Decision-making In Fuzzy Environment Research Articles

Green, resilient, and inclusive supplier selection using enhanced BWM-TOPSIS with scenario-varying Z-numbers and reversed PageRank

Firms need a reliable decision-making framework to select suppliers who are Green, Resilient, and embrace Inclusive Development (GRID). However, GRID supplier selection is challenged by events that are uncertain and often dependent. Such events affect supplier rankings in a fuzzy decision environment. This paper builds a GRID-based criteria system and applies an enhanced Best Worst Method (BWM) and Technique for Order Preferencing using the Similarity to the Ideal Solution (TOPSIS), with Scenario-Varying Z-numbers (SVZs) and reversed PageRank. The decision criteria are extracted from the literature and confirmed by expert judgement. The SVZs and SVZ Preference Relations (SVZPRs) are developed by combining the Concept of Stratification Theory (CST) with Z-numbers, to reflect the impact of Dependent Uncertain Events (DUEs). Reverse PageRank is then used to find the occurrence probability of the scenarios due to DUEs. BWM is applied in an SVZPR environment to obtain the weights of the decision criteria. TOPSIS, incorporating SVZs, is used to rank the suppliers. An example is used to highlight the enhanced BWM-TOPSIS, to find the priority ranking of the GRID weights. Comparative and sensitivity analyses are conducted to validate the proposed method.

Intuitionistic fuzzy normalized weighted geometric Bonferroni means with reducibility and boundedness and application in decision making1

Bonferroni mean (BM) is an important aggregation operator in decision making. The desirable characteristic of the BM is that it can capture the interrelationship between the aggregation arguments or the individual attributes. The optimized weighted geometric Bonferroni mean (OWGBM) and the generalized optimized weighted geometric Bonferroni mean (GOWGBM) proposed by Jin et al in 2016 are the extensions of the BM. However, the OWGBM and the GOWGBM have neither the reducibility nor the boundedness, which will lead to the illogical and unreasonable aggregation results and might make the wrong decision. To overcome these existing drawbacks, based on the normalized weighted Bonferroni mean (NWBM) and the GOWGBM, we propose the normalized weighted geometric Bonferroni mean (NWGBM) and the generalized normalized weighted geometric Bonferroni mean (GNWGBM), which can not only capture the interrelationship between the aggregation arguments, but also have the reducibility and the boundedness. Further, we extend the NWGBM and the GNWGBM to the intuitionistic fuzzy decision environment respectively, and develop the intuitionistic fuzzy normalized weighted geometric Bonferroni mean (IFNWGBM) and the generalized intuitionistic fuzzy normalized weighted geometric Bonferroni mean (GIFNWGBM). Subsequently, we prove some properties of these operators. Moreover, we present a new intuitionistic fuzzy decision method based on the IFNWGBM and the GIFNWGBM. Two application examples and comparisons with other existing methods are used to verify the validity of the proposed method.

Fuzzy Method to Improve Products and Processes Considering the Approach of Sustainable Development (FQE-SD Method)

Assumptions of the concept of sustainable development should include actions towards the development of modern, well-managed enterprises. However, making decisions in this area is difficult as it often results from subjective assessments of environmental problems. Hence, there is a motivation to develop a method of analysing the search for solutions to environmental problems that supports decisions in the area of improving the quality of products or processes while considering their impacts on the natural environment. In view of the specification of this problem, it was considered that this method should be conducted in a fuzzy decision environment. This method is called FQE-SD (fuzzy qualitatively environmentally sustainable development). This method integrated, in a hybrid way, the selected tools or elements of qualitative and multi-criteria decision methods, i.e., using the SMARTER method, brainstorming (BM), a method to select the team of experts, the Pareto-Lorenz analysis, the fuzzy QE-FMEA method, and the fuzzy AHP method. The main contribution of the FQE-SD method is its hybrid methodology, which supports: (i) a coherent and objective approach during the identification, analyses, and ranking of the causes of incompatibility of products or processes and (ii) the realization of the sustainable development of products or processes. The method was tested using the magnetic-powder test (MT). This control was carried out for producers of an outer bearing made fromAMS6470 steel. The results of this work confirmed the practical possibilities of applying the FQE-SD method. This method can also be applied to other production situations, if appropriate assumptions are made.

Solving multi-objective linear fractional decentralized bi-level decision-making problems through compensatory intuitionistic fuzzy mathematical method

This study has focused on developing a new method for solving the multi-objective linear fractional decentralized bi-level decision-making (MOLF-DBLDM) problem in an intuitionistic fuzzy decision environment. The main motivation behind this study is to determine a solution approach algorithm that is effective and simple to apply while taking actual decision-making circumstances into account. Real-world decision-making situations involve a bi-level of several decision-makers who use various decision-making processes such as agree, not certain, and disagree. By considering such real-world conditions here, all uncertain coefficients of objectives, constraint functions, and resources are portrayed as trapezoidal intuitionistic fuzzy numbers, and their crisp form is obtained through the (α,β)-cut (confidence level) concept. In the suggested method, we have transformed the crisp (α,β)-MOLF-DBLDM problem into a single objective linear decentralized bi-level decision-making (SOL-DBLDM) model using a modified linearized approach. Additionally, the upper level specified a tolerance region for its decision variables to regulate the lower levels in order to prevent decision lock. Then, the SOL-DBLDM problem is expressed as a single-level model using a new scalar function of membership and non-membership degree for each objective function at all decision-makers and upper level decision control variables. The key advantage of this method over others is that it produces an alternate, preferable compromise solution to the problem of imprecise hierarchical-level optimization. Finally, comparisons to the current methodology and appropriate numerical examples are used to demonstrate the superiority and effectiveness of the suggested method.

Method of Fuzzy Analysis of Qualitative-Environmental Threat in Improving Products and Processes (Fuzzy QE-FMEA)

Improving products and production processes is necessary to ensure the competitiveness of the organisation. As part of these improvements, the popular approach is to use the FMEA method (Failure Mode and Effect Analysis). In the traditional FMEA approach, only the qualitative aspect is included, i.e., the analysis of the quality level of products or processes, its possible incompatibilities, and then proposing improving actions for them. It seems insufficient in times of expansion of the idea of sustainable development and dynamically changing customer requirements. Hence, the purpose of the research is to develop a fuzzy QE-FMEA method to simultaneously analyze hazards for product quality and the natural environment. This method will be based on a fuzzy decision environment. The main elements of originality of the developed method are: (i) extension of the characteristics of the selection of ratings for indicators with triangular fuzzy numbers and the development of a new characteristics of the selection of ratings for the environmental impact indicator, (ii) development of a selection matrix for the qualitative-environmental indicator (QE) according to the rules of triangular fuzzy numbers, (iii) determination of the method of estimating the value of the threat priority, additionally considering the qualitative-environmental indicator (RQE). The complement of research is developed procedure of the Fuzzy QE-FMEA method. It was shown that it is possible to include the effects of incompatibilities (effects of defects occurring in products or processes), which were simultaneously assessed considering the importance and impact on the natural environment. This method will be useful for any company for analysing defects of any products or processes mainly with significant impact on the natural environment.

Application of fuzzy goal programming approach in the real-life problem of agriculture sector

Goal: The present study aimed to demonstrate the applicability of the fuzzy goal programming to frame the decision support system for the decision-makers to deal with the real-life problem of the agriculture sector namely the apple cultivation planning problem and to obtain an optimum solution. Design / Methodology / Approach: The proposed method occurred within the apple-producing sector in the Kashmir valley of India and included the collection of data through interviews and surveys with various farmers. Also, the results were drawn with the help of LINGO 18.0. Results: The current finding implies that all of the desired objectives have been met, as well as an optimal solution. The proposed model offers a significant approach for designing plans to determine various agricultural activities in a fuzzy decision environment. Finally, the current study conducts a case study in the apple cultivation sector to obtain various competing objectives. Sensitivity analysis was also performed on its preferential weight parameters. Limitations of the investigation: It should be noted that the parameters related to production cost, transportation cost, and cost of material may change over the years. The selling price may also vary according to the quality of apples and over the years. Because of natural factors, the annual production of apples may also vary. Practical implication: The current study shows that using fuzzy goal programming techniques in an apple production process has a huge potential to increase farmers' income. It can be concluded that the current study can help decision-makers to deal with real-life planning issues in the agricultural sector. Future research can also take advantage of other initiation strategies. Originality/Value: The study looks at the applicability of the fuzzy goal programming paradigm in a new field of apple production in the agriculture sector. According to our knowledge, this is the first time the optimization model has been used in the apple cultivation sector.

A decision framework for cultural and creative products based on IF-TODIM method and group consensus reaching model

Along with increasing the emphasis on cultural attributes, product design is not only satisfied with the realization of function and appearance, but also considers the embodiment of human emotion and social style. As a result, the number of creative product based on cultural style is increasing. However, existing product decision studies do not consider this style-oriented product- ranking problem, ignore the influence of cultural style and fuzzy decision-making environment on the limited psychological behavior of decision makers (DMs). Decision is a worthwhile research topic in order to facilitate ranking for cultural and creative products (CCPs). Therefore, this paper provides a decision framework based on intuitionistic fuzzy TODIM (IF-TODIM) method and group consensus reaching (GCR) model to fill this gap. Benefit from the theory of intuitionistic fuzzy set, IF-TODIM method can deal with the limited psychological behavior of DMs and the fuzziness of decision environment. This method bases on CCPs characteristics and cultural hierarchy theory (CHT) to select the decision criteria, and applies IF-TODIM method to quantify the relationship of DMs’ preferences and establish the dominance matrix for the alternatives. Furthermore, the GCP model is introduced to improve the group consensus of DMs, and the modified overall dominance matrixes are adopted to determine the alternatives scores and ranking results. The new Chinese style decorations are used as a case study to demonstrate the practicality and feasibility of the proposed method. Moreover, comparing with IF-TOPSIS method are further to verify its effectiveness and superiority.

METHOD OF PROCESSING CUSTOMERS' EXPECTATIONS TO IMPROVE THE QUALITY OF PRODUCTS

Obtaining and processing customers’ expectations for products is a problem in view of the dynamic changes of these expectations. It refers to precisely determining the most important requirements of customers to improve the product in a standardized way. The purpose of the article was to develop a method to process customers' expectations to improve the quality of products. The motivation for developing this method was to reduce uncertainty in expressing customer expectations. Using the SMART(-ER) method, the goal of the analysis was determined. Then, during brainstorming (BM), the product and the analysis criteria were selected. Next, by using a survey with the Likert scale, the customer expectations were obtained. Later, these expectations were processed by the fuzzy analytic hierarchy process (FAHP). Based on processed weights, the choice of important criteria was compiled. The choice was made according to the Pareto-Lorenz Rule. The originality of the article is based on a method, which simultaneously combines quality management tools with the method in a fuzzy decision environment. This method can be used to verify customers' expectations for any product.

Sustainable Supplier Selection Model in Supply Chains During the COVID-19 Pandemic

As global supply chains become more developed and complicated, supplier quality has become increasingly influential on the competitiveness of businesses during the Covid-19 pandemic. Consequently, supplier selection is an increasingly important process for any business around the globe. Choosing a supplier is a complex decision that can result in lower procurement costs and increased profits without increasing the cost or lowering the quality of the product. However, these decision-making problems can be complicated in cases with multiple potential suppliers. Vietnam's textile and garment industry, for example, has made rapid progress in recent years but is still facing great difficulties as the supply of raw materials and machinery depends heavily on foreign countries. Therefore, it is extremely important for textile and garment manufacturing companies in Vietnam to implement an effective supplier evaluation and selection process. While multicriteria decision-making models are frequently employed to assist with supplier evaluation and selection problems, few of these models consider the problem under the condition of a fuzzy decision-making environment. The aim of this paper is to create a hybrid MCDM model using the Fuzzy Analytical Hierarchy Process (FAHP) model and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to assist the supplier selection process in the garment industry in a fuzzy decision-making environment. In this study, the FAHP method is used to evaluate the performance and the weight of each criterion. TOPSIS is then used to rank all potential suppliers. The proposed model is then applied to a real-world case study to demonstrate both the process of calculation as well as its real-world applicability. The results from the case study provide empirical evidence that the model is feasible. The proposed approach can also be used in combination with other MCDM models to better support decision makers and can be modified to be applied in similar supplier selection processes for different industries.

INTEGRATED APPROACH OF FUZZY MULTI-ATTRIBUTE DECISION MAKING AND DATA MINING FOR CUSTOMER SEGMENTATION

This research work focuses on integrating the multi attribute decision making with data mining in a fuzzy decision environment for customer relationship management. The main objective is to analyse the relation between multi attribute decision making and data mining considering a complex problem of ordering customers segments, which is based on four criteria of customer’s life time value, viz. length (L), recency (R), frequency (F) and monetary value (M). The proposed integrated approach involves fuzzy C-means (FCM) cluster analysis as data mining tool. The experiment conducted using MATLAB 12.0 for identifying eight clusters of customers. The two multi attribute decision making tools i.e., fuzzy AHP (Analytic Hierarchy Process) and fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) are used for ranking these identified clusters. The applicability of the integrated decision making technique is also demonstrated in this paper considering the case of Indian retail sector. This research collected responses from nine experts from Indian retail industry regarding their perception of relative importance of four criteria of customer life value and evaluated weights of each criterion using fuzzy AHP. Transaction data of 18 months of the case retail store was analysed to segment 1,600 customers into eight clusters using fuzzy c-means clustering analysis technique. Finally, these eight clusters were ranked using fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution). The findings of this research could be helpful for firms in identifying the more valuable customers for them and allocate more resources to satisfy them. The findings will be also helpful in developing different loyalty program strategies for customers of different clusters.

Solution procedure for multi-objective fractional programming problem under hesitant fuzzy decision environment

For the three last decades, the multi-objective fractional programming problem has attracted the attention of many researchers due to various applications in production planning, financial field, and inventory management, and so on. The main aim of this study is to introduce a new application of hesitant fuzzy sets in real-life modeling. We intend to model multi-objective linear fractional programming problems under a hesitant fuzzy environment and present a procedure to solve them. the increasing applications of multi-objective linear fractional programming problems and the lack of research papers in this field under a hesitant fuzzy environment are the main motivations of this study. In a hesitant fuzzy set, the membership degree of an element belongs to the set can be represented by several possible values in [0,1]. These values can be chosen by different experts that cannot reach a single opinion in determining a membership degree. so, in our model several evaluations for each of goals established by decision makers based on their attitudes. The generalization of the fuzzy decision-making principle and some new concepts provide an effective solution procedure for the problem. Finally, a practical example is extended to illustrate the applicability of the proposed method.

Model Supporting Development Decisions by Considering Qualitative–Environmental Aspects

Dynamic changes in customers’ expectations and unfavorable climate changes have generated the need to consider such aspects in the process of creating new products and the modernization of existing products. Simultaneously including customers’ expectations and environmental impact is a key element of the sustainable development of products. Enterprises attempt, within their awareness and possibilities, to apply the idea of sustainability; they do this more or less methodically. As such, an instrument to support decision-making in the area of product development is still needed because it would both be desirable for customers and have less of a negative effect on the natural environment. The purpose of this study was to develop a model that supports decision-making in the development of products while considering sustainability. The model determines the key criteria of the product, criteria states (current and future), and their positive correlations (e.g., achieving high levels of product quality and no (or a reduction in) destructive impact on the environment). To reduce the fuzzy decision-making environment, multiplicative decision methods with the fuzzy Saaty scale were implemented. These methods were the fuzzy analytic hierarchy process (FAHP) and the fuzzy technique for order of preference by similarity to ideal solution (FTOPSIS). The model is able to support qualitative–environment decisions in the development of any product.

CPT-TODIM METHOD FOR INTERVAL-VALUED BIPOLAR FUZZY MULTIPLE ATTRIBUTE GROUP DECISION MAKING AND APPLICATION TO INDUSTRIAL CONTROL SECURITY SERVICE PROVIDER SELECTION

In recent years, a series of serious attacks against industrial control system (ICS), such as Stuxnet, Duqu, Flame and Havex, have sounded the alarm for industrial enterprises. For many industrial enterprises, it is a very important part of enterprise management decisions to evaluate ICS security suppliers and choose the appropriate one as a partner. The purpose of this paper is to build the TODIM method based on cumulative prospect theory (CPT-TODIM) to solve multiattribute group decision making problem under interval-valued bipolar fuzzy environment. This extraordinary model not only uses CPT to supplement the traditional TODIM method, but also introduces the entropy weight method to determine the attribute weight so as to avoid the negative influence of subjective weight on the decision result. In addition, in this model, all attribute evaluation information will be interval-valued bipolar fuzzy number (IVBFN) to cope with complex and fuzzy decision environment. Then, the most important part of this paper, we elaborate on the logical structure of the model. What’s more, in the last two sections, we apply this newly constructed model to the selection of ICS security suppliers, and verify the acceptability of this method by sensitivity analysis and comparing with IVBFWA and IVBFWG operator.

Multi-Criteria Decision-Making Methods in Fuzzy Decision Problems: A Case Study in the Frozen Shrimp Industry

The European Union (EU) is the largest shrimp consumer market in the world in terms of requirements for shrimp product imports. Therefore, other enterprises that export frozen shrimp to the EU must consider many criteria when choosing suppliers of raw shrimp. The difficulty of choosing suppliers of raw shrimp makes selecting raw material suppliers in the fisheries sector a multi-criteria decision-making problem. In such problems, the decision makers must review and evaluate many criteria—including qualitative and quantitative factors—to achieve an optimal result. While there have been multiple multi-criteria decision making models developed to support supplier selection processes in different industries, none of these have been developed to solve the particular problems facing the shrimp industry, especially as it concerns a fuzzy decision-making environment. In this research, the authors propose a Multi-Criteria Decision Making model (MCDM) including the Fuzzy Analytical Network Process (FANP) and Weighted Aggregated Sum Product Assessment (WASPAS) for the evaluation and selection process of shrimp suppliers in the fisheries industry. The model is applied to a real-world case study and the results show that Supplier 3 (SA3) is the most optimal supplier of raw shrimp. The contribution of this work is the employment of FANP and WASPAS to propose an MCDM for ranking potential suppliers in the fisheries industry in a fuzzy environment. The proposed approach can also be modified to support complex decision-making processes in fuzzy environments in different industries.

A Model for Selecting a Biomass Furnace Supplier Based on Qualitative and Quantitative Factors

In developing countries, solar energy is the largest source of energy, accounting for 35%–45% of the total energy supply. This energy resource plays a vital role in meeting the energy needs of the world, especially in Vietnam. Vietnam has favorable natural conditions for this energy production. Because it is hot and humid, and it has much rainfall and fertile soil, biomass develops very quickly. Therefore, byproducts from agriculture and forestry are abundant and continuously increasing. However, byproducts that are considered natural waste have become the cause of environmental pollution; these include burning forests, straw, and sawdust in the North; and rice husks dumped into rivers and canals in the Mekong Delta region. Biomass energy is provided in a short cycle, is environmentally safe to use and is encouraged by organizations that support sustainable development. Taking advantage of this energy source provides energy for economic development and ensures environmental protection. Due to the abovementioned favorable conditions, many biomass energy plants are being built in Vietnam. Like other renewable energy investment projects, the selection of the construction contractor, the selection of equipment for the installation of the power plant, and the choice of construction site are complex multi-criteria decisions. In this case, decision-makers must evaluate many qualitative and quantitative factors. These factors interact with each other and it is difficult to use personal experience to choose the optimal solution for such complex decision-making problems, especially in a fuzzy decision-making environment. Therefore, in this study, the authors use a Multi-Criteria Decision-Making (MCDM) model that uses a Fuzzy Analytic Hierarchy Process (FAHP) model and the Combined Compromise Solution (CoCoSo) algorithm to select biomass furnace suppliers utilizing both qualitative and quantitative factors. Furthermore, the results of this work will provide the first look at a hybrid CoCoSo/FAHP method that decision-makers in other fields can use to find the best supplier.

Multi-classification decision-making method for interval-valued intuitionistic fuzzy three-way decisions and its application in the group decision-making

With the introduction of the interval-valued intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy numbers are used instead of precise numbers to provide fuzzy characterization of feature attribute values and misclassification loss function values, which is more in line with the realistic fuzzy decision-making environment. Also, the constructive covering algorithm is introduced into the three-way decisions model, which effectively solves the shortcomings of the traditional decision-theoretic rough sets model in dealing with multi-classification problems, such as too many artificial parameters, complicated computation, redundant decisions, decisional conflicts and excessively large boundary domains. At the same time, in order to avoid the one-sidedness of individual decisions, the group decision-making method is introduced into the preliminarily constructed multi-classification model in this paper to build a multi-classification group decision-making model for interval-valued intuitionistic fuzzy three-way decisions based on the constructive covering algorithm. This model determines the initial weights of feature attributes by the precise weighting method, and determines the expert weights by the grey relational precise weighting method, which effectively achieves the consistency of group decision-making. The decision-making process and rules are also deduced, which expand the model of three-way decisions as well as its practical application value and scope.

Using FQFD and FGRA to Enhance the Advertising Effectiveness of Cross-Regional E-Commerce Platforms

The thriving development of cross-regional e-commerce has gradually increased online marketing activities and consumers’ intention to shop online. The objective of this paper to find solutions for improving the advertising effectiveness has therefore become very important. In this article, an integrating method of fuzzy quality function deployment (FQFD) and fuzzy grey relational analysis (FGRA) was proposed to identify solutions for improving the advertising effectiveness. Based on this method, the house of quality (HoQ) to facilitate investigation of the 17 advertising effectiveness needs and 10 feasible technical improvements were presented. Through the questionnaire survey of platform users, the importance and satisfaction of the attributes of advertising needs were obtained. After that, a fuzzy relationship matrix was constructed to link technical improvements and advertising effectiveness needs in a fuzzy decision environment. Finally, the priority of technical improvements regarding improving advertising effectiveness were obtained. The results show that the proposed method can help decision makers of cross-regional e-commerce to improve advertising effectiveness effectively, so that they can effectively use resources to design advertisements that meet user needs.

Coordination model of supply chain based on deciders’ risk attitudes in a fuzzy decision environment

This paper considers a two-stage supply chain consisting of one manufacturer and one retailer, exploring the impact of the fuzzy uncertainty of product yield and demand and the deciders’ risk attitudes on the optimal order quantity of the retailer. At the same time, this study tries to analyze the coordination problem in the two-stage supply chain with consideration of the retailer and the manufacturer’s risk attitudes. Firstly, this study develops a supply chain optimal decision model in a centralized decision framework. In the proposed model, the L-R fuzzy numbers are used to depict the yield and demand with fuzzy characteristics. Then, the coordination of quantity discount in a supply chain is studied. Consequently, this research further investigates a special case in which the market demand and yield are assumed to be triangular fuzzy numbers, and the optimal solution of the order quantity and the wholesale price are obtained. At last, this paper utilizes several numerical examples to validate the proposed model. The results show that the quantity discount contract can coordinate the supply chain in a fuzzy environment, and the optimal order quantity decreases with the increasing of the risk bias coefficient of the retailer and the manufacturer. It also suggests that risk-seeking retailer will order more products, in addition, the manufacturer tend to choose a risk-seeking retailer as partner and the retailer is more likely to choose a risk-seeking rather than risk-aversion manufacturer as partner.

Novel Transformation Methods Among Intuitionistic Fuzzy Models for Mixed Intuitionistic Fuzzy Decision Making Problems

As an extension of classical fuzzy set, intuitionistic fuzzy set (IFS) has been widely utilized in decision making. Based on IFS, intuitionistic fuzzy value (IFV) and linguistic intuitionistic fuzzy set (LIFS) have also been proposed to represent uncertain assessments of alternatives in different intuitionistic fuzzy decision environment. Theoretically, IFV, IFS or LIFS own themselves processing methods in decision making. However, they may be utilized to represent intuitionistic fuzzy assessments of alternatives in the same decision problem, i.e., multi-decision makers provide IFVs, IFSs or LIFSs to represent uncertain assessments of alternatives in the decision problem. In such case, the significant challenge is to unify IFVs, IFSs and LIFSs in the decision making matrices. In the paper, novel transformation methods are presented to unify IFVs, IFSs and LIFSs. To this end, mixed intuitionistic fuzzy decision matrix is presented to represent IFV, IFS or LIFS assessments of alternatives provided by decision makers, then transformation functions are developed to unify IFVs, IFSs and LIFSs in the decision making matrices, their properties are also analysed. The decision making method is presented to solve the intuitionistic fuzzy decision making problems with IFVs, IFSs or LIFSs, which is consisted of six main phases, accordingly the algorithm is designed to carry out the problems. Illustrative examples show that transformation functions are useful in intuitionistic fuzzy decision environment, the proposed decision making method is an effective and alternative tool for the intuitionistic fuzzy decision making problems with IFVs, IFSs or LIFSs.

A Normalized Weighted Bonferroni Mean Aggregation Operator Considering Shapley Fuzzy Measure Under Interval-valued Neutrosophic Environment for Decision-Making

Previous theories have suggested that Bonferroni mean (BM) and its extensions can fulfil the desired properties to become a good aggregation operator in managing interrelationship between arguments under fuzzy decision-making environment. However, along with this growth, there are concerns regarding the limitation of the BM and its extensions where interrelationship between arguments is limited to a pair of input arguments. In contrast to previous works, a novel normalized weighted Bonferroni mean aggregation operator considering Shapley fuzzy measure has been introduced, where the overall interactions of input arguments are combined. Unlike other extensions of BM that are mostly considered under fuzzy environment, this aggregation operation introduces the Shapley fuzzy measure to deal with interactions of all arguments under the interval-valued neutrosophic environment. Specifically, the interval-valued neutrosophic Shapley normalized weighted Bonferroni mean (INSNWBM) is proposed in this paper. The properties of INSNWBM such as reducibility, idempotency, monotonicity, commutativity, and boundedness are also presented as to reflect the interrelationship among multiple input arguments. In addition, we have also proposed successive steps of a decision-making procedure in which the proposed INSNWBM is included. Together included is a numerical example as to illustrate the applicability of the INSNWBM in decision-making. The consistency of the ranking results is confirmed by variations of parameters of INSNWBM. It is shown that the ranking results are almost consistent despite the different usages of parameter values. Results obtained appear to support the idea that the proposed aggregation operator is able to address the interrelationship between arguments in decision-making.