Triangular Fuzzy Neutrosophic Preference Relations and Their Application in Enterprise Resource Planning Software Selection

  • Fanyong Meng
  • Na WangEmail author
  • Yanwei Xu


Enterprise resource planning (ERP) system selection is one of the most important topics in an ERP implementation program that ensures the success of the system. Because of the inherent complexity of ERP systems, it is difficult and time consuming for the organization to select the suitable ERP software. This paper employs triangular fuzzy neutrosophic preference relations (TFNPRs) to express the recognitions of decision-makers (DMs) for the choice of ERP software. Preference relation is a powerful tool to express complex decision problems, and the triangular fuzzy neutrosophic number (TFNN) is a good choice to represent the recognitions of the DMs; this paper combines preference relation with TFNN to define the concept of triangular fuzzy neutrosophic preference relations (TFNPRs). To rank the evaluated ERP systems logically, a multiplicative consistency concept for TFNPRs is defined. Then, several multiplicative consistency-based 0–1 mixed programming models are established for estimating missing values in incomplete TFNPRs and for deriving multiplicatively consistent TFNPRs from inconsistent ones, respectively. For group decision-making (GDM), a Manhattan distance measure-based consensus index is defined to measure the agreement degrees of the DMs’ opinions. A multiplicative consistency and consensus-based algorithm to GDM with TFNPRs is provided that can cope with incomplete and inconsistent TFNPRs. Meanwhile, an illustrative example about the selection of ERP software is offered to show the utilization of the new method, and comparison analysis is performed with several previous methods about ERP software selection. The new method adopts TFNPRs that can express the fuzzy truth-membership degree, the fuzzy indeterminacy-membership degree and the fuzzy falsity-membership degree of the recognitions of the DMs. It extends the application of preference relations and endows the DMs with more flexibility to denote their recognitions. Furthermore, the new method is based on the multiplicative consistency and consensus analysis that ensures the rational and representative ranking of the considered objects.


ERP software selection Group decision-making Triangular fuzzy neutrosophic preference relation Multiplicative consistency Consensus 



This study was funded by the National Natural Science Foundation of China (Nos. 71571192, 71874112, and 71771025), the Innovation-Driven Project of Central South University (No. 2018CX039), the Beijing Intelligent Logistics System Collaborative Innovation Center (No. 2019KF-09), the Major Project for National Natural Science Foundation of China (Nos. 91846301, 71790615), and the State Key Program of National Natural Science of China (No. 71431006).

Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed Consent

Informed consent was obtained from all individual participants included in the study.


  1. 1.
    Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986;20(1):87–96.Google Scholar
  2. 2.
    Al-Mashari M, Al-Mudimigh A, Zairi M. Enterprise resource planning: a taxonomy of critical factors. Eur J Oper Res. 2003;146(2):352–65.Google Scholar
  3. 3.
    Atanassov KT, Gargov G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989;31(3):343–9.Google Scholar
  4. 4.
    Asl MB, Khalilzadeh A, Youshanlouei HR, Mood MM. Identifying and ranking the effective factors on selecting enterprise resource planning (ERP) system using the combined Delphi and Shannon entropy approach. Procedia-Soc Behav Sci. 2012;41:513–20.Google Scholar
  5. 5.
    Ayag Z, Özdemir RG. An intelligent approach to ERP software selection through fuzzy ANP. Int J Prod Res. 2007;45(10):2169–94.Google Scholar
  6. 6.
    Biswas P, Pramanik S, Giri BC. TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput & Applic. 2016a;27(3):727–37.Google Scholar
  7. 7.
    Biswas P, Pramanik S, Giri BC. Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making. Neutrrosophic Sets Syst. 2016b;12(1):20–40.Google Scholar
  8. 8.
    Bausys R, Zavadskas EK. Multicriteria decision making approach by VIKOR under interval neutrosophic set environment. Econ Comput Econ Cy Stud Res. 2017;49(4):33–48.Google Scholar
  9. 9.
    Çakır S. Selecting appropriate ERP software using integrated fuzzy linguistic preference relations–fuzzy TOPSIS method. Int J Comput Intell Syst. 2016;9(3):433–49.Google Scholar
  10. 10.
    Chen SM, Lin TE, Lee LW. Group decision making using incomplete fuzzy preference relations based on the additive consistency and the order consistency. Inf Sci. 2014;259:14):1–15.Google Scholar
  11. 11.
    Czubenko M, Kowalczuk Z, Ordys A. Autonomous driver based on an intelligent system of decision-making. Cogn Comput. 2015;7(5):569–81.Google Scholar
  12. 12.
    Daher SDFD, Almeida ATD. Group preference aggregation based on ELECTRE methods for ERP system selection. Springer Berlin Heidelberg. 2013;139:215–222.Google Scholar
  13. 13.
    Efe B. An integrated fuzzy multi criteria group decision making approach for ERP system selection. Appl Soft Comput. 2016;38(3):106–17.Google Scholar
  14. 14.
    Gürbüz T, Alptekin SE, Alptekin GI. A hybrid MCDM methodology for ERP selection problem with interacting criteria. Decis Support Syst. 2012;54(1):206–14.Google Scholar
  15. 15.
    Guo YH, Cheng HD, Zhang Y. A new neutrosophic approach to image denoising. New Math Nat Comput. 2009;5(3):653–62.Google Scholar
  16. 16.
    Gülçin B, Da R. Evaluation of software development projects using a fuzzy multi-criteria decision approach. Math Comput Simul. 2008;77(5):464–75.Google Scholar
  17. 17.
    Guo YH, Şengur A. A novel image edge detection algorithm based on neutrosophic set. Comput Electr Eng. 2014;40(8):3–25.Google Scholar
  18. 18.
    Jafarnejad A, Ansari M, Youshanlouei HR, Mood MM. A hybrid MCDM approach for solving the ERP system selection problem with application to steel industry. Int J Enterp Inf Syst. 2012;8(3):54–73.Google Scholar
  19. 19.
    Kacprzyk J. Group decision making with a fuzzy linguistic majority. Fuzzy Sets Syst. 1986;18(2):105–18.Google Scholar
  20. 20.
    Kim SH, Ahn BS. Group decision making procedure considering preference strength under incomplete information. Comput Oper Res. 1997;24(12):1101–12.Google Scholar
  21. 21.
    Kara SS, Cheikhrouhou N. A multi criteria group decision making approach for collaborative software selection problem. J Intell Fuzzy Syst. 2014;26(1):37–47.Google Scholar
  22. 22.
    Karsak EE, Özogul CO. An integrated decision making approach for ERP system selection. Expert Syst Appl. 2009;36(1):660–7.Google Scholar
  23. 23.
    Liao X, Li Y, Lu B. A model for selecting an ERP system based on linguistic information processing. Inf Syst. 2007;32(7):1005–17.Google Scholar
  24. 24.
    Liu P, Tang G. Multi-criteria group decision-making based on interval neutrosophic uncertain linguistic variables and choquet integral. Cogn Comput. 2016;8(6):1036–56.Google Scholar
  25. 25.
    Laurent PA. A neural mechanism for reward discounting: insights from modeling hippocampal–striatal interactions. Cogn Comput. 2013;5(1):152–60.Google Scholar
  26. 26.
    Liang RX, Wang JQ, Zhang HY. Evaluation of e-commerce websites: an integrated approach under a single-valued trapezoidal neutrosophic environment. Knowl-Based Syst. 2017;135:44–59.Google Scholar
  27. 27.
    Millet I. The effectiveness of alternative preference elicitation methods in the analytic hierarchy process. J Multi-Criteria Decis Anal. 1997;6(1):41–51.Google Scholar
  28. 28.
    Meng FY, Chen XH. A robust additive consistency-based method for decision making with triangular fuzzy reciprocal preference relations. Fuzzy Optim Decis Ma. 2018;17(1):49–73.Google Scholar
  29. 29.
    Meng FY, Tang J. New ranking order for linguistic hesitant fuzzy sets. J Oper Res Soc. 2018:1–10.
  30. 30.
    Meng FY, Tang J, An QX, Chen XH. Decision making with intuitionistic linguistic preference relations. Int Trans Oper Res. 2019a.
  31. 31.
    Meng FY, Tang J, Fujita H. Consistency-based algorithms for decision making with interval fuzzy preference relations. IEEE Trans Fuzzy Syst. 2019b:1.
  32. 32.
    Motaki N, Kamach O. ERP selection: a step-by-step application of AHP method. Int J Comput Appl. 2017;176(7):15–21.Google Scholar
  33. 33.
    Meng FY, Lin J, Tan CQ, Zhang Q. A new multiplicative consistency based method for decision making with triangular fuzzy reciprocal preference relations. Fuzzy Sets Syst. 2017a;315:1–25.Google Scholar
  34. 34.
    Méxas MP, Quelhas OLG, Costa HG. Prioritization of enterprise resource planning systems criteria: focusing on construction industry. Int J Prod Econ. 2012;139(1):340–50.Google Scholar
  35. 35.
    Meng FY, Tan CQ, Chen XH. Multiplicative consistency analysis for interval fuzzy preference relations: a comparative study. Omega. 2017b;68:17–38.Google Scholar
  36. 36.
    Meng FY, Chen XH. Correlation coefficients of hesitant fuzzy sets and their application based on fuzzy measures. Cogn Comput. 2015;7(4):445–63.Google Scholar
  37. 37.
    Meng FY, Wang C, Chen XH. Linguistic interval hesitant fuzzy sets and their application in decision making. Cogn Comput. 2016;8(1):52–68.Google Scholar
  38. 38.
    Ma YX, Wang JQ, Wang J, Wu XH. An interval neutrosophic linguistic multi-criteria group decision-making method and its application in selecting medical treatment options. Neural Comput & Applic. 2017;28(9):2745–65.Google Scholar
  39. 39.
    Nie RX, Wang JQ, Li L. 2-tuple linguistic intuitionistic preference relation and its application in sustainable location planning voting system. J Intell Fuzzy Syst. 2017;33(2):885–99.Google Scholar
  40. 40.
    Orlovsky SA. Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1978;1(3):155–67.Google Scholar
  41. 41.
    Pramanik S, Biswas P, Giri BC. Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment. Neural Comput & Applic. 2017;28(5):1163–76.Google Scholar
  42. 42.
    Pramanik S, Chackrabarti S. A study on problems of construction workers in West Bengal based on neutrosophic cognitive maps. Int J Innov Res Sci Eng Technol. 2013;2(11):6387–94.Google Scholar
  43. 43.
    Saaty TL. The analytic hierarchy process. New York: McGraw-Hill; 1980.Google Scholar
  44. 44.
    Smarandache F. A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic. Rehoboth: American Research Press; 1999.Google Scholar
  45. 45.
    Smarandache F. An introduction to the neutrosophic probability applied in quantum physics. Math. 2000;22d(1):13–25.Google Scholar
  46. 46.
    Szmidt E, Kacprzyk J. A consensus-reaching process under intuitionistic fuzzy preference relations. Int J Intell Syst. 2003;18(7):837–52.Google Scholar
  47. 47.
    Sun B, Ma W. An approach to consensus measurement of linguistic preference relations in multi-attribute group decision making and application. Omega. 2015;51(1):83–92.Google Scholar
  48. 48.
    Salama AA, Smarandache F, Kroumov V. Neutrosophic crisp sets and neutrosophic crisp topological spaces. Neutrosophic Sets Syst. 2014;32:24–30.Google Scholar
  49. 49.
    Saaty TL, Vargas LG. Uncertainty and rank order in the analytic hierarchy process. Eur J Oper Res. 1987;32(1):107–17.Google Scholar
  50. 50.
    Turksen I. Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 1986;20(2):191–210.Google Scholar
  51. 51.
    Tang J, Chen SM, Meng FY. Heterogeneous group decision making in the setting of incomplete preference relations. Inf Sci. 2019;483:396–418.Google Scholar
  52. 52.
    Tang J, Meng FY. Ranking objects from group decision making with interval-valued hesitant fuzzy preference relations in view of additive consistency and consensus. Knowl-Based Syst. 2018a;162:46–61.Google Scholar
  53. 53.
    Tang J, Meng FY. An approach to interval-valued intuitionstic fuzzy decision making based on induced generalized symmetrical Choquet-Shapley operator. Sci Iran. 2018b;25:1456–70.Google Scholar
  54. 54.
    Tang J, Meng FY, Li CL, Li CH. A consistency-based approach to group decision making with uncertain multiplicative linguistic fuzzy preference relations. J Intell Fuzzy Syst. 2018a;35:1037–54.Google Scholar
  55. 55.
    Tian ZP, Wang J, Wang JQ, Zhang HY. A likelihood-based qualitative flexible approach with hesitant fuzzy linguistic information. Cogn Comput. 2016;8(4):670–83.Google Scholar
  56. 56.
    Tong XY, Wang ZJ. A group decision framework with intuitionistic preference relations and its application to low carbon supplier selection. Int J Environ Res Public Health. 2016;13(9):923.Google Scholar
  57. 57.
    Tang J, Meng FY, Zhang YL. Decision making with interval-valued intuitionistic fuzzy preference relations based on additive consistency analysis. Inf Sci. 2018b;467:115–34.Google Scholar
  58. 58.
    Van Laarhoven PJM, Pedrycz W. A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst. 1983;11:229–41.Google Scholar
  59. 59.
    Wan SP, Dong JY. Possibility method for triangular intuitionistic fuzzy multi-attribute group decision making with incomplete weight information. Int J Comput Intell Syst. 2014;7(1):65–79.Google Scholar
  60. 60.
    Wang YM, Elhag TMS, Hua Z. A modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process. Fuzzy Sets Syst. 2006;157(23):3055–71.Google Scholar
  61. 61.
    Wang HB, Smarandache F, Zhang YQ, Sunderraman R. Interval neutrosophic sets and logic: theory and applications in computing. Comput Therm Sci. 2005;65(4):87.Google Scholar
  62. 62.
    Wang HB, Smarandache F, Zhang YQ, Sunderraman R. Single valued neutrosophic sets. Multispace Multistruct. 2010;4:410–3.Google Scholar
  63. 63.
    Xu ZS. On compatibility of interval fuzzy preference relations. Fuzzy Optim Decis Ma. 2004;3(3):217–25.Google Scholar
  64. 64.
    Xu ZS. A method for priorities of triangular fuzzy number complementary judgment matrices. Fuzzy Syst Math. 2002;16(1):47–50.Google Scholar
  65. 65.
    Xu ZS. Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst. 2007a;15(6):1179–87.Google Scholar
  66. 66.
    Xu ZS. Intuitionistic preference relations and their application in group decision making. Inf Sci. 2007b;177(11):2363–79.Google Scholar
  67. 67.
    Xu ZS. Consistency of interval fuzzy preference relations in group decision making. Appl Soft Comput. 2011;11(5):3898–909.Google Scholar
  68. 68.
    Xu ZS, Chen J. On geometric aggregation over interval-valued intuitionistic fuzzy information. Pro 4 Int con Fuzzy Syst Knowl Dis. 2007;2:466–71.Google Scholar
  69. 69.
    Xu ZS, Liao HC. A survey of approaches to decision making with intuitionistic fuzzy preference relations. Knowl-Based Syst. 2015;80(5):131–42.Google Scholar
  70. 70.
    Xu ZS, Yager RR. Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. Fuzzy Optim Decis Ma. 2009;8(2):123–39.Google Scholar
  71. 71.
    Ye J. Multiple attribute decision-making methods based on the expected value and the similarity measure of hesitant neutrosophic linguistic numbers. Cogn Comput. 2018;10(3):454–63.Google Scholar
  72. 72.
    Zadeh LA. A fuzzy-algorithmic approach to the definition of complex or imprecise concepts. Int J Man-Mach Stud. 1976;8(3):249–91.Google Scholar
  73. 73.
    Zhao N, Xu Z, Liu F. Group decision making with dual hesitant fuzzy preference relations. Cogn Comput. 2016;8(6):1119–43.Google Scholar
  74. 74.
    Zhang Z, Wu C. A novel method for single-valued neutrosophic multi-criteria decision making with incomplete weight information. Neutrosophic Sets Syst. 2014;4:35–49.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of InformationBeijing Wuzi UniversityBeijingChina
  2. 2.School of BusinessCentral South UniversityChangshaChina

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