Ch14-Multicriteria Decision Making

Introduction to Management Science
(8th Edition, Bernard W. Taylor III)

Chapter 14
Multicriteria Decision Making

Chapter 14 - Multicriteria Decision Making

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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 14 - Multicriteria Decision Making

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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.

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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

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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.

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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.

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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+

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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

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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 -

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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

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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 14.1
Goal Constraints
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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 14.2
The First-Priority Goal: Minimize
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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 14.3
The Second-Priority Goal: Minimize
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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 14.4
The Third-Priority Goal: Minimize
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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 14.5

The Fourth-Priority Goal: Minimize
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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
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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 14.1
Chapter 14 - Multicriteria Decision Making

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Goal Programming
Computer Solution Using QM for Windows (2 of 3)

Exhibit 14.2
Chapter 14 - Multicriteria Decision Making

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Goal Programming
Computer Solution Using QM for Windows (3 of 3)

Exhibit 14.3
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Goal Programming
Computer Solution Using Excel (1 of 3)

Exhibit 14.4

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Goal Programming
Computer Solution Using Excel (2 of 3)

Exhibit 14.5

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Goal Programming
Computer Solution Using Excel (3 of 3)

Exhibit 14.6

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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 14 - Multicriteria Decision Making

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Goal Programming
Solution for Altered Problem Using Excel (2 of 6)

Exhibit 14.7

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Goal Programming
Solution for Altered Problem Using Excel (3 of 6)

Exhibit 14.8
Chapter 14 - Multicriteria Decision Making

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Goal Programming
Solution for Altered Problem Using Excel (4 of 6)

Exhibit 14.9

Chapter 14 - Multicriteria Decision Making

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Goal Programming
Solution for Altered Problem Using Excel (5 of 6)

Exhibit 14.10

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Goal Programming
Solution for Altered Problem Using Excel (6 of 6)

Exhibit 14.11

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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.

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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
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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.

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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 14 - Multicriteria Decision Making

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Analytical Hierarchy Process
Pair-wise Comparisons
In a pair-wise comparison, two alternatives are compared
according to a criterion and one is preferred.
A preference scale assigns numerical values to different
levels of performance.

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Analytical Hierarchy Process
Pair-wise Comparisons (2 of 2)

Table 14.1
Preference Scale for Pair-wise Comparisons
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Analytical Hierarchy Process
Pair-wise Comparison Matrix
A pair-wise comparison matrix summarizes the pair-wise
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 14 - Multicriteria Decision Making

Infrastructure

1 1/3 1

3 1 7













1 1/7 1

Transportation

1 1/3 1/2

3 1
4









2 1/4

1





35

Analytical Hierarchy Process
Developing Preferences Within Criteria (1 of 3)
In synthetization, decision alternatives are prioritized with
each criterion and then normalized:
Site
A
B
C

Site
A
B
C

Customer Market
A
B
C
3
2
1
1
1/5
1/3
5
1
1/2
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

Chapter 14 - Multicriteria Decision Making

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Analytical Hierarchy Process
Developing Preferences Within Criteria (2 of 3)

Table 14.2
The Normalized Matrix with Row Averages

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Analytical Hierarchy Process
Developing Preferences Within Criteria (3 of 3)

Table 14.3
Criteria Preference Matrix

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Analytical Hierarchy Process
Ranking the Criteria (1 of 2)
Pair-wise Comparison Matrix:
Criteria
Market
Income
Infrastructure
Transportation

Market
1
5
1/3
1/4

Income
1/5
1
1/9
1/7

Infrastructure Transportation
3
4
9
7
1
2
1/2
1

Table 14.4
Normalized Matrix for Criteria with Row Averages
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Analytical Hierarchy Process
Ranking the Criteria (2 of 2)
Preference Vector:
Market
Income
Infrastructure
Transportation

Chapter 14 - 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:

Site
Charlotte
Atlanta
Birmingham

Chapter 14 - Multicriteria Decision Making

Score
0.5314
0.3091
0.1595
1.0000
41

Analytical Hierarchy Process
Summary of Mathematical Steps
Develop a pair-wise comparison matrix for each decision alternative for
each criteria.
Synthetization
Sum the values of each column of the pair-wise 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 pair-wise 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.
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Goal Programming
Excel Spreadsheets (1 of 4)

Exhibit 14.12

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Goal Programming
Excel Spreadsheets (2 of 4)

Exhibit 14.13
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Goal Programming
Excel Spreadsheets (3 of 4)

Exhibit 14.14

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Goal Programming
Excel Spreadsheets (4 of 4)

Exhibit 14.15

Chapter 14 - Multicriteria Decision Making

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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.

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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)
0.30
School proximity
0.25
Median income
0.25
Vehicular traffic
0.10
Mall quality, size
0.10
Other shopping

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 14 - Multicriteria Decision Making

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Scoring Model
Excel Solution

Exhibit 14.16

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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.
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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

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Goal Programming Example Problem
Solution (2 of 2)
Step 2: QM for Windows Solution:

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Analytical Hierarchy Process Example Problem
Problem Statement
Purchasing decision, three model alternatives, three
decision criteria.
Pair-wise comparison matrices:
Bike X
X
1
Y
1/3
1/6
Z

Price
Y
3
1
1/2

Z
6
2
1

Bike
X
Y
Z

Gear Action
X
Y
Z
1/3 1/7
1
1/4
1
3
4
1
7

Weight/Durability
Bike
X
Y
Z
1
X
1
3
1/2
Y
1/3
1
2
1
Z
1

Prioritized decision criteria:
C rite ria
P ric e
G e a rs
W e ig h t

P ric e
1
1 /3
1 /5

Chapter 14 - Multicriteria Decision Making

G e a rs
3
1
1 /2

W e ig h t
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 pair-wise 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

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Analytical Hierarchy Process Example Problem
Problem Solution (2 of 4)
Step 1 continued: Develop normalized matrices and
preference vectors for all the pair-wise 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

Chapter 14 - Multicriteria Decision Making

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 14 - 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 14 - Multicriteria Decision Making

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