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

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. IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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) IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

Percent weight estimates 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)   Criterion Percent weight estimates IIMA IIMB IIMC Academic reputation 0.40 0.25 0.35 Placement potential 0.30 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 0.35 0.35 0.30 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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) IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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) IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA AHP (contd..) Alternatives: IIMA IIMB IIMC 0.30 0.40 0.30 (p11) (p12) (p13) (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) IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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. IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA AHP (contd..) Objective: Selecting a car Criteria: Style, Cost, Fuel-economy Alternatives: Civic , i20 , Escort, Alto IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

AHP (contd..) Hierarchy tree Civic i20 Escort Alto Alternative courses of action IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 3

AHP (contd..) Ranking Scale for Criteria & Alternatives IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

AHP (contd..) Ranking of Criteria Style Cost Fuel Economy 1 1/2 3 2 1 4 1/3 1/4 1 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

AHP (contd..) Ranking of Priorities   1 0.5 3 2 1 4 0.33 0.25 1.0 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 sums 3.33 1.75 8.00 1.00 1.00 1.00 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 5

AHP (contd..) Criteria Weights Style 0.32 Cost 0.56 Fuel Economy 0.12 Selecting a New Car 1.00 Style 0.32 Cost 0.56 Fuel Economy 0.12 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 7

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. IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

AHP (contd..) Calculation of consistency ratio The next stage is to calculate max so as to lead to the Consistency Index (CI) and the Consistency Ratio. Consider [Ax = max x] where x is the Eigenvector. A x Ax x 1 0.5 3 2 1 4 0.33 0.25 1.0 0.98 1.68 0.36 0.32 0.56 0.12 0.32 0.56 0.12 = = 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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. IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

AHP (contd..) Ranking alternatives Priority vector Style Civic i20 Escort Alto Civic 1 1/4 4 1/6 0.13 0.24 0.07 0.56 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 8

AHP (contd..) Ranking alternatives Kilometer/litre Priority Vector 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. IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 9

AHP (contd..) Ranking alternatives Selecting a New Car 1.00 Style 0.32 Cost 0.56 Fuel Economy 0.12 Civic 0.13 i20 0.24 Escort 0.07 Alto 0.56 Civic 0.38 i20 0.29 Escort 0.07 Alto 0.26 Civic 0.30 i20 0.24 Escort 0.21 Alto 0.25 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 10

AHP (contd..) Ranking alternatives Style Economy Cost Fuel Civic Escort Alto i20 0.13 0.38 0.30 0.24 0.29 0.24 0.07 0.07 0.21 0.56 0.26 0.25 x 0.32 0.56 0.12 = 0.28 0.25 0.07 0.34 Priority matrix Criteria Weights IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 11

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! Normalized Cost Cost/Benefits Ratio Cost Benefits Civic 620000 0.22 0.28 0.78 i20 900000 0.28 0.25 1.12 Escort 540000 0.17 0.07 2.42 Alto 1080000 0.33 0.34 0.97 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA 13

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA AHP (contd..)   IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA AHP (contd..) Now if the matrix is consistent, then its form will be Such that we have: IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA AHP (contd..) and: IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA

RNSengupta,IME Dept.,IIT Kanpur,INDIA 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 IME634 RNSengupta,IME Dept.,IIT Kanpur,INDIA