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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

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It is not always possible to put all management’s objectives into one overall measure of performance

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Objectives might be so different (contradictory) that no common basis for measuring progress toward these objectives

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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.

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The company is preparing to replace its current product line with the next generation of products—three new power tools.

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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

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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.

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It is not always possible to fit all managerial objectives into a single objective function. Managerial objectives might include: – – – – – – –

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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.

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Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal.

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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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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 = = = = = =