrete per day at plant 3, that plant’s capacity is doubled to 60 loads Find the new optimal solution using the northwest corner rule and stepping-stone method How has changing the third plant’s capacity altered the optimal shipping assignment? Discuss the concepts of degeneracy and multiple optimal solutions with regard to this problem M8-16 The Saussy Lumber Company ships pine flooring to three building supply houses from its mills in ­Pineville, Oak Ridge, and Mapletown Determine the best transportation schedule for the data given in the ­table above Use the northwest corner rule and the stepping-stone method M8-17 The Krampf Lines Railway Company specializes in coal handling On Friday, April 13, Krampf had empty cars at the following towns in the quantities indicated: Town Supply of Cars Morgantown 35 Youngstown 60 Pittsburgh 25 By Monday, April 16, the following towns will need coal cars as follows: Town Demand For Cars Coal Valley 30 Coaltown 45 Coal Junction 25 Coalsburg 20 Using a railway city-to-city distance chart, the dispatcher constructs a mileage table for the preceding towns The result is shown in the table below Minimizing total miles over which cars are moved to new locations, compute the best shipment of coal cars M8-18 An air conditioning manufacturer produces room air conditioners at plants in Houston, Phoenix, and Memphis These are sent to regional distributors in Dallas, Atlanta, and Denver The shipping costs vary, and the company would like to find the least-cost way to meet the demands at each of the distribution centers Dallas needs to receive 800 air conditioners per month, Atlanta needs 600, and Denver needs 200 Houston has 850 air conditioners available each month, Phoenix has 650, and Memphis has 300 The shipping cost per unit from Houston to Dallas is $8, to Atlanta is $12, and to Denver is $10 The cost per unit from Phoenix to Dallas is $10, to Atlanta is $14, and to Denver is $9 The cost per unit from Memphis to Dallas is $11, to Atlanta is $8, and to Denver is $12 How many units should be shipped from each Table for Problem M8-17 From Z03_REND9327_12_SE_M08.indd 35 To Coal Valley Coaltown Coal Junction Coalsburg MORGANTOWN  50 30 60 70 YOUNGSTOWN  20 80 10 90 PITTSBURGH 100 40 80 30 01/03/14 5:33 PM www.downloadslide.net M8-36   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms plant to each regional distribution center? What is the total cost for this? M8-19 Finnish Furniture manufactures tables in facilities ­located in three cities—Reno, Denver, and P ­ ittsburgh The tables are then shipped to three retail stores ­located in Phoenix, Cleveland, and Chicago Management wishes to develop a distribution schedule that will meet the demands at the lowest possible cost The shipping cost per unit from each of the sources to each of the destinations is shown in the following table: To From Phoenix Cleveland Chicago RENO 10 16 19 DENVER 12 14 13 PITTSBURGH 18 12 12 The available supplies are 120 units from Reno, 200 from Denver, and 160 from Pittsburgh Phoenix has a demand of 140 units, Cleveland has a demand of 160 units, and Chicago has a demand of 180 units How many units should be shipped from each manufacturing facility to each of the retail stores if cost is to be minimized? What is the total cost? M8-20 Finnish Furniture has experienced a decrease in the demand for tables in Chicago; the demand has fallen to 150 units (see Problem M8-19) What special condition would exist? What is the minimum-cost solution? Will there be any units remaining at any of the manufacturing facilities? M8-21 Consider the transportation table given below Find an initial solution using the northwest corner rule What special condition exists? Explain how you will proceed to solve the problem M8-22 The B Hall Real Estate Investment Corporation has identified four small apartment buildings in which it would like to invest Mrs Hall has approached three savings and loan companies regarding financing ­Because Hall has been a good client in the past and has maintained a high credit rating in the community, each savings and loan company is willing to consider providing all or part of the mortgage loan needed on each property Each loan officer has set differing interest rates on each property (rates are affected by the neighborhood of the apartment building, condition of the property, and desire by the individual savings and loan to finance various-size buildings), and each loan company has placed a maximum credit ceiling on how much it will lend Hall in total This information is summarized in the table on this page Each apartment building is equally attractive as an investment to Hall, so she has decided to purchase all buildings possible at the lowest total payment of interest From which savings and loan companies should she borrow to purchase which buildings? More than one savings and loan can finance the same property Table for Problem M8-21 To Destination A Destination B Destination C Source $8 $9 $4 SOURCE SOURCE SOURCE From DEMAND 110 34 Supply 72 38 46 19 31 175 Table for Problem M8-22 Property (Interest Rates) (%) Savings and Loan Company Hill St Banks St Park Ave Drury Lane Maximum Credit Line ($) FIRST HOMESTEAD  8 10 11  80,000 COMMONWEALTH 10 12 10 100,000 WASHINGTON FEDERAL 11 10  9 120,000 $60,000 $40,000 $130,000 $70,000 LOAN REQUIRED TO PURCHASE BUILDING Z03_REND9327_12_SE_M08.indd 36 01/03/14 5:33 PM www.downloadslide.net Discussion Questions and Problems   M8-37  M8-23 The J Mehta Company’s production manager is planning for a series of 1-month production periods for stainless steel sinks The demand for the next months is as follows: Month Demand for Stainless Steel Sinks 120 160 240 100 The Mehta firm can normally produce 100 stainless steel sinks in a month This is done during regular production hours at a cost of $100 per sink If demand in any month cannot be satisfied by regular production, the production manager has three other choices: (1) He can produce up to 50 more sinks per month in overtime but at a cost of $130 per sink; (2) he can purchase a limited number of sinks from a friendly competitor for resale (the maximum number of outside purchases over the 4-month period is 450 sinks, at a cost of $150 each); or (3) he can fill the demand from his on-hand inventory The inventory carrying cost is $10 per sink per month Back orders are not permitted Inventory on hand at the beginning of month is 40 sinks Set up this “production smoothing” problem as a transportation problem to minimize cost Use the northwest corner rule to find an initial level for production and outside purchases over the 4-month period M8-24 In a job shop operation, four jobs may be performed on any of four machines The hours required for each job on each machine are presented in the following table The plant supervisor would like to assign jobs so that total time is minimized Find the best solution Machine Job W X Y Z A12 10 14 16 13 A15 12 13 15 12 B2 12 12 11 B9 14 16 18 16 M8-25 Four automobiles have entered Bubba’s Repair Shop for various types of work, ranging from a transmission overhaul to a brake job The experience level of the mechanics is quite varied, and Bubba would like to minimize the time required to complete all of the jobs He has estimated the time in minutes for each mechanic to complete each job Billy can complete job in 400 minutes, job in 90 minutes, job in 60 minutes, and job in 120 minutes Taylor will Z03_REND9327_12_SE_M08.indd 37 finish job in 650 minutes, job in 120 minutes, job in 90 minutes, and job in 180 minutes Mark will finish job in 480 minutes, job in 120 minutes, job in 80 minutes, and job in 180 minutes John will complete job in 500 minutes, job in 110 minutes, job in 90 minutes, and job in 150 minutes Each mechanic should be assigned to just one of these jobs What is the minimum total time required to finish the four jobs? Who should be assigned to each job? M8-26 Baseball umpiring crews are currently in four cities where three-game series are beginning When these are finished, the crews are needed to work games in four different cities The distances (miles) from each of the cities where the crews are currently working to the cities where the new games will begin are shown in the following table: To From Seattle Arlington Oakland Baltimore Kansas City Chicago Detroit Toronto 1,500 1,730 1,940 2,070 460 810 1,020 1,270 1,500 1,850 2,080  X 960 610 400 330 The X indicates that the crew in Oakland cannot be sent to Toronto Determine which crew should be sent to each city to minimize the total distance traveled How many miles will be traveled if these assignments are made? To see how much better this solution is than the assignments that might have been made, find the assignments that would give the maximum distance traveled M8-27 Roscoe Davis, chairman of a college’s business department, has decided to apply a new method in assigning professors to courses next semester As a criterion for judging who should teach each course, Professor Davis reviews the past two years’ teaching evaluations (which were filled out by students) Since each of the four professors taught each of the four courses at one time or another during the two-year period, Davis is able to record a course rating for each instructor These ratings are shown in the table Find the best assignment of professors to courses to maximize the overall teaching rating Course Professor Statistics Management Finance Economics Anderson 90 65 95 40 Sweeney 70 60 80 75 Williams 85 40 80 60 McKinney 55 80 65 55 01/03/14 5:33 PM www.downloadslide.net M8-38   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms M8-28 The hospital administrator at St Charles General must appoint head nurses to four newly established departments: urology, cardiology, orthopedics, and obstetrics In anticipation of this staffing problem, she had hired four nurses: Hawkins, Condriac, ­Bardot, and Hoolihan Believing in the quantitative analysis approach to problem solving, the administrator has interviewed each nurse, considered his or her background, personality, and talents, and developed a cost scale ranging from to 100 to be used in the assignment A for Nurse Bardot being assigned to the cardiology unit implies that she would be perfectly suited to that task A value close to 100, on the other hand, would imply that she is not at all suited to head that unit The accompanying table gives the complete set of cost figures that the hospital administrator felt represented all possible assignments Which nurse should be assigned to which unit? Project Worker Adams $11 $14 $6 Brown 10 11 Cooper 12 10 13 Davis M8-30 The XYZ Corporation is expanding its market to include Texas Each salesperson is assigned to potential distributors in one of five different areas It is anticipated that the salesperson will spend about three to four weeks in the assigned area A statewide marketing campaign will begin once the product has been delivered to the distributors The five sales people who will be assigned to these areas (one person for each area) have rated the areas on the desirability of the assignment as shown in the table below The scale is (least desirable) to (most desirable) Which assignments should be made if the total of the ratings is to be maximized? Department Nurse Urology Cardiology Orthopedics Obstetrics Hawkins 28 18 15 75 Problems* Condriac 32 48 23 38 Bardot 51 36 24 36 Hoolihan 25 38 55 12 M8-31 Bechtold Construction is in the process of installing power lines to a large housing development Steve Bechtold wants to minimize the total length of wire used, which will minimize his costs The housing development is shown as a network in Figure M8.15 Each house has been numbered, and the distances between houses are given in hundreds of feet What you recommend? M8-29 The Fix-It Shop (see Section M8-4) has added a fourth repairman, Davis Solve the accompanying cost table for the new optimal assignment of workers to projects Why did this solution occur? Table for Problem M8-30 Austin/San Antonio Dallas/Ft Worth El Paso/West Texas Houston/ Galveston Corpus Christi/Rio Grande Valley Erica 3 Louis 4 2 Maria 3 Paul 4 Orlando 5 Figure M8.15  Network for Problem M8-31 3 2 4 14 5 12 13 3 10 *Note:  11 means the problem may be solved with QM for Windows Z03_REND9327_12_SE_M08.indd 38 01/03/14 5:33 PM www.downloadslide.net Discussion Questions and Problems   M8-39  Figure M8.16  Network for Problem M8-32 2 2 2 Figure M8.17  Network for Problem M8-33 100 10 50 100 120 13 40 New Office 60 50 40 2 70 10 100 13 200 20 100 100 Old Office 11 50 100 Figure M8.18  Network for Problem M8-35 12 65 56 26 75 41 50 48 37 M8-32 The city of New Berlin is considering making several of its streets one-way What is the maximum number of cars per hour that can travel from east to west? The network is shown in Figure M8.16 M8-33 Transworld Moving has been hired to move the office furniture and equipment of Cohen Properties to their new headquarters What route you recommend? The network of roads is shown in Figure M8.17 M8-34 Because of a sluggish economy, Bechtold Construction has been forced to modify its plans for the housing development in Problem M8-31 The result is that the path from node to now has a distance of What impact does this have on the total length of wire needed to install the power lines? Z03_REND9327_12_SE_M08.indd 39 100 10 53 23 M8-35 The director of security wants to connect security video cameras to the main control site from five potential trouble locations Ordinarily, cable would simply be run from each location to the main control site However, because the environment is potentially explosive, the cable must be run in a special conduit that is continually air purged This conduit is very expensive but large enough to handle five cables (the maximum that might be needed) Use the minimal-spanning tree technique to find a minimum distance route for the conduit between the locations noted in Figure M8.18 (Note that it makes no difference which one is the main control site.) M8-36 One of our best customers has had a major plant breakdown and wants us to make as many widgets 01/03/14 5:33 PM www.downloadslide.net M8-40   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms Figure M8.20  Network for Problem M8-37 Figure M8.19  Network for Problem M8-36 110 55 45 10 127 70 40 32 160 200 15 8 15 Figure M8.21  Network for Problem M8-39 4 2 1 3 6 1 2 Z03_REND9327_12_SE_M08.indd 40 11 10 14 13 14 11 10 Figure M8.22  Network for Problem M8-41 M8-39 Solve the maximal-flow problem presented in the network of Figure M8.21 below The numbers in the network represent thousands of gallons per hour as they flow through a chemical processing plant M8-40 Two terminals in the chemical processing plant, represented by nodes and 7, require emergency repair (see Problem M8-39) No material can flow into or out of these nodes What impact does this have on the capacity of the network? M8-41 Solve the shortest-route problem presented in the network of Figure M8.22 below, going from node to node 16 All numbers represent kilometers between German towns near the Black Forest M8-42 Due to bad weather, the roads going through nodes and have been closed (see Problem M8-41) No traffic can get onto or off of these roads Describe the impact that this will have (if any) on the shortest route through this network M8-43 Grey Construction would like to determine the least expensive way of connecting houses it is building with cable TV It has identified 11 possible branches or routes that could be used to connect the houses 12 10 10 10 16 11 18 10 12 13 10 16 18 14 18 14 16 11 22 16 17 12 15 12 18 1 9 20 0 for him as possible during the next few days, until he gets the necessary repairs done With our generalpurpose equipment there are several ways to make widgets (ignoring costs) Any sequence of activities that takes one from node to node in Figure M8.19 will produce a widget How many widgets can we produce per day? Quantities given are number of widgets per day M8-37 The road system around the hotel complex on International Drive (node 1) to Disney World (node 11) in Orlando, Florida, is shown in the network of ­Figure M8.20 The numbers by the nodes represent the traffic flow in hundreds of cars per hour What is the maximum flow of cars from the hotel complex to Disney World? M8-38 A road construction project would increase the road capacity around the outside roads from International Drive to Disney World by 200 cars per hour (see Problem M8-20) The two paths affected would be 1–2–6–9–11 and 1–5–8–10–11 What impact would this have on the total flow of cars? Would the total flow of cars increase by 400 cars per hour? 25 15 20 15 12 15 01/03/14 5:33 PM www.downloadslide.net Discussion Questions and Problems   M8-41  The cost in hundreds of dollars and the branches are summarized in the following table (a) What is the least expensive way to run cable to the houses? Branch Start Node End Node Distance (In Hundreds of Miles) Branch 1 Start Node End Node Cost ($100s) Branch Branch Branch 1 Branch Branch Branch 5 Branch Branch 6 Branch Branch 5 Branch 7 Branch Branch Branch Branch Branch 7 Branch 10 Branch 8 Branch Branch 10 Branch 11 (b) After reviewing cable and installation costs, Grey Construction would like to alter the costs for installing cable TV between its houses The first branches need to be changed The changes are summarized in the following table What is the impact on total costs? Branch Start Node End Node Cost ($100s) Branch 1 Branch Branch Branch Branch Branch Branch 7 Branch 8 Branch Branch 10 Branch 11 M8-44 In going from Quincy to Old Bainbridge, there are 10 possible roads that George Olin can take Each road can be considered a branch in the shortest-route problem (a) Determine the best way to get from Quincy (node 1) to Old Bainbridge (node 8) that will minimize total distance traveled All distances are in hundreds of miles Z03_REND9327_12_SE_M08.indd 41 (b) George Olin made a mistake in estimating the distances from Quincy to Old Bainbridge The new distances are in the following table What impact does this have on the shortest route from Quincy to Old Bainbridge? Branch Start Node End Node Distance (In Hundreds of Miles) Branch 1 Branch Branch Branch Branch 5 Branch 6 Branch 7 Branch Branch Branch 10 M8-45 South Side Oil and Gas, a new venture in Texas, has developed an oil pipeline network to transport oil from exploration fields to the refinery and other locations There are 10 pipelines (branches) in the network The oil flow in hundreds of gallons and the network of pipelines is given in the table on the following page (a) What is the maximum that can flow through the network? 01/03/14 5:33 PM www.downloadslide.net M8-42   MODULE 8  •  Transportation, Assignment, and ­Network Algorithms M8-46 Northwest University is in the process of completing a computer bus network that will connect computer facilities throughout the university The prime objective is to string a main cable from one end of the campus to the other (nodes 1–25) through underground conduits These conduits are shown in the network of Figure M8.23; the distance between them is in hundreds of feet Fortunately, these underground conduits have remaining capacity through which the bus cable can be placed (a) Given the network for this problem, how far (in hundreds of feet) is the shortest route from node to node 25? (b) In addition to the computer bus network, a new phone system is also being planned The phone system would use the same underground conduits If the phone system were installed, the following paths along the conduit would be at capacity and would not be available for the computer bus network: 6–11, 7–12, and 17–20 What changes (if any) would you have to make to the path used for the computer bus if the phone system were installed? (c) The university did decide to install the new phone system before the cable for the computer network Because of unexpected demand for computer networking facilities, an additional cable is needed for node to node 25 Unfortunately, the cable for the first or original network has completely used up the capacity along its path Given this situation, what is the best path for the second network cable? Start End Reverse Branch Node Node Capacity Capacity Flow 10 10 Branch 12 10 Branch 6 Branch 5 Branch 6 10 10 Branch 10 10 Branch 5 Branch 10 10 Branch 10 10 (b) South Side Oil and Gas needs to modify its pipeline network flow patterns The new data is in the following table What impact does this have on the maximum flow through the network? Start End Reverse Branch Node Node Capacity Capacity Flow Branch 1 10 10 Branch Branch 12 10 Branch 0 Branch 5 Branch 6 10 10 Branch 10 10 Branch 5 Branch 10 10 Branch 10 10 Figure M8.23  Network for Problem M8-46 15 6 10 15 14 11 12 5 15 6 16 13 17 18 19 20 17 22 15 10 1 10 23 10 24 25 20 Branch 20 Branch 10 21 Cases See Chapter for cases relevant to this module Bibliography See Chapter for references relevant to this module Z03_REND9327_12_SE_M08.indd 42 01/03/14 5:33 PM ... entitled Quantitative Analysis for Management, 12th edition, ISBN 978-0-13-350733-1, by Barry Render, Ralph M Stair, Jr., Michael E Hanna, and Trevor S Hale, published by Pearson Education ©... Operations Management, Principles of Operations Management, Service Management, Introduction to Management Science, and Cases and Readings in Management Science More than 100 articles of Dr Render. .. Render on a variety of management topics have appeared in Decision Sciences, Production and Operations Management, Interfaces, Information and Management, Journal of Management Information Systems,