Introduction to management science 10e by bernard taylor chapter 09

Multicriteria
Decision Making
Chapter 9

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9-1


Chapter Topics
■ Goal Programming
■ Graphical Interpretation of Goal Programming
■ Computer Solution of Goal Programming Problems
with QM for Windows and Excel
■ The Analytical Hierarchy Process
■ Scoring Models

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Overview
■ Study of problems with several criteria, multiple
criteria, instead of a single objective when making a
decision.
■ Three techniques discussed: goal programming, the
analytical hierarchy process and scoring models.
■ Goal programming is a variation of linear

programming considering more than one objective
(goals) in the objective function.
■ The analytical hierarchy process develops a score
for each decision alternative based on comparisons
of each under different criteria reflecting the
decision makers preferences.
■ Scoring models are based on a relatively simple
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scoring technique.
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9-3


Goal Programming Example
Problem Data (1 of 2)
Beaver Creek Pottery Company Example:
Maximize Z = $40x1 + 50x2
subject to:
1x1 + 2x2  40 hours of labor
4x1 + 3x2  120 pounds of clay
x1 , x 2  0
Where: x1 = number of bowls produced
x2 = number of mugs produced

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Goal Programming Example
Problem Data (2 of 2)
■ Adding objectives (goals) in order of importance,
the company:
1. Does not want to use fewer than 40 hours of
labor per day.
2. Would like to achieve a satisfactory profit level
of $1,600 per day.
3. Prefers not to keep more than 120 pounds of
clay on hand each day.
4. Would like to minimize the amount of overtime.

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9-5


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 seeks to minimize the
deviation from the respective goals in the order of
the goal priorities.
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9-6


al Programming Model Formulation
al Constraints (1 of 3)
Labor goal:
x1 + 2x2 + d1- - d1+ = 40

(hours/day)

Profit goal:
40x1 + 50 x2 + d2 - - d2

+

= 1,600 ($/day)

Material goal:
4x1 + 3x2 + d3 - - d3
clay/day)

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+

= 120

(lbs of
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Goal Programming Model
Formulation
Objective Function (2 of 3)
1. Labor goals constraint

(priority 1 - less than 40 hours labor; priority 4 minimum overtime):


Minimize P1d1-, P4d1+

2. Add profit goal constraint
(priority 2 - achieve profit of $1,600):


Minimize P1d1-, P2d2-, P4d1+

3. Add material goal constraint
(priority 3 - avoid keeping more than 120 pounds of clay
on hand):
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9-8


Goal Programming Model
Formulation
Complete Model (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

(labor)
(profit)
(clay)

x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0

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9-9


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- “important to
achieve the goal for mugs”:
 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 Added 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
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Figure 9.1 Goal
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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 -,
+
Figure
9.2d3The
Goal:
 First-Priority
0
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-

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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 d2-

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9-14


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 d3+

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9-15


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 d1+

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9-16


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
Solution: x1 = 15 bowls
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x2Publishing
= 20as mugs
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9-17


Goal Programming Computer
Solution
+
+ (1 of 3)
Using
MinimizeQM
P1d1for
, P2dWindows
2 , 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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Exhibit
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Goal Programming Computer
Solution
Using QM for Windows (2 of 3)

Exhibit
9.2
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9-19


Goal Programming Computer
Solution
Using QM for Windows (3 of 3)

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Exhibit

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Goal Programming Computer
Solution
Using Excel (1 of 3)

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Exhibit
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Goal Programming
Computer Solution Using Excel (2 of
3)

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Exhibit 9.5

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Goal Programming
Computer Solution Using Excel (3 of
3)

Exhibit

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9-23


Goal Programming
Solution for Altered Problem Using
Excel (1 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d4+, 4P5d5- + 5P5d6-

subject 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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9-24


Goal Programming
Solution for Altered Problem Using
Excel (2 of 6)

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Exhibit
9.7

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