International Journal of Industrial Mathematics
Volume:9 Issue: 2, Spring 2017

The importance as well as the difficulty of the problem of ranking fuzzy numbers is pointed out. Here we consider approaches to the ranking of fuzzy numbers based upon the idea of associating with a fuzzy number a scalar value, its signal/noise ratios, where the signal and the noise are defined as the middle-point and the spread of each $\gamma$-cut of a fuzzy number, respectively. We use the value of a as the weight of the signal/noise ratio of each $\gamma$-cut of a fuzzy number to calculate the ranking index of each fuzzy number. The proposed method can rank any kinds of fuzzy numbers with different kinds of membership functions.ý

In this paper, we introduce a new concept in set-valued mappings which we have called condition $(UHS)$. Then, adding this condition to a new type of contractive set-valued mappings, recently has been introduced by Amini-Harandi [Fixed and coupled fixed points of a new type contractive set-valued mapping in complete metric spaces, Fixed point theory and applications, 215 (2012)], we prove that this mapping have a unique end point. Then, we state and prove a result about existence of coupled fixed point of this type of contractive set-valued mappings defined on $M \times M$, where $M$ is a complete metric space (Recently, Amini- Harandi proved existence of coupled fixed point only for self mappings). Finally, we introduce one another new concept, which we have called condition $(UHS)^*$. Then, adding this condition we state and prove existence of coupled endpoint for such contractive set-valued mappings. Some examples are given to illustrate the results.ý

ýIn this paper we propose a method for computing approximations of solution of fuzzy fractional differential equations using fuzzy variational iteration method. Defining a fuzzy fractional derivative, we verify the utility of the method through two illustrative ýexamples.ý

ýThis paper has two aims. The first is forecasting inflation in Iran using Macroeconomic variables data in Iran (Inflation rate, liquidity, GDP, prices of imported goods and exchange rates) , and the second is comparing the performance of forecasting vector auto regression (VAR), Bayesian Vector-Autoregressive (BVAR), GARCH, time series and neural network models by which Iran's inflation is forecasted. The comparison of performance of forecasting models used to forecast Iran's inflation has been done based on the Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) of the models. Due to the annual values of Inflation, liquidity, GDP, prices of imported goods and exchange rates at free market to estimate different models in this paper and compare root mean square error and Mean Absolute Percentage Error of models by which inflation has been forecasted, neural network model had better performance than others models in forecasting Iran's inflation. Indeed root mean square error and Mean Absolute Percentage Error of neural network model have less value rather than root mean square error and Mean Absolute Percentage Error of other forecasting ýmodels.ý

This paper proposes two methods to predict the efficiency of photochemical removal of AY23 by UV/Ag-TiO$_{2}$ process. In this work the potential of the particle swarm optimization (PSO) and imperialist competitive algorithm (ICA) modeling approaches are presented to forecast the photocatalytic removal of AY23 in the presence of Ag-TiO$_{2}$ nanoparticles prepared under desired conditions. To validate the techniques, a total of 100 data are used that randomly splitted in two parts, 80 samples for the training the models and 20 for testing of the models. Experimental results on datasets show that ICA approach is better than PSO approach. Remarkable analysis results reveal that AY23 initial concentration is the most significant factors that influence on the AY23 removal ýefficiency.ý

One of the considerable discussions for solving the nonlinear equations is to find the optimal iteration, and to use a proper termination criterion which is able to obtain a high accuracy for the numerical solution. In this paper, for a certain class of the family of optimal two-point methods, we propose a new scheme based on the stochastic arithmetic to find the optimal number of iterations in the given iterative solution and obtain the optimal solution with its accuracy. For this purpose, a theorem is proved to illustrate the accuracy of the iterative method and the CESTAC$^1$\footnote{$^1$Controle et Estimation Stochastique des Arrondis de Calculs} method and CADNA$^2$\footnote{$^2$Control of Accuracy and Debugging for Numerical Application} library are applied which allows us to estimate the round-off error effect on any computed result. The classical criterion to terminate the iterative procedure is replaced by a criterion independent of the given accuracy ($\epsilon$) such that the best solution is evaluated numerically, which is able to stop the process as soon as a satisfactory informatical solution is obtained. Some numerical examples are given to validate the results and show the efficiency and importance of using the stochastic arithmetic in place of the floating-point ýarithmetic.ý

ýThe importance of Problem Solving (PS) has been realized for such a long time that in a direct or indirect way affects our daily lives in many waysý. ýAssessment cases appear frequently for PS skills which involve a degree of uncertainty and (or) ambiguityý. ýFuzzy logicý, ýdue to its nature of characterizing such cases with multiple valuesý, ýoffers rich resources for dealing with themý. ýOn the other handý, ýFuzzy Numbers (FNs) play a fundamental role in fuzzy mathematicsý, ýanalogous to the role played by the ordinary numbers in classical mathematicsý. ýIn the present paper we utilize the two simplest forms of themý, ýi.eý. ýthe Triangular and Trapezoidal FNsý, ýtogether with the Centre of Gravity (COG) defuzzification technique as assessment tools for PS skillsý. ýOur results are illustrated by three examples in which the assessment outcomes of FNs are validated through their comparison with the corresponding outcomes of assessment methods of the bi-valued and fuzzy logic already tested in author's earlier worksý.

ýýIn this paper, Solvability nonlinear Volterra integral equations with general vanishing delays is stated. So far sinc methods for approximating the solutions of Volterra integral equations have received considerable attention mainly due to their high accuracy. These approximations converge rapidly to the exact solutions as number sinc points increases. Here the numerical solution of nonlinear delay Volterra integral equations is considered by two methods. The methods are developed by means of the sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. The existence and uniqueness of numerical solutions for these equations are provided. Also an error analysis for the methods is given. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Numerical results are included to confirm the efficiency and accuracy of the ýmethods.ý