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Analytic Hierarchy Process (AHP) for Decision Making

Understand the Analytic Hierarchy Process (AHP) for complex decisions, developed by Thomas L. Saaty. Learn how to apply AHP in group decision making with uncertain data to reach meaningful conclusions.

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Analytic Hierarchy Process (AHP) for Decision Making

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  1. IME634: Management Decision Analysis Raghu Nandan Sengupta Industrial & Management Department Indian Institute of Technology Kanpur RNSengupta,IME Dept.,IIT Kanpur,INDIA

  2. Analytic Hierarchy Process (AHP) • The Analytic Hierarchy Process (AHP) is a structured technique for organizing and analyzing complex decisions • It was developed by Thomas L. Saaty in the 1970s • Application in group decision making. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  3. AHP (contd..) • Decision analysis problems involving finite number of alternatives arise frequently in practical situations • One must remember that the type of data available for analysis, based on which one has to draw some conclusions can be deterministic, probabilistic or uncertain • When the data is uncertain, then one of the many tools used for analysis is Analytical Hierarchy Process (AHP) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  4. AHP (contd..) • In AHP, subjective judgement is quantified in logical manner and then utilized to reach some meaningful conclusions • One must remember that the decision makers assessment towards risk and his/her attitude towards return or average benefit reflects the decision makers overall outlook about any decision process RNSengupta,IME Dept.,IIT Kanpur,INDIA

  5. AHP (contd..) • Consider Ram has received the final calls from IIMA, IIMB and IIMC. His main criterion based on which he will take the decision is • 1. Academic reputation • 2. Placement potential • For his academic reputation is two (2) more important than placement potential. Thus placement potential is 1/3, while academic reputation is 2/3 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  6. AHP (contd..) • Thus Ram ranks this as • IIMA: (0.30*1/3+0.40*2/3) • IIMB: (0.40*1/3+0.25*2/3) • IIMC: (0.30*1/3+0.35*2/3) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  7. AHP (contd..) • Now consider Rams brother Shyam, also has got calls from the same three institutes and both want to be in the same place, so that their parents can reduce their overall cost of expenditure Decision: Select IIM Hierarchy # 1: Placement potential Academic Reputation ¼ ¾ Placement potential Alternatives: IIMA IIMB IIMC 0.25 0.25 0.50 Academic Reputation Alternatives: IIMA IIMB IIMC 0.35 0.35 0.30 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  8. AHP (contd..) • IIMA: (0.25*1/4+0.35*3/4) • IIMB: (0.25*1/4+0.35*3/4) • IIMC: (0.50*1/4+0.30*3/4) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  9. AHP (contd..) So Rams and Shyams collective hierarchy is as given Decision Select IIM Hierarchy # 1 Ram Shyam 0.5 (p) 0.5 (q) Hierarchy # 2: PP AR PP AR 1/3 2/3 ¼ ¾ (p1) (p2) (q1) (q2) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  10. AHP (contd..) Alternatives: IIMA IIMB IIMC 0.30 0.40 0.30 (p11) (p12) (p13) Alternatives: IIMA IIMB IIMC 0.30 0.40 0.30 (p21) (p22) (p23) Alternatives: IIMA IIMB IIMC 0.25 0.25 0.50 (q11) (q12) (q13) Alternatives: IIMA IIMB IIMC 0.35 0.35 0.30 (q21) (q22) (q23) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  11. AHP (contd..) So • IIMA: p*p1*p11 + p*p2*p21+q*q1*q11+q*q2*q21 • IIMB: p*p1*p12 + p*p2*p22+q*q1*q12+q*q2*q22 • IIMC: p*p1*p13 + p*p2*p23+q*q1*q13+q*q2*q23 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  12. AHP (contd..) 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 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  13. AHP (contd..) AHP algorithm is basically composed of two steps: • Determine the relative weights of the decision criteria • Determine the relative rankings (priorities) of alternatives Both qualitative and quantitative information can be compared by using informed judgments to derive weights and priorities. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  14. AHP (contd..) • Objective: Selecting a car • Criteria: Style, Cost, Fuel-economy • Alternatives: Civic , i20 , Escort, Alto RNSengupta,IME Dept.,IIT Kanpur,INDIA

  15. AHP (contd..) Hierarchy tree Civici20EscortAlto Alternative courses of action RNSengupta,IME Dept.,IIT Kanpur,INDIA

  16. AHP (contd..)Ranking Scale for Criteria & Alternatives RNSengupta,IME Dept.,IIT Kanpur,INDIA

  17. Style Cost Fuel Economy Style 1 1/2 3 2 1 4 Cost 1/3 1/4 1 Fuel Economy AHP (contd..)Ranking of Criteria RNSengupta,IME Dept.,IIT Kanpur,INDIA

  18. 1 0.5 3 2 1 4 0.33 0.25 1.0 AHP (contd..)Ranking of Priorities Row averages Normalized Column Sums 0.30 0.28 0.37 0.60 0.57 0.51 0.10 0.15 0.12 0.32 0.56 0.12 A= X= Priority vector Column sums3.33 1.75 8.00 1.00 1.00 1.00 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  19. AHP (contd..)Criteria Weights Criteria weights • Style 0.32 • Cost 0.56 • Fuel Economy 0.12 Selecting a New Car 1.00 Cost 0.56 Style 0.32 Fuel Economy 0.12 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  20. AHP (contd..)Checking for consistency • The next stage is to calculate a Consistency Ratio (CR) to measure how consistent the judgments have been relative to large samples of purely random judgments. • AHP evaluations are based on the assumption 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 judgments are untrustworthy because they are too close for comfort to randomness and the exercise is valueless or must be repeated. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  21. 1 0.5 3 2 1 4 0.33 0.25 1.0 0.32 0.56 0.12 0.32 0.56 0.12 0.98 1.68 0.36 AHP (contd..)Calculation of consistency ratio • The next stage is to calculate max so as to lead to the Consistency Index (CI) and theConsistency Ratio. • Consider [Ax = max x] where x is the Eigenvector. A x Ax x =max = • λmax=average{0.98/0.32, 1.68/0.56, 0.36/0.12}=3.04 • CI = (λmax-n)/(n-1)=(3.04-3)/(3-1)= 0.02 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  22. AHP (contd..) • CR = CI/RI where RI is the random index n 1 2 3 4 5 6 7 R.I. 0 0 0 0.52 0.88 1.11 1.25 1.35 • C.I. = 0.02 n = 3 RI = 0.50 (from table) • Hence: CR = (CI/RI) = 0.02/0.52 = 0.04 • CR ≤ 0.1 indicates sufficient consistency for decision. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  23. 0.13 0.24 0.07 0.56 AHP (contd..) Ranking alternatives Priority vector Style Civic i20 Escort Alto Civic 1 1/4 4 1/6 i20 4 1 4 1/4 Escort 1/4 1/4 1 1/5 Alto 6 4 5 1 Cost Civic i20 Escort Alto Civic 1 2 5 1 0.38 0.29 0.07 0.26 i20 1/2 1 3 2 Escort 1/5 1/3 1 1/4 Alto 1 1/2 4 1 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  24. AHP (contd..) Ranking alternatives Priority Vector Kilometer/litre Civic 34 0.30 Fuel Economy i20 27 0.24 Escort 24 0.21 Alto 28 113 0.25 1.0 Since fuel economy is a quantitative measure, fuel consumption ratios can be used to determine the relative ranking of alternatives. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  25. AHP (contd..) Ranking alternatives Selecting a New Car 1.00 Style 0.32 Fuel Economy 0.12 Cost 0.56 Civic 0.30 i20 0.24 Escort 0.21 Alto 0.25 Civic 0.38 i20 0.29 Escort 0.07 Alto 0.26 Civic 0.13 i20 0.24 Escort 0.07 Alto 0.56 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  26. Civic 0.13 0.38 0.30 0.24 0.29 0.24 0.07 0.07 0.21 0.56 0.26 0.25 0.32 0.56 0.12 0.28 0.25 0.07 0.34 i20 = x Escort Alto AHP (contd..) Ranking alternatives Style Cost Fuel Economy Criteria Weights Priority matrix RNSengupta,IME Dept.,IIT Kanpur,INDIA

  27. AHP (contd..)Including Cost as a Decision Criteria • 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! Civic 620000 0.22 0.28 0.78 i20 9000000.28 0.25 1.12 Escort 540000 0.17 0.07 2.42 Alto 1080000 0.33 0.34 0.97 Cost/Benefits Ratio Normalized Cost Cost Benefits RNSengupta,IME Dept.,IIT Kanpur,INDIA

  28. AHP (contd..) • Escort is the winner with the highest benefit to Cost Ratio, hence it is 1st • 2nd position is that of i20 • At 3rd is Alto • While 4th position goes to Civic RNSengupta,IME Dept.,IIT Kanpur,INDIA

  29. AHP (contd..) Pros • It allows multi criteria decision making. • 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 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  30. AHP (contd..) Cons • There are hidden assumptions like consistency. • Repeating evaluations is cumbersome • It is difficult to use when the number of criteria or alternatives is high, i.e., more than 7 • It is difficult to add a new criterion or alternative • It is difficult to take out an existing criterion or alternative, since the best alternative might differ if the worst one is excluded RNSengupta,IME Dept.,IIT Kanpur,INDIA

  31. AHP (contd..) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  32. AHP (contd..) Now if the matrix is consistent, then its form will be Such that we have: RNSengupta,IME Dept.,IIT Kanpur,INDIA

  33. AHP (contd..) and: RNSengupta,IME Dept.,IIT Kanpur,INDIA

  34. AHP (contd..) Thus we have: Hence: A(nXn)w(nX1) = nw(nX1) iff A is consistent and in case of inconsistency we try to find RNSengupta,IME Dept.,IIT Kanpur,INDIA

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