Novel fusion strategies for continuous interval-valued q-rung orthopair fuzzy information: a case study in quality assessment of SmartWatch appearance design


The notion of Yager’s q-rung orthopair fuzzy set (QROFS) have gained considerable and continuously increasing attention as a useful tool for imprecision and uncertainty representation due to its capability to discard the constraints on the membership and nonmembership functions as generally required by its intuitionistic fuzzy counterpart. Among the generalizations and variants established in the past few years, the interval-valued QROFSs (IVQROFSs) have been diffusely considered to be a powerful generalization of the interval-valued fuzzy sets. The continuous ordered weighted averaging (COWA) operator has been extended successfully to some special cases of IVQROFSs, including interval-valued intuitionistic and Pythagorean fuzzy sets. Thus, to expand on previous studies, several continuous IVQROF (C-IVQROF) aggregation operators are proposed in this study. First, the dual C-GOWA operator is defined on the basis of the continuous generalized ordered weighted averaging (C-GOWA) operator and Yager class of fuzzy negation. Subsequently, the C-IVQROFOWA operator with two independent parameters is constructed, and the weighted C-IVQROFOWA operator is then proposed for aggregating a collection of IVQROFSs. The C-IVQROFOWA operator and its weighted version can model commendably the attitudinal characteristics of the decision-maker. Second, a parameter optimization model and its algorithm-solving strategy driven by consensus measures are built to develop a group decision-making method. Finally, a case study to evaluate the SmartWatch design alternatives is provided to demonstrate the proposed approach, and the results of a comparative analysis verify the rationality and efficiency of the proposed operators.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    Yager RR (2016) Generalized orthopair fuzzy sets. IEEE Trans Fuzzy Syst 25(5):1222–1230

    Article  Google Scholar 

  2. 2.

    Yager RR, Alajlan N (2017) Approximate reasoning with generalized orthopair fuzzy sets. Inf Fusion 38:65–73

    Article  Google Scholar 

  3. 3.

    Yager RR, Alajlan N, Bazi Y (2018) Aspects of generalized orthopair fuzzy sets. Int J Intell Syst 33(11):2154–2174

    Article  Google Scholar 

  4. 4.

    Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    MATH  Article  Google Scholar 

  5. 5.

    Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    MATH  Article  Google Scholar 

  6. 6.

    Yager RR (2013) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965

    Article  Google Scholar 

  7. 7.

    Wei G, Gao H, Wei Y (2018) Some \(q\)-Rung orthopair fuzzy Heronian mean operators in multiple attribute decision making. Int J Intell Syst 33(7):1426–1458

    Article  Google Scholar 

  8. 8.

    Liu Z, Wang S, Liu P (2018a) Multiple attribute group decision making based on \(q\)-Rung orthopair fuzzy Heronian mean operators. Int J Intell Syst 33(12):2341–2363

    Article  Google Scholar 

  9. 9.

    Xing Y, Zhang R, Wang J, Bai K, Xue J (2020) A new multi-criteria group decision-making approach based on \(q\)-Rung orthopair fuzzy interaction Hamy mean operators. Neural Comput Appl 32:7465–7488

    Article  Google Scholar 

  10. 10.

    Liu P, Ali Z, Mahmood T (2019a) A method to multi-attribute group decision-making problem with complex \(q\)-rung orthopair linguistic information based on heronian mean operators. Int J Comput Intell Syst 12(2):1465–1496

    Google Scholar 

  11. 11.

    Yang W, Pang Y (2019) New \(q\)-Rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making. Int J Intell Syst 34(3):439–476

    Article  Google Scholar 

  12. 12.

    Wei G, Wei C, Wang J, Gao H, Wei Y (2019) Some \(q\)-Rung orthopair fuzzy Maclaurin symmetric mean operators and their applications to potential evaluation of emerging technology commercialization. Int J Intell Syst 34(1):50–81

    Article  Google Scholar 

  13. 13.

    Liu P, Ali Z, Mahmood T, Hassan N (2020) Group decision-making using complex \(q\)-rung orthopair fuzzy Bonferroni mean. Int J Comput Intel Syst 13(1):822–851

    Google Scholar 

  14. 14.

    Ju Y, Luo C, Ma J, Wang A (2019a) A novel multiple-attribute group decision-making method based on \(q\)-Rung orthopair fuzzy generalized power weighted aggregation operators. Int J Intell Syst 34(9):2077–2103

    Article  Google Scholar 

  15. 15.

    Du WS (2019a) Weighted power means of \(q\)-Rung orthopair fuzzy information and their applications in multiattribute decision making. Int J Intell Syst 34(11):2835–2862

    Article  Google Scholar 

  16. 16.

    Liu Z, Liu P, Liang X (2018b) Multiple attribute decision-making method for dealing with heterogeneous relationship among attributes and unknown attribute weight information under \(q\)-Rung orthopair fuzzy environment. Int J Intell Syst 33(9):1900–1928

    Article  Google Scholar 

  17. 17.

    Liu P, Wang P (2018) Some \(q\)-Rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int J Intell Syst 33(2):259–280

    Article  Google Scholar 

  18. 18.

    Peng X, Dai J, Garg H (2018) Exponential operation and aggregation operator for \(q\)-Rung orthopair fuzzy set and their decision-making method with a new score function. Int J Intell Syst 33(11):2255–2282

    Article  Google Scholar 

  19. 19.

    Du WS (2018) Minkowski-type distance measures for generalized orthopair fuzzy sets. Int J Intell Syst 33(4):802–817

    Article  Google Scholar 

  20. 20.

    Liu D, Chen X, Peng D (2019b) Some cosine similarity measures and distance measures between \(q\)-Rung orthopair fuzzy sets. Int J Intell Syst 34(7):1572–1587

    Article  Google Scholar 

  21. 21.

    Peng X, Dai J (2019) Research on the assessment of classroom teaching quality with \(q\)-Rung orthopair fuzzy information based on multiparametric similarity measure and combinative distance-based assessment. Int J Intell Syst 34(7):1588–1630

    Article  Google Scholar 

  22. 22.

    Du WS (2019b) Correlation and correlation coefficient of generalized orthopair fuzzy sets. Int J Intell Syst 34(4):564–583

    Article  Google Scholar 

  23. 23.

    Peng X, Liu L (2019) Information measures for \(q\)-Rung orthopair fuzzy sets. Int J Intell Syst 34(8):1795–1834

    Article  Google Scholar 

  24. 24.

    Gao J, Liang Z, Shang J, Xu Z (2018) Continuities, derivatives, and differentials of \(q\)-rung orthopair fuzzy functions. IEEE Trans Fuzzy Syst 27(8):1687–1699

    Article  Google Scholar 

  25. 25.

    Gao J, Liang Z, Xu Z (2020) Additive integrals of \(q\)-Rung orthopair fuzzy functions. IEEE Trans Cybern 50(10):4406–4419

    Article  Google Scholar 

  26. 26.

    Ye J, Ai Z, Xu Z (2019) Single variable differential calculus under \(q\)-Rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application. Int J Intell Syst 34(7):1387–1415

    Article  Google Scholar 

  27. 27.

    Shu X, Ai Z, Xu Z, Ye J (2019) Integrations of \(q\)-Rung orthopair fuzzy continuous information. IEEE Trans Fuzzy Syst 27(10):1974–1985

    Article  Google Scholar 

  28. 28.

    Li H, Yin S, Yang Y (2019) Some preference relations based on \(q\)-Rung orthopair fuzzy sets. Int J Intell Syst 34(11):2920–2936

    Article  Google Scholar 

  29. 29.

    Zhang C, Liao H, Luo L (2019a) Additive consistency-based priority-generating method of \(q\)-Rung orthopair fuzzy preference relation. Int J Intell Syst 34(9):2151–2176

    Article  Google Scholar 

  30. 30.

    Banerjee D, Dutta B, Guha D, Martínez L (2020) SMAA-QUALIFLEX methodology to handle multicriteria decision-making problems based on \(q\)-rung fuzzy set with hierarchical structure of criteria using bipolar Choquet integral. Int J Intell Syst 35(3):401–431

    Article  Google Scholar 

  31. 31.

    Chen Z-S, Chin K-S, Tsui K-L (2019a) Constructing the geometric Bonferroni mean from the generalized Bonferroni mean with several extensions to linguistic 2-tuples for decision-making. Appl Soft Comput 78:595–613

    Article  Google Scholar 

  32. 32.

    Chen Z-S, Yang Y, Wang X-J, Chin K-S, Tsui K-L (2019b) Fostering linguistic decision-making under uncertainty: a proportional interval type-2 hesitant fuzzy TOPSIS approach based on Hamacher aggregation operators and andness optimization models. Inf Sci 500:229–258

    Article  Google Scholar 

  33. 33.

    Yang Y, Chen Z-S, Chen Y-H, Chin K-S (2018) Interval-valued Pythagorean fuzzy Frank power aggregation operators based on an isomorphic Frank dual triple. Int J Comput Intell Syst 11(1):1091–1110

    Article  Google Scholar 

  34. 34.

    Joshi BP, Singh A, Bhatt PK, Vaisla KS (2018) Interval valued \(q\)-Rung orthopair fuzzy sets and their properties. J Intell Fuzzy Syst 35(5):5225–5230

    Article  Google Scholar 

  35. 35.

    Ju Y, Luo C, Ma J, Gao H, Santibanez Gonzalez E D, Wang A (2019) Some interval-valued \(q\)-Rung orthopair weighted averaging operators and their applications to multiple-attribute decision making. Int J Intell Syst 34(10):2584–2606

    Article  Google Scholar 

  36. 36.

    Wang J, Wei G, Wang R, Alsaadi FE, Hayat T, Wei C, Zhang Y, Wu J (2019) Some \(q\)-Rung interval-valued orthopair fuzzy Maclaurin symmetric mean operators and their applications to multiple attribute group decision making. Int J Intell Syst 34(11):2769–2806

    Article  Google Scholar 

  37. 37.

    Jan N, Mahmood T, Zedam L, Ullah K, Alcantud JCR, Davvaz B (2019) Analysis of social networks, communication networks and shortest path problems in the environment of interval-valued q-Rung orthopair fuzzy graphs. Int J Fuzzy Syst 21(6):1687–1708

    MathSciNet  Article  Google Scholar 

  38. 38.

    Yager RR (2004a) OWA aggregation over a continuous interval argument with applications to decision making. IEEE Trans Syst Man Cybern Part B (Cybern) 34(5):1952–1963

    Article  Google Scholar 

  39. 39.

    Yager RR (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18(1):183–190

    MATH  Article  Google Scholar 

  40. 40.

    Yager RR (1996) Quantifier guided aggregation using OWA operators. Int J Intell Syst 11(1):49–73

    Article  Google Scholar 

  41. 41.

    Chen H, Zhou L (2011) An approach to group decision making with interval fuzzy preference relations based on induced generalized continuous ordered weighted averaging operator. Expert Syst Appl 38(10):13432–13440

    Article  Google Scholar 

  42. 42.

    Zhou H, Ma X, Zhou L, Chen H, Ding W (2018) A novel approach to group decision-making with interval-valued intuitionistic fuzzy preference relations via shapley value. Int J Fuzzy Syst 20(4):1172–1187

    MathSciNet  Article  Google Scholar 

  43. 43.

    Zhou L, Wu J, Chen H (2014a) Linguistic continuous ordered weighted distance measure and its application to multiple attributes group decision making. Appl Soft Comput 25:266–276

    Article  Google Scholar 

  44. 44.

    Jin F, Ni Z, Chen H, Li Y, Zhou L (2016) Multiple attribute group decision making based on interval-valued hesitant fuzzy information measures. Comput Ind Eng 101:103–115

    Article  Google Scholar 

  45. 45.

    Jin F, Pei L, Chen H, Zhou L (2014) Interval-valued intuitionistic fuzzy continuous weighted entropy and its application to multi-criteria fuzzy group decision making. Knowl-Based Syst 59:132–141

    Article  Google Scholar 

  46. 46.

    Wu J, Chiclana F (2014) A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions. Appl Soft Comput 22:272–286

    Article  Google Scholar 

  47. 47.

    Zhou L, Tao Z, Chen H, Liu J (2014b) Continuous interval-valued intuitionistic fuzzy aggregation operators and their applications to group decision making. Appl Math Model 38(7–8):2190–2205

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Lin J, Zhang Q (2017) Note on continuous interval-valued intuitionistic fuzzy aggregation operator. Appl Math Model 43:670–677

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Yang Y, Lv H-X, Li Y-L (2017) WIC-IVIFOWA operator based on standard negation and its application. Control Decis 32(11):2021–2033

    MATH  Google Scholar 

  50. 50.

    Chen Z-S, Yu C, Chin K-S, Martínez L (2019c) An enhanced ordered weighted averaging operators generation algorithm with applications for multicriteria decision making. Appl Math Model 71:467–490

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Liu J, Lin S, Chen H, Zhou L (2013) The continuous quasi-OWA operator and its application to group decision making. Group Decis Negot 22(4):715–738

    Article  Google Scholar 

  52. 52.

    Wang L, Li N (2019) Continuous interval-valued Pythagorean fuzzy aggregation operators for multiple attribute group decision making. J Intell Fuzzy Syst 36(6):6245–6263

    Article  Google Scholar 

  53. 53.

    Yager RR (2004b) Generalized OWA aggregation operators. Fuzzy Optim Decis Mak 3(1):93–107

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Yager RR (1979) On the measure of fuzziness and negation part I: membership in the unit interval. Int J Gener Syst 5:221–229

    MATH  Article  Google Scholar 

  55. 55.

    Yager RR (1980) On the measure of fuzziness and negation. II. Lattices. Inf Control 44(3):236–260

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Beliakov G, Pradera A, Calvo T et al (2007) Aggregation functions: a guide for practitioners, vol 221. Springer, Berlin

    Google Scholar 

  57. 57.

    Rodríguez RM, Labella Á, De Tré G, Martínez L (2018) A large scale consensus reaching process managing group hesitation. Knowl-Based Syst 159:86–97

    Article  Google Scholar 

  58. 58.

    Dutta B, Labella Á, Rodríguez RM, Martínez L (2019) Aggregating interrelated attributes in multi-attribute decision-making with ELICIT information based on Bonferroni mean and its variants. Int J Comput Intell Syst 12(2):1179–1196

    Article  Google Scholar 

  59. 59.

    Chen Z-S, Liu X-L, Rodríguez RM, Wang X-J, Chin K-S, Tsui K-L, Martínez L (2020) Identifying and prioritizing factors affecting in-cabin passenger comfort on high-speed rail in China: a fuzzy-based linguistic approach. Appl Soft Comput 95:106558

    Article  Google Scholar 

  60. 60.

    Chen Z-S, Liu X-L, Chin K-S, Pedrycz W, Tsui K-L, Skibniewski MJ (2021) Online-review analysis based large-scale group decision-making for determining passenger demands and evaluating passenger satisfaction: case study of high-speed rail system in China. Inf Fusion 69:22–39

    Article  Google Scholar 

  61. 61.

    Labella Á, Liu Y, Rodríguez R, Martínez L (2018) Analyzing the performance of classical consensus models in large scale group decision making: A comparative study. Appl Soft Comput 67:677–690

    Article  Google Scholar 

  62. 62.

    Zhang L, Li JT, Zhao YY, Tian ZQ (2019b) Evaluation method for product design based on users’ emotional needs. Oper Res Manag Sci 28(1):152–157

    Google Scholar 

  63. 63.

    Yu C, Shao Y, Wang K, Zhang L (2019) A group decision making sustainable supplier selection approach using extended TOPSIS under interval-valued Pythagorean fuzzy environment. Expert Syst Appl 121:1–17

    Article  Google Scholar 

  64. 64.

    Tao Z, Liu X, Chen H, Zhou L (2016) Using new version of extended \(t\)-norms and \(s\)-norms for aggregating interval linguistic labels. IEEE Trans Syst Man Cybern Syst 47(12):3284–3298

    Article  Google Scholar 

  65. 65.

    Tao Z, Shao Z, Liu J, Zhou L, Chen H (2020) Basic uncertain information soft set and its application to multi-criteria group decision making. Eng Appl Artif Intell 95:103871

    Article  Google Scholar 

Download references


This work was supported by the National Natural Science Foundation of China (Grant nos. 71801175, 71871171, 71971182, and 72031009), The Ministry of Education of Humanities and Social Science Foundation of China (Grant no. 20YJCZH210), the Natural Science Foundation of Hunan Province, China (Grant no. 2020JJ5112), the Theme-based Research Projects of the Research Grants Council (Grant no. T32-101/15-R), the Spanish Ministry of Economy and Competitiveness through the Spanish National Research Project (Grant no. PGC2018-099402-B-I00) and the postdoctoral fellowship Ramón y Cajal (Grant no. RyC-2017-21978), the Ger/HKJRS project (Grant no. G-CityU103/17), and partly by the City University of Hong Kong SRG (Grant no. 7004969).

Author information



Corresponding author

Correspondence to Zhen-Song Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yang, Y., Chen, ZS., Rodríguez, R.M. et al. Novel fusion strategies for continuous interval-valued q-rung orthopair fuzzy information: a case study in quality assessment of SmartWatch appearance design. Int. J. Mach. Learn. & Cyber. (2021).

Download citation


  • Interval-valued q-rung orthopair fuzzy sets
  • Aggregation operators
  • Group decision making
  • Product appearance design evaluation