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IME634: Management Decision Analysis

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1 IME634: Management Decision Analysis
Raghu Nandan Sengupta Industrial & Management Department Indian Institute of Technology Kanpur VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

2 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

3 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

4 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

5 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

6 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR (contd..) Assume you have ๐‘š decisions/alternatives, ๐ด ๐‘– , ๐‘–=1,โ‹ฏ,๐‘š and ๐‘› attributes/decision criteria/goals ๐ถ ๐‘— , ๐‘—=1,โ‹ฏ,๐‘› Consider ๐ถ ๐‘— ๐ด ๐‘– as the value of the ๐‘— ๐‘กโ„Ž attributes/decision criteria/goals for the ๐‘– ๐‘กโ„Ž alternative such that ๐ฟ ๐‘,๐‘– = ๐‘—=1 ๐‘› ๐‘ค ๐‘— ๐ถ ๐‘— ๐ด + โˆ’ ๐ถ ๐‘— ๐ด ๐‘– ๐ถ ๐‘— ๐ด + โˆ’ ๐ถ ๐‘— ๐ด โˆ’ ๐‘ ๐‘ , ๐‘–=1,โ‹ฏ,๐‘š; 0โ‰ค๐‘โ‰คโˆž Here: ๐ถ ๐‘— ๐ด + = max ๐‘– ๐ถ ๐‘— ๐ด ๐‘– and ๐ถ ๐‘— ๐ด โˆ’ = min ๐‘– ๐ถ ๐‘— ๐ด ๐‘– VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

7 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR (contd..) Remember ๐ฟ ๐‘,๐‘– is used to formulate the ranking measure, where ๐‘=1,2,โ‹ฏ, is integer and denotes the distance measure norm used for ๐‘–= 1,โ‹ฏ,๐‘š ๐‘=1 signifies the Manhattan norm while ๐‘= โˆž denotes the infinity norm VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

8 VIKOR (contd..): Distance
The ๐ฟ 1 norm or Manhattan distance between vector/points ๐’™= ๐‘ฅ 1 ,โ‹ฏ, ๐‘ฅ ๐‘› and ๐’š= ๐‘ฆ 1 ,โ‹ฏ, ๐‘ฆ ๐‘› is ๐‘–=1 ๐‘› ๐‘ฅ ๐‘– โˆ’ ๐‘ฆ ๐‘– . The name relates to the distance a taxi has to drive in a rectangular street grid The ๐ฟ ๐‘ norm between vector/points ๐’™= ๐‘ฅ 1 ,โ‹ฏ, ๐‘ฅ ๐‘› and ๐’š= ๐‘ฆ 1 ,โ‹ฏ, ๐‘ฆ ๐‘› is ๐‘–=1 ๐‘› ๐‘ฅ ๐‘– โˆ’ ๐‘ฆ ๐‘– ๐‘ ๐‘ The ๐ฟ โˆž norm between vector/points ๐’™= ๐‘ฅ 1 ,โ‹ฏ, ๐‘ฅ ๐‘› and ๐’š= ๐‘ฆ 1 ,โ‹ฏ,๐‘ฆ is max โˆ€๐‘– ๐‘ฆ ๐‘– โˆ’ ๐‘ฆ ๐‘– VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

9 VIKOR (contd..): Distance (Example)
๐’™= 2,โˆ’5,20 and ๐’š= โˆ’12,15,0 ๐ฟ 1 = ๐‘–=1 3 ๐‘ฅ ๐‘– โˆ’ ๐‘ฆ ๐‘– = 2โˆ’ โˆ’ โˆ’5 โˆ’ โˆ’0 = =54 ๐ฟ 2 = ๐‘–= ๐‘ฅ ๐‘– โˆ’ ๐‘ฆ ๐‘– = 2โˆ’ โˆ’ โˆ’5 โˆ’ โˆ’ โˆ’ โˆ’ โˆ’5 โˆ’ โˆ’ = = =31.55 ๐ฟ โˆž = max โˆ€๐‘– ๐‘ฅ ๐‘– โˆ’ ๐‘ฆ ๐‘– =๐‘š๐‘Ž๐‘ฅ 2โˆ’ โˆ’12 , โˆ’5 โˆ’15 , 20โˆ’0 = ๐‘š๐‘Ž๐‘ฅ 14,20,20 =20 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

10 VIKOR (contd..): Distance (Example)
RNSengupta,IME Dept.,IIT Kanpur,INDIA

11 VIKOR (contd..): Distance
RNSengupta,IME Dept.,IIT Kanpur,INDIA

12 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

13 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR Algorithm Assume decisions/alternatives as ๐ด ๐‘– , ๐‘–= 1,โ‹ฏ,๐‘š Assume attributes/decision criteria/goals are ๐ถ ๐‘— , ๐‘—=1,โ‹ฏ,๐‘› We state the pseudo-codes for the working principle of VIKOR VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

14 VIKOR Algorithm (contd..)
1: DEFINE: ๐‘ฟ ๐‘šร—๐‘› (matrix consisting of priority scores assigned to decisions/alternatives), ๐ด ๐‘– ,based on attributes/decision criteria/goals, ๐ถ ๐‘— ; ๐‘ค ๐‘— (weight for the attributes/decision criteria/goals) such that ๐‘—=1 ๐‘› ๐‘ค ๐‘— =1; ๐ถ ๐‘— ๐ด ๐‘– (function relationship between attributes/decision criteria/goals for each decisions/alternatives); ๐ถ ๐‘— ๐ด + = max ๐‘– ๐ถ ๐‘— ๐ด ๐‘– ; ๐ถ ๐‘— ๐ด โˆ’ = min ๐‘– ๐ถ ๐‘— ๐ด ๐‘– ; ๐ฟ ๐‘,๐‘– = ๐‘—=1 ๐‘› ๐‘ค ๐‘— ๐ถ ๐‘— ๐ด + โˆ’ ๐ถ ๐‘— ๐ด ๐‘– ๐ถ ๐‘— ๐ด + โˆ’ ๐ถ ๐‘— ๐ด โˆ’ ๐‘ 1 ๐‘ . Here ๐‘–=1,โ‹ฏ,๐‘š;๐‘—=1,โ‹ฏ,๐‘› and ๐‘=1,2,..,โˆž (distance norm) 2: INPUT: ๐‘ฟ ๐‘šร—๐‘› (matrix consisting of priority scores assigned to decisions/alternatives), ๐ด ๐‘– ,based on attributes/decision criteria/goals, ๐ถ ๐‘— ; ๐‘ค ๐‘— (weight for the attributes/decision criteria/goals) such that ๐‘—=1 ๐‘› ๐‘ค ๐‘— =1; ๐ถ ๐‘— ๐ด ๐‘– (function relationship between attributes/decision criteria/goals for each decisions/alternatives). Here ๐‘–=1,โ‹ฏ,๐‘š and ๐‘—=1,โ‹ฏ,๐‘›. 3: START if: ๐‘–=1:๐‘š 4: START if: ๐‘—=1:๐‘› 5: CALCULATE: ๐ถ ๐‘— ๐ด ๐‘– ; ๐ถ ๐‘— ๐ด + = max ๐‘– ๐ถ ๐‘— ๐ด ๐‘– ; ๐ถ ๐‘— ๐ด โˆ’ = min ๐‘– ๐ถ ๐‘— ๐ด ๐‘– ; ๐ฟ ๐‘,๐‘– = ๐‘—=1 ๐‘› ๐‘ค ๐‘— ๐ถ ๐‘— ๐ด + โˆ’ ๐ถ ๐‘— ๐ด ๐‘– ๐ถ ๐‘— ๐ด + โˆ’ ๐ถ ๐‘— ๐ด โˆ’ ๐‘ 1 ๐‘ where ๐‘–=1,โ‹ฏ,๐‘š;๐‘—=1,โ‹ฏ,๐‘› and ๐‘=1,2,..,โˆž (distance norm) 6: END if 7: END if 8: CALCULATE: ๐ถ ๐‘— ๐ด + ; ๐ถ ๐‘— ๐ด โˆ’ ; ๐ฟ ๐‘,๐‘– 9: REPORT: ๐ถ ๐‘— ๐ด + ; ๐ถ ๐‘— ๐ด โˆ’ ; ๐ฟ ๐‘,๐‘– 10: END VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

15 VIKOR: Steps # 01 (Construct the normalized decision matrix)
Assume the decision matrix, ๐‘ฟ= ๐‘ฅ 11 โ‹ฏ ๐‘ฅ 1๐‘› โ‹ฎ โ‹ฑ โ‹ฎ ๐‘ฅ ๐‘š1 โ‹ฏ ๐‘ฅ ๐‘š๐‘› ๐‘šร—๐‘› , m is number of Alternatives and n is number of Criterion VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

16 VIKOR: Steps # 01 (Construct the normalized decision matrix)
๐‘ฟ= ร—4 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

17 VIKOR: Steps # 01 (Construct the normalized decision matrix)
Convert the entries in X into scaled normalized values, where ๐‘Ÿ ๐‘–๐‘— = ๐‘ฅ ๐‘–๐‘— 2 ๐‘˜=1 ๐‘š ๐‘ฅ ๐‘˜๐‘— 2 , which has no dimension Thus ๐‘น= ๐‘Ÿ 11 โ‹ฏ ๐‘Ÿ 1๐‘› โ‹ฎ โ‹ฑ โ‹ฎ ๐‘Ÿ ๐‘š1 โ‹ฏ ๐‘Ÿ ๐‘š๐‘› ๐‘šร—๐‘› = ๐‘ฅ ๐‘˜=1 ๐‘š ๐‘ฅ ๐‘˜ โ‹ฏ ๐‘ฅ 1๐‘› 2 ๐‘˜=1 ๐‘š ๐‘ฅ ๐‘˜๐‘› 2 โ‹ฎ โ‹ฑ โ‹ฎ ๐‘ฅ ๐‘š1 2 ๐‘˜=1 ๐‘š ๐‘ฅ ๐‘˜ โ‹ฏ ๐‘ฅ ๐‘š๐‘› 2 ๐‘˜=1 ๐‘š ๐‘ฅ ๐‘˜๐‘› ๐‘šร—๐‘› VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

18 VIKOR: Steps # 01 (Construct the normalized decision matrix)
๐‘น= ร—4 , where ๐‘Ÿ ๐‘–๐‘— = ๐‘ฅ ๐‘–,๐‘— ๐‘˜=1 ๐‘š ๐‘ฅ ๐‘˜,๐‘— VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

19 VIKOR: Steps # 01 (Construct the normalized decision matrix)
๐‘น= ร—4 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

20 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, ๐‘พ= ๐‘ค 1 โ‹ฏ 0 โ‹ฎ โ‹ฑ โ‹ฎ 0 โ‹ฏ ๐‘ค ๐‘› ๐‘›ร—๐‘› , such that ๐‘—=1 ๐‘› ๐‘ค =1 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

21 VIKOR: Step # 02 (Construct the weighted normalized decision matrix)
๐‘พ= ร—4 , such that ๐‘—=1 ๐‘› ๐‘ค =1 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

22 VIKOR: Step # 02 (Construct the weighted normalized decision matrix)
Calculate ๐‘ญ=๐‘น๐‘พ= ๐‘“ 1,1 โ‹ฏ ๐‘“ 1,๐‘› โ‹ฎ โ‹ฑ โ‹ฎ ๐‘“ ๐‘š,1 โ‹ฏ ๐‘“ ๐‘š,๐‘› ๐‘šร—๐‘› VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

23 VIKOR: Step # 02 (Construct the weighted normalized decision matrix)
๐‘ญ=๐‘น๐‘พ= ร— ร—4 = ร—4 VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

24 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the maximum/best from the criterion values) Determine the maximum/best: ๐‘“ ๐‘— โˆ— = max โˆ€๐‘– ๐‘“ ๐‘–,๐‘— , ๐‘—=1,โ‹ฏ,๐‘› ๐‘“ 1 โˆ— =๐‘š๐‘Ž๐‘ฅ ๐‘“ 1,1 , ๐‘“ 2,1 ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,1 , ๐‘“ ๐‘š,1 ๐‘“ 2 โˆ— =๐‘š๐‘Ž๐‘ฅ ๐‘“ 1,2 , ๐‘“ 2,2 ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,2 , ๐‘“ ๐‘š,2 . ๐‘“ ๐‘›โˆ’1 โˆ— =๐‘š๐‘Ž๐‘ฅ ๐‘“ 1,๐‘›โˆ’1 , ๐‘“ 2,๐‘›โˆ’1 ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,๐‘›โˆ’1 , ๐‘“ ๐‘š,๐‘›โˆ’1 ๐‘“ ๐‘› โˆ— =๐‘š๐‘Ž๐‘ฅ ๐‘“ 1,๐‘› , ๐‘“ 2,๐‘› ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,๐‘› , ๐‘“ ๐‘š,๐‘› VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

25 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the maximum/best from the criterion values) ๐‘“ 1 โˆ— =๐‘š๐‘Ž๐‘ฅ ,0.0598,0.0765,0.0233, = ๐‘“ 2 โˆ— =๐‘š๐‘Ž๐‘ฅ ,0.1793,0.1682,0.2214, = ๐‘“ 3 โˆ— =๐‘š๐‘Ž๐‘ฅ ,0.0004,0.0008,0.0006, = ๐‘“ 4 โˆ— =๐‘š๐‘Ž๐‘ฅ ,0.0105,0.0046,0.0047, = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

26 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the minimum/worst from the criterion values) Determine the minimum/worst ๐‘“ ๐‘— โˆ’ = min โˆ€๐‘– ๐‘“ ๐‘–,๐‘— , ๐‘—=1,โ‹ฏ,๐‘› ๐‘“ 1 โˆ’ =๐‘š๐‘–๐‘› ๐‘“ 1,1 , ๐‘“ 2,1 ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,1 , ๐‘“ ๐‘š,1 ๐‘“ 2 โˆ’ =๐‘š๐‘–๐‘› ๐‘“ 1,2 , ๐‘“ 2,2 ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,2 , ๐‘“ ๐‘š,2 . ๐‘“ ๐‘›โˆ’1 โˆ’ =๐‘š๐‘–๐‘› ๐‘“ 1,๐‘›โˆ’1 , ๐‘“ 2,๐‘›โˆ’1 ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,๐‘›โˆ’1 , ๐‘“ ๐‘š,๐‘›โˆ’1 ๐‘“ ๐‘› โˆ’ =๐‘š๐‘–๐‘› ๐‘“ 1,๐‘› , ๐‘“ 2,๐‘› ,โ‹ฏ, ๐‘“ ๐‘šโˆ’1,๐‘› , ๐‘“ ๐‘š,๐‘› VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

27 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 03 (Determine the minimum/worst from the criterion values) ๐‘“ 1 โˆ’ =๐‘š๐‘–๐‘› ,0.0598,0.0765,0.0233, = ๐‘“ 2 โˆ’ =๐‘š๐‘–๐‘› ,0.1793,0.1682,0.2214, = ๐‘“ 3 โˆ’ =๐‘š๐‘–๐‘› ,0.0004,0.0008,0.0006, = ๐‘“ 4 โˆ’ =๐‘š๐‘–๐‘› ,0.0105,0.0046,0.0047, = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

28 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

29 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐‘ณ ๐Ÿ norm utilizing maximum/best and minimum/worst ratios) Compute: ๐ฟ 1,๐‘– = ๐‘—=1 ๐‘› ๐‘ค ๐‘— ๐‘“ ๐‘— โˆ— โˆ’ ๐‘“ ๐‘–,๐‘— ๐‘“ ๐‘— โˆ— โˆ’ ๐‘“ ๐‘— โˆ’ , ๐‘–=1,โ‹ฏ,๐‘š ๐ฟ 1,1 = ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ + ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 1,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ +โ‹ฏ+ ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ 1,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ + ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ 1,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ ๐ฟ 1,2 = ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 2,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ + ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ +โ‹ฏ+ ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ 2,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ + ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ 2,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ . ๐ฟ 1,๐‘šโˆ’1 = ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ + ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ +โ‹ฏ+ ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ + ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ ๐ฟ 1,๐‘š = ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ ๐‘š,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ + ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ ๐‘š,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ +โ‹ฏ+ ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘š,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ + ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘š,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

30 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐‘ณ ๐Ÿ norm utilizing maximum/best and minimum/worst ratios) ๐ฟ 1,1 =0.25ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ = ๐ฟ 1,2 =0.25ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ = ๐ฟ 1,3 =0.25ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ = ๐ฟ 1,4 =0.25ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ = ๐ฟ 1,5 =0.25ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ ร— โˆ’ โˆ’ = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

31 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐‘ณ โˆž norm utilizing maximum/best and minimum/worst ratios) Compute: ๐ฟ โˆž,๐‘– = max โˆ€๐‘— ๐‘—=1 ๐‘› ๐‘ค ๐‘— ๐‘“ ๐‘— โˆ— โˆ’ ๐‘“ ๐‘–,๐‘— ๐‘“ ๐‘— โˆ— โˆ’ ๐‘“ ๐‘— โˆ’ , ๐‘–=1,โ‹ฏ,๐‘š ๐ฟ โˆž,1 =๐‘š๐‘Ž๐‘ฅ ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ , ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 1,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ ,โ‹ฏ, ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ 1,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ , ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ 1,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ ๐ฟ โˆž,2 =๐‘š๐‘Ž๐‘ฅ ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 2,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ , ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ ,โ‹ฏ, ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ 2,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ , ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ 2,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ . ๐ฟ โˆž,๐‘šโˆ’1 =๐‘š๐‘Ž๐‘ฅ ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ , ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ ,โ‹ฏ, ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ , ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘šโˆ’1,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ ๐ฟ โˆž,๐‘š =๐‘š๐‘Ž๐‘ฅ ๐‘ค 1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ ๐‘š,1 ๐‘“ 1 โˆ— โˆ’ ๐‘“ 1 โˆ’ , ๐‘ค 2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ ๐‘š,2 ๐‘“ 2 โˆ— โˆ’ ๐‘“ 2 โˆ’ ,โ‹ฏ, ๐‘ค ๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘š,๐‘›โˆ’1 ๐‘“ ๐‘›โˆ’1 โˆ— โˆ’ ๐‘“ ๐‘›โˆ’1 โˆ’ , ๐‘ค ๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘š,๐‘› ๐‘“ ๐‘› โˆ— โˆ’ ๐‘“ ๐‘› โˆ’ VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

32 RNSengupta,IME Dept.,IIT Kanpur,INDIA
VIKOR: Step # 04 (Determine the relative ratios based on ๐‘ณ โˆž norm utilizing maximum/best and minimum/worst ratios) ๐ฟ โˆž,1 =๐‘š๐‘Ž๐‘ฅ 0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ = ๐ฟ โˆž,2 =๐‘š๐‘Ž๐‘ฅ 0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ = ๐ฟ โˆž,3 =๐‘š๐‘Ž๐‘ฅ 0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ = ๐ฟ โˆž,4 =๐‘š๐‘Ž๐‘ฅ 0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ = ๐ฟ โˆž,5 =๐‘š๐‘Ž๐‘ฅ 0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ ,0.25ร— โˆ’ โˆ’ = VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

33 VIKOR: Step # 05 (Determine the maximum/minimum based on ๐‘ณ ๐Ÿ norm)
Find: ๐ฟ 1,โˆ€๐‘– ๐‘š๐‘Ž๐‘ฅ = max โˆ€๐‘– ๐ฟ 1,๐‘– , i.e., ๐ฟ 1,โˆ€๐‘– ๐‘š๐‘Ž๐‘ฅ = max ๐ฟ 1,1 , ๐ฟ 1,2 ,โ‹ฏ, ๐ฟ 1,๐‘šโˆ’1 , ๐ฟ 1,๐‘š Find: ๐ฟ 1,โˆ€๐‘– ๐‘š๐‘–๐‘› = min โˆ€๐‘– ๐ฟ 1,๐‘– , ๐‘–=1,โ‹ฏ,๐‘š, i.e., ๐ฟ 1,โˆ€๐‘– ๐‘š๐‘–๐‘› =min ๐ฟ 1,1 , ๐ฟ 1,2 ,โ‹ฏ, ๐ฟ 1,๐‘šโˆ’1 , ๐ฟ 1,๐‘š VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

34 VIKOR: Step # 05 (Determine the maximum/minimum based on ๐‘ณ ๐Ÿ norm)
RNSengupta,IME Dept.,IIT Kanpur,INDIA

35 VIKOR: Step # 05 (Determine the maximum/minimum based on ๐‘ณ โˆž norm)
Find: ๐ฟ โˆž,โˆ€๐‘– ๐‘š๐‘Ž๐‘ฅ = max โˆ€๐‘– ๐ฟ โˆž,๐‘– , ๐‘–=1,โ‹ฏ,๐‘š, i.e., ๐ฟ โˆž,โˆ€๐‘– ๐‘š๐‘Ž๐‘ฅ =max ๐ฟ โˆž,1 , ๐ฟ โˆž,2 ,โ‹ฏ, ๐ฟ โˆž,๐‘šโˆ’1 , ๐ฟ โˆž,๐‘š Find: ๐ฟ โˆž,โˆ€๐‘– ๐‘š๐‘–๐‘› = min โˆ€๐‘– ๐ฟ โˆž,๐‘– , ๐‘–=1,โ‹ฏ,๐‘š, i.e., ๐ฟ โˆž,โˆ€๐‘– ๐‘š๐‘–๐‘› =min ๐ฟ โˆž,1 , ๐ฟ โˆž,2 ,โ‹ฏ, ๐ฟ โˆž,๐‘šโˆ’1 , ๐ฟ โˆž,๐‘š VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

36 VIKOR: Step # 05 (Determine the maximum/minimum based on ๐‘ณ โˆž norm)
RNSengupta,IME Dept.,IIT Kanpur,INDIA

37 VIKOR: Step # 06 (Calculate relative ranking)
Compute: ๐‘„ ๐‘– =๐‘ฃ ๐ฟ 1,๐‘– โˆ’ ๐ฟ 1,๐‘– ๐‘š๐‘–๐‘› ๐ฟ 1,๐‘– ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ 1,๐‘– ๐‘š๐‘–๐‘› โˆ’๐‘ฃ ๐ฟ โˆž,๐‘– โˆ’ ๐ฟ โˆž,๐‘– ๐‘š๐‘–๐‘› ๐ฟ โˆž,๐‘– ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ โˆž,๐‘– ๐‘š๐‘–๐‘› , ๐‘–=1,โ‹ฏ,๐‘š ๐‘„ 1 =๐‘ฃ ๐ฟ 1,1 โˆ’ ๐ฟ 1,1 ๐‘š๐‘–๐‘› ๐ฟ 1,1 ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ 1,1 ๐‘š๐‘–๐‘› โˆ’๐‘ฃ ๐ฟ โˆž,1 โˆ’ ๐ฟ โˆž,1 ๐‘š๐‘–๐‘› ๐ฟ โˆž,1 ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ โˆž,1 ๐‘š๐‘–๐‘› ๐‘„ 2 =๐‘ฃ ๐ฟ 1,2 โˆ’ ๐ฟ 1,2 ๐‘š๐‘–๐‘› ๐ฟ 1,2 ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ 1,2 ๐‘š๐‘–๐‘› โˆ’๐‘ฃ ๐ฟ โˆž,2 โˆ’ ๐ฟ โˆž,2 ๐‘š๐‘–๐‘› ๐ฟ โˆž,2 ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ โˆž,2 ๐‘š๐‘–๐‘› . ๐‘„ ๐‘šโˆ’1 =๐‘ฃ ๐ฟ 1,๐‘šโˆ’1 โˆ’ ๐ฟ 1,๐‘šโˆ’1 ๐‘š๐‘–๐‘› ๐ฟ 1,๐‘šโˆ’1 ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ 1,๐‘šโˆ’1 ๐‘š๐‘–๐‘› โˆ’๐‘ฃ ๐ฟ โˆž,๐‘šโˆ’1 โˆ’ ๐ฟ โˆž,๐‘šโˆ’1 ๐‘š๐‘–๐‘› ๐ฟ โˆž,๐‘šโˆ’1 ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ โˆž,๐‘šโˆ’1 ๐‘š๐‘–๐‘› ๐‘„ ๐‘š =๐‘ฃ ๐ฟ 1,๐‘š โˆ’ ๐ฟ 1,๐‘š ๐‘š๐‘–๐‘› ๐ฟ 1,๐‘š ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ 1,๐‘š ๐‘š๐‘–๐‘› โˆ’๐‘ฃ ๐ฟ โˆž,๐‘š โˆ’ ๐ฟ โˆž,๐‘š ๐‘š๐‘–๐‘› ๐ฟ โˆž,๐‘š ๐‘š๐‘Ž๐‘ฅ โˆ’ ๐ฟ โˆž,๐‘š ๐‘š๐‘–๐‘› VIKOR RNSengupta,IME Dept.,IIT Kanpur,INDIA

38 VIKOR: Step # 06 (Calculate relative ranking)
๐‘ฃ is introduced as weight of the strategy of โ€˜โ€˜the majority of criteriaโ€ (or โ€˜โ€˜the maximum group utilityโ€), here one can consider ๐‘ฃ=0.5 Rank the alternatives, sorting by the values ๐ฟ 1,๐‘– , ๐ฟ โˆž,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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