Preemptive Open Shop Research Articles

The stage shop scheduling problem: lower bound and metaheuristic

Remarkable efforts are made to develop the job shop scheduling problem up to now. As a novel generalization, the stage shop can be defined as an environment, in which each job is composed of some stages and each stage may include one operation or more. A stage can be defined a subset of operations of a job, such that these operations can be done in any arbitrary relative order while the stages should be processed in a predetermined order. In other words, the operations of a stage cannot be initiated until all operations of the prior stage are completed. In this paper, an innovative lower bound based on solving the preemptive open shop (using a linear programming model in polynomial time) is devised for the makespan in a stage shop problem. In addition, three metaheuristics, including firefly, harmony search and water wave optimization algorithms are applied to the problem. The results of the algorithms are compared with each other, the proposed lower bound, and a commercial solver.

A fast heuristic to minimize number of tardy jobs in preemptive open shops

This article considers the problem of scheduling preemptive open shops to minimize the number of tardy jobs. A mixed integer program is presented for solving small-sized problems. An efficient heuristic is constructed for solving large-sized problems. This heuristic can be viewed as a generalization of the well-known Moore’s algorithm for the single machine non-preemptive scheduling problem to the preemptive open shop scheduling problem. A lower bound based on assignment problem is proposed for performance evaluation of the heuristic for large-sized problems. Computational results for randomly generated problems are reported. For small-sized problems, the solution obtained by the heuristic has an average deviation of about 4.02% from the optimal value. The lower bound has an average deviation of about 21.03% from the optimal value. For large-sized problems, the solution obtained by the heuristic has an average deviation of less than 54% from the lower bound.

An improved branch-and-bound algorithm for the preemptive open shop total completion time scheduling problem

This article examines the problem of scheduling preemptive open shops to minimize the total completion time. The problem is known to be NP-hard. An efficient constructive heuristic is developed for solving large sized problems. A new lower bound scheme based on the solution of another special type of preemptive open shop problem is presented. Both of the proposed upper and lower bounds are incorporated into a branch-and-bound algorithm to solve medium sized problems. Computational results are reported. The branch-and-bound algorithm can handle problems of up to 14 jobs and 14 machines in size within a reasonable amount of time. The solution obtained by the heuristic has an average deviation of less than 0.14% from the optimal value, while the initial lower bound has an average deviation of less than 2.12% from the optimal value. Moreover, the heuristic finds approved optimal solutions for over 70% of the problem instances completely solved.

Preemptive Open-Shop Scheduling: Network Flow Based Algorithm

Preemptive open-shop scheduling problem was studied, and a network flow based algorithm was presented. Firstly, based on the characteristics of the preemptive open-shop, the scheduling problem was formulated as a mixed-integer programming model with the objective to minimize the make-span. The maximum flow model of the preemptive open-shop was developed to model the machine resource allocation and time constraints. Moreover a new preflow push algorithm for the maximum flow model was put forward. Based on the solution of machine resource allocation problem got by preflow push algorithm, the sequences of the tasks processed by each machine were determined by calculating the matrix of the processing times and decrementing set. Finally, the validity of the developed scheduling algorithm is illustrated by randomly generated example.

Adapting a Main-Stream Internet Switch Architecture for Multihop Real-Time Industrial Networks

As real-time industrial control systems scale up, single real-time local area network (LAN) is no longer sufficient; instead, we need real-time switches to merge many real-time LANs into real-time wide area networks (WANs). However, nowadays commercially-off-the-shelf WAN switches are designed for best-effort Internet traffic rather than real-time traffic. To address this problem, we propose a real-time crossbar switch design that minimally modifies, and even simplifies the de facto industrial standard switch design of iSLIP. Specifically, we change the iSLIP request-grant-accept negotiation to deterministic grant. The switch runs periodically with an M cell-time clock-period. Every input port runs per-flow queueing, and every output port deterministically grants input port per-flow queues according to its own M cell-time clock-period schedule. The schedules are created offline. We prove that the global scheduling can be reduced to a preemptive open shop scheduling problem; as long as every input/output needs to send/fetch no more than M cells per M cell-time clock-period, all outputs schedules do not conflict; and the scheduling algorithm takes O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) time (N is the number of input/output ports). Such design serves real-time periodic/aperiodic traffic in a time-division multiple-access (TDMA) fashion. This simplifies analysis, provides isolation, and results in a close-form end-to-end delay bound. We implemented the proposed real-time switch using Xilinx field programmable gate arrays (FPGAs), and built a distributed control test bed upon the switched networks. Using the test bed, we carried out experiments to compare the implemented real-time switches and iSLIP switches. The results prove the necessity of using real-time switches for real-time industrial control.

A projective algorithm for preemptive open shop scheduling with two multiprocessor groups

We study a multiprocessor extension of the preemptive open shop scheduling problem, where the set of processors is partitioned into processor groups. We show that the makespan minimization problem is polynomially solvable for two multiprocessor groups even if preemptions are restricted to integral times.

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

In this paper, we consider the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. For this strongly NP-hard problem, we develop and discuss different constructive heuristic algorithms. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is evaluated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimate of mean flow time. We observe that the recommendation of an appropriate constructive algorithm strongly depends on the ratio n/ m.

Simulated annealing and genetic algorithms for minimizing mean flow time in an open shop

This paper considers the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. It continues recent work by Bräsel et al. [H. Bräsel, A. Herms, M. Mörig, T. Tautenhahn, T. Tusch, F. Werner, Heuristic constructive algorithms for open shop scheduling to minmize mean flow time, European J. Oper. Res., in press ( doi.10.1016/j.ejor.2007.02.057)] on constructive algorithms. For this strongly NP-hard problem, we present two iterative algorithms, namely a simulated annealing and a genetic algorithm. For the simulated annealing algorithm, several neighborhoods are suggested and tested together with the control parameters of the algorithm. For the genetic algorithm, new genetic operators are suggested based on the representation of a solution by the rank matrix describing the job and machine orders. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The algorithms are compared relative to each other, and the quality of the results is also estimated partially by a lower bound for the corresponding preemptive open shop problem. For most of the problems, the genetic algorithm is superior when fixing the same number of 30 000 generated solutions for each algorithm. However, in contrast to makespan minimization problems, where the focus is on problems with an equal number of jobs and machines, it turns out that problems with a larger number of jobs than machines are the hardest problems.

Preemptive open shop scheduling with multiprocessors: polynomial cases and applications

This paper addresses a multiprocessor generalization of the preemptive open-shop scheduling problem. The set of processors is partitioned into two groups and the operations of the jobs may require either single processors in either group or simultaneously all processors from the same group. We consider two variants depending on whether preemptions are allowed at any fractional time points or only at integer time points. We reduce the former problem to solving a linear program in strongly polynomial time, while a restricted version of the second problem is solved by rounding techniques. Applications to course scheduling and hypergraph edge coloring are also discussed.

A COMPARISON OF HEURISTICS FOR MEAN FLOW TIME OPEN SHOP SCHEDULING

In this paper, the problem of scheduling n jobs on m machines in an open shop environment is considered so that mean flow time becomes minimal. Since this problem is strongly NP-hard, different constructive and iterative heuristic algorithms are developed and discussed. Computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is estimated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimation of mean flow time.

An efficient tabu search approach for the two-machine preemptive open shop scheduling problem

This article considers the problem of scheduling two-machine preemptive open shops to minimize total tardiness. The problem is known to be NP-hard. An optimal timing algorithm is presented to determine the completion time of each job in a given job completion sequence. Then a tabu search (TS) approach is adopted together with the optimal timing algorithm to generate job completion sequences and final schedules. An efficient heuristic is developed to obtain an initial solution for the TS approach. Diversification and intensification strategies are suggested. Finally, computational experiments are conducted to demonstrate the performance of the proposed approach. The results show that the proposed TS approach finds extremely high-quality solutions within a reasonable amount of time. Scope and purpose Shop scheduling problems, such as flow, job and open shop problems, are widely used in the modeling of industrial production processes and are receiving an increasing amount of attention from researchers. In an open shop scheduling problem, the order of processing the operations of a job on different machines is immaterial. Examples of open shop scheduling include teacher-class assignment, examination scheduling, and testing/repair operation scheduling. The purpose of this paper is to examine the two-machine total tardiness open shop scheduling problem with the assumption that the processing of an operation can be arbitrarily preempted. An approximate solution method based on tabu search is proposed. Computational results show that the proposed solution method performs well in terms of both solution quality and computation time.

Preemptive scheduling with rejection

We consider the problem of preemptively scheduling a set of n jobs on m (identical, uniformly related, or unrelated) parallel machines. The scheduler may reject a subset of the jobs and thereby incur job-dependent penalties for each rejected job, and he must construct a schedule for the remaining jobs so as to optimize the preemptive makespan on the m machines plus the sum of the penalties of the jobs rejected. We provide a complete classification of these scheduling problems with respect to complexity and approximability. Our main results are on the variant with an arbitrary number of unrelated machines. This variant is APX-hard, and we design a 1.58-approximation algorithm for it. All other considered variants are weakly -hard, and we provide fully polynomial time approximation schemes for them. Finally, we argue that our results for unrelated machines can be carried over to the corresponding preemptive open shop scheduling problem with rejection.

A (2+ ε)-approximation algorithm for the generalized preemptive open shop problem with minsum objective

In this paper we consider a generalized version of the classical preemptive open shop problem with sum of weighted job completion times objective. The main result is a (2+ ε)-approximation algorithm for this problem. In the last section we also discuss the possibility of improving our algorithm.

Feasible edge colorings of trees with cardinality constraints

A variation of preemptive open shop scheduling corresponds to finding a feasible edge coloring in a bipartite multigraph with some requirements on the size of the different color classes. We show that for trees with fixed maximum degree, one can find in polynomial time an edge k-coloring where for i=1,…, k the number of edges of color i is exactly a given number h i , and each edge e gets its color from a set ϕ( e) of feasible colors, if such a coloring exists. This problem is NP-complete for general bipartite multigraphs. Applications to open shop problems with costs for using colors are described.

An open shop scheduling problem with fuzzy allowable time and fuzzy resource constraint

A model for a preemptive open shop scheduling problem is presented and constraints due to fuzzy resource and allowable time are introduced. This problem is a fuzzy version of the model by Werra et al. and has bi-criteria to be maximized, i.e., minimal satisfaction degree with respect to the processing intervals of jobs and, minimal satisfaction degree of resource amounts used in the processing intervals. We present an efficient algorithm for this problem.

A two-machine preemptive openshop scheduling problem: An elementary proof of NP-completeness

This paper deals with a two-machine preemptive openshop system processing a set of jobs with different ready dates. The problem of minimizing the mean flow time is known to be strongly NP-complete. We reprove this fact in an elementary way.

Restrictions and preassignments in preemptive open shop scheduling

Preemptive open shop scheduling can be viewed as an edge coloring problem in a bipartite multigraph. In some applications, restrictions of colors (in particular preassignments) are made for some edges. We give characterizations of graphs where some special preassignments can be embedded in a minimum coloring (number of colors = maximum degree). The case of restricted colorings of trees is shown to be solvable in polynomial time.

Dual criteria preemptive open-shop problems with minimum makespan

In this article we present an algorithm for the minimum makespan preemptive open shop, which is superior to existing algorithms in both space and time requirements. We define the more complex generalized open shop and flexible open shop and address the minimum makespan problem on these shops. We show how we can use the algorithm for the minimum makespan open shop to achieve load balancing in simple and generalized open shops without increasing the complexity of the algorithm. Load balancing dictates that the number of busy machines in each period is as even as possible. We also consider preventive maintenance issues in the open shop, and makespan retains its minimum value. In particular we consider the scenario where a machine can be maintained during any period that it happens to be idle. Also we consider the case that a maintenance schedule is prespecified. We show that this problem can be solved via a linear programming formulation that can also take into account release times for the jobs and ready times for the machines. Faster algorithms are presented for open shops with three machines or less. © 1995 John Wiley & Sons, Inc.

On the complexity of preemptive openshop scheduling problems

We investigate the complexity status of the preemptive openshop scheduling problem. After reviewing recent studies of the shop with various objective functions, we define the concept of preempting job w.r. to a schedule, and state some general results for the preemptive openshop environment. These results are then used to show by reduction from the 3-Partition, that the n job-2 machine mean flow time problem with ready times and job preemption is unary NP-hard.

Some preemptive open shop scheduling problems with a renewable or a nonrenewable resource

Some preemptive open shop scheduling problems with a renewable or a nonrenewable resource