Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 9
Multicriteria Decision Making
Chapter 9 - Multicriteria Decision Making
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Chapter Topics
Goal Programming
Graphical Interpretation of Goal Programming
Computer Solution of Goal Programming Problems with
QM for Windows and Excel
Overview
Study of problems with several criteria, multiple criteria, instead of a
single objective when making a decision.
Goal programming is a variation of linear programming considering
more than one objective (goals) in the objective function.
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Goal Programming
Model Formulation (1 of 2)
Beaver Creek Pottery Company Example:
Maximize Z = $40x1 + 50x2
subject to:
1x1 + 2x2  40 hours of labor
4x2 + 3x2  120 pounds of clay
x1, x2  0
Where: x1 = number of bowls produced
x2 = number of mugs produced
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Goal Programming
Model Formulation (2 of 2)
Adding objectives (goals) in order of importance (i.e.
priorities), the company:
Does not want to use fewer than 40 hours of labor per
day.
Would like to achieve a satisfactory profit level of
$1,600 per day.
Prefers not to keep more than 120 pounds of clay on
hand each day.
Would like to minimize the amount of overtime.
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Goal Programming
Goal Constraint Requirements
All goal constraints are equalities that include deviational
variables d- and d+.
A positive deviational variable (d+) is the amount by which a
goal level is exceeded.
A negative deviation variable (d-) is the amount by which a
goal level is underachieved.
At least one or both deviational variables in a goal
constraint must equal zero.
The objective function in a goal programming model seeks
to minimize the deviation from goals in the order of the goal
priorities.
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Goal Programming: Goal Constraints (1 of 3)
x1 + 2x2 = 40 - d1- + d1+
40x1 + 50 x2 = 1,600 - d2- + d2+
4x1 + 3x2 = 120 - d3- + d3+
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
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Goal Programming: Objective Function (2 of 3)
Let Pi= Priority i, where i = 1, 2, 3, and 4.
Labor goals constraint (1, less than 40 hours labor; 4,
minimum overtime):
Minimize P1d1-, P4d1+
Add profit goal constraint (2, achieve profit of $1,600):
Minimize P1d1-, P2d2-, P4d1+
Add material goal constraint (3, avoid keeping more than
120 pounds of clay on hand):
Minimize P1d1-, P2d2-, P3d3+, P4d1+
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Goal Programming
Goal Constraints and Objective Function (3 of 3)
Complete Goal Programming Model:
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
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Goal Programming
Alternative Forms of Goal Constraints (1 of 2)
Changing fourth-priority goal limits overtime to 10 hours
instead of minimizing overtime:
d1- + d4 - - d4+ = 10
minimize P1d1 -, P2d2 -, P3d3 +, P4d4 +
Addition of a fifth-priority goal- due to limited warehouse
space, cannot produce more than 30 bowls and 20 mugs
daily.
x1 + d5 - = 30 bowls
x2 + d6 - = 20 mugs
minimize P1d1 -, P2d2 -, P3d3 -, P4d4 -, 4P5d5 -, 5P5d6 -
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Goal Programming
Alternative Forms of Goal Constraints (2 of 2)
Complete Model with New Goals:
Minimize P1d1-, P2d2-, P3d3-, P4d4-, 4P5d5-, 5P5d6subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50x2 + d2- - d2+ = 1,600
4x1 + 3x2 + d3- - d3+ = 120
d1+ + d4- - d4+ = 10
x1 + d5- = 30
x2 + d6- = 20
x1, x2, d1-, d1+, d2-, d2+, d3-, d3+, d4-, d4+, d5-, d6-  0
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Goal Programming
Graphical Interpretation (1 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.1
Goal Constraints
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Goal Programming
Graphical Interpretation (2 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.2
The First-Priority Goal: Minimize
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Goal Programming
Graphical Interpretation (3 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.3
The Second-Priority Goal: Minimize
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Goal Programming
Graphical Interpretation (4 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.4
The Third-Priority Goal: Minimize
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Goal Programming
Graphical Interpretation (5 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.5
The Fourth-Priority Goal: Minimize
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Goal Programming
Graphical Interpretation (6 of 6)
Goal programming solutions do not always achieve all goals
and they are not optimal, they achieve the best or most
satisfactory solution possible.
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
x1 = 15 bowls
x2 = 20 mugs
d1- = 15 hours
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Goal Programming
Computer Solution Using QM for Windows (1 of 3)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50 x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Exhibit 9.1
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Goal Programming
Computer Solution Using QM for Windows (2 of 3)
Exhibit 9.2
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Goal Programming
Computer Solution Using QM for Windows (3 of 3)
Exhibit 9.3
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Goal Programming
Computer Solution Using Excel (1 of 3)
Exhibit 9.4
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Goal Programming
Computer Solution Using Excel (2 of 3)
Exhibit 9.5
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Goal Programming
Computer Solution Using Excel (3 of 3)
Exhibit 9.6
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Goal Programming
Solution for Alternate Problem Using Excel (1 of 6)
Minimize P1d1-, P2d2-, P3d3-, P4d4-, 4P5d5-, 5P5d6subject to:
x1 + 2x2 + d1- - d1+ = 40
40x1 + 50x2 + d2- - d2+ = 1,600
4x1 + 3x2 + d3- - d3+ = 120
d1+ + d4- - d4+ = 10
x1 + d5- = 30
x2 + d6- = 20
x1, x2, d1-, d1+, d2-, d2+, d3-, d3+, d4-, d4+, d5-, d6-  0
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Goal Programming
Solution for Alternate Problem Using Excel (2 of 6)
Exhibit 9.7
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Goal Programming
Solution for Alternate Problem Using Excel (3 of 6)
Exhibit 9.8
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Goal Programming
Solution for Alternate Problem Using Excel (4 of 6)
Exhibit 9.9
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Goal Programming
Solution for Alternate Problem Using Excel (5 of 6)
Exhibit 9.10
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Goal Programming
Solution for Alternate Problem Using Excel (6 of 6)
Exhibit 9.11
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Goal Programming
Excel Spreadsheets (1 of 4)
Exhibit 9.12
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Goal Programming
Excel Spreadsheets (2 of 4)
Exhibit 9.13
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Goal Programming
Excel Spreadsheets (3 of 4)
Exhibit 9.14
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Goal Programming
Excel Spreadsheets (4 of 4)
Exhibit 9.15
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Goal Programming Example Problem
Problem Statement
Public relations firm survey interviewer staffing requirements
determination.
One person can conduct 80 telephone interviews or 40 personal
interviews per day.
$50/ day for telephone interviewer; $70 for personal interviewer.
Goals (in priority order):
At least 3,000 total interviews.
Interviewer conducts only one type of interview each day. Maintain
daily budget of $2,500.
At least 1,000 interviews should be by telephone.
Formulate a goal programming model to determine number of
interviewers to hire in order to satisfy the goals, and then solve the
problem.
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