Supplier Selection Model Based on D Numbers and Transformation Function


  • Leihui Xiong Shanghai University of Electric Power, China
  • Xiaoyan Su Shanghai University of Electric Power, China
  • Hong Qian Shanghai University of Electric Power, China



D numbers, transformation functions, analytic hierarchy process, fuzzy preference relation


Selecting reasonable suppliers can effectively improve the efficiency of enterprise supply chain management. Among them, expert evaluation is an important part of supplier selection problem, but the uncertainty, fuzziness and incompleteness of expert opinions make supplier selection problem difficult to solve. In order to systematically and effectively solve the uncertainty, ambiguity and incompleteness in supplier selection problem, this paper presents a new supplier selection method based on D numbers and transformation function. First, fuzzy preference relation is generated based on the decision matrix of pairwise comparisons given by experts. D numbers which can effectively deal with uncertain information extend fuzzy preference relation (D matrix). Second, the D matrix is converted into a crisp matrix form based on the integration representation of D numbers according to different situations whether or not the information in D matrix is complete. Third, the crisp matrix is converted into judgement matrix by using the transformation functions. Finally, analytic hierarchy process (AHP) method is applied based on the judgment matrix to give a priority weights for decision making. Three numerical examples and application of the supplier selection are used to show the feasibility and effectiveness of the proposed method.


Pelissari, R.; Khan, S.; Ben-Amor, S.(2021). Application of Multi-Criteria Decision-Making Methods in Sustainable Manufacturing Management: A Systematic Literature Review and Analysis of the Prospects, International Journal of Information Technology & Decision Making, 2(21),493- 515, 2021.

Zheng, Y.; Zheng, J.(2021). A novel portfolio optimization model via combining multi-objective optimization and multi-attribute decision making, Applied Intelligence, 52(5),5684-5695, 2021.

Fu, S.; Xiao, Y. Z.; Zhou, H. J.; Liu, S. Z.(2021). Venture capital project selection based on interval number grey target decision model, Soft Computing, 25(6),4865-4874, 2021.

Deng X.; Kong Z.(2021). Humanitarian rescue scheme selection under the COVID-19 crisis in China: Based on group decision-making method, Symmetry, 13(4),668, 2021.

Wu Y.; Deng Z.; Tao Y.(2021). Site selection decision framework for photovoltaic hydrogen production project using BWM-CRITIC-MABAC: A case study in Zhangjiakou, Journal of Cleaner Production, 324, 2021.

Emeksiz C, Yüksel A.(2022). A suitable site selection for sustainable bioenergy production facility by using hybrid multi-criteria decision making approach, case study: Turkey, Fuel, 315, 2022.

Konneh K.V., Masrur H., Othman M.L.(2021). Multi-Attribute decision-Making approach for a cost-Effective and sustainable energy system considering weight assignment analysis, Sustainability, 13(10),5615,2021.

Bai C.; Kusi-Sarpong S.; Badri A.H.(2019). Social sustainable supplier evaluation and selection: a group decision-support approach, International Journal of Production Research, 57(22), 7046- 7067, 2019.

Schramm V.B.; Cabral L.; Schramm F.(2020). Approaches for supporting sustainable supplier selection-A literature review, Journal of Cleaner Production, 273, 2020.

Hruška R.; Pruša P.; Babic D.(2014). The use of AHP method for selection of supplier, Transport, 29(2), 195-203, 2014.

Liu M.; Wang X.; Li Y.(2022). Service supplier selection under fuzzy and stochastic uncertain environments, Journal of Intelligent & Fuzzy Systems, 3(42),1301-1315,2022.

Pelissari R.; Oliveira M.C.; Abackerli A.J.(2021). Techniques to model uncertain input data of multi-criteria decision-making problems: a literature review, International Transactions in Operational Research, 28(2): 523-559,2021.

Rashidi, K.; Cullinane, K. (2018). A Comparison of Fuzzy DEA and Fuzzy TOPSIS in Sustainable Supplier Selection: Implications for Sourcing Strategy, Expert Systems with Applications, 121, 2018.

Chen, Z.; Ming, X.; Zhou, T.; Chang, Y. (2020). Sustainable supplier selection for smart supply chain considering internal and external uncertainty: An integrated rough-fuzzy approach, Applied Soft Computing, 87, 2020.

Xing, Y.;Cao M.; Liu Y.; Zhou M. ; Jian Wu (2022). A Choquet integral based interval Type-2 trapezoidal fuzzy multiple attribute group decision making for Sustainable Supplier Selection, Computers & Industrial Engineering, 165, 2022.

Hekmat, S.; Amiri, M.; Madraki, G.;Shi, Y.(2021). Strategic Supplier Selection in Payment Industry: A Multi-Criteria Solution for Insufficient and Interrelated Data Sources, International Journal of Information Technology & Decision Making, 6(20),1711-1745, 2021.

Babak A. ; Meysam R. ; Madjid T.(2021). An integrated information fusion and grey multicriteria decision-making framework for sustainable supplier selection, International Journal of Systems Science: Operations & Logistics, 4(8),348-370, 2021.

Sureeyatanapas, P.;Waleekhajornlert, N.; Arunyanart, S.; Niyamosoth, T.(2020). Resilient supplier selection in electronic components procurement: An integration of evidence theory and rule-based transformation into TOPSIS to tackle uncertain and incomplete information, Symmetry, 7(12),1109, 2020.

Zhang, X.; Deng, Y.; Chan, F.; Adamatzky, A.; Mahadevan, S.(2016). Supplier selection based on evidence theory and analytic network process, Proceedings of the institution of Mechanical Engineers, Part B: Journal of Engineering manufacture, 3(230),562-573, 2016.

Fei, L.; Xia, J.;Feng, Y.; Liu, L.(2019). An ELECTRE-based multiple criteria decision making method for supplier selection using Dempster-Shafer theory, IEEE Access, 7,2019.

Liu, PD.; Zhang, XH.(2019). A novel approach to multi-criteria group decision-making problems based on linguistic D numbers, Computational & Applied Mathematics, 39(2), 132, 2020.

Dempster, A. P. (1967). Upper and Lower Probabilities Induced by a Multivalued Mapping, The Annals of Mathematical Statistics, 38(2), 325 - 339, 1967.

Shafer, G. (1976). A mathematical theory of evidence, Princeton university Press, 1976.

Guan, X.; Liu, HQ.; Yi, X.; Zhao, J.(2018). The improved combination rule of D numbers and its application in radiation source identification, Mathematical Problems in Engineering, 15(4), 15501477198, 2018.

Deng, XY.; Jiang, W.(2019). A total uncertainty measure for D numbers based on belief intervals, International Journal of Intelligent Systems, 34(12), 3302-3316, 2019.

Xia, J; Feng, YQ.; Liu, LN.; Liu, DJ.; Fei, LG.(2019). On entropy function and reliability indicator for D numbers, Applied Intelligence, 49(9), 3248-3266, 2019.

Li, XH.; Chen, XH.(2018). D-intuitionistic hesitant fuzzy sets and their application in multiple attribute decision making, Cognitive Computation, 10(3), 496-505, 2018.

Zhang, J.; Zhong, DH.; Zhao, MQ.; Yu, J.; Lv, F.(2019). An optimization model for construction stage and zone plans of rockfill dams based on the enhanced whale optimization algorithm, Energies, 12(3), 466, 2019.

Mo, HM.(2021). A SWOT method to evaluate safety risks in life cycle of wind turbine extended by D number theory, Journal of Intelligent & Fuzzy Systems, 40(3), 4439-4452, 2021.

Mo, HM.(2020). A New Evaluation Methodology for Quality Goals Extended by D Number Theory and FAHP, Journal of Intelligent & Fuzzy Systems, 11(4), 206, 2020.

Mo, HM.(2020). An Emergency Decision-Making Method for Probabilistic Linguistic Term Sets Extended by D Number Theory, Symmetry, 12(3), 380, 2020.

Deng, XY.; Jiang, W.(2019). Evaluating green supply chain management practices under fuzzy environment: a novel method based on D number theory, International Journal of Fuzzy Systems, 21(5), 1389-1402, 2019.

Deng, XY.; Jiang, W.(2019). D number theory based game-theoretic framework in adversarial decision making under a fuzzy environment, International Journal of Approximate Reasoning, 106(1), 194-213, 2019.

Mi, XJ.; Tian, Y.; Kang, BY.(2021). A hybrid multi-criteria decision making approach for assessing health-care waste management technologies based on soft likelihood function and D-numbers, Applied Intelligence, 15(2), 1-20, 2021.

Mousavi-Nasab, S.H.; Sotoudeh-Anvari, A.(2020). An extension of best-worst method with D numbers: Application in evaluation of renewable energy resources, Sustainable Energy Technologies and Assessments, 40,2020.

Seiti, H.; Hafezalkotob, A.; Herrera-Viedma, E.(2020). A novel linguistic approach for multigranular information fusion and decision-making using risk-based linguistic D numbers, Information Sciences, 530, 43-65, 2020.

Tian, Y.; Mi, XJ.; Liu, LL.; Kang, BY.(2020). A New Soft Likelihood Function Based on D Numbers in Handling Uncertain Information, International Journal of Fuzzy Systems, 22(7), 31- 34, 2020.

Xiao, FY. (2019). A Multiple-Criteria Decision-Making Method Based on D Numbers and Belief Entropy, International Journal of Fuzzy Systems, 21(4), 1144-1153, 2019.

Deng, XY.; Hu, Y.; Deng, Y.; Sankaran M.(2014). Supplier selection using AHP methodology extended by D numbers, Expert Systems with Application, 41(1), 156-167, 2014.

Deng, XY.; Deng, Y.(2019). D-AHP method with different credibility of information, Soft Computing, 23(2), 683-691, 2019.

Fei, L.(2020). D-ANP: a multiple criteria decision making method for supplier selection, Applied Intelligence, 8(50),2537-2554,2020.

Herrera Viedma, E.; Herrera, F.; Chiclana, F.; Luque, M.(2004). Some issues on consistency of fuzzy preference relations, European journal of operational research, 154(1), 98-109, 2004.

Tanino, T.(1984). Fuzzy preference orderings in group decision making, Fuzzy sets and systems, 12(2), 117-131, 1984.

Felix, T.S.C.; Niraj, K.(2007). Global supplier development considering risk factors using fuzzy extended AHP-based approach, Omega: The international journal of management science, 35(4), 417-431, 2007.

Saaty, T.L.(1982). The analytic hierarchy process: A new approach to deal with fuzziness in architecture, Architectural Science Review, 25(3), 64-69, 1982.

Additional Files



Most read articles by the same author(s)

Obs.: This plugin requires at least one statistics/report plugin to be enabled. If your statistics plugins provide more than one metric then please also select a main metric on the admin's site settings page and/or on the journal manager's settings pages.