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

2. 9-2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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

3. 9-3 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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

4. 9-4 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Beaver Creek Pottery Company Example: Maximize Z = $40x 1 + 50x 2 subject to: 1x 1 + 2x 2  40 hours of labor 4x 1 + 3x 2  120 pounds of clay x 1, x 2  0 Where: x 1 = number of bowls produced x 2 = number of mugs produced Goal Programming Example Problem Data (1 of 2)

5. 9-5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 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 Example Problem Data (2 of 2)

6. 9-6 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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 Programming Goal Constraint Requirements

7. 9-7 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Labor goal: x 1 + 2x 2 + d 1 - - d 1 + = 40 (hours/day) Profit goal: 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 ($/day) Material goal: 4x 1 + 3x 2 + d 3 - - d 3 + = 120 (lbs of clay/day) Goal Programming Model Formulation Goal Constraints (1 of 3)

8. 9-8 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 1.Labor goals constraint (priority 1 - less than 40 hours labor; priority 4 - minimum overtime):  Minimize P 1 d 1 -, P 4 d 1 + 2.Add profit goal constraint (priority 2 - achieve profit of $1,600):  Minimize P 1 d 1 -, P 2 d 2 -, P 4 d 1 + 3.Add material goal constraint (priority 3 - avoid keeping more than 120 pounds of clay on hand):  Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + Goal Programming Model Formulation Objective Function (2 of 3)

9. 9-9 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Complete Goal Programming Model: Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 (labor) 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 (profit) 4x 1 + 3x 2 + d 3 - - d 3 + = 120 (clay) x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Model Formulation Complete Model (3 of 3)

10. 9-10 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■Changing fourth-priority goal “limit overtime to 10 hours” instead of minimizing overtime:  d 1 - + d 4 - - d 4 + = 10  minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 4 + ■Addition of a fifth-priority goal- “important to achieve the goal for mugs”:  x 1 + d 5 - = 30 bowls  x 2 + d 6 - = 20 mugs  minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 4 +, 4P 5 d 5 - + 5P 5 d 6 - Goal Programming Alternative 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.

11. 9-11 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Goal Programming Alternative Forms of Goal Constraints (2 of 2) Complete Model with Added New Goals: Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 4 +, 4P 5 d 5 - + 5P 5 d 6 - subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 d 1 + + d 4 - - d 4 + = 10 x 1 + d 5 - = 30 x 2 + d 6 - = 20 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 +, d 5 -, d 6 -  0

12. 9-12 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 9.1 Goal constraints Goal Programming Graphical Interpretation (1 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0

13. 9-13 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 9.2 The first-priority goal: minimize d 1 - Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Graphical Interpretation (2 of 6)

14. 9-14 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 9.3 The second-priority goal: Minimize d 2 - Goal Programming Graphical Interpretation (3 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0

15. 9-15 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 9.4 The third-priority goal: minimize d 3 + Goal Programming Graphical Interpretation (4 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0

16. 9-16 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 9.5 The Fourth-Priority Goal: Minimize d 1 + Goal Programming Graphical Interpretation (5 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0

17. 9-17 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Goal programming solutions do not always achieve all goals and they are not optimal; they achieve the best or most satisfactory solution possible. Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Solution:x 1 = 15 bowls x 2 = 20 mugs d 1 - = 15 hours Goal Programming Graphical Interpretation (6 of 6)

18. 9-18 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.1 Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50 x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Computer Solution Using QM for Windows (1 of 3)

19. 9-19 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.2 Goal Programming Computer Solution Using QM for Windows (2 of 3)

20. 9-20 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.3 Goal Programming Computer Solution Using QM for Windows (3 of 3)

21. 9-21 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.4 Goal Programming Computer Solution Using Excel (1 of 3) Goal constraint for cell G5 Decision variables— B10:B11 Deviational variables— E5:F7

22. 9-22 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.5 Goal Programming Computer Solution Using Excel (2 of 3) Minimize deviational variable for first-priority goal in E5 Decision and deviational variables Goal constraints

23. 9-23 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.6 Goal Programming Computer Solution Using Excel (3 of 3)

24. 9-24 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 4 +, 4P 5 d 5 - + 5P 5 d 6 - subject to: x 1 + 2x 2 + d 1 - - d 1 + = 40 40x 1 + 50x 2 + d 2 - - d 2 + = 1,600 4x 1 + 3x 2 + d 3 - - d 3 + = 120 d 1 + + d 4 - - d 4 + = 10 x 1 + d 5 - = 30 x 2 + d 6 - = 20 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 +, d 5 -, d 6 -  0 Goal Programming Solution for Altered Problem Using Excel (1 of 6) This model includes goals for overtime and maximum storage levels for bowls and mugs.

25. 9-25 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.7 Goal Programming Solution for Altered Problem Using Excel (2 of 6) Goal constraint for labor Goal constraint for overtime; =F5+E8-F8 =C9*B13+E9

26. 9-26 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.8 Goal Programming Solution for Altered Problem Using Excel (3 of 6)

27. 9-27 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Goal Programming Solution for Altered Problem Using Excel (4 of 6) Exhibit 9.9 First two priority goals, minimizing and, achieved Third-priority goal to minimize not achieved

28. 9-28 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.10 Goal Programming Solution for Altered Problem Using Excel (5 of 6) First- and second-priority goals achieved; add E5=0 and E6=0

29. 9-29 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.11 Goal Programming Solution for Altered Problem Using Excel (6 of 6) Fourth-priority goal to minimize overtime,, not achieved

30. 9-30 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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

31. 9-31 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 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 Process Example Problem Statement

32. 9-32 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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 Process Hierarchy Structure

33. 9-33 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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 Process General Mathematical Process

34. 9-34 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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 Process Pairwise Comparisons (1 of 2)

35. 9-35 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Table 9.1 Preference scale for pairwise comparisons Analytical Hierarchy Process Pairwise Comparisons (2 of 2)

36. 9-36 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Income Level Infrastructure Transportation A B C Analytical Hierarchy Process Pairwise Comparison Matrix A pairwise comparison matrix summarizes the pairwise comparisons for a criteria.

37. 9-37 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Analytical Hierarchy Process Developing Preferences Within Criteria (1 of 3) In synthesization, decision alternatives are prioritized within each criterion

38. 9-38 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Table 9.2 The normalized matrix with row averages Analytical Hierarchy Process Developing Preferences Within Criteria (2 of 3) The row average values represent the preference vector

39. 9-39 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Table 9.3 Criteria preference matrix Analytical Hierarchy Process Developing Preferences Within Criteria (3 of 3) Preference vectors for other criteria are computed similarly, resulting in the preference matrix

40. 9-40 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Table 9.4 Normalized matrix for criteria with row averages Analytical Hierarchy Process Ranking the Criteria (1 of 2) Pairwise Comparison Matrix:

41. 9-41 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Analytical Hierarchy Process Ranking the Criteria (2 of 2) Preference Vector for Criteria: Market Income Infrastructure Transportation

42. 9-42 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Overall Score: Analytical Hierarchy Process Developing 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)=.5314 Overall Ranking:

43. 9-43 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Analytical Hierarchy Process Summary of Mathematical Steps 1.Develop a pairwise comparison matrix for each decision alternative for each criteria. 2.Synthesization a.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 alternative 7.Rank the decision alternatives.

44. 9-44 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 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 4 site selection criteria 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 MarketIncomeInfrastructureTransportationCriteria Market15-Jan340.1993 Income51970.6535 Infrastructure3-Jan9-Jan120.086 Transportation4-Jan7-Jan2-Jan10.0612

45. 9-45 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 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 = 4.1786 2.8524/0.6535 = 4.3648 0.3474/0.0860 = 4.0401 0.2473/0.0612 = 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)

46. 9-46 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 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 n2345678910 RI00.580.901.121.241.321.411.451.51

47. 9-47 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.12 Analytical Hierarchy Process Excel Spreadsheets (1 of 4) Click on “Format,” then “Cells,” then “Fraction” to enter fractions Row average formula for F14 =SUM(B5:B7) =B5/B8

48. 9-48 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.13 Analytical Hierarchy Process Excel Spreadsheets (2 of 4) =SUM(B14:E14)/4 =SUM(B5:B8) =B5/B9

49. 9-49 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.14 Analytical Hierarchy Process Excel Spreadsheets (3 of 4) Atlanta score in cell C12 Cells G14:G17 from Exhibit 9.13 Cells F14:F16 from Exhibit 9.12

50. 9-50 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.15 Analytical Hierarchy Process Excel Spreadsheets (4 of 4) =I5/G5 =SUM(B11:B14) =(4.1564-4)/3 =G12/0.90

51. 9-51 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Each decision alternative is graded in terms of how well it satisfies the criterion according to following formula: S i =  g ij w j where: w j = a weight between 0 and 1.00 assigned to criterion j; 1.00 important, 0 unimportant; sum of total weights equals one. g ij = a grade between 0 and 100 indicating how well alternative i satisfies criteria j; 100 indicates high satisfaction, 0 low satisfaction. Scoring Model Overview

52. 9-52 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Mall selection with four alternatives and five criteria: S 1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) = 62.75 S 2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) = 73.50 S 3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) = 76.00 S 4 = (.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. Scoring Model Example Problem

53. 9-53 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 9.16 Scoring Model Excel Solution

54. 9-54 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 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/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

55. 9-55 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 1: Model Formulation: Minimize P 1 d 1 -, P 2 d 2 +, P 3 d 3 - subject to: 80x 1 + 40x 2 + d 1 - - d 1 + = 3,000 interviews 50x 1 + 70x 2 + d 2 - - d 2 + = $2,500 budget 80x 1 + d 3 - - d 3 + = 1,000 telephone interviews where: x 1 = number of telephone interviews x 2 = number of personal interviews Goal Programming Example Problem Solution (1 of 2)

56. 9-56 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Goal Programming Example Problem Solution (2 of 2) Step 2: QM for Windows Solution:

57. 9-57 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Purchasing decision, three model alternatives, three decision criteria. Pairwise comparison matrices: Prioritized decision criteria: Analytical Hierarchy Process Example Problem Problem Statement

58. 9-58 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 1: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Analytical Hierarchy Process Example Problem Problem Solution (1 of 4)

59. 9-59 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 1 continued: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Analytical Hierarchy Process Example Problem Problem Solution (2 of 4) Preference vectors are summarized

60. 9-60 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 2: Rank the criteria. Price Gears Weight Analytical Hierarchy Process Example Problem Problem Solution (3 of 4)

61. 9-61 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 3: Develop an overall ranking. Bike X Bike Y Bike Z Bike X score =.6667(.6479) +.0853(.2299) +.4429(.1222) =.5057 Bike Y score =.2222(.6479) +.2132(.2299) +.1698(.1222) =.2138 Bike 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). Analytical Hierarchy Process Example Problem Problem Solution (4 of 4)

62. 9-62 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.