© The McGraw-Hill Companies, Inc., 200311.1McGraw-Hill/Irwin

Table of ContentsChapter 11 (Goal Programming)

A Case Study: Dewright Co. Goal Programming (Section 11.1)

11.2–11.4Weighted Goal Programming (Section 11.2)

11.5–11.8Preemptive Goal Programming (Section 11.3)

11.9–11.18

© The McGraw-Hill Companies, Inc., 200311.2McGraw-Hill/Irwin

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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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

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Weighted Goal Programming Formulation forthe 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 GoalGoal 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

andPi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

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Weighted Goal Programming Spreadsheet

3456789101112131415

B C D E F G H I J K L M N OGoals

Contribution per Unit Produced Level Amount Amount BalanceProduct 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) Goal

Goal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 48.333333 = 40 8.333333 0 40 = 40Goal 3 (Investment) 5 7 8 55 <= 55 0 0 55 = 55

Product 1 Product 2 Product 3 Penalty Over Under Weighted SumUnits Produced 8.33333333 0 1.66666667 Weights Goal Goal of Deviations

Profit 5 16.66666667Employment 2 4Investment 3

Deviations Constraints

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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).

© The McGraw-Hill Companies, Inc., 200311.10McGraw-Hill/Irwin

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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Preemptive Goal Programming for Dewright

The goals in the order of importance are:1. Achieve a total profit (net present value) of at least $125 million.

2. Avoid decreasing the employment level below 4,000 employees.

3. Hold the capital investment down to no more than $55 million.

4. 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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Preemptive Goal Programming Formulation forthe 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 GoalGoal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = 40Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = 55

andPi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

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Preemptive Goal Programming Formulation forthe 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 GoalGoal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = 40Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = 55

(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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Preemptive Goal Programming Formulation forthe 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 GoalGoal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = 40Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = 55

(Under Goal 1) = (Level Achieved in Step 1)(Under Goal 2) = (Level Achieved in Step 2)

andPi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

© The McGraw-Hill Companies, Inc., 200311.15McGraw-Hill/Irwin

Preemptive Goal Programming Formulation forthe 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 GoalGoal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = 40Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = 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)

andPi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

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Preemptive Goal Programming SpreadsheetStep 1: Minimize (Under Goal 1)

1

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A B C D E F G H I J K L M N O

Dewright Co. Goal Programming (Preemptive Priority 1: Minimize Under Goal 1)

GoalsContribution per Unit Produced Level Amount Amount Balance

Product 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) GoalGoal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 40 = 40 0 0 40 = 40Goal 3 (Investment) 5 7 8 61.481 <= 55 6.481 0 55 = 55

Minimize (Under Goal 1)Product 1 Product 2 Product 3

Units Produced 3.7037 0 5.3704

Deviations Constraints

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Preemptive Goal Programming SpreadsheetStep 3: Minimize (Over Goal 3)

1

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A B C D E F G H I J K L M N O

Dewright Co. Goal Programming (Preemptive Priority 3: Minimize Over Goal 3)

GoalsContribution per Unit Produced Level Amount Amount Balance

Product 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) GoalGoal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 48.333 = 40 8.333333 0 40 = 40Goal 3 (Investment) 5 7 8 55 <= 55 0 0 55 = 55

Minimize (Over Goal 3)Product 1 Product 2 Product 3 (Under Goal 1) = 0

Units Produced 8.333 0 1.667 (Under Goal 2) = 0

Deviations Constraints

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Preemptive Goal Programming SpreadsheetStep 4: Minimize (Over Goal 2)

1

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A B C D E F G H I J K L M N O

Dewright Co. Goal Programming (Preemptive Priority 4: Minimize Over Goal 2)

GoalsContribution per Unit Produced Level Amount Amount Balance

Product 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) GoalGoal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 48.333 = 40 8.333 0 40 = 40Goal 3 (Investment) 5 7 8 55 <= 55 0 0 55 = 55

Minimize (Over Goal 2)Product 1 Product 2 Product 3 (Under Goal 1) = 0

Units Produced 8.333 0 1.667 (Under Goal 2) = 0(Over Goal 3) = 0

Deviations Constraints

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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