Total Weighted Completion Time Research Articles

Scheduling problems with a weight-modifying-activity

We study single machine scheduling problems with an additional option of performing a weight-modifying-activity. If such an activity is performed, the cost of subsequent jobs is reduced, as reflected by smaller job-weights. We focus first on minimizing total weighted completion time. A pseudo-polynomial dynamic programming algorithm is introduced for this problem. Several special cases of unit processing time jobs are solved in polynomial time. We also solve in polynomial time (an extension of) the minmax version of the problem, by adapting the well-known Lawler’s Algorithm for minimizing maximum cost on a single machine.

Flow shop scheduling with general position weighted learning effects to minimise total weighted completion time

This article considers the flow shop problem of minimising the total weighted completion time in which the processing times of jobs are variable according to general position weighted learning effects. Two simple heuristics are proposed, and their worst-case error bounds are analysed. In addition, some complex heuristics (including simulated annealing algorithms) and a branch-and-bound algorithm are proposed as solutions to this problem. Finally, computational experiments are performed to examine the effectiveness and efficiency of the proposed algorithms.

Two-agent scheduling of unit processing time jobs to minimize total weighted completion time and total weighted number of tardy jobs

In this paper, we revisit a two-agent scheduling problem on a single machine with jobs of unit processing times. In this problem, we have two agents: agent A and agent B. Each agent has his/her own disjoint job set. The objective of agent A is to minimize the total weighted completion time, and the objective of agent B is to minimize the total weighted number of tardy jobs. The complexity of the decision version of the problem was proposed as open in Oron et al. (European Journal of Operational Research 244:86–99, 2015). We construct a reduction from a variant of Even-Odd Partition to show that this problem is NP-complete. Finally, we also propose a pseudo-polynomial algorithm and a dual FPTAS for the problem.

Single machine batch scheduling with two non-disjoint agents and splitable jobs

We investigate the scheduling problem on a single bounded parallel-batch machine where jobs belong to two non-disjoint agents (called agent A and agent B) and are of equal length but different size. Each job’s size can be arbitrarily split into two parts and processed in the consecutive batches. It is permitted to process the jobs from different agents in a common batch. We show that it is unary NP-hard for the problem of minimizing the total weighted completion time of the jobs of agent A subject to the maximum cost of the jobs of agent B being upper bounded by a given threshold. For the case of the jobs of agent A having identical weights, we study the version of Pareto problem, and give a polynomial-time algorithm to generate all Pareto optimal points and a Pareto optimal schedule corresponding to each Pareto optimal point.

Scheduling proportionate flow shops with preventive machine maintenance

We examine machine scheduling problems that have been inspired by the production environment of semiconductor manufacturing. In semiconductor manufacturing a job processing machine is often enforced to go through a preventive maintenance operation with an objective of minimizing the wafer quality risk. In this research such periodic machine maintenance is considered for single machine and flow shop scheduling models. For the flow shop model we focus on an important special case with proportionate processing times. We first present that the maximum lateness and the total completion time in a single machine can be minimized by a monotone scheduling rule. For the maximum lateness it is shown that the scheduling rule of a single machine special case is also optimal for a proportionate flow shop. However, for the total completion time we observe that the scheduling algorithm of a single machine is not necessarily optimal for its extension to a proportionate flow shop. We prove that the total completion time in a proportionate flow shop is solvable by an O(n2) algorithm, and even the total weighted completion time is minimized with the same computational complexity. Lastly, we discuss how these results are further generalized when setup operations are additionally considered.

An approximation scheme for rejection-allowed single-machine rescheduling

We study a novel rescheduling problem in which a set of jobs has been assigned an original schedule to minimize the total weighted completion time on a single machine, with the assumption but not with 100% certainty that all of them will be available when the planned processing begins. The need for rescheduling arises due to some jobs could not arrive in time; the decision-maker has to adjust the original schedule to account for the delayed jobs without causing excessive time disruption to the original schedule and to minimize their operational cost. While the decision-maker can choose to reject any of the delayed or non-delayed jobs, the total rejection cost and the tardiness of each accepted job in the adjusted schedule are strictly upper bounded by given thresholds, respectively. The total operational cost includes three components: the total weighted completion time of the accepted jobs, the total rejection cost of the rejected jobs, and the penalty on the maximum tardiness for the accepted jobs. We study this novel rescheduling problem from approximation algorithm perspective, as it generalizes several classic NP-hard scheduling problems; we design a pseudo-polynomial time dynamic programming exact algorithm and, when the total rejection cost is unbounded, we develop the exact algorithm into a fully polynomial time approximation scheme.

Scheduling with partial rejection

We study a single machine scheduling problem with partial rejection. Each job is with an integer processing time. Partial rejection occurs when a job is only partly processed with penalty for the rejected part. We focus on integer size rejection. The objective is to minimize the total weighted completion time of processed jobs plus the total rejection cost. We develop a polynomial time optimal algorithm to solve the problem. We also present an easy-to-implement pseudopolynomial time optimal algorithm.

Scheduling jobs with release dates on identical parallel machines by minimizing the total weighted completion time

This paper addresses the problem of scheduling a set of jobs that are released over the time on a set of identical parallel machines, aiming at the minimization of the total weighted completion time. This problem, referred to as P|rj|∑wjCj, is of great importance in practice, because it models a variety of real-life applications. Despite its importance, the P|rj|∑wjCj has not received much attention in the recent literature. In this work, we fill this gap by proposing mixed integer linear programs and a tailored branch-and-price algorithm. Our branch-and-price relies on the decomposition of an arc-flow formulation and on the use of efficient exact and heuristic methods for solving the pricing subproblem. Computational experiments carried out on a set of randomly generated instances prove that the proposed methods can solve to the proven optimality instances with up to 200 jobs and 10 machines, and provide very low gaps for larger instances.

Job Shop and Flexible Job Shop Scheduling Problems: Critical Analysis of Literature Review and their Basic Technology

The job shop scheduling problems (JSSPs) is a significant decision point in a production scheduling problem for those who are concerned with the fields of industry, economics in addition to management. This kind of problem is a class of combinational optimization problem which is known as the NP-hard problem. JSSPs deal amid a set of machines and a set of jobs with a variety of predetermined routes through the machines, where the objective is to assemble a schedule of jobs that diminishes the certain criterion such as make span, total weighted completion time, maximum lateness, and total weighted tardiness. Over the precedent several decades, curiosity in meta-heuristic approaches, advancement of heuristic approaches, to address JSSPs has augmented due to the knack of these approaches to breed feasible resolutions which are superior to those breezed from heuristics algorithms independently. This article presents the classification, constraints and objective functions requisite on JSSPs that are accessible in the literature review.

Parallel-batch scheduling with deterioration and rejection on a single machine

The single machine parallel-batch scheduling with deteriorating jobs and rejection is considered in this paper. A job is either rejected, in which a rejection penalty should be paid, or accepted and processed on the machine. Each job’s processing time is an increasing linear function of its starting time. The machine can process any number of jobs simultaneously as a batch. The processing time of a batch is equal to the largest processing time of the jobs in the batch. The objectives are to minimize the makespan and the total weighted completion time, respectively, under the condition that the total rejection penalty cannot exceed a given upper bound Q. We show that both problems are NP-complete and present dynamic programming algorithms and fully polynomial time approximation schemes (FPTASs) for the considered problems.

A Competitive Online Algorithm for Minimizing Total Weighted Completion Time on Uniform Machines

We consider the classic online scheduling problem on m uniform machines in the online setting where jobs arrive over time. Preemption is not allowed. The objective is to minimize total weighted completion time. An online algorithm based on the directly waiting strategy is proposed. Its competitive performance is proved to be max2smax1−1/2∑si,2smax/1+smax2.5−1/2m by the idea of instance reduction, where sm is the fastest machine speed after being normalized by the slowest machine speed.

Parameterized Multi-Scenario Single-Machine Scheduling Problems

We study a class of multi-scenario single-machine scheduling problems. In this class of problems, we are given a set of scenarios with each one having a different realization of job characteristics. We consider these multi-scenario problems where the scheduling criterion can be any one of the following three: The total weighted completion time, the weighted number of tardy jobs, and the weighted number of jobs completed exactly at their due-date. As all the resulting problems are NP-hard, our analysis focuses on whether any one of the problems becomes tractable when some specific natural parameters are of limited size. The analysis includes the following parameters: The number of jobs with scenario-dependent processing times, the number of jobs with scenario-dependent weights, and the number of different due-dates.

Single-machine lot scheduling with variable lot processing times

This research addresses single-machine lot scheduling problems, with a focus on minimizing the total weighted completion time. The following properties for the lots are assumed: the lots have the same capacity, each lot may contain several orders of different sizes, the size of each order is less than or equal to the identical capacity, and orders must be processed at most in two consecutive lots. To date, lot scheduling researchers have assumed that all lots have identical processing times. In this study, variable lot processing times are considered; specifically, the setting where the processing time of the lots is position dependent, reflecting general position-dependent processing time properties, e.g. learning/ageing effects. Three special cases are examined: minimizing the total completion time, minimizing the makespan, and minimizing the linear combination of the makespan and the total completion time. All problems are shown to be solved in polynomial time.

Randomized selection algorithm for online stochastic unrelated machines scheduling

We consider an online stochastic unrelated machines scheduling problem. Specifically, a set of jobs arriving online over time must be randomly scheduled on the unrelated machines, which implies that the information of each job, including the release date and the weight, is not known until it is released. Furthermore, the actual processing time of each job is disclosed upon completion of this job. In addition, we focus on unrelated machines, which means that each job has a processing speed on every machine. Our goal is to minimize the expected total weighted completion time of all jobs. In this paper, we present a randomized selection algorithm for this problem and prove that the competitive ratio is a constant. Moreover, we show that it is asymptotic optimal for the online stochastic uniform machines scheduling problem when some parameters are bounded. Moreover, our proof does not require any probabilistic assumption on the job parameters.

Group Scheduling Problems with Time-Dependent and Position-Dependent DeJong’s Learning Effect

In this paper, we introduce a group scheduling model with time-dependent and position-dependent DeJong’s learning effect. The objectives of scheduling problems are to minimize makespan, the total completion time, and the total weighted completion time, respectively. We show that the problems remain solvable in polynomial time under the proposed model.

Scheduling customer orders on unrelated parallel machines to minimise total weighted completion time

This paper addresses the scheduling problem for customer orders on a set of unrelated parallel machines. Each order consists of multiple product types with various workloads that can be assigned to and processed by the machines. The objective is to minimise the total weighted completion time of all customer orders. Several optimality properties are developed, and an easily computable lower bound is established. Besides, for two important special cases, the corresponding optimal schedules are further provided. Inspired by these results, three heuristic algorithms are proposed, and their worst case performances are proved to be bounded. The effectiveness of the lower bound and the proposed algorithms are demonstrated through numerical experiments. This study brings new perspectives to the management of differentiated customers in complicated production environment.

Ant colony optimization algorithm for total weighted completion time minimization on non-identical batch machines

This paper aims at the problem of scheduling a set of jobs with arbitrary sizes on parallel batch processing machines with arbitrary capacities. The optimization objective is to minimize the total weighted completion time of jobs. After describing the studied problem, we analyze its complexity and present a lower bound of the problem. A heuristic is provided to solve the problem firstly. Then, with the proposed first job selection strategy based on the weights of jobs, two algorithms based on the ant system and the max-min ant system, respectively, are designed to address the problem. Through extensive experiments, the performance of the proposed algorithms is compared with several state-of-the-art algorithms. The comparative results verify effectiveness and efficiency of the proposed algorithms.

Randomized mechanism design for decentralized network scheduling

ABSTRACT In the network scheduling, jobs (tasks) must be scheduled on uniform machines (processors) connected by a complete graph so as to minimize the total weighted completion time. This setting can be applied in distributed multi-processor computing environments and also in operations research. In this paper, we study the design of randomized decentralized mechanism in the setting where a set of non-preemptive jobs select randomly a machine from a set of uniform machines to be processed on, and each machine can process at most one job at a time. We introduce a new concept of myopic Bayes–Nash incentive compatibility which weakens the classical Bayes–Nash incentive compatibility and derive a randomized decentralized mechanism under the assumption that each job is a rational and selfish agent. We show that our mechanism can induce jobs to report truthfully their private information referred to myopic Bayes–Nash implementability by using a graph theoretic interpretation of the incentive compatibility constraints. Furthermore, we prove that the performance of this mechanism is asymptotically optimal.

Scheduling to Minimize Total Weighted Completion Time via Time-Indexed Linear Programming Relaxations

We study approximation algorithms for scheduling problems with the objective of minimizing total weighted completion time, under identical and related machine models with job precedence constraints. We give algorithms that improve upon many previous 15 to 20-year-old state-of-art results. A major theme in these results is the use of time-indexed linear programming relaxations. These are natural relaxations for their respective problems, but surprisingly are not studied in the literature. We also consider the scheduling problem of minimizing total weighted completion time on unrelated machines. The recent breakthrough result of [Bansal-Srinivasan-Svensson, STOC 2016] gave a $(1.5-c)$-approximation for the problem, based on some lift-and-project SDP relaxation. Our main result is that a $(1.5 - c)$-approximation can also be achieved using a natural and considerably simpler time-indexed LP relaxation for the problem. We hope this relaxation can provide new insights into the problem.

A Heuristic’s Job Order Gain in Pyramidal Preemptive Job Scheduling Problems for Total Weighted Completion Time Minimization

A possibility of speeding up the job scheduling by a heuristic based on the shortest processing period approach is studied in the paper. The scheduling problem is such that the job volume and job priority weight are increasing as the job release date increases. Job preemptions are allowed. Within this model, the input for the heuristic is formed by either ascending or descending job order. Therefore, an estimator of relative difference in duration of finding an approximate schedule by these job orders is designed. It is ascertained that the job order results in different time of computations when scheduling at least a few hundred jobs. The ascending-order solving becomes on average by 1 % to 2.5 % faster when job volumes increase steeply. As the steepness of job volumes decreases, this gain vanishes and, eventually, the descending-order solving becomes on average faster by up to 4 %. The gain trends of both job orders slowly increase as the number of jobs increases.