Lecture 07 MGT1107 Management Science Integer Programming PDF

MGT1107 MANAGEMENT SCIENCE USING BINARY INTEGER PROGRAMMING TO DEAL WITH YES OR NO DECISIONS / INTEGER PROGRAMMING  What is a Binary Variable? A binary variable is a variable with only two values. For example: 1 / 0. Yes / No. Success / Failure. Male / Female. Bl...

MGT1107 MANAGEMENT SCIENCE USING BINARY INTEGER PROGRAMMING TO DEAL WITH YES OR NO DECISIONS / INTEGER PROGRAMMING  What is a Binary Variable? A binary variable is a variable with only two values. For example: 1 / 0. Yes / No. Success / Failure. Male / Female. Black / White Binary Decision Variables are variables that have only two values, 0 and 1. They are widely used in integer programming models to represent yes-or-no decisions, where a value of 1 corresponds to a yes decision and a value of 0 corresponds to a no decision. Such models are called Binary Integer Programming (BIP) models.  Applications of Binary Variables Since binary variables only provide two choices, they are ideally suited to be the decision variables when dealing with yes-or-no decisions. Examples: Should we undertake a particular fixed project? Should we make a particular fixed investment? Should we locate a facility in a particular site? Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior 7.3 written consent of McGraw-Hill Education. California Manufacturing Company The California Manufacturing Company is a diversified company with several factories and warehouses throughout California, but none yet in Los Angeles or San Francisco. A basic issue is whether to build a new factory in Los Angeles or San Francisco, or perhaps even both. Management is also considering building at most one new warehouse, but will restrict the choice to a city where a new factory is being built. Question: Should the California Manufacturing Company expand with factories and/or warehouses in Los Angeles and/or San Francisco? Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior 7.9 written consent of McGraw-Hill Education.  Data for California Manufacturing Net Present Capital Decision Yes-or-No Decision Value Required Number Question Variable (Millions) (Millions) 1 Build a factory in Los Angeles? x1 $8 $6 2 Build a factory in San Francisco? x2 5 3 3 Build a warehouse in Los Angeles? x3 6 5 4 Build a warehouse in San Francisco? x4 4 2 Capital Available: $10 million Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 7.10  Binary Decision Variables Decision Decision Possible Interpretation Interpretation Number Variable Value of a Value of 1 of a Value of 0 Build a factory in Do not build 1 x1 0 or 1 Los Angeles this factory Build a factory in Do not build 2 x2 0 or 1 San Francisco this factory Build a warehouse in Do not build 3 x3 0 or 1 Los Angeles this warehouse Build a warehouse in Do not build 4 x4 0 or 1 San Francisco this warehouse Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 7.11  Algebraic Formulation Let x1 = 1 if build a factory in L.A.; 0 otherwise x2 = 1 if build a factory in S.F.; 0 otherwise x3 = 1 if build a warehouse in Los Angeles; 0 otherwise x4 = 1 if build a warehouse in San Francisco; 0 otherwise Maximize NPV = 8x1 + 5x2 + 6x3 + 4x4 ($millions) subject to Capital Spent: 6x1 + 3x2 + 5x3 + 2x4 ≤ 10 ($millions) Max 1 Warehouse: x3 + x4 ≤ 1 Warehouse only if Factory: x3 ≤ x1 x4 ≤ x2 and x1, x2, x3, x4 are binary variables. Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior 7.12 written consent of McGraw-Hill Education. Spreadsheet Model Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior 7.14 written consent of McGraw-Hill Education. Spreadsheet Model  Management’s Conclusion Management’s initial tentative decision had been to make $10 million of capital available. With this much capital, the best plan would be to build a factory in both Los Angeles and San Francisco, but no warehouses. An advantage of this plan is that it only uses $9 million of this capital, which frees up $1 million for other projects. A heavy penalty (a reduction of $4 million in total net present value) would be paid if the capital made available were to be reduced below $9 million. Increasing the capital made available by $1 million (to $11 million) would enable a substantial ($4 million) increase in the total net present value. Management decides to do this. With this much capital available, the best plan is to build a factory in both cities and a warehouse in San Francisco. Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior 7.16 written consent of McGraw-Hill Education.  Some Other Applications Investment Analysis Should we make a certain fixed investment? Examples: Turkish Petroleum Refineries (1990), South African National Defense Force (1997), Grantham, Mayo, Van Otterloo and Company (1999) Site Selection Should a certain site be selected for the location of a new facility? Example: AT&T (1990) Designing a Production and Distribution Network Should a certain plant remain open? Should a certain site be selected for a new plant? Should a distribution center remain open? Should a certain site be selected for a new distribution center? Should a certain distribution center be assigned to serve a certain market area? Examples: Ault Foods (1994), Digital Equipment Corporation (1995) Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior 7.17 written consent of McGraw-Hill Education.  Some Other Applications Dispatching Shipments Should a certain route be selected for a truck? Should a certain size truck be used? Should a certain time period for departure be used? Examples: Quality Stores (1987), Air Products and Chemicals, Inc. (1983), Reynolds Metals Co. (1991), Sears, Roebuck and Company (1999) Scheduling Interrelated Activities Should a certain activity begin in a certain time period? Examples: Texas Stadium (1983), China (1995) Scheduling Asset Divestitures Should a certain asset be sold in a certain time period? Example: Homart Development (1987) Airline Applications: Should a certain type of airplane be assigned to a certain flight leg? Should a certain sequence of flight legs be assigned to a crew? Examples: American Airlines (1989, 1991), Air New Zealand (2001) Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior 7.18 written consent of McGraw-Hill Education. PROBLEM ANSWER ANSWER ANSWER ANSWER PROBLEM PROBLEM PROBLEM ANSWER ANSWER ANSWER ANSWER PROBLEM ANSWER ANSWER ANSWER ANSWER PLEASE MAKE SURE TO READ THE NEXT TOPIC: NON-LINEAR PROGRAMMING