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# THE ANALYTIC HIERARCHY PROCESS CAR PURCHASE EXAMPLE

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### THE ANALYTIC HIERARCHY PROCESS CAR PURCHASE EXAMPLE

1. 1. THE ANALYTIC HIERARCHY PROCESS CAR PURCHASE EXAMPLE
2. 2. CAR PURCHASE EXAMPLE <ul><li>We now consider a motivating example. </li></ul><ul><li>After completing this example, you will have an understanding of the basics of AHP and its application through Expert Choice (www.expertchoice.com). </li></ul><ul><li>We want to apply the AHP to help a couple decide which car they should purchase. </li></ul>
3. 3. CAR PURCHASE EXAMPLE <ul><li>The couple is considering three criteria: cost, safety, and appearance. </li></ul><ul><li>They have narrowed their alternatives to three specific cars: Honda, Mazda, and Volvo. </li></ul><ul><li>We demonstrate how to build the AHP hierarchy in Expert Choice. </li></ul>
4. 4. <ul><li>After launching Expert Choice, select the F ile, N ew option, and after selecting a destination folder, enter a file name such as CARS. (Expert Choice add the AHP file extension.) </li></ul><ul><li>Next, enter a description for your goal, such as, “Select the best car.” </li></ul>EXPERT CHOICE: FILE SETUP
5. 5. <ul><li>To enter the criteria, for example, cost, safety, and appearance, use the E dit, and Insert C hild of Current Node commands. </li></ul><ul><li>Use the Esc key or hit an extra enter when finished entering the criteria. </li></ul><ul><li>To add the alternative cars select the E dit, A lternative, and I nsert commands. </li></ul>EXPERT CHOICE: FILE SETUP
6. 6. <ul><li>You can also use the “Add Alternative” button in the upper right hand corner of the model window. </li></ul><ul><li>Repeat for all alternatives. </li></ul><ul><li>Additional details can be found in the Expert Choice tutorial provided with the software. </li></ul>EXPERT CHOICE: FILE SETUP
7. 7. ANALYZING THE HIERARCHY <ul><li>1. Determine the weights of the alternatives for each criterion. </li></ul><ul><li>2. Determine the priorities or weights of the criteria in achieving the goal. </li></ul><ul><li>3. Determine the overall weight of each alternative in achieving the goal. This is accomplished by combining the results of the first two stages and is called synthesis. </li></ul>
8. 8. ANALYZING THE HIERARCHY <ul><li>To complete the first stage, the couple can base their judgments on the following (hypothetical) performance information. </li></ul><ul><li>All alternative pairwise comparisons should be based on data. </li></ul>
9. 9. HYPOTHETICAL DATA FOR CAR PURCHASE EXAMPLE Car Cost Safety* Appearance Honda \$22,000 28 Sporty Mazda 28,500 39 Slick Volvo 33,000 52 Dull * Safety Rating from a consumer testing service - the higher the number, the safer the car.
10. 10. DETERMINING PRIORITIES <ul><li>The couple begins by making pairwise comparison judgments between each pair of cars for the cost criterion. </li></ul><ul><li>In our example, three judgments are needed: Honda to Mazda, Mazda to Volvo, and Honda to Volvo. </li></ul><ul><li>The scale on the next page is the standard one. </li></ul>
11. 11. STANDARD 1 - 9 MEASUREMENT SCALE <ul><li>Intensity of Importance Definition Explanation </li></ul><ul><li>1 Equal importance Two activities contribute equally </li></ul><ul><li>3 Moderate importance Experience and judgment slightly favor one </li></ul><ul><li>activity over another </li></ul><ul><li>5 Strong importance Experience and judgment strongly favor one </li></ul><ul><li>activity over another </li></ul><ul><li>7 Very strong An activity is favored very strongly over </li></ul><ul><li>another </li></ul><ul><li>9 Extreme importance The evidence favoring one activity over </li></ul><ul><li>another is of the highest possible order </li></ul><ul><li>of affirmation </li></ul><ul><li> 2, 4, 6, 8 For compromise Sometimes one needs to interpolate a </li></ul><ul><li>values compromise between the above judgment </li></ul><ul><li>numerically because there is no good </li></ul><ul><li>word to describe it </li></ul><ul><li> 1.1 - 1.9 For tied activities When elements are close and nearly </li></ul><ul><li>indistinguishable; moderate is 1.3 and </li></ul><ul><li>extreme is 1.9 </li></ul><ul><li>Reciprocals of above If activity A has For example, if the pairwise comparison of </li></ul><ul><li>one of the above A to B is 3.0, then the pairwise comparison </li></ul><ul><li>numbers assigned of B to A is 1/3 </li></ul><ul><li>to it when compared </li></ul><ul><li>with activity B, </li></ul><ul><li>then B has the </li></ul><ul><li>reciprocal value </li></ul><ul><li>when compared to A. </li></ul>
12. 12. COST PAIRWISE COMPARISONS <ul><li>The pairwise comparisons are represented in the form of pairwise comparison matrices. </li></ul><ul><li>The computation of the weights are also shown. </li></ul><ul><li>Consider the pairwise comparison matrix to compare the cars for the cost criterion. </li></ul><ul><li>Remember that the costs of the three cars are: \$22000, \$28500, and \$33000, respectively. </li></ul>
13. 13. <ul><li>If we compare the Honda to the Honda, obviously they are equal. </li></ul><ul><li>Therefore, a 1 (equal preferred) is placed in the first row, first column entry of the matrix. </li></ul>COST PAIRWISE COMPARISONS
14. 14. <ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 </li></ul><ul><li>28.5K Mazda </li></ul><ul><li>33K Volvo </li></ul>COST PAIRWISE COMPARISONS
15. 15. <ul><li>The other entries along the main diagonal of the matrix are also 1. </li></ul><ul><li>This simply means that everything is equally preferred to itself. </li></ul>COST PAIRWISE COMPARISONS
16. 16. <ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 </li></ul><ul><li>28.5K Mazda 1 </li></ul><ul><li>33K Volvo 1 </li></ul>COST PAIRWISE COMPARISONS
17. 17. <ul><li>Suppose we believe the Honda (\$22000) is equally to moderately preferred to the Mazda (\$28500). Place a 2 in the row 1, column 2 entry. </li></ul><ul><li>Some might argue that the Honda should be 1.295 times better than the Mazda (28,500/22,000). </li></ul>COST PAIRWISE COMPARISONS
18. 18. <ul><li>Do you agree? </li></ul><ul><li>It depends! </li></ul><ul><li>For some, \$28,500 is significantly greater than \$22,000, implying a judgments greater than 1.295. </li></ul><ul><li>Others with a lot of money may perceive virtually no difference between the two costs, implying a judgment somewhere between 1 and 1.295. </li></ul>COST PAIRWISE COMPARISONS
19. 19. <ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 </li></ul><ul><li>28.5K Mazda 1 </li></ul><ul><li>33K Volvo 1 </li></ul>COST PAIRWISE COMPARISONS
20. 20. <ul><li>If the Honda is 2 times better than the Mazda, this implies that the Mazda (\$28500) is one half as good as the Honda (\$22000). </li></ul><ul><li>The reciprocal judgment, (1/2), should be placed in the row 2, column 1 entry of the matrix. </li></ul>COST PAIRWISE COMPARISONS
21. 21. <ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 </li></ul><ul><li>28.5K Mazda 1/2 1 </li></ul><ul><li>33K Volvo 1 </li></ul>COST PAIRWISE COMPARISONS
22. 22. <ul><li>Suppose that we judge the Mazda (\$28500) to be equally to moderately preferred to the Volvo (\$33000). </li></ul><ul><li>The following judgments would be entered in the matrix. </li></ul>COST PAIRWISE COMPARISONS
23. 23. <ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/2 1 </li></ul>COST PAIRWISE COMPARISONS
24. 24. <ul><li>Assuming perfect consistency of judgments, we would expect that the Honda (\$22000) is 4 times (that is, moderately to strongly) preferred to the Volvo (\$33000). </li></ul><ul><li>We will relax this assumption later. </li></ul>COST PAIRWISE COMPARISONS
25. 25. <ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/4 1/2 1 </li></ul>COST PAIRWISE COMPARISONS
26. 26. <ul><li>The matrix is now complete and the weights for each car (for the cost criterion) can be computed. </li></ul><ul><li>The exact computational procedure is implemented in Expert Choice. </li></ul><ul><li>For details see Expert Choice homepage and download AHPDEMO.EXE. </li></ul>COST PAIRWISE COMPARISONS
27. 27. <ul><li>A simple three step procedure can be used to approximate the weights for each alternative. </li></ul><ul><li>Essentially, this procedure normalizes the ratios of the judgments between any pair of alternatives. </li></ul>COST PAIRWISE COMPARISONS
28. 28. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/4 1/2 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS </li></ul>COST PAIRWISE COMPARISONS
29. 29. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/4 1/2 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 7/2 7 </li></ul>COST PAIRWISE COMPARISONS
30. 30. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/4 1/2 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 7/2 7 </li></ul>COST PAIRWISE COMPARISONS
31. 31. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2 . DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/4 1/2 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 7/2 7 </li></ul><ul><li>B. ADJUSTED COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>Honda 4/7* 4/7 4/7 </li></ul><ul><li>Mazda 2/7 2/7 2/7 </li></ul><ul><li>Volvo 1/7 1/7 1/7 </li></ul><ul><li>* This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). </li></ul>COST PAIRWISE COMPARISONS
32. 32. <ul><li>Notice that no variation is seen across the rows because the judgments are perfectly consistent. </li></ul><ul><li>For the third column, judgments totaling 7 were awarded. The Honda received 4 of 7 (57.1%), the Mazda 2 of 7 (28.6%), and the Volvo 1 of 7 (14.3%) of the weight. </li></ul><ul><li>Similar comparisons can be made for the other two columns. </li></ul>COST PAIRWISE COMPARISONS
33. 33. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/4 1/2 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 7/2 7 </li></ul><ul><li>B. ADJUSTED COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>Honda 4/7* 4/7 4/7 </li></ul><ul><li>Mazda 2/7 2/7 2/7 </li></ul><ul><li>Volvo 1/7 1/7 1/7 </li></ul><ul><li>* This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). </li></ul>COST PAIRWISE COMPARISONS
34. 34. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 2 </li></ul><ul><li>33K Volvo 1/4 1/2 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 7/2 7 </li></ul><ul><li>B. ADJUSTED COST PAIRWISE COMPARISON MATRIX WEIGHTS </li></ul><ul><li> Honda Mazda Volvo (ROW AVG.) </li></ul><ul><li>Honda 4/7* 4/7 4/7 0.571 </li></ul><ul><li>Mazda 2/7 2/7 2/7 0.286 </li></ul><ul><li>Volvo 1/7 1/7 1/7 0.143 </li></ul><ul><li> --------- </li></ul><ul><li> TOTAL 1.000 </li></ul><ul><li>* This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). </li></ul>COST PAIRWISE COMPARISONS
35. 35. <ul><li>Expert Choice offers a variety of modes for entering the judgments. </li></ul><ul><li>Highlight the cost node and select the Pairwise Numerical comparison button (3:1). </li></ul><ul><li>This button appears on the top left-hand side of the toolbar to the right of the model view button. </li></ul>EXPERT CHOICE: Entering Judgments
36. 36. <ul><li>Sliding the bar between Honda and Mazda to the left so that it rests on the 2 means that the Honda is two times better than the Mazda when considering cost. </li></ul><ul><li>If the Mazda were 2 times better than the Honda, the bar would be slid to the 2 on the right. </li></ul><ul><li>The other comparisons are entered in a similar fashion. </li></ul>EXPERT CHOICE: Entering Judgments
37. 37. <ul><li>For our problem, Expert Choice only displays three judgments. </li></ul><ul><li>1’s along the main diagonal and reciprocal judgments do not appear. </li></ul>EXPERT CHOICE: Entering Judgments
38. 38. <ul><li>There are different modes for entering judgments. </li></ul><ul><li>The Pairwise Verbal Comparisons (ABC) and the Pairwise Graphical Comparisons (the button that looks like a bar graph) are available. </li></ul><ul><li>The only difference between these modes is how the pairwise comparison questions are displayed. </li></ul>EXPERT CHOICE: Entering Judgments
39. 39. <ul><li>A 1-9 scale is used for numerical comparisons. </li></ul><ul><li>The verbal comparisons are: equal, moderate, strong, very strong, and extreme. </li></ul><ul><li>The graphical mode makes comparisons based on the length of two bars. </li></ul><ul><li>The user selects the desired mode. </li></ul>EXPERT CHOICE: Entering Judgments
40. 40. <ul><li>After entering all pairwise comparisons, record judgments by clicking Yes. </li></ul><ul><li>The model view will be displayed with alternative weights for the cost criterion now appearing. </li></ul>EXPERT CHOICE: Entering Judgments
41. 41. INCONSISTENCY OF JUDGMENTS <ul><li>Since our pairwise comparisons were perfectly consistent, Expert Choice reports Incon: 0.00. </li></ul><ul><li>If this ratio is greater than 0.1 some revision of judgments is required. </li></ul><ul><li>Select Inconsis t ency (within any Pairwise Comparison mode) to identify the most inconsistent judgments. </li></ul>
42. 42. INCONSISTENCY OF JUDGMENTS <ul><li>Inconsistency of judgments may result from: </li></ul><ul><li>problems of estimation; </li></ul><ul><li>errors between the comparisons; </li></ul><ul><li>or, the comparisons may be naturally inconsistent. </li></ul>
43. 43. INCONSISTENCY OF JUDGMENTS <ul><li>One example of natural inconsistency is in a sporting contest. </li></ul><ul><li>If team A is twice as likely to beat team B, and if team B is three times as likely to beat team C, this does not necessarily imply that team A is six times as likely to beat team C. </li></ul><ul><li>This inconsistency may result because of the way that the teams “match-up” overall. </li></ul>
44. 44. INCONSISTENCY OF JUDGMENTS <ul><li>The point is not to stop inconsistency from occurring. </li></ul><ul><li>Make sure that the level of inconsistency remains within some reasonable limit. </li></ul>
45. 45. INCONSISTENCY OF JUDGMENTS <ul><li>How does a judgment change affect the car weights? </li></ul><ul><li>Suppose the Mazda to Volvo changes from 2 to 3. </li></ul><ul><li>This obviously changes the comparison for Volvo to Mazda from (1/2) to (1/3). </li></ul><ul><li>The judgments are now somewhat inconsistent. </li></ul>
46. 46. <ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 3 </li></ul><ul><li>33K Volvo 1/4 1/3 1 </li></ul>COST PAIRWISE COMPARISONS
47. 47. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 3 </li></ul><ul><li>33K Volvo 1/4 1/3 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 10/3 8 </li></ul>COST PAIRWISE COMPARISONS
48. 48. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 3 </li></ul><ul><li>33K Volvo 1/4 1/3 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 10/3 8 </li></ul><ul><li>B. ADJUSTED COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>Honda 4/7* 6/10 4/8 </li></ul><ul><li>Mazda 2/7 3/10 3/8 </li></ul><ul><li>Volvo 1/7 1/10 1/8 </li></ul><ul><li>* This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). </li></ul>COST PAIRWISE COMPARISONS
49. 49. <ul><li>1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. </li></ul><ul><li>2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM. </li></ul><ul><li>THIS RESULTS IN THE ADJUSTED MATRIX. </li></ul><ul><li>3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. </li></ul><ul><li>A. ORIGINAL COST PAIRWISE COMPARISON MATRIX </li></ul><ul><li> Honda Mazda Volvo </li></ul><ul><li>22K Honda 1 2 4 </li></ul><ul><li>28.5K Mazda 1/2 1 3 </li></ul><ul><li>33K Volvo 1/4 1/3 1 </li></ul><ul><li> ------- ------- ------- </li></ul><ul><li>COLUMN TOTALS 7/4 10/3 8 </li></ul><ul><li>B. ADJUSTED COST PAIRWISE COMPARISON MATRIX WEIGHTS </li></ul><ul><li> Honda Mazda Volvo (ROW AVG.) </li></ul><ul><li>Honda 4/7* 6/10 4/8 0.557 </li></ul><ul><li>Mazda 2/7 3/10 3/8 0.320 </li></ul><ul><li>Volvo 1/7 1/10 1/8 0.123 </li></ul><ul><li>-------- </li></ul><ul><li>TOTAL 1.000 </li></ul><ul><li>* This entry is obtained by dividing the Honda entry in the original matrix (1) by the Honda column total (7/4). </li></ul>COST PAIRWISE COMPARISONS
50. 50. INCONSISTENCY OF JUDGMENTS <ul><li>The new weights are: 0.557, 0.320, and 0.123. The inconsistency resulted in some change in the original weights of 0.571, 0.286, and 0.143. </li></ul><ul><li>As expected, the weight for the Mazda increased while the weight for the Volvo decreased. </li></ul><ul><li>The weights now vary across each row. Essentially, inconsistency measures the degree of variation across the rows. </li></ul>
51. 51. <ul><li>To make this change in Expert Choice, highlight cost node and select any Pairwise Comparison mode . </li></ul><ul><li>Within the numerical mode, slide the comparison bar to the left from 2 to 3, select the Model View, and record the judgments to see the new weights. </li></ul><ul><li>The weights of 0.558, 0.320, and 0.122 are slightly different from the three-step procedure weights. </li></ul><ul><li>This is not due to rounding -- Expert Choice gives the exact results. </li></ul>EXPERT CHOICE: Revising Judgments
52. 52. INCONSISTENCY OF JUDGMENTS <ul><li>The inconsistency ratio is now 0.02. </li></ul><ul><li>The weights can also be used to measure the effectiveness of the alternatives. </li></ul><ul><li>For example, based on all comparisons, the Honda is 1.74 (0.558/0.320) times better than the Mazda. </li></ul>
53. 53. INCONSISTENCY OF JUDGMENTS <ul><li>We knew that a \$22,000 car is better than a \$28,500 car, but now we know how much better. </li></ul><ul><li>Why is this ratio 1.74 and not the pairwise comparison of 2? </li></ul><ul><li>Inconsistency in the judgments! </li></ul>
54. 54. REMAINING COMPUTATIONS <ul><li>Next, the cars must be pairwise compared for the safety criterion and then for the appearance criterion. </li></ul><ul><li>These judgments are shown on the next page. </li></ul><ul><li>The safety comparisons are all inverted, that is, for each comparison, the top bar was moved to the left. </li></ul><ul><li>This means that the Mazda is 2 times more preferred than the Honda, with respect to safety. </li></ul>
55. 55. SAFETY & APPEARANCE JUDGMENTS <ul><li>Safety Pairwise Comparison Matrix </li></ul><ul><li>Honda Mazda Volvo </li></ul><ul><li>28 Honda 1 1/2 1/5 </li></ul><ul><li>39 Mazda 2 1 1/4 </li></ul><ul><li>52 Volvo 5 4 1 </li></ul><ul><li>Appearance Pairwise Comparison Matrix </li></ul><ul><li>Honda Mazda Volvo </li></ul><ul><li>Sporty Honda 1 5 9 </li></ul><ul><li>Slick Mazda 1/5 1 2 </li></ul><ul><li>Dull Volvo 1/9 1/2 1 </li></ul>
56. 56. REMAINING COMPUTATIONS <ul><li>Next, the criteria must be pairwise compared. </li></ul><ul><li>These judgments are shown on the next page. </li></ul><ul><li>There are no data to support these judgments since they are purely a reflection of your preferences. </li></ul>
57. 57. CRITERIA JUDGMENTS <ul><li>Original Criteria Pairwise Comparison Matrix </li></ul><ul><li>Cost Safety Appearance </li></ul><ul><li>Cost 1 1/2 3 </li></ul><ul><li>Safety 2 1 5 </li></ul><ul><li>Appearance 1/3 1/5 1 </li></ul>
58. 58. REMAINING COMPUTATIONS <ul><li>The last stage computes the final weights for each car. </li></ul><ul><li>Multiply the criteria weight by the car weight for each criterion and then sum over all criteria. </li></ul><ul><li>This is nothing more than a weighted average. </li></ul><ul><li>The computational results are shown next. </li></ul>
59. 59. FINAL CAR WEIGHTS <ul><li>CRITERIA WEIGHTS </li></ul><ul><li>COST SAFETY APPEARANCE </li></ul><ul><li>0.309 0.582 0.109 </li></ul><ul><li>CARS FINAL WEIGHTS </li></ul><ul><li>Honda 0.558 0.117 0.761 </li></ul><ul><li>Mazda 0.320 0.200 0.158 </li></ul><ul><li>Volvo 0.122 0.683 0.082 </li></ul>
60. 60. FINAL CAR WEIGHTS <ul><li>CRITERIA WEIGHTS </li></ul><ul><li>COST SAFETY APPEARANCE </li></ul><ul><li>0.309 0.582 0.109 </li></ul><ul><li>CARS FINAL WEIGHTS </li></ul><ul><li>Honda 0.558 0.117 0.761 0.324 </li></ul><ul><li>Mazda 0.320 0.200 0.158 </li></ul><ul><li>Volvo 0.122 0.683 0.082 </li></ul><ul><li>Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324 </li></ul><ul><li> 0.173 0.068 0.083 </li></ul>
61. 61. FINAL CAR WEIGHTS <ul><li>CRITERIA WEIGHTS </li></ul><ul><li>COST SAFETY APPEARANCE </li></ul><ul><li>0.309 0.582 0.109 </li></ul><ul><li>CARS FINAL WEIGHTS </li></ul><ul><li>Honda 0.558 0.117 0.761 0.324 </li></ul><ul><li>Mazda 0.320 0.200 0.158 0.232 </li></ul><ul><li>Volvo 0.122 0.683 0.082 </li></ul><ul><li>Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324 </li></ul><ul><li> 0.173 0.068 0.083 </li></ul><ul><li>Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232 </li></ul><ul><li> 0.099 0.116 0.017 </li></ul>
62. 62. FINAL CAR WEIGHTS <ul><li>CRITERIA WEIGHTS </li></ul><ul><li>COST SAFETY APPEARANCE </li></ul><ul><li>0.309 0.582 0.109 </li></ul><ul><li>CARS FINAL WEIGHTS </li></ul><ul><li>Honda 0.558 0.117 0.761 0.324 </li></ul><ul><li>Mazda 0.320 0.200 0.158 0.232 </li></ul><ul><li>Volvo 0.122 0.683 0.082 0.444 </li></ul><ul><li>Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324 </li></ul><ul><li> 0.173 0.068 0.083 </li></ul><ul><li>Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232 </li></ul><ul><li> 0.099 0.116 0.017 </li></ul><ul><li>Volvo: (0.122)(0.309) + (0.683)(0.582) + (0.082)(0.109) = 0.444 </li></ul><ul><li> 0.038 0.397 0.009 </li></ul>
63. 63. LOCAL VS GLOBAL WEIGHTS <ul><li>For cost, the local weights for the cars are 0.558, 0.320, and 0.122 and sum to 1.000. </li></ul><ul><li>The global weights are computed by multiplying the cost criterion weight by the local car weights. </li></ul><ul><li>The global weights are 0.173, 0.099, and 0.038 and sum to the cost criterion weight of 0.309. </li></ul>
64. 64. <ul><li>The final weights are shown in Expert Choice after all comparisons are entered and when the Model View is displayed and the goal is highlighted. </li></ul><ul><li>Choose Dis t ributive Mode. </li></ul><ul><li>The difference between the Distributive and Ideal modes will be discussed later. </li></ul>EXPERT CHOICE: Synthesis
65. 65. INTERPRETING THE RESULTS <ul><li>The final weights provide a measure of the relative performance of each alternative. </li></ul><ul><li>It is important to properly interpret the meaning of these numbers. </li></ul><ul><li>The Volvo is ranked first, the Honda second, and Mazda third. </li></ul><ul><li>The Volvo is preferred 1.37 (0.444/0.324) times more than the Honda. </li></ul>
66. 66. INTERPRETING THE RESULTS <ul><li>Should we buy the Volvo? </li></ul><ul><li>The output is a decision-making aid and cannot replace the decision-maker. </li></ul><ul><li>The results can be used to support discussion and possibly the judgments will be revised. </li></ul><ul><li>This iterative process is quite normal. </li></ul><ul><li>AHP can help to facilitate communication and generate consensus between different groups. </li></ul>
67. 67. SYNTHESIS MODES <ul><li>The process used to compute the final weights is called distributive synthesis . </li></ul><ul><li>This method works well when there is a fixed amount of resources that must be distributed to a fixed set of alternatives. </li></ul>
68. 68. SYNTHESIS MODES <ul><li>In some cases after completing an AHP analysis, an additional alternative may need to be considered. </li></ul><ul><li>It is possible that a rank reversal could occur. </li></ul><ul><li>Our rankings are: Volvo, Honda, and Mazda. </li></ul><ul><li>If another Volvo is added that is similar to the original Volvo, it is possible that the Honda will be ranked higher than the original Volvo. </li></ul>
69. 69. SYNTHESIS MODES <ul><li>In some cases this is acceptable, in others it is not. </li></ul><ul><li>Distributive synthesis should not be used if preservation of rank is important. </li></ul><ul><li>Ideal Synthesis should be used to prevent rank reversal. </li></ul>
70. 70. IDEAL MODE <ul><li>The ideal mode gives the full weight of the criterion to the alternative that ranks highest under that criterion. </li></ul><ul><li>The other alternatives are given a portion of the criterion weight based on their local weight. </li></ul>
71. 71. IDEAL MODE <ul><li>The local weights for the three cars with respect to cost are: 0.558, 0.320, and 0.122, respectively. The cost criterion weight is 0.309. </li></ul><ul><li>Since the Honda has the highest cost weight it is initially assigned the full cost weight of 0.309. </li></ul><ul><li>Mazda would be (0.320 / 0.558)*(0.309) = 0.177. </li></ul><ul><li>Volvo would be (0.122 / 0.558)*(0.309) = 0.068. </li></ul>
72. 72. IDEAL MODE <ul><li>Using the same approach, the weights for the three cars with respect to safety are: 0.100, 0.170, and 0.582, respectively. </li></ul><ul><li>The weights for the three cars with respect to appearance are: 0.109, 0.023, and 0.012, respectively. </li></ul>
73. 73. IDEAL MODE <ul><li>For each car, add the three criteria weights: </li></ul><ul><li>Honda Mazda Volvo </li></ul><ul><li>Cost 0.309 0.177 0.068 </li></ul><ul><li>Safety 0.100 0.170 0.582 </li></ul><ul><li>Appearance 0.109 0.023 0.012 </li></ul><ul><li>Total 0.518 0.370 0.662 </li></ul>
74. 74. IDEAL MODE <ul><li>For each car, add the three criteria weights: </li></ul><ul><li>Honda Mazda Volvo </li></ul><ul><li>Cost 0.309 0.177 0.068 </li></ul><ul><li>Safety 0.100 0.170 0.582 </li></ul><ul><li>Appearance 0.109 0.023 0.012 </li></ul><ul><li>Total 0.518 0.370 0.662 </li></ul>Since the sum of the three weights is 1.550, we divide each weight by 1.550 to normalize the results.
75. 75. IDEAL MODE <ul><li>For each car, add the three criteria weights: </li></ul><ul><li>Honda Mazda Volvo </li></ul><ul><li>Cost 0.309 0.177 0.068 </li></ul><ul><li>Safety 0.100 0.170 0.582 </li></ul><ul><li>Appearance 0.109 0.023 0.012 </li></ul><ul><li>Total 0.518 0.370 0.662 </li></ul><ul><li>Total/1.550 0.335 0.239 0.427 </li></ul><ul><li>These are the ideal weights reported in </li></ul><ul><li>Expert Choice. </li></ul>Since the sum of the three weights is 1.550, we divide each weight by 1.550 to normalize the results.
76. 76. SENSITIVITY ANALYSIS <ul><li>Sensitivity analysis is an important aspect of any decision-making process. </li></ul><ul><li>Sensitivity analysis determines whether small changes in judgments affects the final weights and rankings of the alternatives. </li></ul><ul><li>If so, the decision-maker may want to review the sensitive judgments. </li></ul>
77. 77. EXPERT CHOICE: Sensitivity Analysis <ul><li>In Expert Choice sensitivity analysis from the GOAL shows how the weights and the rankings of the alternatives change if some or all of the criteria weights change. </li></ul><ul><li>There are five graphical sensitivity analysis modes available: Performance, Dynamic, Gradient, Two-Dimensional, and Difference. </li></ul><ul><li>The first three show how a change in a criterion weight affects the final weights of the alternatives. </li></ul>
78. 78. <ul><li>The last two show how the alternatives perform with respect to any two criteria. </li></ul><ul><li>Performance : places all sensitivity information on a single chart with horizontal line graphs for the alternatives linked to vertical bars for the criteria. </li></ul><ul><li>Dynamic : two sets of dynamically linked horizontal bar graphs: one for criteria and one for alternatives. </li></ul>EXPERT CHOICE: Sensitivity Analysis
79. 79. <ul><li>Gradient : a line graph that shows how the weights of the alternatives vary according to the weight assigned to a specific criterion. (Use the X -Axis to change the selected criterion.) </li></ul><ul><li>Two-Dimensional : shows how well the alternatives perform with respect to any two criteria. </li></ul><ul><li>Difference : a graph that shows the differences between any two alternatives for any criterion. </li></ul>EXPERT CHOICE: Sensitivity Analysis
80. 80. <ul><li>An important use of sensitivity analysis is to determine how much a given criterion weight must change before there is a change in the rankings of the two highest alternatives. </li></ul><ul><li>This type of breakeven analysis can be easily done in Expert Choice. </li></ul>EXPERT CHOICE: Sensitivity Analysis
81. 81. <ul><li>Choose D ynamic from the Sensitivit y -Graphs option. </li></ul><ul><li>Drag the cost criterion bar 30.9% to approximately 45.9%, and see that the Volvo and Honda have the same highest final weight. </li></ul><ul><li>The final rankings are relatively insensitive to a change in the cost weight since it had to be increased by almost 50% to get a change in the final rankings. </li></ul><ul><li>The sensitivity results are different for the ideal mode. </li></ul>EXPERT CHOICE: Sensitivity Analysis