ABSTRACT
It has been noticed that in almost all the MCDM techniques, there is a requirement to determine weights or relative importance of the considered criteria as the final ranking of the alternatives greatly depends of these criteria weights. But, in the PSI method, proposed by Maniya & Bhatt [1], there is no need to estimate criteria weights, because the overall preference value of the considered criteria is computed using the concepts of statistics. This method seems to be quite attractive when there is a conflict in deciding the relative importance between the participating criteria. While calculating the overall preference values, the candidate alternatives are ranked based on their preference selection indexes (PSIs). The procedural steps of PSI method are highlighted as below:
Step 1: Identify the Objective: For a given decision-making problem, identify the feasible alternatives and the corresponding set of evaluation criteria.
Step 2: Develop the Decision Matrix: Considering the performance scores of all the alternatives with respect to different criteria, the corresponding decision matrix, X = [xij ] m×n is developed, where m is the number of alternatives, n is the number of criteria and xij is the performance of i th alternative against j th criterion.
Step 3: Normalization of the Decision Matrix: In order to make all the elements of the developed decision matrix dimensionless and comparable, the matrix is normalized using the following equations:
For Beneficial Criterion: https://www.w3.org/1998/Math/MathML"> r i j = x i j x j max https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math17_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
For Non-Beneficial Criterion: https://www.w3.org/1998/Math/MathML"> r i j = x j min x i j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math17_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where; rij is the normalized value of xij .
Step 4: Computation of the Preference Variation Value: The preference variation value (PVj ) for each criterion is now estimated using the sample variance concept based on the following expression: https://www.w3.org/1998/Math/MathML"> P V j = ∑ i = 1 m ( r i j − r ¯ j ) 2 ( j = 1 , 2 , ... , n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math17_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where; r̅j is the mean of the normalized values of criterion j.
Step 5: Determination of the Overall Preference Value: The overall preference value (Ψj ) for each criterion is calculated in this step. For this purpose, the deviation (Φj ) in the preference variation value (PVj ) for each criterion is computed based on the following equation: https://www.w3.org/1998/Math/MathML"> Φ j = 1 − P V j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math17_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Now, the overall preference value (Ψj ) is calculated using the following expression: https://www.w3.org/1998/Math/MathML"> Φ j = 1 − P V j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math17_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
It is worthwhile to mention here that the total overall preference value for all the considered criteria should always be one.
Step 6: Derive the Preference Selection Index (PSI): The PSI (Ii ) for i th alternative is determined based on the following expression: https://www.w3.org/1998/Math/MathML"> I i = ∑ j = 1 n ( r i j × Φ j ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math17_6_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Step 7: Ranking of the Alternatives: Based on the descending values of the calculated PSI, all the candidate alternatives are finally ranked from the best to the worst. Thus, the most preferred alternative should have the highest PSI.
