## Chapter 11 (Goal Programming)

Chapter 11 (Goal Programming) McGraw-Hill/Irwin 11.١ © The McGraw-Hill Companies, Inc., 2003 Introduction: what goal programming is? • All probl...
Author: Posy Poole
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

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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.

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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),

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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)

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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

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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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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

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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)

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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

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Product 2 0

Product 3 1.66666667

F

G Goals

H

Level Achieved 125 >= 48.333333 = 55 = = = = = =