1. MIS 463 Analytic Hierarchy Process

2. 2 The Analytic Hierarchy Process (AHP) It is popular and widely used method for multi-criteria decision making. Allows the use of qualitative, as well as quantitative criteria in evaluation. Founded by Saaty in 1980. Wide range of applications exists:  Selecting a car for purchasing  Deciding upon a place to visit for vacation  Deciding upon an MBA program after graduation.

3. 3 AHP-General Idea Develop an hierarchy of decision criteria and define the alternative courses of actions. AHP algorithm is basically composed of two steps: 1. Determine the relative weights of the decision criteria 2. Determine the relative rankings (priority) of alternatives ! Both qualitative and quantitative information can be compared using informed judgements to derive weights and priorities.

4. 4 Example: Car Selection Objective  Selecting a car Criteria  Style, Reliability, Fuel-economyCost? Alternatives  Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata

5. 5 Hierarchy tree CivicSaturnEscortMiata Alternative courses of action

6. 6 Ranking of Criteria and Alternatives Pairwise comparisons are made with the grades ranging from 1-9. A basic, but very reasonable, assumption: If attribute A is absolutely more important than attribute B and is rated at 9, then B must be absolutely less important than A and is valued at 1/9. These pairwise comparisons are carried out for all factors to be considered, usually not more than 7, and the matrix is completed.

7. 7 Ranking Scale for Criteria and Alternatives

8. 8 Ranking of criteria StyleReliabilityFuel Economy Style Reliability Fuel Economy 1/11/23/1 2/11/14/1 1/31/41/1

9. 9 Ranking of priorities Consider [Ax = max x] where  A is the comparison matrix of size n×n, for n criteria.  x is the Eigenvector of size n×1  max  is the Eigenvalue, max  > n. To find the ranking of priorities, namely the Eigen Vector X: Initialization: Take the squared power of matrix A, i.e., A 2 =A.A Find the row sums of A 2 and normalize this array to find E 0. Set A:=A 2 Main: 1. Take the squared power of matrix A, i.e., A 2 =A.A 2. Find the row sums of A 2 and normalize this array to find E 1. 3. Find D= E 1 - E 0. 4. IF the elements of D are close to zero, then X= E 1, STOP. ELSE set A:=A 2, set E 0 :=E 1 and go to Step 1.

10. 10 3.001.75 8.00 5.33 3.00 14.0 1.17 0.67 3.00 A2=A2= Row sums 12.75 22.33 4.83 39.92 Normalized Row Sums 0.3194 0.5595 0.1211 1.0 Iteration 1: Initialization: A= A 2 xA 2 = 27.6715.8372.50 48.3327.67126.67 10.566.0427.67 1 0.5 3 2 14 0.33 0.251.0 Row sums 12.75 22.33 4.83 39.92 Normalized Row Sums 0.3196 0.5584 0.1220 0.0002 -0.0011 0.0009 E 1 -E 0 = - 0.3194 0.5595 0.1211 0.3196 0.5584 0.1220 = Almost zero, so Eigen Vector, X = E 1. E0E0 E1E1

11. 11 Criteria weights Style.3196 Reliability.5584 Fuel Economy.1220

12. 12 Checking for Consistency The next stage is to calculate a Consistency Ratio (CR) to measure how consistent the judgements have been relative to large samples of purely random judgements. AHP evaluations are based on the aasumption that the decision maker is rational, i.e., if A is preferred to B and B is preferred to C, then A is preferred to C. If the CR is greater than 0.1 the judgements are untrustworthy because they are too close for comfort to randomness and the exercise is valueless or must be repeated.

13. 13 Calculation of Consistency Ratio The next stage is to calculate max so as to lead to the Consistency Index and the Consistency Ratio. Consider [Ax = max x] where x is the Eigenvector. 0.3196 0.5584 0.1220 1 0.5 3 2 14 0.333 0.251.0 0.9648 1.6856 0.3680 = = max λmax=average{0.9648/0.3196, 1.6856/0.5584, 0.3680/0.1220}=3.0180 0.3196 0.5584 0.1220 A x x Consistency index is found by CI=(λmax-n)/(n-1)=(3.0180-3)/(3-1)= 0.009

14. 14 Consistency Ratio The final step is to calculate the Consistency Ratio, CR by using the table below, derived from Saaty’s book, in which the upper row is the order of the random matrix, and the lower is the corresponding index of consistency for random judgements. Each of the numbers in this table is the average of CI’s derived from a sample of randomly selected reciprocal matrices using the AHP scale. An inconsistency of 10% or less implies that the adjustment is small compared to the actual values of the eigenvector entries. A CR as high as, say, 90% would mean that the pairwise judgements are just about random and are completely untrustworthy! In the above example: CR=CI/0.58=0.0090/0.58=0.01552 (less than 0.1, so the evaluations are consistent)

15. 15 Ranking alternatives Style Civic Saturn Escort 1/1 1/44/1 1/6 4/1 1/14/1 1/4 1/4 1/4 1/11/5 Miata6/1 4/1 5/1 1/1 CivicSaturnEscortMiata Reliability Civic Saturn Escort 1/1 2/15/1 1/1 1/2 1/1 3/1 2/1 1/5 1/3 1/11/4 Miata1/1 1/2 4/1 1/1 CivicSaturnEscortMiata.1160.2470.0600.5770 Eigenvector.3790.2900.0740.2570

16. 16 Fuel Economy Civic Saturn Escort Miata 34 27 24 28 113 Miles/gallon Normalized.3010.2390.2120.2480 1.0 Ranking alternatives ! Since fuel economy is a quantitative measure, fuel consumption ratios can be used to determine the relative ranking of alternatives; however this is not obligatory. Pairwise comparisons may still be used in some cases.

17. 17 - Civic.1160 - Saturn.2470 - Escort.0600 - Miata.5770 - Civic.3790 - Saturn.2900 - Escort.0740 - Miata.2570 - Civic.3010 - Saturn.2390 - Escort.2120 - Miata.2480

18. 18 Ranking of alternatives Style Reliability Fuel Economy Civic Escort Miata Saturn.1160.3790.3010.2470.2900.2390.0600.0740.2120.5770.2570.2480 *.3196.5584.1220 =.2854.2700.0864.3582 Criteria Weights

19. 19 Including Cost as a Decision Criteria CIVIC$12K.2220.778 SATURN$15K.27781.028 ESCORT$9K.16671.929 MIATA$18K.3330.930 Cost Normalized Cost Cost/Benefits Ratio Adding “cost” as a a new criterion is very difficult in AHP. A new column and a new row will be added in the evaluation matrix. However, whole evaluation should be repeated since addition of a new criterion might affect the relative importance of other criteria as well! Instead one may think of normalizing the costs directly and calculate the cost/benefit ratio for comparing alternatives!

20. Methods for including cost criterion Using graphical representations to make trade-offs. cost Calculate benefit/cost ratios Use linear programming Use seperate benefit and cost trees and then combine the results 20 benefit

21. 21 Complex decisions Many levels of criteria and sub-criteria exists for complex problems.

22. 22 Professional commercial software Expert Choice developed by Expert Choice Inc. is available which simplifies the implementation of the AHP’s steps and automates many of its computations  computations  sensitivity analysis  graphs, tables AHP Software:

23. Ex 2: Evaluation of Job Offers 23 Ex: Peter is offered 4 jobs from Acme Manufacturing (A), Bankers Bank (B), Creative Consulting (C), and Dynamic Decision Making (D). He bases his evaluation on the criteria such as location, salary, job content, and long-term prospects. Step 1: Decide upon the relative importance of the selection criteria: Location Content Long-term Salary 11/51/31/2 5124 31/2 13 21/21/31 LocationSalaryContentLong-term

24. A Different Way of Calculating Priority Vectors: 24 1) Normalize the column entries by dividing each entry by the sum of the column. 2) Take the overall row averages Location Content Long-term Salary 0.0910.1020.0910.059 0.4550.5130.5450.471 0.2730.256 0.2730.353 0.1820.1280.0910.118 LocationSalaryContentLong-term Average 0.086 0.496 0.289 0.130 + 1 1 1 1 1

25. Example 2: Evaluation of Job Offers 25 Step 2: Evaluate alternatives w.r.t. each criteria ABCDABCD 11/21/35 211/27 3219 1/51/71/91 A B C D Relative Location Scores Location Scores ABCDABCD 0.1610.1370.1710.227 0.3220.2750.2570.312 0.4840.5490.5140.409 0.032 0.040 0.057 0.045 A B C D Avg. 0.174 0.293 0.489 0.044

26. Example 2: Calculation of Relative Scores 26 Relative Scores for Each Criteria ABCDABCD 0.174 0.050 0.210 0.510 0.293 0.444 0.038 0.012 0.489 0.312 0.354 0.290 0.044 0.194 0.398 0.188 Location Salary Content Long-Term 0.086 0.496 0.289 0.130 Relative weights for each criteria x= Relative scores for each alternative 0.164 0.256 0.335 0.238

27. More about AHP: Pros and Cons 27 AHP is technique for formalizing decision making such that It is applicable when it is difficult to formulate criteria evaluations, i.e., it allows qualitative evaluation as well as quantitative evaluation. It is applicable for group decision making environments However There are hidden assumptions like consistency Difficult to use when there are large number of evaluations Use GDSS Use constraints to eliminate some alternatives Difficult to add a new criterion or alternative Use cost/benefit ratio if applicable Difficult to take out an existing criterion or alternative, since the best alternative might differ if the worst one is excluded.

28. Group Decision Making 28 The AHP allows group decision making, where group members can use their experience, values and knowledge to break down a problem into a hierarchy and solve. Doing so provides:  Understand the conflicting ideas in the organization and try to reach a consensus.  Minimize dominance by a strong member of the group.  Members of the group may vote for the criteria to form the AHP tree. (Overall priorities are determined by the weighted averages of the priorities obtained from members of the group.) However; The GDSS does not replace all the requirements for group decision making. Open meetings with the involvement of all members are still an asset.

29. Example 3: AHP in project management 29 Prequalification of contractors aims at the elimination of incompetent contractors from the bidding process. It is the choice of the decision maker to eliminate contractor E from the AHP evalution since it is not “feasible” at all !!

30. Example 3: AHP in project management 30 Step 1: Evaluation of the weights of the criteria Step 2: a) Pairwise comparison matrix for experience

31. Example 3: AHP in project management 31 Calculation of priority vector: x = Note that a DSS supports the decision maker, it can not replace him/her. Thus, an AHP Based DSS should allow the decision maker to make sensitivity analysis of his judgements on the overall priorities ! Probably Contractor-E should have been eliminated. It appears to be the worst.

32. References 32 Al Harbi K.M.A.S. (1999), Application of AHP in Project Management, International Journal of Project Management, 19, 19-27. Haas R., Meixner, O., (2009) An Illustrated Guide to the Analytic Hierarchy Process, Lecture Notes, Institute of Marketing & Innovation, University of Natural Resources and http://www.boku.ac.at/mi/ Saaty, T.L., Vargas, L.G., (2001), Models, Methods, Concepts & Applications of the Analytic Hierarchy Process, Kluwer’s Academic Publishers, Boston, USA.