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Chapter 9 - Multicriteria Decision Making 1 Chapter 9 Multicriteria Decision Making Introduction to Management Science 8th Edition by Bernard W. Taylor III
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Chapter 9 - Multicriteria Decision Making 1
Chapter 9
Multicriteria Decision Making
Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 9 - Multicriteria Decision Making 2
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.
Chapter Topics
Chapter 9 - Multicriteria Decision Making 3
Beaver Creek Pottery Company Example:
Maximize Z = $40x1 + 50x2
subject to:1x1 + 2x2 40 hours of labor4x2 + 3x2 120 pounds of clayx1, x2 0
Where: x1 = number of bowls produced x2 = number of mugs produced
Goal ProgrammingModel Formulation (1 of 2)
Chapter 9 - Multicriteria Decision Making 4
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.
Goal ProgrammingModel Formulation (2 of 2)
Chapter 9 - Multicriteria Decision Making 5
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.
Goal ProgrammingGoal Constraint Requirements
Chapter 9 - Multicriteria Decision Making 6
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
Chapter 9 - Multicriteria Decision Making 7
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
+
Goal Programming: Objective Function (2 of 3)
Chapter 9 - Multicriteria Decision Making 8
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
Goal ProgrammingGoal Constraints and Objective Function (3 of 3)
Chapter 9 - Multicriteria Decision Making 9
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
-
Goal ProgrammingAlternative Forms of Goal Constraints (1 of 2)
Chapter 9 - Multicriteria Decision Making 10
Goal ProgrammingAlternative Forms of Goal Constraints (2 of 2)
Complete Model with New Goals:
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
Chapter 9 - Multicriteria Decision Making 11
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 9.1Goal Constraints
Goal ProgrammingGraphical Interpretation (1 of 6)
Chapter 9 - Multicriteria Decision Making 12
Figure 9.2The First-Priority Goal: Minimize
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
Goal ProgrammingGraphical Interpretation (2 of 6)
Chapter 9 - Multicriteria Decision Making 13
Figure 9.3The Second-Priority Goal: Minimize
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
Goal ProgrammingGraphical Interpretation (3 of 6)
Chapter 9 - Multicriteria Decision Making 14
Figure 9.4The Third-Priority Goal: Minimize
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
Goal ProgrammingGraphical Interpretation (4 of 6)
Chapter 9 - Multicriteria Decision Making 15
Figure 9.5The Fourth-Priority Goal: Minimize
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
Goal ProgrammingGraphical Interpretation (5 of 6)
Chapter 9 - Multicriteria Decision Making 16
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 bowlsx2 = 20 mugsd1
- = 15 hours
Goal ProgrammingGraphical Interpretation (6 of 6)
Chapter 9 - Multicriteria Decision Making 17
Exhibit 9.1
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
Goal ProgrammingComputer Solution Using QM for Windows (1 of 3)
Chapter 9 - Multicriteria Decision Making 18
Exhibit 9.2
Goal ProgrammingComputer Solution Using QM for Windows (2 of 3)
Chapter 9 - Multicriteria Decision Making 19
Exhibit 9.3
Goal ProgrammingComputer Solution Using QM for Windows (3 of 3)
Chapter 9 - Multicriteria Decision Making 20
Exhibit 9.4
Goal ProgrammingComputer Solution Using Excel (1 of 3)
Chapter 9 - Multicriteria Decision Making 21
Exhibit 9.5
Goal ProgrammingComputer Solution Using Excel (2 of 3)
Chapter 9 - Multicriteria Decision Making 22
Exhibit 9.6
Goal ProgrammingComputer Solution Using Excel (3 of 3)
Chapter 9 - Multicriteria Decision Making 23
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
Goal ProgrammingSolution for Alternate Problem Using Excel (1 of 6)
Chapter 9 - Multicriteria Decision Making 24
Exhibit 9.7
Goal ProgrammingSolution for Alternate Problem Using Excel (2 of 6)
Chapter 9 - Multicriteria Decision Making 25
Exhibit 9.8
Goal ProgrammingSolution for Alternate Problem Using Excel (3 of 6)
Chapter 9 - Multicriteria Decision Making 26
Exhibit 9.9
Goal ProgrammingSolution for Alternate Problem Using Excel (4 of 6)
Chapter 9 - Multicriteria Decision Making 27
Exhibit 9.10
Goal ProgrammingSolution for Alternate Problem Using Excel (5 of 6)
Chapter 9 - Multicriteria Decision Making 28
Exhibit 9.11
Goal ProgrammingSolution for Alternate Problem Using Excel (6 of 6)
Chapter 9 - Multicriteria Decision Making 29
Exhibit 9.12
Goal ProgrammingExcel Spreadsheets (1 of 4)
Chapter 9 - Multicriteria Decision Making 30
Exhibit 9.13
Goal ProgrammingExcel Spreadsheets (2 of 4)
Chapter 9 - Multicriteria Decision Making 31
Exhibit 9.14
Goal ProgrammingExcel Spreadsheets (3 of 4)
Chapter 9 - Multicriteria Decision Making 32
Exhibit 9.15
Goal ProgrammingExcel Spreadsheets (4 of 4)
Chapter 9 - Multicriteria Decision Making 33
Goal Programming Example ProblemProblem 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.
