9-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Multicriteria Decision...

Multicriteria Decision Making

Chapter 9

■ 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 Topics

■ Study of problems with several criteria, i.e., 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.

Overview

Beaver Creek Pottery Company Example:

Maximize Z = $40x1 + 50x2

subject to:1x1 + 2x2 40 hours of labor4x1 + 3x2 120 pounds of clayx1, x2 0

Where: x1 = number of bowls produced x2 = number of mugs produced

Goal Programming ExampleProblem Data (1 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.

Goal Programming ExampleProblem Data (2 of 2)

■ 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.

Goal ProgrammingGoal Constraint Requirements

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)

Goal Programming Model FormulationGoal Constraints (1 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):

-, P4d1+

3. Add material goal constraint

(priority 3 - avoid keeping more than 120 pounds of clay on hand):

-, P3d3+, P4d1

Goal Programming Model FormulationObjective Function (2 of 3)

Complete Goal Programming Model:

-, P3d3+, P4d1

subject to:

x1 + 2x2 + d1- - d1

+ = 40 (labor)

40x1 + 50 x2 + d2 - - d2

+ = 1,600 (profit)

4x1 + 3x2 + d3 - - d3

+ = 120 (clay)

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

Goal Programming Model FormulationComplete Model (3 of 3)

■ Changing fourth-priority goal “limit 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 -

Goal ProgrammingAlternative Forms of Goal Constraints (1 of 2)

Two or more goals at the same priority level can be assigned weights to indicate their relative importance.

Goal ProgrammingAlternative Forms of Goal Constraints (2 of 2)

Complete Model with Added New Goals:

-, 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

Figure 9.1 Goal constraints

Goal ProgrammingGraphical Interpretation (1 of 6)

-, P3d3+,

- - d1+ =

40 40x1 + 50 x2 + d2

- - d2 +

= 1,600 4x1 + 3x2 + d3

- - d3 + =

120x1, x2, d1

-, d1 +, d2

-, d2 +, d3

Figure 9.2 The first-priority goal: minimize d1

-, P3d3+,

- - d1+ =

40 40x1 + 50 x2 + d2

- - d2 + =

1,600 4x1 + 3x2 + d3

- - d3 + =

120x1, x2, d1

-, d1 +, d2

-, d2 +, d3

Figure 9.3 The second-priority goal: Minimize d2

-, P3d3+,

- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + =

120x1, x2, d1

-, d1 +, d2

-, d2 +, d3

Figure 9.4 The third-priority goal: minimize d3

-, P3d3+,

- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + =

120x1, x2, d1

-, d1 +, d2

-, d2 +, d3

-, d3 +

Figure 9.5 The Fourth-Priority Goal: Minimize d1

-, P3d3+,

- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + =

120x1, x2, d1

-, d1 +, d2

-, d2 +, d3

-, d3 + 0

Goal programming solutions do not always achieve all goals and they are not optimal; they achieve the best or most satisfactory solution possible.

-, P3d3+, P4d1

- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

Solution: x1 = 15 bowlsx2 = 20 mugsd1

- = 15 hours

Exhibit 9.1

-, P3d3+, P4d1

- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,600 4x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

Goal Programming Computer SolutionUsing QM for Windows (1 of 3)

Exhibit 9.2

Exhibit 9.3

Exhibit 9.4

Goal Programming Computer SolutionUsing Excel (1 of 3)

Goal constraint for cell G5

Decision variables—B10:B11

Deviational variables—E5:F7

Exhibit 9.5

Goal ProgrammingComputer Solution Using Excel (2 of 3) Minimize deviational

variable for first-priority goal in E5

Decision and deviational variables

Goal constraints

Exhibit 9.6

Goal ProgrammingComputer Solution Using Excel (3 of 3)

-, P3d3+, P4d4

+, 4P5d5- + 5P5d6

- - 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

Goal ProgrammingSolution for Altered Problem Using Excel (1 of 6)

This model includes goals for overtime and maximum storage levels for bowls and mugs.

Exhibit 9.7

Goal constraint for labor

Goal constraint for overtime; =F5+E8-F8

=C9*B13+E9

Exhibit 9.8

Exhibit 9.9

First two priority goals, minimizing and , achieved

Third-priority goal to minimize not achieved3d

Exhibit 9.10

First- and second-priority goals achieved; add E5=0 and E6=0

Exhibit 9.11

Fourth-priority goal to minimize overtime, , not achieved

■ 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.

Analytical Hierarchy Process (AHP)Overview

Southcorp Development Company shopping mall site selection.

■ Three potential sites: Atlanta Birmingham Charlotte.

■ Criteria for site comparisons: Customer market base. Income level Infrastructure

Analytical Hierarchy ProcessExample Problem Statement

■ 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.

Analytical Hierarchy ProcessHierarchy Structure

■ 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

■ Select the site with the highest score.

Analytical Hierarchy ProcessGeneral Mathematical Process

■ 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.

Analytical Hierarchy ProcessPairwise Comparisons (1 of 2)

Table 9.1 Preference scale for pairwise comparisons

Analytical Hierarchy ProcessPairwise Comparisons (2 of 2)

Income Level Infrastructure Transportation

1 16 9

1 13 2

Customer Market Site A B C A B C

1 1/3 1/2

2 1/5 1

Analytical Hierarchy ProcessPairwise Comparison Matrix

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

Customer Market Site A B C

1/2 11/6

3 1 5 9

2 1/5 1

Customer Market Site A B C

6/11 2/11 3/11

3/9 1/9 5/9

5/8 1/16 5/16

Analytical Hierarchy ProcessDeveloping Preferences Within Criteria (1 of 3)In synthesization, decision alternatives are prioritized

within each criterion

Table 9.2 The normalized matrix with row averages

Analytical Hierarchy ProcessDeveloping Preferences Within Criteria (2 of 3)The row average values represent the preference vector

Table 9.3 Criteria preference matrix

Analytical Hierarchy ProcessDeveloping Preferences Within Criteria (3 of 3)Preference vectors for other criteria are computed similarly,

resulting in the preference matrix

Criteria Market Income Infrastructure Transportation Market Income Infrastructure Transportation

1/3 1/4

1/9 1/7

4 7 2 1

Table 9.4 Normalized matrix for criteria with row averages

Analytical Hierarchy ProcessRanking the Criteria (1 of 2)

Pairwise Comparison Matrix:

0.1993

0.6535

0.0860

0.0612

Analytical Hierarchy ProcessRanking the Criteria (2 of 2)

Preference Vector for Criteria:

MarketIncomeInfrastructureTransportation

Overall Score:

Site Score Charlotte Atlanta Birmingham

0.5314 0.3091 0.1595 1.0000

Analytical Hierarchy ProcessDeveloping an Overall Ranking

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)

= .5314Overall

Ranking:

Analytical Hierarchy ProcessSummary of Mathematical Steps

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

2. Synthesizationa. 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

alternative7. Rank the decision alternatives.

Analytical Hierarchy Process: Consistency (1 of 3)

Consistency Index (CI): Check for consistency and validity of multiple pairwise comparisonsExample: Southcorp’s consistency in the pairwise comparisons

of the 4 site selection criteria

(1)(0.1993) +

(1/5)(0.6535) +

(3)(0.0860)

(4)(0.0612

= 0.8328

(5)(0.1993) +

(1)(0.6535) +

(9)(0.0860)

(7)(0.0612

= 2.8524

(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

Market Income Infrastructure Transportation Criteria

Market 1 5-Jan 3 4 0.1993

Income 5 1 9 7 0.6535

Infrastructure 3-Jan 9-Jan 1 2 0.086

Transportation 4-Jan 7-Jan 2-Jan 1 0.0612

Step 2: Divide each value by the corresponding weight from the

preference vector and compute the average 0.8328/0.1993 = 4.17862.8524/0.6535 = 4.36480.3474/0.0860 = 4.04010.2473/0.0612 = 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 number of items

compared

CI = (4.1564-4)/(4-1) = 0.0521 (CI = 0 indicates perfect consistency)

Step 4: Compute the Ratio CI/RI where RI is a random index value obtained from Table

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

n 2 3 4 5 6 7 8 9 10

RI 0 0.58

Exhibit 9.12

Analytical Hierarchy ProcessExcel Spreadsheets (1 of 4)

Click on “Format,” then “Cells,” then “Fraction” to enter fractionsRow average formula for F14

=SUM(B5:B7)

=B5/B8

Exhibit 9.13

=SUM(B14:E14)/4

=SUM(B5:B8)

=B5/B9

Exhibit 9.14

Analytical Hierarchy Process Excel Spreadsheets (3 of 4)

Atlanta score in cell C12

Cells G14:G17 from Exhibit 9.13Cells F14:F16

from Exhibit 9.12

Exhibit 9.15

=I5/G5

=SUM(B11:B14)

=(4.1564-4)/3

=G12/0.90

Each decision alternative is 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 criterion 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.

Scoring Model Overview

Mall selection with four alternatives and five criteria:

S1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) = 62.75S2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) = 73.50S3 = (.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.

Grades for Alternative (0 to 100) Decision Criteria

Weight (0 to 1.00)

Mall 1

Mall 2

Mall 3

Mall 4

School proximity Median income Vehicular traffic Mall quality, size Other shopping

0.30 0.25 0.25 0.10 0.10

40 75 60 90 80

60 80 90

100 30

90 65 79 80 50

60 90 85 90 70

Scoring ModelExample Problem

Exhibit 9.16

Scoring ModelExcel Solution

Goal Programming Example ProblemProblem 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/day for personal interviewer.

■ Goals (in priority order):

1. At least 3,000 total interviews conducted.

2. Interviewer conducts only one type of interview each day; maintain daily budget of $2,500.

3. At least 1,000 interviews should be by telephone.

Formulate and solve a goal programming model to determine number of interviewers to hire in order to satisfy the goals

Step 1: Model Formulation:

+, P3d3-

subject to:80x1 + 40x2 + d1

- - d1+ = 3,000 interviews

50x1 + 70x2 + d2- - d2

+ = $2,500 budget80x1 + d3

- - d3 + = 1,000 telephone interviews

where: x1 = number of telephone interviews x2 = number of personal interviews

Goal Programming Example ProblemSolution (1 of 2)

Goal Programming Example ProblemSolution (2 of 2)

Step 2: QM for Windows Solution:

Purchasing decision, three model alternatives, three decision criteria.

Pairwise comparison matrices:

Prioritized decision criteria:

Price Bike X Y Z

1 1/3 1/6

Gear Action Bike X Y Z

1/3 1 4

1/7 1/4 1

Weight/Durability Bike X Y Z

1 1/3 1

1 1/2 1

Criteria Price Gears Weight Price Gears Weight

1 1/3 1/5

Analytical Hierarchy Process Example ProblemProblem Statement

Step 1: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria.

Bike X Y Z Row Averages

0.6667 0.2222 0.1111

0.6667 0.2222 0.1111 1.0000

Gear Action

Bike X Y Z Row Averages X Y Z

0.0909 0.2727 0.6364

0.0625 0.1875 0.7500

0.1026 0.1795 0.7179

0.0853 0.2132 0.7014 1.0000

Analytical Hierarchy Process Example ProblemProblem Solution (1 of 4)

Weight/Durability

Bike X Y Z Row Averages X Y Z

0.4286 0.1429 0.4286

0.5000 0.1667 0.3333

0.4000 0.2000 0.4000

0.4429 0.1698 0.3873 1.0000

Criteria Bike Price Gears Weight

0.6667 0.2222 0.1111

0.0853 0.2132 0.7014

0.4429 0.1698 0.3873

Step 1 continued: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria.

Preference vectors are summarized

Step 2: Rank the criteria.

Price Gears Weight

0.6479

0.2299

0.1222

Criteria Price Gears Weight Row Averages Price Gears Weight

0.6522 0.2174 0.1304

0.6667 0.2222 0.1111

0.6250 0.2500 0.1250

0.6479 0.2299 0.1222 1.0000

Step 3: Develop an overall ranking. Bike X Bike Y Bike Z

Bike X score = .6667(.6479) + .0853(.2299) + .4429(.1222) = .5057Bike Y score = .2222(.6479) + .2132(.2299) + .1698(.1222) = .2138Bike 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).

0.6667 0.0853 0.4429 0.6479

0.2222 0.2132 0.1698 0.2299

0.1111 0.7014 0.3837 0.1222

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