The Hierarchical VIKOR Method with Incomplete Information: Supplier Selection Problem
Abstract
:1. Introduction
2. The Hierarchical VIKOR Method with Incomplete Information
2.1. The VIKOR Method
2.2. The Incomplete Information
2.3. The Hierarchical VIKOR with Incomplete Information
2.3.1. The Hierarchical Structure
2.3.2. The Hierarchical VIKOR Method
- Step 0.
- Structurize the problem at hand to the hierarchical one when it is deemed appropriate to do so.
- Step 1.
- Compute the extreme points of incomplete alternatives’ values and determine the lower and upper consequences of the ith alternative.Moreover, note that and where is a collection of elements associated with alternative in the set of extreme points of , i.e., , .
- Step 2.
- Compute S and R intervals.In general, it is an efficient way to obtain and intervals via LPs for the hierarchical problem since the multi-leveled criteria require enormous calculations to obtain the extreme points of the criteria weights. Below are LPs for obtaining the intervals and :Calculating is troublesome work requiring repeated calculations and thus we modify Equations (14)–(16) by Foroughi and Aouni [45].The third constraint in Equation (16) means that and thus results in the mini-max of for all in view of the objective function.
- Step 3.
- Compute Q intervals.Compute the value for each alternative using the relation.The constant is the weight introduced to support the strategy of maximum group utility while is used to weigh the individual regret, usually, .
- Step 4.
- Determine the final ranking.Further computations are needed to obtain the final ranking of alternatives in the face of the intervals in Step 3. The methods for ranking intervals can be classified into three categories. One of the most widely-used methods is to consider each location of intervals (i.e., the gaps, overlapping, etc.) and the distributions of intervals [17,41]. Xu and Da [46] presented formulas for comparing intervals based on the degree of possibility. Ahn [47] presented a method to prioritize intervals by taking into account the strength of preference based on a probabilistic measure. The similarity between two intervals can be gauged by two measures that characterize the intervals: The ratio of the overlapping portion of two intervals and the level of closeness of the midpoints between two intervals [48].
3. Numerical Example
4. Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Hierarchy | Incomplete Criteria Weights |
---|---|
Goal ( | |
Sub-criteria of | |
Sub-criteria of | |
Sub-criteria of | |
Sub-criteria of | |
Sub-criteria of | |
Sub-criteria of | |
Sub-criteria of |
Type | Incomplete Alternatives’ Values |
---|---|
Precise values | |
Weak preference | |
Strict preference | |
Weak preference | |
Interval values | |
Weak preference | |
Verbal | |
Interval values | |
Verbal | |
Weak preference | |
Ratio preference | . |
Weak differences of preference | |
Weak preference | |
Weak preference | |
Weak preference | |
Verbal | |
Weak preference | |
Verbal | |
Weak preference |
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Forms | Extreme Points |
---|---|
Strict preference } | , , , , ) |
Weak preference | , , |
Weak differences in preference | , , , |
Ratio preference | , , , |
Level 1 | |||||
Sub-criteria | |||||
[0, 0] | [0, 1] | [0.4286, 0.7143] | |||
[0, 0.9] | [0, 1] | [0.1429, 0.7143] | |||
[0, 0.9] | [0, 1] | [0.8571, 1] | |||
[0, 1] | [0, 1] | [0.1429, 0.4296] | |||
[0, 1] | [1, 1] | [0, 0.5714] | |||
Level 1 | |||||
Sub-criteria | |||||
Level 1 | |||||
Sub-criteria | |||||
. | |||||
. | |||||
Level 1 | |||||
Sub-criteria | |||||
S Interval | R Interval | Q Interval | |||||
---|---|---|---|---|---|---|---|
0.1366 | |||||||
0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |
---|---|---|---|---|---|---|---|---|---|---|---|
2.2173 | 2.2279 | 2.2381 | 2.248 | 2.2574 | 2.2666 | 2.2756 | 2.2843 | 2.2927 | 2.3009 | 2.3087 | |
3.2207 | 3.1801 | 3.1406 | 3.1021 | 3.0645 | 3.0279 | 2.9921 | 2.9571 | 2.923 | 2.8898 | 2.8571 | |
2.3846 | 2.3598 | 2.3358 | 2.3124 | 2.2895 | 2.2669 | 2.2446 | 2.2225 | 2.2007 | 2.1789 | 2.1573 | |
2.5992 | 2.5829 | 2.5661 | 2.549 | 2.5317 | 2.5141 | 2.4963 | 2.4783 | 2.4599 | 2.4414 | 2.4229 | |
2.3423 | 2.3606 | 2.3779 | 2.3941 | 2.4097 | 2.4245 | 2.4386 | 2.4522 | 2.4652 | 2.4777 | 2.4899 | |
Ranking | * | ** | ** | ** | ** | ** | *** | *** | **** | **** | **** |
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Kim, J.H.; Ahn, B.S. The Hierarchical VIKOR Method with Incomplete Information: Supplier Selection Problem. Sustainability 2020, 12, 9602. https://doi.org/10.3390/su12229602
Kim JH, Ahn BS. The Hierarchical VIKOR Method with Incomplete Information: Supplier Selection Problem. Sustainability. 2020; 12(22):9602. https://doi.org/10.3390/su12229602
Chicago/Turabian StyleKim, Jong Hyen, and Byeong Seok Ahn. 2020. "The Hierarchical VIKOR Method with Incomplete Information: Supplier Selection Problem" Sustainability 12, no. 22: 9602. https://doi.org/10.3390/su12229602