TOPSIS and fuzzy multi-objective model integration for supplier selection problem

TOPSIS for supplier selection problem.Submitted By :

Pranav Mishra11ID60R20Important termsThere can be n number of decision makers.Example: criteria rating scaleEntertainment-10 very good -1 noneFacilities-10 very good -1 noneTravel cost-10 low-1 very highAccomodation-10 low-1 very highDecision makers These are experts who are assigned with the task of weighting each attribute.

CriteriaDecision makersAttributes weightsHarryRonHermoineentertainment426== 4Facilities231== 2

Let us considerEach option can be evaluated against certain criteria.Criteria for vacation destinations can be:EntertainmentFacilitiesAccommodation costTravel costSimilarly Selecting a source of information (library, internet, etc) involves various criteria such as:

Reliability of informationTime to gather informationCost of acquiring information

These are examples of MULTI-CRITERIA problems and requires MCDM approach.MCDMSelection of the best, from a set of alternatives, each of which is evaluated against multiple criteria.Some problem solving techniques are :SAW (Simple Additive Weighting)TOPSIS (Technique for Order Preference by Similarity to the Ideal Solution)ELECTRE (Elimination et Choice Translating Reality)AHP (The Analytical Hierarchy Process)SMART (The Simple Multi Attribute Rating Technique )ANP (Analytic network process)

Multiple Criteria Decision MakingImportant termsAlternatives These are the options which are to be evaluated for selection of the best.Example: (for vacation problem)Hogwarts, Hogsmeade, Azkaban

Criteria or Attributes These will impact the selection of alternatives.Example: (for vacation problem) entertainment, facilities, travel cost, accommodation cost

Completeness: It is important to ensure that all of the important criteria are included. Redundancy: In principle, criteria that have been judged relatively unimportant or to be duplicates should be removed at a very early stage. Operationality: It is important that each alternative can be judged against each criterion.

Important termsWeights These estimates relative importance of criteria.

Each attribute is given certain points on 0-10 or 0-100 rating scale by a team of experts or decision makers.Example: criteria weightrating scaleEntertainment-410 very good -1 noneFacilities-210 very good -1 noneTravel cost-610 low-1 very highAccomodation-810 low-1 very highImportant termsDecision maker rates each attribute of each alternativeExample: for alternate 1 ( hogwarts)Decision matrix A table that is used to objectively make decision about making selection from a range of options.

CriteriaDecision makersAttributes weightsHarryRonHermoineentertainment1089== 9Facilities786== 7Travel cost567== 6Acc. Cost957== 7

Similarlyfor alternate 2 ( hogsmeade)CriteriaDecision makersAttributes weightsHarryRonHermoineentertainment969== 8Facilities894== 7Travel cost71010== 9Acc. Cost576== 6

CriteriaDecision makersAttributes weightsHarryRonHermoineentertainment948== 7Facilities798== 8Travel cost657== 6Acc. Cost765== 6for alternate 3 ( azkaban)

Decision matrixCriteriaHogwartshogsmeadeAzkabanentertainment987Facilities778Travel cost696Acc. Cost766

Hogwarts (Table 1)Hogsmeade (Table 2)Azkaban (Table 3)TOPSIS In this method two artificial alternatives are hypothesized:

Ideal alternative: One which has the best attributes values (i.e. max. benefit attributes and min. cost attributes)Negative ideal alternative: One which has the worst attribute values. (i.e. min. benefit attributes and max. cost attributes)

TOPSIS selects the alternative that is the closest to the ideal solution and farthest from negative ideal solution.

Technique for Order Preference by Similarity to Ideal SolutionSteps involved in TOPSISStep 1 standardize the decision matrix.This step transforms various attribute dimensions into non-dimensional attributes, which allows comparisons across criteria.For standardizing, each column of decision matrix, is divided by root of sum of square of respective columns.CriteriaHogwartsHogsmeadeAzkabanentertainment987 = = 13.93Facilities778 = =12.73Travel cost696 = =12.37Acc. Cost766 = = 11.00

DECISION MATRIXSteps involved in TOPSISStep 1 standardize the decision matrix.This step transforms various attribute dimensions into non-dimensional attributes, which allows comparisons across criteria.For standardizing, each column of decision matrix, is divided by root of sum of square of respective columns.CriteriaHogwartsHogsmeadeAzkabanentertainment987 = = 13.93Facilities778 = =12.73Travel cost696 = =12.37Acc. Cost766 = = 11.00

Steps involved in TOPSISStep 1 standardize the decision matrix.This step transforms various attribute dimensions into non-dimensional attributes, which allows comparisons across criteria.For standardizing, each column of decision matrix, is divided by root of sum of square of respective columns.CriteriaHogwartsHogsmeadeAzkabanentertainment987 = = 13.93Facilities778 = =12.73Travel cost696 = =12.37Acc. Cost766 = = 11.00

CriteriaHogwartsHogsmeadeAzkabanEntertainment9 / 13.93FacilitiesTravel costAcc. CostCriteriaHogwartsHogsmeadeAzkabanEntertainment0.65FacilitiesTravel costAcc. CostStandardized decision matrix.

CriteriaHogwartsHogsmeadeAzkabanEntertainment9 / 13.938 / 13.93FacilitiesTravel costAcc. CostCriteriaHogwartsHogsmeadeAzkabanEntertainment0.650.57FacilitiesTravel costAcc. CostStandardized decision matrix.

CriteriaHogwartsHogsmeadeAzkabanEntertainment9 / 13.938 / 13.937 / 13.93FacilitiesTravel costAcc. CostCriteriaHogwartsHogsmeadeAzkabanEntertainment0.650.570.50FacilitiesTravel costAcc. CostStandardized decision matrix.

CriteriaHogwartsHogsmeadeAzkabanEntertainment0.650.570.50FacilitiesTravel costAcc. CostCriteriaHogwartsHogsmeadeAzkabanEntertainment0.650.570.50Facilities0.550.550.63Travel cost0.490.730.49Acc. Cost0.640.550.55Standardized decision matrix.Similarly.

Steps involved in TOPSIS Step 2 - Construct weighted standardized decision matrix by multiplying attributes weight to each rating. CriteriaHogwartsHogsmeadeAzkabanEntertainment0.650.570.50Facilities0.550.550.63Travel cost0.490.730.49Acc. Cost0.640.550.55CriteriaHogwartsHogsmeadeAzkabanEntertainment2.6FacilitiesTravel costAcc. CostAttributes weights Weighted Standardized decision matrixXStandardized decision matrix

Steps involved in TOPSISCriteriaHogwartsHogsmeadeAzkabanEntertainment0.650.570.50Facilities0.550.550.63Travel cost0.490.730.49Acc. Cost0.640.550.55CriteriaHogwartsHogsmeadeAzkabanEntertainment2.62.28FacilitiesTravel costAcc. CostAttributes weights Standardized decision matrixXWeighted Standardized decision matrix Step 2 - Construct weighted standardized decision matrix by multiplying attributes weight to each rating.

Steps involved in TOPSIS Step 2 - construct weighted standardized decision matrix by multiplying attributes weight to each rating. CriteriaHogwartsHogsmeadeAzkabanEntertainment0.650.570.50Facilities0.550.550.63Travel cost0.490.730.49Acc. Cost0.640.550.55CriteriaHogwartsHogsmeadeAzkabanEntertainment2.62.28Facilities1.1Travel costAcc. CostAttributes weights Standardized decision matrixXWeighted Standardized decision matrixCriteriaHogwartsHogsmeadeAzkabanEntertainment2.62.28Facilities1.1Travel costAcc. CostSteps involved in TOPSIS Step 2 - construct weighted standardized decision matrix by multiplying attributes weight to each rating. CriteriaHogwartsHogsmeadeAzkabanEntertainment2.62.282Facilities1.11.11.26Travel cost2.944.382.94Acc. Cost5.124.44.4Similarly.Weighted Standardized decision matrixSteps involved in TOPSIS Step 3 Determine ideal solution and negative ideal solution. CriteriaHogwartsHogsmeadeAzkabanEntertainment2.62.282Facilities1.11.11.26Travel cost2.944.382.94Acc. Cost 5.124.44.4Ideal solution = {2.6, 1.26, 4.38, 5.12}2.61.264.385.12A set of maximum values for each criteria is Ideal solution.Max.Max.Max.Max. CriteriaHogwartsHogsmeadeAzkabanEntertainment2.62.282Facilities1.11.11.26Travel cost2.944.382.94Acc. Cost5.124.44.4Steps involved in TOPSIS Step 3 Determine ideal solution and negative ideal solution. Negative Ideal solution = {2.6, 1.26, 2.94, 4.4}2.01.12.944.4A set of minimum values for each criteria is Negative Ideal solution.Min.Min.Min.Min. Steps involved in TOPSIS Step 4 Determine separation from ideal solution. Si*CriteriaHogwartsHogsmeadeAzkabanEntertainment(2.6-2.6)2(2.28-2.6) 2(2.0-2.6) 2Facilities(1.1-1.26) 2(1.1-1.26) 2(1.26-1.26) 2Travel cost(2.94-2.94) 2(4.38-2.94) 2(2.94-2.94) 2Acc. Cost(5.12-4.4) 2(4.4-4.4) 2(4.4-4.4) 2CriteriaHogwartsHogsmeadeAzkabanEntertainment0.00.100.36Facilities0.020.020.0Travel cost0.02.070.0Acc. Cost0.510.00.00.74 1.48 0.6Si* = (0.+0.02+0+0.51)1/2 (.1+.02+2.07+0)1/2 (0.36+.0+0+0)1/2= = =Steps involved in TOPSIS Step 5 Determine separation from negative ideal solution.CriteriaHogwartsHogsmeadeAzkabanEntertainment(2.6-2.0)2(2.28-2.0) 2(2.0-2.0) 2Facilities(1.1-1.1) 2(1.1-1.1) 2(1.26-1.1) 2Travel cost(2.94-4.38) 2(4.38-4.38) 2(2.94-4.38) 2Acc. Cost(5.12-5.12) 2(4.4-5.12) 2(4.4-5.12) 2CriteriaHogwartsHogsmeadeAzkabanEntertainment0.360.070.0Facilities0.00.00.02Travel cost2.070.02.07Acc. Cost0.00.510.51(0.36+0+2+0)1/2 (.07+0+0+.51)1/2 (0+.02+2+.51)1/21.56 0.773 1.618Si = = = =Steps involved in TOPSIS Step 6 Determine relative closeness to ideal solution.CriteriaHogwartsHogsmeadeAzkabanSi*0.741.480.6Si 1.560.7731.618Si*+Si 2.32.2532.218Si /(Si*+Si )1.56/2.30.77/2.251.62/2.210.680.3430.729BESTMax.TOPSIS for supplier selection problem.Consider 3 suppliersS1S2S33 suppliers are evaluated against 4 attributesSpecial factor (Q1)On time delivery (Q2)Performance history (Q3)Technical capability (Q4)BENEFIT ATTRIBUTESCOST ATTRIBUTE TOPSIS for supplier selection problem.There are 4 decision makersD1D2D3D4to express their preferences and ratings to select the best supplier.TOPSIS for supplier selection problem.Step 1 finding ATTRIBUTE WEIGHTS.rating given to each attribute by each decision maker is shown in table.AttributesD1D2D3D4WNormalized WQ1H(.375)M(.275)ML(.225)VL(.125).25.25/.91 = 0.275Q2VH(.425)VL(.125)VVL(.05)ML(.225).21.21/.91 = 0.230Q3L(.175)VL(.125)VVL(.05)MH(.325).17.17/.91 = 0.187Q4VH(.425)L(.175)M(.275)ML(.225).28.28/.91 = 0.308Total0.911.00AVERAGEAVERAGEAVERAGEAVERAGEATTRIBUTE SCALEVVLVLLMLMMHHVHVVHWEIGHTS.05.125.175.225.275.325.375.425.475scale of attribute weightsTOPSIS for supplier selection problem.Step 1 finding ATTRIBUTE WEIGHTS.rating given to each attribute by each decision maker is shown in table.AttributesD1D2D3D4WNormalized WQ1H(.375)M(.275)ML(.225)VL(.125).25.25/.91 = 0.275Q2VH(.425)VL(.125)VVL(.05)ML(.225).21.21/.91 = 0.230Q3L(.175)VL(.125)VVL(.05)MH(.325).17.17/.91 = 0.187Q4VH(.425)L(.175)M(.275)ML(.225).28.28/.91 = 0.308Total0.911.00ATTRIBUTE SCALEVVLVLLMLMMHHVHVVHWEIGHTS.05.125.175.225.275.325.375.425.475scale of attribute weightsTOPSIS for supplier selection problem.Step 2 finding ATTRIBUTE RATINGS.Rating given to each supplier by each decision maker for attribute Q1.SuppliersD1D2D3D4G1S1.03.03.03.03.03S2.05.05.05.05.05S3.01.01.01.01.01AVERAGEAVERAGEAVERAGERating given to each supplier by each decision maker for attribute Q2.SuppliersD1D2D3D4G2S1.95.95.95.95.95S2.98.98.98.98.98S3.85.85.85.85.85AVERAGEAVERAGEAVERAGETOPSIS for supplier selection problem.Step 2 finding ATTRIBUTE RATINGS.rating given to each supplier by each decision maker for attribute Q3.SuppliersD1D2D3D4G3S1G(9)P(1)MP(3)MP(3)4S2MP(3)MP&F(4)MP&F(4)MP&F(4)3.75S3F(5)F(5)MP&F(4)F(5)4.75ATTRIBUTE SCALEPMPFMGGRATINGS13579scale of attribute ratingsrating given to each supplier by each decision maker for attribute Q4.SuppliersD1D2D3D4G3S1G(9)MP(3)P(1)MP(3)4S2MP(3)MP&F(4)F(5)F(5)4.25S3G(9)G(9)MP&G(6)MP&G(6)8.5AVERAGEAVERAGEAVERAGEAVERAGEAVERAGEAVERAGETOPSIS for supplier selection problem.Step 3 construct DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.03.9544S2.05.983.754.25S3.01.854.758.5

Q1Q4

Q2

Q3TOPSIS for supplier selection problem.Step 4 STANDARDIZE DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.03.9544S2.05.983.754.25S3.01.854.758.5(.03+.05+.01)(.95+.98+.85)(4+3.75+4.75)(4+4.25+8.5)= 0.059= 1.608= 7.254= 10.31DECISION TABLETOPSIS for supplier selection problem.Step 4 STANDARDIZE DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.03.9544S2.05.983.754.25S3.01.854.758.5(.03+.05+.01)(.95+.98+.85)(4+3.75+4.75)(4+4.25+8.5)= 0.059= 1.608= 7.254= 10.31AttributesSuppliersQ1Q2Q3Q4S1.03/.059 = .508S2S3DECISION TABLESTANDARD DECISION TABLETOPSIS for supplier selection problem.Step 4 STANDARDIZE DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.03.9544S2.05.983.754.25S3.01.854.758.5(.03+.05+.01)(.95+.98+.85)(4+3.75+4.75)(4+4.25+8.5)= 0.059= 1.608= 7.254= 10.31AttributesSuppliersQ1Q2Q3Q4S1.03/.059 = .508S2.05/.059 = .845S3DECISION TABLESTANDARD DECISION TABLETOPSIS for supplier selection problem.Step 4 STANDARDIZE DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.03.9544S2.05.983.754.25S3.01.854.758.5(.03+.05+.01)(.95+.98+.85)(4+3.75+4.75)(4+4.25+8.5)= 0.059= 1.608= 7.254= 10.31AttributesSuppliersQ1Q2Q3Q4S1.03/.059 = .508.95/1.608= .591S2.05/.059 = .845S3DECISION TABLESTANDARD DECISION TABLETOPSIS for supplier selection problem.Step 4 STANDARDIZE DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.03.9544S2.05.983.754.25S3.01.854.758.5(.03+.05+.01)(.95+.98+.85)(4+3.75+4.75)(4+4.25+8.5)= 0.059= 1.608= 7.254= 10.31AttributesSuppliersQ1Q2Q3Q4S1.03/.059 = .508.95/1.608= .591S2.05/.059 = .845.98/1.608= .609S3DECISION TABLESTANDARD DECISION TABLETOPSIS for supplier selection problem.Step 4 STANDARDIZE DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.03.9544S2.05.983.754.25S3.01.854.758.5(.03+.05+.01)(.95+.98+.85)(4+3.75+4.75)(4+4.25+8.5)= 0.059= 1.608= 7.254= 10.31AttributesSuppliersQ1Q2Q3Q4S1.03/.059 = .508.95/1.608= .5914/7.254= .5514/10.31= .388S2.05/.059 = .845.98/1.608= .6093.75/7.254= .5174.25/10.31= .412S3.01/.059 = .169.85/1.608= .5294.75/7.254= .6558.5/10.31= .824DECISION TABLESTANDARD DECISION TABLETOPSIS for supplier selection problem.Step 4 STANDARD DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.508.591.551.388S2.845.609.517.412S3.169.529.655.824TOPSIS for supplier selection problem.Step 5 WEIGHTED STANDARD DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.508X.275S2S3Multiplying attribute weights as obtained in step-1, with respective attribute values in standard decision matrix.ATTRIBUTESQ1Q2Q3Q4WEIGHTS0.2750.2300.1870.308attribute weights as obtained from Step-1

Standard decision table as obtained from Step-4TOPSIS for supplier selection problem.Step 5 WEIGHTED STANDARD DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.508X.275S2.845X.275S3Multiplying attribute weights as obtained in step-1, with respective attribute values in standard decision matrix.ATTRIBUTESQ1Q2Q3Q4WEIGHTS0.2750.2300.1870.308attribute weights as obtained from Step-1

Standard decision table as obtained from Step-4TOPSIS for supplier selection problem.Step 5 WEIGHTED STANDARD DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.508X.275.591X.230S2.845X.275S3Multiplying attribute weights as obtained in step-1, with respective attribute values in standard decision matrix.ATTRIBUTESQ1Q2Q3Q4WEIGHTS0.2750.2300.1870.308attribute weights as obtained from Step-1

Standard decision table as obtained from Step-4TOPSIS for supplier selection problem.Step 5 WEIGHTED STANDARD DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.508X.275.591X.230.551X.187S2.845X.275S3Multiplying attribute weights as obtained in step-1, with respective attribute values in standard decision matrix.ATTRIBUTESQ1Q2Q3Q4WEIGHTS0.2750.2300.1870.308attribute weights as obtained from Step-1

Standard decision table as obtained from Step-4TOPSIS for supplier selection problem.Step 5 WEIGHTED STANDARD DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S1.508X.275.591X.230.551X.187.388X.308S2.845X.275.609X.230.517X.187.412X.308S3.169X.275.529X.230.655X.187.824X.308Multiplying attribute weights as obtained in step-1, with respective attribute values in standard decision matrix.ATTRIBUTESQ1Q2Q3Q4WEIGHTS0.2750.2300.1870.308attribute weights as obtained from Step-1

Standard decision table as obtained from Step-4TOPSIS for supplier selection problem.Step 5 WEIGHTED STANDARD DECISION TABLE.AttributesSuppliersQ1Q2Q3Q4S10.140.1350.1030.119S20.2320.140.0970.126S30.0460.1210.1220.253TOPSIS for supplier selection problem.Step 6 Construct the IDEAL SOLUTION & NEGATIVE IDEAL SOLUTION.IDEAL SOLTIONCost attributeBenefit attributesSuppliersQ1Q2Q3Q4S10.140.1350.1030.119S20.232(Max) 0.140.0970.126S3(min.) 0.0460.121(Max) 0.122(Max) 0.253Ideal solution = {0.046, 0.14, 0.122, 0.253} Minimum value of Cost Attributes are Ideal.Maximum value of Benefit Attributes are Ideal.TOPSIS for supplier selection problem.Step 6 Construct the IDEAL SOLUTION & NEGATIVE IDEAL SOLUTION. NEGATIVE IDEAL SOLTIONCost attributeBenefit attributesSuppliersQ1Q2Q3Q4S10.140.1350.103(min.) 0.119S2(max.) 0.2320.14(min.) 0.0970.126S30.046(min.) 0.1210.1220.253Negative Ideal solution = {0.232, 0.121, 0.097 0.119} Maximum value of Cost Attributes are Negative Ideal.Minimum value of Benefit Attributes are Negative Ideal.TOPSIS for supplier selection problem.Step 7 Construct the SEPARATION from Ideal solution & negative ideal solution. SEPARATION from Ideal solution Si*Cost attributeBenefit attributesSuppliersQ1Q2Q3Q4S1(0.14 - .046)2 = 0.009(0.135 - 0.14)2 = 0.000(0.103 - 0.122)2 = 0.0004(0.119 - 0.253)2 = 0.018S20.2320.140.0970.126S30.0460.1210.1220.253Ideal solution = {0.046, 0.14, 0.122, 0.253}S1* = (0.009+0.00+0.0004+0.018)1/2 = 0.166TOPSIS for supplier selection problem.Step 7 Construct the SEPARATION from Ideal solution & negative ideal solution. SEPARATION from Ideal solution Si*Cost attributeBenefit attributesSuppliersQ1Q2Q3Q4S1(0.14 - .046)2 = 0.009(0.135 - 0.14)2 = 0.000(0.103 - 0.122)2 = 0.0004(0.119 - 0.253)2 = 0.018S2(0.232 - .046)2 = 0.035(0.14 - 0.14)2 = 0.000(0.097 - 0.122)2 = 0.0006(0.126 - 0.253)2 = 0.016S30.0460.1210.1220.253Ideal solution = {0.046, 0.14, 0.122, 0.253}S1* = (0.009+0.00+0.0004+0.018)1/2 = 0.166S2* = (0.035+0.00+0.0006_0.016)1/2 = 0.227TOPSIS for supplier selection problem.Step 7 Construct the SEPARATION from Ideal solution & negative ideal solution. SEPARATION from Ideal solution Si*Cost attributeBenefit attributesSuppliersQ1Q2Q3Q4S1(0.14 - .046)2 = 0.009(0.135 - 0.14)2 = 0.000(0.103 - 0.122)2 = 0.0004(0.119 - 0.253)2 = 0.018S2(0.232 - .046)2 = 0.035(0.14 - 0.14)2 = 0.000(0.097 - 0.122)2 = 0.0006(0.126 - 0.253)2 = 0.016S3(0.046 - .046)2 = 0.000(0.121- 0.14)2 = 0.0004(0.122 - 0.122)2 = 0.000(0.253 - 0.253)2 = 0.000Ideal solution = {0.046, 0.14, 0.122, 0.253}S1* = (0.009+0.00+0.0004+0.018)1/2 = 0.166S2* = (0.035+0.00+0.0006_0.016)1/2 = 0.227S3* = (0.00+0.0004+0.00+0.0000)1/2 = 0.02TOPSIS for supplier selection problem.Cost attributeBenefit attributesSuppliersQ1Q2Q3Q4S1(0.14 - .232)2 = 0.008(0.135 - .121)2 = 0.0002(0.103 - .097)2 = 0.0000(0.119 - .119)2 = 0.000S2(0.232 - .232)2 = 0.000(0.14 - .121)2 = 0.0004(0.097 - .097)2 = 0.0000(0.126 - .119)2 = 0.0001S3(0.046 - .232)2 = 0.035(0.121 - .121)2 = 0.000(0.122 - .097)2 = 0.0006(0.253 - .119)2 = 0.0179Negative Ideal solution = {0.232, 0.121, 0.097 0.119}Step 7 Construct the SEPARATION from Ideal solution & negative ideal solution. SEPARATION from Negative ideal solution SiS1 = (0.008+0.0002+0.000+0.000)1/2 = 0.09S2 = (0.00+0.0004+0.000+0.0001)1/2 = 0.022S3 = (0.035+0.00+0.0006+0.0179)1/2 = 0.231TOPSIS for supplier selection problem.Step 8 Calculate the RELATIVE CLOSENESS to Ideal solution. CLOSENESS to ideal solution Ci* = Si /(Si* +Si )CriteriaS1S2S3Si*0.1660.2270.02Si 0.090.0220.231Si*+Si 0.2560.2490.251Si /(Si*+Si )0.09/0.2560.022/0.2490.231/0.2510.3510.0880.920C3* > C1* > C2*C1* = 0.351C2* = 0.088C3* = 0.920TOPSIS for supplier selection problem.Step 8 RANK THE ORDER of suppliers based on STEP 8 C3* > C1* > C2*Supplier3 > Supplier1 > Supplier2THANK YOU