Single-valued bipolar neutrosophic (SVbN) numbers are a distinctive set of the neutrosophic set on the real line. In decision making problem, SVbN-number has a great importance when human decision is based on both positive and negative thought. All membership functions (truth, indeterminacy, and falsity) of SVbN number are contained the positive and negative parts within the range of −1 to 0 and 0 to 1. In this article, we have defined the possibilistic ranking method of single valued bipolar neutrosophic numbers. The notions of possibilistic mean and SD of SVbN number are introduced here. A new approach possibilistic multiattribute decision making is developed according to the possibilistic mean and SD. In addition, we have invented possibilistic multiattribute decision making on water resource management problem under bipolar neutrosophic environment. Finally, a numerical example is considered with single-valued trapezoidal bipolar neutrosophic informations to show the pertinence and implementation of the proposed possibilistic decision making method.

The aim of the paper is to present the concept of a multiplicative preference relation (MPR) with the features of the single-valued neutrosophic (SVN) set and named as SVN multiplicative preference relation (SVNMPR). The SVN set (SVNS) handles uncertainties more broadly than the intuitionistic fuzzy set by considering the three independent degree systems with a scale rating of 0–1 and is symmetrical about 0.5. However, for asymmetrical distribution, Saaty discusses the 1–9 scale to display the information. Driven by the advantages of the scale of 1–9 and the independence nature of degrees in SVNS, in this paper, we introduce the concept of SVNMPR with ratings values as SVN numbers with range 1 ∕ q − q with q > 1. The main advantages of the proposed relation are the asymmetric and nonuniform distributions of about 1. To complete address this, we divide the stated work into three parts. First, an idea of SVNMPR is added wherein pairwise evaluation values are represented the usage of SVN numbers over a scale of 1 ∕ q − q with q > 1. Also, we outline the idea of the SVN multiplicative preference set (SVNMPS). Second, to rank the proposed SVNMPS, we define a new ranking method with improved score functions by adding the expert's attitude to the analysis. The relationship between the proposed and the existing functions is also derived. Third, we are defining some new operation laws for SVNMPS to preserve the closeness property. On the basis of these laws, we define a number of operators to aggregate the different pairs of the SVNMPSs and to derive a number of relations and inequalities. Finally, a group decision-making approach is established and the practicality of it is demonstrated with a numerical example and compared with a number of existing approaches.

This paper explores the multi-item vendor's inventory (VI) model in which the vendor replenishes the products from various suppliers and satisfies the retailers' demand. Different types of products scrutinize this model under two cases. The first case considers the products are replenished in bulk and packed in small quantities by the vendor and sent to the retailers. The second case presumes that the vendor replenishes the packed items from the supplier and sent to retailers without the packaging process. The wastage of products during the packaging process occurs in the first case and the deterioration of items occurs in both cases. The unpredictable parameters such as the proportion of wastage products and the proportion of deteriorated products lead the model under fuzzy logic. The VI model and the Vendor's fuzzy inventory (VFI) are framed. VFI model is formulated by assuming the wastage rate and the deterioration rate as triangular fuzzy numbers (TFN) as well as trapezoidal fuzzy numbers. The defuzzification method of TFN and trapezoidal fuzzy numbers by the combination of Beta distribution and a uniform distribution is utilized in this paper. For both cases, the optimal ordering and purchasing period, the optimal retailer's replenishment period, and the optimal packaging period for case (i) is computed which minimizes the vendor's total cost. The computational example and sensitivity analysis are performed to flourish the study more effectively.

Complex q-rung orthopair fuzzy sets (CQROFSs) are proposed to convey vague material in decision-making problems. The CQROFSs can enthusiastically modify the region of proof by altering the factor q ≥ 1 for real and imaginary parts based on the variation degree and, therefore, favor further uncountable options. Consequently, this set reverses over the existing theories, such as complex intuitionistic fuzzy sets (CIFSs) and complex Pythagorean fuzzy sets (CPFS). In everyday life, there are repeated situations that can occur, which involve an impartial assertiveness of the decision-makers. To determine the best decision to handle such situations, in this study, we propose modern operational laws by joining the characteristics of the truth factor sum and collaboration between the truth degrees into the analysis for CQROFSs. Based on these principles, we determined several weighted averaging neutral aggregation operators (AOs) to collect the CQROF knowledge. Subsequently, we established an original multiattribute decision-making (MADM) procedure by using the demonstrated AOs based on CQROFS. To evaluate the effectiveness, in terms of reliability and consistency, of the proposed operators, they were applied to some numerical examples. A comparative analysis of the investigated operators and other existing operators was also performed to find the dominance and validity of the introduced MADM method.

Harris hawks optimization (HHO) is one of the newest metaheuristic algorithms (MHAs) which mimic the interdependent behaviour and hunting style of Harris hawks in nature. It is an efficient swarm optimization technique that has been used to solve various kinds of optimization problems. However, for some optimization cases, it has a tendency to be trapped into local search space and it endures an improper balance between exploitation and exploration. To get rid of this situation and to explore the global searching ability of HHO, an effective hybrid method improved teaching–learning HHO (ITLHHO) has been developed using improved teaching–learning-based optimization for solving different kinds of engineering design and numerical optimization problems. The performance of ITLHHO has been demonstrated by 33 well-known benchmark functions, including IEEE Congress of Evolutionary Computation Benchmark Test Functions (CEC-C06, 2019 Competition) and 10 multidisciplinary challenging engineering optimization problems. After illustration, the outcomes of the proposed ITLHHO are compared with several recently developed competitive MHAs. Additionally, the ITLHHO results are statistically investigated with the Wilcoxon rank-sum test and multiple comparison test to show the significance of the results. The experimental results suggest that ITLHHO significantly outperforms other algorithms and becomes a remarkable and promising tool for solving various kinds of optimization problems.

As a generalization of the interval-valued intuitionistic fuzzy sets, a consciousness of interval-valued q-rung orthopair fuzzy sets (IV q-ROFSs) is a robust and trustworthy tool to fulfill the imprecise information with an adaptation of the manageable parameter q ≥ 1. However, the ranking of any interval-valued numbers is very valuable for interval-valued decision-making problems. Possibility degree measure is a worthy tool to manage the degree of possibility of one object over the other. Driven by these requisite characteristics, it is fascinating to manifest the possibility degree of comparison between two IV q-ROFSs, and an innovative method is then encouraged to rank the given numbers. Few properties are checked to explain their features and exhibited the advantages of it over the existing possibility measures with some counterintuitive examples. Later on, we consider the multiattribute group decision making (MAGDM) method and embellish it with numerical examples, to rank the alternatives. Several numerical examples are implemented to test the superiority of the stated MAGDM method and to confer its more manageable and adaptable nature.

The aim of the paper is to introduce a novel concept of connection number-based q-rung orthopair fuzzy set (CN- q-ROFS) and thus to develop a method for solving the multiattribute group decision-making (MAGDM) problem. The q-rung orthopair fuzzy set ( q-ROFS) has outstanding characteristics to deal with the uncertain information, while the connection number resolves with the uncertainties and certainty as a three-degree system, namely identity, contrary, and discrepancy. By combining these two ideas, this paper is divided into three parts. First, the CN- q-ROFS concept is introduced and its features are read. Moreover, some operation laws are defined to combine the different elements of the proposed set. Second, a novel measure of the degree of possibility is presented to measure the degree of possibility within the given objects. Finally, a group decision-making method for solving problems with uncertain information is propagated and illustrated with a series of examples. Finally, a group decision-making method is promoted to solve the problems of uncertain information and illustrates a number of examples. The advantages, comparative as well as superiority analysis of the proposed framework are provided to confirm the approach.

As a useful tool, probabilistic hesitant fuzzy set is an enhanced version for hesitant fuzzy set. It could be used to model the uncertainty very effectively. However, in probabilistic hesitant fuzzy risky multiple attribute decision making problems, the occurrence probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status are often difficult to obtain by subjective evaluation of a decision maker. This paper aims to propose two nonlinear programming models for calculating the probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status respectively. First, a nonlinear programming model using maximum entropy principle is established for determining the probabilities of elements in a probabilistic hesitant fuzzy element. Second, by introducing the water-filling theory, we put forward its extension and design a novel mathematical programming model to determine the probability of risk status. Moreover, we have proved that both the two mathematical programming models are convex programming models and their global optimal solutions can be found. Thirdly, the collective overall expected values of alternatives are calculated and the ranking order can be derived. Then, the selection of investment project is investigated, and comparison analysis shows the superiority of the presented approach.

This paper aspires to present single and group decision-making methods based on complex intuitionistic fuzzy (CIF) preference relations (CIFPRs). The presented work is divided into three folds. The first fold is that the concept of CIFPRs is introduced in this study in which the pairwise comparison values are represented using CIF numbers which have the characteristic of portraying membership and nonmembership degrees over the unit disc of the complex plane. The conditions for additive consistent CIFPR are defined and the related results are explored. In addition, an algorithm for rectifying inconsistent CIFPR is also developed. The second fold is that two goal programming models are established for generating CIF normalized priority weights corresponding to individual and group decision-making problems. The third fold is to develop the algorithms for handling individual and group decision-making problems with CIFPRs. The practicality of the proposed algorithms is shown by applying them to real-life decision-making problems. The results of the presented methods are compared with several existing studies.

In this paper, we develop a multicriteria group decision-making methodology with probabilistic linguistic q-rung orthopair fuzzy sets (PLqROFSs). The benefit of choosing PLqROFSs is that they consider the simultaneous occurrence of stochastic and nonstochastic uncertainty and so are superior to probabilistic hesitant fuzzy sets, linguistic intuitionistic fuzzy sets, and linguistic Pythagorean fuzzy sets. To develop the methodology, we first propose two types of operators, namely, probabilistic linguistic q-rung orthopair fuzzy weighted Generalized Dombi operator which gives the flexibility of choosing the parameter values and probabilistic linguistic q-rung orthopair fuzzy weighted Generalized Dombi Bonferroni mean operator which can consider the interrelationships between criteria. Since both the subjective and objective weights of experts are hardly considered in the current PLqROFSs-based research, so in our proposed methodology, we deploy the thought of consistency and similarity between the experts to calculate the subjective and objective weights, respectively, of the experts and as a result the evaluation results do not get malformed. For measuring the weights of criteria, the gray correlation coefficient of the assessment value of criteria is used to reflect the similarity between the criteria and its reference value. To exhibit the applicability of the proposed operators, we provide a case study on biomass feedstock selection. Moreover, to make sure that our model is stable, we have investigated a sensitivity analysis of parameters. At the end, we make a comparison of our approach to different existing schemes.

Fuzzy linear programming (FLP) problems with a wide varietyof applications in sciencesand engineering allow working with imprecise data and constraints, leading to more realistic models. The main contribution of this study is to deal with the formulation of a kind of FLP problems, known as bounded interval-valued fuzzy numbers linear programming (BIVFNLP) problems, with coefficients of decision variables in the objective function, resource vector, and coefficients of the technological matrix represented as interval-valued fuzzy numbers (IVFNs), and crisp decision variables limited to lower and upper bounds. Here, based on signed distance ranking to order IVFNs, the bounded simplex method is extended to obtain an interval-valued fuzzy optimal value for the BIVFNLP problem under consideration. Finally, one illustrative example is given to show the superiority of the proposed algorithm over the existing ones.

The aim of this paper is to develop some new logarithm operational laws (LOL) with real number base for the Pythagorean fuzzy sets. Some properties of LOL have been studied and based on these, various weighted averaging and geometric operators have been developed. Then, we utilized it to solve the decision-making problems. The validity of the proposed method is demonstrated with a numerical example and compared the results with the several existing approaches result. Finally, the influences of logarithmic operation and the selection of the logarithmic base in practice are discussed.

Hesitant Pythagorean fuzzy (HPF) sets can easily express the uncertain information while Maclaurin symmetric mean (MSM) operator, can capture the interrelationship among the multiattributes, and are suitable for aggregating the information into a single number. By taking the advantages of both, in this paper, we extend the traditional MSM to HPF environment. For this, we develop the HPFMSM operator for aggregating the HPF information. The desirable characteristics, such as idempotency, monotonicity, and boundedness, are studied. Then, we discussed some special cases with respect to the parameter value of the HPFMSM operators and showed that it generalizes the various existing operators. Furthermore, we studied the weighted HPFMSM operator to aggregate the HPF information with different preferences to the input arguments. On the basis of these operators, we solved the multiattribute decision-making problems with HPF information. The practicality and effectiveness of the developed approach are demonstrated through a numerical example.

Pythagorean fuzzy sets (PFSs) accommodate more uncertainties than Lx the intuitionistic fuzzy sets and hence its applications are more extensive. Under the PFS, the objective of this paper is to develop some new operational laws and their corresponding weighted geometric aggregation operators. For it, we define some new neutral multiplication and power operational laws by including the feature of the probability sum and the interaction coefficient into the analysis to get a neutral or a fair treatment to the membership and nonmembership functions of PFSs. Associated with these operational laws, we define some novel Pythagorean fuzzy weighted, ordered weighted, and hybrid neutral geometric operators for Pythagorean fuzzy information, which can neutrally treat the membership and nonmembership degrees. The desirable relations and the characteristics of the proposed operators are studied in details. Furthermore, a multiple attribute group decision-making approach based on the proposed operators under the Pythagorean fuzzy environment is developed. Finally, an illustrative example is provided to show the practicality and the feasibility of the developed approach.

The aim of this paper is to develop some new power aggregation operators for intuitionistic fuzzy (IF) soft numbers. The aggregation operators are named as IF soft power averaging (IFSPA) operator, weighted IFSPA (WIFSPA) operator, ordered WIFSPA operator, IF soft power geometric (IFSPG) operator, and weighted and ordered weighted IFSPG aggregation operators. The salient features of these operators are discussed in detail. Further, these operators are extended to its generalized version and called generalized IFSPA or geometric aggregation operators. Then, we utilized these operators to develop an approach to solve the decision-making problem under IF soft set environment and demonstrated with an illustrative example. A comparative analysis of existing approaches has been done for showing the validity of the proposed work.

Complex intuitionistic fuzzy sets (CIFSs), modeled by complex-valued membership and nonmembership functions with codomain the unit disc in a complex plane, handle two-dimensional information in a single set. Under this environment, the primary objective of the present study is to introduce some novel formulae of information measures (similarity measures, distance measures, entropies, and inclusion measures) and discuss the transformation relationships among them. To demonstrate the efficiency of the proposed similarity measures, we apply it to pattern recognition problem and a detailed comparative analysis is conducted with some of the existing measures. Further, algorithms based on proposed measures are developed for handing multicriteria decision-making problems and their working is illustrated with the help of an example. Besides this, the practicality of the proposed similarity measure is demonstrated by developing a clustering algorithm under CIFS environment.

In this article, we define two new exponential operational laws about the interval-valued Pythagorean fuzzy set (IVPFS) and their corresponding aggregation operators. However, the exponential parameters (weights) of all the existing operational laws of IVPFSs are crisp values in IVPFS decision-making problems. As a supplement, this paper first introduces new exponential operational laws of IVPFS, where bases are crisp values or interval numbers and exponents are interval-valued Pythagorean fuzzy numbers. The prominent characteristic of these proposed operations is studied. Based on these laws, we develop some new weighted aggregation operators, namely the interval-valued Pythagorean fuzzy weighted exponential averaging operator and the dual interval-valued Pythagorean fuzzy weighted exponential averaging. Finally, a decision-making approach is presented based on these operators and illustrated with some numerical examples to validate the developed approach.

In this article, a new decision-making model with probabilistic information and using the concept of immediate probabilities has been developed to aggregate the information under the Pythagorean fuzzy set environment. In it, the existing probabilities have been modified by introducing the attitudinal character of the decision maker by using an ordered weighted average operator. Based on it, we have developed some new probabilistic aggregation operator with Pythagorean fuzzy information, namely probabilistic Pythagorean fuzzy weighted average operator, immediate probability Pythagorean fuzzy ordered weighted average operator, probabilistic Pythagorean fuzzy ordered weighted average, probabilistic Pythagorean fuzzy weighted geometric operator, immediate probability Pythagorean fuzzy ordered weighted geometric operator, probabilistic Pythagorean fuzzy ordered weighted geometric, etc. Furthermore, we extended these operators by taking interval-valued Pythagorean fuzzy information and developed their corresponding aggregation operators. Few properties of these operators have also been investigated. Finally, an illustrative example about the selection of the optimal production strategy has been given to show the utility of the developed method.

In this article, a new linguistic Pythagorean fuzzy set (LPFS) is presented by combining the concepts of a Pythagorean fuzzy set and linguistic fuzzy set. LPFS is a better way to deal with the uncertain and imprecise information in decision making, which is characterized by linguistic membership and nonmembership degrees. Some of the basic operational laws, score, and accuracy functions are defined to compare the two or more linguistic Pythagorean fuzzy numbers and their properties are investigated in detail. Based on the norm operations, some series of the linguistic Pythagorean weighted averaging and geometric aggregation operators, named as linguistic Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric with linguistic Pythagorean fuzzy information are proposed. Furthermore, a multiattribute decision-making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.

q-Rung orthopair fuzzy set (q-ROFS) is a powerful tool that attracts the attention of many scholars in dealing with uncertainty and vagueness. The aim of paper is to present a new score function of q-rung orthopair fuzzy number (q-ROFN) for solving the failure problems when comparing two q-ROFNs. Then a new exponential operational law about q-ROFNs is defined, in which the bases are positive real numbers and the exponents are q-ROFNs. Meanwhile, some properties of the operational law are investigated. Later, we apply them to derive the q-rung orthopair fuzzy weighted exponential aggregation operator. Additionally, an approach for multicriteria decision-making problems under the q-rung orthopair fuzzy data is explored by applying proposed aggregation operator. Finally, an example is investigated to illustrate the feasibility and validity of the proposed approach. The salient features of the proposed method, compared to the existing q-rung orthopair fuzzy decision-making methods, are (1) it can obtain the optimal alternative without counterintuitive phenomena; (2) it has a great power in distinguishing the optimal alternative.

The objective of this paper is to present some series of geometric-aggregated operators under Pythagorean fuzzy environment by relaxing the condition that the sum of the degree of membership functions is less than one with the square sum of the degree of membership functions is less than one. Under these environments, aggregator operators, namely, Pythagorean fuzzy Einstein weighted geometric, Pythagorean fuzzy Einstein ordered weighted geometric, generalized Pythagorean fuzzy Einstein weighted geometric, and generalized Pythagorean fuzzy Einstein ordered weighted geometric operators, are proposed in this paper. Some of its properties have also been investigated in details. Finally, an illustrative example for multicriteria decision-making problems of alternatives is taken to demonstrate the effectiveness of the approach.

The objective of this work is to present an improved accuracy function for the ranking order of interval-valued Pythagorean fuzzy sets (IVPFSs). Shortcomings of the existing score and accuracy functions in interval-valued Pythagorean environment have been overcome by the proposed accuracy function. In the proposed function, degree of hesitation between the element of IVPFS has been taken into account during the analysis. Based on it, multicriteria decision-making method has been proposed for finding the desirable alternative(s). Finally, an illustrative example for solving the decision-making problem has been presented to demonstrate application of the proposed approach.

The objective of this article is to extend and present an idea related to weighted aggregated operators from fuzzy to Pythagorean fuzzy sets (PFSs). The main feature of the PFS is to relax the condition that the sum of the degree of membership functions is less than one with the square sum of the degree of membership functions is less than one. Under these environments, aggregator operators, namely, Pythagorean fuzzy Einstein weighted averaging (PFEWA), Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA), generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA), and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA), are proposed in this article. Some desirable properties corresponding to it have also been investigated. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean fuzzy environment to show the validity, practicality, and effectiveness of the new approach. Finally, a systematic comparison between the existing work and the proposed work has been given.

Pythagorean fuzzy set (PFS) is one of the most successful in terms of representing comprehensively uncertain and vague information. Considering that the correlation coefficient plays an important role in statistics and engineering sciences, in this paper, after pointing out the weakness of the existing correlation coefficients between intuitionistic fuzzy sets (IFSs), we propose a novel correlation coefficient and weighted correlation coefficient formulation to measure the relationship between two PFSs. Pairs of membership, nonmembership, and hesitation degree as a vector representation with the two elements have been considered during formulation. Numerical examples of pattern recognition and medical diagnosis have been taken to demonstrate the efficiency of the proposed approach. Results computed by the proposed approach are compared with the existing indices.