One-machine Sequencing Problem Research Articles

Exact resolution of the one-machine sequencing problem with no machine idle time

This paper investigates the one-machine sequencing problem in a workshop where the machine has to satisfy the no-idle constraint, that is, the machine must process jobs without interruption. The objective is to minimize the makespan. Each job has a release date for which it is available for processing on the machine and a delivery time during which it must remain in the system after being processed by the machine. This problem has been studied without adding the no-idle constraint. It is solved in polynomial time, when the preemption of jobs is allowed, applying Jackson’s rule. But, when the preemption of jobs is not allowed, it becomes strongly NP-hard. Although, it can be solved in a very short time using Carlier’s branch and bound algorithm. Below, we propose to adapt Carlier’s branch and bound method in order to calculate an optimal nonpreemptive schedule for the problem when adding the no-idle constraint.

On the representation of the one machine sequencing problem in the shifting bottleneck heuristic

The Shifting Bottleneck heuristic decomposes the Job Shop problem into a series of One Machine Sequencing Problems (OMSPs) with release and due dates, precedence constraints and the minimization of maximum lateness objective. It is well known that delayed precedence constraints may exist between two operations to be performed on the same machine. We identify a new type of precedence relationship that may exist in an OMSP between the predecessor of an operation and the successor of another. The premise that an OMSP captures the sequencing relationships on other machines in the release and due date information is not valid when such precedence relationships exist. A modification of the OMSP representation is proposed based on a generalized lateness objective defined on a due window. The implications of such a representation for the OMSP solution procedure have been explored.

Applying machine based decomposition in 2-machine flow shops

The shifting bottleneck (SB) heuristic is among the most successful approximation methods for solving the job shop problem. It is essentially a machine based decomposition procedure where a series of one machine sequencing problems (OMSPs) are solved. However, such a procedure has been reported to be highly ineffective for the flow shop problems. In particular, we show that for the 2-machine flow shop problem, the SB heuristic will deliver the optimal solution in only a small number of instances. We examine the reason behind the failure of the machine based decomposition method for the flow shop. An optimal machine based decomposition procedure is formulated for the 2-machine flow shop, the time complexity of which is worse than that of the celebrated Johnson’s rule. The contribution of the present study lies in showing that the same machine based decomposition procedures which are so successful in solving complex job shops can also be suitably modified to optimally solve the simpler flow shops.

Branch-and-bound algorithms for solving hard instances of the one-machine sequencing problem

Despite being strongly NP -hard, the one-machine sequencing problem that minimizes the makespan was thought to be computationally inexpensive to solve using Carlier’s algorithm (CA). CA achieved excellent results on random instances with 50–10,000 jobs. However, a thorough computational study reveals that hard instances do exist for CA. In this paper, we present three improved branch-and-bound algorithms based on CA and other existing techniques. We also develop a new lower bound and a problem size reduction technique. The new algorithms are demonstrated to substantially outperform CA in terms of CPU time and robustness.

An application of the neural network energy function to machine sequencing

We apply a neural network approach for solving a one-machine sequencing problem to minimize either single- or multi-objectives, namely the total tardiness, total flowtime, maximimum tardiness, maximum flowtime, and number of tardy jobs. We formulate correspondingly nonlinear integer models, for each of which we derive a quadratic energy function, a neural network, and a system of differential equations. Simulation results based on solving the nonlinear differential equations demonstrate that our approach can effectively solve the sequencing problems to optimality in most cases and near optimality in a few cases. The neural network approach can also be implemented on a parallel computing network, resulting in significant runtime savings over the optimization approach.

A procedure for the one-machine sequencing problem with dependent jobs

The problem of minimizing the makespan on one machine with possible dependencies between jobs is considered. We enlarge the notion of precedence constraint by considering a minimal duration between the start times of each pair of dependent jobs. We propose an exact algorithm where this duration is taken into account. This algorithm is based on Carlier's algorithm [2], closely related to that of McMahon and Florian [10], for the One-Machine Sequencing (OMS) problem with independent jobs. Computational results are reported showing the efficiency of our procedure.

Heuristics for scheduling a single machine subject to unequal job release times

Solving realistic scheduling problems in a reasonable amount of computer time almost inevitably requires the use of heuristic methods. Such a problem is the one-machine sequencing problem where the jobs are subject to unequal release times. Here a recently published heuristic is generalised and these generalisations are shown to provide very good solutions to some large problems in a modest amount of computer time. The heuristic is further extended by applying the method of tabu search, which is used for investigating the quality of the solutions obtained by the original heuristics.

The one-machine sequencing problem with dependent jobs

The problem of minimizing the makespan on one machine with possible dependencies between jobs is considered. We enlarge the notion of precedence constraint by considering a minimal duration between the start times of each pair of dependent jobs. We propose an exact algorithm where this duration is taken into account. This algorithm is based on Carlier's algorithm [Carlier 82] for the One-Machine Sequencing (OMS) problem with independent jobs.

Optimal single-machine sequencing and assignment of common due-dates

This paper considers an n-job one-machine sequencing problem with common due-dates. The objective is to determine the optimal common due-date value and the optimal job sequence that jointly minimize a cost function which is dependent on the individual job earliness and tardiness values. Using Kuhn-Tucker's optimality conditions for constrained convex programming problems, we show that for a given job sequence, the optimal due-date is a simple function of the number of jobs. This result allows separation of the due-date assignment problem from the job sequencing problem. A well-known theorem in algebra can be applied to solve the latter problem, which in turn yields the optimal solution to the overall problem.

A note on minimizing maximum lateness in a one-machine sequencing problem with release dates

This paper deals with the problem of minimizing maximum lateness on one machine with release dates. For this problem Carlier [1] suggests that if a schedule obtained by Schrage's algorithm is optimal, then there exists a set of jobs J such that the makespan of the Schrage schedule is equal to min i ε J a i + Σ iεJ d i + min; iϵJ q i where a i , d i and q i is the release date, processing time and the tail of job i, respectively. In this paper a counterexample is given for this assertion. It is also shown that the assertion is true if the processing times of all the jobs are equal to one.

The one-machine sequencing problem

Let M 1 and M 3 be non-bottleneck machines and M 2 a bottleneck machine processing only one job at a time. Suppose that n jobs have to be processed on M 1 , M 2 and M 1 (in that order) and job i has to spend a time a , on M 1 , d 1 on M 2 and q 1 on M 3 : we want to minimize the makespan. This problem is important since its resolution provides a bound on the makespan of complicated systems such as job shops. It is NP-hard in the strong sense. However, efficient branch and bound methods exist and we describe one of them. Our bound for the tree-search is very close to the bound used by Florian et al., but the principle of branching is quite different. At every node, we construct by an O( n log n ) algorithm a Schrage schedule; then we define a critical job c , a critical set J and consider two subsets of schedules; the schedules when c procedes every job of J and the schedules when c follows every job of J . We give the results of this method and prove that the difference between the optimum and the Schrage schedule is less than d 1 .

Dynamic Programming Solution of Sequencing Problems with Precedence Constraints

Consider a set of tasks that are partially ordered by precedence constraints. A subset of tasks is called feasible if, for every task in the subset, all predecessors are also in the subset. The major results are (1) a method for enumerating all feasible subsets and (2) a method for assigning to each feasible subset an easily computed label that can be used as a physical address for storing information about the subset. These two results permit a very compact computer implementation of a dynamic programming algorithm for solving one-machine sequencing problems with precedence constraints. This algorithm appears to be much more efficient than previous ones for certain one-machine sequencing problems.

The One-Machine Sequencing Problem with Early Starts and Due Dates

Abstract An optimal procedure is developed for the sequencing of n-jobs on one machine to minimize the maximum job lateness when the n-jobs have unequal early start times and due dates. The decision rule derived for the procedure primarily sequences the jobs in order of their nondecreasing early starts plus due dates. A branch-and-bound approach is employed by the procedure which moves to opti-mality in a finite number of steps.