Agenda for This Week Monday, April 25 AHP Wednesday, April 27 Friday, April 29 Monday, May 2 Exam 2

Analytical Hierarchy Process – Part 1 Chapter 15 Analytical Hierarchy Process – Part 1

Analytical Hierarchy Process Designed to solve multicriteria decision problems Model under conditions of certainty where we know all of the info but we need to trade-off many factors. We are comparing several alternatives on the basis of the same set of attributes.

Developing the Hierarchy Develop a visual representation of the problem in terms of the overall goal, the criteria to be used and the decision alternatives GOAL Attributes Alternatives

AHP Example Decision Hierarchy Select Best Apartment Criteria Rent Size Proximity Apt. 1 Apt. 2 Apt. 3 Apt. 1 Apt. 2 Apt. 3 Apt. 1 Apt. 2 Apt. 3 Alternatives

AHP Major question is how to assign relative weights across alternatives, as well as for the attributes (assigning weights is the crux of AHP). People have been found to be more consistent when they do pairwise comparisons than when they just try to assign relative weights.

AHP - Steps 1. Make pairwise comparisons 2. Synthesize judgments 3. Check for consistency

AHP – Pairwise Comparison Scale (Always Use) Verbal Judgement of Preferences Numerical Rating Extremely preferred 9 Very strongly to extremely 8 Very strongly preferred 7 Strongly to very strongly 6 Strongly preferred 5 Moderately to strongly 4 Moderately preferred 3 Equally to moderately 2 Equally preferred 1

Step 1 – Pairwise Comparison Make pairwise comparison for CRITERIA CRITERIA Rent Size Proximity 1 5 6 1/5 1/3 1/6 3

Step 1 – Pairwise Comparison Make pairwise comparison for each attribute Apt 1 compared to Apt 2 Apt 2 compared to Apt 3 Apt 1 compared to Apt 3 For RENT

Step 1 – Pairwise Comparison RENT Apt 1 Apt 2 Apt 3 1 4 1/3 ¼ 1/7 3 7

Step 1 – Pairwise Comparison SIZE Apt 1 Apt 2 Apt 3 1 1/6 6

Step 1 – Pairwise Comparison PROXIMITY Apt 1 Apt 2 Apt 3 1 5 8 1/5 3 1/8 1/3

AHP - Steps 1. Make pairwise comparisons  2. Synthesize judgments 3. Check for consistency

AHP – Step 2 Synthesization – the process of calculating the priority of each criterion in terms of its contribution to the overall goal of achieving your goal

Step 2 - Synthesization Step 1: Sum values in each column of pairwise comparison matrix Step 2: Divide each element by its column total (gives normalized pairwise comparison matrix) Step 3: Compute average of elements in each row (gives estimate of relative priorities of elements being compared)

Step 2 - Synthesization RENT Apt 1 Apt 2 Apt 3 1 4 1/3 ¼ 1/7 3 7 SUM 17/4 12 31/21 Next… divide each element by column total

Step 2 - Synthesization RENT Relative priorities Apt 1 Apt 2 Apt 3 Avg .235 .334 .226 .266 .059 .083 .097 .080 .706 .583 .677 .654 Normalized pairwise comparison matrix

AHP – Relative Priorities Relative Priorities for Rent .266 .080 .654 Apt. 1 Apt. 2 Apt. 3 What do these mean?

AHP – Relative Priorities Relative Priorities for Rent .266 .080 .654 Apt. 1 Apt. 2 Apt. 3 These mean that with respect to Rent, Apt. 3 is preferred first (65%), then Apt. 1 (27%), then Apt. 2 (8%)

Step 2 - Synthesization SIZE Apt 1 Apt 2 Apt 3 1 1/6 6 SUM 8 8/6 Next… divide each element by column total

Step 2 - Synthesization SIZE Relative priorities Apt 1 Apt 2 Apt 3 Avg .125 .75 Normalized pairwise comparison matrix

Step 2 - Synthesization PROXIMITY Apt 1 Apt 2 Apt 3 1 5 8 1/5 3 1/8 1/3 SUM 53/40 19/3 12 Next… divide each element by column total

Step 2 - Synthesization PROXIMITY Relative priorities Apt 1 Apt 2 Apt 3 Avg .755 .790 .667 .737 .151 .158 .25 .186 .094 .053 .083 .077 Normalized pairwise comparison matrix

AHP - Steps 1. Make pairwise comparisons  2. Synthesize judgments  3. Check for consistency

AHP – Step 3 A key step in the making of several pairwise comparisons is considering the consistency of the pairwise judgements. Example: If A compared to B = 3 and B compared to C = 2 then A compared to C should = 3x2 = 6. If it wasn’t, some inconsistency would occur. With AHP, we can measure the degree of consistency; and if unacceptable, we can revise pairwise comparisons