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Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 9
Multicriteria Decision Making
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
2
Overview
Study of problems with several criteria, multiple criteria,
instead of a single objective when making a decision.
Two techniques discussed: goal programming, and the
analytical hierarchy process.
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.
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
4
Goal Programming
Model Formulation (2 of 2)
Adding objectives (goals) in order of importance, 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 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 and Objective Function (1 of 2)
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
7
Goal Programming
Goal Constraints and Objective Function (2 of 2)
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
Chapter 9 - Multicriteria Decision Making
8
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 -
Chapter 9 - Multicriteria Decision Making
9
Goal Programming
Alternative Forms of Goal Constraints (2 of 2)
Complete Model with 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
Chapter 9 - Multicriteria Decision Making
10
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
Chapter 9 - Multicriteria Decision Making
11
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
Chapter 9 - Multicriteria Decision Making
12
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
Chapter 9 - Multicriteria Decision Making
13
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
Chapter 9 - Multicriteria Decision Making
14
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
Chapter 9 - Multicriteria Decision Making
15
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
Chapter 9 - Multicriteria Decision Making
16
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
Chapter 9 - Multicriteria Decision Making
17
Goal Programming
Computer Solution Using QM for Windows (2 of 3)
Exhibit 9.2
Chapter 9 - Multicriteria Decision Making
18
Goal Programming
Computer Solution Using QM for Windows (3 of 3)
Exhibit 9.3
Chapter 9 - Multicriteria Decision Making
19
Goal Programming
Computer Solution Using Excel (1 of 3)
Exhibit 9.4
Chapter 9 - Multicriteria Decision Making
20
Goal Programming
Computer Solution Using Excel (2 of 3)
Exhibit 9.5
Chapter 9 - Multicriteria Decision Making
21
Goal Programming
Computer Solution Using Excel (3 of 3)
Exhibit 9.6
Chapter 9 - Multicriteria Decision Making
22
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
Chapter 9 - Multicriteria Decision Making
23
Goal Programming
Solution for Altered Problem Using Excel (2 of 6)
Exhibit 9.7
Chapter 9 - Multicriteria Decision Making
24
Goal Programming
Solution for Altered Problem Using Excel (3 of 6)
Exhibit 9.8
Chapter 9 - Multicriteria Decision Making
25
Goal Programming
Solution for Altered Problem Using Excel (4 of 6)
Exhibit 9.9
Chapter 9 - Multicriteria Decision Making
26
Goal Programming
Solution for Altered Problem Using Excel (5 of 6)
Exhibit 9.10
Chapter 9 - Multicriteria Decision Making
27
Goal Programming
Solution for Altered Problem Using Excel (6 of 6)
Exhibit 9.11
Chapter 9 - Multicriteria Decision Making
28
Analytical Hierarchy Process
Overview
AHP is a method for ranking several decision alternatives
and selecting the best one when the decision maker has
multiple objectives, or criteria, on which to base the
decision.
The decision maker makes a decision based on how the
alternatives compare according to several criteria.
The decision maker will select the alternative that best
meets his or her decision criteria.
AHP is a process for developing a numerical score to rank
each decision alternative based on how well the alternative
meets the decision maker’s criteria.
Chapter 9 - Multicriteria Decision Making
29
Analytical Hierarchy Process
Example Problem Statement
Southcorp Development Company shopping mall site
selection.
Three potential sites:
Atlanta
Birmingham
Charlotte.
Criteria for site comparisons:
Customer market base.
Income level
Infrastructure
Chapter 9 - Multicriteria Decision Making
30
Analytical Hierarchy Process
Hierarchy Structure
Top of the hierarchy: the objective (select the best site).
Second level: how the four criteria contribute to the
objective.
Third level: how each of the three alternatives contributes
to each of the four criteria.
Chapter 9 - Multicriteria Decision Making
31
Analytical Hierarchy Process
General Mathematical Process
Mathematically determine preferences for each site for
each criteria.
Mathematically determine preferences for criteria (rank
order of importance).
Combine these two sets of preferences to mathematically
derive a score for each site.
Select the site with the highest score.
Chapter 9 - Multicriteria Decision Making
32
Analytical Hierarchy Process
Pairwise Comparisons
In a pairwise comparison, two alternatives are compared
according to a criterion and one is preferred.
A preference scale assigns numerical values to different
levels of performance.
Chapter 9 - Multicriteria Decision Making
33
Analytical Hierarchy Process
Pairwise Comparisons (2 of 2)
Table 9.1
Preference Scale for Pairwise Comparisons
Chapter 9 - Multicriteria Decision Making
34
Analytical Hierarchy Process
Pairwise Comparison Matrix
A pairwise comparison matrix summarizes the pairwise
comparisons for a criteria.
Customer Market
A
B
C
1
3
2
1/3
1
1/5
1/2
5
1
Site
A
B
C
Income Level
A
B
C







1
6 1/3

1/6 1 1/9
3
9
1



Chapter 9 - Multicriteria Decision Making
Infrastructure







1 1/3 1
3
1

7



1 1/7 1
Transportation







1 1/3 1/2
3
1
2 1/4
4
1





35
Analytical Hierarchy Process
Developing Preferences Within Criteria (1 of 3)
In synthesization, decision alternatives are prioritized with
each criterion and then normalized:
Site
A
B
C
Site
A
B
C
Chapter 9 - Multicriteria Decision Making
Customer Market
A
B
C
1
3
2
1/3
1
1/5
1/2
5
1
11/6
9
16/5
Customer Market
A
B
C
6/11
3/9
5/8
2/11
1/9
1/16
3/11
5/9
5/16
36
Analytical Hierarchy Process
Developing Preferences Within Criteria (2 of 3)
Table 9.2
The Normalized Matrix with Row Averages
Chapter 9 - Multicriteria Decision Making
37
Analytical Hierarchy Process
Developing Preferences Within Criteria (3 of 3)
Table 9.3
Criteria Preference Matrix
Chapter 9 - Multicriteria Decision Making
38
Analytical Hierarchy Process
Ranking the Criteria (1 of 2)
Pairwise Comparison Matrix:
Criteria
Market
Income
Infrastructure
Transportation
Market
1
5
1/3
1/4
Income Infrastructure Transportation
1/5
3
4
1
9
7
1/9
1
2
1/7
1/2
1
Table 9.4
Normalized Matrix for Criteria with Row Averages
Chapter 9 - Multicriteria Decision Making
39
Analytical Hierarchy Process
Ranking the Criteria (2 of 2)
Preference Vector:
Market
Income
Infrastructure
Transportation
Chapter 9 - Multicriteria Decision Making











0.1993
0.6535
0.0860









0.0612
40
Analytical Hierarchy Process
Developing an Overall Ranking
Overall Score:
Site A score = .1993(.5012) + .6535(.2819) +
.0860(.1790) + .0612(.1561) = .3091
Site B score = .1993(.1185) + .6535(.0598) +
.0860(.6850) + .0612(.6196) = .1595
Site C score = .1993(.3803) + .6535(.6583) +
.0860(.1360) + .0612(.2243) = .5314
Overall Ranking:
Chapter 9 - Multicriteria Decision Making
Site
Charlotte
Atlanta
Birmingham
Score
0.5314
0.3091
0.1595
1.0000
41
Analytical Hierarchy Process
Summary of Mathematical Steps
Develop a pairwise comparison matrix for each decision alternative for
each criteria.
Synthesization
Sum the values of each column of the pairwise comparison
matrices.
Divide each value in each column by the corresponding column
sum.
Average the values in each row of the normalized matrices.
Combine the vectors of preferences for each criterion.
Develop a pairwise comparison matrix for the criteria.
Compute the normalized matrix.
Develop the preference vector.
Compute an overall score for each decision alternative
Rank the decision alternatives.
Chapter 9 - Multicriteria Decision Making
42
Goal Programming
Excel Spreadsheets (1 of 4)
Exhibit 9.12
Chapter 9 - Multicriteria Decision Making
43
Goal Programming
Excel Spreadsheets (2 of 4)
Exhibit 9.13
Chapter 9 - Multicriteria Decision Making
44
Goal Programming
Excel Spreadsheets (3 of 4)
Exhibit 9.14
Chapter 9 - Multicriteria Decision Making
45
Goal Programming
Excel Spreadsheets (4 of 4)
Exhibit 9.15
Chapter 9 - Multicriteria Decision Making
46
Scoring Model
Overview
Each decision alternative graded in terms of how well it
satisfies the criterion according to following formula:
Si = gijwj
where:
wj = a weight between 0 and 1.00 assigned to criteria j;
1.00 important, 0 unimportant; sum of total weights
equals one.
gij = a grade between 0 and 100 indicating how well
alternative i satisfies criteria j; 100 indicates high
satisfaction, 0 low satisfaction.
Chapter 9 - Multicriteria Decision Making
47
Scoring Model
Example Problem
Mall selection with four alternatives and five criteria:
Grades for Alternative (0 to 100)
Weight
Decision Criteria (0 to 1.00)
School proximity
0.30
Median income
0.25
Vehicular traffic
0.25
Mall quality, size
0.10
Other shopping
0.10
Mall 1
40
75
60
90
80
Mall 2
60
80
90
100
30
Mall 3
90
65
79
80
50
Mall 4
60
90
85
90
70
S1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) = 62.75
S2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) = 73.50
S3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) = 76.00
S4 = (.30)(60) + (.25)(90) + (.25)(85) + (.10)(90) + (.10)(70) = 77.75
Mall 4 preferred because of highest score, followed by malls 3, 2, 1.
Chapter 9 - Multicriteria Decision Making
48
Scoring Model
Excel Solution
Exhibit 9.16
Chapter 9 - Multicriteria Decision Making
49
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
50
Goal Programming Example Problem
Solution (1 of 2)
Step 1: Model Formulation:
Minimize P1d1-, P2d2-, P3d3subject to:
80x1 + 40x2 + d1- - d1+ = 3,000 interviews
50x1 + 70x2 + d2- - d2 + = $2,500 budget
80x1 + d3- - d3 + = 1,000 telephone interviews
where:
x1 = number of telephone interviews
x2 = number of personal interviews
Chapter 9 - Multicriteria Decision Making
51
Goal Programming Example Problem
Solution (2 of 2)
Step 2: QM for Windows Solution:
Chapter 9 - Multicriteria Decision Making
52
Analytical Hierarchy Process Example Problem
Problem Statement
Purchasing decision, three model alternatives, three
decision criteria.
Pairwise comparison matrices:
Bike X
X
1
Y
1/3
Z
1/6
Price
Y
3
1
1/2
Z
6
2
1
Bike
X
Y
Z
Gear Action
X
Y
Z
1
1/3 1/7
3
1
1/4
7
4
1
Weight/Durability
Bike
X
Y
Z
X
1
3
1
Y
1/3
1
1/2
Z
1
2
1
Prioritized decision criteria:
Criteria
Price
Gears
Weight
Price
1
1/3
1/5
Chapter 9 - Multicriteria Decision Making
Gears
3
1
1/2
Weight
5
2
1
53
Analytical Hierarchy Process Example Problem
Problem Solution (1 of 4)
Step 1: Develop normalized matrices and preference
vectors for all the pairwise comparison matrices for criteria.
Price
Bike
X
Y
Z
Bike
X
Y
Z
X
0.6667
0.2222
0.1111
Y
0.6667
0.2222
0.1111
Z
0.6667
0.2222
0.1111
Row Averages
0.6667
0.2222
0.1111
1.0000
X
0.0909
0.2727
0.6364
Gear Action
Y
0.0625
0.1875
0.7500
Z
0.1026
0.1795
0.7179
Row Averages
0.0853
0.2132
0.7014
1.0000
Chapter 9 - Multicriteria Decision Making
54
Analytical Hierarchy Process Example Problem
Problem Solution (2 of 4)
Step 1 continued: Develop normalized matrices and
preference vectors for all the pairwise comparison matrices
for criteria.
Bike
X
Y
Z
Weight/Durability
X
Y
Z
0.4286
0.5000
0.4000
0.1429
0.1667
0.2000
0.4286
0.3333
0.4000
Bike
X
Y
Z
Chapter 9 - Multicriteria Decision Making
Price
0.6667
0.2222
0.1111
Criteria
Gears
0.0853
0.2132
0.7014
Row Averages
0.4429
0.1698
0.3873
1.0000
Weight
0.4429
0.1698
0.3873
55
Analytical Hierarchy Process Example Problem
Problem Solution (3 of 4)
Step 2: Rank the criteria.
Criteria
Price
Gears
Weight
Price
0.6522
0.2174
0.1304
Gears
0.6667
0.2222
0.1111
Price
Gears
Weight
Chapter 9 - Multicriteria Decision Making









Weight
0.6250
0.2500
0.1250
Row Averages
0.6479
0.2299
0.1222
1.0000
0.6479







0.2299
0.1222
56
Analytical Hierarchy Process Example Problem
Problem Solution (4 of 4)
Step 3: Develop an overall ranking.
Bike X
Bike Y
Bike Z









0.6667 0.0853 0.4429 0.6479





















0.2222 0.2132 0.1698  0.2299
0.1111 0.7014 0.3837 0.1222
Bike X score = .6667(.6479) + .0853(.2299) + .4429(.1222) = .5057
Bike Y score = .2222(.6479) + .2132(.2299) + .1698(.1222) = .2138
Bike Z score = .1111(.6479) + .7014(.2299) + .3873(.1222) = .2806
Overall ranking of bikes: X first followed by Z and Y (sum of
scores equal 1.0000).
Chapter 9 - Multicriteria Decision Making
57
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