Abstract
Evaluating and selecting suitable sustainable recycling partners is a key work in sustainable supply chain management. In order to deal with the probabilistic linguistic influence relations between criteria and obtain the key factors that influence the evaluation results of sustainable recycling partners, we propose a new decision-making trial and evaluation laboratory (DEMATEL) method. First, we propose a new generalized weighted ordered weighted averaging (GWOWA) operator and discuss its properties. Second, we use probabilistic linguistic term sets (PLTSs) to aggregate the experts’ hesitant fuzzy linguistic decision-making information and develop a novel method of transforming PLTSs into triangular fuzzy numbers (TFNs) based on the proposed GWOWA operator and the characteristics of PLTSs. Furthermore, we propose a method of making criteria relation analysis based on DEMATEL with TFNs. With the method, we not only access the importance weights of criteria but also obtain the influence relation among the criteria and cluster the criteria into two groups: cause group and effect group. Finally, we apply our method to a real case of sustainable recycling partner selection.
Similar content being viewed by others
References
Addae, B. A., Zhang, L., Zhou, P., & Wang, F. D. (2019). Analyzing barriers of Smart Energy City in Accra with two-step fuzzy DEMATEL. Cities, 89, 218–227.
Cui, L., Chan, H. K., Zhou, Y. Z., Dai, J., & Lim, J. J. (2019). Exploring critical factors of green business failure based on Grey-Decision Making Trial and Evaluation Laboratory (DEMATEL). Journal of Business Research, 98, 450–461.
Gou, X. J., & Xu, Z. S. (2016). Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets. Information Sciences, 372, 407–427.
Han, W., Sun, Y. H., Xie, H., & Che, Z. M. (2017). Hesitant fuzzy linguistic group DEMTEL method with multi-granular evaluation scales. International Journal of Fuzzy Systems, 20, 1–15.
Kumar, A., & Dash, M. K. (2017). Causal modelling and analysis evaluation of online reputation management using fuzzy Delphi and DEMATEL. International Journal of Strategic Decision Sciences, 8, 27–45.
Li, S. P., Tang, D. B., & Wang, Q. (2019). Rating engineering characteristics in open design using a probabilistic language method based on fuzzy QFD. Computers & Industrial Engineering, 135, 348–358.
Li, P., & Wei, C. P. (2018). A case-based reasoning decision-making model for hesitant fuzzy linguistic information. International Journal of Fuzzy Systems, 20(7), 2175–2186.
Li, P., & Wei, C. P. (2019). An emergency decision-making method based on D–S evidence theory for probabilistic linguistic term sets. International Journal of Disaster Risk Reduction, 37, 101178.
Liao, H. C., Mi, X. M., & Xu, Z. S. (2019). A survey of decision-making methods with probabilistic linguistic information: Bibliometrics, preliminaries, methodologies, applications and future directions. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-019-09309-5.
Liao, H. C., Xu, Z. S., & Zeng, X. J. (2015). Hesitant fuzzy linguistic VIKOR method and its application in qualitative multiple criteria decision making. IEEE Transactions on Fuzzy Systems, 23(5), 1343–1355.
Liu, H. B., & Rodríguez, R. M. (2014). A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making. Information Sciences, 258, 220–238.
Pang, Q., Wang, H., & Xu, Z. (2016). Probabilistic linguistic term sets in multi-attribute group decision making. Information Sciences, 369, 128–143.
Ramos, T. R. P., Gomes, M. I., & Barbosa-Póvoa, A. P. (2014). Planning a sustainable reverse logistics system: balancing costs with environmental and social concerns. Omega, 48, 60–74.
Rodríguez, R. M., Martínez, L., & Herrera, F. (2012). Hesitant fuzzy linguistic terms sets for decision making. IEEE Transactions on Fuzzy Systems, 20, 109–119.
Sharma, V., Kumar, R., & Kumar, R. (2017). QUAT-DEM: Quaternion-DEMATEL based neural model for mutual coordination between UAVs. Information Sciences, 418–419, 74–90.
Song, W. Y., & Cao, J. T. (2017). A rough DEMATEL-based approach for evaluating interaction between requirements of product-service system. Computers & Industrial Engineering, 110, 353–363.
Torra, V. (1997). The weighted OWA operator. International Journal of Intelligent Systems, 12, 153–166.
Wang, Y. M., Chin, K. S., Poon, G. K. K., & Yang, J. B. (2009). Risk evaluation in failure mode and effects analysis using fuzzy weighted geometric mean. Expert Systems with Applications, 36, 1195–1207.
Xu, Z. S., He, Y., & Wang, X. Z. (2019). An overview of probabilistic-based expressions for qualitative decision-making: Techniques, comparisons and developments. International Journal of Machine Learning and Cybernetics, 10(6), 1513–1528.
Yager, R. R. (1981). A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, 24, 143–161.
Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems, Man, and Cybernetics, 18, 183–190.
Yager, R. R., & Alajlan, N. (2016). Some issues on the OWA aggregation with importance weighted arguments. Knowledge-Based Systems, 100, 89–96.
Zhang, Y. X., Xu, Z. S., & Liao, H. C. (2017). A consensus process for group decision making with probabilistic linguistic preference relations. Information Sciences, 414, 260–275.
Zhang, Z., Yu, W., Martinez, L., & Gao, Y. (2020). Managing multigranular unbalanced hesitant fuzzy linguistic information in multiattribute large-scale group decision making: A linguistic distribution-based approach. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/tfuzz.2019.2949758.
Zhou, F. L., Wang, X., Lim, M. K., He, Y. D., & Li, L. X. (2018). Sustainable recycling partner selection using fuzzy DEMATEL-AEWFVIKOR: A case study in small-and-medium enterprises (SMEs). Journal of Cleaner Production, 196, 489–500.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Project No. 71971190) and Ministry of Education Foundation of Humanities and Social Sciences (No. 19YJA630039).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Theorem 1
Let \( a_{1} ,a_{2} , \ldots ,a_{n} \) be a series of positive numbers. If \( a_{1} = a_{2} = \cdots = a_{n} = a \), then
Proof
Since \( a_{1} = a_{2} = \cdots = a_{n} = a \), we have \( f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) = (\sum\nolimits_{i = 1}^{n} {v_{i} a_{{}}^{\lambda } } )^{{\frac{1}{\lambda }}} \). Owing to \( \sum\nolimits_{i = 1}^{n} {v_{i} } = 1 \), we can obtain \( (\sum\nolimits_{i = 1}^{n} {v_{i} a_{{}}^{\lambda } } )^{{\frac{1}{\lambda }}} = a \).□
Theorem 2
Let \( a_{1} ,a_{2} , \ldots ,a_{n} \) and \( b_{1} ,b_{2} , \ldots ,b_{n} \) be two arbitrary series of positive numbers. If \( a_{j} \ge b_{j} \)(\( j = 1,2, \ldots ,n \)), then \( f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \ge f_{GWOWA}^{P,W} (b_{1} ,b_{2} , \ldots ,b_{n} ) \).
Proof
Since \( a_{j} \ge b_{j} \)(\( j = 1,2, \ldots ,n \)), we have \( a_{j}^{\lambda } \ge b_{j}^{\lambda } \). According to Torra (1997), we can obtain \( f_{WOWA}^{P,W} (a_{1}^{\lambda } ,a_{2}^{\lambda } , \ldots ,a_{n}^{\lambda } ) \ge f_{WOWA}^{P,W} (b_{1}^{\lambda } ,b_{2}^{\lambda } , \ldots ,b_{n}^{\lambda } ) \). Therefore, we have \( [f_{WOWA}^{P,W} (a_{1}^{\lambda } ,a_{2}^{\lambda } , \ldots ,a_{n}^{\lambda } )]^{{\frac{1}{\lambda }}} \ge [f_{WOWA}^{P,W} (b_{1}^{\lambda } ,b_{2}^{\lambda } , \ldots ,b_{n}^{\lambda } )]^{{\frac{1}{\lambda }}} \). In other words, \( f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \ge f_{GWOWA}^{P,W} (b_{1} ,b_{2} , \ldots ,b_{n} ) \) holds.□
Theorem 3
Let \( a_{1} ,a_{2} , \ldots ,a_{n} \) be a series of positive numbers, then
Proof
Let \( \hbox{min} \{ a_{1} ,a_{2} , \ldots ,a_{n} \} = a^{ - } \) and \( \hbox{max} \{ a_{1} ,a_{2} , \ldots ,a_{n} \} = a^{ + } \). We can easily obtain \( f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \le (\sum\nolimits_{i = 1}^{n} {v_{i} (a^{ + } )^{\lambda } } )^{{\frac{1}{\lambda }}} = a^{ + } \). Similarly, we can obtain \( f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \ge (\sum\nolimits_{i = 1}^{n} {v_{i} (a^{ - } )^{\lambda } } )^{{\frac{1}{\lambda }}} = a^{ - } \).
Therefore, \( \hbox{min} \{ a_{1} ,a_{2} , \ldots a_{n} \} \le f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \le \hbox{max} \{ a_{1} ,a_{2} , \ldots ,a_{n} \} \).□
Theorem 4
(1) If \( \lambda = 1 \), then the GWOWA operator reduces to the WOWA operator.
Furthermore, if \( p_{1} = p_{2} = \cdots = p_{n} = \frac{1}{n} \), then the GWOWA operator reduces to the OWA operator; and if \( w_{1} = w_{2} = \cdots = w_{n} = \frac{1}{n} \), then the GWOWA operator reduces to the weighted average operator.
(2) If \( \lambda \to + \infty \), then the GWOWA operator reduces to the max operator, that is,
Proof
(1) When \( \lambda = 1 \), then \( f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) = (\sum\nolimits_{i = 1}^{n} {v_{i} b_{i}^{\lambda } } )^{{\frac{1}{\lambda }}} = \sum\nolimits_{i = 1}^{n} {v_{i} b_{i} } = f_{WOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \).
Furthermore, when \( p_{1} = p_{2} = \cdots = p_{n} = \frac{1}{n} \), then \( v_{i} = w^{*} (\sum\nolimits_{j = 1}^{i} {p_{\sigma (j)} } ) - w^{*} (\sum\nolimits_{j = 1}^{i - 1} {p_{\sigma (j)} } ) = \sum\nolimits_{j \ge i}^{{}} {w_{j} } - \sum\nolimits_{j < i}^{{}} {w_{j} } = w_{i} \). Therefore, the GWOWA operator reduces to the OWA operator.
When \( w_{1} = w_{2} = \cdots = w_{n} = \frac{1}{n} \), then \( w^{*} (x) = x \). Therefore, we have \( v_{i} = w^{*} (\sum\nolimits_{j = 1}^{i} {p_{\sigma (j)} } ) - w^{*} (\sum\nolimits_{j = 1}^{i - 1} {p_{\sigma (j)} } ) = p_{\sigma (i)} \).
Then the GWOWA operator reduces to the weighted average operator.
(2) When \( \lambda \to + \infty \), \( f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) = \mathop {\lim }\nolimits_{\lambda \to \infty } (\sum\nolimits_{i = 1}^{n} {v_{i} (b_{i} )^{\lambda } )^{{\frac{1}{\lambda }}} } = \mathop {\hbox{max} }\nolimits_{j \in N} \{ a_{j} \} \).□
Theorem 5
Let \( a_{1} ,a_{2} , \ldots ,a_{n} \) be a series of positive numbers, \( \eta \) be an arbitrary sequence in \( N = \{ 1,2,3 \ldots ,n\} \), and \( f_{\eta } (a_{1} ,a_{2} , \ldots ,a_{n} ) = (\sum\nolimits_{j \in N}^{{}} {v_{j} a_{{_{\eta (j)} }}^{\lambda } } )^{{\frac{1}{\lambda }}} \). If \( w_{1} \ge w_{2} \ge \cdots \ge w_{n} \), then
Proof
(1) Without loss of generality we assume that \( a_{1} \ge a_{2} \ge \cdots \ge a_{n} \). Given two elements \( a_{\eta (i)} \) and \( a_{\eta (i + 1)} \), we assume that \( a_{\eta (i)} \le a_{\eta (i + 1)} \) in sequence \( \eta \). We interchange the two elements \( a_{\eta (i)} \) and \( a_{\eta (i + 1)} \), and denote this by sequence \( \eta^{'} \). We easily obtain \( f_{\eta } (a_{1} , \ldots ,a_{n} ) - f_{{\eta^{'} }} (a_{1} , \ldots ,a_{n} ) = v_{i} a_{{^{{_{\eta (i)} }} }}^{\lambda } + v_{i + 1} a_{\eta (i + 1)}^{\lambda } - v_{i}^{'} a_{\eta (i + 1)}^{\lambda } - v_{i + 1}^{'} a_{{^{{_{\eta (i)} }} }}^{\lambda } = a_{\eta (i + 1)}^{\lambda } (v_{i + 1} - v_{i}^{'} ) - a_{{^{{_{\eta (i)} }} }}^{\lambda } (v_{i + 1}^{'} - v_{i} ) \).
Because \( v_{i} + v_{i + 1} = v_{i}^{'} + v_{i + 1}^{'} \) holds, we have \( v_{i + 1} - v_{i}^{'} = v_{i + 1}^{'} - v_{i} \).
Because \( w_{1} \ge w_{2} \ge \cdots \ge w_{n} \), function \( w^{*} \) is concave. Then we obtain \( \frac{{v_{i + 1} }}{{p_{\eta (i + 1)} }} \le \frac{{v_{i}^{'} }}{{p_{\eta (i + 1)} }} \). Because \( p_{\eta (i + 1)} > 0 \), we obtain \( v_{i + 1} - v_{i}^{'} \le 0 \). Therefore, we have \( f_{\eta } (a_{1} , \ldots ,a_{n} ) - f_{{\eta^{'} }} (a_{1} , \ldots ,a_{n} )_{{^{{_{\eta (i)} }} }}^{\lambda } \ge 0 \) and \( f_{\eta } (a_{1} ,a_{2} , \ldots ,a_{n} ) \le f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \).
(2) Because function \( w^{*} \) is concave, we have the slope \( \frac{{w_{1} }}{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 n}}\right.\kern-0pt} \!\lower0.7ex\hbox{$n$}}}} \ge \frac{{v_{i} }}{{p_{\sigma (i)} }} \). Hence,
Therefore, \( f_{\eta } (a_{1} ,a_{2} , \ldots ,a_{n} ) \le f_{GWOWA}^{P,W} (a_{1} ,a_{2} , \ldots ,a_{n} ) \le (nw_{1} \sum\nolimits_{i = 1}^{n} {p_{i} a_{i}^{\lambda } } )^{{\frac{1}{\lambda }}} \) holds.□
Rights and permissions
About this article
Cite this article
Li, P., Liu, J. & Wei, C. Factor relation analysis for sustainable recycling partner evaluation using probabilistic linguistic DEMATEL. Fuzzy Optim Decis Making 19, 471–497 (2020). https://doi.org/10.1007/s10700-020-09326-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-020-09326-9