1. Introduction to Management Science
8th Edition by
Bernard W. Taylor III
Chapter 9
Multicriteria Decision Making
Chapter 9 - Multicriteria Decision Making

2. 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.
Chapter 9 - Multicriteria Decision Making

3. Model Formulation (1 of 2)
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
Chapter 9 - Multicriteria Decision Making

4. Model Formulation (2 of 2)
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.
Chapter 9 - Multicriteria Decision Making

5. Goal Constraint Requirements
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.
Chapter 9 - Multicriteria Decision Making

6. Goal Programming: Goal Constraints (1 of 3)
x1 + 2x2 = 40 - d1- + d1+
40x x2 = 1,600 - d2- + d2+
4x1 + 3x2 = d3- + d3+
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Chapter 9 - Multicriteria Decision Making

7. 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+
Chapter 9 - Multicriteria Decision Making

8. Goal Constraints and Objective Function (3 of 3)
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
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Chapter 9 - Multicriteria Decision Making

9. Alternative Forms of Goal Constraints (1 of 2)
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 -
Chapter 9 - Multicriteria Decision Making

10. Alternative Forms of Goal Constraints (2 of 2)
Goal Programming
Alternative 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. Graphical Interpretation (1 of 6)
Goal Programming
Graphical Interpretation (1 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.1
Goal Constraints
Chapter 9 - Multicriteria Decision Making

12. The First-Priority Goal: Minimize
Goal Programming
Graphical Interpretation (2 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x 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
Chapter 9 - Multicriteria Decision Making

13. The Second-Priority Goal: Minimize
Goal Programming
Graphical Interpretation (3 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x 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
Chapter 9 - Multicriteria Decision Making

14. The Third-Priority Goal: Minimize
Goal Programming
Graphical Interpretation (4 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x 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
Chapter 9 - Multicriteria Decision Making

15. The Fourth-Priority Goal: Minimize
Goal Programming
Graphical Interpretation (5 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x 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
Chapter 9 - Multicriteria Decision Making

16. Graphical Interpretation (6 of 6)
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
40x 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
Chapter 9 - Multicriteria Decision Making

17. Computer Solution Using QM for Windows (1 of 3)
Goal Programming
Computer Solution Using QM for Windows (1 of 3)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Exhibit 9.1
Chapter 9 - Multicriteria Decision Making

18. Computer Solution Using QM for Windows (2 of 3)
Goal Programming
Computer Solution Using QM for Windows (2 of 3)
Exhibit 9.2
Chapter 9 - Multicriteria Decision Making

19. Computer Solution Using QM for Windows (3 of 3)
Goal Programming
Computer Solution Using QM for Windows (3 of 3)
Exhibit 9.3
Chapter 9 - Multicriteria Decision Making

20. Computer Solution Using Excel (1 of 3)
Goal Programming
Computer Solution Using Excel (1 of 3)
Exhibit 9.4
Chapter 9 - Multicriteria Decision Making

21. Computer Solution Using Excel (2 of 3)
Goal Programming
Computer Solution Using Excel (2 of 3)
Exhibit 9.5
Chapter 9 - Multicriteria Decision Making

22. Computer Solution Using Excel (3 of 3)
Goal Programming
Computer Solution Using Excel (3 of 3)
Exhibit 9.6
Chapter 9 - Multicriteria Decision Making

23. Solution for Alternate Problem Using Excel (1 of 6)
Goal Programming
Solution for Alternate 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
Chapter 9 - Multicriteria Decision Making

24. Solution for Alternate Problem Using Excel (2 of 6)
Goal Programming
Solution for Alternate Problem Using Excel (2 of 6)
Exhibit 9.7
Chapter 9 - Multicriteria Decision Making

25. Solution for Alternate Problem Using Excel (3 of 6)
Goal Programming
Solution for Alternate Problem Using Excel (3 of 6)
Exhibit 9.8
Chapter 9 - Multicriteria Decision Making

26. Solution for Alternate Problem Using Excel (4 of 6)
Goal Programming
Solution for Alternate Problem Using Excel (4 of 6)
Exhibit 9.9
Chapter 9 - Multicriteria Decision Making

27. Solution for Alternate Problem Using Excel (5 of 6)
Goal Programming
Solution for Alternate Problem Using Excel (5 of 6)
Exhibit 9.10
Chapter 9 - Multicriteria Decision Making

28. Solution for Alternate Problem Using Excel (6 of 6)
Goal Programming
Solution for Alternate Problem Using Excel (6 of 6)
Exhibit 9.11
Chapter 9 - Multicriteria Decision Making

29. Excel Spreadsheets (1 of 4)
Goal Programming
Excel Spreadsheets (1 of 4)
Exhibit 9.12
Chapter 9 - Multicriteria Decision Making

30. Excel Spreadsheets (2 of 4)
Goal Programming
Excel Spreadsheets (2 of 4)
Exhibit 9.13
Chapter 9 - Multicriteria Decision Making

31. Excel Spreadsheets (3 of 4)
Goal Programming
Excel Spreadsheets (3 of 4)
Exhibit 9.14
Chapter 9 - Multicriteria Decision Making

32. Excel Spreadsheets (4 of 4)
Goal Programming
Excel Spreadsheets (4 of 4)
Exhibit 9.15
Chapter 9 - Multicriteria Decision Making

33. 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.
Chapter 9 - Multicriteria Decision Making