Resource optimal control in some single-machine scheduling problems (original) (raw)
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Single-machine scheduling with deteriorating jobs
International Journal of Systems Science, 2010
In this article, we study a single-machine scheduling problem in which the processing time of a job is a nonlinear function of its basic processing time and starting time. The objectives are to minimise the makespan, the sum of weighted completion times and the sum of the kth powers of completion times. We show that the makespan minimisation problem can be solved in polynomial time. However, the total completion time and the sum of the kth powers of completion times minimisation problems can be solved in polynomial time in some cases. Besides, some useful properties are also provided for the sum of weighted completion times problem under certain conditions.
IFAC Proceedings Volumes (IFAC-PapersOnline), 2012
We consider the problem of scheduling jobs with release times and due dates on a single machine to minimize the maximal job lateness. This problem is NP-hard, and its version when the job processing times are restricted to p, 2p, 3p, 4p, . . ., for an integer p, is also NPhard. We consider the case when the maximal job processing time is kp, for any constant k, and propose its polynomial-time solution. We easily establish that the version of this problem with unrestricted k is NP-hard. Moreover, it is strongly NP-hard if p has no exponential-time dependence on the maximal job due date. From a practical point of view, this is a realistic assumption.
Computers & Operations Research, 2006
This paper presents a bicriterion analysis of time/cost trade-offs for the single-machine scheduling problem where both job processing times and release dates are controllable by the allocation of a continuously nonrenewable resource. Using the bicriterion approach, we distinguish between our sequencing criterion, namely the makespan, and the cost criterion, the total resource consumed, in order to construct an efficient time/cost frontier. Although the computational complexity of the problem of constructing this frontier remains an open question, we show that the optimal job sequence is independent of the total resource being used; thereby we were able to reduce the problem to a sequencing one. We suggest an exact dynamic programming algorithm for solving small to medium sizes of the problem, while for large-scale problems we present some heuristic algorithms that turned out to be very efficient. Five different special cases that are solvable by using polynomial time algorithms are also presented.
Single machine scheduling with discretely controllable processing times
Operations Research Letters, 1997
In the field of machine scheduling problems with controllable processing times, it is often assumed that the possible processing time of a job can be continuously controlled, i.e. it can be any number in a given interval. In this paper, we introduce a discrete model in which job processing times are discretely controllable, i.e. the possible processing time of a job can only be controlled to be some specified lengths. Under this discrete model, we consider a class of single machine problems with the objective of minimizing the sum of the total processing cost and the cost measured by a standard criterion. We investigate most common criteria, e.g. makespan, maximum tardiness, total completion time, weighted number of tardy jobs, and total earliness-tardiness penalty. The computational complexity of each problem is analyzed, and pseudo-polynomial dynamic programming algorithms are proposed for hard problems. © 1997 Elsevier Science B.V.
Single machine scheduling of unit-time jobs with controllable release dates
Journal of Global Optimization, 2003
The paper presents a bicriterion approach to solve the single-machine scheduling problem in which the job release dates can be compressed while incurring additional costs. The two criteria are the makespan and the compression cost. For the case of equal job processing times, an O(n 4 ) algorithm is developed to construct integer Pareto optimal points. We discuss how the algorithm developed can be modified to construct an ε-approximation of noninteger Pareto optimal points. The complexity status of the problem with total weighted completion time criterion is also established.
International Journal of …, 2009
We extend the classical scheduling problem of minimizing the number of tardy jobs in a single-machine to the case where the job processing times are controllable by allocating a continuous and non-renewable resource to the processing operations. Our aim is to construct an efficient trade-off curve between the number of tardy jobs and total resource consumption using a bicriteria approach. Most of the research on scheduling with controllable resource-dependent processing times is done either to a linear resource consumption function or to a specific type of convex resource consumption function. We, in contrast, analyze the problem for a more general type of convex decreasing resource consumption function, which guarantees a very robust analysis that can be applied to a wide range of problems. We present four different variations of the problem and prove them to be NPNP-hard. We then present a polynomial time algorithm to solve an important special case of the problem and also suggest and compare the performances of three different heuristic algorithms.
Maximization Problems in Single Machine Scheduling
Annals of Operations Research, 2004
Problems of scheduling n jobs on a single machine to maximize regular objective functions are studied. Precedence constraints may be given on the set of jobs and the jobs may have different release times. Schedules of interest are only those for which the jobs cannot be shifted to start earlier without changing job sequence or violating release times or precedence constraints. Solutions to the maximization problems provide an information about how poorly such schedules can perform. The most general problem of maximizing maximum cost is shown to be reducible to n similar problems of scheduling n−1 jobs available at the same time. It is solved in O(mn+n 2) time, where m is the number of arcs in the precedence graph. When all release times are equal to zero, the problem of maximizing the total weighted completion time or the weighted number of late jobs is equivalent to its minimization counterpart with precedence constraints reversed with respect to the original ones. If there are no precedence constraints, the problem of maximizing arbitrary regular function reduces to n similar problems of scheduling n−1 jobs available at the same time.
Mathematics
We study a single-machine scheduling problem to minimize the total completion time of the given set of jobs, which have to be processed without job preemptions. The lower and upper bounds on the job duration is the only information that is available before scheduling. Exact values of the job durations remain unknown until the completion of the jobs. We use the optimality region for the job permutation as an optimality measure of the optimal schedule. We investigate properties of the optimality region and derive O ( n ) -algorithm for calculating a quasi-perimeter of the optimality set (i.e., the sum of lengths of the optimality segments for n given jobs). We develop a fast algorithm for finding a job permutation having the largest quasi-perimeter of the optimality set. The computational results in constructing such permutations show that they are close to the optimal ones, which can be constructed for the factual durations of all given jobs.
Single machine scheduling with a restricted rate-modifying activity
Naval Research Logistics, 2005
In this paper we consider the problem of scheduling a set of jobs on a single machine on which a rate-modifying activity may be performed. The rate-modifying activity is an activity that changes the production rate of the machine. So the processing time of a job is a variable, which depends on whether it is scheduled before or after the rate-modifying activity. We assume that the rate-modifying activity can take place only at certain predetermined time points, which is a constrained case of a similar problem discussed in the literature. The decisions under consideration are whether and when to schedule the rate-modifying activity, and how to sequence the jobs in order to minimize some objectives. We study the problems of minimizing makespan and total completion time. We first analyze the computational complexity of both problems for most of the possible versions. The analysis shows that the problems are NP-hard even for some special cases. Furthermore, for the NP-hard cases of the makespan problem, we present a pseudo-polynomial time optimal algorithm and a fully polynomial time approximation scheme. For the total completion time problem, we provide a pseudo-polynomial time optimal algorithm for the case with agreeable modifying rates.
Single machine scheduling with resource dependent release times and processing times
European Journal of Operational Research, 2005
We consider the single machine scheduling problem with resource dependent release times and processing times, in which both the release times and processing times are strictly linear decreasing functions of the amount of resources consumed. The objective is to minimize the makespan plus the total resource consumption costs. We propose a heuristic algorithm for the general problem by utilizing some derived optimal properties and analyze its performance bound. For some special cases, we propose another heuristic algorithm that achieves a tighter performance bound. 2 1 t J J J 2 1 t J J J is in decreasing order of their processing times. From (10), we have ), , ( * ) 1 ( l J J J