Multiattribute decision-making based on TOPSIS technique and novel correlation coefficient of q-rung orthopair fuzzy sets

<abstract> <p>Green supplier selection has been an important technique for environmental sustainability and reducing the harm of ecosystems. In the current climate, green supply chain management (GSCM) is imperative for maintaining environmental compliance and commercial growth. To handle the change related to environmental concern and how the company manages and operates, they are integrated the GSCM into traditional supplier selection process. The main aims of this study were to outline both traditional and environmental criteria for selecting suppliers, providing a comprehensive framework to assist decision-maker in prioritizing green supplier effectively. In order to address issue to simulate decision-making problems and manage inaccurate data. A useful technique of fuzzy set was proposed to handle uncertainty in various real-life problems, but all types of data could not be handled such as incomplete and indeterminate. However, several extensions of fuzzy set were considered, such as intuitionistic fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, and q-rung orthopair fuzzy soft set considering membership and nonmember ship grade to handle the uncertainty problem. However, there was a lack of information about the neutral degree and parameterization axioms lifted by existing approaches, so to fill this gap and overcome the difficulties Ali et al. proposed a generalized structure by combining the structure of picture fuzzy set and q-rung orthopair fuzzy soft set, known as q-rung orthopair picture fuzzy soft sets, characterized by positive, neutral and negative membership degree with parameterization tools and aggregation operator to solve the multi criteria group decision-making problem. Additionally, the TOPSIS method is a widely utilized to assist individuals and organizations in selecting the most appropriate option from a range of choices, taking into account various criteria. Finally, we demonstrate an illustrative example related to GSCM to enhance competitiveness, based on criteria both in general and with a focus on environmental consideration, accompanied by an algorithm and flow chart.</p> </abstract>

The complex q-rung orthopair fuzzy (CQ-ROF) set can describe the complex uncertain information. In this manuscript, we develop the Yager operational laws based on the CQ-ROF information and Yager t-norm and t-conorm. Furthermore, in aggregating the CQ-ROF values, the power, averaging, and geometric aggregation operators have played a very essential and critical role in the environment of fuzzy set. Inspired from the discussed operators, we propose the CQ-ROF power Yager averaging (CQ-ROFPYA), CQ-ROF power Yager ordered averaging (CQ-ROFPYOA), CQ-ROF power Yager geometric (CQ-ROFPYG), and CQ-ROF power Yager ordered geometric (CQ-ROFPYOG) operators. These operators are the modified version of the Power, Yager, averaging, geometric, and the combination of these all based on fuzzy set (FS), intuitionistic FS, Pythagorean FS, q-rung orthopair FS, complex FS, complex intuitionistic FS, and complex Pythagorean FS. Moreover, we also discuss the main properties of the proposed operators. Additionally, we develop a multi-attribute decision-making (MADM) method based on the developed operators. To show the supremacy and validity of the proposed method, the comparison between the proposed method and some existing methods is done by some examples, and results show that the proposed method is better than the others in terms of generality and effectiveness.

An effective technique for dealing with multiple-criteria decision-making (MCDM) problems of real world is the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). q-Rung Orthopair Fuzzy Sets (q-ROFs) were introduced by Yager as a generalization of intuitionistic fuzzy sets, in which the sum of the qth powers of the (membership and non-membership) degrees is restricted to one. As the value of q increases, the feasible region for orthopair also increases. This results in more orthopairs satisfying the limitations and hence broadening the scope of representation of fuzzy information. This chapter makes an attempt to integrate the TOPSIS technique with q-ROF sets with some of its generalizations and extensions. Initially, we explore the concept of TOPSIS technique to solve MCDM problems under q-Rung Orthopair Fuzzy (q-ROF) environment and illustrate it with its application in solving a transport policy problem. Then we consider the TOPSIS technique to solve MCDM problems under q-Rung Orthopair Hesitant Fuzzy (q-ROHF) settings. To explain it we have mentioned an illustration of military aircraft overhaul effectiveness evaluation. In addition to the above-mentioned methods, we present the TOPSIS technique for solving decision-making problems under the newly introduced q-rung orthopair fuzzy soft set (q-ROFS\(_f\)S). Here, we tackle a problem of selection of a medical clinic utilizing the q-ROFS\(_f\) TOPSIS method.KeywordsTOPSISq-ROFSq-ROFNsq-ROHFSMCDM

In recent years, q-rung orthopair fuzzy sets have been appeared to deal with an increase in the value of q > 1 , which allows obtaining membership and nonmembership grades from a larger area. Practically, it covers those membership and nonmembership grades, which are not in the range of intuitionistic fuzzy sets. The hybrid form of q-rung orthopair fuzzy sets with soft sets have emerged as a useful framework in fuzzy mathematics and decision-makings. In this paper, we presented group generalized q-rung orthopair fuzzy soft sets (GGq-ROFSSs) by using the combination of q-rung orthopair fuzzy soft sets and q-rung orthopair fuzzy sets. We investigated some basic operations on GGq-ROFSSs. Notably, we initiated new averaging and geometric aggregation operators on GGq-ROFSSs and investigated their underlying properties. A multicriteria decision-making (MCDM) framework is presented and validated through a numerical example. Finally, we showed the interconnection of our methodology with other existing methods.

Fuzzy set theory is a mathematical method for dealing with uncertainty and imprecision in decision-making. Some of the challenges and complexities involved in medical diagnosis can be addressed with the help of fuzzy set theory. Ovarian cancer is a disease that affects the female reproductive system's ovaries, which also make the hormones progesterone and estrogen. The ovarian cancer stages demonstrate how far the disease has spread from the ovaries to other organs. The TOPSIS technique (Technique for Order Preference by Similarity to Ideal Solution) aids in selecting the best option from a selection of choices by taking into account a number of variables. It provides a ranking or preference order after weighing the benefits and drawbacks of each solution. Intuitionistic fuzzy soft set (IFSS) is the framework to deal with the uncertain information with the help of the parameters. The goal of this article is to develop some basic aggregation operators (AOs) based on the IFSS and then use them to diagnose the stages of the ovarian cancer using the TOPSIS technique. Furthermore, the variation of the parameters used in the developed model AOs is also observed and graphically represented.

Green Supply Chain Management (GSCM) is essential to ensure environmental compliance and commercial growth in the current climate. Businesses constantly look for fresh concepts and techniques for ensuring environmental sustainability. To keep up with the new trends in environmental concerns related to company management and procedures, Green Supplier Selection (GSS) criteria are added to the traditional supplier selection processes. This study aims to identify general and environmental supplier selection criteria to provide a framework that can assist decision-makers in choosing and prioritizing appropriate green supplier selection. The development and implementation of decision support systems aimed to solve these difficulties at a rapid rate. In order to manage inaccurate data and simulate decision-making problems. Fuzzy sets introduced by Zadeh, are a useful technique to handle the imperfectness and uncertainty in different problems. Although fuzzy sets can handle incomplete information in different real worlds problems, but its cannot handle all type of uncertainty such as incomplete and indeterminate data. Therefore different extensions of fuzzy sets such as intuitionistic fuzzy, pythagorean fuzzy and q-rung orthopair fuzzy sets introduced to address the problems of uncertainty by considering the membership and non-membership grade. However, these concepts have some shortcomings in the handling uncertainty with sub-attributes. To overcome this difficulties Khan et al. developed the structure of q-rung orthopair fuzzy hypersoft sets by combining q-rung orthopair fuzzy sets with hypersoft sets. A remarkable and beneficial research work is done in the field of q-rung orthopair fuzzy hypersoft sets, and then we think about the application. In this paper, we use the structure of q-rung orthopair fuzzy hypersoft in multi-criteria supplier selection problems. For this, we present aggregation operator to solve multi-criteria decision-making (MCDM) problems with q-rung orthopair fuzzy hypersoft (q-ROFH) information, known as ordered weighted geometric aggregation operator. Since the uncertainty and vagueness is an unavoidable feature of multi-criteria decision-making problems, the proposed structure can be a useful tool for decision making in an uncertain environment. Further, the expert opinions were investigated using the multi-criteria decision-making (MCDM) technique, which helped identify interrelationship and causal preference of green supplier evaluation aspects that used aggregation operators. Finally, a numerical example of the proposed method for the task of Green Supplier Selection is presented.

In this study, the authors are interested in exploring the existing concepts of fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets, hesitant fuzzy sets, and fuzzy rough sets in order to reduce the difficulty of their capabilities and will point out the possible limitations and advantages in order to develop new structures of q-rung orthopair hesitant fuzzy rough sets, complex q-rung orthopair hesitant fuzzy rough sets, and complex q-rung orthopair probabilistic hesitant fuzzy rough sets and their basic results. The authors will also develop some new extensions to the existing literature and then compare the results with the existing notions. They will also do some work on the graphical features of existing literature and try to introduce some novel graph theoretic concepts for new generalities.

The q-rung picture fuzzy sets serve the fuzzy set theory as a competent, broader and accomplished extension of q-rung orthopair fuzzy sets and picture fuzzy sets which exhibit excellent performance in modeling the obscure data beyond the limits of existing approaches owing to the parameter q and three real valued membership functions. The accomplished strategy of VIKOR method is established on the major concepts of regret measure and group utility measure to specify the compromise solution. Further, TOPSIS method is another well established multi-criteria decision-making strategy that finds out the best solution with reference to the distances from ideal solutions. In this research study, we propose the innovative and modified versions of VIKOR and TOPSIS techniques using the numerous advantages of q-rung picture fuzzy information for obtaining the compromise results and rankings of alternatives in decision-making problems with the help of two different point-scales of linguistic variables. The procedure for the entropy weighting information is adopted to compute the normal weights of attributes. The q-rung picture fuzzy VIKOR (q-RPF VIKOR) method utilizes ascending order to rank the alternatives on the basis of maximum group utility and minimum individual regret of opponent. Moreover, a compromise solution is established by scrutinizing the acceptable advantage and the stability of decision. In the case of TOPSIS technique, the distances of alternatives to ideal solutions are determined by employing the Euclidean distance between q-rung picture fuzzy numbers. The TOPSIS method provides the ranking of alternatives by considering the descending order of closeness coefficients. For explanation, the presented methodologies are practiced to select the right housing society and the suitable industrial robot. The comparative results of the proposed techniques with four existing approaches are also presented to validate their accuracy and effectiveness.

The notion of a q-rung orthopair fuzzy soft rough set ( $^{q}ROFSRS$ ) appeared as an extension of q-rung orthopair fuzzy set ( $^{q}ROFS$ ) and q-rung orthopair fuzzy soft set ( $^{q}ROFSS$ ) with the aid of rough set (RS) definition. Thus, $^{q}ROFSRS$ and m-polar fuzzy set ( $_{m}PFS$ ) are convenient to deal with uncertain knowledge which helps us to solve many problems in decision making. In this paper, we define the soft rough q-rung orthopair m-polar fuzzy sets ( $^{\text {q}}RO_{\text {m}}PFS$ ) and q-rung orthopair m-polar fuzzy soft rough sets ( $^{q}RO_{\text {m}}PFSRS$ ) through crisp soft and q- rung orthopair (q-RO) m-polar fuzzy soft approximation space. The related characteristics of these models are also studied. Then, we construct two new algorithms for these models to solve MADM issues. The successful application and corresponding comparative analyses proves that our proposed models are rational and effective.

The q-rung orthopair fuzzy sets accommodate more uncertainties than the Pythagorean fuzzy sets and hence their applications are much extensive. Under the q-rung orthopair fuzzy set, the objective of this paper is to develop new types of q-rung orthopair fuzzy lower and upper approximations by applying the tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can get only the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. Finally a technique is devloped to rank the alternatives in a q-rung orthopair fuzzy information system based on similarity relation. We find that the proposed method/technique is more efficient when compared with other existing techniques.

A q-rung orthopair fuzzy set (q-ROFS) is an effective tool for describing uncertainty and fuzziness, and is the promotion of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs). This paper extends q-rung orthopair fuzzy environment to probabilistic q-rung orthopair hesitant fuzzy environment, and proposes the concept of probabilistic q-rung orthopair hesitant fuzzy sets (P-q-ROHFSs). Some desired operational laws and properties of P-q-ROHFSs are studied. And we develop the some aggregation operators for probabilistic q-rung orthopair hesitant fuzzy information. Then the relationship among these operators is discussed by comparing the score function. In order to measure the uncertain information, this paper proposes the four distance measures between two P-q-ROHFSs. Especially, ranking and expansion of P-q-ROHFSs are discussed in detail. Furthermore, we apply them to pattern recognition and multi-attribute group decision making (MAGDM) under probabilistic q-rung orthopair hesitant fuzzy environment. Finally, examples are given to show the rationality and practicability of the proposed method.

The objective of the paper is to present a new concept named as Cubic q-rung orthopair fuzzy linguistic set (Cq-ROFLS) to quantify the uncertainty in the information. The proposed Cq-ROFLS is qualitative form of cubic q-rung orthopair fuzzy set (Cq-ROFS), where membership degrees (MDs) and non-membership degrees (NMDs) are represented in terms of linguistic variables. The basic notions of Cq-ROFLS have been introduced and study their basic operations and properties. Further, to aggregate the different pairs of the preferences, we introduce the Cq-ROFL Muirhead mean, weighted Muirhead mean, dual Muirhead mean based operators. The major advantages of considering the Muirhead mean is that it considers the interrelationship between more than two arguments at a time. On the other hand, the Cq-ROFLS has the ability to describe the qualitative information in terms of linguistic variables. Several properties and relation of the derived operators are argued. Besides, we also investigate multi attribute decision making (MADM) problems under the Cq-ROFLS environment and illustrate with a numerical examples. Finally, the effectiveness and advantages of the work are established by comparing with other methods.

Optimizing the allocation of preschool education resources and improving the efficiency of resource allocation is of great strategic significance for the universal and inclusive development of preschool education and the realization of “education for young children". In recent years, the shift from high-speed development to high-quality development of the social economy has significantly improved the balanced development level of China’s preschool education industry. However, preschool education remains the weakest link in China’s education system and the most unfavorable aspect of educational resource allocation. Problems such as shortage of preschool education resources, insufficient investment, uneven regional development, imbalanced supply and demand structure, low resource allocation efficiency, and “difficult to enter, expensive to enter” are still prominent. How to optimize resource allocation and improve resource utilization efficiency in the limited resources of preschool education is the key to achieving balanced, fair, coordinated, and high-quality development of preschool education. The county preschool education resource allocation level evaluation is MAGDM problems. Recently, the TODIM and TOPSIS technique was employed to cope with MAGDM issues. The interval-valued Pythagorean fuzzy sets (IVPFSs) are employed as a tool for characterizing uncertain information during the county preschool education resource allocation level evaluation. In this manuscript, the interval-valued Pythagorean fuzzy TODIM-TOPSIS (IVPF-TODIM-TOPSIS) technique is built to solve the MAGDM under IVPFSs. Finally, a numerical case study for county preschool education resource allocation level evaluation is given to validate the proposed technique. The main contribution of this paper is managed: (1) the TODIM and TOPSIS technique was extended to IVPFSs; (2) Information Entropy is employed to manage the weight values under IVPFSs. (3) the IVPF-TODIM-TOPSIS technique is founded to manage the MAGDM under IVPFSs; (4) Algorithm analysis for county preschool education resource allocation level evaluation and comparison analysis are constructed based on one numerical example to verify the feasibility and effectiveness of the IVPF-TODIM-TOPSIS technique.

Q-rung orthopair fuzzy set (q-ROFS) is a powerful tool to describe uncertain information in the process of subjective decision-making, but not express vast objective phenomenons that obey normal distribution. For this situation, by combining the q-ROFS with the normal fuzzy number, we proposed a new concept of q-rung orthopair normal fuzzy (q-RONF) set. Firstly, we defined the conception, the operational laws, score function, and accuracy function of q-RONF set. Secondly, we presented some new aggregation operators to aggregate the q-RONF information, including the q-RONF weighted operators, the q-RONF ordered weighted operators, the q-RONF hybrid operator, and the generalized form of these operators. Furthermore, we discussed some desirable properties of the above operators, such as monotonicity, commutativity, and idempotency. Meanwhile, we applied the proposed operators to the multi-attribute decision-making (MADM) problem and established a novel MADM method. Finally, the proposed MADM method was applied in a numerical example on enterprise partner selection, the numerical result showed the proposed method can effectively handle the objective phenomena with obeying normal distribution and complicated fuzzy information, and has high practicality. The results of comparative and sensitive analysis indicated that our proposed method based on q-RONF aggregation operators over existing methods have stronger information aggregation ability, and are more suitable and flexible for MADM problems.

The decision-making technique is used to evaluate the best optimal among the collection of finite alternatives. Further, the technique of complex p, q-rung orthopair fuzzy (CPQROF) set is very reliable and dominant due to parameters “p” and “q”. In contrast, the technique of simple q-rung orthopair fuzzy sets is the special case of the CPQROF set. The major theme of this article is to expose the novel theory of fairly operational laws based on CPQROF numbers (CPQROFNs). Further, we evaluate the weighted fairly aggregation operators based on CPQROF information, called CPQROF weighted fairly averaging (CPQROFWFA) operator and CPQROF ordered weighted fairly averaging (CPQROFOWFA) operator. Some properties are also discussed for the above operators. Additionally, we validate the system of multi-attribute decision-making (MADM) difficulties based on initiated operators. Finally, we associate our suggested ranking consequences with approximately prevailing techniques to demonstrate the proposed theory's sovereignty and validity.