Interval Fuzzy Numbers Research Articles

A Hybrid Multi-Attribute Decision-Making Method Considering Tourists' Behavioral Preferences for Customized Tours

A decision-making analysis method considering tourists' behavioral preferences is proposed to address the problem of hybrid multi-attribute decision making (HMDM) for customized tours in which attribute values are intuitionistic fuzzy numbers, interval numbers and real numbers. This method is presented through step-by-step instructions:1) construct an evaluation index system of the HMDM for customized tours; 2) convert the attribute values of different data types into normalized intuitionistic fuzzy numbers, and obtain their objective weights by calculating their intuitionistic fuzzy entropies; 3) use the intuitionistic fuzzy weighted average operator to aggregate all the converted attribute values of each customized scheme;4) calculate scores of aggregation values for customized schemes and rank these schemes accordingly under the scenarios where tourists express such behavioral preferences as caution, mildness and recklessness; and 5) apply the proposed method to an example case to prove its feasibility.

A Novel Data-Envelopment Analysis Interval-Valued Fuzzy-Rough-Number Multi-Criteria Decision-Making (DEA-IFRN MCDM) Model for Determining the Efficiency of Road Sections Based on Headway Analysis

The capacity of transport infrastructure is one of the very important tasks in transport engineering, which depends mostly on the geometric characteristics of road and headway analysis. In this paper, we have considered 14 road sections and determined their efficiency based on headway analysis. We have developed a novel interval fuzzy-rough-number decision-making model consisting of DEA (data envelopment analysis), IFRN SWARA (interval-valued fuzzy-rough-number stepwise weight-assessment-ratio analysis), and IFRN WASPAS (interval-valued fuzzy-rough-number weighted-aggregate sum–product assessment) methods. The main contribution of this study is a new extension of WASPAS method with interval fuzzy rough numbers. Firstly, the DEA model was applied to determine the efficiency of 14 road sections according to seven input–output parameters. Seven out of the fourteen alternatives showed full efficiency and were implemented further in the model. After that, the IFRN SWARA method was used for the calculation of the final weights, while IFRN WASPAS was applied for ranking seven of the road sections. The results show that two sections are very similar and have almost equal efficiency, while the other results are very stable. According to the results obtained, the best-ranked is a measuring segment of the Ivanjska–Šargovac section, with a road gradient = −5.5%, which has low deviating values of headways according to the measurement classes from PC-PC to AT-PC, which shows balanced and continuous traffic flow. Finally, verification tests such as changing the criteria weights, comparative analysis, changing the λ parameter, and reverse rank analysis have been performed.

On SIR-based MCDM approach: Selecting a charcoal firm using hybrid fuzzy number on a Triple Vague structure

Multi-Criteria Decision Making (MCDM) is a widely used method across various sectors to help decision-makers assess and choose the most suitable solutions from a range of available options. This research introduces an innovative framework called the Triple Vague (TV) structure for Superiority and Inferiority Ranking (SIR) within the context of MCDM. This framework incorporates interval fuzzy numbers and hesitant fuzzy numbers, including negative interval fuzzy numbers for different vague ideas. The main goal of this research is to present an effective solution to the challenge of selecting a firm, specifically focusing on the manufacturing of charcoal from waste coconut shells. The SIR approach is employed to establish two complete rankings, and a TV value is used to articulate the assessment information. In this article, our contribution lies in the proposed Triple Vague approach, which utilizes the superiority and inferiority ranking methodology to identify the best firm for charcoal manufacturing. Further, it demonstrates through an application example involving the selection of the most suitable firm among four different options based on specified criteria for charcoal production from coconut shell waste. Also, comparative analysis has been done with other MCDM methods like TOPSIS, Grey method and PROMETHEE method. This analysis examines that the proposed structure is better than other MCDM technique. Moreover, the results highlight the framework's capability to handle the hesitation, vagueness, and like other undesirable characteristics. Hence it provides valuable insights for decision-makers to thoroughly evaluate and consider the best alternatives.

Matrix games involving interval-valued hesitant fuzzy linguistic sets and its application to electric vehicles

Game theory has been successfully applied in a variety of domains to deal with competitive environments between individuals or groups. The matrix games involving fuzzy, interval fuzzy, and intuitionistic fuzzy numbers exclusively examine the numeric components of an issue. However, several researchers have also examined various extensions of conventional game theory, considering the ambiguous situations for payoffs and goals. In many real-life scenarios, qualitative information is often critical in expressing the payoffs of a matrix game. Thus, the present work contributes to the field of matrix games where the payoffs have been quantified via qualitative variables, termed interval-valued hesitant fuzzy linguistic sets. The mathematical formulation and solution concept for matrix games involving interval-valued hesitant fuzzy linguistic numbers is designed by utilizing an aggregation operator supported by linguistic scale function and solving them by employing score function. Finally, the proposed approach is validated by applying it to electric vehicle sales.

Regret Theory-Based Three-Way Decision Method on Incomplete Multiscale Decision Information Systems With Interval Fuzzy Numbers

In realistic decision-making environments, human behaviors bring influences to various decision-making procedures, and classic multiattribute decision-making approaches based on utility decisions own some deviations from actual situations. The behavioral decision theory modifies classic decision-making theories to make the new method more applicable. Regret theory, as one of the important components in behavioral decision theories, has been widely used in theories and applications. Based on regret theory, we establish a generalized three-way decision method on incomplete multiscale decision information systems with interval fuzzy numbers. First, we select an incomplete optimal subsystem for the incomplete multiscale information system, and convert the multiscale evaluation information into an interval fuzzy number by using a linguistic term set. Second, based on the probability distribution of evaluation values and tradeoff factors, we propose a target-dependent approximation estimation method for the incomplete interval fuzzy subsystem. Then, the regret–rejoicing preferences between objects are obtained. Finally, a tripartition and ranking method based on a max-bipartition and preference index is established. Moreover, the incompleteness experiments show that the decision-making results of our method can still maintain more than 97% consistency in an incomplete information system with a missing rate of at most 20%. In addition, the parameter analysis shows that the proposed method maintains an accuracy rate of more than 98% and a classification error rate of no more than 0.6%.

A Novel Definition of Fuzzy Difference on Non-increasing Fuzzy Real Numbers

In this paper, firstly, a novel definition of a fuzzy difference based on non-increasing fuzzy real numbers is introduced, which is different from the previous definitions by using fuzzy interval-numbers and Zadeh’s extension principle. Then we give some important conclusions of fuzzy difference from the view of two cuts of fuzzy sets. Moreover, a definition of a opposite fuzzy real number is given so that we show the connection between the fuzzy difference and fuzzy addition on fuzzy real numbers. Finally, we provide several examples for the sake of illustrating the fuzzy difference on non-increasing fuzzy real numbers is reasonable generalization of classical difference. In addition, we give the prospect that we want to use this difference to research fuzzy derivatives.

An improved MCDM combined with GIS for risk assessment of multi-hazards in Hong Kong

Hong Kong frequently suffers from multi-hazards such as floods, muddy-water flows and landslides induced by rainstorms. This study presents an improved multi criteria decision making (MCDM) approach with integrating the interval numbers into fuzzy analytical hierarchy process (FAHP) to assess multi-hazard risks. To illustrate the efficiency of the improved MCDM method, the AHP, interval-FAHP and analytical network process (ANP) were incorporated into a geographical information system (GIS, abbreviated as AHP-GIS, interval-FAHP-GIS, and ANP-GIS) to assess the risks of multi-hazards (e.g., floods, muddy-water flows, and landslides) in Hong Kong. The assessed multi-risks indicated that the percentages of areas with a high risk of flood, muddy-water flow, and landslide were more than 15%, 17%, and 18%, respectively. The results demonstrated that MCDM methods considered multi-criteria contributions on multi hazards. Different assessment factors contributed different importance on different multi-hazard risks. The comparison indicates that interval-FAHP-GIS perform better than AHP-GIS and ANP-GIS in capturing high-risk areas. The interval-FAHP-GIS method adopts interval fuzzy numbers instead of crisp numbers in AHP-GIS to reflect the degree of importance of assessment factors, which increases the accuracy of the assessed multi-risks.

A New ELECTRE Method Based on Left and Right Score for Multicriteria Decision-Making.

The procedure for ranking the performance of systems using the method Elimination and Choice Translating Reality (ELECTRE) is one of the most important and practical methods in multicriteria decision-making (MCDM). The classical ELECTRE technique uses the concept of dominance implicitly. In this method, all options are evaluated using a comparison of unexpected rankings, thus eliminating ineffective options. All stages of the electrification technique are based on a coordinated set and an uncoordinated set, which is why it is known as "coordination analysis." The ELECTRE method has two important weaknesses, first, the use of thresholds, which are used to calculate the coordination and noncoordination matrix, and second, this method does not give us a complete and final ranking and is limited to the top options. In order to reduce these weaknesses in this research, interval fuzzy numbers have been used. In the proposed process, the numbers to the left and right of the intervals are considered, and a better method is presented with fuzzy techniques based on the ELECTRE technique. In the research results, using the examples taken from previous researches, the ability of the proposed method is shown.

An integrated decision-making approach to evaluating design schemes under heterogeneous fuzzy environment

A few design schemes can be derived using the modularisation strategy, which can effectively reduce the assembling cost and delivery time to the development of product design. Therefore, it is important to select modular architecture design schemes (MADSs) according to evaluation information (fuzzy and uncertain) provided by design experts. As a classic multiple attributes group decision-making problem, the fuzzy theory exhibits a distinct advantage in dealing with the heterogeneous fuzzy information. In this study, an integrated VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) approach was proposed to evaluate MADSs based on a similarity degree model and modified convergent iterative algorithm (MCIA). First, the evaluation measures of MADSs were provided using various types of fuzzy (interval and real) numbers according to the characteristics of the evaluation attributes. Then, the similarity degree between the heterogeneous fuzzy data of individual judgements was used to construct a collective decision matrix, which was obtained by the MCIA. Finally, the weighted VIKOR approach integrated with the Euclidean distance of heterogeneous fuzzy data was presented to derive the compromise solutions of MADSs. A real-world case study on liquid crystal display, method comparisons, and sensitivity analysis demonstrated the effectiveness of the proposed approach for evaluating MADSs under heterogeneous fuzzy environment.

Failure mode risk assessment methodology for controlling multi-uncertainties in the evaluation process

The failure mode risk evaluation results of FMEA are affected by multi-uncertainties. This paper proposes a risk evaluation methodology for controlling multi-uncertainties in the assessment process. First, the fuzzy confidence interval number (FCIN) evaluation model is provided to control the uncertainty in assessing the severity (S), occurrence (O), and detectability (D). Then, the FCINs are converted into generalized trapezoidal fuzzy numbers (GTrFNs), and the GTrFNs’ scalar characteristic distances modified by the non-membership are used as the evaluation results of S, O, D and their synthesizer or risk priority number (RPN) to control the risk evaluation model uncertainty. Furthermore, the evaluation parameter value criteria of S, O, D are formulated based on the sensitivity analysis results, more precise than the general value guidelines introduced by industrial FMEA standards. The case study results show that the proposed methodology can significantly improve the risk assessment results and the risk discrimination of failure modes.

Interval-Fuzzy Fixed Charge Transportation Problems

The fixed charge transportation problem (FCTP) is a variant of the well-known transportation problem in which the price of conveying products from a source to a sink is made up of a set cost and a continuous cost proportionate to the number of goods sent. The transportation system and FCTP with uncertain parameters are inextricably linked. Fuzzy numbers or interval numbers can be used to successfully handle these imprecise values. The cost coefficients, supplies, and demands are all represented as interval numbers in this paper. A modified interval version of Vogel's Approximation process is used to obtain the initial basic feasible solution (IBFS) of the interval fixed charge transportation problem (IFCTP), which is then optimized using interval version of MODI method. A numerical example is provided to show the efficiency of the proposed method.

Fuzzy preference ranking organization method for enrichment evaluation‐based node cooperation technique for improving QoS in MANETs

SummaryCooperation between mobile nodes is the predominant factor required for achieving reliable data dissemination in mobile ad hoc networks (MANETs), since non‐cooperative nodes crumbles the routing process and degrades network performance. The degree of non‐cooperation rendered by mobile nodes need to be accurately identified by exploring feasible and comprehensive number of criteria to achieve maximized cooperation in uncertain MANET. In this paper, fuzzy preference ranking organization method for enrichment evaluation (FPROMETHEE)‐based node cooperation technique is proposed with the merits of trapezoidal fuzzy interval numbers that aids in the selection of high trustworthy mobile nodes for attaining maximized data delivery. This FPROMETHEE technique utilizes complex and multiple criteria for potential ranking in the presence of finite number of mobile nodes. It prevented the bias of information that is more common during the judgmental process associated with the cooperation estimation of mobile nodes. It includes information about the relative weights of criteria and decision maker's preferences for estimating the trust of mobile nodes to include or isolate them from the routing path. It is also useful in handling the degree of uncertainly involved in the process of routing without compromising Quality of Service (QoS) level in the network. The simulation experiments of the proposed FPROMETHEE technique conducted using ns‐2 simulator confirmed its predominance in enhancing the packet dissemination rate, throughput, energy consumptions, total overhead, and packet loss rate for different number of mobile nodes, malicious nodes, and pairs of source and destination nodes.

COPRAS method based on interval-valued hesitant Fermatean fuzzy sets and its application in selecting desalination technology

This paper firstly introduces the idea of interval-valued hesitant Fermatean fuzzy set (IVHFFS). It is one of the significant theoretical and practical features, which derive from the integration of hesitant Fermatean fuzzy set (HFFSs) and interval number. In this study, we firstly present the basic concept, various operations and distance measures of IVHFFSs. Then, based on the operations of IVHFFSs, some aggregation operators (AOs) are proposed for aggregating the interval-valued hesitant Fermatean fuzzy information. Furthermore, some elegant properties of these operators are discussed in detail. To show the applicability, we discuss a decision analysis process on IVHFFSs environment with the help of conventional complex proportional assessment (COPRAS) methodology. Finally, the selection process of desalination technology for treating the feed water is also taken to validate the developed decision analysis procedure on IVHFFSs. This paper considers various criteria and sub-criteria in the assessment procedure and determines the most significant criteria influencing the desalination technology selection process are technical, social, environment and economic criteria, with significance weights of 0.3972, 0.289, 0.1653 and 0.1486, respectively. The work concludes that the reverse osmosis (RO) is the suitable desalination technology under the considered criteria followed by electrodialysis (ED), and multiple-effect distillation (MED). Moreover, the sensitivity investigation and comparative study are presented to verify the robustness and stability of the proposed method. The findings of this study show that the proposed model can suggest more feasible performance while facing several input uncertainties and influencing factors.

Solving Discounting Problem Using Piece-Wise Quadratic Fuzzy Numbers

The discounting problem is one of the important aspects in investment, portfolio selection, purchasing with credit, and many other financial operations. In this paper, a discounting problem using piecewise quadratic fuzzy numbers is proposed. The implementation of piecewise quadratic fuzzy numbers is described based on such operations. Fuzzy arithmetic and interval number arithmetic are used for computation. The close interval approximation of piecewise quadratic fuzzy numbers is used for solving the proposed discounting problem. This research article addresses the discounted investment for Year 1, Year 2, and Year 3. Additionally, the authors determine the cumulative discounted investment for different values of the parameter α ranging from 0 to 1. A discounting problem using piecewise quadratic fuzzy numbers is solved as a numerical example to illustrate the proposed procedure.

Analysis and Construction of EEMD Smart Model and Fuzzy Forecasting through Improved Bayesian Estimation

Different portfolio models will have a great impact on the model rate of return. In order to meet the data needs of portfolio investment forecasting, a big data generation method based on incremental Bayesian network model is proposed. Firstly, the fuzzy interval number is constructed by using the global GM (1pare 1) prediction model, then the fuzzy Mmurv model is established based on the interval fuzzy number, and the model is optimized based on the midpoint, radius and acceptability of the interval number, thus a single objective programming model with parameters is obtained. The future data is partially generated by using the time series generation algorithm, and the Bayesian network model trained by historical data is updated with the newly generated data. So that the updated Bayesian network can reflect the relationship between the variables and the laws contained in the new and old financial data in this period of time. And use the model to predict the amount of foreign direct investment absorbed by China in 2020, test the robustness and analyze the impact of major random events to show the reliability of the prediction. The analysis is carried out on multiple scales. The results show that the integration of IMF1 and IMF2 components through the white noise test can be regarded as the random impact of external factors; the cycle length of IMF3 component is about 3 years, which can be regarded as the inventory cycle caused by the change of enterprise inventory; the cycle length of IMF4 component is 5 years, which can be regarded as political cycle; the cycle length of IMF5 component is about 20 years, which can be regarded as construction cycle. The set of portfolio paths is generated by path search algorithm in Bayesian network, and the big data set with real data characteristics is generated according to the probability distribution of each path. The experimental results show that the method is feasible and ensures a certain accuracy. The forecast results show that the amount of foreign direct investment in China will continue to fluctuate seasonally and in a trend in recent years.

Investment Decision of New Energy Vehicle Enterprises Using the Interval Basic Probability Assignment-Based Intuitionistic Fuzzy Set

To evaluate the investment decisions of new energy vehicle enterprises scientifically and reasonably and improve the investment efficiency and accuracy of decision-makers, this paper proposes an investment decision method based on interval intuitionistic fuzzy sets. In the investment decision-making process of new energy vehicle enterprises, first, based on the characteristics of new energy vehicle enterprise investment projects, an index system is constructed to comprehensively cover the influencing factors of investment decisions. Second, we obtain the interval fuzzy number of the decision index through a questionnaire survey, use the structural entropy method to empower decision indicators, and comprehensively evaluate the decision index by the organic combination of the interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operator and structural entropy method. Finally, interval BPA is used to express the value of each decision index interval intuitionistic fuzzy number. Based on the conversion relationship between interval evidence and the intuitionistic fuzzy set, the orthogonal sum operation results of the intuitionistic fuzzy set converted from the normalized interval BPA is replaced by the interval evidence combination result, and the final decision is determined by comparing the fusion result. Applying the investment decision method based on interval intuitionistic fuzzy sets to the field of new energy vehicle investment decision-making can provide a reference for investment decision-makers to make efficient and accurate decisions, and it has application value and practical significance to promote the effective development of new energy investment decisions.

Development of a multi-period multi-attribute group decision-making method using type- 2 fuzzy set of linguistic variables

Recently, fuzzy linguistic variables have gained a great deal of attention from researchers in many decision-making problems. In this problems, type-2 fuzzy sets have been used to better cover linguistic data uncertainty. However, many of research works in this regard have been performed in type-2 fuzzy domain and in static mode. Since the decision making problems in the real world usually fluctuate over time, so it need to use decision making models in multi period of time. In the present research, a Multi-Period (Dynamic) Multi-Attribute Group Decision-Making (MPMADM) method is presented based on type-2 fuzzy sets where decision-making attributes are first expressed in linguistic terms and then incorporated, as interval type-2 fuzzy numbers, into problem solving where a new integrating operator called Multi-Period Trapezoidal Interval Type-2 Fuzzy Number Weighted Arithmetic averaging (MPTIT2FNWA) is defined on type-2 interval fuzzy numbers to integrate decision-making information in multiple periods of time. Once finished with explaining the proposed method, a numerical example is given to evaluate the proposed method in terms of effectiveness and applicability, with the results compared to those of other methods.

A Mathematical Model for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers via Interval Linear Programming Problems

We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

MULTI-CRITERIONAL CHOICE OF AN ALTERNATIVE UNDER THE RULES OF FUZZY PRODUCTS WITH SOME RELIABILITY DEGREE

One of the main problems of decision-making tasks is the need to take into account subjective expert assessments, the complete consistency of which is rare, and the choice of the best alternative. The complexity of the connections between the many-sided aspects of the decision-making situation and the lack of an accurate forecast of the consequences leads to the fact that when assessing and choosing alternatives, it is possible, and often necessary, to use and process qualitatively fuzzy estimates. In decision-making situations, when at least one of the elements (outcomes, criteria, preferences, expert opinions, etc.) is described qualitatively, indistinctly, there are problems of multi-criteria decision-making with fuzzy initial information.
 Let’s consider the solution to the problem of multi-criteria choice based on the rules of fuzzy conditional inference, which have the form of fuzzy statements, the conditions and conclusions of which, along with expert assessments of the criteria, are presented in the form of interval fuzzy numbers of the second type (IT2FN). The convolution of private implications in each statement is made according to Lukasiewicz's rule. To reduce the type and defuzzify the resulting IT2FN, the Karrnik-Mendel algorithm was used to construct the minimum and maximum centroids of nested fuzzy sets of the first type, which give an estimate of the utility interval for each alternative. To refine the obtained utility estimates, under conditions of incomplete definiteness of statements, using the generalized Bayesian inference mechanism, adjusted estimates of the utility intervals of alternatives are constructed. By comparing these intervals, a larger interval is determined and the corresponding alternative is taken as a solution to the problem under consideration.
 The application of the proposed approach to solving the problem of multicriteria selection of the most corroded section of a gas pipeline with ambiguous expert opinions is shown. To date, specific practical and theoretical results have been obtained for decision-making problems with fuzzy initial information

Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM

Numerous research papers and several engineering applications have proved that the fuzzy set theory is an intelligent effective tool to represent complex uncertain information. In fuzzy multi-criteria decision-making (fuzzy MCDM) methods, intelligent information system and fuzzy control-theoretic models, complex qualitative information are extracted from expert’s knowledge as linguistic variables and are modeled by linear/non-linear fuzzy numbers. In numerical computations and experiments, the information/data are fitted by nonlinear functions for better accuracy which may be little hard for further processing to apply in real-life problems. Hence, the study of non-linear fuzzy numbers through triangular and trapezoidal fuzzy numbers is very natural and various researchers have attempted to transform non-linear fuzzy numbers into piecewise linear functions of interval/triangular/trapezoidal in nature by different methods in the past years. But it is noted that the triangular/trapezoidal approximation of nonlinear fuzzy numbers has more loss of information. Therefore, there is a natural need for a better piecewise linear approximation of a given nonlinear fuzzy number without losing much information for better intelligent information modeling. On coincidence, a new notion of Generalized Hexagonal Fuzzy Number has been introduced and its applications on Multi-Criteria Decision-Making problem (MCDM) and Generalized Hexagonal Fully Fuzzy Linear System (GHXFFLS) of equations have been studied by Lakshmana et al. in 2020. Therefore, in this paper, approximation of nonlinear fuzzy numbers into the hexagonal fuzzy numbers which includes trapezoidal, triangular and interval fuzzy numbers as special cases of Hexagonal fuzzy numbers with less loss/gain of information than other existing methods is attempted. Since any fuzzy information is satisfied fully by its modal value/core of that concept, any approximation of that concept is expected to be preserved with same modal value/core. Therefore, in this paper, a stepwise procedure for approximating a non-linear fuzzy number into a new Hexagonal Fuzzy Number that preserves the core of the given fuzzy number is proposed using constrained nonlinear programming model and is illustrated numerically by considering a parabolic fuzzy number. Furthermore, the proposed method is compared for its efficiency on accuracy in terms of loss of information. Finally, some properties of the new hexagonal fuzzy approximation are studied and the applicability of the proposed method is illustrated through the Group MCDM problem using an index matrix (IM).