Trapezoidal Fuzzy Numbers Research Articles

Trapezoidal Intuitionistic Fuzzy Power Heronian Aggregation Operator and Its Applications to Multiple-Attribute Group Decision-Making

Decision-making problems involve imprecise and incomplete information that can be modelled well using intuitionistic fuzzy numbers (IFNs). Various IFNs are available in the literature for modelling such problems. However, trapezoidal intuitionistic fuzzy numbers (TrIFNs) are widely used. It is mainly because of the flexibility in capturing the incompleteness that occurs in the data. Aggregation operators play a vital role in real-life decision-making problems (modelled under an intuitionistic fuzzy environment). Different aggregation operators are available in the literature for better decision-making. Various aggregation operators are introduced in the literature as a generalization to the conventional aggregation functions defined on the set of real numbers. Each aggregation operator has a specific purpose in solving the problems effectively. In recent years, the power average (PA) operator has been used to reduce the effect of biased data provided by decision-makers. Similarly, the Heronian mean (HM) operator has a property that considers the inter-relationship among various criteria available in the decision-making problem. In this paper, we have considered both the operators (HM, PA) to introduce a new aggregation operator (on the set of TrIFNs), which takes advantage of both properties of these operators. In this study, firstly, we propose the idea of a trapezoidal intuitionistic fuzzy power Heronian aggregation (TrIFPHA) operator and a trapezoidal intuitionistic fuzzy power weighted Heronian aggregation (TrIFPWHA) operator by combining the idea of the Heronian mean operator and power average operator in real numbers. Secondly, we study different mathematical properties of the proposed aggregation operators by establishing a few essential theorems. Thirdly, we discuss a group decision-making algorithm for solving problems modelled under a trapezoidal intuitionistic fuzzy environment. Finally, we show the applicability of the group decision-making algorithm by solving a numerical case problem, and we compare the proposed method’s results with existing methods.

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.

A New Approach of Ranking of Generalized Trapezoidal Fuzzy Numbers and Application in Fuzzy Risk Analysis

In this present work, a new approach for ranking generalized trapezoidal fuzzy numbers has been introduced. This new proposed method is based on a left and right deviation and fuzzyness measure of the generalized trapezoidal fuzzy numbers. Some propositions have been introduced corresponding to the new proposed method. The proposed method has been compared with the different existing methods of ranking generalized trapezoidal fuzzy numbers. Also, it has been shown that the proposed technique succeeds in overcoming the drawbacks of the existing methods. The proposed method has been applied to a risk analysis problem in the selection of a production house by a retailer.

Solving Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation with Necessary Arithmetic Multiplication Operations

A couple of Sylvester matrix equations (CSME) are required to be solved simultaneously in many applications, especially in analysing the stability of control systems. However, there are some situations in which the crisp CSME are not well equipped to deal with the uncertainty problem during the stability analysis of control systems. Thus, this paper proposes a new method for solving Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation (ACTrFFSME). New arithmetic fuzzy multiplication operations are developed and applied to convert the ACTrFFSME to a reduced system of non-linear equations. Then it is converted to a system of absolute equations where the arbitrary fuzzy solution is obtained by solving that system. The proposed method can solve many arbitrary fuzzy equations, such as fully fuzzy matrix equations, Sylvester and Lyapunov fully fuzzy matrix equations with triangular and trapezoidal fuzzy numbers without any restrictions. We illustrate the proposed method by solving two numerical examples.

Semi Parametric Logistic Regression Model with the Outputs Representing Trapezoidal Intuitionistic Fuzzy Number

In this paper, the fuzzy logic and the trapezoidal fuzzy intuitionistic number were presented, as well as some properties of the trapezoidal fuzzy intuitionistic number and semi- parametric logistic regression model when using the trapezoidal fuzzy intuitionistic number. The output variable represents the dependent variable sometimes cannot be determined in only two cases (response, non-response)or (success, failure) and more than two responses, especially in medical studies; therefore so, use a semi parametric logistic regression model with the output variable (dependent variable) representing a trapezoidal fuzzy intuitionistic number.
 the model was estimated on simulation data when sample sizes 25,50 and 100, as the parametric part was estimated by two methods of estimation, are fuzzy ordinary least squares estimators FOLSE method and suggested fuzzy weighted least squares estimators SFWLSE , while the non-parametric part is estimated by Nadaraya Watson estimation and Nearest Neighbor estimator. The results were the fuzzy ordinary least squares estimators method was better than the suggested fuzzy weighted least squares estimators while, in the non-parametric portion, the Nadaraya Watson estimators had better than Nearest Neighbor estimators to estimate the model

A DESIGN OF SPECIAL PURPOSE DOUBLE SAMPLING PLAN OF TYPE DSP(0, 1) USING FUZZY PARAMETER

In traditional sampling plan, the proportion defective is generally assumed to be crisp value. But, in real life parameters are vague and assumptions are too rigid. So traditional methods are inaccurate. In this paper Fuzzy set theory is applied to acceptance sampling. A procedure for designing Special Purpose Double Sampling Plan of Type DSP(0, 1) using trapezoidal fuzzy number is developed and it is based on fuzzy binomial distribution. The OC curve values are determined using fuzzy parameter while fixing the sample size. Fuzzy probability of acceptance values are calculated for different AQL ] and LQL ] values and are presented in Tables for the selection of DSP(0, 1) Plan. Optimum value of the sample size is obtained such that it satisfies both the condition of producer’s risk and consumer’s risk for given AQL ] and LQL ] values. The sample size is obtained such that it minimizes the sum of risks. Numerical examples are provided for the illustrations.

Sustainable Supplier Selection and Evaluation for the Effective Supply Chain Management System

With increased environmental protection awareness, sustainability has been incorporated into supply chain management. Sustainable supplier selection and evaluation have become an acritical part of supply chain management. They can significantly improve the supply chain’s operational performance and enhance enterprises’ competitiveness. Based on trapezoidal interval type-2 fuzzy numbers (TIT2FNs) and cloud probability dominance relations (PDR), manufacturers can make more tangible and environmentally friendly decisions in the SSSE process. In this paper, a SSSE indicator system is first established using the necessary economic, environmental, and social factors. The importance of the indicators described in linguistic terms is transformed into TIT2FNs, and the weight of each indicator is calculated. In order to prevent candidate suppliers from promoting performance maliciously, different weights are given according to the impact of the enterprise’s historical performance on the present. Finally, the cloud PDR method is used to determine the optimal sustainable supplier. A case study and analysis are provided to show the feasibility and superiority of the proposed method.

A computational method for normalization of Trapezoidal Fuzzy Numbers

A computational method for normalization of Trapezoidal Fuzzy Numbers

Numerical Treatment of Hybrid Fuzzy Differential Equations Subject to Trapezoidal and Triangular Fuzzy Initial Conditions Using Picard’s and the General Linear Method

We study hybrid fuzzy differential equations (HFDEs) under the Hukuhara derivative numerically using Picard’s and the general linear method (GLM). We use trapezoidal and triangular fuzzy numbers as the initial conditions. To demonstrate the efficiency of the proposed methods, the exact as well as the numerical solutions are presented numerically and graphically. In addition, a comparison is made between the results from applying the GLM and those obtained when applying the fifth order Runge–Kutta method as reported in the literature.

A New Method for Ranking Fuzzy Numbers Based on Principal Component Analysis

It is difficult to rank fuzzy numbers because of their ambiguous values. A few numbers of ranking techniques have been encountered in last few decades. However, existing techniques are situation-dependent which have drawbacks. In this paper we introduced a statistical technique to rank two types of fuzzy numbers, triangular and trapezoidal fuzzy numbers. This technique is the multi-dimensional scaling, more precisely the principal component analysis. Here, we presented each fuzzy numbers as a row in a matrix, then found the scale points in a low dimensions. The scale points are then reconfigured to have a unique representation. The results from this approach are obtained by comparison to the other ranking methods. The validity has been established by comparison with existing works.

Optimization of Cold Chain Distribution Route with Mixed Time Window considering Customer Priority

In order to study the mixed time window vehicle routing optimization problem based on customer priority, a customer differentiation management strategy based on customer priority is proposed. Combined with the main factors affecting customer priority evaluation and the characteristics of vehicle routing problem with mixed time windows, a comprehensive evaluation index affecting customer priority was first established and DBSCAN clustering algorithm was used for clustering analysis of customer priority to solve the optimization problem of cold chain distribution route considering customer priority. Fuzzy time window of refrigerated vehicles was then constructed with trapezoidal fuzzy number, and a mathematical programming model was built with an objective function for minimizing the sum of fixed, green, penalty, refrigeration, and cargo damage costs. Two scenarios of out-of-stock and not-out-of-stock were designed. Finally, an improved genetic algorithm was used to solve the model, and the rationality of the model was verified through a case of imported fruit distribution in Xiamen City. Results showed that the proposed method can effectively solve the routing problem of refrigerated trucks considering customer priority. Moreover, the findings of this study can provide a new approach for solving the routing optimization problem of refrigerated trucks considering customer priority.

A fairness-concern-based LINMAP method for heterogeneous multi-criteria group decision making with hesitant fuzzy linguistic truth degrees

Heterogeneous multi-criteria group decision making (MCGDM) is a hot topic in the decision analysis field. This paper proposes a fairness-concern-based LINMAP (Linear Programming Technique for Multidimensional Analysis of Preference) method for heterogeneous MCGDM with hesitant fuzzy linguistic (HFL) truth degrees. Heterogeneous evaluation information includes crisp numbers, interval numbers, intuitionistic fuzzy values (IFVs), trapezoidal fuzzy numbers (TrFNs) and hesitant fuzzy sets (HFSs). This paper introduces the fairness concern to calculate the HFL consistency and the HFL inconsistency indices. Based on the framework of LINMAP, a bi-objective HFL programming model is built to derive the criteria weights, the positive ideal fairness vector (PIFV) and the negative ideal fairness vector (NIFV) for each decision maker (DM) simultaneously. Based on the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution), a multi-objective programming model is built to obtain DMs’ weights. The alternatives ranking is derived by comprehensive collective relative closeness degrees. Finally, a real example is applied to verify effectiveness and superiority of this heterogeneous MCGDM method. The proposed heterogeneous MCGDM method provides a very useful approach for MCGDM with heterogeneous information.

Multicriteria group decision making via generalized trapezoidal intuitionistic fuzzy number-based novel similarity measure and its application to diverse COVID-19 scenarios.

Havoc, brutality, economic breakdown, and vulnerability are the terms that can be rightly associated with COVID-19, for the kind of impact it is having on the whole world for the last two years. COVID-19 came as a nightmare and it is still not over yet, changing its form factor with each mutation. Moreover, each unpredictable mutation causes more severeness. In the present article, we outline a decision support algorithm using Generalized Trapezoidal Intuitionistic Fuzzy Numbers (GTrIFNs) to deal with various facets of COVID-19 problems. Intuitionistic fuzzy sets (IFSs) and their continuous counterparts, viz., the intuitionistic fuzzy numbers (IFNs), have the flexibility and effectiveness to handle the uncertainty and fuzziness associated with real-world problems. Although a meticulous amount of research works can be found in the literature, a wide majority of them are based mainly on normalized IFNs rather than the more generalized approach, and most of them had several limitations. Therefore, we have made a sincere attempt to devise a novel Similarity Measure (SM) which considers the evaluation of two prominent features of GTrIFNs, which are their expected values and variances. Then, to establish the superiority of our approach we present a comparative analysis of our method with several other established similarity methods considering ten different profiles of GTrIFNs. The proposed SM is then validated for feasibility and applicability, by elaborating a Fuzzy Multicriteria Group Decision Making (FMCGDM) algorithm and it is supportedby a suitable illustrative example. Finally, the proposed SM approach is applied to tackle some significant concerns due to COVID-19. For instance, problems like the selection of best medicine for COVID-19 infected patients; proper healthcare waste disposal technique; and topmost government intervention measures to prevent the COVID-19 spread, are some of the burning issues which are handled with our newly proposed SM approach.Graphical abstract

A Game Theory-based Approach to Fuzzy Linear Transportation Problem

Background: Transport models have wide application areas in the real world and play an important role in reducing transportation costs, increasing service quality, etc. These models may have uncertain transportation costs and supply or demand capacities of the product. Hence, it would be effective to model the vagueness of customer demands, economic conditions, and technical or non-technical uncertainties because of uncontrollable factors. Therefore, we focus on developing a mathematical solution approach to the fuzzy transportation problems. Objective: In this paper, an integrated approach is proposed for the solution of the fuzzy linear transportation problem that has fuzzy cost coefficients in the objective function. Since transportation problem is encountered frequently in the national and international environment, it is considered that proposing a new solution method to this problem will be useful. Methods: Fuzzy cost coefficients are taken as trapezoidal fuzzy numbers due to their widespread use in the literature. Firstly, the fuzziness is removed by converting the original single objective fuzzy transportation problem into a crisp Multi-Objective Linear Programming Problem (MOLPP). After the classical payoff matrix is constructed, ratio matrices are obtained to scale the objectives. Then, an approach based on game theory is implemented to solve the MOLPP, which is handled as a zero-sum game. Results: Creating different ratio matrices in the game theory part of the approach can generate compromise solutions for the decision-makers. To demonstrate the effectiveness of the proposed approach, two numerical examples from the literature are solved. While the same solution is obtained in one of the examples, a different compromise solution set is generated, which could be presented to the decision-maker in the other example. Conclusion: In this paper, we developed a novel game theory-based approach to the fuzzy transportation problem. The proposed approach overcomes the non-linear structure due to the uncertainty in the cost coefficients. The greatest advantage of the proposed approach is that it can generate more than one optimal solution for the decision-maker.

Construction of fuzzy X¯ - S control chart using trapezoidal fuzzy number with unbalanced data

ABSTRACT Statistical quality control is the most useful tool to monitor the process, whether it is the control or not under certain conditions, quality control technique for monitoring the production process through balanced data. In some cases, there are some uncertainties, and unbalanced data are presented. In these circumstances, obtaining the control limit is ineffective for the control charts. This paper deals with obtaining a fuzzy chart by using trapezoidal fuzzy number (TpFN). The unbalanced TpFN information for each sample is converted and deals with the fuzzy chart. The proposed technique is considered how to deal with the unbalanced control limits. Instead, we conducted a fuzzy process capability performance to measure the process performance. Lastly, we illustrate with an example to show the efficiency of the proposed fuzzy chart for uncertain information.

Not-ordered quasi-distance measures of generalised trapezoidal hesitant fuzzy numbers and their application to decision making problems

As an extension of the trapezoidal fuzzy number, the generalised trapezoidal hesitant fuzzy number is an effective mathematical tool for handling uncertainty and vagueness in decision-making problems. Considering that the quasi-distance measure has a strong ability to process and analyse data, we initiated some novel quasi-distance measures to measure the strength of the relationship between generalised trapezoidal hesitant fuzzy numbers in this paper. Moreover, based on the proposed measures, a new multi-criteria decision-making approach is proposed to address uncertain real-life situations. Finally, a practical application of the proposed approach is also illustrated to demonstrate the effectiveness and applicability.

Risk Assessment of Constructing Deep Foundation Pits for Metro Stations Based on Fuzzy Evidence Reasoning and Two-tuple Linguistic Analytic Network Process

Accidents occur frequently while constructing deep foundation pits for metro stations, thereby risking substantial economic losses and casualties. To subject the construction of such pits to scientific and rational risk assessment and overcome the limitations of existing risk evaluation models and risk fusion problems, proposed here is a risk-assessment model for such pits based on fuzzy evidential reasoning and the two-tuple linguistic analytic network process (TL-ANP). First, the risk loss indicators are optimized, the weights of different risk events and of each risk loss indicator in the metro-station deep-foundation-pit construction project are calculated using TL-ANP, and trapezoidal fuzzy numbers are used to describe the occurrence probability of each risk event and loss. Second, relying on a table of expert weight indices, the best–worst method based on generalized interval-valued trapezoidal fuzzy numbers is used to determine the weights of experts. Finally, the overall risk grade of the construction project is evaluated by aggregating the risk levels of all risk events through an evidence-reasoning algorithm. The analysis results for a deep foundation pit for a station on Line 5 of Nanning Metro show that the model provides a quantitative basis for determining expert weights and risk loss weights reasonably and improving the reliability of the evaluation system. Also, not only does applying the method show that such a construction project can be judged as having a certain risk grade, but more importantly it can identify the key factors and loss indicators affecting the overall risk grade of the pit, whereupon risk control measures can be adopted in a targeted manner. In comparison with traditional methods, the proposed method is shown to be practical and effective, providing a reference basis for analyzing the risks of similar projects in the future and guaranteeing construction safety.

An extensive operational law for monotone functions of LR fuzzy intervals with applications to fuzzy optimization

The operational law proposed by Zhou et al. (J Intell Fuzzy Syst 30(1): 71–87, 2016) contributes to developing fuzzy arithmetic, while its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which are hindering their operational law from dealing with more general cases, such as problems formulated as monotone functions and problems with fuzzy variables represented as fuzzy intervals (e.g., trapezoidal fuzzy numbers). In order to handle such cases we generalize the operational law of Zhou et al. in both the monotonicity of function and fuzzy variables in this paper and then apply the extensive operational law to the cases with monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs) (of which regular LR fuzzy numbers are special cases). Specifically, we derive the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs, respectively. On the other hand, we develop a solution scheme to dispose of fuzzy optimization problems with regular LR-FIs, in which a fuzzy programming is converted to a deterministic equivalent one and a newly devised solution algorithm is utilized to get the deterministic programming solved. The numerical experiments are conducted using our proposed solution scheme and the traditional fuzzy simulation-based genetic algorithm in the context of a purchasing planning problem. Computational results show that our method is much more efficient, yielding high-quality solutions.

A new maximal flow algorithm for solving optimization problems with linguistic capacities and flows

The maximal flow problems (MFPs) are among the most significant optimization problems in network flow theory with widespread and diverse applications. To represent qualitative aspects of uncertainty in the maximal flow model, which asks for the largest amount of flow transported from one vertex to another, the use of linguistic variables has effective means for experts in expressing their views. In this paper, we first define trapezoidal Pythagorean fuzzy numbers (TrPFNs) along with some new arithmetic operations which cover the gaps in previously defined operations. For defuzzification of TrPFNs, we introduce a ranking procedure based on value and ambiguity indices. This work puts forward a theoretical framework for a new Pythagorean fuzzy maximal flow algorithm (PFMFA), which helps to solve different optimization problems with PF information by considering linguistic capacities and flows. The implementation of the algorithm is elaborated by considering two case studies. Firstly, we examine the maximum flow of a water distribution pipeline network in Pyigyitagon Township, Mandalay, Myanmar. Secondly, we compute maximum PF power flow in a 14-bus electricity network provided by the IEEE working group, concerning the example data from the University of Washington. The results illustrate the superiority of the proposed method and give a detailed analysis of flow connected with several practical performances. In addition, the Pythagorean fuzzy optimal flows corresponding to each network arc are compared and performance comparison of our method is investigated which shows the increasing and decreasing trends of backward and forward arcs of the network, respectively. Moreover, the runtime analysis of existing well-known maximal flow algorithms is provided. Finally, we present the advantages of our technique to promote its cogency.

Mari's algorithm to solve a fuzzy transportation problem by using Trapezoidal Fuzzy numbers

Mari's algorithm to solve a fuzzy transportation problem by using Trapezoidal Fuzzy numbers