### No Slide Title

```Introduction to Management Science
9th 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
Scoring Models
Chapter 9 - Multicriteria Decision Making
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.
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
4
Goal Programming Example
Problem Data (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 the respective goals in the
order of the goal priorities.
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
(lbs of clay/day)
7
Goal Programming Model Formulation
Objective Function (2 of 3)
Labor goals constraint (priority 1 - less than 40 hours labor;
priority 4 - minimum overtime):
Minimize P1d1-, P4d1+
Add profit goal constraint (priority 2 - achieve profit of \$1,600):
Minimize P1d1-, P2d2-, P4d1+
Add material goal constraint (priority 3 - avoid keeping more
than 120 pounds of clay on hand):
Minimize P1d1-, P2d2-, P3d3+, P4d1+
Chapter 9 - Multicriteria Decision Making
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
(labor)
40x1 + 50 x2 + d2 - - d2 + = 1,600
(profit)
4x1 + 3x2 + d3 - - d3 + = 120
(clay)
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Chapter 9 - Multicriteria Decision Making
9
Goal Programming
Alternative Forms of Goal Constraints (1 of 2)
Changing fourth-priority goal “limits overtime to 10 hours”
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
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
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
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
Chapter 9 - Multicriteria Decision Making
17
Goal Programming
Computer Solution Using Excel (1 of 3)
Exhibit 9.4
Chapter 9 - Multicriteria Decision Making
18
Goal Programming
Computer Solution Using Excel (2 of 3)
Exhibit 9.5
Chapter 9 - Multicriteria Decision Making
19
Goal Programming
Computer Solution Using Excel (3 of 3)
Exhibit 9.6
Chapter 9 - Multicriteria Decision Making
20
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
21
Goal Programming
Solution for Altered Problem Using Excel (2 of 6)
Exhibit 9.7
Chapter 9 - Multicriteria Decision Making
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Goal Programming
Solution for Altered Problem Using Excel (3 of 6)
Exhibit 9.8
Chapter 9 - Multicriteria Decision Making
23
Goal Programming
Solution for Altered Problem Using Excel (4 of 6)
Exhibit 9.9
Chapter 9 - Multicriteria Decision Making
24
Goal Programming
Solution for Altered Problem Using Excel (5 of 6)
Exhibit 9.10
Chapter 9 - Multicriteria Decision Making
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Goal Programming
Solution for Altered Problem Using Excel (6 of 6)
Exhibit 9.11
Chapter 9 - Multicriteria Decision Making
26
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
27
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
28
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
29
Analytical Hierarchy Process
General Mathematical Process
Mathematically determine preferences for sites with respect
to each criterion.
Mathematically determine preferences for criteria (rank
order of importance).
Combine these two sets of preferences to mathematically
derive a composite score for each site.
Select the site with the highest score.
Chapter 9 - Multicriteria Decision Making
30
Analytical Hierarchy Process
Pairwise Comparisons (1 of 2)
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
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Analytical Hierarchy Process
Pairwise Comparisons (2 of 2)
Table 9.1 Preference Scale for Pairwise Comparisons
Chapter 9 - Multicriteria Decision Making
32
Analytical Hierarchy Process
Pairwise Comparison Matrix
A pairwise comparison matrix summarizes the pairwise
comparisons for a criteria.
Customer Market
A
B
1
3
1/3
1
1/2
5
Site
A
B
C
Income Level
A
B
C







1
6 1/3

1/6 1 1/9
3
9
1



Infrastructure







1 1/3 1
3
1

7



1 1/7 1
Chapter 9 - Multicriteria Decision Making
C
2
1/5
1
Transportation







1 1/3 1/2
3
1
2 1/4
4
1





33
Analytical Hierarchy Process
Developing Preferences Within Criteria (1 of 3)
In synthesization, decision alternatives are prioritized with
each criterion and then normalized:
Customer Market
Site
A
B
C
A
1
3
2
B
1/3
1
1/5
C
1/2
5
1
11/6
9
16/5
Site
A
B
C
A
6/11
2/11
3/11
Customer Market
B
C
3/9
5/8
1/9
1/16
5/9
5/16
Chapter 9 - Multicriteria Decision Making
34
Analytical Hierarchy Process
Developing Preferences Within Criteria (2 of 3)
The row average values represent the preference vector
Table 9.2 The Normalized Matrix with Row Averages
Chapter 9 - Multicriteria Decision Making
35
Analytical Hierarchy Process
Developing Preferences Within Criteria (3 of 3)
Preference vectors for other criteria are computed similarly,
resulting in the preference matrix
Table 9.3 Criteria Preference Matrix
Chapter 9 - Multicriteria Decision Making
36
Analytical Hierarchy Process
Ranking the Criteria (1 of 2)
Pairwise Comparison Matrix:
Criteria
Market Income Infrastructure Transportation
Market
1
1/5
3
4
Income
5
1
9
7
Infrastructure
1/3
1/9
1
2
Transportation
1/4
1/7
1/2
1
Table 9.4 Normalized Matrix for Criteria with Row Averages
Chapter 9 - Multicriteria Decision Making
37
Analytical Hierarchy Process
Ranking the Criteria (2 of 2)
Preference Vector for Criteria:
Market
Income
Infrastructure
Transportation











0.1993
0.6535
0.0860









0.0612
Chapter 9 - Multicriteria Decision Making
38
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:
Site
Charlotte
Atlanta
Birmingham
Chapter 9 - Multicriteria Decision Making
Score
0.5314
0.3091
0.1595
1.0000
39
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
40
Analytical Hierarchy Process: Consistency (1 of 3)
Consistency Index (CI): Check for consistency and validity of
multiple pairwise comparisons
Example: Southcorp’s consistency in the pairwise comparisons of the 4
site selection criteria
Step 1: Multiply the pairwise comparison matrix of the 4 criteria
by its preference vector
Market Income Infrastruc. Transp.
Criteria
Market
1
1/5
3
4
0.1993
Income
5
1
9
7
X
0.6535
Infrastructure 1/3
1/9
1
2
0.0860
Transportation 1/4
1/7
1/2
1
0.0612
(1)(.1993)+(1/5)(.6535)+(3)(.0860)+(4)(.0612) = 0.8328
(5)(.1993)+(1)(.6535)+(9)(.0860)+(7)(.0612) = 2.8524
(1/3)(.1993)+(1/9)(.6535)+(1)(.0860)+(2)(.0612) = 0.3474
(1/4)(.1993)+(1/7)(.6535)+(1/2)(.0860)+(1)(.0612) = 0.2473
Chapter 9 - Multicriteria Decision Making
41
Analytical Hierarchy Process: Consistency (2 of 3)
Step 2: Divide each value by the corresponding weight from the
preference vector and compute the average
0.8328/0.1993 = 4.1786
2.8524/0.6535 = 4.3648
0.3474/0.0860 = 4.0401
0.2473/0.0612 = 4.0422
16.257
Average = 16.257/4
= 4.1564
Step 3: Calculate the Consistency Index (CI)
CI = (Average – n)/(n-1), where n is no. of items compared
CI = (4.1564-4)/(4-1) = 0.0521
(CI = 0 indicates perfect consistency)
Chapter 9 - Multicriteria Decision Making
42
Analytical Hierarchy Process: Consistency (3 of 3)
Step 4: Compute the Ratio CI/RI
where RI is a random index value obtained from Table 9.5
Table 9.5 Random Index Values for n Items Being Compared
CI/RI = 0.0521/0.90 = 0.0580
Note: Degree of consistency is satisfactory if CI/RI < 0.10
Chapter 9 - Multicriteria Decision Making
43
Analytical Hierarchy Process
Exhibit 9.12
Chapter 9 - Multicriteria Decision Making
44
Analytical Hierarchy Process
Exhibit 9.13
Chapter 9 - Multicriteria Decision Making
45
Analytical Hierarchy Process
Exhibit 9.14
Chapter 9 - Multicriteria Decision Making
46
Analytical Hierarchy Process
Exhibit 9.15
Chapter 9 - Multicriteria Decision Making
47
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 criterion 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
48
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
S1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80)
S2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30)
S3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50)
S4 = (.30)(60) + (.25)(90) + (.25)(85) + (.10)(90) + (.10)(70)
Mall 3
90
65
79
80
50
=
=
=
=
Mall 4
60
90
85
90
70
62.75
73.50
76.00
77.75
Mall 4 preferred because of highest score, followed by malls 3, 2, 1.
Chapter 9 - Multicriteria Decision Making
49
Scoring Model
Excel Solution
Exhibit 9.16
Chapter 9 - Multicriteria Decision Making
50
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
51
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
Gears
3
1
1/2
Chapter 9 - Multicriteria Decision Making
Weight
5
2
1
52
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
X
0.0909
0.2727
0.6364
Y
0.6667
0.2222
0.1111
Gear Action
Y
0.0625
0.1875
0.7500
Z
0.6667
0.2222
0.1111
Z
0.1026
0.1795
0.7179
Chapter 9 - Multicriteria Decision Making
Row Averages
0.6667
0.2222
0.1111
1.0000
Row Averages
0.0853
0.2132
0.7014
1.0000
53
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
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
Chapter 9 - Multicriteria Decision Making
54
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









Weight
0.6250
0.2500
0.1250
Row Averages
0.6479
0.2299
0.1222
1.0000
0.6479







0.2299
0.1222
Chapter 9 - Multicriteria Decision Making
55
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
56
End of chapter
Chapter 9 - Multicriteria Decision Making
57
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