1. 5-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Integer Programming Chapter 5

2. 5-2 Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With QM for Windows 0-1 Integer Programming Modeling Examples Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3. 5-3 Integer Programming Models Types of Models Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Total Integer Model:All decision variables required to have integer solution values. 0-1 Integer Model:All decision variables required to have integer values of zero or one. Mixed Integer Model:Some of the decision variables (but not all) required to have integer values.

4. 5-4 A Total Integer Model (1 of 2) ■Machine shop wants to expand its capacity by purchasing new presses and lathes. ■Marginal profitability: each press $100/day; each lathe $150/day. ■Resource constraints: $40,000 budget, 200 sq. ft. floor space. ■Machine purchase prices and space requirements: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5. 5-5 A Total Integer Model (2 of 2) x 1 = number of presses purchased x 2 = number of lathes purchased Integer Programming Model: Maximize Z = $100x 1 + $150x 2 subject to: $8,000x 1 + 4,000x 2  $40,000 15x 1 + 30x 2  200 ft 2 x 1, x 2  0 and integer Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

6. 5-6 ■Community council wants to decide which recreation facilities to construct so that daily usage by residents can be maximized. ■Resource constraints: $120,000 budget; 12 acres of land. ■Selection constraint: either swim. pool or tennis center can be cons.(not both). 0 - 1 Integer Model (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7. 5-7 x 1 = construction of a swimming pool (1 if constructed, 0 otherwise) x 2 = construction of a tennis center x 3 = construction of an athletic field x 4 = construction of a gymnasium 0-1 Integer Programming Model: Maximize Z = 300x 1 + 90x 2 + 400x 3 + 150x 4 subject to: $35,000x 1 + 10,000x 2 + 25,000x 3 + 90,000x 4  $120,000 4x 1 + 2x 2 + 7x 3 + 3x 4  12 acres x 1 + x 2  1 facility (either pool or tennis center noth both) x 1, x 2, x 3, x 4 = 0 or 1 0 - 1 Integer Model (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

8. 5-8 0 - 1 Integer Model to decide which recreation facility to construct What if we had the following statements: 1. Either swim. pool or tennis center must be built. x 1 + x 2 = 1 2. Exactly two of the recreation facilities must be built. x 1 + x 2 + x 3 + x 4 = 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

9. 5-9 0 - 1 Integer Model to decide which recreation facility to construct 3. No more than two facilities must be constructed. x 1 + x 2 + x 3 + x 4 <= 2 4. Tennis center can not be built unless the pool is constructed. x 2 <= x 1 5. If the pool is accepted, the tennis center must also be selected. x 2 = x 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

10. 5-10 A 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. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

11. 5-11 x 1 = # of condominiums purchased (integer) x 2 = acres of land purchased (continuous) x 3 = # of bonds purchased (integer) Mixed Integer Programming Model: Maximize Z = $9,000x 1 + 1,500x 2 + 1,000x 3 subject to: 50,000x 1 + 12,000x 2 + 8,000x 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 A Mixed Integer Model (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

12. 5-12 ■Rounding non-integer solution values to the nearest integer value can result in an infeasible solution. ■A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution. Integer Programming Graphical Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

13. 5-13 Integer Programming Example Graphical Solution of Machine Shop Model Maximize Z = $100x 1 + $150x 2 subject to: 8,000x 1 + 4,000x 2  $40,000 15x 1 + 30x 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

14. 5-14 ■Rounding non-integer solution values to the nearest integer value can result in an infeasible solution. x 1 = 2.22 presses  2 x 2 = 5.55 lathes  6 15x 1 + 30x 2  200 15(2) + 30(6) = 210  Violates constraint  Infeasible ■A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution. x 1 = 2.22 presses  2 x 2 = 5.55 lathes  5 Z = 950. But there is a better solution! (x 1 = 1 and x 2 = 6). Integer Programming Graphical Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

15. 5-15 Branch and Bound Method ■Traditional approach to solving integer programming problems.  Feasible solutions can be partitioned into smaller subsets  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 book’s companion website – “Integer Programming: the Branch and Bound Method” for detailed description of this method. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

16. 5-16 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

17. 5-17 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

18. 5-18 Computer Solution of IP Problems 0 – 1 Model with QM for Windows (1 of 3) Recreational Facilities Example: Maximize Z = 300x 1 + 90x 2 + 400x 3 + 150x 4 subject to: $35,000x 1 + 10,000x 2 + 25,000x 3 + 90,000x 4  $120,000 4x 1 + 2x 2 + 7x 3 + 3x 4  12 acres x 1 + x 2  1 facility x 1, x 2, x 3, x 4 = 0 or 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

19. 5-19 Exhibit 5.6 Computer Solution of IP Problems 0 – 1 Model with QM for Windows (2 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

20. 5-20 Exhibit 5.7 Computer Solution of IP Problems 0 – 1 Model with QM for Windows (3 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

21. 5-21 Exhibit 5.14 Computer Solution of IP Problems Mixed Integer Model with QM for Windows (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

22. 5-22 Exhibit 5.15 Computer Solution of IP Problems Mixed Integer Model with QM for Windows (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

23. 5-23 ■University bookstore expansion project. ■Not enough space available for both a computer department and a clothing department. 0 – 1 Integer Programming Modeling Examples Capital Budgeting Example (1 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

24. 5-24 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 x i = 1 if project “i” is selected, 0 if project “i” is not selected Maximize Z = $120x 1 + $85x 2 + $105x 3 + $140x 4 + $70x 5 subject to: 55x 1 + 45x 2 + 60x 3 + 50x 4 + 30x 5  150 40x 1 + 35x 2 + 25x 3 + 35x 4 + 30x 5  110 25x 1 + 20x 2 + 30x 4  60 x 3 + x 4  1 x i = 0 or 1 0 – 1 Integer Programming Modeling Examples Capital Budgeting Example (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

25. 5-25 Solution for Capital Budgeting Example X 1 = 1 (Web Site) X 2 = 1 (Computer Dept.) X 3 = 1 (ATMs) Z = $330000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

26. 5-26 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? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

27. 5-27 y i = 0 if farm i is not selected, and 1 if farm i is selected;i = 1,2,3,4,5,6 x ij = potatoes (1000 tons) shipped from farm I to plant j;j = A,B,C. Minimize Z =18x 1A + 15x 1B + 12x 1C + 13x 2A + 10x 2B + 17x 2C + 16x 3 + 14x 3B +18x 3C + 19x 4A + 15x 4b + 16x 4C + 17x 5A + 19x 5B +12x 5C + 14x 6A + 16x 6B + 12x 6C + 405y 1 + 390y 2 + 450y 3 + 368y 4 + 520y 5 + 465y 6 subject to: x 1A + x 1B + x 1B - 11.2y 1 ≤ 0x 2A + x 2B + x 2C -10.5y 2 ≤ 0 x 3A + x 3A + x 3C - 12.8y 3 ≤ 0x 4A + x 4b + x 4C - 9.3y 4 ≤ 0 x 5A + x 5B + x 5B - 10.8y 5 ≤ 0x 6A + x 6B + X 6C - 9.6y 6 ≤ 0 x 1A + x 2A + x 3A + x 4A + x 5A + x 6A = 12 x 1B + x 2B + x 3B + x 4B + x 5B + x 6B = 10 x 1C + x 2C + x 3C + x 4C + x 5C + x 6C = 14 x ij ≥ 0 y i = 0 or 1 0 – 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

28. 5-28 SOLUTION For Fixed Charge and Facility Example X 1C = 11200 tons y 1 =1 (farm 1) X 2A = 2400 tons y 2 =1 (farm 2) X 2B = 8100 tons y 4 =1 (farm 4) X 4B = 1900 tons y 6 =1 (farm 6) X 4C = 2800 tons X 6A = 9600 tons Z = $2 082 300 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

29. 5-29 Cities Cities within 300 miles 1. AtlantaAtlanta, Charlotte, Nashville 2. BostonBoston, New York 3. CharlotteAtlanta, Charlotte, Richmond 4. CincinnatiCincinnati, Detroit, Indianapolis, Nashville, Pittsburgh 5. DetroitCincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh 6. IndianapolisCincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St. Louis 7. MilwaukeeDetroit, Indianapolis, Milwaukee 8. NashvilleAtlanta, Cincinnati, Indianapolis, Nashville, St. Louis 9. New YorkBoston, New York, Richmond 10. PittsburghCincinnati, Detroit, Pittsburgh, Richmond 11. RichmondCharlotte, New York, Pittsburgh, Richmond 12. St. Louis Indianapolis, Nashville, St. Louis APS wants to construct the min. set of package distribution hubs in these 12 cities such that there is a hub within 300 miles of every city: 0 – 1 Integer Programming Modeling Examples Set Covering Example (1 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

30. 5-30 x i = city i, i = 1 to 12; x i = 0 if city is not selected as a hub and x i = 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:x 1 + x 3 + x 8  1 Boston:x 2 + x 10  1 Charlotte:x 1 + x 3 + x 11  1 Cincinnati:x 4 + x 5 + x 6 + x 8 + x 10  1 Detroit:x 4 + x 5 + x 6 + x 7 + x 10  1 Indianapolis: x 4 + x 5 + x 6 + x 7 + x 8 + x 12  1 Milwaukee:x 5 + x 6 + x 7  1 Nashville: x 1 + x 4 + x 6 + x 8 + x 12  1 New York:x 2 + x 9 + x 11  1 Pittsburgh:x 4 + x 5 + x 10 + x 11  1 Richmond: x 3 + x 9 + x 10 + x 11  1 St Louis: x 6 + x 8 + x 12  1 x ij = 0 or 1 0 – 1 Integer Programming Modeling Examples Set Covering Example (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

31. 5-31 SOLUTION FOR Set Covering Example X 2 = 1 (Boston) X 3 = 1 (Charlotte) X 5 = 1 (Detroit) X 12 =1 (St. Louis) Z = 4 Hubs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

32. 5-32 Total Integer Programming Modeling Example Problem Statement (1 of 3) ■Textbook company developing two new sales regions. ■The company has at most 10 sales people to transfer to the new regions. ■Average annual expenses for sales person: ▪Region 1 - $10,000/salesperson ▪Region 2 - $7,000/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 sales? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

33. 5-33 Step 1: Formulate the Integer Programming Model Maximize Z = $85,000x 1 + 60,000x 2 subject to: x 1 + x 2  10 salespeople $10,000x 1 + 7,000x 2  $72,000 expense budget x 1, x 2  0 or integer Step 2: Solve the Model using QM for Windows Total Integer Programming Modeling Example Model Formulation (2 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

34. 5-34 Total Integer Programming Modeling Example Solution with QM for Windows (3 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall