1. Table of Contents CD Chapter 17 (Goal Programming)
A Case Study: Dewright Co. Goal Programming (Section 17.1) 17.2–17.4
Weighted Goal Programming (Section 17.2) 17.5–17.8
Preemptive Goal Programming (Section 17.3) 17.9–17.18
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2. The Dewright Company
The Dewright Company is one of the largest producers of power tools in the United States.
The company is preparing to replace its current product line with the next generation of products—three new power tools.
Management needs to determine the mix of the company’s three new products to best meet the following three goals:
Achieve a total profit (net present value) of at least $125 million.
Maintain the current employment level of 4,000 employees.
Hold the capital investment down to no more than $55 million.
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3. Penalty Weight for Missing Goal
Penalty Weights
Goal
Factor
Penalty Weight for Missing Goal 1
Total profit
5 (per $1 million under the goal) 2
Employment level
4 (per 100 employees under the goal) 2 (per 100 employees over the goal) 3
Capital investment
3 (per $1 million over the goal)
Table Penalty weights that measure the relative seriousness of missing the goals for the Dewright Company problem
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4. Data for Contribution to the Goals
Unit Contribution of Product
Factor 1 2 3
Goal
Total profit (millions of dollars) 12 9 15
≥ 125
Employment level (hundreds of employees) 5 4
= 40
Capital investment (millions of dollars) 7 8
≤ 55
Table Contribution to the goals per unit rate of production of each product for the Dewright Co. problem.
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5. Weighted Goal Programming
A common characteristic of many management science models (linear programming, integer programming, nonlinear programming) is that they have a single objective function.
It is not always possible to fit all managerial objectives into a single objective function. Managerial objectives might include:
Maintain stable profits.
Increase market share.
Diversify the product line.
Maintain stable prices.
Improve worker morale.
Maintain family control of the business.
Increase company prestige.
Weighted goal programming provides a way of striving toward several objectives simultaneously.
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6. Weighted Goal Programming
With weighted goal programming, the objective is to
Minimize W = weighted sum of deviations from the goals.
The weights are the penalty weights for missing the goal.
Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal.
The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints: Level Achieved – Amount Over + Amount Under = Goal
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7. Weighted Goal Programming Formulation for the Dewright Co. Problem
Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize W = 5(Under Goal 1) + 2Over Goal 2) + 4 (Under Goal 2) + 3 (Over Goal 3) subject to Level Achieved Deviations Goal Goal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
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8. Weighted Goal Programming Spreadsheet
Figure A spreadsheet model for the Dewright Co. weighted goal-programming problem formulated as a linear programming problem, where the changing cells Units Produced (C12:E12) show the optimal production rates and the changing cells Deviations (J6:K8) show the optimal amounts over and under the goals. The target cell Weighted Sum of Deviations (M13) gives the resulting weighted sum of deviations from the goals.
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9. Weighted vs. Preemptive Goal Programming
Weighted goal programming is designed for problems where all the goals are quite important, with only modest differences in importance that can be measured by assigning weights to the goals.
Preemptive goal programming is used when there are major differences in the importance of the goals.
The goals are liested in the order of their importance.
It begins by focusing solely on the most important goal.
It next does the same for the second most important goal (as is possible without hurting the first goal).
It continues the the following goals (as is possible without hurting the previous more important goals).
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10. Preemptive Goal Programming
Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal.
The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints: Level Achieved – Amount Over + Amount Under = Goal
Start with the objective of achieving the first goal (or coming as close as possible):
Minimize (Amount Over/Under Goal 1)
Continue with the next goal, but constrain the previous goals to not get any worse:
Minimize (Amount Over/Under Goal 2) subject to Amount Over/Under Goal 1 = (amount achieved in previous step)
Repeat the previous step for all succeeding goals.
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11. Preemptive Goal Programming for Dewright
The goals in the order of importance are:
Achieve a total profit (net present value) of at least $125 million.
Avoid decreasing the employment level below 4,000 employees.
Hold the capital investment down to no more than $55 million.
Avoid increasing the employment level above 4,000 employees.
Start with the objective of achieving the first goal (or coming as close as possible):
Minimize (Under Goal 1)
Then, if for example goal 1 is achieved (i.e., Under Goal 1 = 0), then
Minimize (Under Goal 2) subject to (Under Goal 1) = 0
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12. Preemptive Goal Programming Formulation for the Dewright Co
Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 1)
Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Under Goal 1) subject to Level Achieved Deviations Goal Goal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
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13. Preemptive Goal Programming Formulation for the Dewright Co
Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 2)
Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Under Goal 2) subject to Level Achieved Deviations Goal Goal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = (Under Goal 1) = (Level Achieved in Step 1) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
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14. Preemptive Goal Programming Formulation for the Dewright Co
Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 3)
Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Over Goal 3) subject to Level Achieved Deviations Goal Goal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = (Under Goal 1) = (Level Achieved in Step 1) (Under Goal 2) = (Level Achieved in Step 2) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
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15. Preemptive Goal Programming Formulation for the Dewright Co
Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 4)
Let Pi = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Over Goal 2) subject to Level Achieved Deviations Goal Goal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = (Under Goal 1) = (Level Achieved in Step 1) (Under Goal 2) = (Level Achieved in Step 2) (Over Goal 3) = (Level Achieved in Step 3) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
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16. Preemptive Goal Programming Spreadsheet Step 1: Minimize (Under Goal 1)
Figure A spreadsheet model formulated as a linear programming model for the first step of the Dewright Co. preemptive goal-programming problem. Since Priority 1 is to minimize the deviation under Goal 1, the target cell is Under Goal 1 (K6) for this step. The changing cells Units Produced (C12:E12) show the resulting production rates and the other changing cells Deviations (J6:K8) show the resulting amounts over and under the goals after running the Solver. Since Priority 2 is to minimize Under Goal 2 (K7), which already has a value of 0, the procedure will bypass step 2 and go directly to step 3.
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17. Preemptive Goal Programming Spreadsheet Step 3: Minimize (Over Goal 3)
Figure The revision of the spreadsheet model in Figure 17.2 needed to perform step 3 of the preemptive goal-programming procedure. Since Priority 3 is to minimize the deviation over Goal 3, the target cell is Over Goal 3 (J8) for this step. Constraints that Under Goal 1 (K6) = 0 and Under Goal 2 (K7) = 0 also have been added to the model. The changing cells show the results after clicking on the Solve button.
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18. Preemptive Goal Programming Spreadsheet Step 4: Minimize (Over Goal 2)
Figure The revision of the spreadsheet model in Figure 17.3 needed to perform step 4 of the preemptive goal-programming procedure. Since Priority 4 is to minimize the deviation over Goal 2, the target cell is Over Goal 2 (J7) for this step. One more constraint, Over Goal 3 (J8) = 0, also has been added to the model. Since this is the final step, the changing cells show the optimal solution obtained for Dewright’s preemptive goal-programming problem by clicking on the Solve button.
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19. Multi-Objective Decision Making
Many problems have multiple objectives:
Planning the national budget
save social security, reduce debt, cut taxes, build national defense
Admitting students to college
high SAT or GMAT, high GPA, diversity
Planning an advertising campaign
budget, reach, expenses, target groups
Choosing taxation levels
raise money, minimize tax burden on low-income, minimize flight of business
Planning an investment portfolio
maximize expected earnings, minimize risk
Techniques
Preemptive goal programming
Weighted goal programming
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