© The McGraw-Hill Companies, Inc., 200311.١McGraw-Hill/Irwin

Chapter 11 (Goal Programming)

© The McGraw-Hill Companies, Inc., 200311.٢McGraw-Hill/Irwin

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

© The McGraw-Hill Companies, Inc., 200311.٣McGraw-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.

© The McGraw-Hill Companies, Inc., 200311.٤McGraw-Hill/Irwin

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),

© The McGraw-Hill Companies, Inc., 200311.٥McGraw-Hill/Irwin

Penalty Weights

3 (per $1 million over the goal)Capital investment3

4 (per 100 employees under the goal)2 (per 100 employees over the goal)

Employment level2

5 (per $1 million under the goal)Total profit1

Penalty Weight for Missing GoalFactorGoal

© The McGraw-Hill Companies, Inc., 200311.٦McGraw-Hill/Irwin

Data for Contribution to the Goals

≤ 55875Capital investment (millions of dollars)

= 40435Employment level (hundreds of employees)

≥ 12515912Total profit (millions of dollars)

Goal321Factor

Unit Contribution of Product

© The McGraw-Hill Companies, Inc., 200311.٧McGraw-Hill/Irwin

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.

© The McGraw-Hill Companies, Inc., 200311.٨McGraw-Hill/Irwin

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, thatwill 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

© The McGraw-Hill Companies, Inc., 200311.٩McGraw-Hill/Irwin

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) + 2(Over 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)

© The McGraw-Hill Companies, Inc., 200311.١٠McGraw-Hill/Irwin

Weighted Goal Programming Spreadsheet

3456789

101112131415

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

© The McGraw-Hill Companies, Inc., 200311.١١McGraw-Hill/Irwin

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 listed 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 with the following goals (as is possible without hurting the previous

more important goals).

© The McGraw-Hill Companies, Inc., 200311.١٢McGraw-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.

© The McGraw-Hill Companies, Inc., 200311.١٣McGraw-Hill/Irwin

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

© The McGraw-Hill Companies, Inc., 200311.١٤McGraw-Hill/Irwin

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)

© The McGraw-Hill Companies, Inc., 200311.١٥McGraw-Hill/Irwin

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)

© The McGraw-Hill Companies, Inc., 200311.١٦McGraw-Hill/Irwin

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., 200311.١٧McGraw-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)

© The McGraw-Hill Companies, Inc., 200311.١٨McGraw-Hill/Irwin

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

123456789

101112

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

© The McGraw-Hill Companies, Inc., 200311.١٩McGraw-Hill/Irwin

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

123456789

101112

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

© The McGraw-Hill Companies, Inc., 200311.٢٠McGraw-Hill/Irwin

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

123456789

10111213

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

© The McGraw-Hill Companies, Inc., 200311.٢١McGraw-Hill/Irwin

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