Integer Programming. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall

Transcription

1 Integer Programming Chapter 5 5-1

2 Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With Excel and QM for Windows 0-1 Integer Programming Modeling Examples 5-2

3 Integer Programming Models Types of Models 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. 5-3

4 A 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 budget, 200 sq. ft. floor space. Machine purchase prices and space requirements: Machine Press Lathe Required Floor Space (ft. 2 ) Purchase Price $8,000 4,

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

6 A 0-1 Integer Model (1 of 2) Recreation facilities selection to maximize daily usage by residents. Resource constraints: $120,000 budget; 12 acres of land. Selection constraint: either swimming pool or tennis center (not both). Recreation Facility Swimming pool Tennis Center Athletic field Gymnasium Expected Usage (people/day) Cost ($) ,000 10,000 25,000 90,000 Land Requirement (acres)

7 A 0-1 Integer Model (2 of 2) Integer Programming Model: Maximize Z = 300x x x x 4 subject to: $35,000x ,000x ,000x ,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 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 5-7

8 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. 5-8

9 A Mixed Integer Model (2 of 2) Integer Programming Model: Maximize Z = $9,000x 1 + 1,500x 2 + 1,000x 3 subject to: 50,000x ,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 x 1 = condominiums purchased x 2 = acres of land purchased x 3 = bonds purchased 5-9

10 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 non-integer solution values but may result in a less than optimal (sub-optimal) solution. 5-10

11 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 x ft 2 x 1, x 2 0 and integer Optimal Solution: Z = $1, x 1 = 2.22 presses x 2 = 5.55 lathes Figure 5.1 Feasible Solution Space with Integer Solution Points 5-11

12 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. 5-12

13 Computer Solution of IP Problems 0 1 Model with Excel (1 of 5) Recreational Facilities Example: Maximize Z = 300x x x x 4 subject to: $35,000x ,000x ,000x ,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

14 Computer Solution of IP Problems 0 1 Model with Excel (2 of 5) Exhibit

15 Computer Solution of IP Problems 0 1 Model with Excel (3 of 5) Exhibit

16 Computer Solution of IP Problems 0 1 Model with Excel (4 of 5) Exhibit

17 Computer Solution of IP Problems 0 1 Model with Excel (5 of 5) Exhibit

18 Computer Solution of IP Problems 0 1 Model with QM for Windows (1 of 3) Recreational Facilities Example: Maximize Z = 300x x x x 4 subject to: $35,000x ,000x ,000x ,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

19 Computer Solution of IP Problems 0 1 Model with QM for Windows (2 of 3) Exhibit

20 Computer Solution of IP Problems 0 1 Model with QM for Windows (3 of 3) Exhibit

21 Computer Solution of IP Problems Total Integer Model with Excel (1 of 5) Integer Programming Model of Machine Shop: Maximize Z = $100x 1 + $150x 2 subject to: 8,000x 1 + 4,000x 2 $40,000 15x x ft 2 x 1, x 2 0 and integer 5-21

22 Computer Solution of IP Problems Total Integer Model with Excel (2 of 5) Exhibit

23 Computer Solution of IP Problems Total Integer Model with Excel (4 of 5) Exhibit

24 Computer Solution of IP Problems Total Integer Model with Excel (3 of 5) Exhibit

25 Computer Solution of IP Problems Total Integer Model with Excel (5 of 5) Exhibit

26 Computer Solution of IP Problems Mixed Integer Model with Excel (1 of 3) Integer Programming Model for Investments Problem: Maximize Z = $9,000x 1 + 1,500x 2 + 1,000x 3 subject to: 50,000x ,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 5-26

27 Computer Solution of IP Problems Total Integer Model with Excel (2 of 3) Exhibit

28 Computer Solution of IP Problems Solution of Total Integer Model with Excel (3 of 3) Exhibit

29 Computer Solution of IP Problems Mixed Integer Model with QM for Windows (1 of 2) Exhibit

30 Computer Solution of IP Problems Mixed Integer Model with QM for Windows (2 of 2) Exhibit

31 0 1 Integer Programming Modeling Examples Capital Budgeting Example (1 of 4) University bookstore expansion project. Not enough space available for both a computer department and a clothing department. Project NPV Return ($1,000s) Project Costs per Year ($1000) Web site 2. Warehouse 3. Clothing department 4. Computer department 5. ATMs $ $ $ $ Available funds per year

32 0 1 Integer Programming Modeling Examples Capital Budgeting Example (2 of 4) 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 x x x x x x x x x x x x 4 60 x 3 + x 4 1 x i = 0 or

33 0 1 Integer Programming Modeling Examples Capital Budgeting Example (3 of 4) Exhibit

34 0 1 Integer Programming Modeling Examples Capital Budgeting Example (4 of 4) Exhibit

35 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? Farms Plant A B C Annual Fixed Costs ($1000) Available Capacity (tons,1000s) Projected Annual Harvest (tons, 1000s) Farm Plant ($/ton shipped) A B C

36 0 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (2 of 4) 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; Minimize Z = j = A,B,C. 18x 1A + 15x 1B + 12x 1C + 13x 2A + 10x 2B + 17x 2C + 16x x 3B +18x 3C + 19x 4A + 15x 4b + 16x 4C + 17x 5A + 19x 5B +12x 5C + 14x 6A + 16x 6B + 12x 6C + 405y y y y y y 6 subject to: x 1A + x 1B + x 1B y 1 0 x 2A + x 2B + x 2C -10.5y 2 0 x 3A + x 3A + x 3C y 3 0 x 4A + x 4b + x 4C - 9.3y 4 0 x 5A + x 5B + x 5B y 5 0 x 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

37 0 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (3 of 4) Exhibit

38 0 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (4 of 4) Exhibit

39 0 1 Integer Programming Modeling Examples Set Covering Example (1 of 4) APS wants to construct the minimum set of new hubs in these twelve cities such that there is a hub within 300 miles of every city: Cities Cities within 300 miles 1. Atlanta Atlanta, Charlotte, Nashville 2. Boston Boston, New York 3. Charlotte Atlanta, Charlotte, Richmond 4. Cincinnati Cincinnati, Detroit, Indianapolis, Nashville, Pittsburgh 5. Detroit Cincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh 6. Indianapolis Cincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St. Louis 7. Milwaukee Detroit, Indianapolis, Milwaukee 8. Nashville Atlanta, Cincinnati, Indianapolis, Nashville, St. Louis 9. New York Boston, New York, Richmond 10. Pittsburgh Cincinnati, Detroit, Pittsburgh, Richmond 11. Richmond Charlotte, New York, Pittsburgh, Richmond 12. St. Louis Indianapolis, Nashville, St. Louis 5-39

40 0 1 Integer Programming Modeling Examples Set Covering Example (2 of 4) 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

41 0 1 Integer Programming Modeling Examples Set Covering Example (3 of 4) Exhibit

42 0 1 Integer Programming Modeling Examples Set Covering Example (4 of 4) Exhibit

43 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? 5-43

44 Total Integer Programming Modeling Example Model Formulation (2 of 3) Step 1: Formulate the Integer Programming Model Maximize Z = $85,000x ,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 5-44

45 Total Integer Programming Modeling Example Solution with QM for Windows (3 of 3) 5-45