1. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-1 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Analytic Hierarchy Process

2. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-2 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458Introduction  AHP was developed by Thomas L. Saaty and published in his 1980 book, The Analytic Hierarchy Process.  Analytic hierarchy process (AHP) is an approach designed to quantify the preferences for various factors and alternatives.  This process involves pairwise comparisons.  The decision maker starts by laying out the overall hierarchy of the decision.  This hierarchy reveals the factors to be considered as well as the various alternatives in the decision. Then, a number of pairwise comparisons are done, which result in the determination of factor weights and factor evaluations.

3. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-3 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Analytic Hierarchy Process  Break decision into stages or levels.  Starting at the lowest level, for each level, make pairwise comparison of the factors.  9-step scale: 1.equally preferred 2.equally to moderately preferred 3.moderately preferred 4.moderately to strongly preferred 5.strongly preferred 6.strongly to very strongly preferred 7.very strongly preferred 8.very to extremely preferred 9.extremely preferred

4. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-4 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Analytic Hierarchy Process  Develop the matrix representation:  Comparison matrix  Normalized matrix  Priority matrix  Develop and the consistency ratio.  Determine factor weights.  Perform a multifactor evaluation.

5. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-5 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Judy Grim's Computer Decision As an example of this process, we take the case of Judy Grim, who is looking for a new computer systems for her small business. She has determined that the most important overall factors hardware, software, and vendor support. Furthermore, Judy has narrowed down her alternatives to three possible computer systems. She has labeled these SYSTEM-1, SYSTEM-2, and SYSTEM-3. To begin, Judy has placed these factors and alternatives into a decision hierarchy (see Figure 1).

6. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-6 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Decision Hierarchy for Computer System Selection Select Computer System Hardware Software Vendor Support System: System: System: 1 2 3 1 2 3 1 2 3 Figure (1) The key to using AHP is pairwise comparisons. The decision maker, Judy Grim, needs to compare two different alternatives using a scale that ranges from equally preferred to extremely preferred.

7. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-7 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Using Pairwise Comparisons Judy begins by looking at the hardware factor and by comparing computer SYSTEM-1 with computer SYSTEM-2. Using the 9-step scale. Judy determines that the hardware for computer SYSTEM-1 is moderately preferred to computer SYSTEM-2. Thus, Judy uses the number 3, representing moderately preferred. She believes that the hardware for computer SYSTEM-1 is extremely preferred to computer SYSTEM-3. This is a numerical score of 9. She believes that the hardware for computer SYSTEM-2 is strongly to very strongly preferred to the hardware for computer SYSTEM-3, a score of 6.

8. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-8 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Beginning Comparison Matrix Judy Grim has used the 9-point scale for pairwise comparison to evaluate each system on hardware capabilities Hardware System-1 System-3 System-1 System-2 System-3 System-2 39 6 1 1

9. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-9 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Comparison Matrix (continued) Hardware System-1 System-3 System-1 System-2 System-3 System-2 39 6 1 1 1 1/3 1/91/6

10. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-10 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hardware System-1 System-3 System-1 System-2 System-3 System-2 39 6 1 1 1 1/3 1/91/6 1.4444.16716.0 Column Totals Normalizing the Matrix The totals are used to create a normalized matrix

11. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-11 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hardware System-1 System-3 System-1 System-2 System-3 System-2 0.69230.7200 0.23000.24000.3750 0.07690.04000.0625 Normalized Matrix 0.5625 = 1/ 1.444 =.333/ 1.444

12. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-12 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Final Matrix for Hardware FactorSystem-1System-2System-3 Hardware0.65830.28190.0598 To determine the priorities for hardware for the three computer systems, we simply find the average of the various rows from the matrix of numbers as follows:

13. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-13 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Weighted Sum Vector F = [ 0.6583 0.2819 0.0598] 39 0.3316 0.110.1671 1 (0.6583)(1) + (0.2819)(3) +(0.0598)(9) = 2.0423 0.6583)(0.33) + (0.2819)(1) + (0.0598)(6) = 0.8602 (0.6583)(0.11) + (0.2819)(0.167) + (0.0598)(1) = 0.1799

14. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-14 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Consistency Vector 2.0423 / 0.6583 3.1025 = 0.8602 0.2819 = 3.0512 0.1799/ 0.0598 3.0086

15. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-15 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Computing Lambda Lambda is the average value of the consistency vectors. = 3.1025 + 3.0512 + 3.0086 3 = 3.0541

16. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-16 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Consistency Index The consistency index is: CI = 3.0541 – 3 3 – 1 = 0.0270

17. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-17 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Consistency Ratio The consistency ratio (CR) tells how consistent the decision maker is with her answers. A higher number means less consistency. In general, a number of 0.10 or greater suggests the decision maker should reevaluate her responses during the pairwise comparison. CR = CI RI (random index) = 0.0270 0.58 = 0.0466 This is a table value Is Judy consistent in her answers regarding hardware??

18. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-18 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Random Index Table

19. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-19 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Achieving a Final Ranking  We must now perform a second pairwise comparison regarding the relative importance of each of the remaining two factors. Factor Evaluation System 1System 2System 3 Hardware Software Vendor Support 0.6583 0.2819 0.0598 0.0874 0.1622 0.7504 0.4967 0.3967 0.1066 Table (1): Factor Evaluations

20. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-20 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Achieving a Final Rank (continued) Determining Factor Weights Next, we need to determine the factor weights. In comparing the three factors, Judy determines that software is the most important. Software is very to extremely strongly preferred over hardware (number 8). Software is moderately preferred over vendor support (number 3). In comparing vendor support to hardware, we decide that the vendor support is more important. Vendor support is moderately preferred to hard ware (number 3). With these values, we can construct the pairwise comparison matrix and then compute the weights for hardware, software, and support.

21. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-21 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Achieving a Final Rank (continued) After making the appropriate calculations, the factor weights for hardware, software, and vendor support are shown in the next table: Table (2): Factor weights

22. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-22 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Judy Grim’s Final Decision Overall Ranking After the factor weights have been determined, we can multiply the factor evaluations in table (1) times the factor weights in table (2). It will give us the overall ranking for the three computer systems, which is shown in next table.

23. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-23 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458Example Select the Best Car Cost Cost SafetySafetyAppearanceAppearance HondaMazda Volvo VolvoHondaMazda HondaMazda HondaMazda HondaMazda HondaMazda Overall Goal Criteria DecisionAlternatives

24. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-24 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example (continued) Cost Honda Volvo Honda Mazda Volvo Mazda 24 3 1 1 1 1/2 1/41/3

25. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-25 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example (continued) Safety Honda Volvo Honda Mazda Volvo Mazda 1/21/5 1/4 1 1 1 2 54

26. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-26 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example (continued) Appearance Honda Volvo Honda Mazda Volvo Mazda 59 2 1 1 1 1/5 1/91/2

27. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-27 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example (continued) Criteria Cost Appear. Cost Safety Appear. Safety 1/23 5 1 1 1 2 1/31/5

28. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-28 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Factor Evaluation HondaMazdaVolvo Cost0.5570.3200.123 Safety0.1170.2000.683 Appear0.7610.1580.082

29. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-29 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 FactorFactor Weight Cost0.309 SAFETY0.582 APPEARANCE0.109

30. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M1-30 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Overall Ranking System or Alternative Total Weighted Evaluation Honda0.324 Mazda0.232 Volvo0.444 Best Decision!!