Chapter 6 Multicriteria Decision Making · Chapter 6 Multicriteria Decision Making 2 Goal...

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Chapter 6Multicriteria Decision

Making

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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

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Study of problems with several criteria, multiple criteria, instead of a single objective when making a decision.Three techniques discussed: goal programming, theanalytical 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

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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 producedx2 = number of mugs produced

Goal Programming ExampleProblem Data (1 of 2)

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Adding objectives (goals) in order of importance, thecompany:

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.

Goal Programming ExampleProblem Data (2 of 2)

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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.

Goal ProgrammingGoal Constraint Requirements

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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)

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

Minimize P1d1-, P4d1

+

Add profit goal constraint (priority 2 - achieve profit of $1,600):


-, P4d1+

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)

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Complete Goal Programming Model:Minimize P1d1

-, P2d2-, 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

+ ≥ 0

Goal Programming Model FormulationComplete Model (3 of 3)

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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

-

Goal ProgrammingAlternative Forms of Goal Constraints (1 of 2)

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Goal ProgrammingAlternative Forms of Goal Constraints (2 of 2)

Complete Model with Added New Goals:Minimize P1d1

-, P2d2-, P3d3

+, P4d4+, 4P5d5

- + 5P5d6-


- - d1+ = 40

40x1 + 50x2 + d2- - d2

+ = 1,6004x1 + 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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-, P3d3+, P4d1

+

subject to: x1 + 2x2 + d1

- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,6004x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ ≥ 0

Figure 9.1 Goal Constraints

Goal ProgrammingGraphical Interpretation (1 of 6)

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13Figure 9.2 The First-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,6004x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ ≥ 0


14Figure 9.3 The Second-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,6004x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ ≥ 0


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15Figure 9.4 The Third-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,6004x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ ≥ 0


16Figure 9.5 The Fourth-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,6004x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ ≥ 0


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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+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

+ = 1,6004x1 + 3x2 + d3

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ ≥ 0

Solution: x1 = 15 bowlsx2 = 20 mugsd1

- = 15 hours


18Exhibit 9.4

Goal ProgrammingComputer Solution Using Excel (1 of 3)

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19Exhibit 9.5


20Exhibit 9.6


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-, P3d3+, P4d4

+, 4P5d5- + 5P5d6

-


- - d1+ = 40

40x1 + 50x2 + d2- - d2

+ = 1,6004x1 + 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

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

22Exhibit 9.7


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23Exhibit 9.8


24Exhibit 9.9


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25Exhibit 9.10


26Exhibit 9.11


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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.

Analytical Hierarchy ProcessOverview

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Southcorp Development Company shopping mall site selection.Three potential sites:

AtlantaBirmingham Charlotte.

Criteria for site comparisons:Customer market base.Income levelInfrastructure

Analytical Hierarchy ProcessExample Problem Statement

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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

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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.

Analytical Hierarchy ProcessGeneral Mathematical Process

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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)

32Table 9.1 Preference Scale for Pairwise Comparisons

Analytical Hierarchy ProcessPairwise Comparisons (2 of 2)

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Income Level Infrastructure Transportation

A

B

C ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

1931/911/61/361

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

11/7171311/31

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

11/42413

1/21/31

Customer Market Site A B C A B C

1 1/3 1/2

3 1 5

2 1/5 1

Analytical Hierarchy ProcessPairwise Comparison Matrix

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

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Customer Market Site A B C

A B C

1 1/3

1/2 11/6

3 1 5 9

2 1/5 1

16/5

Customer Market Site A B C

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 with each criterion and then normalized:

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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

36Table 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

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Criteria Market Income Infrastructure Transportation Market Income Infrastructure Transportation

1 5

1/3 1/4

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

Analytical Hierarchy ProcessRanking the Criteria (1 of 2)Pairwise Comparison Matrix:

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⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

0.06120.08600.65350.1993

Analytical Hierarchy ProcessRanking the Criteria (2 of 2)

Preference Vector for Criteria:MarketIncomeInfrastructureTransportation

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Overall Score:Site A score = .1993(.5012) + .6535(.2819) + .0860(.1790) + .0612(.1561) = .3091Site B score = .1993(.1185) + .6535(.0598) + .0860(.6850) + .0612(.6196) = .1595Site C score = .1993(.3803) + .6535(.6583) + .0860(.1360) + .0612(.2243) = .5314

Overall Ranking:

Site Score Charlotte Atlanta

Birmingham

0.5314 0.3091 0.1595 1.0000

Analytical Hierarchy ProcessDeveloping an Overall Ranking

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Analytical Hierarchy ProcessSummary 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 alternativeRank 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 comparisonsExample: 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. CriteriaMarket 1 1/5 3 4 0.1993Income 5 1 9 7 X 0.6535

Infrastructure 1/3 1/9 1 2 0.0860Transportation 1/4 1/7 1/2 1 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

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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 = 4.17862.8524/0.6535 = 4.36480.3474/0.0860 = 4.04010.2473/0.0612 = 4.0422

16.257Average = 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/RIwhere 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

44Exhibit 9.12

Analytical Hierarchy ProcessExcel Spreadsheets (1 of 4)

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45Exhibit 9.13


46Exhibit 9.14


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47Exhibit 9.15


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Each decision alternative graded in terms of how well it satisfies the criterion according to following formula:

Si = Σgijwj

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 ModelOverview

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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) = 76.00S4 = (.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

50Exhibit 9.16

Scoring ModelExcel Solution

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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 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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Purchasing decision, three model alternatives, three decision criteria.Pairwise comparison matrices:

Prioritized decision criteria:

Price Bike X Y Z

X Y Z

1 1/3 1/6

3 1

1/2

6 2 1

Gear Action Bike X Y Z

X Y Z

1 3 7

1/3 1 4

1/7 1/4 1

Weight/Durability Bike X Y Z

X Y Z

1 1/3 1

3 1 2

1 1/2 1

Criteria Price Gears Weight Price Gears Weight

1 1/3 1/5

3 1

1/2

5 2 1

Analytical Hierarchy Process Example ProblemProblem Statement

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

Price Bike X Y Z Row Averages

X Y Z

0.6667 0.2222 0.1111

0.6667 0.2222 0.1111

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)

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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

X Y Z

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.


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Step 2: Rank the criteria.

Price GearsWeight ⎥

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

0.12220.22990.6479

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


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Step 3: Develop an overall ranking.

Bike XBike YBike 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).

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

•1222.02299.06479.0

3837.07014.01111.01698.02132.02222.04429.00853.06667.0


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