Multiple Attributes Group Decision-Making under Interval-Valued Dual Hesitant Fuzzy Unbalanced Linguistic Environment with Prioritized Attributes and Unknown Decision-Makers’ Weights
Abstract
:1. Introduction
2. Literature Review on Hesitant Fuzzy Linguistic MADM Approaches
3. Preliminaries for IVDHFUBLS
- (1)
- (2)
- (3)
- (4)
- (1)
- If , then .
- (2)
- If , then
- (a)
- If , then ;
- (b)
- If , then .
4. Proposed Approach for MAGDM Based on IVDHFUBLS
4.1. Prioritized Average Aggregation Operator for IVDHFUBLS
- (1)
- When , obviously, it is right.
- (2)
- When ,So, when , Theorem 1 also is right.
- (3)
- Suppose that when , Theorem 1 is right; then, we haveThen, when ,So, when , Theorem 1 is right too.According to steps (1), (2), and (3), we get that Theorem 1 is right for all . ☐
- (1)
- Commutativity: Letbe any permutation of, then
- (2)
- Idempotency: Let, for all, then
- (3)
- Boundedness: the IVDHFUBLPWA operator lies between the max and min operators,
- (1)
- Assume that is any permutation of ; then, for each , there exists one and only one , such that and vice versa. Additionally, also we have . Thus, based on Theorem 1, we have
- (2)
- Since for all , then
- (3)
- Suppose , , in whichObviously,Additionally, for all , we haveMeanwhile, we haveThenAccording to Definition 4 and Theorem 1, we have
4.2. A Hybrid Model for Determining the Unknown Experts’ Weights
4.3. Algorithm for MAGDM Based on IVDHFUBLS with Prioritization Relation among Evaluative Attributes and Unknown Decision-Makers’ Weights
- Step I-1.
- Compute the weight vector for decision-makers by applying Equation (17).
- Step I-2.
- According to the prioritization relation, , among attributes, transform each individual decision matrix to the prioritized individual decision matrix ,, in which , .
- Step I-3.
- Calculate prioritized levels in prioritized individual IVDHFUBL decision matrices: , .Calculate the score values of according to Equation (3) in Definition 4, then compute the numerical prioritized levels in each prioritized individual IVDHFUBL decision matrix, in which
- Step I-4.
- Obtain aggregated results in prioritized individual decision matrices, , , by applying operator IVDHFUBLPWA.Utilize the IVDHFUBLPWA operator described in Definition 6 to aggregate so that we get the decision-maker’s decision result on the alternative , in which
- Step I-5.
- Obtain collective results of all alternatives by applying decision-makers’ weighting vector.Given the weighting vector for decision-makers, which has been determined in Step 1, we now aggregate all the individual overall decision values into the overall group decision values by use of the IVDHFUBLWA operator described in Equation (10), in which
- Step I-6.
- According to Definition 4, calculate the score value of the group overall assessments to alternatives , then rank all the alternatives and select the most desirable one(s).
5. Illustrative Examples
5.1. Applied Case Study on Green Supplier Selection Problem
- Step I-1.
- Compute the weight vector for decision-makers. Firstly, by solving the programming model (M-2), we obtain deviation-based weighting vector asThen, by applying Equation (11), we get the accuracy-measure based experts’ weighting vector asFinally, according to Equation (17), we here suppose ; then, the hybrid experts’ weighting vector is obtained as
- Step I-2.
- Step I-3.
- Calculate prioritized levels in each prioritized individual IVDHFUBL decision matrix by Equations (18) and (19), and then we have
- , ; , , , , , , ,
- , , , , , ; , , , , , , ,
- , , , , , , ; , , , , , , , ; , , , , ,
- Step I-4.
- Utilize the IVDHFUBLPWA operator described in Definition 6 to aggregate , so that we get the k th expert’s decision result on alternatives , in which
- = (,{[0.439,0.5428],[0.4392,0.5429],[0.4397,0.5438],[0.4399,0.5439]},{[0.1633,0.2744],[0.1633,0.2744],[0.1633,0.2745],[0.1633,0.2746],[0.1675,0.2744],[0.1675,0.2744],[0.1675,0.2745],[0.1675,0.2746],[0.2598,0.373],[0.2598,0.373],[0.2598,0.3731],[0.2598,0.3732],[0.2665,0.373],[0.2665,0.373],[0.2665,0.3731],[0.2665,0.3732]}});
- = (,{[0.5479,0.6638],[0.5483,0.6642],[0.5517,0.6676],[0.5521,0.6681],[0.5538,0.6676],[0.5542,0.668],[0.5576,0.6714],[0.558,0.6718],[0.5579,0.6728],[0.5583,0.6732],[0.5616,0.6766],[0.562,0.677],[0.5637,0.6765],[0.5641,0.677],[0.5673,0.6802],[0.5677,0.6806]},{[0.1152,0.2172],[0.1225,0.2252],[0.1152,0.225],[0.1225,0.2333]});
- = (,{[0.2778,0.383],[0.2789,0.3841],[0.2778,0.3927],[0.2789,0.3938],[0.3647,0.475],[0.3656,0.476],[0.3647,0.4832],[0.3656,0.4842]},{[0.2911,0.4718],[0.2948,0.4753],[0.3186,0.5093],[0.3227,0.5131]});
- = (,{[0.1186,0.3686],[0.1431,0.3885]},{[0.1649,0.2682],[0.165,0.2683],[0.2188,0.3278],[0.2189,0.3279]});
- = (,{[0.4817,0.6085],[0.483,0.6099]},{[0.1246,0.231],[0.1256,0.2326]});
- = (,{[0.4365,0.5779]},{[0.2377,0.3445],[0.2646,0.3745]});
- = (,{[0.4134,0.5925],[0.4139,0.5925]},{[0.235,0.3833],[0.2487,0.3997]});
- = (,{[0.4072,0.5418],[0.4106,0.5452],[0.4294,0.5685],[0.4327,0.5718],[0.4646,0.5954],[0.4676,0.5985],[0.4846,0.6191],[0.4876,0.6219]},{[0.1748,0.2911],[0.1777,0.2932]});
- = (,{[0.6027,0.7085]},{[0.1053,0.2225],[0.1284,0.2522]}).
- Step I-5.
- Aggregate all the individual overall decision values into the overall group decision values by use of the IVDHFUBLWA operator described in Equation (10) and experts’ weighting vector determined in Step 1. Taking as an example, we have
- ,{[0.4597,0.5809],[0.46,0.5811],[0.4597,0.583],[0.46,0.5833], [0.4815,0.6021],[0.4817,0.6023],[0.4815,0.6041],[0.4817,0.6043]},{[0.191,0.3273],[0.2047,0.3419],[0.1979,0.3364],[0.2121,0.3514],[0.1918,0.328],[0.2055,0.3427],[0.1987,0.3372],[0.2129,0.3522],[0.1966,0.3354],[0.2107,0.3504],[0.2037,0.3448],[0.2183,0.3601],[0.1974,0.3362],[0.2116,0.3512],[0.2046,0.3456],[0.2192,0.361]}}).
- Step I-6.
- Calculating scores of the alternatives , we haveAccordingly, then the ranking order of all the alternatives is determined asTherefore, solution is the most desirable green supplier.
5.2. Comparison with IVDHFUBLS-Based TOPSIS Method
- Step II-1.
- Obtaining individual decision matrices from decision-makers, we get .
- Step II-2.
- Aggregate individual decision matrices into individual overall evaluation values corresponding to each alternative according to IVDHFULWA operator. Here, assume , and
- Step II-3.
- Calculate separating measure from positive and negative ideal solutions.Determine positive ideal solution (PIS) and negative ideal solution (NIS) , in which , .Then, we calculate the separating measure from the PIS and NIS for each alternative according to the distance measure introduced in Equation (5), in whichNext, we can obtain
- Step II-4.
- Calculate the relative closeness to the ideal solution by
- Step II-5.
- Rank the green suppliers according to the descending order of ; then, we get the most desirable supplier.
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Authors | Methodology Properties | |||||
---|---|---|---|---|---|---|
Linguistic Variable | Description of Hesitancy | Prioritized Attributes | Unknown Decision-Makers’ Weights | |||
Balanced | Unbalanced | Hesitant Fuzzy Set | Dual Hesitant Fuzzy Set | |||
Lin, et al. [37] | √ | × | √ | × | × | Single-person MADM |
Wang, et al. [36] | √ | × | √ | × | √ | Single-person MADM |
Yang and Ju [40] | √ | × | × | √ | √ | Single-person MADM |
Qi, et al. [41] | √ | × | × | √ | × | √ |
Wang, et al. [63] | √ | × | √ | × | × | × |
Qi, et al. [47] | √ | √ | √ | √ | × | Power aggregation-based method |
This paper | √ | √ | √ | √ | √ | Deviation-maximizing method |
(VH,{[0.2,0.3]}, {[0.2,0.4],[0.3,0.4]}) | (H,{[0.5,0.6]}, {[0.2,0.3]}) | (M,{[0.3,0.4]}, {[0.4,0.5],[0.5,0.6]}) | (T,{[0.2,0.4]}, {[0.5,0.6]}) | |
(AT,{[0.4,0.5],[0.5,0.6]}, {[0.2,0.3],[0.2,0.4]}) | (T,{[0.6,0.7]}, {[0.1,0.2]}) | (L,{[0.6,0.7], [0.7,0.8]},{[0.1,0.2]}) | (QH,{[0.3,0.5]}, {[0.2,0.3]}) | |
(L,{[0.1,0.2],[0.1,0.3]}, {[0.6,0.7]}) | (H,{[0.2,0.3]}, {[0.5,0.6],[0.6,0.7]}) | (AH,{[0.4,0.5]}, {[0.2,0.3]}) | (VH,{[0.6,0.7]}, {[0.1,0.2],[0.2,0.3]}) | |
(L,{[0.4,0.5]}, {[0.1,0.2],[0.4,0.5]}) | (M,{[0.1,0.2],[0.3,0.5]}, {[0.3,0.5]}) | (AH,{[0.4,0.5],[0.5,0.6]}, {[0.2,0.3],[0.2,0.4]}) | (L,{[0.1,0.3]}, {[0.4,0.6]}) | |
(M,{[0.6,0.7]}, {[0.1,0.2]}) | (VH,{[0.2,0.4],[0.5,0.6]}, {[0.2,0.3]}) | (L,{[0.5,0.6],[0.7,0.8]}, {[0.1,0.2]}) | (H,{[0.5,0.7]}, {[0.1,0.2],[0.2,0.3]}) | |
(M,{[0.4,0.5],[0.6,0.7]}, {[0.1,0.3]}) | (H,{[0.3,0.4]}, {[0.4,0.5]}) | (QH,{[0.4,0.5],[0.5,0.6]}, {[0.3,0.4]}) | (VH,{[0.4,0.6]}, {[0.3,0.4]}) |
(M,{[0.3,0.5]}, {[0.1,0.2]}) | (AH,{[0.1,0.4]}, {[0.2,0.3],[0.3,0.4]}) | (H,{[0.2,0.4]}, {[0.4,0.5]}) | (QH,{[0.2,0.4]}, {[0.5,0.6]}) | |
(AT,{[0.4,0.7]}, {[0.2,0.3] }) | (L,{[0.5,0.6]}, {[0.1,0.2] }) | (AH,{[0.6,0.7],[0.7,0.8]}, {[0.1,0.2]}) | (M,{[0.2,0.3]}, {[0.5,0.6],[0.6,0.7]}) | |
(AT,{[0.6,0.8]}, {[0.1,0.2]}) | (H,{[0.4,0.5]}, {[0.3,0.4],[0.4,0.5]}) | (H,{[0.4,0.5]}, {[0.2,0.3]}) | (AH,{[0.4,0.5]}, {[0.2,0.3]}) | |
(M,{[0.1,0.2],[0.2,0.3]}, {[0.1,0.2]}) | (L,{[0.6,0.7]}, {[0.1,0.2]}) | (H,{[0.3,0.4]}, {[0.2,0.3],[0.4,0.5]}) | (M,{[0.5,0.7]}, {[0.2,0.3]}) | |
(VH,{[0.6,0.7]}, {[0.1,0.2]}) | (AT,{[0.2,0.3]}, {[0.5,0.7]}) | (H,{[0.5,0.8]}, {[0.1,0.2]}) | (AT,{[0.3,0.5]}, {[0.3,0.4]}) | |
(QH,{[0.4,0.5]}, {[0.3,0.4]}) | (L,{[0.7,0.8]}, {[0.1,0.2]}) | (VH,{[0.2,0.5]}, {[0.3,0.4]}) | (H,{[0.3,0.5]}, {[0.3,0.4]}) |
(M,{[0.6,0.8]}, {[0.1,0.2]}) | (H,{[0.4,0.5]}, {[0.4,0.5]}) | (H,{[0.2,0.4],[0.3,0.4]}, {[0.2,0.3]}) | (M,{[0.7,0.8]}, {[0.1,0.2]}) | |
(T,{[0.3,0.4]}, {[0.4,0.6]}) | (M,{[0.4,0.5],[0.5,0.6]}, {[0.2,0.3] }) | (VH,{[0.6,0.7]}, {[0.1,0.3]}) | (H,{[0.1,0.3],[0.2,0.4]}, {[0.3,0.5]}) | |
(VH,{[0.4,0.5]}, {[0.1,0.2],[0.3,0.4]}) | (VH,{[0.7,0.8]}, {[0.1,0.2]}) | (H,{[0.6,0.8]}, {[0.1,0.2]}) | (M,{[0.6,0.7]}, {[0.1,0.3]}) | |
(T,{[0.2,0.5]}, {[0.3,0.5]}) | (VH,{[0.6,0.7]}, {[0.2,0.3]}) | (M,{[0.1,0.2]}, {[0.5,0.8]}) | (VH,{[0.3,0.4]}, {[0.1,0.3],[0.2,0.5]}) | |
(M,{[0.4,0.6], [0.5,0.7]},{[0.1,0.2]}) | (VH,{[0.3,0.6]}, {[0.1,0.3],[0.2,0.4]}) | (H,{[0.4,0.6]}, {[0.3,0.4]}) | (VH,{[0.7,0.8]}, {[0.1,0.2]}) | |
(VH,{[0.6,0.7]}, {[0.1,0.2]}) | (M,{[0.5,0.6]}, {[0.3,0.4]}) | (H,{[0.3,0.5]}, {[0.4,0.5]}) | (H,{[0.6,0.7]}, {[0.1,0.3]}) |
(H,{[0.5,0.6]}, {[0.2,0.3]}) | (L,{[0.4,0.5]}, {[0.1,0.2],[0.4,0.5]}) | (VH,{[0.2,0.3 0.2,0.4],[0.3,0.4]}) | (L,{[0.1,0.3]}, {[0.4,0.6]}) | |
(T,{[0.6,0.7]}, {[0.1,0.2]}) | (M,{[0.6,0.7]}, {[0.1,0.2]}) | (AT,{[0.4,0.5],[0.5,0.6]}, {[0.2,0.3],[0.2,0.4]}) | (H,{[0.5,0.7]}, {[0.1,0.2],[0.2,0.3]}) | |
(H,{[0.2,0.3]}, {[0.5,0.6],[0.6,0.7]}) | (M,{[0.4,0.5],[0.6,0.7]}, {[0.1,0.3]}) | (L,{[0.1,0.2],[0.1,0.3]}, {[0.6,0.7]}) | (VH,{[0.4,0.6]}, {[0.3,0.4]}) | |
(T,{[0.2,0.4]}, {[0.5,0.6]}) | (M,{[0.1,0.2],[0.3,0.5]}, {[0.3,0.5]}) | (AH,{[0.4,0.5],[0.5,0.6]}, {[0.2,0.3],[0.2,0.4]}) | (M,{[0.3,0.4]}, {[0.4,0.5],[0.5,0.6]}) | |
(QH,{[0.3,0.5]}, {[0.2,0.3]}) | (VH,{[0.2,0.4],[0.5,0.6]}, {[0.2,0.3]}) | (L,{[0.5,0.6],[0.7,0.8]}, {[0.1,0.2]}) | (L,{[0.6,0.7],[0.7,0.8]}, {[0.1,0.2]}) | |
(VH,{[0.6,0.7]}, {[0.1,0.2],[0.2,0.3]}) | (H,{[0.3,0.4]}, {[0.4,0.5]}) | (QH,{[0.4,0.5],[0.5,0.6]}, {[0.3,0.4]}) | (AH,{[0.4,0.5]}, {[0.2,0.3]}) |
(AH,{[0.1,0.4]}, {[0.2,0.3],[0.3,0.4]}) | (M,{[0.1,0.2], [0.2,0.3]},{[0.1,0.2]}) | (M,{[0.3,0.5]}, {[0.1,0.2]}) | (M,{[0.5,0.7]}, {[0.2,0.3]}) | |
(L,{[0.5,0.6]}, {[0.1,0.2] }) | (VH,{[0.6,0.7]}, {[0.1,0.2]}) | (AT,{[0.4,0.7]}, {[0.2,0.3] }) | (AT,{[0.3,0.5]}, {[0.3,0.4]}) | |
(H,{[0.4,0.5]}, {[0.3,0.4],[0.4,0.5]}) | (QH,{[0.4,0.5]}, {[0.3,0.4]}) | (AT,{[0.6,0.8]}, {[0.1,0.2]}) | (H,{[0.3,0.5]}, {[0.3,0.4]}) | |
(QH,{[0.2,0.4]}, {[0.5,0.6]}) | (L,{[0.6,0.7]}, {[0.1,0.2]}) | (H,{[0.3,0.4]}, {[0.2,0.3],[0.4,0.5]}) | (H,{[0.2,0.4]}, {[0.4,0.5]}) | |
(M,{[0.2,0.3]}, {[0.5,0.6],[0.6,0.7]}) | (AT,{[0.2,0.3]}, {[0.5,0.7]}) | (H,{[0.5,0.8]}, {[0.1,0.2]}) | (AH,{[0.6,0.7],[0.7,0.8]}, {[0.1,0.2]}) | |
(AH,{[0.4,0.5]}, {[0.2,0.3]}) | (L,{[0.7,0.8]}, {[0.1,0.2]}) | (VH,{[0.2,0.5]}, {[0.3,0.4]}) | (H,{[0.4,0.5]}, {[0.2,0.3]}) |
(H,{[0.4,0.5]}, {[0.4,0.5]}) | (T,{[0.2,0.5]}, {[0.3,0.5]}) | (M,{[0.6,0.8]}, {[0.1,0.2]}) | (VH,{[0.3,0.4]}, {[0.1,0.3],[0.2,0.5]}) | |
(M,{[0.4,0.5],[0.5,0.6]}, {[0.2,0.3] }) | (M,{[0.4,0.6], [0.5,0.7]},{[0.1,0.2]}) | (T,{[0.3,0.4]}, {[0.4,0.6]}) | (VH,{[0.7,0.8]}, {[0.1,0.2]}) | |
(VH,{[0.7,0.8]}, {[0.1,0.2]}) | (VH,{[0.6,0.7]}, {[0.1,0.2]}) | (VH,{[0.4,0.5]}, {[0.1,0.2],[0.3,0.4]}) | (H,{[0.6,0.7]}, {[0.1,0.3]}) | |
(M,{[0.7,0.8]}, {[0.1,0.2]}) | (VH,{[0.6,0.7]}, {[0.2,0.3]}) | (M,{[0.1,0.2]}, {[0.5,0.8]}) | (H,{[0.2,0.4],[0.3,0.4]}, {[0.2,0.3]}) | |
(H,{[0.1,0.3],[0.2,0.4]}, {[0.3,0.5]}) | (VH,{[0.3,0.6]}, {[0.1,0.3],[0.2,0.4]}) | (H,{[0.4,0.6]}, {[0.3,0.4]}) | (VH,{[0.6,0.7]}, {[0.1,0.3]}) | |
(M,{[0.6,0.7]}, {[0.1,0.3]}) | (M,{[0.5,0.6]}, {[0.3,0.4]}) | (H,{[0.3,0.5]}, {[0.4,0.5]}) | (H,{[0.6,0.8]}, {[0.1,0.2]}) |
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Qi, X.-W.; Zhang, J.-L.; Liang, C.-Y. Multiple Attributes Group Decision-Making under Interval-Valued Dual Hesitant Fuzzy Unbalanced Linguistic Environment with Prioritized Attributes and Unknown Decision-Makers’ Weights. Information 2018, 9, 145. https://doi.org/10.3390/info9060145
Qi X-W, Zhang J-L, Liang C-Y. Multiple Attributes Group Decision-Making under Interval-Valued Dual Hesitant Fuzzy Unbalanced Linguistic Environment with Prioritized Attributes and Unknown Decision-Makers’ Weights. Information. 2018; 9(6):145. https://doi.org/10.3390/info9060145
Chicago/Turabian StyleQi, Xiao-Wen, Jun-Ling Zhang, and Chang-Yong Liang. 2018. "Multiple Attributes Group Decision-Making under Interval-Valued Dual Hesitant Fuzzy Unbalanced Linguistic Environment with Prioritized Attributes and Unknown Decision-Makers’ Weights" Information 9, no. 6: 145. https://doi.org/10.3390/info9060145
APA StyleQi, X. -W., Zhang, J. -L., & Liang, C. -Y. (2018). Multiple Attributes Group Decision-Making under Interval-Valued Dual Hesitant Fuzzy Unbalanced Linguistic Environment with Prioritized Attributes and Unknown Decision-Makers’ Weights. Information, 9(6), 145. https://doi.org/10.3390/info9060145