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Chapter 6 Integer and Goal Programming Models

Chapter 6 Integer and Goal Programming Models. Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu. Variations of Basic Linear Programming. Integer Programming Goal Programming

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Chapter 6 Integer and Goal Programming Models

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  1. Chapter 6Integer and Goal Programming Models Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu

  2. Variations of Basic Linear Programming • Integer Programming • Goal Programming • Nonlinear Programming (skip)

  3. Integer Programming (IP) Where some or all decision variables are required to be whole numbers. • General Integer Variables (0,1,2,3,etc.) Values that count how many • Binary Integer Variables (0 or 1) Usually represent a Yes/No decision

  4. General Integer Example:Harrison Electric Co. Produce 2 products (lamps and ceiling fans) using 2 limited resources Decision: How many of each product to make? (must be integers) Objective: Maximize profit

  5. Decision Variables L = number of lamps to make F = number of ceiling fans to make

  6. LP Model Summary Max 600 L + 700 F ($ of profit) Subject to the constraints: 2L + 3F < 12 (wiring hours) 6L + 5F < 30 (assembly hours) L, F> 0

  7. Graphical Solution

  8. Properties of Integer Solutions • Rounding off the LP solution might not yield the optimal IP solution • The IP objective function value is usually worse than the LP value • IP solutions are usually not at corner points

  9. Using Solver for IP • IP models are formulated in Excel in the same way as LP models • The additional integer restriction is entered like an additional constraint int - Means general integer variables bin - Means binary variables Go to file 6-1.xls

  10. Go to file 6-1.xls

  11. Binary Integer Example:Portfolio Selection Choosing stocks to include in portfolio Decision: Which of 7 stocks to include? Objective: Maximize expected annual return (in $1000’s)

  12. Stock Data

  13. Decision Variables Use the first letter of each stock’s name Example for Trans-Texas Oil: T = 1 if Trans-Texas Oil is included T = 0 if not included

  14. Restrictions • Invest up to $3 million • Include at least 2 Texas companies • Include no more than 1 foreign company • Include exactly 1 California company • If British Petro is included, then Trans-Texas Oil must also be included

  15. Objective Function (in $1000’s return) Max 50T + 80B + 90D + 120H + 110L + 40S + 75C Subject to the constraints: Invest up to $3 Million 480T + 540B + 680D + 1000H + 700L + 510S + 900C < 3000

  16. Include At Least 2 Texas Companies T + H + L > 2 Include No More Than 1 Foreign Company B + D < 1 Include Exactly 1 California Company S + C = 1

  17. If British Petro is included (B=1), then Trans-Texas Oil must also be included (T=1) Combinations of B and T allows the 3 acceptable combinations and prevents the unacceptable one B < T

  18. IP Model for Portfolio Selection Max $50T + $80B + $90D + $120H + $110L + $40S + $75C Subject to the constraints: 480T + 540B + 680D + 1000H + 700L + 510S + 900C < 3000 (investment limit) T + H + L > 2 (Texas companies) B + D < 1 (foreign companies) S + C = 1 (California companies) B < T (Trans-Texas and British petro) All variables = 0 or 1 Go to file 6-3.xls

  19. Go to file 6-3.xls

  20. Goal Programming Models • Permit multiple objectives • Try to “satisfy” goals rather than optimize • Objective is to minimize underachievement of goals

  21. Goal Programming Example:Wilson Doors Co. Makes 3 types of doors from 3 limited resources Decision: How many of each of 3 types of doors to make? Objective: Minimize total underachievement of goals

  22. Data

  23. LP Model Maximize $70E+ $110I + $110C St. 4E + 3I + 7 C < 9,000 (steel usage) 2E + 4I + 3C < 6,000 (forming time) 2E + 3I + 4C < 5,200 (assembly time) E, I, C > 0 Go to file 6-6.xls

  24. LP Solution (File: 6-6.xls) E: 1400, I=800, and C=0 with a total sales of $186,000

  25. Goals • Total sales at least $180,000 • Exterior door sales at least $70,000 • Interior door sales at lest $60,000 • Commercial door sales at least $35,000

  26. Regular Decision Variables E = number of exterior doors made I = number of interior doors made C = number of commercial doors made Deviation Variables di+ = amount by which goal i is overachieved di-= amount by which goal i is underachieved

  27. Goal Constraints Goal 1: Total sales at least $180,000 70E + 110I + 110C + dT- - dT+ = 180,000 Goal 2: Exterior door sales at least $70,000 70E + dE- - dE+ = 70,000 Note: Each highlighted deviation variable measures goal underachievement

  28. Goal 3: Interior door sales at least $60,000 110 I + dI- - dI+ = 60,000 Goal 4: Commercial door sales at least $35,000 110C+ dC- - dC+ = 35,000

  29. Goals • Total sales at least $180,000 • Exterior door sales at least $70,000 • Interior door sales at lest $60,000 • Commercial door sales at least $35,000 Goal 1: 70E + 110I + 110C + dT- - dT+ = 180,000 Goal 2: 70E + dE- - dE+ = 70,000 Goal 3: 110 I + dI- - dI+ = 60,000 Goal 4: 110C+ dC- - dC+ = 35,000

  30. Objective Function Minimize total goal underachievement Min dT- + dE- + dI- + dC- Subject to the constraints: • The 4 goal constraints • The “regular” constraints (3 limited resources) • nonnegativity

  31. Objective Function Minimize dT- + dE- + dI- + dC- Subject to the constraints: 70E + 110I + 110C + dT- - dT+ = 180,000 (total sales goal) 70E + dE- - dE+ = 70,000 (exterior door sales goal) 110 I + dI- - dI+ = 60,000 (interior door sales goal) 110C+ dC- - dC+ = 35,000 (comm. door sales goal) 4E + 3I + 7 C < 9,000 (steel usage) 2E + 4I + 3C < 6,000 (forming time) 2E + 3I + 4C < 5,200 (assembly time) E, I, C, dT-, dT+, dE- , dE+, dI- , dI+, dC- , dC+> 0 Go to file 6-6.xls

  32. Weighted Goals • When goals have different priorities, weights can be used • Suppose that Goal 1 is 5 times more important than each of the others Objective Function Min 5dT- + dE- + dI- + dC- Go to file 6-6.xls, sheet:6-6A

  33. Go to file 6-6.xls, sheet:6-6A GP#1

  34. Properties of Weighted Goals • Solution may differ depending on the weights used • Appropriate only if goals are measured in the same units • What if Goal 1 is only 2.5 times important than each of the others? Objective Function Min 2.5dT- + dE- + dI- + dC- Go to file 6-6.xls, sheet:6-6B GP#2, 6-6B IP

  35. Go to file 6-6.xls, sheet:6-6B GP#2

  36. Go to file 6-6.xls, sheet:6-6B IP

  37. Ranked Goals • Lower ranked goals are considered only if all higher ranked goals are achieved • Suppose they added a 5th goal Goal 5: Steel usage as close to 9000 lb as possible 4E + 3I + 7C + dS- = 9000 (lbs steel) (no dS+ is needed because we cannot exceed 9000 pounds)

  38. Rank R1: Goal 1 • Rank R2: Goal 5 • Rank R3: Goals 2, 3, and 4 A series of LP models must be solved • Solve for the R1 goal while ignoring the other goals Objective Function: Min dT-

  39. Objective Function Objective Function: Min dT- Subject to the constraints: 70E + 110I + 110C + dT- - dT+ = 180,000 (total sales goal) 4E + 3I + 7C + dS- = 9000 (steel usage goal) 70E + dE- - dE+ = 70,000 (exterior door sales goal) 110 I + dI- - dI+ = 60,000 (interior door sales goal) 110C+ dC- - dC+ = 35,000 (comm. door sales goal) 4E + 3I + 7 C < 9,000 (steel usage) 2E + 4I + 3C < 6,000 (forming time) 2E + 3I + 4C < 5,200 (assembly time) E, I, C, dT-, dT+, dE- , dE+, dI- , dI+, dC- , dC+> 0 Go to file 6-7.xls

  40. Go to file 6-7A R1.xls

  41. 2) If the R1 goal can be achieved (dT- = 0), then this is added as a constraint and we attempt to satisfy the R2 goal (Goal 5) Objective Function: Min dS- 3) If the R2 goal can be achieved (dS- = 0), then this is added as a constraint and we solve for the R3 goals (Goals 2, 3, and 4) Objective Function: Min dE- + dI- + dC- Go to file 6-7.xls

  42. Go to file 6-7B R2.xls

  43. Go to file 6-7B R2 IP.xls

  44. Go to file 6-7C R3.xls

  45. Go to file 6-7C R3 IP.xls

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