Chapter 5 Integer Programming

Chapter 5 Integer Programming Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With Excel 2 1

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. 3 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, 200 sq. ft. floor space. Machine purchase prices and space requirements: Machine Press Lathe Required Floor Space (sq. ft.) 15 30 Purchase Price $8,000 4,000 4 2

A Total Integer Model (2 of 2) Integer Programming Model: Maximize Z = $100x 1 + $150x 2 8,000x 1 + 4,000x 2 $40,000 15x 1 + 30x 2 200 ft 2 x 1, x 2 0 and integer x 1 = number of presses x 2 = number of lathes 5 A 0-1 Integer Model (1 of 2) Recreation facilities selection to maximize daily usage by residents. Resource constraints: $120,000 budget; 12 acres of land. Selection constraint: either swimming pool or tennis center (not both). Data: Recreation Facility Expected Usage (people/day) Cost ($) Land Requirement (acres) Swimming pool Tennis Center Athletic field Gymnasium 300 90 400 150 35,000 10,000 25,000 90,000 4 2 7 3 6 3

A 0-1 Integer Model (2 of 2) Integer Programming Model: Maximize Z = 300x 1 + 90x 2 + 400x 3 + 150x 4 $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 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 7 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. 8 4

A Mixed Integer Model (2 of 2) Integer Programming Model: Maximize Z = $9,000x 1 + 1,500x 2 + 1,000x 3 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 x 1 = condominiums purchased x 2 = acres of land purchased x 3 = bonds purchased 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. 10 5

Integer Programming Example Graphical Solution of Maximization Model Maximize Z = $100x 1 + $150x 2 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 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 used in this book. 12 6

Computer Solution of IP Problems 0 1 Model with Excel (1 of 5) Recreational Facilities Example: Maximize Z = 300x 1 + 90x 2 + 400x 3 + 150x 4 $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 13 Computer Solution of IP Problems 0 1 Model with Excel (2 of 5) Exhibit 5.2 14 7

Computer Solution of IP Problems 0 1 Model with Excel (3 of 5) Exhibit 5.3 15 Computer Solution of IP Problems 0 1 Model with Excel (4 of 5) Exhibit 5.4 16 8

Computer Solution of IP Problems 0 1 Model with Excel (5 of 5) Exhibit 5.5 17 Computer Solution of IP Problems Total Integer Model with Excel (1 of 5) Integer Programming Model: Maximize Z = $100x 1 + $150x 2 8,000x 1 + 4,000x 2 $40,000 15x 1 + 30x 2 200 ft 2 x 1, x 2 0 and integer 18 9

Computer Solution of IP Problems Total Integer Model with Excel (2 of 5) Exhibit 5.8 19 Computer Solution of IP Problems Total Integer Model with Excel (3 of 5) Exhibit 5.10 20 10

Computer Solution of IP Problems Total Integer Model with Excel (4 of 5) Exhibit 5.9 21 Computer Solution of IP Problems Total Integer Model with Excel (5 of 5) Exhibit 5.11 22 11

Computer Solution of IP Problems Mixed Integer Model with Excel (1 of 3) Integer Programming Model: Maximize Z = $9,000x 1 + 1,500x 2 + 1,000x 3 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 23 Computer Solution of IP Problems Total Integer Model with Excel (2 of 3) Exhibit 5.12 24 12

Computer Solution of IP Problems Solution of Total Integer Model with Excel (3 of 3) Exhibit 5.13 25 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. Data: Project NPV Return ($1000) Project Costs per Year ($1000) 1 2 3 1. Website 2. Warehouse 3. Clothing department 4. Computer department 5. ATMs 120 85 105 140 75 55 45 60 50 30 40 35 25 35 30 25 20 -- 30 -- Available funds per year 150 110 60 26 13

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 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 27 0 1 Integer Programming Modeling Examples Capital Budgeting Example (3 of 4) Exhibit 5.16 28 14

0 1 Integer Programming Modeling Examples Capital Budgeting Example (4 of 4) Exhibit 5.17 29 0 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (1 of 4) Farms 1 2 3 4 5 6 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: Annual Fixed Costs ($1000) 405 390 450 368 520 465 Shipping Costs Projected Annual Harvest (tons, 1000s) 11.2 10.5 12.8 9.3 10.8 9.6 Plant A B C Farm 1 2 3 4 5 6 Available Capacity (tons,1000s) 12 10 14 Plant A B C 18 15 12 13 10 17 16 14 18 19 15 16 17 19 12 14 16 12 30 15

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 (tons, 1000s) shipped from farm i, i = 1,2,3,4,5,6 to plant j, j = A,B,C. Minimize Z = 18x 1A + 15x 1B + 12x 1C + 13x 2A + 10x 2B + 17x 2C + 16x 3A + 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 x 1A + x 1B + x 1B - 11.2y 1 0 x 2A + x 2B + x 2C -10.5y 2 0 x 3A + x 3A + x 3C - 12.8y 3 0 x 4A + x 4b + x 4C -9.3y 4 0 x 5A + x 5B + x 5B - 10.8y 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 1 31 0 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (3 of 4) Exhibit 5.18 32 16

0 1 Integer Programming Modeling Examples Fixed Charge and Facility Example (4 of 4) Exhibit 5.19 33 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 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 34 17

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 = 1if 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 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 35 0 1 Integer Programming Modeling Examples Set Covering Example (3 of 4) Exhibit 5.20 36 18

0 1 Integer Programming Modeling Examples Set Covering Example (4 of 4) Exhibit 5.21 37 19