Table of Contents Chapter 11 Goal Programming A

Table of Contents Chapter 11 (Goal Programming) A Case Study: Dewright Co. Goal Programming (Section 11. 1) Weighted Goal Programming (Section 11. 2) Preemptive Goal Programming (Section 11. 3) Mc. Graw-Hill/Irwin 1 11. 2– 11. 4 11. 5– 11. 8 11. 9– 11. 18 © The Mc. Graw-Hill Companies, Inc. , 2003

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: 1. Achieve a total profit (net present value) of at least $125 million. 2. Maintain the current employment level of 4, 000 employees. 3. Hold the capital investment down to no more than $55 million. Mc. Graw-Hill/Irwin 2 © The Mc. Graw-Hill Companies, Inc. , 2003

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) Mc. Graw-Hill/Irwin 3 © The Mc. Graw-Hill Companies, Inc. , 2003

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 3 4 = 40 Capital investment (millions of dollars) 5 7 8 ≤ 55 Mc. Graw-Hill/Irwin 4 © The Mc. Graw-Hill Companies, Inc. , 2003

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. Mc. Graw-Hill/Irwin 5 © The Mc. Graw-Hill Companies, Inc. , 2003

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 Mc. Graw-Hill/Irwin 6 © The Mc. Graw-Hill Companies, Inc. , 2003

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) + 2 Over Goal 2) + 4 (Under Goal 2) + 3 (Over Goal 3) subject to Level Achieved Deviations Goal 1: 12 P 1 + 9 P 2 + 15 P 3 – (Over Goal 1) + (Under Goal 1) = 125 Goal 2: 5 P 1 + 3 P 2 + 4 P 3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3: 5 P 1 + 7 P 2 + 8 P 3 – (Over Goal 3) + (Under Goal 3) = 55 and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) Mc. Graw-Hill/Irwin 7 © The Mc. Graw-Hill Companies, Inc. , 2003

Weighted Goal Programming Spreadsheet Mc. Graw-Hill/Irwin 8 © The Mc. Graw-Hill Companies, Inc. , 2003

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 following goals (as is possible without hurting the previous more important goals). Mc. Graw-Hill/Irwin 9 © The Mc. Graw-Hill Companies, Inc. , 2003

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. Mc. Graw-Hill/Irwin 10 © The Mc. Graw-Hill Companies, Inc. , 2003

Preemptive Goal Programming for Dewright The goals in the order of importance are: 1. 2. 3. 4. • 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 Mc. Graw-Hill/Irwin 11 © The Mc. Graw-Hill Companies, Inc. , 2003

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 1: 12 P 1 + 9 P 2 + 15 P 3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5 P 1 + 3 P 2 + 4 P 3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5 P 1 + 7 P 2 + 8 P 3 – (Over Goal 3) + (Under Goal 3) = 125 40 55 and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) Mc. Graw-Hill/Irwin 12 © The Mc. Graw-Hill Companies, Inc. , 2003

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 1: 12 P 1 + 9 P 2 + 15 P 3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5 P 1 + 3 P 2 + 4 P 3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5 P 1 + 7 P 2 + 8 P 3 – (Over Goal 3) + (Under Goal 3) = 125 40 55 (Under Goal 1) = (Level Achieved in Step 1) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) Mc. Graw-Hill/Irwin 13 © The Mc. Graw-Hill Companies, Inc. , 2003

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 1: 12 P 1 + 9 P 2 + 15 P 3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5 P 1 + 3 P 2 + 4 P 3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5 P 1 + 7 P 2 + 8 P 3 – (Over Goal 3) + (Under Goal 3) = 125 40 55 (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) Mc. Graw-Hill/Irwin 14 © The Mc. Graw-Hill Companies, Inc. , 2003

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 1: 12 P 1 + 9 P 2 + 15 P 3 – (Over Goal 1) + (Under Goal 1) = Goal 2: 5 P 1 + 3 P 2 + 4 P 3 – (Over Goal 2) + (Under Goal 2) = Goal 3: 5 P 1 + 7 P 2 + 8 P 3 – (Over Goal 3) + (Under Goal 3) = 125 40 55 (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) Mc. Graw-Hill/Irwin 15 © The Mc. Graw-Hill Companies, Inc. , 2003

Preemptive Goal Programming Spreadsheet Step 1: Minimize (Under Goal 1) Mc. Graw-Hill/Irwin 16 © The Mc. Graw-Hill Companies, Inc. , 2003

Preemptive Goal Programming Spreadsheet Step 3: Minimize (Over Goal 3) Mc. Graw-Hill/Irwin 17 © The Mc. Graw-Hill Companies, Inc. , 2003

Preemptive Goal Programming Spreadsheet Step 4: Minimize (Over Goal 2) Mc. Graw-Hill/Irwin 18 © The Mc. Graw-Hill Companies, Inc. , 2003

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 Mc. Graw-Hill/Irwin 19 © The Mc. Graw-Hill Companies, Inc. , 2003