Three-machine shop scheduling with partially ordered processing routes

This paper considers the problem of sequencing n jobs in a three-machine shop with the objective of minimising the maximum completion time. The shop consists of three machines, M1,M2 and M_{3}. A job is first processed on M1 and then is assigned either the route (M2,M_{3}) or the route (M_{3},M2). Thus, for our model the processing route is given by a partial order of machines, as opposed to the linear order of machines for a job shop, or to an arbitrary sequence of machines for an open shop. The main result is on O(nlog n) time heuristic, which generates a schedule with the makespan that is at most 5/3 times the optimum value.

Image denoise by fourth-order PDE based on the changes of laplacian

Fourth-order partial differential equation (PDE) proposed by You and Kaveh (You-Kaveh fourth-order PDE), which replaces the gradient operator in classical second-order nonlinear diffusion methods with a Laplacian operator, is able to avoid blocky effects often caused by second-order nonlinear PDEs. However, the equation brought forward by You and Kaveh tends to leave the processed images with isolated black and white speckles. Although You and Kaveh use median filters to filter these speckles, median filters can blur the processed images to some extent, which weakens the result of You-Kaveh fourth-order PDE. In this paper, the reason why You-Kaveh fourth-order PDE can leave the processed images with isolated black and white speckles is analyzed, and a new fourth-order PDE based on the changes of Laplacian (LC fourth-order PDE) is proposed and tested. The new fourth-order PDE preserves the advantage of You-Kaveh fourth-order PDE and avoids leaving isolated black and white speckles. Moreover, the new fourth-order PDE keeps the boundary from being blurred and preserves the nuance in the processed images, so, the processed images look very natural.