CH_09_MULTICRITERIA_DECISION_MAKING

Multicriteria Decision
Making

Chapter 9 9-1

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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
■ Scoring Models

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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.

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Goal Programming Example 9-4
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

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Goal Programming Example
Problem Data (2 of 2)

■ Adding objectives (goals) in order of importance, the company:

1. Does not want to use fewer than 40 hours of labor per day.
2. Would like to achieve a satisfactory profit level of $1,600 per

day.
3. Prefers not to keep more than 120 pounds of clay on hand

each day.
4. 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 seeks to minimize the deviation from

the respective goals in the order of the goal priorities.

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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 (lbs of clay/day)

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Goal Programming Model Formulation
Objective Function (2 of 3)

1. Labor goals constraint
(priority 1 - less than 40 hours labor; priority 4 - minimum overtime):

Minimize P1d1-, P4d1+
2. Add profit goal constraint
(priority 2 - achieve profit of $1,600):

Minimize P1d1-, P2d2-, P4d1+
3. Add material goal constraint
(priority 3 - avoid keeping more than 120 pounds of clay on hand):

Minimize P1d1-, P2d2-, P3d3+, P4d1+

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Goal Programming Model Formulation
Complete Model (3 of 3)

Complete Goal Programming Model: (labor)
(profit)
Minimize P1d1-, P2d2-, P3d3+, P4d1+ (clay)
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 Added 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

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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 9.1 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 9.2 The First-Priority Goal: Minimize d1-

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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 9.3 The Second-Priority Goal: Minimize d2-

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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 9.4 The Third-Priority Goal: Minimize d3+

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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 9.5 The Fourth-Priority Goal: Minimize d1+

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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

Solution: 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 9.1

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

Exhibit 9.2

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

Exhibit 9.3

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Goal Programming Computer Solution
Using Excel (1 of 3)

Exhibit 9.4

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Goal Programming
Computer Solution Using Excel (2 of 3)

Exhibit 9.5

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Goal Programming
Computer Solution Using Excel (3 of 3)

Exhibit 9.6 9-23

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Goal Programming
Solution for Altered 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

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

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9-25

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

Exhibit 9.8

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

Exhibit 9.9

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

Exhibit 9.10

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

Exhibit 9.11

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Analytical Hierarchy Process (AHP)
Overview

■ 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 the
decision criteria.

■ 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 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.

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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.

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Analytical Hierarchy Process
Pairwise Comparisons (2 of 2)

Table 9.1 Preference Scale for Pairwise Comparisons 9-35

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Analytical Hierarchy Process
Pairwise Comparison Matrix

A pairwise comparison matrix summarizes the pairwise
comparisons for a criteria.

Customer Market

Site A B C

A1 3 2
B 1/3 1 1/5
C 1/2 5 1

Income Level Infrastructure Transportation
1 1/3 1
A  1 6 1/3 3 1 7 1 1/3 1/2

B 1/6 1 1/9  3 1 
4 
 1 1/7 1 
  2 1/4 
C  3 9 1  1 

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Analytical Hierarchy Process
Developing Preferences Within Criteria (1 of 3)

In synthesization, decision alternatives are prioritized within each criterion

Customer Market

Site A B C

A1 3 2

B 1/3 1 1/5

C 1/2 5 1

11/6 9 16/5

Customer Market

Site A B C

A 6/11 3/9 5/8

B 2/11 1/9 1/16

C 3/11 5/9 5/16

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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 9-38

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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

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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 9-40

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Analytical Hierarchy Process
Ranking the Criteria (2 of 2)

Preference Vector for Criteria:

Market 0.1993
Income
Infrastructure 
Transportation
0.6535



0.0860



0.0612

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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 Site Score

Charlotte 0.5314

Overall Ranking: Atlanta 0.3091
Birmingham 0.1595

1.0000

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Analytical Hierarchy Process
Summary of Mathematical Steps

1. Develop a pairwise comparison matrix for each decision alternative
for each criteria.

2. Synthesization

a. Sum each column value of the pairwise comparison matrices.

b. Divide each value in each column by its column sum.

c. Average the values in each row of the normalized matrices.

d. Combine the vectors of preferences for each criterion.

3. Develop a pairwise comparison matrix for the criteria.

4. Compute the normalized matrix.

5. Develop the preference vector.

6. Compute an overall score for each decision alternative

7. Rank the decision alternatives.

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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

Marke Income Infrastructure Transportatio Criteria
0.1993
Market t 1 1/5 3 n4

Income 5 1 9 7X 0.6535
Infrastructure 1/3 1/9 1 2 0.0860

Transportatio 1/4 1/7 1/2 1 0.0612
n
(1/5)(0.6535) + (3)(0.0860) + (4)(0.0612) = 0.8328
(1)(0.1993) + (1)(0.6535) + (9)(0.0860) + (7)(0.0612) = 2.8524

(5)(0.1993) +

(1/3)(0.1993) + (1/9)(0.6535) + (1)(0.0860) + (2)(0.0612) = 0.3474
(¼)(0.1993) + (1/7)(0.6535) + (½)(0.0860) + (1)(0.0612) = 0.2473

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Analytical Hierarchy Process: Consistency (2 of 3)

Step 2: Divide each value by the corresponding weight from the 9-45
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)

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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

n 2 3 4 5 6 7 8 9 10
RI 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.51

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

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Analytical Hierarchy Process
Excel Spreadsheets (1 of 4)

Exhibit 9.12

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Analytical Hierarchy Process
Excel Spreadsheets (2 of 4)

Exhibit 9.13 9-48

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Analytical Hierarchy Process
Excel Spreadsheets (3 of 4)

Exhibit 9.14

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Analytical Hierarchy Process
Excel Spreadsheets (4 of 4)

Exhibit 9.15 9-50

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