# SJSU MATH 108 - TOPSIS (21 pages)

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## TOPSIS

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## TOPSIS

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Pages:
21
School:
San Jose State University
Course:
Math 108 - Introduction to Abstract Mathematics and Proofs
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1 Multi Criteria Decision Making TOPSIS METHOD TOPSIS METHOD Technique of Order Preference by Similarity to Ideal Solution This method considers three types of attributes or criteria Qualitative benefit attributes criteria Quantitative benefit attributes Cost attributes or criteria 2 TOPSIS METHOD In this method two artificial alternatives are hypothesized Ideal alternative the one which has the best level for all attributes considered Negative ideal alternative the one which has the worst attribute values TOPSIS selects the alternative that is the closest to the ideal solution and farthest from negative ideal alternative 3 Input to TOPSIS TOPSIS assumes that we have m alternatives options and n attributes criteria and we have the score of each option with respect to each criterion Let xij score of option i with respect to criterion j We have a matrix X xij m n matrix Let J be the set of benefit attributes or criteria more is better Let J be the set of negative attributes or criteria less is better 4 Steps of TOPSIS Step 1 Construct normalized decision matrix This step transforms various attribute dimensions into non dimensional attributes which allows comparisons across criteria Normalize scores or data as follows rij xij x2ij for i 1 m j 1 n i 5 Steps of TOPSIS Step 2 Construct the weighted normalized decision matrix Assume we have a set of weights for each criteria wj for j 1 n Multiply each column of the normalized decision matrix by its associated weight An element of the new matrix is vij wj rij 6 Steps of TOPSIS Step 3 Determine the ideal and negative ideal solutions Ideal solution A v1 vn where vj max vij if j J min vij if j J i i Negative ideal solution A v1 vn where v min vij if j J max vij if j J i i 7 Steps of TOPSIS Step 4 Calculate the separation measures for each alternative The separation from the ideal alternative is Si vj vij 2 i 1 m j Similarly the separation from the negative ideal alternative is S i vj vij 2 i 1 m j 8 Steps of TOPSIS Step 5 Calculate the relative closeness to the ideal solution Ci Ci S i Si S i 0 C i 1 Select the option with Ci closest to 1 WHY 9 Applying TOPSIS Method to Example Weight 0 1 Style 0 4 Reliability 0 3 0 2 Fuel Eco Cost Civic 7 9 9 8 Saturn 8 7 8 7 Ford 9 6 8 9 Mazda 6 7 8 6 10 Applying TOPSIS to Example m 4 alternatives car models n 4 attributes criteria xij score of option i with respect to criterion j X xij 4 4 score matrix J set of benefit attributes style reliability fuel economy more is better J set of negative attributes cost less is better 11 Steps of TOPSIS Step 1 a calculate x2ij 1 2 for each column Style Rel Fuel Cost Civic 49 81 81 64 Saturn 64 49 64 49 Ford 81 36 64 81 Mazda 36 49 64 36 xij2i 230 215 273 230 x2 1 2 15 17 14 66 16 52 15 17 12 Steps of TOPSIS Step 1 b divide each column by x2ij 1 2 to get rij Style Rel Fuel Cost Civic 0 46 0 61 0 54 0 53 Saturn 0 53 0 48 0 48 0 46 Ford 0 59 0 41 0 48 0 59 Mazda 0 40 0 48 0 48 0 40 13 Steps of TOPSIS Step 2 b multiply each column by wj to get vij Style Rel Fuel Cost Civic 0 046 0 244 0 162 0 106 Saturn 0 053 0 192 0 144 0 092 Ford 0 059 0 164 0 144 0 118 Mazda 0 040 0 192 0 144 0 080 14 Steps of TOPSIS Step 3 a determine ideal solution A A 0 059 0 244 0 162 0 080 Style Rel Fuel Cost Civic 0 046 0 244 0 162 0 106 Saturn 0 053 0 192 0 144 0 092 Ford 0 059 0 164 0 144 0 118 Mazda 0 040 0 192 0 144 0 080 15 Steps of TOPSIS Step 3 a find negative ideal solution A A 0 040 0 164 0 144 0 118 Style Rel Fuel Cost Civic 0 046 0 244 0 162 0 106 Saturn 0 053 0 192 0 144 0 092 Ford 0 059 0 164 0 144 0 118 Mazda 0 040 0 192 0 144 0 080 16 Steps of TOPSIS Step 4 a determine separation from ideal solution A 0 059 0 244 0 162 0 080 Si vj vij 2 for each row j Civic Style Rel Fuel Cost 046 059 2 244 244 2 0 2 026 2 Saturn 053 059 2 192 244 2 018 2 012 2 Ford 053 059 2 164 244 2 018 2 038 2 Mazda 053 059 2 192 244 2 018 2 0 2 17 Steps of TOPSIS Step 4 a determine separation from ideal solution Si vj vij 2 Si vj vij 2 Civic 0 000845 0 029 Saturn 0 003208 0 057 Ford 0 008186 0 090 Mazda 0 003389 0 058 18 Steps of TOPSIS Step 4 b find separation from negative ideal solution A 0 040 0 164 0 144 0 118 Si vj vij 2 for each row j Civic Style Rel Fuel Cost 046 040 2 244 164 2 018 2 012 2 Saturn 053 040 2 192 164 2 0 2 026 2 Ford 053 040 2 164 164 2 0 2 0 2 Mazda 053 040 2 192 164 2 0 2 038 2 19 Steps of TOPSIS Step 4 b determine separation from negative ideal solution Si vj vij 2 Si vj vij 2 Civic 0 006904 0 083 Saturn 0 001629 0 040 Ford 0 000361 0 019 Mazda 0 002228 0 047 20 Steps of TOPSIS Step 5 Calculate the relative closeness to the ideal solution Ci S i Si S i S i Si S i Ci Civic 0 083 0 112 0 74 Saturn 0 040 0 097 0 41 Ford 0 019 0 109 0 17 Mazda 0 047 0 105 0 45 BEST 21

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