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IME634: Management Decision Analysis
Raghu Nandan Sengupta Industrial & Management Department Indian Institute of Technology Kanpur VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje)
The idea of VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) i.e., Multicriteria Optimization and Compromise Solution), a MCDM technique, was developed by Serafim Opricovic during his Ph.D. work VIKOR method was introduced as one applicable technique to be implemented within MCDM problem and it as developed as a multi attribute decision making method to solve a discrete decision making problem with non-commensurable (different units) and conflicting criteria VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR (contd..) In this method the decision maker likes a solution that is closest to the ideal, and hence the decisions/alternatives are evaluated/compared/ranked accordingly While ranking, the decisions/alternatives, rather than the best solution, is the target as finding out the ideal solution is not always feasible, but rather the closest to the ideal is what is practically possible. VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR (contd..) Two of the MCDM methods, i.e., VIKOR and TOPSIS are based on an aggregating function which represents the concept of closeness of the solution to the ideal solution In VIKOR we follow linear normalization while in TOPSIS it is vector normalization VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR (contd..) The use of normalization is used to eliminate the units of criterion functions and thus ensure a level playing field for different criterion In VIKOR we determine a maximum group utility for the majority and a minimum of an individual regret for the opponent In TOPSIS a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution is required to be found. While doing this we do not consider the relative importance of these distances. VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR (contd..) Assume you have ๐ decisions/alternatives, ๐ด ๐ , ๐=1,โฏ,๐ and ๐ attributes/decision criteria/goals ๐ถ ๐ , ๐=1,โฏ,๐ Consider ๐ถ ๐ ๐ด ๐ as the value of the ๐ ๐กโ attributes/decision criteria/goals for the ๐ ๐กโ alternative such that ๐ฟ ๐,๐ = ๐=1 ๐ ๐ค ๐ ๐ถ ๐ ๐ด + โ ๐ถ ๐ ๐ด ๐ ๐ถ ๐ ๐ด + โ ๐ถ ๐ ๐ด โ ๐ ๐ , ๐=1,โฏ,๐; 0โค๐โคโ Here: ๐ถ ๐ ๐ด + = max ๐ ๐ถ ๐ ๐ด ๐ and ๐ถ ๐ ๐ด โ = min ๐ ๐ถ ๐ ๐ด ๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR (contd..) Remember ๐ฟ ๐,๐ is used to formulate the ranking measure, where ๐=1,2,โฏ, is integer and denotes the distance measure norm used for ๐= 1,โฏ,๐ ๐=1 signifies the Manhattan norm while ๐= โ denotes the infinity norm VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR (contd..): Distance
The ๐ฟ 1 norm or Manhattan distance between vector/points ๐= ๐ฅ 1 ,โฏ, ๐ฅ ๐ and ๐= ๐ฆ 1 ,โฏ, ๐ฆ ๐ is ๐=1 ๐ ๐ฅ ๐ โ ๐ฆ ๐ . The name relates to the distance a taxi has to drive in a rectangular street grid The ๐ฟ ๐ norm between vector/points ๐= ๐ฅ 1 ,โฏ, ๐ฅ ๐ and ๐= ๐ฆ 1 ,โฏ, ๐ฆ ๐ is ๐=1 ๐ ๐ฅ ๐ โ ๐ฆ ๐ ๐ ๐ The ๐ฟ โ norm between vector/points ๐= ๐ฅ 1 ,โฏ, ๐ฅ ๐ and ๐= ๐ฆ 1 ,โฏ,๐ฆ is max โ๐ ๐ฆ ๐ โ ๐ฆ ๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR (contd..): Distance (Example)
๐= 2,โ5,20 and ๐= โ12,15,0 ๐ฟ 1 = ๐=1 3 ๐ฅ ๐ โ ๐ฆ ๐ = 2โ โ โ5 โ โ0 = =54 ๐ฟ 2 = ๐= ๐ฅ ๐ โ ๐ฆ ๐ = 2โ โ โ5 โ โ โ โ โ5 โ โ = = =31.55 ๐ฟ โ = max โ๐ ๐ฅ ๐ โ ๐ฆ ๐ =๐๐๐ฅ 2โ โ12 , โ5 โ15 , 20โ0 = ๐๐๐ฅ 14,20,20 =20 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR (contd..): Distance (Example)
RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR (contd..): Distance
RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR (contd..): Distance
The green line (L2-norm) is the unique shortest path, while the red, blue, yellow (L1-norm) are all same length (=12) for the same route This is why L2-norm has unique solution while L1- norm does not have any unique solution One can generalize this to n-dimension case VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR Algorithm Assume decisions/alternatives as ๐ด ๐ , ๐= 1,โฏ,๐ Assume attributes/decision criteria/goals are ๐ถ ๐ , ๐=1,โฏ,๐ We state the pseudo-codes for the working principle of VIKOR VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR Algorithm (contd..)
1: DEFINE: ๐ฟ ๐ร๐ (matrix consisting of priority scores assigned to decisions/alternatives), ๐ด ๐ ,based on attributes/decision criteria/goals, ๐ถ ๐ ; ๐ค ๐ (weight for the attributes/decision criteria/goals) such that ๐=1 ๐ ๐ค ๐ =1; ๐ถ ๐ ๐ด ๐ (function relationship between attributes/decision criteria/goals for each decisions/alternatives); ๐ถ ๐ ๐ด + = max ๐ ๐ถ ๐ ๐ด ๐ ; ๐ถ ๐ ๐ด โ = min ๐ ๐ถ ๐ ๐ด ๐ ; ๐ฟ ๐,๐ = ๐=1 ๐ ๐ค ๐ ๐ถ ๐ ๐ด + โ ๐ถ ๐ ๐ด ๐ ๐ถ ๐ ๐ด + โ ๐ถ ๐ ๐ด โ ๐ 1 ๐ . Here ๐=1,โฏ,๐;๐=1,โฏ,๐ and ๐=1,2,..,โ (distance norm) 2: INPUT: ๐ฟ ๐ร๐ (matrix consisting of priority scores assigned to decisions/alternatives), ๐ด ๐ ,based on attributes/decision criteria/goals, ๐ถ ๐ ; ๐ค ๐ (weight for the attributes/decision criteria/goals) such that ๐=1 ๐ ๐ค ๐ =1; ๐ถ ๐ ๐ด ๐ (function relationship between attributes/decision criteria/goals for each decisions/alternatives). Here ๐=1,โฏ,๐ and ๐=1,โฏ,๐. 3: START if: ๐=1:๐ 4: START if: ๐=1:๐ 5: CALCULATE: ๐ถ ๐ ๐ด ๐ ; ๐ถ ๐ ๐ด + = max ๐ ๐ถ ๐ ๐ด ๐ ; ๐ถ ๐ ๐ด โ = min ๐ ๐ถ ๐ ๐ด ๐ ; ๐ฟ ๐,๐ = ๐=1 ๐ ๐ค ๐ ๐ถ ๐ ๐ด + โ ๐ถ ๐ ๐ด ๐ ๐ถ ๐ ๐ด + โ ๐ถ ๐ ๐ด โ ๐ 1 ๐ where ๐=1,โฏ,๐;๐=1,โฏ,๐ and ๐=1,2,..,โ (distance norm) 6: END if 7: END if 8: CALCULATE: ๐ถ ๐ ๐ด + ; ๐ถ ๐ ๐ด โ ; ๐ฟ ๐,๐ 9: REPORT: ๐ถ ๐ ๐ด + ; ๐ถ ๐ ๐ด โ ; ๐ฟ ๐,๐ 10: END VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Steps # 01 (Construct the normalized decision matrix)
Assume the decision matrix, ๐ฟ= ๐ฅ 11 โฏ ๐ฅ 1๐ โฎ โฑ โฎ ๐ฅ ๐1 โฏ ๐ฅ ๐๐ ๐ร๐ , m is number of Alternatives and n is number of Criterion VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Steps # 01 (Construct the normalized decision matrix)
๐ฟ= ร4 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Steps # 01 (Construct the normalized decision matrix)
Convert the entries in X into scaled normalized values, where ๐ ๐๐ = ๐ฅ ๐๐ 2 ๐=1 ๐ ๐ฅ ๐๐ 2 , which has no dimension Thus ๐น= ๐ 11 โฏ ๐ 1๐ โฎ โฑ โฎ ๐ ๐1 โฏ ๐ ๐๐ ๐ร๐ = ๐ฅ ๐=1 ๐ ๐ฅ ๐ โฏ ๐ฅ 1๐ 2 ๐=1 ๐ ๐ฅ ๐๐ 2 โฎ โฑ โฎ ๐ฅ ๐1 2 ๐=1 ๐ ๐ฅ ๐ โฏ ๐ฅ ๐๐ 2 ๐=1 ๐ ๐ฅ ๐๐ ๐ร๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Steps # 01 (Construct the normalized decision matrix)
๐น= ร4 , where ๐ ๐๐ = ๐ฅ ๐,๐ ๐=1 ๐ ๐ฅ ๐,๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Steps # 01 (Construct the normalized decision matrix)
๐น= ร4 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 02 (Construct the weighted normalized decision matrix)
If the decision maker decides on the set of weights, depending on his/her preference, then the weight, ๐พ= ๐ค 1 โฏ 0 โฎ โฑ โฎ 0 โฏ ๐ค ๐ ๐ร๐ , such that ๐=1 ๐ ๐ค =1 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 02 (Construct the weighted normalized decision matrix)
๐พ= ร4 , such that ๐=1 ๐ ๐ค =1 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 02 (Construct the weighted normalized decision matrix)
Calculate ๐ญ=๐น๐พ= ๐ 1,1 โฏ ๐ 1,๐ โฎ โฑ โฎ ๐ ๐,1 โฏ ๐ ๐,๐ ๐ร๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 02 (Construct the weighted normalized decision matrix)
๐ญ=๐น๐พ= ร ร4 = ร4 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the maximum/best from the criterion values) Determine the maximum/best: ๐ ๐ โ = max โ๐ ๐ ๐,๐ , ๐=1,โฏ,๐ ๐ 1 โ =๐๐๐ฅ ๐ 1,1 , ๐ 2,1 ,โฏ, ๐ ๐โ1,1 , ๐ ๐,1 ๐ 2 โ =๐๐๐ฅ ๐ 1,2 , ๐ 2,2 ,โฏ, ๐ ๐โ1,2 , ๐ ๐,2 . ๐ ๐โ1 โ =๐๐๐ฅ ๐ 1,๐โ1 , ๐ 2,๐โ1 ,โฏ, ๐ ๐โ1,๐โ1 , ๐ ๐,๐โ1 ๐ ๐ โ =๐๐๐ฅ ๐ 1,๐ , ๐ 2,๐ ,โฏ, ๐ ๐โ1,๐ , ๐ ๐,๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the maximum/best from the criterion values) ๐ 1 โ =๐๐๐ฅ ,0.0598,0.0765,0.0233, = ๐ 2 โ =๐๐๐ฅ ,0.1793,0.1682,0.2214, = ๐ 3 โ =๐๐๐ฅ ,0.0004,0.0008,0.0006, = ๐ 4 โ =๐๐๐ฅ ,0.0105,0.0046,0.0047, = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the minimum/worst from the criterion values) Determine the minimum/worst ๐ ๐ โ = min โ๐ ๐ ๐,๐ , ๐=1,โฏ,๐ ๐ 1 โ =๐๐๐ ๐ 1,1 , ๐ 2,1 ,โฏ, ๐ ๐โ1,1 , ๐ ๐,1 ๐ 2 โ =๐๐๐ ๐ 1,2 , ๐ 2,2 ,โฏ, ๐ ๐โ1,2 , ๐ ๐,2 . ๐ ๐โ1 โ =๐๐๐ ๐ 1,๐โ1 , ๐ 2,๐โ1 ,โฏ, ๐ ๐โ1,๐โ1 , ๐ ๐,๐โ1 ๐ ๐ โ =๐๐๐ ๐ 1,๐ , ๐ 2,๐ ,โฏ, ๐ ๐โ1,๐ , ๐ ๐,๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the minimum/worst from the criterion values) ๐ 1 โ =๐๐๐ ,0.0598,0.0765,0.0233, = ๐ 2 โ =๐๐๐ ,0.1793,0.1682,0.2214, = ๐ 3 โ =๐๐๐ ,0.0004,0.0008,0.0006, = ๐ 4 โ =๐๐๐ ,0.0105,0.0046,0.0047, = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Graphical representation of maximum/best and minimum/worst for each criterion) VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐ณ ๐ norm utilizing maximum/best and minimum/worst ratios) Compute: ๐ฟ 1,๐ = ๐=1 ๐ ๐ค ๐ ๐ ๐ โ โ ๐ ๐,๐ ๐ ๐ โ โ ๐ ๐ โ , ๐=1,โฏ,๐ ๐ฟ 1,1 = ๐ค 1 ๐ 1 โ โ ๐ 1,1 ๐ 1 โ โ ๐ 1 โ + ๐ค 2 ๐ 2 โ โ ๐ 1,2 ๐ 2 โ โ ๐ 2 โ +โฏ+ ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ 1,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ + ๐ค ๐ ๐ ๐ โ โ ๐ 1,๐ ๐ ๐ โ โ ๐ ๐ โ ๐ฟ 1,2 = ๐ค 1 ๐ 1 โ โ ๐ 2,1 ๐ 1 โ โ ๐ 1 โ + ๐ค 2 ๐ 2 โ โ ๐ 2,2 ๐ 2 โ โ ๐ 2 โ +โฏ+ ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ 2,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ + ๐ค ๐ ๐ ๐ โ โ ๐ 2,๐ ๐ ๐ โ โ ๐ ๐ โ . ๐ฟ 1,๐โ1 = ๐ค 1 ๐ 1 โ โ ๐ ๐โ1,1 ๐ 1 โ โ ๐ 1 โ + ๐ค 2 ๐ 2 โ โ ๐ ๐โ1,2 ๐ 2 โ โ ๐ 2 โ +โฏ+ ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ + ๐ค ๐ ๐ ๐ โ โ ๐ ๐โ1,๐ ๐ ๐ โ โ ๐ ๐ โ ๐ฟ 1,๐ = ๐ค 1 ๐ 1 โ โ ๐ ๐,1 ๐ 1 โ โ ๐ 1 โ + ๐ค 2 ๐ 2 โ โ ๐ ๐,2 ๐ 2 โ โ ๐ 2 โ +โฏ+ ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ ๐,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ + ๐ค ๐ ๐ ๐ โ โ ๐ ๐,๐ ๐ ๐ โ โ ๐ ๐ โ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐ณ ๐ norm utilizing maximum/best and minimum/worst ratios) ๐ฟ 1,1 =0.25ร โ โ ร โ โ ร โ โ ร โ โ = ๐ฟ 1,2 =0.25ร โ โ ร โ โ ร โ โ ร โ โ = ๐ฟ 1,3 =0.25ร โ โ ร โ โ ร โ โ ร โ โ = ๐ฟ 1,4 =0.25ร โ โ ร โ โ ร โ โ ร โ โ = ๐ฟ 1,5 =0.25ร โ โ ร โ โ ร โ โ ร โ โ = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐ณ โ norm utilizing maximum/best and minimum/worst ratios) Compute: ๐ฟ โ,๐ = max โ๐ ๐=1 ๐ ๐ค ๐ ๐ ๐ โ โ ๐ ๐,๐ ๐ ๐ โ โ ๐ ๐ โ , ๐=1,โฏ,๐ ๐ฟ โ,1 =๐๐๐ฅ ๐ค 1 ๐ 1 โ โ ๐ 1,1 ๐ 1 โ โ ๐ 1 โ , ๐ค 2 ๐ 2 โ โ ๐ 1,2 ๐ 2 โ โ ๐ 2 โ ,โฏ, ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ 1,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ , ๐ค ๐ ๐ ๐ โ โ ๐ 1,๐ ๐ ๐ โ โ ๐ ๐ โ ๐ฟ โ,2 =๐๐๐ฅ ๐ค 1 ๐ 1 โ โ ๐ 2,1 ๐ 1 โ โ ๐ 1 โ , ๐ค 2 ๐ 2 โ โ ๐ 2,2 ๐ 2 โ โ ๐ 2 โ ,โฏ, ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ 2,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ , ๐ค ๐ ๐ ๐ โ โ ๐ 2,๐ ๐ ๐ โ โ ๐ ๐ โ . ๐ฟ โ,๐โ1 =๐๐๐ฅ ๐ค 1 ๐ 1 โ โ ๐ ๐โ1,1 ๐ 1 โ โ ๐ 1 โ , ๐ค 2 ๐ 2 โ โ ๐ ๐โ1,2 ๐ 2 โ โ ๐ 2 โ ,โฏ, ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ , ๐ค ๐ ๐ ๐ โ โ ๐ ๐โ1,๐ ๐ ๐ โ โ ๐ ๐ โ ๐ฟ โ,๐ =๐๐๐ฅ ๐ค 1 ๐ 1 โ โ ๐ ๐,1 ๐ 1 โ โ ๐ 1 โ , ๐ค 2 ๐ 2 โ โ ๐ ๐,2 ๐ 2 โ โ ๐ 2 โ ,โฏ, ๐ค ๐โ1 ๐ ๐โ1 โ โ ๐ ๐,๐โ1 ๐ ๐โ1 โ โ ๐ ๐โ1 โ , ๐ค ๐ ๐ ๐ โ โ ๐ ๐,๐ ๐ ๐ โ โ ๐ ๐ โ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐ณ โ norm utilizing maximum/best and minimum/worst ratios) ๐ฟ โ,1 =๐๐๐ฅ 0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ = ๐ฟ โ,2 =๐๐๐ฅ 0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ = ๐ฟ โ,3 =๐๐๐ฅ 0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ = ๐ฟ โ,4 =๐๐๐ฅ 0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ = ๐ฟ โ,5 =๐๐๐ฅ 0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ ,0.25ร โ โ = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 05 (Determine the maximum/minimum based on ๐ณ ๐ norm)
Find: ๐ฟ 1,โ๐ ๐๐๐ฅ = max โ๐ ๐ฟ 1,๐ , i.e., ๐ฟ 1,โ๐ ๐๐๐ฅ = max ๐ฟ 1,1 , ๐ฟ 1,2 ,โฏ, ๐ฟ 1,๐โ1 , ๐ฟ 1,๐ Find: ๐ฟ 1,โ๐ ๐๐๐ = min โ๐ ๐ฟ 1,๐ , ๐=1,โฏ,๐, i.e., ๐ฟ 1,โ๐ ๐๐๐ =min ๐ฟ 1,1 , ๐ฟ 1,2 ,โฏ, ๐ฟ 1,๐โ1 , ๐ฟ 1,๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 05 (Determine the maximum/minimum based on ๐ณ ๐ norm)
RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 05 (Determine the maximum/minimum based on ๐ณ โ norm)
Find: ๐ฟ โ,โ๐ ๐๐๐ฅ = max โ๐ ๐ฟ โ,๐ , ๐=1,โฏ,๐, i.e., ๐ฟ โ,โ๐ ๐๐๐ฅ =max ๐ฟ โ,1 , ๐ฟ โ,2 ,โฏ, ๐ฟ โ,๐โ1 , ๐ฟ โ,๐ Find: ๐ฟ โ,โ๐ ๐๐๐ = min โ๐ ๐ฟ โ,๐ , ๐=1,โฏ,๐, i.e., ๐ฟ โ,โ๐ ๐๐๐ =min ๐ฟ โ,1 , ๐ฟ โ,2 ,โฏ, ๐ฟ โ,๐โ1 , ๐ฟ โ,๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 05 (Determine the maximum/minimum based on ๐ณ โ norm)
RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 06 (Calculate relative ranking)
Compute: ๐ ๐ =๐ฃ ๐ฟ 1,๐ โ ๐ฟ 1,๐ ๐๐๐ ๐ฟ 1,๐ ๐๐๐ฅ โ ๐ฟ 1,๐ ๐๐๐ โ๐ฃ ๐ฟ โ,๐ โ ๐ฟ โ,๐ ๐๐๐ ๐ฟ โ,๐ ๐๐๐ฅ โ ๐ฟ โ,๐ ๐๐๐ , ๐=1,โฏ,๐ ๐ 1 =๐ฃ ๐ฟ 1,1 โ ๐ฟ 1,1 ๐๐๐ ๐ฟ 1,1 ๐๐๐ฅ โ ๐ฟ 1,1 ๐๐๐ โ๐ฃ ๐ฟ โ,1 โ ๐ฟ โ,1 ๐๐๐ ๐ฟ โ,1 ๐๐๐ฅ โ ๐ฟ โ,1 ๐๐๐ ๐ 2 =๐ฃ ๐ฟ 1,2 โ ๐ฟ 1,2 ๐๐๐ ๐ฟ 1,2 ๐๐๐ฅ โ ๐ฟ 1,2 ๐๐๐ โ๐ฃ ๐ฟ โ,2 โ ๐ฟ โ,2 ๐๐๐ ๐ฟ โ,2 ๐๐๐ฅ โ ๐ฟ โ,2 ๐๐๐ . ๐ ๐โ1 =๐ฃ ๐ฟ 1,๐โ1 โ ๐ฟ 1,๐โ1 ๐๐๐ ๐ฟ 1,๐โ1 ๐๐๐ฅ โ ๐ฟ 1,๐โ1 ๐๐๐ โ๐ฃ ๐ฟ โ,๐โ1 โ ๐ฟ โ,๐โ1 ๐๐๐ ๐ฟ โ,๐โ1 ๐๐๐ฅ โ ๐ฟ โ,๐โ1 ๐๐๐ ๐ ๐ =๐ฃ ๐ฟ 1,๐ โ ๐ฟ 1,๐ ๐๐๐ ๐ฟ 1,๐ ๐๐๐ฅ โ ๐ฟ 1,๐ ๐๐๐ โ๐ฃ ๐ฟ โ,๐ โ ๐ฟ โ,๐ ๐๐๐ ๐ฟ โ,๐ ๐๐๐ฅ โ ๐ฟ โ,๐ ๐๐๐ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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VIKOR: Step # 06 (Calculate relative ranking)
๐ฃ is introduced as weight of the strategy of โโthe majority of criteriaโ (or โโthe maximum group utilityโ), here one can consider ๐ฃ=0.5 Rank the alternatives, sorting by the values ๐ฟ 1,๐ , ๐ฟ โ,๐ and ๐ ๐ in decreasing order The results are three ranking lists. Propose as a compromise solution the alternatives obtained above VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA
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