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The flow shop scheduling problems with fuzzy processing times are investigated in this paper. For some special kinds of fuzzy numbers, the analytic formulas of the fuzzy compltion time can be obtained. For the general bell-shaped fuzzy numbers, we present a computational procedure to obtain the approximated membership function of the fuzzy completion time. We define a defuzzification function to rank the fuzzy numbers. Under this ranking concept among fuzzy numbers, we plan to minimize the fuzzy makespan and total weighted fuzzy completion time. Because the ant colony algorithm has been successfully used to solve the scheduling problems with real-valued processing times, we shall also apply the ant colony algorithm to search for the best schedules when the processing times are assumed as fuzzy numbers. Numerical examples are also provided and solved by using the commercial software MATLAB.

In this paper, we give some refinements of Simpson’s rule in cases when it is not applicable in its classical form i.e., when the target function is not four times differentiable on a given interval. Some sharp two-sided inequalities for an extended form of Simpson’s rule are also proven.

In this paper, a new method named Improved Finite Integration Method (IFIM) is proposed for solving Time Fractional Diffusion Equations (TFDEs). In the IFIM, the Extended Simpson’s Rule (ESR) is employed for numerical quadrature in spatial discretization. Besides, the Piecewise Quadratic Interpolation (PQI) in sense of the Hadamard finite-part integral is utilized for time discretization. Compared with the primary Finite Integration Method (FIM) with Trapezoidal rule which uses the finite difference scheme to address the time discretization, the combination of ESR and PQI in IFIM will lead to a better performance in solving TFDEs. Numerical examples are performed and compared to show the superiority of IFIM. It can also be found that the IFIM is able to obtain a higher accuracy without losing the stability and efficiency.