Chapter 11 (Goal Programming)
McGraw-Hill/Irwin
11.١
© The McGraw-Hill Companies, Inc., 2003
Introduction: what goal programming is? •
All problems studied in the previous chapters share one common characteristic: a single objective function that express the overall measure of performance
•
It is not always possible to put all management’s objectives into one overall measure of performance
•
Objectives might be so different (contradictory) that no common basis for measuring progress toward these objectives
•
Management might instead set numeric goals for the various objectives and seek a solution that make as much progress as possible toward all these goals
McGraw-Hill/Irwin
11.٢
© The McGraw-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.
McGraw-Hill/Irwin
11.٣
© The McGraw-Hill Companies, Inc., 2003
Relative Importance of the Goals •
It will not be possible to attain all these goals simultaneously
•
All are important but by small margins their order of importance is: – Goal 1, part of Goal 2 (avoid decreasing the employment level), Goal 3, part of Goal 2 (avoid increasing the employment level),
McGraw-Hill/Irwin
11.٤
© The McGraw-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)
McGraw-Hill/Irwin
11.٥
© The McGraw-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
McGraw-Hill/Irwin
11.٦
© The McGraw-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.
McGraw-Hill/Irwin
11.٧
© The McGraw-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
•
This way of formulating the model makes it a LP model
McGraw-Hill/Irwin
11.٨
© The McGraw-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 Goal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125 Goal 2: 5P1 + 3P2 + 4P3
– (Over Goal 2) + (Under Goal 2) =
40
Goal 3: 5P1 + 7P2 + 8P3
– (Over Goal 3) + (Under Goal 3) =
55
and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
McGraw-Hill/Irwin
11.٩
© The McGraw-Hill Companies, Inc., 2003
Weighted Goal Programming Spreadsheet B 3 4 5 6 7 8 9 10 11 12 13 14 15
Goal 1 (Profit) Goal 2 (Employment) Goal 3 (Investment)
Units Produced
C
D
E
Contribution per Unit Produced Product 1 Product 2 Product 3 12 9 15 5 3 4 5 7 8
Product 1 8.33333333
McGraw-Hill/Irwin
Product 2 0
Product 3 1.66666667
F
G Goals
H
Level Achieved 125 >= 48.333333 = 55 = = = = = =
