Lecture 08 Analytic Hierarchy Process (Module 1) Industrial Systems Engineering Dept.- IU Office: Room 508

Learning Objectives Students will be able to: Use the multifactor evaluation process in making decisions that involve a number of factors, where importance weights can be assigned. Understand the use of the analytic hierarchy process in decision making. Contrast multifactor evaluation with the analytic hierarchy process.

Module Outline M1.1 Introduction M1.2 Multifactor Evaluation Process M1.3 Analytic Hierarchy Process

Introduction Multifactor decision making involves individuals subjectively and intuitively considering various factors prior to making a decision. Multifactor evaluation process (MFEP) is a quantitative approach that gives weights to each factor and scores to each alternative. Analytic hierarchy process (AHP) is an approach designed to quantify the preferences for various factors and alternatives. Example: Buying a computer or laptop, smart phone: have to consider some factor: cost, brand name, specifications, … Give the weights for each of these factor (cost 60%, specification 30% and brand name 10%) Give the scores of each factor for the alternatives you want to buy.

Multifactor Evaluation Process Example: Steve. M.: considering employment with 3 companies. determined 3 factors important to him, assigned each factor a weight. Steve evaluated the various factors on a 0 to 1 scale for each of these jobs. Factor Importance (weight) AA Co. EDS, LTD. PW, Inc. Salary 0.3 0.7 0.8 0.9 Career Advancement 0.6 Location 0.1 Score Table Weights should sum to 1

Evaluation of AA Co. Factor Factor Factor Weighted Weight Evaluation Evaluation X = Factor Factor Factor Weighted Name Weight Evaluation Evaluation Salary 0.3 0.7 0.21 Career 0.6 0.9 0.54 Location 0.1 0.6 0.06 Total 0.81

Comparison of Results Decision is AA Co: Highest weighted evaluation Factor AA Co. EDS,LTD. PW,Inc. Salary 0.21 0.24 0.27 Career 0.54 0.42 0.36 Location 0.06 0.08 0.09 Weighted Evaluation 0.81 0.74 0.72 Decision is AA Co: Highest weighted evaluation

The Analytic Hierarchy Process (AHP) Founded by Saaty in 1980. It is a popular and widely used method for multi-criteria decision making. Allows the use of qualitative, as well as quantitative criteria in evaluation. 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. … Dr. Thomas L. Saaty Distinguished Prof. at U. of Pittsburgh In many situations one may not be able to assign weights to the different decision factors. Therefore one must rely on a technique that will allow the estimation of the weights. What is a solution? One such process, The Analytical Hierarchy Process (AHP), involves pairwise comparisons between the various factors. Method for ranking decision alternatives and selecting the best one when the decision maker has multiple objectives, or criteria

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 (priorities) of alternatives Both qualitative and quantitative information can be compared by using informed judgments to derive weights and priorities.

Steps Step 0: Construction of Hierarchy Structure (including: Goal, Factors, Criteria, and Alternatives) Step 1: Calculation of Factor Weight Step 1-1: Pairwise Comparison Matrix Step 1-2: Eigenvalue and Eigenvector (Priority vector) Step 1-3:Consistency Test Consistency Index Consistency Ratio Step 2:Calculation of Level Weight Step 3: Calculation of Overall Ranking

Hierarchy Tree More General Goal More Specific Alternatives C1 C2 C3 Sub-criteria at the lowest level Level 0 Level 1 (factors) Level 2 (criteria) Level .. Tom Saaty suggests that hierarchies be limited to six levels and nine items per level. This is based on the psychological result that people can consider 7 +/- 2 items simultaneously (Miller, 1956).

Pairwise Comparisons Size Apple A Apple B Apple C Resulting Priority Eigenvector Relative Size of Apple Apple A 1 2 6 6/10 A Apple B 1/2 1 3 3/10 B Apple C 1/6 1/3 1 1/10 C A compare with A = 1 consider the apple A and B about the size. A = 2B => B= ½A Explain Eigenvector later Criteria #1 Criteria #2 1 Intensity of Importance 2 3 4 5 6 8 7 9

Pairwise Comparison Matrix Ranking of Criteria and Alternatives Pairwise Comparison Matrix a33 a32 a31 A3 a22 a21 A2 a13 a12 a11 A1 to Pairwise Comparison Matrix A = ( aij ) (a) aii = 1 A comparison of criterion i with itself: equally important (b) aij = 1/ aji aji are reverse comparisons and must be the reciprocals of aij Ranking Scale for Criteria and Alternatives Values for aij : Numerical values Verbal judgment of preferences 1 equally important 3 weakly more important 5 strongly more important 7 very strongly more important 9 absolutely more important If compare through verbal judgment, the judgments will be transform to numerical values. intermediate values 2,4,6,8 => reverse comparisons reciprocals =>

Example 1: Car Selection (1/15) Objective Selecting a car Criteria Style, Reliability, Fuel-economy Cost? Alternatives Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata In this example, Cost is not a factor. It will be consider later. 14 2

Example 1: Car Selection (2/15) Hierarchy tree Civic Saturn Escort Miata 3

Example 1: Car Selection (3/15) Ranking of Criteria Style Reliability Fuel Economy 1/1 1/2 3/1 2/1 1/1 4/1 1/3 1/4 1/1 16 4

Example 1: Car Selection (4/15) Example 1: Car Selection (4/15) Ranking of Priorities Consider [Ax = x] where A is the comparison matrix of size n×n, for n criteria, also called the priority matrix. x is the Eigenvector of size n×1, also called the priority vector.  is the Eigenvalue,   > n. To find the ranking of priorities, namely the Eigen Vector X: 1) Normalize the column entries by dividing each entry by the sum of the column. 2) Take the overall row averages. Normalize is to transform the values to proportion (sum will be 1) Pairwise Comp. Matrix Norm. Pairwise Comp. Matrix Priority vector 1 0.5 3 2 1 4 0.33 0.25 1.0 Row Averages Normalized Column Sums 0.30 0.29 0.38 0.60 0.57 0.50 0.10 0.14 0.13 0.3196 0.5584 0.1220 A= X= Column sums 3.33 1.75 8.00 1.00 1.00 1.00 5

Example 1: Car Selection (5/15) Ranking of Priorities (cont.) Criteria weights Style .3196 ≈ .3 Reliability .5584 ≈ .6 Fuel Economy .1220 ≈ .1 Second most important criterion First important criterion The least important criterion Here is the tree of criteria with the criteria weights 18 7

Checking for Consistency Example 1: Car Selection (6/15) Checking for Consistency Consistency Ratio (CR): measure how consistent the judgments have been relative to large samples of purely random judgments. AHP evaluations are based on the asumption 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. Suppose we judge apple A to be twice as large as apple B and apple B to be three times as large as apple C. To be perfectly consistent, apple A must be six times as large as apple 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.

Calculation of Consistency Ratio Example 1: Car Selection (7/15) Calculation of Consistency Ratio The next stage is to calculate , Consistency Index (CI) and the Consistency Ratio (CR). Consider [Ax = x] where x is the Eigenvector. A x Ax x A x Ax x A x Ax x 1 0.5 3 2 1 4 0.333 0.25 1.0 0.90 1.60 0.35 0.30 0.60 0.10 0.30 0.60 0.10 =  = = = 0.90/0.30 1.60/0.60 0.35/0.10 3.00 2.67 3.50 Consistency Vector = = Consistency index (CI) is found by Note: This is just an approximate method to determine value of λ

Example 1: Car Selection (8/15) Consistency Index reflects the consistency of one’s judgement Random Index (RI) the CI of a randomly-generated pairwise comparison matrix Tabulated by size of matrix (n): (given by author) n RI 2 0.0 3 0.58 4 0.90 5 1.12 6 1.24 7 1.32 8 1.41 9 1.45 10 1.51 Prof. Saaty suggest the compare CI with the Random Consistency Index RI

Example 1: Car Selection (9/15) Consistency Ratio In practice, a CR of 0.1 or below is considered acceptable. Any higher value at any level indicate that the judgements warrant re-examination. In the above example: so, the evaluations are consistent

Example 1: Car Selection (10/15) Ranking Alternatives Priority vector Style Civic Saturn Escort Miata Civic 1 1/4 4 1/6 0.13 0.24 0.07 0.56 Saturn 4 1 4 1/4 Escort 1/4 1/4 1 1/5 Miata Miata 6 4 5 1 Reliability Civic Saturn Escort Miata 0.38 0.29 0.07 0.26 Civic Similar with Factor Ranking, the Priority vectors of Alternatives are determined. The corresponding CR should be determined to ensure the consistency of pair-wise judgment 1 2 5 1 Saturn 1/2 1 3 2 Escort 1/5 1/3 1 1/4 Miata 1 1/2 4 1 8

Example 1: Car Selection (11/15) Example 1: Car Selection (11/15) Ranking Alternatives (cont.) Ranking Alternatives (cont.) Miles/gallon Normalized Civic 34 .30 Fuel Economy Saturn 27 .24 Escort 24 .21 Miata Miata 28 113 .25 1.0 ! 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. 9

Example 1: Car Selection (12/15) Ranking Alternatives (cont.) Selecting a New Car 1.00 Style 0.30 Reliability 0.60 Fuel Economy 0.10 Civic 0.30 Saturn 0.24 Escort 0.21 Miata 0.25 Civic 0.13 Saturn 0.24 Escort 0.07 Miata 0.56 Civic 0.38 Saturn 0.29 Escort 0.07 Miata 0.26 Choose the largest value. Car Style(0.3) Reliability(0.6) Fuel Economy(0.1) Total Civic 0.13 0.38 0.30 0.30 Saturn 0.24 0.29 0.24 0.27 Escort 0.07 0.07 0.21 0.08 Miata 0.56 0.26 0.25 0.35 largest 10

Ranking of Alternatives (cont.) Example 1: Car Selection (13/15) Ranking of Alternatives (cont.) Reliability Style Economy Fuel Civic Escort Miata Saturn .13 .38 .30 .24 .29 .24 .07 .07 .21 .56 .26 .25 x .30 .60 .10 = .27 .08 .35 Priority matrix Factor Weights 11

Including Cost as a Decision Criteria Example 1: Car Selection (14/15) 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! Cost/Benefits Ratio Normalized Cost Cost Benefits CIVIC $12K .22 .30 0.73 SATURN $15K .28 .27 1.04 ESCORT $ 9K .17 .08 2.13 MIATA $18K .33 .35 0.94 13

Methods for including Cost Criterion Example 1: Car Selection (15/15) Methods for including Cost Criterion Use graphical representations to make trade-offs. Calculate cost/benefit ratios Use linear programming Use seperate benefit and cost trees and then combine the results Miata Miata Civic Civic Saturn Saturn Escort Escort

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

Example 2: Buying the best car *Goal: Buying the best car *There are three criteria: Cost Quality Maintenance Insurance Services *Three alternatives: Honda, Mercedes, Hyundai 30

The Hierarchy for problem Buying the best car Example 2: Buying the best car Level 0 Level 1 Criteria Level 2 Sub-criteria Alternatives The Hierarchy for problem Buying the best car 31

Example 2: Buying the best car Step 1: Criterion comparison Criterion comparison Normalize values: Find Column vector The process is repeated for the sub-criteria until the evaluation for all other alternatives. This example will be supported by Expert Choice software 32

Example 2: Buying the best car Step 2: Determining the Consistency Ratio - CR 2.1. Determining the Consistency vector We begin by determining the weighted sum vector. This is done by multiplying the column vector times the pairwise comparison matrix. Column vector: Pairwise comparison matrix: 1 3 5 1/3 1 2 1/5 1/2 1 X Demo on board Column vector = priority vector Consistency vector Weighted sum vector 1.948 0.690 0.366 Consistency vector = Weighted sum vector/ Column vector 33

Example 2: Buying the best car 2.2. Determining  and the Consistency Index-CI  = (3.006+3.0+3.0) / 3 = 3.002 The CI is: CI = (3.002 - 3) / (3 - 1) = 0.001 2.3. Determining the Consistency Ratio-CR with n = 3, we get RI = 0.58 CR = 0.001 / 0.58 = 0.0017 Since 0< CR < 0.1, we accept this result and move to the lower level. The procedure is repeated till the lowest level. 34

For Cost For Service For Insurance: For Quality Continue for other levels: For subcriteria Insurance – Service: For Cost For Service For Insurance: For Quality And make your final evaluation (students self develop this evaluation) 35

Demo on board

Other way:

1) Weights are defined for each hierarchical level... 2) ...and multiplied down to get the final lower level weights. 0.6 0.4 0.6 0.4 Multiply 0.7 0.3 0.2 0.6 0.2 0.7 0.3 0.2 0.6 0.2 0.42 0.18 0.08 0.24 0.08 Elicitation: suy luan

The AHP solving is computer-aided by Expert Choice (EC) software. Notes: In general, the evaluation scores are collected from many experts and the average scores is used in the pairwise comparison matrix. The AHP solving is computer-aided by Expert Choice (EC) software. - Building structure of problem !!! - Enter judgments (Pairwise Comparisons) - Analysis the weights - Sensitivity Analysis - Advantages and disadvantages - Miscellaneous 40

More about AHP: Pros and Cons 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 Pros There are hidden assumptions like consistency. Repeating evaluations is cumbersome. Difficult to use when the number of criteria or alternatives is high, i.e., more than 7. Difficult to add a new criterion or alternative Difficult to take out an existing criterion or alternative, since the best alternative might differ if the worst one is excluded. Users should be trained to use AHP methodology. Cons Pros and Cons = advantages & disadvantage Use cost/benefit ratio if applicable

Homework 08 (Due: next class) M 1.4, M 1.10, M 1.11