Picture Fuzzy Numbers Research Articles

Career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory

Since the future of the society depends upon the role of students, so suitable career selection methods for the students are considered to be an important problem to explore. It is assumed that if a student has the required capability and positive attitudes towards a subject, then the student will achieve more in that subject. To consider the uncertain issues involved with students’ career selection, picture fuzzy set (PFS) and rough set based approaches are proposed in this study as they are found to be appropriate due to their inherent characteristics to deal with incomplete and imprecise information. For the purpose of selecting a suitable career, the article analyzes student's features in terms of career, memory, interest, knowledge, environment and attitude. We propose two hybridized distance measures using Hausdorff, Hamming and Euclidian distances under picture fuzzy environment where the evaluating information regarding students, subjects and student's features are given in picture fuzzy numbers. Then we present an algorithmic approach using the proposed distance measures and rough set theory. We apply rough set theory to determine whether a particular subject is suitable for a student even if there is controversy to select a stream. Lower and higher approximation with boundary region of rough set theory is used to manage the inconsistent situations. Finally, two case studies are demonstrated to validate the applicability of the proposed idea.

A Cognitive Information-Based Decision-Making Algorithm Using Interval-Valued q-Rung Picture Fuzzy Numbers and Heronian Mean Operators

The complexity of the socioeconomic environment means that it is challenging to make decisions that rely on cognitive information. Decision makers normally cannot obtain a precise or sufficient level of knowledge about the problem domain and hence must provide multiple answers with interval values to depict them. This makes cognizing and decision making very difficult. To address this issue, this paper proposes a novel cognitive information-based decision-making algorithm with interval-valued q-rung picture fuzzy (IVq-RPtF) numbers. We first define the concept of the IVq-RPtF set, including the basic definition, operational laws, a score function, and an accuracy function. Considering the interrelationship between attributes, we then present the IVq-RPtF Heronian mean (IVq-RPtFHM) operators using the new operational laws. Moreover, we discuss the properties of IVq-RPtFHM operators, such as monotonicity, commutativity, and idempotency. Finally, we use a numerical example to verify the viability of the proposed method. The results show that the proposed method effectively expresses multiple types of interval cognitive information. The sensitivity analysis of the parameters shows that the ranking results are susceptible to parameter changes, but regardless of how the parameters change, the score values of the four alternatives in our example are in the range of [1.27, 1.66], within the basic scoring range of [1.352–1.472] for the four alternatives. Therefore, our proposed method based on IVq-RPtFHM operators has a stronger information aggregation ability than other methods. Compared with other methods, the proposed cognitive information-based decision-making algorithm is more widely applicable, avoids loss of cognitive information, and conducts a reasonable decision-making process.

A Novel Approach Based on Sine Trigonometric Picture Fuzzy Aggregation Operators and Their Application in Decision Support System

Picture fuzzy sets (PFSs) are one of the fundamental concepts for addressing uncertainties in decision problems, and they can address more uncertainties compared to the existing structures of fuzzy sets; thus, their implementation was more substantial. The well-known sine trigonometric function maintains the periodicity and symmetry of the origin in nature and thus satisfies the expectations of the decision-maker over the multiple parameters. Taking this feature and the significances of the PFSs into consideration, the main objective of the article is to describe some reliable sine trigonometric laws STLs for PFSs. Associated with these laws, we develop new average and geometric aggregation operators to aggregate the picture fuzzy numbers. Also, we characterized the desirable properties of the proposed operators. Then, we presented a group decision-making strategy to address the multiple attribute group decision-making (MAGDM) problem using the developed aggregation operators and demonstrated this with a practical example. To show the superiority and the validity of the proposed aggregation operations, we compared them with the existing methods and concluded from the comparison and sensitivity analysis that our proposed technique is more effective and reliable.

A hybrid TODIM approach with unknown weight information for the performance evaluation of cleaner production

In the practical decision-making process, it is common that more than one type of evaluation information is adopted for comprehensive assessment. For example, decision-makers are accustomed to evaluating quantitative evaluation information with real numbers, whereas evaluating qualitative evaluation information with fuzzy numbers. In this study, a hybrid multi-criteria decision-making (MCDM) method based on an extended acronym in Portuguese of interactive and multiple attribute decision-making (TODIM) approach with unknown weight information is proposed. First, different criteria values are denoted as real numbers and picture fuzzy numbers (PFNs), respectively. Given that weight information is completely unknown, the best–worst method is modified with PFNs to determine subjective weights. Meanwhile, the entropy weight method is recommended to calculate objective weights. Then, a combined criteria weight vector can be obtained by considering subjective preferences and objective facts, simultaneously. Thereafter, the traditional TODIM approach is extended with hybrid evaluation information to obtain final ranking order. Finally, an example of evaluating the performance of cleaner production for coal mines is illustrated to demonstrate the feasibility of the proposed method. Furthermore, the robustness and strengths of this method are verified by sensitivity and comparison analyses. The results show that the proposed method is feasible and provides references for solving hybrid MCDM problems with unknown weight information.

A Fuzzy System of Operation Safety Assessment Using Multimodel Linkage and Multistage Collaboration for In-Wheel Motor

To simultaneously monitor some electrical or mechanical faults of an in-wheel motor and intelligently evaluate the operation safety, in this article, a fuzzy system of operation safety assessment (OSA) is proposed in this article. This method first uses many symptom parameters (SPs), such as root mean square, crest factor, temperature rise, and current covariance to express the features of the electrical and mechanical faults from different perspectives, such as vibration, noise, temperature, current, and voltage; possibility theory is employed to translate the probability density function of each SP into the possibility function, and sample data are gradually updated to optimize the possibility function for obtaining the SPs’ membership functions that are evaluation models. Second, the probabilities of the current operation state, that is safety, attention, or danger, are obtained from each evaluation model in a stage. Picture fuzzy set (PFS) is used to define a basic picture fuzzy number (PFN); thus, many PFNs from multiple models and multiple stages are used to establish an OSA's decision matrix. Third, Mahalanobis distance is reintegrated into PFS's theory for objectively judging the real-time evaluation information, and the best–worst method is used to estimate subjectively the initial evaluation experience; thus, the multimodel linkage mechanism is designed. Finally, TODIM is modified to define the relative safety ratio, and prospect theory is employed to structure the global index for formulating the multistage collaboration approach; thus, a fuzzy OSA's system is established. The effectiveness of the proposed method was verified by the experimental analysis for the operation safety of in-wheel motor with electrical and mechanical faults.

Picture Fuzzy Geometric Aggregation Operators Based on a Trapezoidal Fuzzy Number and Its Application

The picture fuzzy set is a generation of an intuitionistic fuzzy set. The aggregation operators are important tools in the process of information aggregation. Some aggregation operators for picture fuzzy sets have been proposed in previous papers, but some of them are defective for picture fuzzy multi-attribute decision making. In this paper, we introduce a transformation method for a picture fuzzy number and trapezoidal fuzzy number. Based on this method, we proposed a picture fuzzy multiplication operation and a picture fuzzy power operation. Moreover, we develop the picture fuzzy weighted geometric (PFWG) aggregation operator, the picture fuzzy ordered weighted geometric (PFOWG) aggregation operator and the picture fuzzy hybrid geometric (PFHG) aggregation operator. The related properties are also studied. Finally, we apply the proposed aggregation operators to multi-attribute decision making and pattern recognition.

CPT-TODIM method for picture fuzzy multiple attribute group decision making and its application to food enterprise quality credit evaluation

With the development of society, people’s living standard is constantly improving. Meanwhile, people need various food to satisfy their needs in daily life. Under this situation, more and more food enterprises are appearing in the market. However, some issues about food safety come out. Because of the huge number of food company, managers are difficult in achieving profitability. Therefore, some of the managers try to use some unhealthy materials to produce food in the society. So, it is important for people to distinguish healthy and unhealthy food enterprises in their daily life. In order to help government discern and control the quality of healthy food enterprises in the market, we need to propose an effective evaluation system in estimating food enterprises. In this paper, we introduce a method of evaluating the quality degree of food enterprises which can help us to distinguish enterprises effectively. As we all know, the method of TODIM is widely used in multiple attribute decision making (MADM). In this article, we describe the extended TODIM which based on the cumulative prospect theory (CPT) with picture fuzzy numbers (PF-CPT-TODIM) and use it to evaluate food companies. What’s more, we use entropy method to decide the weights of various attributes. Finally, we select optimal enterprise by using the PF-CPT-TODIM method. Furthermore, we use the comparison of the results of classical PF-TODIM method and PFWA operators to test the availability of PF-CPT-TODIM. It not only can enrich decision-making methods but also make up for the traditional PF-TODIM method in considering the psychological aspects of decision makers.

ERP selection using picture fuzzy CODAS method

The purpose of this study is to develop an extension of CODAS method using picture fuzzy sets. In this respect, a new methodology is introduced to figure out how picture fuzzy numbers can be applied to CODAS method. COmbinative Distance-based Assessment (CODAS) is a new MCDM method proposed by Ghorabaee et al. Picture fuzzy sets (PFSs) are a new extension of ordinary fuzzy sets for representing human’s judgments having possibility more than two answers such as yes, no, refusal and neutral. Compared with other studies, the proposed method integrates multi-criteria decision analysis with picture fuzzy uncertainty based on Euclidean and Taxicab distances and negative ideal solution. ERP system selection problem is handled as the application area of the developed method, picture fuzzy CODAS. Results indicate that the new proposed method finds meaningful rankings through picture fuzzy sets. Comparative analyzes show that the presented method gives successful and robust results for the solutions of MCDM problems under fuzziness.

Multi-criteria decision-making methods based on q-rung picture fuzzy information

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.

A sustainability evaluation framework for WET-PPP projects based on a picture fuzzy similarity-based VIKOR method

Abstract This paper aims to construct a sustainability evaluation framework based on the extended VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) method under a picture fuzzy environment. First, a sustainability evaluation index system for water environment treatment public-private-partnership (WET-PPP) projects is constructed from the perspectives of the economy, engineering, environment, management, and society, including five criteria and 20 sub-criteria. The constructed index system considers engineering sustainability and management sustainability, and can thus more comprehensively evaluate the “sustainability” of the project and meet the requirements of the new economic environment. Second, a corresponding method for the sustainability evaluation of WET-PPP projects is proposed based on the picture fuzzy similarity-based VIKOR method. The main advantages of the proposed method include the following: (1) the sustainability evaluation process for WET-PPP projects involves linguistic terms and picture fuzzy numbers (PFNs), which can not only more comprehensively describe the uncertainty of human cognition and decision-making processes in the form of positive, neutral, and negative membership functions, but can also reduce the difficulty for decision-makers (DMs) to provide evaluation information; (2) the proposed evaluation method combines picture fuzzy similarity and the traditional VIKOR method, and can therefore not only obtain compromise solutions but can also ensure the stability and feasibility of the evaluation results. Finally, a case study of WET-PPP projects in China is provided to testify the feasibility and effectiveness of the constructed framework. The results indicate that the proposed framework can improve the accuracy of decision-making and can be applied for handling evaluation problems.

Multi-period evaluation and selection of rural wastewater treatment technologies: a case study.

Rapid population growth and agricultural development are generating a considerable amount of effluents, which poses threats to the quality of rural water resources as well as sanitary conditions. However, with a range of rural wastewater treatment (WT) technologies available, one major problem facing the practitioners is which to choose as the most favorable option suited to specific areas. In this study, a novel decision-making framework is proposed to evaluate and select the optimal alternative in rural areas of Xi'an within multiple consecutive time periods. Firstly, an evaluation index system is constructed and picture fuzzy numbers (PFNs) are used to represent both evaluation levels and experts' refusal due to limitation of knowledge. Secondly, fuzzy analytical hierarchy process (FAHP) is applied to derive weights of criteria, which enables experts to assign fuzzy numbers to express their preferences for comparison judgments. Thirdly, evidence theory is utilized to obtain the aggregated values from multiple time periods. Finally, based on the belief intervals obtained, sequencing batch reactor (A4) is determined as the optimal rural WT technology in Xi'an from 2006 to 2020, whereas the membrane bio-reactor (A2) is the last option. The effectiveness of the proposed framework is further validated by comparative analysis. This research can hopefully serve as useful guidance for the assessment of rural WT technologies in various regions.

Multiple-attribute decision-making based on picture fuzzy Archimedean power Maclaurin symmetric mean operators

In this paper, a novel multiple-attribute decision-making method based on a set of Archimedean power Maclaurin symmetric mean operators of picture fuzzy numbers is proposed. The Maclaurin symmetric mean operator, power average operator, and operational rules based on Archimedean T-norm and T-conorm are introduced into picture fuzzy environment to construct the aggregation operators. The formal definitions of the aggregation operators are presented. Their general and specific expressions are established. The properties and special cases of the aggregation operators are, respectively, explored and discussed. Using the presented aggregation operators, a method for solving the multiple-attribute decision-making problems based on picture fuzzy numbers is designed. The method is illustrated through example and experiments and validated by comparisons. The results of the comparisons show that the proposed method is feasible and effective that can provide the generality and flexibility in aggregation of values of attributes and consideration of interactions among attributes and the capability to lower the negative effect of biased attribute values on the result of aggregation.

A new picture fuzzy information measure based on shannon entropy with applications in opinion polls using extended VIKOR–TODIM approach

Picture fuzzy set theory developed by Cuong and Kreinovich is an extension of the fuzzy set theory and intuitionistic fuzzy set theory. In this paper, we proposed a new framework for picture fuzzy entropy from a probabilistic viewpoint. The justification of the proposed axiomatic structure is established by offering a new information measure based on Shannon entropy under picture fuzzy environment and also studied its mathematical properties. Besides, we developed an algorithm for picture fuzzy set with the help of TODIM (a Portuguese acronym for Interactive Multi-Criteria Decision Making) and VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) methods to explain the multi-criteria decision-making (MCDM) problems with picture fuzzy numbers. Finally, two numerical examples are given to verify the proposed approach based on opinion surveys to anticipate the election results and the output is reasonably compared with other MCDM method existing in the literature, what’s more, the viable experiment results are gotten.

An optimization study based on Dijkstra algorithm for a network with trapezoidal picture fuzzy numbers

Path finding models attempt to provide efficient approaches for finding shortest paths in networks. A well-known shortest path algorithm is the Dijkstra algorithm. This paper redesigns it in order to tackle situations in which the parameters of the networks may be uncertain. To be precise, we allow that the parameters take the form of special picture fuzzy numbers. We use this concept so that it can flexibly fit the vague character of subjective decisions. The main contributions of this article are fourfold: $$\mathrm{(i)}$$ The trapezoidal picture fuzzy number along with its graphical representation and operational laws is defined. $$\mathrm{(ii)}$$ The comparison of trapezoidal picture fuzzy numbers on the basis of their expected values is proposed in terms of their score and accuracy functions. $$\mathrm{(iii)}$$ Based on these elements, we put forward an adapted form of the Dijkstra algorithm that works out a picture fuzzy shortest path problem, where the costs associated with the arcs are captured by trapezoidal picture fuzzy numbers. Also, a pseudocode for the application of our solution is provided. $$\mathrm{(iv)}$$ The proposed algorithm is numerically evaluated on a transmission network to prove its practicality and efficiency. Finally, a comparative analysis of our proposed method with the fuzzy Dijkstra algorithm is presented to support its cogency.

Hybrid PSO-WDBA method for the site selection of tailings pond

The disposal of tailings is a vital process in mines. This study aims to address site selection issue of tailings pond using a suitable method. First, as different types of evaluation criteria exist, hybrid decision making information is considered. That is, crisp numbers are used to express quantitative values, while picture fuzzy numbers (PFNs) are suggested to describe qualitative information. Moreover, new comparison methods of PFNs are introduced to overcome the existing limitation. Considering the situation where weight information is partly known, a nonlinear optimization model is constructed. Then, to calculate criteria weights, the particle swarm optimization (PSO) algorithm is modified by introducing penalty functions. Afterwards, the weighted distance based approximation (WDBA) method is extended with hybrid evaluation information to rank alternatives. Last, the hybrid PSO-WDBA technique is applied in selecting the best tailings pond site in a gold mine. The effectiveness and advantages of this method are also verified through detailed discussions. The results show that the proposed evaluation model is dependable and effective, and can provide references for the site selection and management of tailings pond.

Some novel aggregation operators for cubic picture fuzzy information: application in multi-attribute decision support problem

Cubic picture fuzzy set (CPFS) is the hybrid version of the interval-valued picture fuzzy set (IPVFS) and the picture fuzzy set (PFS) simultaneously. Clearly, the CPFS contains significantly more information than the general IVPFS and PFS in order to address the vulnerability of decision-making problems. Lamentably, the aggregation operators have not been researched so far on the basis of CPFS. In the techniques of group decision-making, aggregation operators are an important mathematical method to resolve ambiguity in decision-making problems. The present paper proposes few novel geometric mean aggregation operators, in particular, cubic picture fuzzy geometric and cubic picture fuzzy geometric star aggregation operators between cubic picture fuzzy numbers to aggregate the different predispositions of the decision maker. Then, based on the proposed operators in the cubic picture fuzzy environment, we developed a decision-making method and illustrated a numerical example. Finally, in order to demonstrate the supremacy of the proposed technique, classify its comparison with the existing approach. The results showed that the established decision-making technique was superior, accurate and effective.

AN INTEGRATED MULTI-CRITERIA DECISION-MAKING FRAMEWORK FOR SUSTAINABLE SUPPLIER SELECTION UNDER PICTURE FUZZY ENVIRONMENT

Sustainable supplier selection (SSS) is an important part of sustainable supply chain management (SSCM). In this paper, an integrated multi-criteria decision-making (MCDM) framework, based on the picture fuzzy exponential entropy, and the VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) method, is proposed to manage SSS problems. Firstly, the evaluation criteria of SSS, including economic, environmental and social, is established. This can be evaluated in the form of the actual data or linguistic terms provided by suppliers and experts respectively in an actual decision-making process. Then, according to the translated scales, all the evaluation information can be converted into picture fuzzy numbers (PFNs). Secondly, the picture fuzzy exponential entropy is defined. Moreover, based on the entropy’s minimization principle, the defined picture fuzzy exponential entropy is used to determine the weight of the SSS’s criteria. Thirdly, the extended VIKOR method, which combines the grey correlation coefficient, is utilized to select a suitable supplier. This method avoids the shortcomings of the traditional VIKOR method in data mining and solves the conflict between SSS criteria. Finally, the feasibility and effectiveness of the proposed integrated decision framework are verified by an experiment, as well as a sensitivity analysis and comparative analysis.

Picture fuzzy normalized projection and extended VIKOR approach to software reliability assessment

An extended VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) method in group decision-making (GDM) setting is developed in this paper. The decision information is characterized by picture fuzzy number. A new GDM model is established and applied to software reliability assessment. First, this research finds that the current projection measure is not always reasonable in picture fuzzy setting. To solve this problem, a new normalization projection measure is developed in picture fuzzy setting. In addition, this research also finds that the most VIKOR-based GDM method requires a collective decision as a carrier to get an optimal solution. However, if a collective decision is aggregated by all the individual decisions, the positive and negative information might be canceled each other out in the decision process. To solve this problem, this paper proposes a direct VIKOR-based GDM method. The values of three VIKOR indexes are determined by the original decision matrices. The feasibility and practicability developed method in this work are illustrated by an example of software reliability assessment and some experimental analyses.

Decision support framework for the risk ranking of agroforestry biomass power generation projects with picture fuzzy information

With the rapid growth of the global population and economy, energy consumption and demad are increasing sharply. As an essential renewable energy, biomass energy can promote the reform of energy production and consumption. Considering the characteristics of long investment cycle and large investment scale of agroforestry biomass power generation (AFBPG) projects, this study establishes a decision support framework for risk ranking of AFBPG project under picture fuzzy environment. The proposed framework considers not only the fuzziness and uncertainty of decision-making problems but also the decision-makers’ (DMs) psychological behavior. First, given the integrity of information representation, DMs provide risk assessment information expressed with picture fuzzy numbers, and then gives the distance of the picture fuzzy set (PFS) to maximize the PFS information. Second, the entropy weight method is used to compute the objective weight. Third, the VIKOR (Vlse Kriterijumska Optimizacija I Kompromisno Resenje) – TODIM (an acronym in Portuguese for an interactive multi-criteria decision making) method is suggested for ranking risk factors, which reflects the behavioral psychology of DMs. Moreover, the proposed evaluation model is successfully applied in a practical case. The results show that the model is valid for ranking risk factors under picture fuzzy environment. Last but not least, comparison and sensitivity analysis are implemented to verify the effectiveness and applicability of the proposed method and some suggestions for practical application are put forward.

Geometric picture fuzzy numbers and three-dimensional copulas with the non-linear programming approach

This study presents a possible relationship between two main objects, which are three-dimensional copulas (3D-Cs) and geometric picture fuzzy numbers (GPFNs). This opens up a potential field for future studies for these two objects that three-dimensional copulas can become useful tools for handling uncertainty information in the form of a picture fuzzy set (PFS). Specifically, we define a GPFN as a base element of the PFS and a defined domain of three-dimensional copulas that contains a set of GPFNs, then we show some examples of three-dimensional copulas identified on this domain. In this framework, we present the theorems related to these two objects. At the same time, we provide some examples for three-dimensional semi-copulas, three-dimensional quasi-copulas, and three-dimensional empirical copulas defined on D, which is a defined domain of a three-dimensional copula and contains a set of GPFNs D g * . In addition, we also introduce a new approach to non-linear programming problems.