Extended VIKOR-QUALIFLEX Method Based on Trapezoidal Fuzzy Two-Dimensional Linguistic Information for Multiple Attribute Decision-Making with Unknown Attribute Weight
Abstract
:1. Introduction
- (1)
- Some basic concepts about TrF2DLVs are introduced, including expectation value based on scale function, distance measure, and trapezoidal fuzzy two-dimensional linguistic preference relation (TrF2DLPR).
- (2)
- A combination weight model based on trapezoidal fuzzy two-dimensional linguistic information is proposed to solve the MADM problems with unknown attribute weights, which includes proposing a trapezoidal fuzzy two-dimensional linguistic best-worst method (TrF2DL-BWM) to determine the subjective attribute weight, proposing a CRITIC method to determine the objective attribute weight, and proposing a method based on the maximum comprehensive evaluation value to determine the combination weight.
- (3)
- An extended VIKOR-QUALIFLEX method based on TrF2DLVs is presented to solve MADM problems, which can measure the concordance index of each ranking combination by means of group utility and individual maximum regret value of each evaluation alternative, and get more stable evaluation result.
- (4)
- A practical application of lean management assessment for industrial residential projects is solved by the proposed method.
2. Preliminaries
2.1. Trapezoidal Fuzzy Numbers
2.2. Linguistic Term Sets
- (i)
- The set is ordered: if and only if ;
- (ii)
- There is a negation operator: Neg () = ;
- (iii)
- If , then ;
- (iv)
- If , then .
2.3. Trapezoidal Fuzzy Two-Dimensional Linguistic Variables
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- ;
- (v)
- ;
- (vi)
- ;
- (vii)
- .
- (i)
- If , the ;
- (ii)
- If , the ;
- (iii)
- If , the .
- (i)
- ;
- (ii)
- If , then ;
- (iii)
- ;
- (iv)
- .
2.4. Trapezoidal Fuzzy Two-Dimensional Linguistic Preference Relation
3. Models of the Weights of Attributes for MADM Based on Trapezoidal Fuzzy Two-Dimensional Linguistic Information
3.1. Subjective Weight Model Based on TrF2DL-BWM
- Step 1
- determine the best and the worst attributes.
- Step 2
- obtain the Best-to-Others vector and the Others-to-Worst vector .
- Step 3
- convert the vector and to and , respectively.
- Step 4
- obtain the subjective weight vector by solving the model (28).
3.2. Objective Weight Model Based on CRITIC Method
- Step 1
- Normalize the decision matrix into () by the following formulas:For benefit attributes, we haveFor cost attributes, we have
- Step 2
- Calculate the correlation coefficient between the jth and the lth attributes by the following Equation (31) and obtain the correlation coefficient matrix :
- Step 3
- Calculate the standard deviation of the attribute by the following formula:
- Step 4
- Calculate the index () of the attribute by the following formula:
- Step 5
- Obtain the objective weight vector , where
3.3. Combination Weight Model Based on Maximum Expected Value of Evaluation
4. Extended VIKOR-QUALIFLEX Method Based on TrF2DLVs for MADM with Unknown Attribute Weight
4.1. Problem Description of MADM with Unknown Attribute Weight
4.2. Extended VIKOR-QUALIFLEX Method Based on TrF2DLVs
- Step 1
- Establish decision matrix , where is the evaluation value of the alternative according to the attribute represented by a TrF2DLV . Then, normalize the decision-making unit according to the attribute of benefit type or cost type. The normalized matrix can be obtained from Equations (29) and (30) in Section 3.2.
- Step 2
- Calculate the weight vector of attributes using the weight determination model.
- Step 2.1
- Calculate the subjective weight vector using TrF2DL-BWM in Section 2.1, where ,;
- Step 2.2
- Calculate the objective weight vector using the CRITIC method in Section 2.2, where , ;
- Step 2.3
- Calculate the combination weight vector by a mathematical programming model in Section 2.3, where , .
- Step 3
- Determine positive and negative ideal solutions of each subattribute.The positive ideal solution and the negative ideal solution .Note: the comparison rules of are described in Section 2.3.
- Step 4
- Calculate the group utility value and individual regret value of each evaluation alternative, where
- Step 5
- List all possible ranking of m evaluation alternatives .
- Step 6
- The alternatives in each possible permutation are compared in pairs in order, and the concordance index of each comparison ( over ) is calculated by the following Equation (43). Then, the concordance index matrix is established.
- Step 7
- Calculate the overall concordance index for each possible ranking .
- Step 8
- The ranking with the largest overall concordance index is selected as the final ranking result.
5. Practical Case Application
5.1. Method Implementation Process
- Step 1.
- Since all attribute types are benefit type, the normalization can be omitted.
- Step 2.1.
- Calculate the subjective weight vector using TrF2DL-BWM.
- Step 2.1.1.
- First, the best attribute and the worst attribute are determined among all the primary attributes. Then, the best attribute and the worst attribute among the subattributes of each primary attribute are determined.For the five primary attributes, the DMs determine that the best attribute is and the worst attribute is . For the four subattributes under the primary attribute , DMs determine that the best attribute is , and the worst subattribute is . Similarly, for other primary attributes, the best subattributes are , , , and , and the worst subattributes are , , , and .
- Step 2.1.2.
- For the five primary attributes, DMs give the Best-to-Others vector:= = {([0.500, 0.500, 0.500, 0.500],), ([0.667, 0.733, 0.800, 0.867],), ([0.533, 0.600, 0.667, 0.733],), ([0.733, 0.800, 0.867, 0.933],), ([0.800, 0.867, 0.933, 1.000],)}, and the Others-to-Worst vector:= = {([0.800, 0.867, 0.933, 1.000],), ([0.533, 0.600, 0.667, 0.733],), ([0.667, 0.733, 0.800, 0.867],), ([0.667, 0.733, 0.800, 0.867],), ([0.500, 0.500, 0.500, 0.500],)}.For the four subattributes under the primary attribute , DMs give the Best-to-Others vector:= = {([0.500, 0.500, 0.500, 0.500],), ([0.533, 0.600, 0.667, 0.733],), ([0.800, 0.867, 0.933, 1.000],), ([0.667, 0.733, 0.800, 0.867],)}, and the Others-to-Worst vector:= {([0.800, 0.867, 0.933, 1.000],), ([0.667, 0.733, 0.800, 0.867],), ([0.500, 0.500, 0.500, 0.500],), ([0.667, 0.733, 0.800, 0.867],)}.Similarly, DMs can give the Best-to-Others vectors and the Others-to-Worst vectors of other subattributes under the primary attribute , , , and . For the convenience of reading, they are omitted here.
- Step 2.1.3.
- Convert the vector and to and , respectively.Here, we only take the primary attributes as an example to give the experimental results. = = {0.500, 0.767, 0.633, 0.833, 0.930},= {0.900, 0.633, 0.767, 0.575,0.500}.
- Step 2.1.4.
- For the five primary attributes, we can build the following model based on Equation (28):
- Step 2.2.
- Calculate the objective weight vector using the CRITIC method in Section 2.2.
- Step 2.2.1.
- For the four subattributes under the primary attribute , we calculate the correlation coefficient (j, l = 1, 2, 3, 4) between the jth and the lth attributes by the Equations (31) and (32), and the following correlation coefficient matrix: can be determined:
- Step 2.2.3.
- For the primary attributes, calculate the standard deviation of the (j = A, B, C, D, E) primary attribute by Equation (33). Similarly, the standard deviation (j = 1, 2, 3, 4 when and j = 1, 2, 3 when ) of subattributes under each primary attribute can also be obtained.
- Step 2.2.4.
- According to Equations (34) and (35), the objective weight vector for primary attributes and the objective weight vector can be obtained as below:
- Step 2.3.
- Calculate the combination weight vector of all subattributes by the mathematical programming model in Section 2.3 and Equation (40). The result is as follows:ϖ = (0.092, 0.074, 0.032, 0.046, 0.074, 0.091, 0.045, 0.045, 0.053, 0.065, 0.035, 0.067, 0.051, 0.044, 0.023, 0.031, 0.051, 0.050, 0.031)T.
- Step 3.
- Determine positive and negative ideal solutions of each subattribute.
- Step 4.
- Calculate the group utility value and individual regret value of each evaluation alternative based on Equations (41) and (42),
- Step 5.
- All possible ranking of 3 evaluation alternatives can be listed:
- Step 6.
- The concordance index matrix is established by Equation (43),
- Step 7.
- Calculate the overall concordance index for each possible ranking by Equation (44),
- Step 8.
- The largest overall concordance index is , thus the ranking result is .
5.2. Parameter Sensitivity Analysis
5.3. Weight Sensitivity Analysis
5.4. Comparative Analysis and Discussion
- (1)
- Compared with the method in [14]. The TF2DLPGWA operator in [14] can assign a corresponding weight to each attribute value, and the allocation principle is to adjust the weight of extreme value to a smaller weight. The result of this operation is to reconcile the evaluation value of the alternatives, reduce the differentiation between the alternatives, and make it difficult to distinguish some close alternatives. It is not suitable for those DMs with poor discrimination ability. In addition, the proposed method, which uses a CRITIC method to deal with the objective weight, can measure the contrast intensity and conflict, get a more distinct overall correlation coefficient, and then get a more clear objective weight. In addition, although the method based on TF2DLPGWA operator is simple and easy to operate, it is easy to cause the loss of decision-making information. Our method in this paper can calculate the overall concordance index of different ranking combinations by the group utility value and individual regret value, and then get more scientific and reliable ranking results.
- (2)
- Compared with the method in [15]. The WTF2DLPGHM operator in [15] also uses the power mean operator to change the weight of extreme evaluation value to eliminate the effect of extreme value. However, in this way, it may be difficult to distinguish the priority among the alternatives. In addition, although the method based on the WTF2DLPGHM operator uses the Hamy mean operator which can deal with the relationship between attributes, and contains variable parameters which can adapt to different DMs with different preferences. It also brings the corresponding disadvantage of obtaining unstable evaluation result. However, the ranking obtained by our method are easier to identify and distinguish, and more stable in response to the weights of attributes. More importantly, the results are more convincing. It is worth noting that this paper also proposes a combination weight model to determine the attribute weights, which not only considers the subjective weight of the DMs but also considers the objective weight information contained in the evaluation data; this makes the weights of attribute more real and effective, thus improving the accuracy of decision-making. In addition, the proposed weight model can be flexibly applied to different index systems and DMs with different distinguishing ability based on the flexible parameter .
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Primary Attributes | Subattributes |
---|---|
Design lean management A | Assembling design management A1 |
Standardized design management A2 | |
Personalized design management A3 | |
Design suitability management A4 | |
Component production and logistics lean management B | Component production standardization management B1 |
Component quality control management B2 | |
Logistics time management B3 | |
Component lossless transport management B4 | |
Construction lean management C | Construction mechanization management C1 |
Environmental construction management C2 | |
Construction process management C3 | |
Construction safety management C4 | |
Organizational collaborative lean management D | Organizational integration management D1 |
Organizational trust management D2 | |
Cooperative intention management D3 | |
Organizational collaboration technology management D4 | |
Information collaborative lean management E | Information accuracy management E1 |
Information delivery time management E2 | |
Information transfer cost management E3 |
([0.245, 0.267, 0.298, 0.321],) | ([0.256, 0.276, 0.281, 0.285],) | ([0.203, 0.237, 0.271, 0.305],) | |
([0.305, 0.343, 0.381, 0.419],) | ([0.312, 0.343, 0.365, 0.392],) | ([0.076, 0.114, 0.153, 0.191],) | |
([0.276, 0.322, 0.368, 0.414],) | ([0.356, 0.367, 0.373, 0.401],) | ([0.184, 0.230, 0.276, 0.322],) | |
([0.309, 0.347, 0.386, 0.425],) | ([0.398, 0.401, 0.412, 0.433],) | ([0.155, 0.193, 0.231, 0.270],) | |
([0.175, 0.219, 0.263, 0.307],) | ([0.350, 0.394, 0.438, 0.482],) | ([0.175, 0.219, 0.263, 0.307],) | |
([0.345, 0.389, 0.432, 0.475],) | ([0.234, 0.245, 0.259, 0.302],) | ([0.086, 0.130, 0.173, 0.216],) | |
([0.253, 0.284, 0.316, 0.348],) | ([0.312, 0.334, 0.345, 0.356],) | ([0.253, 0.284, 0.316, 0.348],) | |
([0.231, 0.270, 0.309, 0.347],) | ([0.356, 0.367, 0.381, 0.392],) | ([0.309, 0.347, 0.386, 0.424],) | |
([0.305, 0.343, 0.381, 0.419],) | ([0.305, 0.343, 0.381, 0.419],) | ([0.076, 0.114, 0.153, 0.191],) | |
([0.277, 0.312, 0.346, 0.381],) | ([0.346, 0.381, 0.381, 0.381],) | ([0.139, 0.173, 0.208, 0.242],) | |
([0.162, 0.203, 0.243, 0.284],) | ([0.162, 0.203, 0.243, 0.284],) | ([0.324, 0.365, 0.405, 0.446],) | |
([0.276, 0.322, 0.368, 0.414],) | ([0.184, 0.230, 0.276, 0.322],) | ([0.184, 0.230, 0.276, 0.322],) | |
([0.222, 0.259, 0.296, 0.333],) | ([0.296, 0.333, 0.370, 0.408],) | ([0.222, 0.259, 0.296, 0.333],) | |
([0.253, 0.284, 0.316, 0.348],) | ([0.253, 0.284, 0.316, 0.348],) | ([0.253, 0.284, 0.316, 0.348],) | |
([0.250, 0.313, 0.374, 0.437],) | ([0.250, 0.313, 0.374, 0.437],) | ([0.116, 0.174, 0.233, 0.291],) | |
([0.324, 0.365, 0.405, 0.446],) | ([0.162, 0.203, 0.243, 0.284],) | ([0.162, 0.203, 0.243, 0.284],) | |
([0.198, 0.248, 0.296, 0.346],) | ([0.245, 0.286, 0.327, 0.367],) | ([0.010, 0.050, 0.099, 0.149],) | |
([0.327, 0.367, 0.408, 0.449],) | ([0.209, 0.261, 0.312, 0.365],) | ([0.082,0.123,0.164,0.204],) | |
([0.364, 0.455, 0.545, 0.636],) | ([0.395, 0.445, 0.494, 0.544],) | ([0.209, 0.261, 0.312, 0.365],) |
Weights | Ranking | ||
---|---|---|---|
Subjective weights | |||
Combination weights | |||
Subjective weights | |||
Combination weights | |||
Subjective weights | |||
Combination weights |
Weights | Ranking | |
---|---|---|
Subjective weights | ||
Objective weights | ||
Combination weights |
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Li, Y.; Liu, Y. Extended VIKOR-QUALIFLEX Method Based on Trapezoidal Fuzzy Two-Dimensional Linguistic Information for Multiple Attribute Decision-Making with Unknown Attribute Weight. Mathematics 2021, 9, 37. https://doi.org/10.3390/math9010037
Li Y, Liu Y. Extended VIKOR-QUALIFLEX Method Based on Trapezoidal Fuzzy Two-Dimensional Linguistic Information for Multiple Attribute Decision-Making with Unknown Attribute Weight. Mathematics. 2021; 9(1):37. https://doi.org/10.3390/math9010037
Chicago/Turabian StyleLi, Ye, and Yisheng Liu. 2021. "Extended VIKOR-QUALIFLEX Method Based on Trapezoidal Fuzzy Two-Dimensional Linguistic Information for Multiple Attribute Decision-Making with Unknown Attribute Weight" Mathematics 9, no. 1: 37. https://doi.org/10.3390/math9010037
APA StyleLi, Y., & Liu, Y. (2021). Extended VIKOR-QUALIFLEX Method Based on Trapezoidal Fuzzy Two-Dimensional Linguistic Information for Multiple Attribute Decision-Making with Unknown Attribute Weight. Mathematics, 9(1), 37. https://doi.org/10.3390/math9010037