Total Late Work Research Articles

Minimizing the number of late jobs and total late work with step-learning

We study single-machine scheduling problems with step-learning, where an improvement in processing time is experienced if a job is started at, or after, a job-dependent learning-date. We consider minimizing two functions: the number of late jobs and the total late work, and we show that when at least a common due-date or common learning-date is assumed, the problem is NP-hard in the ordinary sense; however, when both are arbitrary, the problem becomes strongly NP-hard. For each of the problems where at least one of the dates is assumed to be common, we analyze the structure of an optimal job schedule with and without idle time and propose pseudo-polynomial time dynamic programming algorithms. We also show that the problem of minimizing the weighted number of late jobs with step-learning can be solved with a minor change to the algorithms for the unweighted case. In addition to this, we show that when a common due-date is assumed and no idle time is allowed, the problem of minimizing the total late work is equivalent to that of minimizing the makespan. Furthermore, we provide a more efficient algorithm to solve the problem of minimizing makespan under the assumption of a common learning-date than the one in the existing literature. Lastly, we show that our analysis can also be applied to the case of step-deterioration, where instead, the processing times of jobs increase at a given date.

Solving tri-criteria: total completion time, total late work, and maximum earliness by using exact, and heuristic methods on single machine scheduling problem

The presented study investigated the scheduling regarding jobs on a single machine. Each job will be processed with no interruptions and becomes available for the processing at time 0. The aim is finding a processing order with regard to jobs, minimizing total completion time , total late work , and maximal tardiness which is an NP-hard problem. In the theoretical part of the present work, the mathematical formula for the examined problem will be presented, and a sub-problem of the original problem of minimizing the multi-objective functions is introduced. Also, then the importance regarding the dominance rule (DR) that could be applied to the problem to improve good solutions will be shown. While in the practical part, two exact methods are important; a Branch and Bound algorithm (BAB) and a complete enumeration (CEM) method are applied to solve the three proposed MSP criteria by finding a set of efficient solutions. The experimental results showed that CEM can solve problems for up to jobs. Two approaches of the BAB method were applied: the first approach was BAB without dominance rule (DR), and the BAB method used dominance rules to reduce the number of sequences that need to be considered. Also, this method can solve problems for up to , and the second approach BAB with dominance rule (DR), can solve problems for up to jobs in a reasonable time to find efficient solutions to this problem. In addition, to find good approximate solutions, two heuristic methods for solving the problem are proposed, the first heuristic method can solve up to jobs, while the second heuristic method can solve up to jobs. Practical experiments prove the good performance regarding the two suggested approaches for the original problem. While for a sub-problem the experimental results showed that CEM can solve problems for up to jobs, the BAB without dominance rule (DR) can solve problems for up to , and the second approach BAB with dominance rule (DR), can solve problems for up to jobs in a reasonable time to find efficient solutions to this problem. Finally, the heuristic method can solve up to jobs. Arithmetic results are calculated by coding (programming) algorithms using (MATLAB 2019a)

Solving the Multi-Criteria Problem: Total Completion Time, Total Late Work, Total Earliness Time, Maximum Earliness, and Maximum Tardine

Solving the Multi-Criteria Problem: Total Completion Time, Total Late Work, Total Earliness Time, Maximum Earliness, and Maximum Tardine

Two-agent single-machine scheduling with a rate-modifying activity

We study single-machine scheduling problems involving a rate-modifying activity and two competing agents with due-date-related functions. Classical scheduling models assume that job processing times remain constant over time; however, in real-world settings, processing times may change due to factors such as technological upgrades or machine maintenance. We complement this with the notion of multiple independent agents competing over the use of a shared resource, each with their own motives. These considerations allow us to model the upcoming trend of the sharing economy, where resources are shared amongst independent competitors in the market. We aim to model these scenarios by considering a variety of scheduling criteria for each agent, including the makespan, the number of late jobs, and the total late work. To account for the change in processing times, we consider an optional rate-modifying activity that once completed, results in a reduction in subsequent job processing times. We show that problems involving the total late work are binary NP-hard and propose efficient pseudo-polynomial dynamic programming algorithms for solving these problems. We also show that the remaining problems are solvable in polynomial time.

Exact and Near Pareto Optimal Solutions for Total Completion Time and Total Late Work Problem

In this paper, the bi-criteria machine scheduling problems (BMSP) are solved, where the discussed problem is represented by the sum of completion and the sum of late work times simultaneously. In order to solve the suggested BMSP, some metaheurisitc methods are suggested which produce good results. The suggested local search methods are simulated annulling and bees algorithm. The results of the new metaheurisitc methods are compared with the complete enumeration method, which is considered an exact method, then compared results of the heuristics with each other to obtain the most efficient method.

제어가능한 공정시간과 납기 지연 조건을 고려한 선점적 단일 머신 일정계획 문제

In classical scheduling problems, jobs are usually assumed to be non-preemptive with constant processing times. In many practical cases, however, restrictions on preemption can be released by outsourcing and the job processing times are controllable by allocating additional resources to the job operations. In this paper, we consider a single- machine scheduling problem with preemption such that each job processing time can be reduced linearly by using additional resource. The objective is to minimize the sum of total late work and total compression cost. We show that the problem can be solved in polynomial time.

Solving the multi-criteria: Total completion time, total late work, and maximum earliness problem

Solving the multi-criteria: Total completion time, total late work, and maximum earliness problem

Bicriterion Pareto‐scheduling of equal‐length jobs on a single machine related to the total weighted late work

AbstractIn this article, we study bicriterion Pareto‐scheduling on a single machine of equal‐length jobs, where one of the criteria is the total weighted late work. Motivated by two Pareto‐scheduling open problems where one criterion is the total (weighted) late work and the other criterion is the weighted number of tardy jobs, we show that 12 constrained scheduling problems unaddressed in the literature are binary ‐hard, implying that the Pareto‐scheduling versions of these problems are also binary ‐hard. Moreover, we introduce the concept of dummy due dates (DDD) for equal‐length jobs to be scheduled in equal‐length intervals. Intriguingly, we find that a DDD‐based technique outperforms the existing solution methods and improves the known time complexities of the related problems. In addition, we extend our research to the two‐agent scheduling model under the assumption of equal‐length or partially equal‐length jobs by including the total weighted late work as the criterion of one agent. For these problems, our results also improve the known time complexity results.

Revisit the scheduling problem with assignable or generalized due dates to minimize total weighted late work

ABSTRACT We revisit the single-machine scheduling for minimising the total weighted late work with assignable due dates (ADD-scheduling) and generalised due dates (GDD-scheduling). In particular, we consider the following three problems: (i) the GDD-scheduling problem for minimising the total weighted late work, (ii) the ADD-scheduling problem for minimising the total weighted late work, and (iii) the ADD-scheduling problem for minimising the total late work. In the literature, the above three problems are proved to be NP-hard, but their exact complexity (unary NP-hardness or pseudo-polynomial-time solvability) are unknown. In this paper, we address these open problems by showing that the first two problems are unary NP-hard and the third problem admits pseudo-polynomial-time algorithms. For the third problem, we also present a 2-approximation solution and a fully polynomial-time approximation scheme. Computational experiments show that our algorithms and solutions are efficient. When the jobs have identical processing times, we further present more efficient polynomial-time algorithms.

Minimizing tardiness scheduling measures with generalized due-dates and a maintenance activity

We study single machine scheduling problems with the following features: (i) Generalized due-dates are considered, i.e., the j-th due-date is assigned to the j-th completed job; (ii) A fixed maintenance activity during which no production is feasible is assumed; (iii) The objective functions are minimizing various tardiness measures. Specifically, four objective functions are considered: total tardiness, maximum tardiness, the number of tardy jobs and total late work. We introduce pseudo-polynomial solution algorithms for these NP-hard problems. Our numerical tests indicate that instances of medium size of all four problems are solved in very reasonable running times.

Single machine scheduling with assignable due dates to minimize maximum and total late work

A single machine scheduling problem with assignable job due dates to minimize total late work has recently been introduced by Mosheiov, Oron, and Shabtay (2021). The problem was proved NP-hard in the ordinary sense, and no solution algorithm was proposed. In this note, we present two pseudo-polynomial dynamic programming algorithms and an FPTAS for this problem. Besides, we introduce a new single machine scheduling problem to minimize maximum late work of jobs with assignable due dates. We develop an O(nlogn) time algorithm for it, where n is the number of jobs. An optimal solution value of this new problem is a lower bound for the optimal value of the total late work minimization problem, and it is used in the FPTAS.

작업 지연과 통제 가능한 공정시간을 고려한 단일공정 스케줄링 문제

We consider a single-machine scheduling problem such that the processing time of each job is inversely proportional to the power of the amount of resource consumption. The objective is to minimize the sum of the total resource consumption cost and the total late work, or minimize the total late work with a constraint on the total resource consumption cost. Under any objective, we prove the NP-hardness of the case with more than or equal to two different due dates, and the polynomiality of the case with a common due date.

Single-machine scheduling with total late work and job rejection

We study single-machine scheduling problems with rejection, where we can refuse to process jobs in the shop at a certain cost. Solutions to the problems investigated in this study consist of (i) partitioning the set of jobs into a set of accepted and a set of rejected jobs, and (ii) scheduling the set of accepted jobs on the single machine. The quality of a solution is measured by two criteria. The first is the total late work of the set of accepted jobs, and the second is the total rejection cost. We study two problems. The goal in the first is to find a solution minimizing the sum of the total late work and the total rejection cost, while the goal in the second is to find a solution minimizing the total rejection cost, given a bound on the total late work. As both problems are NP -hard, we focus on the design of pseudo-polynomial time algorithms and approximation schemes.

Pareto-scheduling of two competing agents with their own equal processing times

We consider the Pareto-scheduling of two competing agents on a single machine, in which the jobs of each agent have their “own equal processing times” (shortly, OEPT). In the literature, two special versions of the OEPT model, in which the jobs have either unit or equal processing times, have been well studied, where the criteria are given by various regular objective functions without including the late work criteria. However, for equal processing times, the exact complexity of three problems is still unaddressed. Two-agent scheduling related to late work criteria is also a hot topic in recent years. This inspires our research by also including the total (weighted) late work as criteria. We show that, for equal processing times, all the problems are binary NP-hard if the criterion of one agent is the total tardiness or the total late work and the criterion of the other agent is either the total tardiness or the total late work or the weighted number of tardy jobs or the total weighted completion time. As a result, complexity classification for equal processing times is completely addressed. We further show that all the problems under the OEPT model are either polynomially solvable or ordinary NP-hard, which results in a complete complexity classification for the OEPT model.

Approximation Solutions For Multicriteria Scheduling Problems

This paper presents local search algorithms for finding approximation solutions of the multicriteria scheduling problems within the single machine context, where the first problem is the sum of maximum tardiness and maximum late work and the second problem is the sum of total late work and maximum late work. Late work criterion estimates the quality of a schedule based on durations of late parts of jobs. Local search algorithms (descent method (DM), simulated annealing (SA) and genetic algorithm (GA))are implemented. Based on results of computational experiments, conclusions are formulated on the efficiency of the local search algorithms.

Bicriteria scheduling to minimize total late work and maximum tardiness with preemption

We consider single-machine bicriteria scheduling of n jobs with preemption to minimize the total late work and maximum tardiness. Given the earliest due date order of the jobs in advance, we present an O(n)-time algorithm for the hierarchical scheduling problem and an O(nlogn)-time algorithm for the constrained scheduling problem. For the Pareto-scheduling problem, when all the processing times and due dates are integral, we construct the trade-off curve in O(nlognP) time, where P is the total processing time of the jobs.

Neural network for minimizing tricriteria objective function for machine scheduling problem

In this paper, neural networks (NN) are implemented to manipulate a problem of machine scheduling for a single machine to minimize multicriteria objective function: maximum tardiness, maximum late work and total late work simultaneously (Tmax, Vmax, ΣVj). Also the results of applying NN are compared with some exact methods, some heuristic methods and local search methods [7,14]. The used NN is learned by pack propagation algorithm for n = 3: 500 jobs. NN technique was be found to be effective and used to select the best efficient and optimal schedule which minimizes the objective function.

Scheduling with competing agents, total late work and job rejection

We study a single-machine scheduling problem with two competing agents. The objective function of one agent is minimizing total late work, whereas the maximum cost of the jobs of the second agent cannot exceed a given upper bound. We introduce and test a pseudo-polynomial dynamic programming algorithm for this NP-hard problem. Our tests indicate that the algorithm can handle large-size instances (containing hundreds of jobs) fairly quickly. We then extend the setting to that of job rejection, where the scheduler may decide to process only a subset of the jobs. The dynamic programming algorithm is modified accordingly, and our numerical tests verify that medium-size instances (containing less than 100 jobs) are solved in reasonable time.

Two competitive agents to minimize the weighted total late work and the total completion time

This paper studies deterministic constraint optimization problem with two competitive agents in which the following objective functions on a single machine: the total weighted late work and the total completion time. We show that the constraint optimization problem is the binary NP-hard by Knapsack problem reduction. Furthermore, we present a pseudo-polynomial time algorithm by early due date maximum not-late sequence, and an approximation Pareto curve by dynamic programming algorithm and two eliminated states, which time complexity of the two approximation algorithms are O(nA2nBQ∑(pjA+pjB)) and O(n4θ2logUBAlogUBB), where pj,θare processing time of job Jj, a given positive constant, and UBx an upper bound of the objective function of agent x,x∈{A,B}. Finally, we present a simple approximation algorithm by the earliest due date (EDD) rule, which jobs of agent B are assigned an dummy due date.

A note on scheduling a rate modifying activity to minimize total late work

We study a single machine scheduling problem with a rate modifying activity. In most cases, performing such an activity (e.g., a maintenance procedure) improves the system performance, as reflected by shorter processing times of the jobs processed after it. The scheduling measure assumed is total late work: each job is considered as a collection of unit-time jobs, and the late work is defined as the number of tardy units. The problem is NP-hard, and we introduce and test numerically a pseudo-polynomial dynamic programming algorithm. Our results indicate that the algorithm solves large problem instances efficiently.