Triangular Fuzzy Variables Research Articles

An Alternative Method for Finding the Critical Path of the Network in Fuzzy Time Cost Trade off Problem

Background : The critical path approach is used to determine the network's longest path, according to historical records. This study examines a different approach to determining the construction network's longest path. Method: Here, the network is viewed as a directed acyclic graph, and the critical path of the network is found using the longest path algorithm of the network. To find the best answer for a building project, the longest path that was found was integrated into a linear programming issue. The triangle fuzzy variable defines all of the project's inputs. The 992 square foot building area is incorporated, and three project manager’s quotes are used as a triangular fuzzy variable. Findings : This work has the options of getting quotation from the project managers, convert them as the fuzzy variables such as triangular fuzzy variable, Trapezoidal fuzzy variable and pentagonal fuzzy variable. After the network is converted into a linear programming problem using the fully fuzzy mathematical model, the best possible solution is found. Novelty and applications : Alternative method for critical path of the network has been incorporated. It has been found that the proposed method reduces the time to find the critical path of the larger networks. Keywords: Directed Acyclic Graph, Longest Path Algorithm, Triangular Fuzzy Variable, Fuzzy Linear Programming Problem, Fully Fuzzy Mathematical Model

Fuzzy Programming Approach to Solve Multi-Objective Fully Fuzzy Transportation Problem

The aim of this study is presenting the solution methodology of multi-objective fuzzy transportation problem with fuzzy decision variables, where all the input parameters and decisions variables of the programming problems are assumed to be triangular fuzzy number and triangular fuzzy decision variables respectively. Moreover the objectives under considerations are minimization of cost of transportation and minimization of shipping time under fuzzy environment. The fuzziness of the objective functions and the fuzzy constraints of the programming problem are defuzzified using the ranking function and the equality property between two fuzzy numbers, respectively. The consequent crisp multi-objective fuzzy transportation problem is tackled by employing fuzzy mathematical programming approach. Finally fuzzy decision is made after solving the resultant mathematical programming problem using LINGO(Schrage and LINDO Systems (1997)) software. Illustrative numerical example is presented in support of the proposed methodology.

Catastrophe Bond Diversification Strategy Using Probabilistic–Possibilistic Bijective Transformation and Credibility Measures in Fuzzy Environment

The variety of catastrophe bond issuances can be used for portfolio diversification. However, the structure of catastrophe bonds differs from traditional bonds in that the face value and coupons depend on triggering events. This study aims to build a diversification strategy model framework using probabilistic–possibilistic bijective transformation (PPBT) and credibility measures in fuzzy environments based on the payoff function. The stages of modeling include identifying the trigger distribution; determining the membership degrees for the face value and coupons using PPBT; calculating the average face value and coupons using the fuzzy quantification theory; formulating the fuzzy variables for the yield; defining the function of triangular fuzzy membership for the yield; defining the credibility distribution for the triangular fuzzy variables for the yield; determining the expectation and total variance for the yield; developing a model of the catastrophe bond diversification strategy; the numerical simulation of the catastrophe bond strategy model; and formulating a solution to the simulation model of the diversification strategy using the sequential method, quadratic programming, transformation, and linearization techniques. The simulation results show that the proposed model can overcome the self-duality characteristic not possessed by the possibilistic measures in the fuzzy variables. The results obtained are expected to contribute to describing the yield uncertainty of investing in catastrophe bond assets so that investors can make wise decisions.

On generating fuzzy Pareto solutions in fully fuzzy multiobjective linear programmingviaa compromise method

In the present paper, it is unified and extended recent contributions on fully fuzzy multiobjective linear programming, and it is proposed a new method for obtaining fuzzy Pareto solutions of a fully fuzzy multiobjective linear programming problem. For its formulation, triangular fuzzy numbers and variables are combined with fuzzy partial orders and fuzzy arithmetic, and no ranking functions are required. By means of solving related crisp multiobjective linear problems, it is provided algorithms to generate fuzzy Pareto solutions; in particular, to generate compromise fuzzy Pareto solutions, what is a novelty in this field.

Fuzzy closed-loop supply chain models with quality and marketing effort-dependent demand

While previous literature studies the market demand affected by quality or marketing effort in a supply chain, this paper explores the pricing and collection strategies of different closed-loop supply chain models in regard to a quality and marketing effort-dependent demand in a fuzzy environment. Specifically, a manufacturer produces new and remanufactured products with three collection scenarios, namely, a manufacturer undertaking collection operations, a retailer undertaking collection operations, and a third party undertaking collection operations. Considering the uncertainty of many parameters in a supply chain, we apply triangular fuzzy variables to describe some parameters, such as, the changeable collection cost of used products, the manufacturing/remanufacturing cost of products, and the market demand (e.g., the market base, the elastic coefficient). By adopting a game theoretical approach, we conduct three collection modes in a deterministic environment as benchmark models and three fuzzy mathematical models, then obtain the equilibrium results and the optimal expected profit of each supply chain member. Furthermore, we conduct numerical experiments to compare the optimal outcomes of three collection modes considering a fuzzy environment and a deterministic environment, then analyze the impacts of fuzziness of parameters on the expected profits. Among three collection modes in a fuzzy environment, we observe when the manufacturer is responsible for the collection business, the whole supply chain system has the highest efficiency, and the expected profit of each chain member is the highest. Simultaneously, with the same collection mode, the performance of supply chain considering the fuzzy environment is better than that in the deterministic environment. The expected profits of the manufacturer and the retailer descend with the increase of the quality effort cost coefficient. Similarly, when the cost coefficient of the marketing effort increases, the expected profits of the manufacturer and the retailer have downward trends. When the fuzziness of the marketing/quality effort coefficient changes, the retailer’s expected profit is more affected than that of the manufacturer.

A fuzzy quantile method for AR time series model based on triangular fuzzy random variables

A fuzzy quantile method for AR time series model based on triangular fuzzy random variables

Variance and Semi-Variances of Regular Interval Type-2 Fuzzy Variables

In this paper, we define the variance and semi-variances of regular interval type-2 fuzzy variables (RIT2-FVs) as well as derive a calculation formula of them based on the credibility distribution. Following the relationship between the variance and the semi-variances of the regular symmetric triangular interval type-2 fuzzy variables (RSTIT2-FVs), a special type of interval type-2 fuzzy variable is discovered and proved. Furthermore, for applying the two measures, we propose the operational law for the variance and semi-variances of the linear function of mutually independent RSTIT2-FVs. Some numerical examples are illustrated. The consequences of examples prove that the formulas we proposed can be effectively applied to the calculation of the variance of RSTIT2-FVs. The results indicate that they play a great role in the application of variance of type-2 fuzzy sets in various fields.

A novel mathematical programming model for multi-mode project portfolio selection and scheduling with flexible resources and due dates under interval-valued fuzzy random uncertainty

Resource management plays a pivotal role in project implementation success. The profitability of project-based companies not only relies on the right selection of project portfolio but also depends on accurate scheduling and resource management. This issue is addressed by optimal project portfolio selection and scheduling using flexible resource availability and projects’ due date. The motivation of flexible resources is to maximize cash flow by resource management and adding more projects to a portfolio by reinvesting the gain to increase resource availability during planning horizon. Flexible due date leads to perform more projects besides increasing the robustness of multiple projects. This study proposes a new linear structure of mixed-integer programming (MIP) model of project portfolio selection and scheduling considering the resource management, cash flow, tardiness cost, and robustness of multiple projects. Moreover, to a better description of real-life project situations, a new solution approach is introduced based on triangular interval-valued fuzzy random variables to incorporate both fuzziness and randomness uncertainties into mathematical programming models. To solve the proposed interval-valued fuzzy random model, a generalized credibility-based chance constraint measure is applied. The application and performance of proposed model are evaluated through different small to large-sized test problems and a real case study to tackle uncertainties and complexities explicitly.

Fuzzy linear programming problem with fuzzy decision variables: A geometrical approach

Decision making problems force the decision maker to consider fuzzy decision variables in a fuzzy linear programming problem. Therefore, the proposed fuzzy linear programming problem considers a triangular fuzzy decision variable with triangular fuzzy parameters. A defuzzyfication method based on incenter of a triangle, which is formed by joining the vertices of the triangular fuzzy parameters and fuzzy decision variables. The coordinate of the incenter is obtained by using the concept of geometry. In next step, the incenter is considered as a vector quantity. Using the concept of obtaining the magnitude of a vector, the final mathematical model is formulated. The final mathematical programming problem is a crisp non-linear programming problem. The resultant crisp mathematical programming problem is solved by appropriate mathematical software. A numerical example is presented to illustrate the methodology.

(2005-5916) A novel defuzzification approach of Type-2 fuzzy variable to solving matrix games: An application to plastic ban problem

The novelty of the type-2 fuzzy variable (T2FV) is its fuzzy secondary possibility distribution. Here, we aim to formulate a new defuzzification model of triangular type-2 fuzzy variables (TT2FVs) based on type reduction. We also show its superiority compared to other existing models by providing some numerical examples. Later, we develop a matrix game problem in the type-2 fuzzy environment to show the applicability of the proposed defuzzification method in a real-world situation. Based on this defuzzification method, two crisp linear programming models are derived, which are subsequently solved by the simplex method using LINGO 17.0 software. Also, the expected pay-off for the maximizing player is calculated as TT2FV, which is desirable. Finally, to show the validity and applicability, the proposed methodology is illustrated with the plastic ban problem.

Solution of Fully Bipolar Fuzzy Linear Programming Models

The Yin-Yang bipolar fuzzy set is a powerful mathematical tool for depicting fuzziness and vagueness. We first extend the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We first define arithmetic operations of unrestricted bipolar fuzzy numbers and multiplication of an unrestricted trapezoidal bipolar fuzzy number (TrBFN) with non-negative TrBFN. We then propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.

Approaches for Solving Fully Fuzzy Rough Multi-Objective Nonlinear Programming Problems

Practical nonlinear programming problem often encounters uncertainty and indecision due to various factors that cannot be controlled. To overcome these limitations, fully fuzzy rough approaches are applied to such a problem. In this paper, an effective two approaches are proposed to solve fully fuzzy rough multi-objective nonlinear programming problems (FFRMONLP) where all the variables and parameters are fuzzy rough triangular numbers. The first, based on a slice sum technique, a fully fuzzy rough multi-objective nonlinear problem has turned into five equivalent multi-objective nonlinear programming (FFMONLP) problems. The second proposed method for solving FFRMONLP problems is α-cut approach, where the triangular fuzzy rough variables and parameters of the FFRMONLP problem are converted into rough interval variables and parameters by α-level cut, moreover the rough MONLP problem turns into four MONLP problems. Furthermore, the weighted sum method is used in both proposed approaches to convert multi-objective nonlinear problems into an equivalent nonlinear programming problem. Finally, the effectiveness of the proposed procedure is demonstrated by numerical examples.

A Simplified Novel Technique for Solving Linear Programming Problems with Triangular Fuzzy Variables via Interval Linear Programming Problems

This study proposes a novel technique for solving Linear Programming Problems with triangular fuzzy variables. A modified version of the well-known simplex method and the Existing Method for Solving Interval Linear Programming problems are used for solving linear programming problems with triangular fuzzy variables. Furthermore, for illustration, some numerical examples and one real problem are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

Reliability analysis of station autonomous computer system based on fuzzy dynamic fault tree and Markov model

AbstractStation autonomous computer system is the core of the centralized traffic control (CTC) system. The working mode is dual computer hot standby, and its reliability is very important. Due to the dynamic and fuzzy issue with the reliability analysis of station autonomous computer system, a reliability analysis method is formulated based on fuzzy dynamic fault tree (DFT) and Markov model. First, the structure and working principle of the dual computer hot standby autonomous computer system and the double 2‐vote‐2 autonomous computer system are analyzed, and two autonomous computer systems based on DFT are formulated. Then, the failure rates and the fault detection probabilities are considered as fuzzy variables and represented as triangular fuzzy variables. The fuzzy state transition matrix of the system is obtained by conversion of DFT to the Markov model. Finally, the fuzzy reliability of the system is calculated by Laplace transform. Simulation results show that the reliability of the dual computer hot standby autonomous computer system is higher than that of the double 2‐vote‐2 autonomous computer system. Besides, the reliability of the hardware system is not only related to the structure of the system, but also affected by the uncertain unit failure rate, fault detection probabilities, and so on.

Heterogeneous wireless network selection by combining GRA and VIKOR under a fuzzy environment

In a heterogeneous wireless environment, network selection is a strategic issue and has a significant impact on providing the best quality of service (QoS) to the users. The selection of an apt network among various alternatives is a kind of multi-criteria decision-making (MCDM) problem. This paper proposes a model based on VlseKriterijumskaOptimizacija I KompromisnoResenje (VIKOR) under fuzzy environment and grey relational analysis (GRA) for the selection of suitable network in a heterogeneous wireless environment. Here, triangular fuzzy linguistic variables are used to handle the vagueness and subjectivity of the decision-making process. This study focuses on four alternatives such as universal mobile telecommunications system (UMTS), long term evolution (LTE), worldwide interoperability for microwave access (WiMAX) and wireless local area network (WLAN). A six evaluation criteria such as throughput, delay, jitter, packet loss, cost, security are considered for choosing the suitable network in heterogeneous wireless environment.

A New Method for Solving Multi-Criteria Multi-Attributes Decision Making Based on the Ranking of Type-2 Fuzzy Variables

In this paper, the authors have proposed the concept of interval type-2 triangular fuzzy variables. Then, they studied the concepts of value and ambiguity of interval type-2 triangular fuzzy variables and interval type-2 trapezoidal fuzzy variables. They introduced the concept of value and ambiguity in order to define the ranking method for the interval type-2 fuzzy variables. A comparative result of the various other ranking methods is also given in the tabular form. A multi-criteria multi-attributes decision-making problem is provided to explain the ranking method in which the evaluation ratings of the alternatives on the attributes, and the criteria weights as provided by the decision makers are expressed as linguistic terms (e.g., very high, medium, fair, and good). The multi-criteria multi-attributes decision-making problem is then worked out by applying the proposed algorithm.

A group decision making method with intuitionistic triangular fuzzy preference relations and its application

This paper aims to offer an intuitionistic triangular fuzzy group decision making method by preference relations. For this purpose, the concept of intuitionistic triangular fuzzy preference relations (ITFPRs) is first offered. Then, an additive consistency concept for ITFPRs is introduced. Meanwhile, a programming model is built to check the consistency of ITFPRs. Considering the case where incomplete ITFPRs are obtained, two programming models are constructed, which aim at maximizing the consistency and minimizing the uncertainty of missing information. To achieve the goals of the minimum total adjustment and the smallest number of adjusted elements, two programming models are established to repair inconsistent ITFPRs. In addition, the weights of decision makers are considered, and the consensus levels of individual ITFPRs are studied to ensure the representativeness of decision results. When individual ITFPRs do not meet the consensus requirement, a programming model to reach the consensus threshold is constructed, which permits different intuitionistic triangular fuzzy variables (ITFVs) to have different adjustments and minimizes the total adjustment. Finally, a group decision making algorithm with ITFPRs is proposed, and its feasibility and efficiency are demonstrated through an example of evaluating the intelligent traditional Chinese medicine decocting centers.

A Clustering Algorithm for Triangular Fuzzy Normal Random Variables

In view of the fact that most clustering algorithms cannot solve the clustering problem about samples with uncertain information, according to the theory of fuzzy sets and probability, we define the fuzzy-probability binary measure space and triangular fuzzy normal random variables firstly, and then combine the advantages of k-means algorithm, such as simple principle, few parameters, fast convergence rate, good clustering effect and good scalability, etc., a clustering algorithm is proposed for samples containing multiple triangular fuzzy normal random variables, which we call TFNRV-k-means algorithm. The algorithm uses our proposed Euclidean random comprehensive absolute distance (ERCAD for short) as a measurement, under the fuzzy measure, the lower bound, the principal value and the upper bound of the triangular fuzzy normal random variables are iterated, respectively, by means, and then the cluster center is updated until it becomes stable and unchanged. Then we analyze the time complexity of the proposed algorithm, and test the algorithm under different sample sets by random simulation experiments. We get the highest clustering accuracy of 99.00% and the maximum Kappa coefficient of 0.9850, and draw the conclusion that TFNRV-k-means clustering algorithm has good clustering effect. Finally, we summarize the content of the article, list the advantages and disadvantages of TFNRV-k-means clustering algorithm, and propose corresponding improvement methods, which provide ideas for further research on TFNRV-k-means in the future.

Application of random triangular and Gaussian type-2 fuzzy variable to solve fixed charge multi-item four dimensional transportation problem

In this paper, two types of hybrid random type-2 uncertain variables such as random type-2 triangular and random type-2 Gaussian fuzzy variables are used to express the uncertain parameters in a four-dimensional transportation problem. Here selling price, purchasing costs, transportation costs, fixed charges, sources, demands, capacities of conveyances are considered as hybrid random type-2 uncertain parameters. The objective function of the proposed model is de-randomized using the expected value method and the constraints of the model are de-randomized using the probability chance constraint programming method. After that, the proposed model is de-fuzzified by CV based reduction technique. These two reduced crisp problems are solved by Generalized Reduced Gradient (GRG) technique using LINGO 14.0 software. The models are numerically illustrated. Some sensitivity analyses are also presented. Conventional solid (3D) and general (2D) transportation problems are derived as particular cases.

The First Moments and Semi-Moments of Fuzzy Variables Based on an Optimism-Pessimism Measure with Application for Portfolio Selection

Possibility, necessity and credibility measures are used in the literature in order to deal with imprecision. Recently, Yang and Iwamura [L. Yang and K. Iwamura, Applied Mathematical Science 2(46) (2008) 2271–2288] introduced a new measure as convex linear combination of possibility and necessity measures and they determined some of its axioms. In this paper, we introduce characteristics (parameters) of a fuzzy variable based on that measure, namely, expected value, variance, semi-variance, skewness, kurtosis and semi-kurtosis. We determine some properties of these characteristics and we compute them for trapezoidal and triangular fuzzy variables. We display their application for the determination of optimal portfolios when assets returns are described by triangular or trapezoidal fuzzy variables.