Some q-rung orthopair fuzzy Hamacher aggregation operators and their application to multiple attribute group decision making with modified EDAS method☆
Introduction
In solving multi-attribute group decision making (MAGDM) problems, decision makers use some methodologies, including fuzzy sets (Zadeh, 1965) and fuzzy linguistic techniques (Rodriguez et al., 2012). As an extension of fuzzy sets, Atanassov (1986) revealed the concept of intuitionistic fuzzy set (IFS). As a generalization of the fuzzy set, IFS provides decision makers with both membership and non-membership functions to express their fuzzy information. IFS works on the assumption that the sum of the membership and non-membership degrees of each ordered pair is less than or equal to one (Wei et al., 2018a). Since its inception, the IFS has attracted more attention from scholars (Cali and Balaman, 2019, Chen et al., 2019, Iancu, 2019, Liu et al., 2017, Luo et al., 2019, Zhang et al., 2019a). In order to provide more space for decision makers, Yager (2013) and Yager and Abbasov (2013) introduced the Pythagorean fuzzy set (PFS). Unlike the IFS, the PFS is more general and is characterized by the idea that the square sum of the membership and non-membership degrees is less than or equal to one (Yager, 2013). It should be noted that IFS is a subset of PFS, which indicates that PFS is more effective in handling uncertain problems. Zhang and Xu (2014) introduced an elaborated mathematical notation for PFS and proposed the Pythagorean fuzzy number (PFN). Since then, many works have focused on the extensions of PFS to solve MADM problems (Chen, 2018, Liang et al., 2018c, Liang et al., 2018b, Liang et al., 2018a, Liang et al., 2019b, Liang et al., 2019a, Liang et al., 2019c, Tang et al., 2019, Yang et al., 2018, Yang et al., 2019).
Due to the increase in volumes and complexity of recent information, Yager (2017) further presented a novel concept of q-rung orthopair fuzzy set (q-ROFS). The q-ROFS has the condition that the sum of the qth power of the membership and the non-membership degrees is constrained to one, i.e., . Both IFS and PFS are special cases of the q-ROFS. We can argue that the q-ROFS is more general because as the rung increases, the acceptable space of the orthopair increases, and the more orthopairs meet the restricted condition. In effect, the q-ROFS provides greater range for decision makers to express their uncertain information. Liu and Wang (2018) pioneered the q-rung fuzzy weighted average (q-ROFWA) operator and the q-rung fuzzy weighted geometric (q-ROFWG) operator and studied their properties. Based on the Bonferroni mean, Liu and Liu (2018) developed the q-rung orthopair fuzzy BM (q-ROFBM) operator, the q-rung orthopair fuzzy weighted BM (q-ROFWBM) operator, the q-rung orthopair fuzzy geometric BM (q-ROFGBM) operator, and the q-rung orthopair fuzzy weighted geometric BM (q-ROFWGBM) operator. Based on these operators the authors designed some MAGDM methods. Liu et al. (2018) proposed two new aggregation operators for q-ROFS, namely q-rung orthopair fuzzy extended Bonferroni mean (q-ROFEBM) operator and its weighted form (q-ROFEWEBM). Furthermore, Wei et al. (2018a) combined Heronian mean with q-ROFS and presented the q-rung orthopair fuzzy generalized Heronian mean (q-ROFGHM) operator, q-rung orthopair fuzzy geometric Heronian mean (q-ROFGHM) operator, q-rung orthopair fuzzy generalized weighted Heronian mean (q-ROFGWHM) operator, and q-rung orthopair fuzzy weighted geometric Heronian mean (q-ROFWGHM) operator. Also, Du (2019) presented four main operations of arithmetic over q-rung orthopair membership grades. With respect to the cosine function, Wang et al. (2019b) proposed the similarity measures of q-ROFS and applied them to MADM problems. Wei et al. (2019) extended the Maclaurin symmetric mean (MSM) operator into the q-ROFS environment and discussed the q-rung orthopair fuzzy MSM operator, the q-rung orthopair fuzzy dual MSM operator, the q-rung orthopair fuzzy weighted MSM operator, and the q-rung orthopair fuzzy weighted dual MSM operator. Wang et al. (2019a) studied some methods for MAGDM with q-rung interval-valued orthopair fuzzy information and their applications to the selection of green suppliers. Moreover, Peng et al. (2018) introduced an exponential operation and aggregation operator for q-rung orthopair fuzzy set and developed a new score function for decision making. More scholars have focused on the extension of q-rung orthopair fuzzy set to solve many decision making problems (Peng and Dai, 2019, Peng and Lin, 2019, Yang and Pang, 2019).
Maclaurin symmetric mean (MSM), a powerful operator for aggregating input arguments was first introduced by Maclaurin (1729) and later developed by Detemple and Robertson (1979). As a general operator, the MSM can model multiple interrelationship among input arguments. Based on the monotonically decreasing nature of the parameter in the MSM operator, decision makers can accurately reflect their risk preferences (Qin and Liu, 2014, Qin and Liu, 2015). This makes the MSM more preferable to other aggregation operators such as the arithmetic average, the geometric, the power average, the Bonferroni mean, and the harmonic mean. Because of the merits of the MSM operator, numerous researchers have studied its extension in solving MADM problems (Qin and Liu, 2014, Qin et al., 2015, Liu and Wang, 2018, Wei et al., 2018a). For instance, Qin and Liu (2014) proposed a multiple attribute decision making method based on the extension of MSM into the intuitionistic fuzzy environment. Again, Qin et al. (2015) developed hesitant fuzzy Maclaurin symmetric mean (HFMSM) and weighted hesitant fuzzy Maclaurin symmetric mean (WHFMSM) operators. Liu and Wang (2018) studied some single-valued trapezoidal neutrosophic MSM operators and examined some properties and some special cases of these operators.Wei et al. (2018a) detailedly investigated some Pythagorean fuzzy MSM operators. Feng et al. (2019) introduced a method of MAGDM based on 2-Tuple linguistic dependent MSM operators. Peng (2019) examined the single-valued neutrosophic reducible weighted MSM (SVNRWMSM) operator and the single-valued neutrosophic reducible weighted dual MSM (SVNRWDMSM) operator for undergraduate teaching audit and evaluation. In light of the Schweizer–Sklar operations, Wang and Liu (2019) developed some Maclaurin symmetric mean aggregation operators using intuitionistic fuzzy information and discussed their applicability in decision making. In addition, Yang and Pang (2018) proposed some new Pythagorean fuzzy interaction Maclaurin symmetric mean operators and applied the in MADM.
The discussions so far reveal that most of the aggregation operators developed are based on the algebraic product and algebraic sum. However, these are not the only operations for fuzzy sets. Hamacher (1978) introduced the Hamacher operations which include Hamacher product and Hamacher sum. Both the Hamacher product and Hamacher sum are good alternatives to the algebraic product and algebraic sum (Wei et al., 2018b). As a generalization of the algebraic and Einstein t-conorm and t-norm (Deschrijver et al., 2004), the Hamacher t-conorm and the hamacher t-norm are more general and flexible. Review on the q-rung orthopair fuzzy aggregation operators shows that there is little research on utilizing Hamacher operations to develop new operators. Therefore, it is necessary to conduct research on aggregation operators using Hamacher operations with q-rung orthopair fuzzy information. Moreover, the selection of optimal candidate(s) is/are very paramount in decision analysis. Hence, it is appropriate that we develop a new methodology to find solution to MAGDM problems. The EDAS method was first mentioned by Ghorabaee et al. (2015) as a multi-criteria decision making (MCDM) method for ABC inventory classification. Unlike TODIM (Lourenzutti and Krohling, 2013), VIKOR (Opricovic and Tzeng, 2004) and TOPSIS (Kuo, 2017), EDAS method uses average solution for evaluating the alternatives. In other words, the optimal alternative is obtained based on the positive distance from average (PDA) and the negative distance from average (NDA). An advantage of the EDAS method is that it ascertains the compromise solution according to the average solution and this makes the method more stable in ranking alternatives (Feng et al., 2018). Kahraman et al. (2017) designed the intuitionistic fuzzy EDAS method to appraise solid waste disposal site. Feng et al. (2018) developed an EDAS method for extended hesitant fuzzy linguistic MCDM. Also, Gundogdu et al. (2018) proposed a novel hesitant fuzzy EDAS method and examined its application to hospital selection. Using an interval-valued neutrosophic EDAS method, Karasan and Kahraman (2018) prioritized the United Nations national sustainable development goals. Zhang et al. (2019b) utilized picture 2-tuple linguistic information to modify the EDAS method for MCGDM. Also, Peng and Liu (2017) compiled some algorithms for neutrosophic soft decision making based on EDAS. It is clear that EDAS has received much attention from scholars but there has been no work extending EDAS to the q-rung orthopair fuzzy domain. In view of these arguments and motivations, the contributions of this paper are outline below:
(1) We propose some new aggregation operators for q-ROFS, which include the q-rung orthopair fuzzy Hamacher average (q-ROFHA) operator, the weighted q-rung orthopair fuzzy Hamacher average (Wq-ROFHA) operator, the q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (q-ROFHMSM) operator and the weighted q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (Wq-ROFHMSM) operator.
(2) We design a novel q-ROF-EDAS method based on the W-qROFHA and the W-qROFHMSM operators. Specifically, we utilize theW-qROFHA operator to fuse the evaluation preferences of the decision makers. Then, with the integration of the W-qROFHMSM operator and the q-ROF-EDAS method we appraise the alternatives.
(3) Again, we modify the best worst method (BWM) introduced by Rezaei (2015) and develop a q-ROF-BWM to determine the weight information of the attributes.
(4) Finally, we test the applicability of our proposed q-ROF-EDAS method by solving a mobile payment platform selection problem.
The remainder of this paper is organized as follows: Section 2 provides the fundamental concepts of q-ROFS, Hamacher operations and MSM. Under the q-rung orthopair fuzzy environment, we develop some Hamacher aggregation operators and discuss their properties and special cases in Section 3. In Section 4, we design a modified EDAS method for MAGDM with q-ROF information. In addition, we propose the q-ROF-BWM method to determine the weight information of the attributes. Section 5 employs an example of mobile payment platform selection to demonstrate the application of the proposed method. Also, some sensitivity analysis and comparison analysis are conducted. Section 6 concludes the study and elaborates on future studies.
Section snippets
Preliminaries
In this section, certain critical notions of q-ROFS (Yager, 2017), Hamacher operations (Hamacher, 1978) and MSM (Maclaurin, 1729) are concisely reviewed.
q-rung orthopair fuzzy Hamacher aggregation operators
Under this segment, we attentively propose some q-rung orthopair fuzzy Hamacher aggregation operators based on the arithmetic average operator and the Maclaurin symmetric mean operator.
A modified EDAS method for MAGDM with q-ROF information
In this part, we design a novel ranking technique to address the q-ROFMAGDM problem. Firstly, the q-ROFMAGDM problem is elaborated. In view of this, we employ the Wq-ROFHMSM operator to accumulate the input arguments of the decision makers into a comprehensive opinion. Then, we construct a modified EDAS method for solving the q-ROFMAGDM problem. Also, this section presents the decision-making steps of the q-ROFMAGDM.
An illustrative example
The advancement of disruptive technologies has made mobile devices acquire new functionalities supporting several mobile financial services, such as account transfers, bill payments, person to person transfers, proximity payments, remote payments, as well as other kinds of services including mobile ticketing, marketing, discounts and location based (Liana-Cabanillas et al., 2018). Among the various mobile technologies offered today, there is a trend called ’mobile payment technology’ which is
Conclusions
In this paper, we develop a novel MAGDM method to solve q-ROFMAGDM problems. In view of the relationship pattern between input arguments, we propose new aggregation operators. In summary, the following conclusions can be made:
(1) Considering the independent relationship between input arguments, we extend the well-known arithmetic mean into the q-ROF environment and apply Hamacher operational laws to develop some novel aggregation operators, i.e., the q-ROFHA and the Wq-ROFHA operators.
(2)
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China (Nos. 71401026, 71432003, 61773352), the Planning Fund for the Humanities and Social Sciences of Ministry of Education of China (No. 19YJA630042), the Double First-class Construction Research Support Project of UESTC (No. SYLYJ2019210) and the Youth Team Program for Technolgoy Innovation of Sichuan Province (No. 2016TD0013).
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No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.engappai.2019.103259.