Multicriteria Decision Making

1
Introduction to Management Science 8th
Edition by Bernard W. Taylor III
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.

3
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
4
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.

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 in a goal programming
  model seeks to minimize the deviation from goals
  in the order of the goal priorities.

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

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

8
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
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- 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 -

10
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
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
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
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
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
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
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 x1 15 bowls x2 20
mugs d1- 15 hours
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
18
Goal Programming Computer Solution Using QM for
Windows (2 of 3)
Exhibit 9.2
19
Goal Programming Computer Solution Using QM for
Windows (3 of 3)
Exhibit 9.3
20
Goal Programming Computer Solution Using Excel (1
of 3)
Exhibit 9.4
21
Goal Programming Computer Solution Using Excel (2
of 3)
Exhibit 9.5
22
Goal Programming Computer Solution Using Excel (3
of 3)
Exhibit 9.6
23
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
24
Goal Programming Solution for Alternate Problem
Using Excel (2 of 6)
Exhibit 9.7
25
Goal Programming Solution for Alternate Problem
Using Excel (3 of 6)
Exhibit 9.8
26
Goal Programming Solution for Alternate Problem
Using Excel (4 of 6)
Exhibit 9.9
27
Goal Programming Solution for Alternate Problem
Using Excel (5 of 6)
Exhibit 9.10
28
Goal Programming Solution for Alternate Problem
Using Excel (6 of 6)
Exhibit 9.11
29
Goal Programming Excel Spreadsheets (1 of 4)
Exhibit 9.12
30
Goal Programming Excel Spreadsheets (2 of 4)
Exhibit 9.13
31
Goal Programming Excel Spreadsheets (3 of 4)
Exhibit 9.14
32
Goal Programming Excel Spreadsheets (4 of 4)
Exhibit 9.15
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.