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Transcript of 9-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Multicriteria Decision...
9-1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Multicriteria Decision Making
Chapter 9
9-2
■ 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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9-3
■ 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.
Overview
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9-4
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)
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9-5
■ 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 $1,600 per day.
3. Prefers not to keep more than 120 pounds of clay on hand each day.
4. Would like to minimize the amount of overtime.
Goal Programming ExampleProblem Data (2 of 2)
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9-6
■ All goal constraints are equalities that include deviational variables d- and d+.
■ A positive deviational variable (d+) is the amount by which a goal level is exceeded.
■ A negative deviation variable (d-) is the amount by which a goal level is underachieved.
■ At least one or both deviational variables in a goal constraint must equal zero.
■ The objective function seeks to minimize the deviation from the respective goals in the order of the goal priorities.
Goal ProgrammingGoal Constraint Requirements
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9-7
Labor goal:
x1 + 2x2 + d1- - d1
+ = 40 (hours/day)
Profit goal:
40x1 + 50 x2 + d2 - - d2 + = 1,600 ($/day)
Material goal:
4x1 + 3x2 + d3 - - d3 + = 120 (lbs of
clay/day)
Goal Programming Model FormulationGoal Constraints (1 of 3)
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9-8
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):
Minimize P1d1-, P2d2
-, P4d1+
3. Add material goal constraint
(priority 3 - avoid keeping more than 120 pounds of clay on hand):
Minimize P1d1-, P2d2
-, P3d3+, P4d1
+
Goal Programming Model FormulationObjective Function (2 of 3)
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9-9
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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9-10
■ 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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9-11
Goal ProgrammingAlternative Forms of Goal Constraints (2 of 2)
Complete Model with Added 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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9-12Figure 9.1 Goal Constraints
Goal ProgrammingGraphical Interpretation (1 of 6)
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Minimize P1d1-, P2d2
-, P3d3+,
P4d1+
subject to: x1 + 2x2 + d1
- - d1+ =
40 40x1 + 50 x2 + d2
- - d2 +
= 1,600 4x1 + 3x2 + d3
- - d3 + =
120x1, x2, d1
-, d1 +, d2
-, d2 +, d3
-, d3
+ 0
9-13Figure 9.2 The First-Priority Goal:
Minimize d1-
Minimize P1d1-, P2d2
-, P3d3+,
P4d1+
subject to: x1 + 2x2 + d1
- - d1+ =
40 40x1 + 50 x2 + d2
- - d2 + =
1,600 4x1 + 3x2 + d3
- - d3 + =
120x1, x2, d1
-, d1 +, d2
-, d2 +, d3
-, d3
+ 0
Goal ProgrammingGraphical Interpretation (2 of 6)
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9-14
Figure 9.3 The Second-Priority Goal: Minimize d2-
Goal ProgrammingGraphical Interpretation (3 of 6)
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Minimize P1d1-, P2d2
-, P3d3+,
P4d1+
subject to: x1 + 2x2 + d1
- - d1+ = 40
40x1 + 50 x2 + d2 - - d2
+ = 1,600 4x1 + 3x2 + d3
- - d3 + =
120x1, x2, d1
-, d1 +, d2
-, d2 +, d3
-, d3
+ 0
9-15
Figure 9.4 The Third-Priority Goal: Minimize d3+
Goal ProgrammingGraphical Interpretation (4 of 6)
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Minimize P1d1-, P2d2
-, P3d3+,
P4d1+
subject to: x1 + 2x2 + d1
- - d1+ = 40
40x1 + 50 x2 + d2 - - d2
+ = 1,600 4x1 + 3x2 + d3
- - d3 + =
120x1, x2, d1
-, d1 +, d2
-, d2 +, d3
-, d3 +
0
9-16
Figure 9.5 The Fourth-Priority Goal: Minimize d1+
Goal ProgrammingGraphical Interpretation (5 of 6)
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Minimize P1d1-, P2d2
-, P3d3+,
P4d1+
subject to: x1 + 2x2 + d1
- - d1+ = 40
40x1 + 50 x2 + d2 - - d2
+ = 1,600 4x1 + 3x2 + d3
- - d3 + =
120x1, x2, d1
-, d1 +, d2
-, d2 +, d3
-, d3 + 0
9-17
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
Solution: x1 = 15 bowlsx2 = 20 mugsd1
- = 15 hours
Goal ProgrammingGraphical Interpretation (6 of 6)
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9-18Exhibit
9.1
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
Goal Programming Computer SolutionUsing QM for Windows (1 of 3)
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9-19
Exhibit 9.2
Goal Programming Computer SolutionUsing QM for Windows (2 of 3)
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9-20Exhibit
9.3
Goal Programming Computer SolutionUsing QM for Windows (3 of 3)
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9-21
■ 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
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9-22
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
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9-23
■ 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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9-25
■ 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)
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9-26Table 9.1 Preference Scale for Pairwise Comparisons
Analytical Hierarchy ProcessPairwise Comparisons (2 of 2)
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9-27
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 ProcessPairwise Comparison Matrix
A pairwise comparison matrix summarizes the pairwise comparisons for a criteria.
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9-28
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
within each criterion
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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
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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
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9-31
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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9-32
0.0612
0.0860
0.6535
0.1993
Analytical Hierarchy ProcessRanking the Criteria (2 of 2)
Preference Vector for Criteria:
MarketIncomeInfrastructureTransportation
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9-33
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
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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.
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9-35
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
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Market
Income
Infrastructure
Transportation
Criteria
Market 1 1/5 3 4 0.1993
Income 5 1 9 7 0.6535
Infrastructure
1/3 1/9 1 2 0.0860
Transportation
1/4 1/7 1/2 1 0.0612
X
(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
9-36
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 16.257 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)
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9-37
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
n 2 3 4 5 6 7 8 9 10
RI 0 0.58
0.90
1.12
1.24
1.32
1.41
1.45
1.51
9-38
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 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
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9-39
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
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9-40
Exhibit 9.16
Scoring ModelExcel Solution
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9-41
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):
1. At least 3,000 total interviews.
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
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9-42
Step 1: Model Formulation:
Minimize P1d1-, P2d2
+, 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)
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9-43
Goal Programming Example ProblemSolution (2 of 2)
Step 2: QM for Windows Solution:
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9-44
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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9-45
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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9-46
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.
Analytical Hierarchy Process Example ProblemProblem Solution (2 of 4)
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9-47
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
Analytical Hierarchy Process Example ProblemProblem Solution (3 of 4)
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9-48
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
Analytical Hierarchy Process Example ProblemProblem Solution (4 of 4)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall