Introduction to Management Science 8 th Edition by

Introduction to Management Science 8 th Edition by Bernard W. Taylor III Chapter 5 Integer Programming Chapter 5 - Integer Programming 1

Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With Excel and QM for Windows Chapter 5 - Integer Programming 2

Integer Programming Models Types of Models Maximize Z = 10 x 1 + 15 x 2 subject to: 8 x 1 + 4 x 2 40 15 x 1 + 30 x 2 200 Total Integer Model: All decision variables required to have integer solution values (e. g. , x 1, x 2 0 and integer). 0 -1 Integer Model: All decision variables required to have integer values of zero or one (e. g. , x 1, x 2 = 0 or 1). Mixed Integer Model: Some of the decision variables (but not all) required to have integer values (e. g. , x 1 0 and integer, x 2 0 ). Chapter 5 - Integer Programming 3

Total Integer Model (1 of 2) Machine shop obtaining new presses and lathes. Marginal profitability: each press $100/day; each lathe $150/day. Resource constraints: $40, 000, 200 sq. ft. floor space. Machine purchase prices and space requirements: Chapter 5 - Integer Programming 4

Total Integer Model (2 of 2) Integer Programming Model: Define x 1 = number of presses x 2 = number of lathes Maximize Z = $100 x 1 + $150 x 2 subject to: 8, 000 x 1 + 4, 000 x 2 $40, 000 15 x 1 + 30 x 2 200 ft 2 x 1, x 2 0 and integer Chapter 5 - Integer Programming 5

0 - 1 Integer Model (1 of 2) Recreation facilities selection to maximize daily usage by residents. Resource constraints: $120, 000 budget; 12 acres of land. Selection constraint: either swimming pool or tennis center (not both). Data: Chapter 5 - Integer Programming 6

0 - 1 Integer Model (2 of 2) Integer Programming Model: Define: x 1 = construction of a swimming pool x 2 = construction of a tennis center x 3 = construction of an athletic field x 4 = construction of a gymnasium Maximize Z = 300 x 1 + 90 x 2 + 400 x 3 + 150 x subject to: $35, 000 x 1 + 10, 000 x 2 + 25, 000 x 3 + 90, 000 x 4 $120, 000 4 x 1 + 2 x 2 + 7 x 3 + 3 x 3 12 acres x 1 + x 2 1 facility x 1, x 2, x 3, x 4 = 0 or 1 Chapter 5 - Integer Programming 7

Mixed Integer Model (1 of 2) $250, 000 available for investments providing greatest return after one year. Data: Condominium cost $50, 000/unit, $9, 000 profit if sold after one year. Land cost $12, 000/ acre, $1, 500 profit if sold after one year. Municipal bond cost $8, 000/bond, $1, 000 profit if sold after one year. Only 4 condominiums, 15 acres of land, and 20 municipal bonds available. Chapter 5 - Integer Programming 8

Mixed Integer Model (2 of 2) Integer Programming Model: Maximize Z = $9, 000 x 1 + 1, 500 x 2 + 1, 000 x 3 subject to: 50, 000 x 1 + 12, 000 x 2 + 8, 000 x 3 $250, 000 x 1 4 condominiums x 2 15 acres x 3 20 bonds x 2 0 x 1, x 3 0 and integer x 1 = condominiums purchased x 2 = acres of land purchased x 3 = bonds purchased Chapter 5 - Integer Programming 9

Integer Programming Graphical Solution Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution A feasible solution is ensured by rounding down noninteger solution values but may result in a less than optimal (sub-optimal) solution. Chapter 5 - Integer Programming 10

Integer Programming Example Graphical Solution of Maximization Model Maximize Z = $100 x 1 + $150 x 2 subject to: 8, 000 x 1 + 4, 000 x 2 $40, 000 15 x 1 + 30 x 2 200 ft 2 x 1, x 2 0 and integer Optimal Solution: Z = $1, 055. 56 x 1 = 2. 22 presses x 2 = 5. 55 lathes Figure 5. 1 Feasible Solution Space with Integer Solution Points Chapter 5 - Integer Programming 11

Branch and Bound Method Traditional approach to solving integer programming problems. Based on principle that total set of feasible solutions can be partitioned into smaller subsets of solutions. Smaller subsets evaluated until best solution is found. Method is a tedious and complex mathematical process. Excel and QM for Windows used in this book. See CD-ROM Module C – “Integer Programming: the Branch and Bound Method” for detailed description of method. Chapter 5 - Integer Programming 12

Computer Solution of IP Problems 0 – 1 Model with Excel (1 of 5) Recreational Facilities Example: Maximize Z = 300 x 1 + 90 x 2 + 400 x 3 + 150 x 4 subject to: $35, 000 x 1 + 10, 000 x 2 + 25, 000 x 3 + 90, 000 x 4 $120, 000 4 x 1 + 2 x 2 + 7 x 3 + 3 x 3 12 acres x 1 + x 2 1 facility x 1, x 2, x 3, x 4 = 0 or 1 Chapter 5 - Integer Programming 13

Computer Solution of IP Problems 0 – 1 Model with Excel (2 of 5) Exhibit 5. 2 Chapter 5 - Integer Programming 14

Computer Solution of IP Problems 0 – 1 Model with Excel (4 of 5) Exhibit 5. 4 Chapter 5 - Integer Programming 15

Computer Solution of IP Problems 0 – 1 Model with Excel (3 of 5) Exhibit 5. 3 Chapter 5 - Integer Programming 16

Computer Solution of IP Problems 0 – 1 Model with Excel (5 of 5) Exhibit 5. 5 Chapter 5 - Integer Programming 17

Computer Solution of IP Problems 0 – 1 Model with QM for Windows (1 of 3) Recreational Facilities Example: Maximize Z = 300 x 1 + 90 x 2 + 400 x 3 + 150 x 4 subject to: $35, 000 x 1 + 10, 000 x 2 + 25, 000 x 3 + 90, 000 x 4 $120, 000 4 x 1 + 2 x 2 + 7 x 3 + 3 x 3 12 acres x 1 + x 2 1 facility x 1, x 2, x 3, x 4 = 0 or 1 Chapter 5 - Integer Programming 18

Computer Solution of IP Problems 0 – 1 Model with QM for Windows (2 of 3) Exhibit 5. 6 Chapter 5 - Integer Programming 19

Computer Solution of IP Problems 0 – 1 Model with QM for Windows (3 of 3) Exhibit 5. 7 Chapter 5 - Integer Programming 20

0 – 1 Integer Programming Modeling Examples Capital Budgeting Example (1 of 3) University bookstore expansion project. Not enough space available for both a computer department and a clothing department. Data: Chapter 5 - Integer Programming 21

0 – 1 Integer Programming Modeling Examples Capital Budgeting Example (2 of 3) x 1 = selection of web site project x 2 = selection of warehouse project x 3 = selection clothing department project x 4 = selection of computer department project x 5 = selection of ATM project xi = 1 if project “i” is selected, 0 if project “i” is not selected Maximize Z = $120 x 1 + $85 x 2 + $105 x 3 + $140 x 4 + $70 x 5 subject to: 55 x 1 + 45 x 2 + 60 x 3 + 50 x 4 + 30 x 5 150 40 x 1 + 35 x 2 + 25 x 3 + 35 x 4 + 30 x 5 110 25 x 1 + 20 x 2 + 30 x 4 60 x 3 + x 4 1 xi = 0 or 1 Chapter 5 - Integer Programming 22

0 – 1 Integer Programming Modeling Examples Capital Budgeting Example (2 of 3) What additional constraint/s are needed for the following: a) If the website is chosen, then the ATMs must be selected? b) The clothing department can be selected only if the computer department is chosen. c) If both the clothing and computer departments are chosen, then the warehouse must be built? Chapter 5 - Integer Programming 23

0 – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (1 of 4) Which of six farms should be purchased that will meet current production capacity at minimum total cost, including annual fixed costs and shipping costs? Data: Chapter 5 - Integer Programming 24

0 – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (2 of 4) yi = 0 if farm i is not selected, and 1 if farm i is selected, i = 1, 2, 3, 4, 5, 6 xij = potatoes (tons, 1000 s) shipped from farm i, i = 1, 2, 3, 4, 5, 6 to plant j, j = A, B, C. Minimize Z = 18 x 1 A + 15 x 1 B + 12 x 1 C + 13 x 2 A + 10 x 2 B + 17 x 2 C + 16 x 3 A + 14 x 3 B + 18 x 3 C + 19 x 4 A + 15 x 4 b + 16 x 4 C + 17 x 5 A + 19 x 5 B + 12 x 5 C + 14 x 6 A + 16 x 6 B + 12 x 6 C + 405 y 1 + 390 y 2 + 450 y 3 + 368 y 4 + 520 y 5 + 465 y 6 subject to: x 1 A + x 1 B - 11. 2 y 1 < 0 x 2 A + x 2 B + x 2 C -10. 5 y 2 < 0 x 3 A + x 3 C - 12. 8 y 3 < 0 x 4 A + x 4 b + x 4 C - 9. 3 y 4 < 0 x 5 A + x 5 B - 10. 8 y 5 < 0 x 6 A + x 6 B + X 6 C - 9. 6 y 6 < 0 x 1 A + x 2 A + x 3 A + x 4 A + x 5 A + x 6 A =12 x 1 B + x 2 B + x 3 A + x 4 b + x 5 B + x 6 B = 10 x 1 B + x 2 C + x 3 C+ x 4 C + x 5 B + x 6 C = 14 xij = 0 yi = 0 or 1 Chapter 5 - Integer Programming 25

0 – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (3 of 4) Exhibit 5. 18 Chapter 5 - Integer Programming 26

0 – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (4 of 4) Exhibit 5. 19 Chapter 5 - Integer Programming 27

0 – 1 Integer Programming Modeling Examples Set Covering Example (1 of 4) APS wants to construct the minimum set of new hubs in the following twelve cities such that there is a hub within 300 miles of every city: Cities 1. Atlanta 2. Boston 3. Charlotte 4. Cincinnati 5. Detroit 6. Indianapolis 7. Milwaukee 8. Nashville 9. New York 10. Pittsburgh 11. Richmond 12. St. Louis Cities within 300 miles Atlanta, Charlotte, Nashville Boston, New York Atlanta, Charlotte, Richmond Cincinnati, Detroit, Nashville, Pittsburgh Cincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St. Louis Detroit, Indianapolis, Milwaukee Atlanta, Cincinnati, Indianapolis, Nashville, St. Louis Boston, New York, Richmond Cincinnati, Detroit, Pittsburgh, Richmond Charlotte, New York, Pittsburgh, Richmond Indianapolis, Nashville, St. Louis Chapter 5 - Integer Programming 28

0 – 1 Integer Programming Modeling Examples Set Covering Example (2 of 4) xi = city i, i = 1 to 12, xi = 0 if city is not selected as a hub and xi = 1 if it is. Minimize Z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 + x 12 subject to: Atlanta: Boston: Charlotte: Cincinnati: Detroit: Indianapolis: Milwaukee: Nashville: New York: Pittsburgh: Richmond: St Louis: Chapter 5 - Integer Programming x 1 + x 3 + x 8 1 x 2 + x 10 1 x 1 + x 3 + x 11 1 x 4 + x 5 + x 8 + x 10 1 x 4 + x 5 + x 6 + x 7 + x 8 + x 12 1 x 5 + x 6 + x 7 1 x 1 + x 4 + x 6+ x 8 + x 12 1 x 2 + x 9+ x 11 1 x 4 + x 5 + x 10 + x 11 1 x 3 + x 9 + x 10 + x 11 1 x 6 + x 8 + x 12 1 xij = 0 or 1 29

0 – 1 Integer Programming Modeling Examples Set Covering Example (3 of 4) Exhibit 5. 20 Chapter 5 - Integer Programming 30

0 – 1 Integer Programming Modeling Examples Set Covering Example (4 of 4) Exhibit 5. 21 Chapter 5 - Integer Programming 31

Total Integer Programming Modeling Example Problem Statement (1 of 3) Textbook company developing two new regions. Planning to transfer some of its 10 salespeople into new regions. Average annual expenses for sales person: Region 1 - $10, 000/salesperson Region 2 - $7, 500/salesperson Total annual expense budget is $72, 000. Sales generated each year: Region 1 - $85, 000/salesperson Region 2 - $60, 000/salesperson How many salespeople should be transferred into each region in order to maximize increased sales? Chapter 5 - Integer Programming 32

Total Integer Programming Modeling Example Model Formulation (2 of 3) Step 1: Formulate the Integer Programming Model Maximize Z = $85, 000 x 1 + 60, 000 x 2 subject to: x 1 + x 2 10 salespeople $10, 000 x 1 + 7, 000 x 2 $72, 000 expense budget x 1, x 2 0 or integer Step 2: Solve the Model using QM for Windows Chapter 5 - Integer Programming 33

Total Integer Programming Modeling Example Solution with QM for Windows (3 of 3) Chapter 5 - Integer Programming 34