VIKOR Method The VIKOR method was introduced for

VIKOR Method The VIKOR method was introduced for multi-criteria optimization problem. This method focuses on ranking and selecting from a set of alternatives, and determines compromise solution for a problem with conflicting criteria, which can help the decision makers to get a final solution. The compromise solution is a feasible solution which is the closest to the ideal, and a compromise means an agreement established by mutual concessions. It introduces the multi-criteria ranking index on the base of the particular measure of ‘‘closeness” to the ‘‘ideal” solution. 3

VIKOR Method Opricovic developed the basic ideas of VIKOR in his Ph. D. dissertation in 1979, and an application was published in 1980. The name VIKOR appeared in 1990 from Serbian: VIse. Kriterijumska Optimizacija I Kompromisno Resenje, that means: Multicriteria Optimization and Compromise Solution, with pronunciation: vikor. The real applications were presented in 1998. 4

Figure 51. Diagram of modified Vlse. Kriterijuska Optimizacija I Komoromisno Resenje (VIKOR) technique adopted from

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VIKOR is an effective tool in multi-criteria decision making, particularly in a situation where the decision maker is not able, or does not know how to express his/her preference at the beginning of system design. The obtained compromise solution could be accepted by the decision makers because it provides a maximum ‘‘group utility” (represented by min S ) of the ‘‘majority”, and a minimum of the ‘‘individual regret” (represented by min R ) of the ‘‘opponent”. The compromise solutions could be the basis for negotiations, involving the decision maker’s preference by criteria weights. 14

First example: 15

Suppose that we have six candidates which need to be selected based on five criteria, as presented in the table. The weights of the criteria were predetermined and depicted in the table: Criteria C 1 C 2 C 3 C 4 C 5 Wj W 1 W 2 W 3 W 4 W 5 0. 203 0. 201 0. 199 0. 198 A 1 0. 7 0. 6 1 0 1 A 2 0. 25 0 0 0. 436 0 A 3 0 0 0. 667 0. 765 0. 217 A 4 0. 6 0. 8 0. 533 0. 282 0. 783 A 5 0. 55 0. 3 0. 667 1 0. 565 A 6 1 1 0. 8 0. 483 0. 435 16

CAND 1 CAND 2 CAND 3 CAND 4 CAND 5 17 CAND 6 C 1 36 45 50 38 39 30 Decision Matrix C 2 C 3 C 4 14 5 1360 20 20 1102 20 10 907 12 12 1193 17 10 768 1074 C 5 67 90 85 72 77 80 C 1 0. 3 0. 75 1 0. 45 0 C 2 0. 4 1 1 0. 2 0. 7 0 C 3 0 1 0. 333 0. 467 0. 333 0. 2 C 4 1 0. 564 0. 235 0. 718 0 0. 517 C 5 0 1 0. 783 0. 217 0. 435 0. 565

CAND 1 CAND 2 CAND 3 CAND 4 CAND 5 CAND 6 18 C 1 0. 3 0. 75 1 0. 45 0 C 2 0. 4 1 1 0. 2 0. 7 0 C 3 0 1 0. 333 0. 476 0. 333 0. 2 C 4 1 0. 5642 0. 2348 0. 7179 0 0. 569 C 5 0 1 0. 7826 0. 2174 0. 4348 0. 5652 1 0 1 0 1 0

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C 1 W 1 C 2 W 2 0. 20345 0. 2014 C 3 W 3 C 4 W 4 0. 19926 0. 1983 C 5 W 5 Calculation of utility measures (Sij) C 1 0. 19759 C 2 C 3 C 4 C 5 0 0. 19759 0. 086419 0 CAND 1 0. 3 0. 4 0 1 0 CAND 1 0. 142415 0. 12084 0. 19926 CAND 2 0. 75 1 1 0. 5642 1 CAND 2 0. 050863 0 CAND 3 1 1 0. 333 0. 2348 0. 7826 CAND 3 0 0 CAND 4 0. 2 0. 476 0. 7179 0. 2174 CAND 4 0. 12207 0. 16112 0. 104412 0. 05594 0. 154634 CAND 5 0. 45 0. 7 0. 333 0 0. 4348 CAND 5 0. 111898 0. 06042 0. 132906 0. 1983 0. 111678 CAND 6 0 0 0. 2 0. 569 0. 5652 CAND 6 0. 20345 1 0 1 0 1 0 20 0 0. 132906 0. 151739 0. 2014 0. 159408 0. 085467 0. 042956 0. 085912

Wj 21 C 2 C 3 C 4 C 5 0. 20345 0. 2014 0. 19926 0. 1983 0. 19759 0. 142 0. 121 0. 199 0. 000 0. 198 0. 19926 0. 660105 0. 0509 0 0 0. 086419 0. 137282 0 0 0. 1329 0. 1517 0. 043 0. 151739 0. 327602 0. 1221 0. 1611 0. 1044 0. 0559 0. 1546 0. 16112 0. 598177 0. 1119 0. 0604 0. 1329 0. 1983 0. 1117 0. 1983 0. 615202 0. 2035 1 0 0. 2014 1 0 0. 1594 1 0 0. 0855 1 0 0. 0859 1 0 0. 20345 0. 735637

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CAND 1 CAND 2 CAND 3 CAND 4 CAND 5 CAND 6 23 0. 19926 0. 086419 0. 151739 0. 16112 0. 1983 0. 20345 0. 660105 0. 137282 0. 327602 0. 598177 0. 615202 0. 735637 0. 918982 0 0. 438108 0. 704285 0. 877358 1

(e) Rank the alternatives, sorting by the values S, R and Q, from the minimum value to the maximum. The results are three ranking lists. Ranking based on: CAND 1 0. 1993 0. 660 0. 918982 CAND 2 0. 0864 0. 137 0 CAND 3 0. 1517 0. 328 0. 438108 CAND 4 0. 1611 0. 598 0. 704285 CAND 5 CAND 6 0. 1983 0. 615 0. 877358 CAND 1 0. 2035 0. 736 1 CAND 6 24

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An Example for VIKOR Method Ref. : MULTI-CRITERIA DECISION MAKING APPROACH FOR VENDOR SELECTION In this example, all of the criteria are cost, not benefit. Also, Individual attribute 26 weights have been used as 0. 20 (equal priority weight).

Normalized quality loss estimates (NQL) have been determined and shown in Table 2. The ideal and negative-ideal solutions which represent the minimum and maximum NQL of all alternatives are as follows: 27

Note that all of the criteria are negative (cost). 28

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VIKOR INDEX of each alternative (candidate vendor) has been presented in Table presented below. Since all of the criteria used were negative (cost), so the ranking based on VIKOR INDEX are ascending (i. e. , from maximum to minimum). 31

An example for validation of ranking based Q: Ranking based on: Alt. A 0. 4542 0. 5837 D C D B 0. 4546 0. 2 0. 585 A D E C 0. 590829 0. 088 0. 4998 B E C D 0. 426596 0. 139 0. 2274 E A A E 32 0. 468535 0. 164 0. 4667 C B B

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The ranking based on Q is: D≈E>C>A>B 34