The presented study investigated the scheduling regarding jobs on a single machine. Each job will be processed with no interruptions and becomes available for the processing at time 0. The aim is finding a processing order with regard to jobs, minimizing total completion time , total late work , and maximal tardiness which is an NP-hard problem. In the theoretical part of the present work, the mathematical formula for the examined problem will be presented, and a sub-problem of the original problem of minimizing the multi-objective functions is introduced. Also, then the importance regarding the dominance rule (DR) that could be applied to the problem to improve good solutions will be shown. While in the practical part, two exact methods are important; a Branch and Bound algorithm (BAB) and a complete enumeration (CEM) method are applied to solve the three proposed MSP criteria by finding a set of efficient solutions. The experimental results showed that CEM can solve problems for up to jobs. Two approaches of the BAB method were applied: the first approach was BAB without dominance rule (DR), and the BAB method used dominance rules to reduce the number of sequences that need to be considered. Also, this method can solve problems for up to , and the second approach BAB with dominance rule (DR), can solve problems for up to jobs in a reasonable time to find efficient solutions to this problem. In addition, to find good approximate solutions, two heuristic methods for solving the problem are proposed, the first heuristic method can solve up to jobs, while the second heuristic method can solve up to jobs. Practical experiments prove the good performance regarding the two suggested approaches for the original problem. While for a sub-problem the experimental results showed that CEM can solve problems for up to jobs, the BAB without dominance rule (DR) can solve problems for up to , and the second approach BAB with dominance rule (DR), can solve problems for up to jobs in a reasonable time to find efficient solutions to this problem. Finally, the heuristic method can solve up to jobs. Arithmetic results are calculated by coding (programming) algorithms using (MATLAB 2019a)
Abstract
The methods of the Principal Components and Partial Least Squares can be regard very important methods in the regression analysis, whe
... Show MoreThis paper considers approximate solution of the hyperbolic one-dimensional wave equation with nonlocal mixed boundary conditions by improved methods based on the assumption that the solution is a double power series based on orthogonal polynomials, such as Bernstein, Legendre, and Chebyshev. The solution is ultimately compared with the original method that is based on standard polynomials by calculating the absolute error to verify the validity and accuracy of the performance.
Phase change materials are known to be good in use in latent heat thermal energy storage (LHTES) systems, but one of their drawbacks is the slow melting and solidification processes. So that, in this work, enhancing heat transfer of phase change material is studied experimentally for in charging and discharging processes by the addition of high thermal conductive material such as copper in the form of brushes, which were added in both PCM and air sides. The additions of brushes have been carried out with different void fractions (97%, 94% and 90%) and the effect of four different air velocities was tested. The results indicate that the minimum brush void fraction gave the maximum heat transfer in PCM and reduced the time
... Show MoreObjective: To compare the radiological and functional outcomes of patients of Comminuted intraarticular distal radius fractures treated with or without bone marrow injection along with external fixator ligamentotaxis. Methodology: This prospective comparative study on 41 patients with comminuted intra-articular lower end radius fractures distributed patients randomly into two groups depending on whether an aspirate of bone marrow was injected at the fracture site two weeks after fracture management with external fixation ligamentotaxis. Results: The mean age was 35.8 years, with male predominance. The radiological parameters (radial height, radial inclination, and volar tilt) at the 6th month follow up were lower in the group who did not re
... Show MoreTrue random number generators are essential components for communications to be conconfidentially secured. In this paper a new method is proposed to generate random sequences of numbers based on the difference of the arrival times of photons detected in a coincidence window between two single-photon counting modules