Scheduling with multiple servers

In this paper, we consider the problem of scheduling a set of jobs on a set of identical parallel machines. Before the processing of a job can start, a setup is required which has to be performed by a given set of servers. We consider the complexity of such problems for the minimization of the makespan. For the problem with equal processing times and equal setup times we give a polynomial algorithm. For the problem with unit setup times, m machines and m − 1 servers, we give a pseudopolynomial algorithm. However, the problem with fixed number of machines and servers in the case of minimizing maximum lateness is proven to be unary NP-hard. In addition, recent algorithms for some parallel machine scheduling problems with constant precessing times are generalized to the corresponding server problems for the case of constant setup times. Moreover, we perform a worst case analysis of two list scheduling algorithms for makespan minimization.     

Pure cycles in flexible robotic cells

In this study, an m-machine flexible robotic manufacturing cell consisting of CNC machines is considered. The flexibility of the machines leads to a new class of robot move cycles called the pure cycles. We first model the problem of determining the best pure cycle in an m-machine cell as a special travelling salesman problem in which the distance matrix consists of decision variables as well as parameters. We focus on two specific cycles among the huge class of pure cycles. We prove that, in most of the regions, either one of these two cycles is optimal. For the remaining regions we derive worst case performances of these cycles. We also prove that the set of pure cycles dominates the flowshop-type robot move cycles considered in the literature. As a design problem, we consider the number of machines in a cell as a decision variable. We determine the optimal number of machines that minimizes the cycle time for given cell parameters such as the processing times, robot travel times and the loading/unloading times of the machines.     

Robotic Cells with Parallel Machines: Throughput Maximization in Constant Travel-Time Cells

We present a general analysis of the problem of sequencing operations in bufferless robotic cell flow shops with parallel machines. Our focus will be cells that produce identical parts. The objective is to find a cyclic sequence of robot moves that maximizes the steady state throughput. Parallel machines are used in the industry to increase throughput, most typically at bottleneck processes having larger processing times.Efficient use of parallel machines requires that several parts be processed in one cycle of robot movements. We analyze such cycles for constant travel-time robotic cells. The number of cycles that produce several parts is very large, so we focus on a subclass called blocked cycles. In this class, we find a dominating subclass called LCM Cycles.The results and the analysis in this paper offer practitioners (i) guidelines to determine whether parallel machines will be cost-effective for a given implementation, (ii) a simple formula for determining how many copies of each machine are required to meet a particular throughput rate, and (iii) an optimal sequence of robot moves for a cell with parallel machines under a certain common condition on the processing times.     

The complexity of mean flow time scheduling problems with release times

We study the problem of preemptive scheduling of n} jobs with given release times on m identical parallel machines. The objective is to minimize the average flow time. In this paper, show that when all jobs have equal processing times then the problem can be solved in polynomial time using linear programming. Our algorithm can also be applied to the open-shop problem with release times and unit processing times. For the general case (when processing times are arbitrary), we show that the problem is unary NP-hard. P. Baptiste and C. Dürr: Supported by the NSF/CNRS grant 17171 and ANR/Alpage. P. Brucker: Supported by INTAS Project 00-217 and by DAAD PROCOPE Project D/0427360. M. Chrobak: Supported by NSF grants CCR-0208856 and INT-0340752. S. A. Kravchenko: Supported by the Alexander von Humboldt Foundation.     

Scheduling independent jobs on uniform parallel machines to minimize tardiness criteria

The problem of scheduling N jobs on M uniform parallel machines is studied. The objective is to minimize the mean tardiness or the weighted sum of tardiness with weights based on jobs, on periods or both. For the mean tardiness criteria in the preemptive case, this problem is NP-hard but good solutions can be calculated with a transportation problem algorithm. In the nonpreemptive case the problem is therefore NP-hard, except for the cases with equal job processing times or with job due dates equal to job processing times. No dominant heuristic is known in the general nonpreemptive case. The author has developed a heuristic to solve the nonpreemptive scheduling problem with unrelated job processing times. Initially, the algorithm calculates a basic solution. Next, it considers the interchanges of job subsets to equal processing time sum interchanging resources (i.e. a machine for a given period). This paper models the scheduling problem. It presents the heuristic and its result quality, solving 576 problems for 18 problem sizes. An application of school timetable scheduling illustrates the use of this heuristic.     

Scheduling of coupled tasks with unit processing times

The coupled tasks scheduling problem was originally introduced for modeling complex radar devices. It is still used for controlling such devices and applied in similar applications. This paper considers a problem of coupled tasks scheduling on a single processor, under the assumptions that all processing times are equal to 1, the gap has exact integer length L and the precedence constraints are strict. We prove that the general problem, when L is part of the input and the precedence constraints graph is a general graph, is NP-hard in the strong sense. We also show that the special case when L=2 and the precedence constraints graph is an in-tree or an out-tree, can be solved in O(n) time.     

Scheduling jobs with equal processing times and time windows on identical parallel machines

We present a linear programming approach to the problem of scheduling equal processing time jobs with release dates and deadlines on identical parallel machines. The known algorithm with complexity O(n ³log log n) of B. Simons schedules all the jobs while minimizing both the maximum completion time and the mean flow time. Our approach permits also to minimize the weighted sum of completion times and total tardiness in polynomial time for the problems without deadlines. The complexity status of these problems was open. Contract/grant sponsor: Alexander von Humboldt Foundation.     

Efficient Approximation Schemes for Scheduling Problems with Release Dates and Delivery Times

We consider the problem of scheduling n independent jobs on m identical machines that operate in parallel. Each job must be processed without interruption for a given amount of time on any one of the m machines. In addition, each job has a release date, when it becomes available for processing, and, after completing its processing, requires an additional delivery time. The objective is to minimize the time by which all jobs are delivered. In the notation of Graham et al. (1979), this problem is noted P|r _j|L_max. We develop a polynomial time approximation scheme whose running time depends only linearly on n. This linear complexity bound gives a substantial improvement of the best previously known polynomial bound (Hall and Shmoys, 1989). Finally, we discuss the special case of this problem in which there is a single machine and present an improved approximation scheme.     

Scheduling jobs with equal processing times subject to machine eligibility constraints

We consider the problem of nonpreemptively scheduling a set of n jobs with equal processing times on m parallel machines so as to minimize the makespan. Each job has a prespecified set of machines on which it can be processed, called its eligible set. We consider the most general case of machine eligibility constraints as well as special cases of nested and inclusive eligible sets. Both online and offline models are considered. For offline problems we develop optimal algorithms that run in polynomial time, while for online problems we focus on the development of optimal algorithms of a new and more elaborate structure as well as approximation algorithms with good competitive ratios.     

Scheduling in Reentrant Robotic Cells: Algorithms and Complexity

We study the scheduling of m-machine reentrant robotic cells, in which parts need to reenter machines several times before they are finished. The problem is to find the sequence of 1-unit robot move cycles and the part processing sequence which jointly minimize the cycle time or the makespan. When m = 2, we show that both the cycle time and the makespan minimization problems are polynomially solvable. When m = 3, we examine a special class of reentrant robotic cells with the cycle time objective. We show that in a three-machine loop-reentrant robotic cell, the part sequencing problem under three out of the four possible robot move cycles for producing one unit is strongly -hard. The part sequencing problem under the remaining robot move cycle can be solved easily. Finally, we prove that the general problem, without restriction to any robot move cycle, is also intractable.