INTRODUCTION TO
MANAGEMENT
SCIENCE, 13e
Anderson
Sweeney
Williams
Martin
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a
©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slides by
JOHN
LOUCKS
St. Edward’s
University
Slide 1
Chapter 14
Multicriteria Decisions
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

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Goal Programming
Goal Programming: Formulation
and Graphical Solution
Scoring Models
Analytic Hierarchy Process (AHP)
Establishing Priorities Using AHP
Using AHP to Develop an Overall Priority Ranking
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2008 Thomson South-Western. All Rights Reserved
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Slide 2
Goal Programming



Goal programming may be used to solve linear
programs with multiple objectives, with each
objective viewed as a "goal".
In goal programming, di+ and di- , deviation
variables, are the amounts a targeted goal i is
overachieved or underachieved, respectively.
The goals themselves are added to the constraint
set with di+ and di- acting as the surplus and slack
variables.
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2008 Thomson South-Western. All Rights Reserved
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Slide 3
Goal Programming

Problem (p.651)
• Multiple goals :
maximum total return, minimum total risk.
• In LP,
Maximize the total return subject to allowable
maximum risk
Or
Minimizing the total risk subject to guaranteed
minimum return.

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2008 Thomson South-Western. All Rights Reserved
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Slide 4
Goal Programming



Constraint and goals.
• Maximum available fund $80,000
• Maximum allowable risk 700
• Minimum total return $9,000
Decision variables
U = number of shares of U.S. Oil purchased
H = number of shares of Hub properties purchased
d1 + , d1 – : deviation from the goal 700
d2 + , d2 – : deviation from the goal 9000
Constraints
25 U + 50 H <= 80000
0.5U + 0.25H = 700 + d1+ – d1–
3 U + 5H = 9000 + d2 + – d2 –
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2008 Thomson South-Western. All Rights Reserved
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Slide 5
Goal Programming


Objective function
Preemptive priorities : P1, P2, . . ., Pk
Pi should be should be optimized before Pi+1 is
optimized
In most goal programming models deviation variables
are minimized.
Goal programming model
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2008 Thomson South-Western. All Rights Reserved
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Slide 6
Goal Programming

Graphical solution
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2008 Thomson South-Western. All Rights Reserved
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Slide 7
Goal Programming


Computer solution
If a Goal programming
solver is not available, LP solver should be used in a
sequence of P1 goal and then, P2 goal, … etc.
Higher goal achievement will be inserted in the
formulation of lower goal formulation as
a constraint.
For goal 1
Min. d1+
25 U + 50 H <= 80000
0.5U + 0.25H – d1+ + d1– = 700
3 U + 5H – d2 + + d2– = 9000
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 8
Goal Programming

P1 solution
Objective value:
0.000000
Variable
Value
D1P
0.000000
U
0.000000
H
0.000000
D1M
700.0000
D2P
0.000000
D2M
9000.000
Row
1
2
3
4
Reduced Cost
1.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Slack or Surplus
Dual Price
0.000000
-1.000000
80000.00
0.000000
0.000000
0.000000
0.000000
0.000000
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a
©
2008 Thomson South-Western. All Rights Reserved
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Slide 9
Goal Programming

For goal 2
Min. d2–

25 U + 50 H <= 80000
0.5U + 0.25H – d1+ + d1– = 700
3 U + 5H – d2 + + d2– = 9000
d1+ = 0
P2 solution
Objective value:
Variable
Value
D2M
600.0000
U
800.0000
H
1200.000
D1P
0.000000
D1M
0.000000
D2P
0.000000
600.0000
Reduced Cost
0.000000
0.000000
0.000000
0.000000
1.333333
1.000000
Row
1
2
3
4
5
Slack or Surplus Dual Price
600.0000
-1.000000
0.000000
0.9333333E-01
0.000000
1.333333
0.000000
-1.000000
0.000000
1.333333
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a
©
2008 Thomson South-Western. All Rights Reserved
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Slide 10
Scoring Model for Job Selection (p.665)
A graduating college student with a double major
in Finance and Accounting has received
the following three job offers:
• financial analyst for an investment
firm in Chicago
• accountant for a manufacturing
firm in Denver
• auditor for a CPA firm in Houston
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 11
Scoring Model for Job Selection


The student made the following comments:
• “The financial analyst position
provides the best opportunity for my
long-run career advancement.”
• “I would prefer living in Denver
rather than in Chicago or Houston.”
• “I like the management style and
philosophy at the Houston CPA firm
the best.”
Clearly, this is a multicriteria decision.
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2008 Thomson South-Western. All Rights Reserved
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Slide 12
Scoring Model for Job Selection



Considering only the long-run career
advancement criterion:
• The financial analyst position in
Chicago is the best decision alternative.
Considering only the location criterion:
• The accountant position in Denver
is the best decision alternative.
Considering only the style criterion:
• The auditor position in Houston is the best
alternative.
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2008 Thomson South-Western. All Rights Reserved
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Slide 13
Scoring Models

Procedure summary
• Setup the criteria (kind of goals)
• Choose weights of importance and assign the weights
to each criterion (subjective weights)
(p.666 and Table 14.1 on p.667)
• Choose scales to evaluate alternatives for each
criterion (p.667)
• Evaluate each alternative for each criterion one by
one using the chosen scales (Table 14.2 on p668)
• Get the weighted average (scores) for each alternative
(Table 14.3 on p.668)
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2008 Thomson South-Western. All Rights Reserved
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Slide 14
Scoring Model: Step 1

List of Criteria (p.666)
• Career advancement
• Location
• Management
• Salary
• Prestige
• Job Security
• Enjoyable work
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2008 Thomson South-Western. All Rights Reserved
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Slide 15
Scoring Model: Step 2

Five-Point Scale Chosen for criteria (Importance,
p.666)
Importance
Weight
Very unimportant
1
Somewhat unimportant
2
Average importance
3
Somewhat important
4
Very important
5
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 16
Scoring Model: Step 2

Assigning a Weight to Each Criterion (p.667)
Criterion
Career advancement
Location
Management
Salary
Prestige
Job security
Enjoyable work
Importance
Weight
Very important
5
Average importance
3
Somewhat important
4
Average importance
3
Somewhat unimportant 2
Somewhat important
4
Very important
5
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 17
Scoring Model: Step 3

Nine-Point Scale Chosen to evaluate alternatives (p.667)
Level of Satisfaction
Extremely low
Very low
Low
Slightly low
Average
Slightly high
High
Very high
Extremely high
Rating
1
2
3
4
5
6
7
8
9
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 18
Scoring Model: Step 3

Rate how well each decision alternative satisfies
each criterion. (p.668)
Criterion
Career advancement
Location
Management
Salary
Prestige
Job security
Enjoyable work
Decision Alternative
Analyst Accountant Auditor
Chicago
Denver
Houston
8
6
4
3
8
7
5
6
9
6
7
5
7
5
4
4
7
6
8
6
5
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 19
Scoring Model: Step 4

Compute the score for each decision alternative.
Decision Alternative 1 - Analyst in Chicago
Criterion
Career advancement
Location
Management
Salary
Prestige
Job security
Enjoyable work
Weight (wi )
5
3
4
3
2
4
5
Rating (ri1)
x
Score
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©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
8
3
5
6
7
4
8
wiri1
=
40
9
20
18
14
16
40
157
Slide 20
Scoring Model: Step 4

Compute the score for each decision alternative.
s j   wi rij
i
S1 = 5(8)+3(3)+4(5)+3(6)+2(7)+4(4)+5(8) = 157
S2 = 5(6)+3(8)+4(6)+3(7)+2(5)+4(7)+5(6) = 167
S3 = 5(4)+3(7)+4(9)+3(5)+2(4)+4(6)+5(5) = 149
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©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 21
Scoring Model: Step 4

Compute the score for each decision alternative. (p.668)
Criterion
Career advancement
Location
Management
Salary
Prestige
Job security
Enjoyable work
Score
Decision Alternative
Analyst
Accountant
Auditor
Chicago
Denver
Houston
40
9
20
18
14
16
40
30
24
24
21
10
28
30
20
21
36
15
8
24
25
157
167
149
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 22
Scoring Model: Step 5

Order the decision alternatives from highest
score to lowest score. The alternative with the highest
score is the recommended alternative.
• The accountant position in Denver has the highest
score and is the recommended decision alternative.
.
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2008 Thomson South-Western. All Rights Reserved
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Slide 23
Analytic Hierarchy Process
The Analytic Hierarchy Process (AHP), is a
procedure designed to quantify managerial judgments
of the relative importance of each of several
conflicting criteria used in the decision making
process.
Developed by Saaty(1982), but still used a lot.
While scoring model assigns weights to criteria (1~5)
and alternatives(1~9) simultaneously considering
all the criteria or all the alternatives,
AHP assign weights to each pairwise comparison
of criteria or to each pairwise comparison of
alternatives.
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2008 Thomson South-Western. All Rights Reserved
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Slide 24
Analytic Hierarchy Process


Problem (p.670)
To select a car (Accord, Saturn, Cavalier) based on
the characteristics of the cars.
Criteria are Price, MPG, Comfort, Style
Any criterion can be divided into subcriteria
such as Comfort(Interior, Sound system)
and Style(Color, Body type)
Hierarchy (p.671)
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2008 Thomson South-Western. All Rights Reserved
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Slide 25
Analytic Hierarchy Process

Another Hierarchy (from Wikipedia)
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2008 Thomson South-Western. All Rights Reserved
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Slide 26
Analytic Hierarchy Process

Scale for the Importance of Criteria(p.673)

Pairwise comparison of Criteria (p.673)
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Slide 27
Analytic Hierarchy Process

Pairwise Comparison Matrix for Criteria(p.675)
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2008 Thomson South-Western. All Rights Reserved
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Slide 28
Analytic Hierarchy Process

Normalized Matrix and Matrix with Priority (p.676)
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2008 Thomson South-Western. All Rights Reserved
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Slide 29
Analytic Hierarchy Process

Normalized Matrix and Matrix with Priority (p.676)
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2008 Thomson South-Western. All Rights Reserved
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Slide 30
Analytic Hierarchy Process

Scale for Preference of alternatives (p.679)

Pairwise comparison matrix of alternatives for each criterion
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 31
Analytic Hierarchy Process

Priorities for each alternative using each criterion(p.680)

Choosing an alternative by calculating weighted average.
s j   wi rij
i
Overall Priorities
Pr(Accord) = 0.398(0.123)+0.085(0.087)+0.218(0.593)+0.299(0.265) = 0.265
Pr(Saturn) = 0.398(0.320)+0.085(0.274)+0.218(0.341)+0.299(0.656) = 0.421
Pr(Cavalier) = 0.398(0.557)+0.085(0.639)+0.218(0.065)+0.299(0.08) = 0.314
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2008 Thomson South-Western. All Rights Reserved
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Slide 32
Analytic Hierarchy Process

Procedure
• Step 1 : List the Overall Goal, Criteria, and Decision
Alternatives
• Step 2: Develop a Pair-wise Comparison Matrix (Criteria)
Scales for the importance are (1/9, 1/8, . . . 1, 2, . . ., 9)
Perform a consistency test
• Step 3: Develop a Normalized Matrix and the Priority
Vector (Criteria)
• Step 4: Develop a Pairwise Comparison Matrices of
alternatives for each criterion (Alternatives)
Scales are (1/9, 1/8, . . . 1, 2, . . ., 9)
Perform consistency tests
• Step 5: Devlop a Normalized Matrix and the Priority
Vector (Alternatives)
• Step 6: Get the weighted average using s j   wi rij
i
• Step 7: Choose the best alternative
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2008 Thomson South-Western. All Rights Reserved
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Slide 33
Analytic Hierarchy Process


How to handle subcriteria
Perform Steps 2~6 for subcriteria.
Perform Steps 2 ~ 7 for higher level criteria. (see slide 24)
Consistency Test
• Are we sure that pairwise comparison matrix is
consistent?
If A >> B(A is preferred to B) and B >> C, then A >> C
• Perfect consistency
a(i, k) = a(i, j) x a(j, k) for all i, j
ex. 6 = 2 x 3  (C : B : A) = 1 : 3 : 6
see Excel Sheet
If the pairwise comparison matrix is perfectly consistent,
the largest eigen value max  n
• Number of perfect consistent matrix of n = 3 is 43.
(1, 1, 1), (1, 2, 2), . . . (1, 9, 9), (2, 2, 4), (2, 3, 6), (2, 4, 8)
(3, 3, 9) etc.
(8+2)x2x2 + 3 = 43 out of 19^3 = 6859
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2008 Thomson South-Western. All Rights Reserved
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Slide 34
Analytic Hierarchy Process

Consistency Test
• Calculate max (p.677)
Multiply each column of comparison matrix
with the corresponding element of priority vector
Get the row sum of the matrix obtained from above
Divide each element of row sum vector (obtained above)
by the corresponding element of priority vector
max is the average of elements of the vector obtained
above
• Consistency Index
CI 
max  n
n 1
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Slide 35
Analytic Hierarchy Process

Consistency Test
max  n
CI 
• Consistency Ratio
n 1
CI
CR 
RI
Where RI is the value depending on the number of
criteria or alternatives.
n
RI
3
4
5
6
7
8
9
10
11
0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 1.51
If CR <= 0.1, then the pairwise comparison matrix is
considered as consistent.
•see Excel Sheet
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 36
End of Chapter 14
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a
©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 37