This chapter presents a novel idea for the general case of the Common Due-Date (CDD) scheduling problem. The problem is about scheduling a certain number of jobs on a single or parallel machines where all the jobs possess different processing times but a common due-date. The objective of the problem is to minimize the total penalty incurred due to earliness or tardiness of the job completions. This work presents exact polynomial algorithms for optimizing a given job sequence for single and identical parallel machines with the run-time complexities of O(nlogn) for both cases, where n is the number of jobs. Besides, we show that our approach for the parallel machine case is also suitable for non-identical parallel machines. We prove the optimality for the single machine case and the run-time complexities of both. Henceforth, we extend our approach to one particular dynamic case of the CDD and conclude the chapter with our results for the benchmark instances provided in the OR library.
This chapter is an extension of a research paper presented by the same authors in @cite_11 . We extend our approach for a dynamic case of the problem and for non-identical parallel machines. Useful examples for both the single and parallel machines case are presented.
@cite_11
2020196988 2135879303 1982831910 2049060423
This paper considers the problem of nonpreemptively scheduling n independent jobs on m identical, parallel processors with the object of minimizing the “makespan”, or completion time for the entire set of jobs. Coffman, Garey, and Johnson [SIAM J. Comput., 7 (1978), pp. 1–17] described an algorithm MULTIFIT which has a considerably better worst case performance than the largest processing time first algorithm. In this paper we tighten the bounds obtained in that paper on the worst case behavior of this algorithm by giving an example showing that it may be as bad as 13 11 and proving that it can be no worse than 6 5. This paper deals with the experimental identification of the dynamic parameters of parallel machines. The dynamic parameters are estimated by using the weighted least squares solution of an over determined linear system obtained from the sampling of the dynamic model along a closed loop exciting trajectory. Experimental results are exhibited for the H4 robot, a fully parallel structure providing 3 degrees of freedom (DOF) in translation and 1 DOF in rotation. A comparative study is performed depending on the available measurements, i.e., different sensor locations (motor, end effector). In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. We investigate a very basic problem in dynamic speed scaling where a sequence of jobs, each specified by an arrival time, a deadline and a processing volume, has to be processed so as to minimize energy consumption. Previous work has focused mostly on the setting where a single variable-speed processor is available. In this paper we study multi-processor environments with m parallel variable-speed processors assuming that job migration is allowed, i.e. whenever a job is preempted it may be moved to a different processor. We first study the offline problem and show that optimal schedules can be computed efficiently in polynomial time. In contrast to a previously known strategy, our algorithm does not resort to linear programming. We develop a fully combinatorial algorithm that relies on repeated maximum flow computations. The approach might be useful to solve other problems in dynamic speed scaling. For the online problem, we extend two algorithms Optimal Available and Average Rate proposed by [16] for the single processor setting. We prove that Optimal Available is αα-competitive, as in the single processor case. Here α>1 is the exponent of the power consumption function. While it is straightforward to extend Optimal Available to parallel processing environments, the competitive analysis becomes considerably more involved. For Average Rate we show a competitiveness of (3 )α 2 + 2α.