taylor introms10 ppt 09 MEF
Multicriteria Decision
Making
Chapter 9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-2
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 weighted scoring
technique.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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, x2 0
Where: x1 = number of bowls produced
x2 = number of mugs produced
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-4
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 $1,600 per
day.
3.
Prefers not to keep more than 120 pounds of clay on hand
each day.
4.
Would like to minimize the amount of overtime.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-5
Goal Programming
Goal Constraint Requirements
■ All goal constraints are equalities that include deviational
variables d- and d+.
■ A positive deviational variable (d+) is the amount by which a goal
level is exceeded.
■ A negative deviation variable (d-) is the amount by which a goal
level is underachieved.
■ At least one or both deviational variables in a goal constraint must
equal zero.
■ The objective function seeks to minimize the deviation from
the respective goals in the order of the goal priorities.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-6
Goal Programming Model Formulation
Goal Constraints (1 of 3)
Labor goal:
x1 + 2x2 + d1- - d1+ = 40
(hours/day)
Profit goal:
40x1 + 50 x2 + d2 - - d2 + = 1,600 ($/day)
Material goal:
4x1 + 3x2 + d3 - - d3 + = 120
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(lbs of clay/day)
9-7
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):
Minimize P1d1-, P2d2-, P3d3+, P4d1+
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(labor)
(profit)
(clay)
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 -
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-10
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-11
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-12
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 d1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-13
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 d2Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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+
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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+
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
x2 = 20 mugs
d1+ = 15 hours
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-17
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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9-18
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)
Exhibit 9.3
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9-20
Goal Programming Computer Solution
Using Excel (1 of 3)
Exhibit 9.4
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9-21
Goal Programming
Computer Solution Using Excel (2 of 3)
Exhibit 9.5
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9-22
Goal Programming
Computer Solution Using Excel (3 of 3)
Exhibit 9.6
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-23
Goal Programming
Solution for Altered 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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9-24
Goal Programming
Solution for Altered Problem Using Excel (2 of 6)
Exhibit 9.7
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9-25
Goal Programming
Solution for Altered Problem Using Excel (3 of 6)
Exhibit 9.8
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9-26
Goal Programming
Solution for Altered Problem Using Excel (4 of 6)
Exhibit 9.9
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9-27
Goal Programming
Solution for Altered Problem Using Excel (5 of 6)
Exhibit 9.10
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9-28
Goal Programming
Solution for Altered Problem Using Excel (6 of 6)
Exhibit 9.11
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-29
Making
Chapter 9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-2
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 weighted scoring
technique.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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, x2 0
Where: x1 = number of bowls produced
x2 = number of mugs produced
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-4
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 $1,600 per
day.
3.
Prefers not to keep more than 120 pounds of clay on hand
each day.
4.
Would like to minimize the amount of overtime.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-5
Goal Programming
Goal Constraint Requirements
■ All goal constraints are equalities that include deviational
variables d- and d+.
■ A positive deviational variable (d+) is the amount by which a goal
level is exceeded.
■ A negative deviation variable (d-) is the amount by which a goal
level is underachieved.
■ At least one or both deviational variables in a goal constraint must
equal zero.
■ The objective function seeks to minimize the deviation from
the respective goals in the order of the goal priorities.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-6
Goal Programming Model Formulation
Goal Constraints (1 of 3)
Labor goal:
x1 + 2x2 + d1- - d1+ = 40
(hours/day)
Profit goal:
40x1 + 50 x2 + d2 - - d2 + = 1,600 ($/day)
Material goal:
4x1 + 3x2 + d3 - - d3 + = 120
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(lbs of clay/day)
9-7
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):
Minimize P1d1-, P2d2-, P3d3+, P4d1+
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(labor)
(profit)
(clay)
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 -
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-10
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-11
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-12
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 d1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-13
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 d2Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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+
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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+
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
x2 = 20 mugs
d1+ = 15 hours
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-17
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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9-18
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)
Exhibit 9.3
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9-20
Goal Programming Computer Solution
Using Excel (1 of 3)
Exhibit 9.4
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9-21
Goal Programming
Computer Solution Using Excel (2 of 3)
Exhibit 9.5
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-22
Goal Programming
Computer Solution Using Excel (3 of 3)
Exhibit 9.6
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-23
Goal Programming
Solution for Altered 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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9-24
Goal Programming
Solution for Altered Problem Using Excel (2 of 6)
Exhibit 9.7
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9-25
Goal Programming
Solution for Altered Problem Using Excel (3 of 6)
Exhibit 9.8
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9-26
Goal Programming
Solution for Altered Problem Using Excel (4 of 6)
Exhibit 9.9
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9-27
Goal Programming
Solution for Altered Problem Using Excel (5 of 6)
Exhibit 9.10
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9-28
Goal Programming
Solution for Altered Problem Using Excel (6 of 6)
Exhibit 9.11
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
9-29