Chapter 14 Multicriteria Decision Making

Chapter 14 - Multicriteria Decision Making 1

Chapter 14

Multicriteria Decision Making

Introduction to Management Science

8th Edition

by

Bernard W. Taylor III


Goal Programming

Graphical Interpretation of Goal Programming

Computer Solution of Goal Programming Problems with QM for Windows and Excel

The Analytical Hierarchy Process

Chapter Topics


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.

Overview


Beaver Creek Pottery Company Example:

Maximize Z = $40x1 + 50x2

subject to:1x1 + 2x2 40 hours of labor4x2 + 3x2 120 pounds of clayx1, x2 0

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

Goal ProgrammingModel Formulation (1 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.

Goal ProgrammingModel Formulation (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 in a goal programming model seeks to minimize the deviation from goals in the order of the goal priorities.

Goal ProgrammingGoal Constraint Requirements


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):


-, P4d1+

Add material goal constraint (3, avoid keeping more than 120 pounds of clay on hand):


-, P3d3+, P4d1

+

Goal ProgrammingGoal Constraints and Objective Function (1 of 2)


Complete Goal Programming Model:


-, 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

Goal ProgrammingGoal Constraints and Objective Function (2 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

-

Goal ProgrammingAlternative Forms of Goal Constraints (1 of 2)


Goal ProgrammingAlternative Forms of Goal Constraints (2 of 2)

Complete Model with New Goals:


-, 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

- 0



-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

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

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ 0

Figure 14.1Goal Constraints

Goal ProgrammingGraphical Interpretation (1 of 6)


Figure 14.2The First-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

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

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ 0



Figure 14.3The Second-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

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

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ 0



Figure 14.4The Third-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

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

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ 0



Figure 14.5The Fourth-Priority Goal: Minimize


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

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

- - d3 + = 120

x1, 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

+ 0

x1 = 15 bowlsx2 = 20 mugsd1

- = 15 hours



Exhibit 14.1


-, P3d3+, P4d1

+


- - d1+ = 40

40x1 + 50 x2 + d2 - - d2

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

- - d3 + = 120

x1, x2, d1 -, d1

+, d2 -, d2

+, d3 -, d3

+ 0

Goal ProgrammingComputer Solution Using QM for Windows (1 of 3)


Exhibit 14.2



Exhibit 14.3



Exhibit 14.4

Goal ProgrammingComputer Solution Using Excel (1 of 3)


Exhibit 14.5



Exhibit 14.6




-, 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

- 0

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


Exhibit 14.7



Exhibit 14.8



Exhibit 14.9



Exhibit 14.10



Exhibit 14.11



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


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 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.

Analytical Hierarchy ProcessGeneral Mathematical Process


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.

Analytical Hierarchy ProcessPair-wise Comparisons


Table 14.1Preference Scale for Pair-wise Comparisons

Analytical Hierarchy ProcessPair-wise Comparisons (2 of 2)


Income Level Infrastructure Transportation

A

B

C

193

1/911/6

1/361

11/71

713

11/31

11/42

413

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 ProcessPair-wise Comparison Matrix

A pair-wise comparison matrix summarizes the pair-wise comparisons for a criteria.



A B C

1 1/3

1/2 11/6

3 1 5 9

2 1/5 1

16/5


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


Table 14.2The Normalized Matrix with Row Averages



Table 14.3Criteria Preference Matrix



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 14.4Normalized Matrix for Criteria with Row Averages

Analytical Hierarchy ProcessRanking the Criteria (1 of 2)

Pair-wise Comparison Matrix:


0.0612

0.0860

0.6535

0.1993

Analytical Hierarchy ProcessRanking the Criteria (2 of 2)

Preference Vector:

MarketIncomeInfrastructureTransportation


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 Score Charlotte Atlanta

Birmingham

0.5314 0.3091 0.1595 1.0000

Analytical Hierarchy ProcessDeveloping an Overall Ranking


Analytical Hierarchy ProcessSummary 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.


Exhibit 14.12

Goal ProgrammingExcel Spreadsheets (1 of 4)


Exhibit 14.13



Exhibit 14.14



Exhibit 14.15



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.

Scoring ModelOverview


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


Exhibit 14.16

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 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.


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.

Pair-wise 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


Step 1: Develop normalized matrices and preference vectors for all the pair-wise 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)


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 pair-wise comparison matrices for criteria.



Step 2: Rank the criteria.

Price

Gears

Weight

0.1222

0.2299

0.6479

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



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).

1222.0

2299.0

6479.0

3837.07014.01111.0

1698.02132.02222.0

4429.00853.06667.0



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Chapter 14 Multicriteria Decision Making