Fuzzy Quantities Research Articles

Fuzzy Cost Analysis In A Fuzzy Transportation System A Study Of The Supply Chain Management In A General Contractor Company

Transportation models play an important role in logistics and supply chain management for reducing cost and improving services. In this paper, the author presented a fuzzy transportation problem, in which the cost coefficients and supply and demand quantities are fuzzy numbers. The problem is solved in two stages. First, calculating the maximum satisfactory level and achieving balances between fuzzy supplies and demands. Second, the problem is solved by considering the unit of transportation costs and optimal solutions which are connected with fuzzy quantities’ satisfactory level are founded. The author used two different satisfactory levels for the problem: The transportation costs breaking points \((\gamma_p)\) and the values that have violated positive condition of optimal solutions in the intervals of \([\gamma_{p-1},\gamma_p]\). A new method is proposed in this paper to find optimal solutions. The proposed method is then illustrated through a numerical example.

Fuzzy Feature Visualization of 3D Vector Field by Information-Entropy-Based Texture Adaptation

Texture adaptation is a challenging issue in tex-ture-based feature visualization. In order to visualize as more information as we can, this paper presents a texture adaptation technique for fuzzy feature visualization of 3D vector field, taking into account information quantity carried by vector field and texture based on extended information entropy. Two definitions of information measurement for 3D vector field and noise texture, MIE and RNIE, are proposed to quantitatively represent the information carried by them. A noise generation algorithm based on three principles derived from minimal differentia of MIE and RNIE is designed to obtain an approximately optimal distribution of noise fragments which shows more details than those used before. A discussion of results is included to demonstrate our algorithm which leads to a more reasonable visualization results based on fuzzy feature measurement and information quantity.

RM approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Expert Syst Appl 36:6833–6842, 2009) is incorrect. The main aim of this paper is to propose a new approach for the ranking of L–R type generalized fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as RM approach. The main advantage of the proposed approach is that it provides the correct ordering of generalized and normal fuzzy numbers and it is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets Syst 118:375–385, 2001).

Optimal applicator placement in hepatic radiofrequency ablation on the basis of rare data

In this paper, a numerical procedure to determine an optimal applicator placement for hepatic radiofrequency ablation incorporating uncertain material parameters is presented. The main focus is set on the treatment of subjective and rare data-based information. For this purpose, we employ the theory of fuzzy sets and model uncertain parameters as fuzzy quantities. While fuzzy modelling has been established in structural engineering in the recent past, it is novel in biomedical engineering. Incorporating fuzzy quantities within an optimisation task is basically innovative. In our context, fuzzy modelling allows us to determine an optimal applicator placement that maximises the therapy success under the given uncertainty conditions. The applicability of our method is demonstrated by means of an example case.

RM Approach for Ranking of Generalized Trapezoidal Fuzzy Numbers

Ranking of fuzzy numbers play an important role in decision making, optimization and forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples, it is proved that ranking method proposed by Chen and Chen (Expert Systems with Applications 36(3): 6833) is incorrect. The main aim of this paper is to propose a new approach for the ranking of generalized trapezoidal fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as an RM approach. The main advantage of the proposed approach is that the proposed approach provides the correct ordering of generalized and normal trapezoidal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets and Systems 118(3): 375).

Optimizing material procurement planning problem by two-stage fuzzy programming

This paper presents a new class of two-stage fuzzy material procurement planning (MPP) models with minimum-risk criteria, in which the material demand, the spot market material unit price and the spot market material supply quantity are uncertain and assumed to be fuzzy variables with known possibility distributions. We formulate the two-stage MPP model with the objective of maximizing the credibility of the total material procurement costs less than a given allowable investment level, and the credibility can be regarded as the material procurement risk criteria in a fuzzy environment. Since the fuzzy material demand, the fuzzy spot market material unit price and the fuzzy spot market material supply quantity are usually continuous fuzzy variables with infinite supports, the proposed MPP model belongs to an infinite-dimensional optimization problem that cannot be solved directly. To avoid this difficulty, we apply an approximation approach (AA) to the proposed two-stage fuzzy MPP model, and turn it into an approximating finite-dimensional optimization one. The convergence about the objective function of the approximating two-stage MPP model to that of the original two-stage MPP one is also discussed. Since the exact analytical expression for the objective function in the approximating fuzzy MPP model is unavailable, and the approximating MPP model is a mixed-integer program that is neither linear nor convex, traditional optimization algorithms cannot be used to solve it. Therefore, we develop two heuristic algorithms to solve the approximating MPP model. The first is a particle swarm optimization (PSO) algorithm based on the AA, and the second is a hybrid PSO algorithm which based on the AA and a neural network (NN). Finally, we provide an actual optimization problem about the fuel procurement to compare the effectiveness of the designed algorithms.

Enhanced genetic algorithm-based fuzzy multiobjective strategy to multiproduct batch plant design

This paper addresses the problem of the optimal design of batch plants with imprecise demands in product amounts. The design of such plants necessary involves how equipment may be utilized, which means that plant scheduling and production must constitute a basic part of the design problem. Rather than resorting to a traditional probabilistic approach for modeling the imprecision on product demands, this work proposes an alternative treatment by using fuzzy concepts. The design problem is tackled by introducing a new approach based on a multiobjective genetic algorithm, combined wit the fuzzy set theory for computing the objectives as fuzzy quantities. The problem takes into account simultaneous maximization of the fuzzy net present value N P ˜ V and of two other performance criteria, i.e. the production delay/advance and a flexibility index. The delay/advance objective is computed by comparing the fuzzy production time for the products to a given fuzzy time horizon, and the flexibility index represents the additional fuzzy production that the plant would be able to produce. The multiobjective optimization provides the Pareto's front which is a set of scenarios that are helpful for guiding the decision's maker in its final choices. About the solution procedure, a genetic algorithm was implemented since it is particularly well-suited to take into account the arithmetic of fuzzy numbers. Furthermore because a genetic algorithm is working on populations of potential solutions, this type of procedure is well adapted for multiobjective optimization.

Real-Time Constrained Fuzzy Arithmetic

Klir introduced constrained fuzzy arithmetic (CFA) as a solution to the unnecessary precision loss when dealing with fuzzy quantities that represent linguistic variables. Since then, some attempts have been made to make CFA efficient, but none of these solutions is suitable for real-time applications. In this paper, we will propose a new CFA algorithm that can be used in such environments.

Application of fuzzy minimum cost flow problems to network design under uncertainty

This paper deals with fuzzy quantities and relations in multi-objective minimum cost flow problem. When t-norms and t-conorms are available, the goal programming is applied to minimize the deviation among the multiple costs of fuzzy flows and the given targets when the fuzzy supplies and demands are satisfied. To obtain the most optimistic and the most pessimistic satisficing solutions of this problem, two polynomial time algorithms are introduced applying some network transformations. To demonstrate the performance of this approach in actual substances, network design under fuzziness is considered and an efficient scheme is proposed including genetic algorithm together with fuzzy minimum cost flow problem. This scheme is applied on a pilot network for more description.

Using the Hamming distance to extend TOPSIS in a fuzzy environment

Considering the fact that, in some cases, determining precisely the exact value of attributes is difficult and that their values can be considered as fuzzy data, this paper extends the TOPSIS method for dealing with fuzzy data, and an algorithm for determining the best choice among all possible choices when the data are fuzzy is also presented. In this approach, to identify the fuzzy ideal solution and fuzzy negative ideal solution, one of the Yager indices which is used for ordering fuzzy quantities in [0, 1] is applied. Using Yager’s index leads to a procedure for choosing fuzzy ideal and negative ideal solutions directly from the data for observed alternatives. Then, the Hamming distance is proposed for calculating the distance between two triangular fuzzy numbers. Finally, an application is given, to clarify the main results developed in the paper.

Extended triangular norms

The paper is devoted to classical t-norms extended to operations on fuzzy quantities in accordance with the generalized Zadeh extension principle. Such extended t-norms are used for calculating intersection of type-2 fuzzy sets. Analytical expressions for membership functions of some extended t-norms are derived assuming special classes of fuzzy quantities, i.e., fuzzy truth intervals or fuzzy truth numbers. The possibility of applying these results in the construction of type-2 adaptive network fuzzy inference systems is illustrated on several examples.

Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution

This paper discusses full fuzzy linear programming (FFLP) problems of which all parameters and variable are triangular fuzzy numbers. We use the concept of the symmetric triangular fuzzy number and introduce an approach to defuzzify a general fuzzy quantity. For such a problem, first, the fuzzy triangular number is approximated to its nearest symmetric triangular number, with the assumption that all decision variables are symmetric triangular. An optimal solution to the above-mentioned problem is a symmetric fuzzy solution. Every FLP models turned into two crisp complex linear problems; first a problem is designed in which the center objective value will be calculated and since the center of a fuzzy number is preferred to (its) margin. With a special ranking on fuzzy numbers, the FFLP transform to multi objective linear programming (MOLP) where all variables and parameters are crisp.

Multi-item EOQ model with hybrid cost parameters under fuzzy/fuzzy-stochastic resource constraints: A geometric programming approach

In this paper, a multi-item economic order quantity (EOQ) model is considered in which the cost parameters are of fuzzy/hybrid nature under two types of resources — (a) resources as fuzzy quantities; (b) resources as fuzzy and fuzzy-random quantities. The unit cost depends on demand rate. The time horizon is taken to be infinite. We find the average cost for the model, which is a function of order quantity and demand rate and also of some hybrid parameters. When the resources are fuzzy quantities, the problem is transformed into its equivalent unconstrained deterministic form by using a surprise function for the constraints. The problem involving hybrid number is again equivalently rewritten as a multi-objective (minimization of the mean of the objective function and variance function of the distribution) inventory problem. Introducing new variables we transform the terms of the functions into signomial types. Using fuzzy multi-objective solution procedure we solve the problem through Geometric Programming approach. Sensitivity analysis has been performed to study the effect of different weights considered for mean objective function and variance function.

Using fuzzy numbers to propagate uncertainty in matrix-based LCI

Background, aim, and scope Analysis of uncertainties plays a vital role in the interpretation of life cycle assessment findings. Some of these uncertainties arise from parametric data variability in life cycle inventory analysis. For instance, the efficiencies of manufacturing processes may vary among different industrial sites or geographic regions; or, in the case of new and unproven technologies, it is possible that prospective performance levels can only be estimated. Although such data variability is usually treated using a probabilistic framework, some recent work on the use of fuzzy sets or possibility theory has appeared in the literature. The latter school of thought is based on the notion that not all data variability can be properly described in terms of frequency of occurrence. In many cases, it is necessary to model the uncertainty associated with the subjective degree of plausibility of parameter values. Fuzzy set theory is appropriate for such uncertainties. However, the computations required for handling fuzzy quantities has not been fully integrated with the formal matrix-based life cycle inventory analysis (LCI) described by Heijungs and Suh (2002).

Assessing and affording the control of flood risk

Flood is a most serious hazard to life and property. Dams, dikes and levees are often designed to a fuzzy quantity (PMF). Probabilistic design is preferable but requires that hydrological data be translated into a local monoscopic flood probability distribution. This process introduces information that goes beyond the facts. The method of relative entropy with quantile constraints minimizes this information and has a practical approximation, which is used here. Socio-economic optimization provides a design criterion that reflects the societal capacity to commit resources (SCCR) to sustainable risk reduction. Details of financing have an important influence on the design of flood control projects by socio-economic optimization; since future life risk must be discounted like finances, the interest rate and amortization period influence design decisively. The example of a city protected by a levee casts a light on the relative importance of the factors influencing the design.

A production inventory model with fuzzy random demand and with flexibility and reliability considerations

The classical inventory control models assume that items are produced by perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be increased without a price, its set-up cost can be reduced with investment in flexibility improvement. In this paper, a production inventory model with flexibility and reliability (of production process) consideration is developed in an imprecise and uncertain mixed environment. The aim of this paper is to introduce demand as a fuzzy random variable in an imperfect production process. Here, the set-up cost and the reliability of the production process along with the production period are the decision variables. Due to fuzzy-randomness of the demand, expected average profit of the model is a fuzzy quantity and its graded mean integration value (GMIV) is optimized using unconstraint signomial geometric programming to determine optimal decision for the decision maker (DM). A numerical example has been considered to illustrate the model.

HEURISTIC OPTIMIZATION OF NATURAL PRODUCTION INVENTORY MODELS WITH THE PREFERENCE OF A DECISION MAKER

In this paper, two natural production inventory models based on fuzzy total production inventory cost with the preference of a decision maker are introduced, and combined by natural number parameters in which values are linguistic values in natural language, crisp real number variables, and fuzzy number variables. These are the one natural production inventory model for crisp production quantity, and the other natural production inventory model for fuzzy production quantity. The natural arithmetical operations of both natural numbers and fuzzy numbers by Function Principle are used to compute fuzzy total production inventory cost of each natural production inventory model. Graded k-preference integration representation method is discussed for defuzzifying fuzzy total production inventory cost by preference of decision maker. Furthermore, Extension of the Lagrangean method is used to solve inequality constrain problem in our natural production inventory model. We find out that our optimal solutions can be specified to the classical production inventory model when preference 0.5 of decision maker and natural numbers and fuzzy numbers in our models are crisp real numbers.

Solution of a fully fuzzy multi-item economic order quantity problem by using fuzzy ranking functions

In this study, a fuzzy multi-item economic order quantity (EOQ) problem is solved by employing four different fuzzy ranking methods. All of the parameters of the multi-item EOQ problem are defined as triangular fuzzy numbers. Fuzzy ranking methods are used to rank the fuzzy objective values and to handle the constraints in the model. The results obtained by employing different fuzzy ranking methods are also compared.

Designing cellular manufacturing systems under dynamic and uncertain conditions

The paper proposes a fuzzy programming based approach to design a cellular manufacturing system under dynamic and uncertain conditions. The dynamic condition indicates a multi-period planning horizon, in which the product mix and demand in each period can be different. As a result, the best cells designed for one period may not be efficient cells for subsequent periods and some of reconfigurations are required. Uncertain condition implicates to the imprecise nature of the part demand and also the availability of the manufacturing facilities in each period planning. An extended mixed-integer programming model of dynamic cellular manufacturing system, in which some of the coefficients in objective function and constraints are fuzzy quantities, is solved by a developed fuzzy programming based approach. The objective is to determine the optimal cell configuration in each period with maximum satisfaction degree of the fuzzy objective and constraint. To illustrate the behavior of the proposed model and verify the performance of the developed approach, a number of numerical examples are solved and the associated computational results are reported.

REMOVED: Note on “The nearest trapezoidal fuzzy number to a fuzzy quantity”

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