1. Multicriteria Decision Making
Chapter 9

2. 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
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3. 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.
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4. Goal Programming Example Problem Data (1 of 2)
Beaver Creek Pottery Company Example:
Maximize Z = $40x1 + 50x2
subject to:
1x1 + 2x2  40 hours of labor
4x1 + 3x2  120 pounds of clay
x1, x2  0
Where: x1 = number of bowls produced
x2 = number of mugs produced
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5. Goal Programming Example Problem Data (2 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 $1,600 per day.
Prefers not to keep more than 120 pounds of clay on hand each day.
Would like to minimize the amount of overtime.
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6. Goal Constraint Requirements
Goal Programming
Goal Constraint Requirements
All goal constraints are equalities that include deviational variables d- and d+.
A positive deviational variable (d+) is the amount by which a goal level is exceeded.
A negative deviation variable (d-) is the amount by which a goal level is underachieved.
At least one or both deviational variables in a goal constraint must equal zero.
The objective function seeks to minimize the deviation from the respective goals in the order of the goal priorities.
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7. Goal Programming Model Formulation Goal Constraints (1 of 3)
Labor goal:
x1 + 2x2 + d1- - d1+ = (hours/day)
Profit goal:
40x x2 + d2 - - d2 + = 1,600 ($/day)
Material goal:
4x1 + 3x2 + d3 - - d3 + = (lbs of clay/day)
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8. Goal Programming Model Formulation Objective Function (2 of 3)
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):
Minimize P1d1-, P2d2-, P4d1+
Add material goal constraint
(priority 3 - avoid keeping more than 120 pounds of clay on hand):
Minimize P1d1-, P2d2-, P3d3+, P4d1+
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9. Goal Programming Model Formulation Complete Model (3 of 3)
Complete Goal Programming Model:
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = (labor)
40x x2 + d2 - - d2 + = 1, (profit)
4x1 + 3x2 + d3 - - d3 + = 120 (clay)
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
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10. Alternative Forms of Goal Constraints (1 of 2)
Goal Programming
Alternative Forms of Goal Constraints (1 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 +, 4P5d P5d6 -
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11. Alternative Forms of Goal Constraints (2 of 2)
Goal Programming
Alternative 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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12. Graphical Interpretation (1 of 6)
Goal Programming
Graphical Interpretation (1 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.1 Goal Constraints
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13. Figure 9.2 The First-Priority Goal: Minimize d1-
Goal Programming
Graphical Interpretation (2 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure The First-Priority Goal: Minimize d1-
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14. Graphical Interpretation (3 of 6)
Goal Programming
Graphical Interpretation (3 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.3 The Second-Priority Goal: Minimize d2-
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15. Graphical Interpretation (4 of 6)
Goal Programming
Graphical Interpretation (4 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.4 The Third-Priority Goal: Minimize d3+
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16. Graphical Interpretation (5 of 6)
Goal Programming
Graphical Interpretation (5 of 6)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Figure 9.5 The Fourth-Priority Goal: Minimize d1+
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17. Graphical Interpretation (6 of 6)
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 P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Solution: x1 = 15 bowls
x2 = 20 mugs
d1- = 15 hours
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18. Goal Programming Computer Solution Using QM for Windows (1 of 3)
Minimize P1d1-, P2d2-, P3d3+, P4d1+
subject to:
x1 + 2x2 + d1- - d1+ = 40
40x x2 + d2 - - d2 + = 1,600
4x1 + 3x2 + d3 - - d3 + = 120
x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 +  0
Exhibit 9.1
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19. Goal Programming Computer Solution Using QM for Windows (2 of 3)
Exhibit 9.2
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20. Goal Programming Computer Solution Using QM for Windows (3 of 3)
Exhibit 9.3
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21. Goal Programming Computer Solution Using Excel (1 of 3)
Exhibit 9.4
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22. Computer Solution Using Excel (2 of 3)
Goal Programming
Computer Solution Using Excel (2 of 3)
Exhibit 9.5
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23. Computer Solution Using Excel (3 of 3)
Goal Programming
Computer Solution Using Excel (3 of 3)
Exhibit 9.6
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24. Solution for Altered Problem Using Excel (1 of 6)
Goal Programming
Solution for Altered Problem Using Excel (1 of 6)
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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25. Solution for Altered Problem Using Excel (2 of 6)
Goal Programming
Solution for Altered Problem Using Excel (2 of 6)
Exhibit 9.7
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26. Solution for Altered Problem Using Excel (3 of 6)
Goal Programming
Solution for Altered Problem Using Excel (3 of 6)
Exhibit 9.8

27. Solution for Altered Problem Using Excel (4 of 6)
Goal Programming
Solution for Altered Problem Using Excel (4 of 6)
Exhibit 9.9
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28. Solution for Altered Problem Using Excel (5 of 6)
Goal Programming
Solution for Altered Problem Using Excel (5 of 6)
Exhibit 9.10
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29. Solution for Altered Problem Using Excel (6 of 6)
Goal Programming
Solution for Altered Problem Using Excel (6 of 6)
Exhibit 9.11
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30. Analytical Hierarchy Process (AHP) Overview
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.
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31. 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
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32. 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.
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33. 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.
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34. 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.
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35. Analytical Hierarchy Process Pairwise Comparisons (2 of 2)
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Table Preference Scale for Pairwise Comparisons

36. Income Level Infrastructure Transportation
Analytical Hierarchy Process
Pairwise Comparison Matrix
A pairwise comparison matrix summarizes the pairwise comparisons for a criteria.
Income Level Infrastructure Transportation A B C
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37. Analytical Hierarchy Process
Developing Preferences Within Criteria (1 of 3)
In synthesization, decision alternatives are prioritized within each criterion
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38. Analytical Hierarchy Process
Developing Preferences Within Criteria (2 of 3)
The row average values represent the preference vector
Table The Normalized Matrix with Row Averages
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39. Analytical Hierarchy Process
Developing Preferences Within Criteria (3 of 3)
Preference vectors for other criteria are computed similarly,
resulting in the preference matrix
Table Criteria Preference Matrix
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40. Analytical Hierarchy Process Ranking the Criteria (1 of 2)
Pairwise Comparison Matrix:
Table Normalized Matrix for Criteria with Row Averages
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41. Analytical Hierarchy Process Ranking the Criteria (2 of 2)
Preference Vector for Criteria:
Market
Income
Infrastructure
Transportation
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42. Analytical Hierarchy Process Developing an Overall Ranking
Overall Score:
Site A score = .1993(.5012) (.2819) (.1790) (.1561)
= .3091
Site B score = .1993(.1185) (.0598) (.6850) (.6196) = .1595
Site C score = .1993(.3803) (.6583) (.1360) (.2243) = .5314
Overall Ranking:
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43. Analytical Hierarchy Process Summary of Mathematical Steps
Develop a pairwise comparison matrix for each decision alternative for each criteria.
Synthesization
Sum each column value of the pairwise comparison matrices.
Divide each value in each column by its 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.
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44. 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 site selection criteria
Market
Income
Infrastructure
Transportation
Criteria 1
1/5 3 4
0.1993 5 9 7
0.6535
1/3
1/9 2
0.0860
1/4
1/7
1/2
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
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45. 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/ =
2.8524/ =
0.3474/ =
0.2473/ =
16.257
Average = /4 =
Step 3: Calculate the Consistency Index (CI)
CI = (Average – n)/(n-1), where n is no. of items compared
CI = ( )/(4-1) =
(CI = 0 indicates perfect consistency)
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46. 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 n 2 3 4 5 6 7 8 9 10 RI
0.58
0.90
1.12
1.24
1.32
1.41
1.45
1.51
Table 9.5 Random Index Values for n Items Being Compared
CI/RI = /0.90 =
Note: Degree of consistency is satisfactory if CI/RI < 0.10
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47. Analytical Hierarchy Process Excel Spreadsheets (1 of 4)
Exhibit 9.12
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48. Analytical Hierarchy Process Excel Spreadsheets (2 of 4)
Exhibit 9.13
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49. Analytical Hierarchy Process Excel Spreadsheets (3 of 4)
Exhibit 9.14
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50. Analytical Hierarchy Process Excel Spreadsheets (4 of 4)
Exhibit 9.15
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51. Scoring Model Overview
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.
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52. Scoring Model Example Problem
Mall selection with four alternatives and five criteria:
S1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) =
S2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) =
S3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) =
S4 = (.30)(60) + (.25)(90) + (.25)(85) + (.10)(90) + (.10)(70) =
Mall 4 preferred because of highest score, followed by malls 3, 2, 1.
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53. Scoring Model Excel Solution Exhibit 9.16
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54. 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 and solve a goal programming model to determine number of interviewers to hire in order to satisfy the goals
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55. Goal Programming Example Problem Solution (1 of 2)
Step 1: Model Formulation:
Minimize P1d1-, P2d2+, P3d3-
subject to:
80x1 + 40x2 + d1- - d1+ = 3,000 interviews
50x1 + 70x2 + d2- - d2 + = $2,500 budget
80x1 + d3- - d3 + = 1,000 telephone interviews
where:
x1 = number of telephone interviews
x2 = number of personal interviews
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56. Goal Programming Example Problem Solution (2 of 2)
Step 2: QM for Windows Solution:
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57. Analytical Hierarchy Process Example Problem Problem Statement
Purchasing decision, three model alternatives, three decision criteria.
Pairwise comparison matrices:
Prioritized decision criteria:
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58. 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.
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59. 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.
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60. Analytical Hierarchy Process Example Problem Problem Solution (3 of 4)
Step 2: Rank the criteria.
Price
Gears
Weight
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61. Analytical Hierarchy Process Example Problem Problem Solution (4 of 4)
Step 3: Develop an overall ranking.
Bike X
Bike Y
Bike Z
Bike X score = .6667(.6479) (.2299) (.1222) = .5057
Bike Y score = .2222(.6479) (.2299) (.1222) = .2138
Bike Z score = .1111(.6479) (.2299) (.1222) = .2806
Overall ranking of bikes: X first followed by Z and Y (sum of scores equal ).
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62. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall