Triangular Fuzzy Numbers Research Articles

Solving a Multimodal Routing Problem with Pickup and Delivery Time Windows under LR Triangular Fuzzy Capacity Constraints

This study models a container routing problem using multimodal transportation to improve its economy, timeliness, and reliability. Pickup and delivery time windows are simultaneously formulated in optimization to provide the shipper and the receiver with time-efficient services, in which early pickup and delayed delivery can be avoided, and nonlinear storage periods at the origin and the destination can be minimized. Furthermore, the capacity uncertainty of the multimodal network is incorporated into the advanced routing to enhance its reliability in practical transportation. The LR triangular fuzzy number is adopted to model the capacity uncertainty, in which its spread ratio is defined to measure the uncertainty level of the fuzzy capacity. Due to the nonlinearity introduced by the time windows and the fuzziness from the network capacity, this study establishes a fuzzy nonlinear optimization model for optimization problem. A chance-constrained linear reformulation equivalent to the proposed model is then generated based on the credibility measure, which makes the global optimum solution attainable by using Lingo software. A numerical case verification demonstrates that the proposed model can effectively solve the problem. The case analysis points out that the formulation of pickup and delivery time windows can improve the timeliness of the entire transportation process and help to achieve on-time transportation. Furthermore, improving the confidence level and the uncertainty level increases the total costs of the optimal route. Therefore, the shipper and the receiver must prepare more transportation budget to improve reliability and address the increasing uncertainty level. Further analysis draws some insights to help the shipper, receiver, and multimodal transport operator to organize a reliable and cost-efficient multimodal transportation under capacity uncertainty through confidence level balance and transportation service and transfer service selection.

Application of Fuzzy AHP in Priority Based Selection of Financial Indices: A Perspective for Investors

Abstract By providing important indicators, financial indices help investors make educated judgements regarding their assets, much like vital sign monitors for the financial markets. The best way for investors to keep up with the market and make strategic adjustments is to keep an eye on these indexes. Researching the most important financial indexes for making educated investing decisions is, thus, quite relevant. Finding the most essential financial indices from an investing standpoint and assigning a weight to each of those indexes are the main goals of this research. A weighted score is derived by combining four financial indices in a Multi-Criteria Decision-Making (MCDM) technique. These objectives are then pursued. Triangular Fuzzy Numbers (TFNs) and the Fuzzy Analytic Hierarchy Process (F-AHP) are used to determine the weights of criteria in this technique. Using these methods together, the research hopes to provide a thorough analysis of the role that different financial indexes have in informing investment choices. This study emphasizes the paramount importance of considering the Price Earning to Growth (PEG) ratio when making investment decisions, followed by the Debt Equity Ratio. Price to Book Value and Dividend Yield, while relevant, carry comparatively less weightage in the overall assessment. Investors are advised to use these insights as a guideline in their financial analysis and decision-making processes.

Developing a fuzzy logic-based carbon emission cost-incorporated inventory model with memory effects

Inventory control is a widely discussed topic in the real world, and recently, it has become closely linked to concerns about carbon emissions and global warming. Global warming is a pressing issue, mainly due to a lack of awareness and action. Traditional inventory models, which typically use integer-order differential equations, overlook the memory aspect of the system. Addressing inventory management is essential in our efforts to combat global warming. This paper introduces a novel approach by incorporating carbon emission costs within a fuzzy environment. Fractional Calculus, a powerful mathematical tool, is employed to capture the memory effect of the system. This approach distinguishes between long memory, characterized by a strong memory effect, and short memory, associated with a poor memory effect, through the use of fractional derivatives and integrals. Numerical results are analyzed based on these memory concepts. Entrepreneurs often find it difficult to determine exact values for known parameters in the real world. Therefore, this study considers the uncertain nature of ordering costs, the rate of deterioration, and demand rates as triangular fuzzy numbers. The optimal average cost and ordering intervals are determined using a solution methodology. A sensitivity analysis is performed to demonstrate how different system parameters influence the outcomes within a fuzzy environment. Notably, it is observed that profit tends to be higher under conditions of strong memory compared to poor memory effects. Moreover, it's worth emphasizing that profits are notably more favorable when employing the signed distance method compared to the graded mean integration method, especially in situations marked by strong memory effects. This study highlights the importance of considering sensitive parameters in the model, especially under conditions of strong memory effects. Such parameters require careful attention in the pursuit of effective inventory management strategies to mitigate carbon emissions and combat global warming.

The Inverse and General Inverse of Trapezoidal Fuzzy Numbers with Modified Elementary Row Operations

Trapezoidal positive/negative fuzzy numbers have no single definition; instead, various authors define them in relation to different concepts. This means that arithmetic operations for trapezoidal fuzzy numbers also differ. For the operations of addition, subtraction, and scalar multiplication, there are not many differences; for multiplication, however, there are many differences. In general, multiplication is divided into various cases. For the inverse operation, there is not much to define; in general, for any trapezoidal fuzzy number u~, u~⊗1u~=i~=(1,1,0,0) does not necessarily apply. As a result of the different arithmetic operations for multiplication and division employed by various authors, several researchers have tackled the same problem and reached different solutions, meaning that the application will also produce different results. To date, many authors have proposed various alternatives for the algebra of the trapezoidal fuzzy number. In this paper, using the parametric form approach to trapezoidal fuzzy numbers, an alternative to multiplication with only one formula is constructed for various cases. Furthermore, based on the definition of multiplication for any trapezoidal fuzzy number, u~ is constructed 1u~ so that u~⊗1u~=i~=(1,1,0,0). Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. Furthermore, we modify the elementary row operational steps for the trapezoidal fuzzy number matrix, which can be used to determine the inverse of a trapezoidal fuzzy number matrix with the order m×m. We also give the steps and examples necessary to determine the general inverse for a trapezoidal fuzzy number matrix of the order m×n with m ≠n. This ability to easily determine the inverse and general inverse of a trapezoidal fuzzy number matrix has a number of applications, such as solving fully trapezoidal fuzzy number linear systems and fuzzy transportation problems, especially in applications in fields outside of mathematics; for example, the application of triangular fuzzy numbers in medical problems is a topic currently receiving a significant amount of attention.

New distance measure-driven flexible linguistic consensus model with application to urban flooding risk assessment

As a novel linguistic representation tool, flexible linguistic expression (FLE) has substantial flexibility in expressing ambiguous and uncertain information from experts. However, FLEs pose the following challenges to developing information measurement and consensus mechanisms for FLEs-based multi-attribute group decision-making (MAGDM). (1) The current studies primarily use linguistic distribution approximation or triangular fuzzy numbers to measure the difference between FLEs indirectly without proposing an exact distance measure. (2) Few studies have been conducted on direct accurate consensus models incorporating adaptive consensus feedback mechanisms. To deal with these issues, the flexible linguistic ordinal Deng entropy (FLODE) is firstly proposed to measure the uncertainty of FLEs, and by combining it with linguistic scale functions, a new distance measure for FLEs is constructed, which can accurately measure the level of group consensus and the difference in pairwise FLEs. Subsequently, by integrating the defined FLODE with the classical entropy weight method, a new method for determining the experts’ personalized attribute weights is constructed, fully considering the impact of experts’ personalized attribute weights on decision-making results. Also, a new consensus model incorporating a minimum adjustment feedback mechanism and dynamic personalized attribute weights is developed, in which the feedback adjustment coefficients and personalized attribute weights are dynamically updated during the consensus reaching process (CRP). Lastly, an urban flooding risk assessment is applied to confirm the effectiveness of the proposed consensus model.

Applying a Genetic Algorithm to Implement the Fuzzy-MACBETH Method in Decision-Making Processes

This paper describes the development of an evolutionary algorithm for building cardinal scales based on the Fuzzy-MACBETH method. This method uses a triangular fuzzy numbers scale in the MACBETH method to incorporate the subjectivity of a semantic scale into mathematical modeling, which enables circumventing the cardinal inconsistency problem of the classical method, facilitating its application in complex contexts. A genetic algorithm is used in the fuzzy system developed here to build the basic fuzzy scale in a cardinally inconsistent decision matrix. The proposed technique is inspired by crossover and mutation genetic operations to explore potential solutions and obtain a cardinal scale aligned with the decision maker’s preferences. Finally, an illustrative example of the application of the proposed decision support system is presented. The results confirm that the FGA-MACBETH method aligns with the classical method. This study’s primary contribution is that circumventing the problem of cardinal inconsistency in a semantically consistent decision matrix enabled obtaining a cardinal scale without requiring the decision maker to redo his/her initial assessments.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

A fuzzy interval optimization approach for p-hub median problem under uncertain information.

Stochastic and robust optimization approaches often result in sub-optimal solutions for the uncertain p-hub median problem when continuous design parameters are discretized to form different environmental scenarios. To solve this problem, this paper proposes a triangular fuzzy number model for the Non-Strict Uncapacitated Multi-Allocation p-hub Median Problem. To enhance the quality and the speed of optimization, a novel optimization approach, combining the triangular fuzzy number evaluation index with the Genetic-Tabu Search algorithm, is proposed. During the iterations of the Genetic-Tabu Search algorithm for finding the optimal solution, the fitness of fuzzy hub schemes is calculated by considering the relative positional relationships of triangular fuzzy number membership functions. This approach directly addresses the triangular fuzzy number model and ensures the integrity of information in the p-hub problem as much as possible. It is verified by the classic Civil Aeronautics Board and several self-constructed data sets. The results indicate that, compared to the traditional Genetic Algorithm and Tabu Search algorithm, the Genetic-Tabu Search algorithm reduces average computation time by 49.05% and 40.93%, respectively. Compared to traditional random, robust, and real-number-based optimization approaches, the proposed optimization approach reduces the total cost in uncertain environments by 1.47%, 2.80%, and 8.85%, respectively.

Enhancing Multi-Criteria Decision-Making with Fuzzy Logic: An Advanced Defining Interrelationships Between Ranked II Method Incorporating Triangular Fuzzy Numbers

Enhancing Multi-Criteria Decision-Making with Fuzzy Logic: An Advanced Defining Interrelationships Between Ranked II Method Incorporating Triangular Fuzzy Numbers

Probabilistic human health risk assessment of PCDD/Fs near municipal solid-waste incinerator using Monte Carlo analysis coupled with triangular fuzzy numbers

PCDD/Fs are dioxins produced by waste incineration and pose risks to human health. We aimed to detail the health risks of airborne and soil PCDD/Fs near a municipal solid-waste incinerator (MSWI) for the surrounding population and develop a new model that improves upon existing methods. Thus, we conducted field sampling and then investigated a MSWI in the Pearl River Delta (2016–2018). Our results showed that the carcinogenic and non-carcinogenic risk values of PCDD/Fs exposed to residents in nearby areas were acceptable, with hazard index (HI) values lower than 1.0 and a total carcinogenic risk lower than 1.0E-6. Notably, the results raised concerns regarding higher non-carcinogenic risks in children than in adults. Comparative analysis of the frequency accumulation diagram, accumulated probability risk, and the absolute value of error (δ) between the 95% confidence interval (CI) and the 90% CI of the Monte Carlo stochastic simulation-triangular fuzzy number (MCSS-TFN) and the MCSS model, respectively, demonstrated that the MCSS-TFN exhibited less uncertainty than the MCSS model, regardless of the health risk value of PCDD/Fs in ambient air or in soil. This observation underscores the superiority of the MCSS-TFN model over other models in assessing the health risks associated with PCDD/Fs in situations with limited data. Our new method overcomes the limited dataset size and high uncertainty in assessing the health risks of dioxin substances, providing a more comprehensive understanding of their associated health risks than MCSS models.

Thermal Analysis of MHD Hybrid Nanofluid on Stretching/Shrinking Non-Parallel Walls with Uncertain Volume Fractions

This study examines the heat transmission of magneto-hydrodynamic (MHD) hybrid nanofluid (Ag–[Formula: see text]O) flows between two stretchable or shrinkable walls. The nanoparticle volume fraction is considered an uncertain parameter and is represented using triangular and Gaussian fuzzy numbers. The governing radial momentum and energy equations are transformed into nonlinear ordinary differential equations using a similarity transformation and with the required boundary conditions. A generalized homotopy approach is used to analyze the impacts of embedding parameters, including the Reynolds number, stretching/shrinking parameter, opening angle, radiation parameter, Magnetic parameter, and solid volume fraction on velocity and thermal profiles in crisp and uncertain cases. The findings demonstrate that fluid velocity and temperature distribution increase as the stretching parameter increases. The correctness of the suggested technique is validated by comparing the results with the available numerical solution. It has been observed that there is a great agreement between the present results and the numerical ones.

Project Investment Decision Under Uncertain Information

With the change of environment and market, the current investment project decision is facing a more complex environment, the competition of enterprises intensifies, the decision complexity increases, and the demand for scientific decision is increasing. The advantage of multi-attribute decision making in investment project decision is increasingly prominent. This paper studies the application of multi-attribute decision making under triangular fuzzy uncertainty information in investment decision making. Considering the decision makers' risk aversion and preference attitude in multi-stage decision making process, a dynamic reference point setting method based on stage development characteristics is proposed. Considering that the attribute value and attribute value will produce compensation in the decision-making process, the attribute value difference formula is constructed to reduce the decision-making bias, and on this basis, an improved TODIM method is proposed. The triangular fuzzy number uncertainty information is introduced to evaluate the attributes of each scheme in each decision stage, and the information is aggregated by the aggregation operator. Combining the dynamic multi-attribute decision model, the practical problem is solved. The practicability and feasibility of this method are demonstrated by the example analysis and method comparison.

Intelligent decision support system for pulmonary tuberculosis detection using bipolar fuzzy utility matrix and bipolar Mamdani fuzzy inference system

Tuberculosis (TB) stands as the second leading global infectious cause of death, following closely behind the impact of COVID-19. The standard approach to diagnose TB involves skin tests, but these tests can yield inaccurate results due to limited access to healthcare and insufficient diagnostic resources. To enhance diagnostic accuracy, this study introduces a novel approach employing a Bipolar Fuzzy Utility Matrix Inference System (BFUMIS) and a Bipolar Mamdani Fuzzy Inference System (BMFIS) to assess TB disease levels. By considering factors associated with the causation of TB, the study devises suitable membership functions for bipolar fuzzy sets (BFS) using both triangular and trapezoidal fuzzy numbers. Using a point factor scale, the study clusters the rules systematically and assesses the level of uncertainty within these grouped rules by utilizing bipolar triangular fuzzy numbers (BTFN). To handle the BTFN, this study proposes converting bipolar triangular fuzzy into bipolar crisp score (CBTFBCS) algorithm as a defuzzification method. The optimal bipolar fuzzy utility sets (BFUS) are determined from the bipolar fuzzy utility matrix to identify patients’ TB disease levels. These sets play a pivotal role in characterizing the severity of TB disease levels in patients. Additionally, rigorous validation of the utility framework is accomplished through measures of bipolar fuzzy satisfactory factors and sensitivity analyses. Furthermore, the study introduces the BMFIS, which presents a novel perspective on the conventional fuzzy inference system. This innovative system integrates the Mamdani fuzzy inference system (MFIS) into a bipolar fuzzy context, enriching the diagnostic process with enhanced insights. To demonstrate the efficacy of the proposed methods, extensive validation is carried out using actual clinical data. The performance metrics used in this validation effectively demonstrate the superiority of the proposed approach.

A Novel Approach for the Assessment of Logistics Performance Index of EU Countries

Abstract The impact of logistics performance in the era of sustainable mobility on the overall economic development of a country is inevitable. It can even be said to represent an extremely important component in identifying economic conditions and provides the possibility of defining adequate strategies. In this article, the evaluation of the member countries of the European Union was carried out on the basis of the logistics performance index (LPI) according to the latest report of the World Bank (WB). A unique and original Multiple-Criteria Decision Making (MCDM) approach has been created, and it involves the application of four methods: Criteria Importance Through Intercriteria Correlation, Method based on the Removal Effects of Criteria, and Entropy and Fuzzy ROV (Range of Value). The weighting coefficients of six factors were obtained with the first three methods in crisp form, so they were converted into Triangular Fuzzy Number. The Fuzzy ROV method has been created for the first time in the literature and represents a great contribution from the methodological aspect. The results of the developed model and the applied steps show that there are certain differences in the rankings compared to the World Bank report, with a note that the best-ranked countries have maintained their positions. In addition, verification tests of the originally obtained results were created, with an emphasis on the importance of evaluation parameter values and their impact on the LPI ranking.

Enhanced MADM framework for fuzzy comprehensive evaluation of attack ability of basketball defenders with the triangular fuzzy neutrosophic numbers

With the continuous improvement of modern basketball technology, higher requirements have been put forward for the personal abilities of basketball players. As the core of an organization, the offensive ability of a defender largely determines the team’s performance. Therefore, it is necessary to objectively evaluate the attacking ability of defenders. Traditional techniques cannot objectively reflect the true level of players due to their strong subjectivity. Therefore, establishing a scientific evaluation technique is particularly important. The fuzzy comprehensive evaluation of attack ability of basketball defenders is viewed as the multi-attribute decision-making (MADM). In this paper, the triangular fuzzy neutrosophic number cross-entropy (TFNN-CE) technique is designed with help of cross-entropy and triangular fuzzy neutrosophic sets (TFNSs). Furthermore, Then, TFNN-CE technique is addressed to solve the MADM. Finally, a numerical example for fuzzy comprehensive evaluation of attack ability of basketball defenders is given and some comparisons are conducted to r illustrate advantages of the designed technique. The main contribution of this paper is addressed: (1) The TFNN-CE technique is addressed under TFNSs; (2) the TFNN-CE technique is addressed for MADM under TFNSs; (2) the TFNN-CE technique for fuzzy comprehensive evaluation of attack ability of basketball defenders is addressed; (3) Through the several efficient comparisons, it is addressed that TFNN-CE technique is effective for fuzzy comprehensive evaluation of attack ability of basketball defenders.

Generalized fractional model of heat transfer in uncertain hybrid nanofluid with entropy optimization in fuzzy-Caputo sense

In this paper, we present a new fuzzy-fractional (FF) transformation to recover FF differential model of hybrid nanofluid. The current study focuses on FF modeling of nanofluid with engine oil as base fluid, while ferrous oxide Fe2O3 and alumina Al2O3 are considered nanoparticles. In accordance to the real industrial phenomena, the flow is simulated between two squeezing plates with thermal radiation and magnetic effects. A generalized fuzzy-fraction flow problem is modeled by introducing new similarity transforms. Obtained model is validated both theoretically and numerically. At integer order Υ=1, the FF model reduces to the integer order fluid model existing in literature, proving theoretical validity. Fuzzy-valued functions are discriminated through triangular fuzzy numbers using r-cut approach. In order to solve, obtained highly non-linear FF nanofluid system, we apply He–Laplace–Carson (HLC) algorithm. Differential and convolution properties of Laplace–Carson Transform (LCT) are utilized for solution purpose. Error and convergence analysis is performed numerically to verify obtained results. Furthermore, graphical illustrations for upper and lower bound analysis on FF profiles is also presented. Analysis reveals that heat transfer in engine oil enhances with an increase in radiation at upper and lower bound in fuzzy-fractional environment. Moreover, entropy decreases with an increase in nanoparticle concentration of Fe2O3 and Al2O3 in engine oil.

Research on Condition Assessment Method of Transformer Based on Cloud Model

The traditional evaluation method of transformer has the problems of strong subjectivity of threshold value and fuzzy correlation of state quantity, which is difficult to accurately evaluate the operation quality of transformer. In order to solve these problems, this paper proposes a comprehensive evaluation method of transformer performance evaluation and operation quality based on cloud model. The evaluation index system of power transformer operation quality considering performance is constructed, including target layer, performance layer and index layer. The triangular fuzzy number is used to optimize the set of expert opinions, and the indexes are combined and weighted by fuzzy analytic hierarchy process and entropy weight method. After the data is degraded, the membership matrix of transformer operation index is constructed by using the cloud model, and the performance state weighting and membership degree are comprehensively considered to realize the performance evaluation of transformer and the comprehensive evaluation of operation quality based on performance. By using this method to predict 10 transformers in operation for a period of one month, it is possible to accurately identify abnormal situations on site, with a prediction accuracy of 96.28%. Compared with the traditional method of “predicting first and then diagnosing”, it has improved by 14.32%, overcoming the shortcomings of subjectivity, ambiguity, and ignoring the correlation of state variables in traditional evaluation methods. This method provides new ideas and methods for accurate evaluation of transformers.

A Pivotal Operation on Triangular Fuzzy Number for Solving Fuzzy Nonlinear Programming Problems

A Pivotal Operation on Triangular Fuzzy Number for Solving Fuzzy Nonlinear Programming Problems

An approach to assessing the technical condition of electrical equipment using weighted fuzzy rules

To ensure the smooth operation of all existing electric power systems, it is necessary to periodically diagnose electrical equipment using various methods and models that take into account all possible parameters and factors affecting the condition. The paper considers an approach to assessing the technical condition of electrical equipment using weighted fuzzy rules, taking into account different types of information (measuring, expert). The novelty of the approach is the representation of parameters in the form of fuzzy triangular numbers and the formation of importance weight vectors of term sets parameter values. This makes it possible to accurately assess the technical condition of electrical equipment in the context of different types of information to predict the condition of the equipment; to quickly identify parameters which values are outside acceptable limits, thereby determining the preliminary cause of equipment failure, as well as making informed diagnostic decisions regarding the condition of electrical equipment.

Analysis of MOFTP using Modern Zero Suffix Method

Abstract: In this paper, a new algorithm is introduced for dealing with the Multi-Objective Fuzzy Transportation Problem (MOFTP) where all parameters such as transportation cost, demand and supply are in triangular fuzzy numbers. The approach involves modern zero suffix method using Harmonic Mean. First the Multi-objective fuzzy transportation problem is converted into a crisp value by using a Robust’s Ranking Method. Then the crisp value is solved by Modern Zero Suffix method. The effectiveness of the data is examined by a numerical example.