1. 1 1 Slide Chapter 14: Goal Programming Goal programming is used to solve linear programs with multiple objectives, with each objective viewed as a "goal". In goal programming, d i + and d i -, deviation variables, are the amounts a targeted goal i is overachieved or underachieved, respectively. The goals themselves are added to the constraint set with d i + and d i - acting as the surplus and slack variables.

2. 2 2 Slide Goal Programming One approach to goal programming is to satisfy goals in a priority sequence. Second-priority goals are pursued without reducing the first-priority goals, etc. For each priority level, the objective function is to minimize the (weighted) sum of the goal deviations. Previous "optimal" achievements of goals are added to the constraint set so that they are not degraded while trying to achieve lesser priority goals.

3. 3 3 Slide Goal Programming Formulation Step 1: Decide the priority level of each goal. Step 2: Decide the weight on each goal. If a priority level has more than one goal, for each goal i decide the weight, w i, to be placed on the deviation(s), d i + and/or d i -, from the goal. If a priority level has more than one goal, for each goal i decide the weight, w i, to be placed on the deviation(s), d i + and/or d i -, from the goal. Step 3: Set up the initial linear program. Min w 1 d 1 + + w 2 d 2 - s.t. Functional Constraints, s.t. Functional Constraints, and Goal Constraints and Goal Constraints Step 4: Solve the current linear program. If there is a lower priority level, go to step 5. Otherwise, a final solution has been reached. If there is a lower priority level, go to step 5. Otherwise, a final solution has been reached.

4. 4 4 Slide Goal Programming Formulation Step 5: Set up the new linear program. Consider the next-lower priority level goals and formulate a new objective function based on these goals. Add a constraint requiring the achievement of the next-higher priority level goals to be maintained. The new linear program might be: Min w 3 d 3 + + w 4 d 4 - Min w 3 d 3 + + w 4 d 4 - s.t. Functional Constraints, s.t. Functional Constraints, Goal Constraints, and Goal Constraints, and w 1 d 1 + + w 2 d 2 - = k w 1 d 1 + + w 2 d 2 - = k Go to step 4. (Repeat steps 4 and 5 until all priority levels have been examined.)

5. 5 5 Slide GP Example: Conceptual Products Conceptual Products produces CP400 and CP500 computers that use memory modules, external hard drives, and cases. The CP400 model uses two memory modules and no external hard drive, whereas the CP500 uses one memory module and one external hard drive. Both models use one case. Suppliers can provide Conceptual Products with 1000 memory modules, 500 external hard drives, and 600 cases on a weekly basis. It takes one hour to manufacture a CP400 and its profit is $200 and it takes one and one-half hours to manufacture a CP500 and its profit is $500.

6. 6 6 Slide GP Example: Conceptual Products The company has four goals: The company has four goals: Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 3: Make at least $250,000 weekly. (Goal 3) Priority 3: Make at least $250,000 weekly. (Goal 3) Priority 4: Use no more than 400 man-hours per week. (Goal 4) Priority 4: Use no more than 400 man-hours per week. (Goal 4)

7. 7 7 Slide GP Example: Formulation Variables x 1 = number of CP400 computers produced weekly x 1 = number of CP400 computers produced weekly x 2 = number of CP500 computers produced weekly x 2 = number of CP500 computers produced weekly d i - = amount the right hand side of goal i is deficient d i - = amount the right hand side of goal i is deficient d i + = amount the right hand side of goal i is exceeded d i + = amount the right hand side of goal i is exceeded Functional Constraints Availability of memory modules: 2 x 1 + x 2 < 1000 Availability of external hard drives: x 2 < 500 Availability of cases: x 1 + x 2 < 600

8. 8 8 Slide GP Example: Formulation Goals (1) 200 CP400 computers weekly: x 1 + d 1 - - d 1 + = 200 (2) 500 total computers weekly: (2) 500 total computers weekly: x 1 + x 2 + d 2 - - d 2 + = 500 x 1 + x 2 + d 2 - - d 2 + = 500 (3) $250(in thousands) profit: (3) $250(in thousands) profit:.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250 (4) 400 total man-hours weekly: (4) 400 total man-hours weekly: x 1 + 1.5 x 2 + d 4 - - d 4 + = 400 x 1 + 1.5 x 2 + d 4 - - d 4 + = 400 Non-negativity: Non-negativity: x 1, x 2, d i -, d i + > 0 for all i x 1, x 2, d i -, d i + > 0 for all i

9. 9 9 Slide GP Example: Formulation Objective Functions Priority 1: Minimize the amount the state contract is not met: Min d 1 - Priority 1: Minimize the amount the state contract is not met: Min d 1 - Priority 2: Minimize the number under 500 computers produced weekly: Min d 2 - Priority 2: Minimize the number under 500 computers produced weekly: Min d 2 - Priority 3: Minimize the amount under $250,000 earned weekly: Min d 3 - Priority 3: Minimize the amount under $250,000 earned weekly: Min d 3 - Priority 4: Minimize the man-hours over 400 used weekly: Min d 4 + Priority 4: Minimize the man-hours over 400 used weekly: Min d 4 +

10. 10 Slide GP Example: Formulation Formulation Summary Min P 1 ( d 1 - ) + P 2 ( d 2 - ) + P 3 ( d 3 - ) + P 4 ( d 4 + ) s.t. 2 x 1 + x 2 < 1000 s.t. 2 x 1 + x 2 < 1000 + x 2 < 500 + x 2 < 500 x 1 + x 2 < 600 x 1 + x 2 < 600 x 1 + d 1 - - d 1 + = 200 x 1 + d 1 - - d 1 + = 200 x 1 + x 2 + d 2 - - d 2 + = 500 x 1 + x 2 + d 2 - - d 2 + = 500.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250 x 1 +1.5 x 2 + d 4 - - d 4 + = 400 x 1 +1.5 x 2 + d 4 - - d 4 + = 400 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 + > 0 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 + > 0

11. 11 Slide GP Example: Graphical Solution Iteration 1 To solve graphically, first graph the functional constraints. Then graph the first goal: x 1 = 200. Note on the next slide that there is a set of points that exceed x 1 = 200 (where d 1 - = 0).

12. 12 Slide GP Example: Graphical Solution Functional Constraints and Goal 1 Graphed 2 x 1 + x 2 < 1000 Goal 1: x 1 > 200 x 1 + x 2 < 600 x 2 < 500 PointsSatisfying Goal 1 x1x1x1x1 x2x2x2x2 1000800600400200 200 400 600 800 1000 1200

13. 13 Slide GP Example: Graphical Solution Iteration 2 Now add Goal 1 as x 1 > 200 and graph Goal 2: x 1 + x 2 = 500. Note on the next slide that there is still a set of points satisfying the first goal that also satisfies this second goal (where d 2 - = 0).

14. 14 Slide GP Example: Graphical Solution Goal 1 (Constraint) and Goal 2 Graphed 2 x 1 + x 2 < 1000 Goal 1: x 1 > 200 x 1 + x 2 < 600 x 2 < 500 Points Satisfying Both Goals 1 and 2 x1x1x1x1 x2x2x2x2 Goal 2: x 1 + x 2 > 500 200 400 600 800 1000 1200 1000800600400200

15. 15 Slide GP Example: Graphical Solution Iteration 3 Now add Goal 2 as x 1 + x 2 > 500 and Goal 3:.2 x 1 +.5 x 2 = 250. Note on the next slide that no points satisfy the previous functional constraints and goals and satisfy this constraint. Thus, to Min d 3 -, this minimum value is achieved when we Max.2 x 1 +.5 x 2. Note that this occurs at x 1 = 200 and x 2 = 400, so that.2 x 1 +.5 x 2 = 240 or d 3 - = 10.

16. 16 Slide GP Example: Graphical Solution Goal 2 (Constraint) and Goal 3 Graphed 2 x 1 + x 2 < 1000 Goal 1: x 1 > 200 x 1 + x 2 < 600 x 2 < 500 Points Satisfying Both Goals 1 and 2 x1x1x1x1 x2x2x2x2 Goal 2: x 1 + x 2 > 500 Goal 3:.2 x 1 +.5 x 2 = 250 (200,400) 200 400 600 800 1000 1200 1000800600400200

17. 17 Slide Goal Programming: More Complex Problems Now we will formulate and solve a GP problem that involves multiple goals within the same priority level. No tradeoffs occur in the achievement of goals with different priorities. However, tradeoffs can occur in the achievement of goals with the same priority and different weights. A GP problem with p preemptive goal priorities can be solved as a series of p linear programming problems. Solving a GP problem as a series of LPs, the objective functions (always to be minimized) will include only relevant deviation variables and not decision variables.

18. 18 Slide GP Example 2: Conceptual Products Conceptual Products is a computer company that produces the CP400, CP500, and CP600 computers. Many of the components used in the three computer models are produced in abundant supply by the company. However, the memory modules, external hard drives, and cases are bought from suppliers. The CP400 model uses two memory modules and no external hard drive, the CP500 uses one memory module and one external hard drive, and the CP600 uses two memory modules and one external hard drive. All three models use the same case.

19. 19 Slide GP Example 2: Conceptual Products Suppliers can provide Conceptual Products with 1000 memory modules, 500 external hard drives, and 600 cases on a weekly basis. It takes one hour to manufacture a CP400 and its profit is $200; it takes one and one-half hours to manufacture a CP500 and its profit is $500; and it takes two hours to manufacture a CP600 and its profit is $900.

20. 20 Slide GP Example 2: Conceptual Products The company has four goals: The company has four goals: Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 3: Make at least $250,000 profit Priority 3: Make at least $250,000 profit weekly. (Goal 3) weekly. (Goal 3) Priority 3: Use no more than 400 man-hours Priority 3: Use no more than 400 man-hours per week. (Goal 4) per week. (Goal 4) Each $1000 underachieved from its profit goal is five times as important as an extra man-hour.

21. 21 Slide GP Example 2: Formulation Variables x 1 = number of CP400 computers produced weekly x 1 = number of CP400 computers produced weekly x 2 = number of CP500 computers produced weekly x 2 = number of CP500 computers produced weekly x 3 = number of CP600 computers produced weekly d i - = amount the right hand side of goal i is deficient d i - = amount the right hand side of goal i is deficient d i + = amount the right hand side of goal i is exceeded d i + = amount the right hand side of goal i is exceeded Functional Constraints Availability of memory modules: 2 x 1 + x 2 + x 3 < 1000 Availability of external hard drives: x 2 + x 3 < 500 Availability of cases: x 1 + x 2 + x 3 < 600

22. 22 Slide GP Example 2: Formulation Goals (1) 200 CP400 computers weekly: x 1 + d 1 - - d 1 + = 200 (2) 500 total computers weekly: (2) 500 total computers weekly: x 1 + x 2 + x 3 + d 2 - - d 2 + = 500 x 1 + x 2 + x 3 + d 2 - - d 2 + = 500 (3) $250(in thousands) profit: (3) $250(in thousands) profit:.2 x 1 +.5 x 2 +.9 x 3 + d 3 - - d 3 + = 250.2 x 1 +.5 x 2 +.9 x 3 + d 3 - - d 3 + = 250 (4) 400 total man-hours weekly: (4) 400 total man-hours weekly: x 1 + 1.5 x 2 + 2 x 3 + d 4 - - d 4 + = 400 x 1 + 1.5 x 2 + 2 x 3 + d 4 - - d 4 + = 400 Non-negativity: Non-negativity: x 1, x 2, x 3, d i -, d i + > 0 for all i x 1, x 2, x 3, d i -, d i + > 0 for all i

23. 23 Slide GP Example 2: Conceptual Products Objective Functions Priority 1: Minimize the number of CP400 under 200 units: Min d 1 - Priority 1: Minimize the number of CP400 under 200 units: Min d 1 - Priority 2: Minimize the number under 500 computers produced weekly: Min d 2 - Priority 2: Minimize the number under 500 computers produced weekly: Min d 2 - Priority 3: Minimize the amount under $250,000 earned weekly: Min 5 d 3 - Priority 3: Minimize the amount under $250,000 earned weekly: Min 5 d 3 - Priority 3: Minimize the man-hours over 400 used weekly: Min d 4 + Priority 3: Minimize the man-hours over 400 used weekly: Min d 4 +

24. 24 Slide GP Example 2: Conceptual Products Priority 1 Formulation Min d 1 - Min d 1 - s.t. 2 x 1 + x 2 + x 3 < 1000 s.t. 2 x 1 + x 2 + x 3 < 1000 + x 2 + x 3 < 500 + x 2 + x 3 < 500 x 1 + x 2 + x 3 < 600 x 1 + x 2 + x 3 < 600 x 1 + d 1 - - d 1 + = 200 x 1 + d 1 - - d 1 + = 200 x 1 + x 2 + x 3 + d 2 - - d 2 + = 500 x 1 + x 2 + x 3 + d 2 - - d 2 + = 500.2 x 1 +.5 x 2 +.9 x 3 + d 3 - - d 3 + = 250.2 x 1 +.5 x 2 +.9 x 3 + d 3 - - d 3 + = 250 x 1 +1.5 x 2 +2 x 3 + d 4 - - d 4 + = 400 x 1 +1.5 x 2 +2 x 3 + d 4 - - d 4 + = 400 x 1, x 2, x 3, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 + > 0 x 1, x 2, x 3, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 + > 0

25. 25 Slide GP Example 2: Conceptual Products Computer Solution (First LP) Objective Function Value = 0.000 Objective Function Value = 0.000 Variable Value Reduced Cost Variable Value Reduced Cost x 1 200.000 0.000 x 1 200.000 0.000 x 2 0.000 0.000 x 2 0.000 0.000 x 3 233.333 0.000 x 3 233.333 0.000 d 1 - 0.000 1.000 d 1 - 0.000 1.000 d 1 + 0.000 0.000 d 1 + 0.000 0.000 d 2 - 66.667 0.000 d 2 - 66.667 0.000 d 2 + 0.000 0.000 d 2 + 0.000 0.000 d 3 - 0.000 0.000 d 3 - 0.000 0.000 d 3 + 0.000 0.000 d 3 + 0.000 0.000 d 4 - 0.000 0.000 d 4 - 0.000 0.000 d 4 + 266.667 0.000 d 4 + 266.667 0.000

26. 26 Slide GP Example 2: Conceptual Products Priority 2 Formulation Min d 2 - Min d 2 - ( include the previous 7 constraints, ( include the previous 7 constraints, i.e. 3 functional constraints and 4 goal constraints ) ( add the constraint: d 1 - = 0 ) ( add the constraint: d 1 - = 0 )

27. 27 Slide GP Example 2: Conceptual Products Computer Solution (Second LP) Objective Function Value = 0.000 Objective Function Value = 0.000 Variable Value Reduced Cost Variable Value Reduced Cost x 1 285.714 0.000 x 1 285.714 0.000 x 2 0.000 0.000 x 2 0.000 0.000 x 3 214.286 0.000 x 3 214.286 0.000 d 1 - 0.000 0.000 d 1 - 0.000 0.000 d 1 + 85.714 0.000 d 1 + 85.714 0.000 d 2 - 0.000 1.000 d 2 - 0.000 1.000 d 2 + 0.000 0.000 d 2 + 0.000 0.000 d 3 - 0.000 0.000 d 3 - 0.000 0.000 d 3 + 0.000 0.000 d 3 + 0.000 0.000 d 4 - 0.000 0.000 d 4 - 0.000 0.000 d 4 + 314.286 0.000 d 4 + 314.286 0.000

28. 28 Slide GP Example 2: Conceptual Products Priority 3 Formulation Min 5 d 3 - + d 4 + Min 5 d 3 - + d 4 + ( include the previous 8 constraints ( include the previous 8 constraints i.e. 3 functional constraints and 4 goal constraints and d 1 - = 0 ) ( add the constraint: d 2 - = 0 ) ( add the constraint: d 2 - = 0 )

29. 29 Slide GP Example 2: Conceptual Products Computer Solution (Third LP) Objective Function Value = 314.286 Objective Function Value = 314.286 Variable Value Reduced Cost Variable Value Reduced Cost x 1 285.714 0.000 x 1 285.714 0.000 x 2 0.000 0.071 x 2 0.000 0.071 x 3 214.286 0.000 x 3 214.286 0.000 d 1 - 0.000 0.000 d 1 - 0.000 0.000 d 1 + 85.714 0.000 d 1 + 85.714 0.000 d 2 - 0.000 0.000 d 2 - 0.000 0.000 d 2 + 0.000 0.714 d 2 + 0.000 0.714 d 3 - 0.000 3.571 d 3 - 0.000 3.571 d 3 + 0.000 1.429 d 3 + 0.000 1.429 d 4 - 0.000 1.000 d 4 - 0.000 1.000 d 4 + 314.286 0.000 d 4 + 314.286 0.000

30. 30 Slide GP Example 2: Conceptual Products Final Solution Thus the optimal recommendation is to produce 285.714 CP400 computers, 0 CP500 computers, and weekly and 214.286 CP600 computers weekly. Thus the optimal recommendation is to produce 285.714 CP400 computers, 0 CP500 computers, and weekly and 214.286 CP600 computers weekly. All goals will be met except goal 4. 314.286 extra man-hours or a total of 714.286man-hours will be used. All goals will be met except goal 4. 314.286 extra man-hours or a total of 714.286man-hours will be used.