Abstract
The notion of multifuzzy sets (MFSs) or multi-interval-valued fuzzy sets (MIVFSs) provides a new method to represent some problems with a sequence of the different and/or same fuzzy/interval-valued fuzzy membership values of an element to the set. Then, a fuzzy cubic set (FCS) consists of a certain part (a fuzzy value) and an uncertain part (an interval-valued fuzzy value) but cannot represent hybrid information of both MFS and MIVFS. To adequately depict the opinion of several experts/decision-makers by using a union/sequence of the different and/or same fuzzy cubic values for an object assessed in group decision-making (GDM) problems, this paper proposes a multifuzzy cubic set (MFCS) notion as the conceptual extension of FCS to express the hybrid information of both MFS and MIVFS in the fuzzy setting of both uncertainty and certainty. Then, we propose three correlation coefficients of MFCSs and then introduce correlation coefficients of MFSs and MIVFSs as special cases of the three correlation coefficients of MFCSs. Further, the multicriteria GDM methods using three weighted correlation coefficients of MFCSs are developed under the environment of MFCSs, which contains the MFS and MIVFS GDM methods. Lastly, these multicriteria GDM methods are applied in an illustrative example on the selection problem of equipment suppliers; then their decision results and comparative analysis indicate that the developed GDM methods are more practicable and effective and reflect that either different correlation coefficients or different information expressions can also impact on the ranking of alternatives. Therefore, this study indicates the main contribution of the multifuzzy cubic information expression, correlation coefficients, and GDM methods in the multifuzzy setting of both uncertainty and certainty.
Highlights
Fuzzy set (FS) [1] is usually depicted by a membership degree in the interval [0, 1], where only indicate one occurrence of each element
To suitably describe the union/sequence of the different and/or same fuzzy cubic value (FCV), this study presents the concept of multifuzzy cubic set (MFCS) based on the hybrid information of multifuzzy set (MFS) and multi-intervalvalued fuzzy set (MIVFS). e MFCS notion can provide a new expression form to overcome the insufficiency of the hesitant fuzzy cubic set (FCS) and to extend the notion of FCS. us, MFCS can describe the opinions that the several experts or decision-makers propose by using a union/sequence of the different and/or same FCVs in group decision-making (GDM) problems
Regarding the hybrid expression problem of MFS and MIVFS in the multifuzzy setting of uncertainty and certainty, this paper presented a MFCS concept as the extension of FCS in order to adequately depict the opinion of several experts/decision-makers by using a union/sequence of the same and/or different FCVs. en, we proposed three correlation coefficients of MFCSs and after that introduced the correlation coefficients of MFSs and MIVFSs as the special cases of the three correlation coefficients of MFCSs
Summary
Fuzzy set (FS) [1] is usually depicted by a membership degree in the interval [0, 1], where only indicate one occurrence of each element. (i) e proposed MFCS can solve the hybrid information expression problem of both MFS and MIVFS and provide an adequate expression form to depict the opinion of several experts/decisionmakers by using a union/sequence of several different and/or the same FCVs in multicriteria GDM problems (ii) e proposed correlation coefficients of MFCSs can provide effective mathematical models for the multicriteria GDM methods in the environments of MFCSs, MFSs, and MIVFSs (iii) e developed multifuzzy cubic GDM methods can solve GDM problems under the environment of MFCSs and show their superiority and usability in the setting of MFCSs. For this study, this paper is composed of the following structures.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
