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# Multicriteria Decision Making

© 2007 Pearson Education 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 = \$40x 1 + 50x 2 subject to: 1x 1 + 2x 2 4x x 1 1 , x + 3x 2 2  0   40 hours of labor 120 pounds of clay Where: x 1 x 2 = number of bowls produced = number of mugs produced Chapter 9 - Multicriteria Decision Making 4

Goal Programming Example Problem Data (2 of 2)

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: x 1 + 2x 2 + d 1 - d 1 + = 40 (hours/day) Profit goal: 40x 1 + 50 x 2 + d 2 - d 2 + = 1,600 (\$/day) Material goal: 4x 1 + 3x 2 + d 3 - d 3 + = 120 (lbs of clay/day) Chapter 9 - Multicriteria Decision Making 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 P 1 d 1 , P 4 d 1 + Add profit goal constraint (priority 2 - achieve profit of \$1,600): Minimize P 1 d 1 , P 2 d 2 , P 4 d 1 + Add material goal constraint (priority 3 - avoid keeping more than 120 pounds of clay on hand): Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + Chapter 9 - Multicriteria Decision Making 8

Goal Programming Model Formulation Complete Model (3 of 3) Complete Goal Programming Model:

Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - d 1 + = 40 (labor) 40x 1 + 50 x 2 + d 2 - d 2 + = 1,600 (profit) 4x 1 + 3x 2 + d 3 - d 3 + = 120 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +  0 (clay) 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” instead of minimizing overtime: d 1 + d 4 - d 4 + = 10 minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 4 + Addition of a fifth-priority goal goal for mugs”: “important to achieve the x 1 + d 5 = 30 bowls x 2 + d 6 = 20 mugs minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 4 + , 4P 5 d 5 + 5P 5 d 6 Chapter 9 - Multicriteria Decision Making 10

Goal Programming Alternative Forms of Goal Constraints (2 of 2) Complete Model with Added New Goals:

Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 4 + , 4P 5 d 5 + 5P 5 d 6 subject to: x 40x 1 1 + 2x 2 + 50x 2 + d 1 + d 4x 1 d 1 + + 3x 2 + d 4 x 1 + d 5 + d - d = 30 3 4 + 2 - d 1 + - d - d 3 + = 10 2 + = 40 = 1,600 = 120 x 2 + d 6 = 20 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +, d 4 , d 4 + , d 5 , d 6  0 Chapter 9 - Multicriteria Decision Making 11

Goal Programming Graphical Interpretation (1 of 6)

Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + subject to: x 40x 1 1 + 2x 2 + 50 x + d 1 2 - d 1 + + d 2 - d = 40 2 + = 1,600 4x 1 + 3x 2 + d 3 - d 3 + = 120 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +  0

Figure 9.1 Goal Constraints

Chapter 9 - Multicriteria Decision Making 12

Goal Programming Graphical Interpretation (2 of 6)

Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + subject to: x 40x 1 1 + 2x 2 + 50 x + d 1 2 - d 1 + + d 2 - d = 40 2 + = 1,600 4x 1 + 3x 2 + d 3 - d 3 + = 120 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +  0

Figure 9.2 The First-Priority Goal: Minimize

Chapter 9 - Multicriteria Decision Making 13

Goal Programming Graphical Interpretation (3 of 6)

Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + subject to: x 40x 1 1 + 2x 2 + 50 x + d 1 2 - d 1 + + d 2 - d = 40 2 + = 1,600 4x 1 + 3x 2 + d 3 - d 3 + = 120 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +  0

Figure 9.3 The Second-Priority Goal: Minimize

Chapter 9 - Multicriteria Decision Making 14

Goal Programming Graphical Interpretation (4 of 6)

Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + subject to: x 40x 1 1 + 2x 2 + 50 x + d 1 2 - d 1 + + d 2 - d = 40 2 + = 1,600 4x 1 + 3x 2 + d 3 - d 3 + = 120 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +  0

Figure 9.4 The Third-Priority Goal: Minimize

Chapter 9 - Multicriteria Decision Making 15

Goal Programming Graphical Interpretation (5 of 6)

Minimize P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + subject to: x 40x 1 1 + 2x 2 + 50 x + d 1 2 - d 1 + + d 2 - d = 40 2 + = 1,600 4x 1 + 3x 2 + d 3 - d 3 + = 120 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +  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 P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 1 + subject to: x 40x 1 1 + 2x 2 + 50 x + d 1 2 - d 1 + + d 2 - d = 40 2 + = 1,600 4x 1 + 3x 2 + d 3 - d 3 + = 120 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +  0 Solution : x 1 = 15 bowls x 2 d 1 = 20 mugs = 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 P 1 d 1 , P 2 d 2 , P 3 d 3 + , P 4 d 4 + , 4P 5 d 5 + 5P 5 d 6 subject to: x 40x 1 1 4x 1 d 1 + + 2x 2 + 50x 2 + d 1 + d 2 - d 1 + - d 2 + = 40 = 1,600 + 3x 2 + d 3 - d 3 + = 120 + d 4 - d 4 + = 10 x 1 + d 5 = 30 x 2 + d 6 = 20 x 1 , x 2 , d 1 , d 1 + , d 2 , d 2 + , d 3 , d 3 +, d 4 , d 4 + , d 5 , d 6  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 22

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 25

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 31

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 Site A B C A 1 1/3 1/2 B 3 1 5 C 2 1/5 1

Income Level Infrastructure Transportation A B C        1 1/6 3 6 1 9 1/3 1/9 1               1 3 1 1/3 1 1/7 1 7 1               1 3 2 1/3 1 1/4 1/2 4 1        Chapter 9 - Multicriteria Decision Making 33

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

In synthesization, decision alternatives are prioritized with each criterion and then normalized:

Site A B C A 1 1/3 1/2 11/6 Customer Market B 3 1 5 9 C 2 1/5 1 16/5 Customer Market Site A B C A 6/11 2/11 3/11 B 3/9 1/9 5/9 C 5/8 1/16 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 5 1/3 1/4 Income Infrastructure Transportation 1/5 1 1/9 1/7 3 9 1 1/2 4 7 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.

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 Criteria 0.1993

X 0.6535

0.0860

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 Excel Spreadsheets (1 of 4) Exhibit 9.12

Chapter 9 - Multicriteria Decision Making 44

Analytical Hierarchy Process Excel Spreadsheets (2 of 4) Exhibit 9.13

Chapter 9 - Multicriteria Decision Making 45

Analytical Hierarchy Process Excel Spreadsheets (3 of 4) Exhibit 9.14

Chapter 9 - Multicriteria Decision Making 46

Analytical Hierarchy Process Excel Spreadsheets (4 of 4) 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: S i =  g ij w j where: w j = a weight between 0 and 1.00 assigned to criterion j; 1.00 important, 0 unimportant; sum of total weights equals one.

g ij = 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:

Decision Criteria School proximity Median income Vehicular traffic Mall quality, size Other shopping Weight (0 to 1.00) 0.30 0.25 0.25 0.10 0.10 Grades for Alternative (0 to 100) Mall 1 Mall 2 Mall 3 Mall 4 40 75 60 90 80 60 80 90 100 30 90 65 79 80 50 60 90 85 90 70

S 1 S 2 S 3 S 4 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) = 62.75

= (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) = 73.50

= (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) = 76.00

= (.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 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

Problem Statement

Purchasing decision, three model alternatives, three decision criteria.

Bike X Y Z X 1 1/3 1/6 Price Y 3 1 1/2 6 2 1 Z Bike X Y Z Gear Action X 1 3 7 Y 1/3 1 4 Z 1/7 1/4 1 Z Weight/Durability Y 3 1 2 Z 1 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 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 Bike X Y Z 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

Chapter 9 - Multicriteria Decision Making 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 X 0.4286 0.1429 0.4286 Weight/Durability Y 0.5000 0.1667 0.3333 Z 0.4000 0.2000 0.4000 Bike X Y Z Price 0.6667 0.2222 0.1111 Criteria Gears 0.0853 0.2132 0.7014 Weight 0.4429 0.1698 0.3873 Row Averages 0.4429 0.1698 0.3873 1.0000

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 Weight 0.6250 0.2500 0.1250 Row Averages 0.6479 0.2299 0.1222 1.0000

Price Gears Weight          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 0 0 .

.

6667 .

2222 1111 0 .

0853 0 .

2132 0 .

7014 0 0 .

4429 0 .

1698 .

3837                    0 0 0 .

.

6479 .

2299 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