Multi-Criteria Decision MakingTOPSIS METHODSlide 3Input to TOPSISSteps of TOPSISSlide 6Slide 7Slide 8Slide 9Applying TOPSIS Method to ExampleApplying TOPSIS to ExampleSlide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 211Multi-Criteria Decision MakingTOPSIS METHOD2TOPSIS 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 criteria3TOPSIS 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.4Input 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)5Steps 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 i6Steps 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 rij7Steps 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' }, wherev' = { min (vij) if j  J ; max (vij) if j  J' } i i8Steps of TOPSIS Step 4: Calculate the separation measures for each alternative. The separation from the ideal alternative is: Si * = [  (vj*– vij)2 ] ½ i = 1, …, m jSimilarly, the separation from the negative ideal alternative is: S'i = [  (vj' – vij)2 ] ½ i = 1, …, m j9Steps of TOPSIS Step 5: Calculate the relative closeness to the ideal solution Ci* Ci* = S'i / (Si* +S'i ) , 0  Ci*  1 Select the option with Ci* closest to 1. WHY ?10 Applying TOPSIS Method to Example Weight 0.1 0.4 0.3 0.2Style Reliability Fuel Eco.Saturn Ford7 9 9 88 7 8 79 6 8 9CivicMazda 6 7 8 6Cost11Applying TOPSIS to Examplem = 4 alternatives (car models) n = 4 attributes/criteriaxij = score of option i with respect to criterion jX = {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)12Steps of TOPSISStep 1(a): calculate (x2ij )1/2 for each column Style Rel. FuelSaturnFord49 81 81 6464 49 64 4981 36 64 81CivicMazdaCostxij2i(x2)1/236 49 64 36230 215 273 23015.17 14.66 16.52 15.1713Steps of TOPSIS Step 1 (b): divide each column by (x2ij )1/2 to get rij Style Rel. FuelSaturnFord0.46 0.61 0.54 0.530.53 0.48 0.48 0.460.59 0.41 0.48 0.59CivicMazda 0.40 0.48 0.48 0.40Cost14Steps of TOPSIS Step 2 (b): multiply each column by wj to get vij. Style Rel. FuelSaturnFord0.046 0.244 0.162 0.1060.053 0.192 0.144 0.0920.059 0.164 0.144 0.118CivicMazda 0.040 0.192 0.144 0.080Cost15Steps of TOPSIS Step 3 (a): determine ideal solution A*. A* = {0.059, 0.244, 0.162, 0.080} Style Rel. FuelSaturnFord0.046 0.244 0.162 0.1060.053 0.192 0.144 0.0920.059 0.164 0.144 0.118CivicMazda 0.040 0.192 0.144 0.080Cost16Steps of TOPSIS Step 3 (a): find negative ideal solution A'. A' = {0.040, 0.164, 0.144, 0.118} Style Rel. FuelSaturnFord0.046 0.244 0.162 0.1060.053 0.192 0.144 0.0920.059 0.164 0.144 0.118CivicMazda 0.040 0.192 0.144 0.080Cost17Steps of TOPSIS Step 4 (a): determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080} Si* = [  (vj*– vij)2 ] ½for each row j Style Rel. FuelSaturnFord(.046-.059)2(.244-.244)2(0)2 (.026)2 CivicMazdaCost(.053-.059)2 (.192-.244)2(-.018)2 (.012)2(.053-.059)2 (.164-.244)2(-.018)2 (.038)2(.053-.059)2 (.192-.244)2(-.018)2 (.0)218Steps of TOPSIS Step 4 (a): determine separation from ideal solution Si* (vj*–vij)2Si* = [  (vj*– vij)2 ] ½SaturnFord0.000845 0.0290.003208 0.0570.008186 0.090CivicMazda0.003389 0.05819Steps of TOPSIS Step 4 (b): find separation from negative ideal solution A' = {0.040, 0.164, 0.144, 0.118} Si' = [  (vj'– vij)2 ] ½for each row j Style Rel. FuelSaturnFord(.046-.040)2(.244-.164)2(.018)2 (-.012)2CivicMazdaCost(.053-.040)2 (.192-.164)2(0)2 (-.026)2(.053-.040)2 (.164-.164)2(0)2 (0)2(.053-.040)2 (.192-.164)2(0)2 (-.038)220Steps of TOPSIS Step 4 (b): determine separation from negative ideal solution Si' (vj'–vij)2Si' = [  (vj'– vij)2 ] ½SaturnFord0.006904 0.0830.001629 0.0400.000361 0.019CivicMazda0.002228 0.04721Steps of TOPSIS Step 5: Calculate the relative closeness to the ideal solution Ci* = S'i / (Si* +S'i ) S'i /(Si*+S'i) Ci*SaturnFord0.083/0.112 0.74  BEST0.040/0.097 0.410.019/0.109 0.17CivicMazda0.047/0.105