Multicriteria Decision Making

1
Introduction to Management Science 8th
Edition by Bernard W. Taylor III
Chapter 14 Multicriteria Decision Making
2
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

3
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.

4
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.

6
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.

7
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

8
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
9
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 -

10
Goal Programming Alternative Forms of Goal
Constraints (2 of 2)
Complete Model with 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 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
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 14.3 The Second-Priority Goal Minimize
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 14.4 The Third-Priority Goal Minimize
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 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
17
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
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Goal Programming Computer Solution Using QM for
Windows (2 of 3)
Exhibit 14.2
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Goal Programming Computer Solution Using QM for
Windows (3 of 3)
Exhibit 14.3
20
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
23
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)
Exhibit 14.7
25
Goal Programming Solution for Altered Problem
Using Excel (3 of 6)
Exhibit 14.8
26
Goal Programming Solution for Altered Problem
Using Excel (4 of 6)
Exhibit 14.9
27
Goal Programming Solution for Altered Problem
Using Excel (5 of 6)
Exhibit 14.10
28
Goal Programming Solution for Altered Problem
Using Excel (6 of 6)
Exhibit 14.11
29
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 makers
  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.

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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.

Income Level
Infrastructure Transportation
A B
C
36
Analytical Hierarchy Process Developing
Preferences Within Criteria (1 of 3)

• In synthetization, decision alternatives are
  prioritized with each criterion and then
  normalized

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Analytical Hierarchy Process Developing
Preferences Within Criteria (2 of 3)
Table 14.2 The Normalized Matrix with Row Averages
38
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
Table 14.4 Normalized Matrix for Criteria with
Row Averages
40
Analytical Hierarchy Process Ranking the Criteria
(2 of 2)
Preference Vector Market Income Infrastr
ucture Transportation
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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
• Overall Ranking

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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
45
Goal Programming Excel Spreadsheets (3 of 4)
Exhibit 14.14
46
Goal Programming Excel Spreadsheets (4 of 4)
Exhibit 14.15
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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
• 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.

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Scoring Model Excel Solution
Exhibit 14.16
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.

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Goal Programming Example Problem Solution (1 of 2)
Step 1 Model Formulation Minimize P1d1-,
P2d2-, P3d3- subject 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
• Prioritized decision criteria

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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.
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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.
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Analytical Hierarchy Process Example
Problem Problem Solution (3 of 4)
Step 2 Rank the criteria.
Price
Gears Weight
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Analytical Hierarchy Process Example
Problem Problem Solution (4 of 4)
Step 3 Develop an overall ranking.
Bike X
Bike Y Bike Z
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).
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