Arbitrary Release Dates Research Articles

Bicriteria scheduling on an unbounded parallel-batch machine for minimizing makespan and maximum cost

This paper studies the bicriteria problem of scheduling n jobs on an unbounded parallel-batch machine. The goal is to minimize makespan and maximum cost simultaneously. When the jobs have arbitrary processing times and equal release dates, we obtain an O(n3)-time algorithm, improving the previously best known O(n4)-time algorithm. When the jobs have equal processing times and arbitrary release dates with a precedence relation, we also obtain an O(n3)-time algorithm, which is the first polynomial time algorithm for this problem.

An exact algorithm for the preemptive single machine scheduling of equal-length jobs

Many real-world production and service environments can be modeled using the preemptive single machine scheduling problem with equal processing times, arbitrary release dates and weights (priorities) minimizing the total weighted completion time. For this problem we propose a Boolean Linear Programming model, two heuristics based on the model’s relaxation, and an exact Branch and Bound algorithm incorporating the model and heuristics. Our computational study involved more than one million problem instances with up to 350 jobs. Each generated instance has been solved to optimality within 31 min. That improves state-of-the art (at most 20 jobs in 60 min) by more than an order of magnitude. Our benchmark instances and optimal schedules are publicly available at https://data.mendeley.com/datasets/nrkx7467tf/1.

Pareto optimal algorithms for minimizing total (weighted) completion time and maximum cost on a single machine.

This paper studies the Pareto scheduling problem of minimizing total weighted completion time and maximum cost on a single machine. It is known that the problem is strongly NP-hard. Algorithms with running time $ O(n^3) $ are presented for the following cases: arbitrary processing times, equal release dates and equal weights; equal processing times, arbitrary release dates and equal weights; equal processing times, equal release dates and arbitrary weights.

Online heuristic for the preemptive single machine scheduling problem to minimize the total weighted tardiness

In this paper, we refine the ties within decision rules of the Total Weighted Tardiness (TWT) heuristic from Jaramillo and Erkoc (2017) applied by the authors to solving the preemptive single machine scheduling problem with arbitrary release and due dates, priority factors (weights), and equal processing times of a finite number of jobs. The TWT heuristic is intended to minimize the total weighted tardiness of schedules. Our refinements allow to extend equal processing times to arbitrary ones such that we are able to solve benchmark instances online with up to 200 jobs and 20 units of processing times within 47 ms and essentially decrease TWT of returned schedules. The time and space complexities of returned schedules outperform the state of the art for the cases of arbitrary release dates, weighted and unweighted total tardiness, with equal and arbitrary processing times.

Minimizing Total Weighted Tardiness for Scheduling Equal-Length Jobs on a Single Machine

In this paper, we consider the problem of minimizing total weighted tardiness for equal-length jobs with arbitrary release dates on a single machine. This problem is mentioned as a minimal open problem, see http://www2.informatik.uni-osnabrueck.de/knust/class/dateien/classes/ein_ma/ein_ma, i.e., its complexity status is still open. The latest results on this problem were presented in the years 2000 and 2005, namely solution algorithms for special cases. We present some properties of this problem and discuss possible ways for future research.

Minimization of total completion time on a batch processing machine with arbitrary release dates: an effectual teaching–learning based optimization approach

In this research study, a single machine batch-processing problem with release dates to minimize the total completion times of jobs is considered. The machine is able to process at most a certain number of jobs at the same time and the total size of the jobs allocated to a batch cannot exceed the machine capacity. Since the research problem has been shown to be NP-hard, an effective Teaching–Learning Based Optimization (TLBO) is proposed. A constructive heuristic approach is developed to generate initial feasible solutions for the TLBO. In order to enhance the efficiency of the proposed TLBO, a Tabu Search (TS) with three different neighborhood generation mechanisms is incorporated into the teaching phase and learner phase separately. To validate the outcomes of the proposed TLBO, we carry out an experimental study and compare its outcomes with the best-known results obtained by several meta-heuristic methods on a set of benchmark instances derived from the literature. The computational results show that the proposed TLBO with the incorporation of TS in its learning phase is able to come up with very good quality solutions.

Optimize Unrelated Parallel Machines Scheduling Problems With Multiple Limited Additional Resources, Sequence-Dependent Setup Times and Release Date Constraints

This paper considers the problem of scheduling a set of jobs on unrelated parallel machines subject to several constraints which are non-zero arbitrary release dates, limited additional resources, and non-anticipatory sequence-dependent setup times. The objective function is to minimize the maximum completion time. In order to find an optimal solution for this problem, a new mixed-integer linear programming model (MILP) is presented. Moreover, a two-stage hybrid metaheuristic based on variable neighborhood search hybrid and simulated annealing (TVNS_SA) is proposed. In the first stage, a developed heuristic is used to find an initial solution with good quality. At the second stage, the obtained initial solution is used as the first neighborhood structures in the proposed metaheuristic, for further progress different neighborhood structures and effective resolution schemes are also presented. The computational results indicate that the proposed metaheuristic is capable of obtaining optimal solutions for most of the instances when compared to the solution obtained by the developed mixed-integer linear programming model. In addition, the metaheuristic dominated the MILP with respect to computing time. The overall evaluation of the proposed algorithm shows its efficiency and effectiveness when compared with other algorithms. Finally, in order to obtain rigorous and fair conclusions, a paired t-test has been conducted to test the significant differences between the five variants of the TVNS_SA.

Competitive algorithms for multistage online scheduling

We study an online flow shop scheduling problem where each job consists of several tasks that have to be completed in t different stages and the goal is to maximize the total weight of accepted jobs. The set of tasks of a job contains one task for each stage and each stage has a dedicated set of identical parallel machines corresponding to it that can only process tasks of this stage. In order to gain the weight (profit) associated with a job j, each of its tasks has to be executed between a task-specific release date and deadline subject to the constraint that all tasks of job j from stages 1,⋯,i−1 have to be completed before the task of the ith stage can be started. In the online version, jobs arrive over time and all information about the tasks of a job becomes available at the release date of its first task. This model can be used to describe production processes in supply chains when customer orders arrive online.Even the basic version of the offline problem with a single machine in each stage, unit weights, unit processing times, and fixed execution times for all tasks (i.e., deadline minus release date equals processing time) is APX-hard and we show that the approximation ratio of any polynomial-time approximation algorithm for this basic version of the problem must depend on the number t of stages.For the online version of the basic problem, we provide a (2t−1)-competitive deterministic online algorithm and a matching lower bound. Moreover, we provide several (sometimes tight) upper and lower bounds on the competitive ratios of online algorithms for several generalizations of the basic problem involving different weights, arbitrary release dates and deadlines, different processing times of tasks, and several identical machines per stage.

A beam search heuristic for scheduling a single machine with release dates and sequence dependent setup times to minimize the makespan

This paper considers the problem of scheduling a set of jobs subject to arbitrary release dates and sequence-dependent setup times on a single machine with the objective of minimizing the maximum completion of all the jobs, or makespan. This problem is often found in manufacturing processes such as painting and metalworking. A new mixed integer linear program (MILP) is firstly proposed. Because the problem is known to be NP-hard, a beam search heuristic is developed. Computational experiments are carried out using a well-known set of instances from the literature. Our results show that the proposed heuristic is effective in finding high quality solutions at low computational cost.

Single machine scheduling with two competing agents, arbitrary release dates and unit processing times

We study various single machine scheduling problems with two competing agents, unit processing times and arbitrary integer release dates. The problems differ by the scheduling criterion used by each of the two agents, and by the variant of the bicriteria problem that has to be solved. We prove that when the scheduling criterion of either one of the two agents is of a max-type, then all considered variants of the bicriteria problem are solvable in polynomial time. However, when the two agents have a sum-type of scheduling criterion, several variants of the bicriteria problem become \(\mathcal {NP}\)-hard.

A two-agent single-machine scheduling problem to minimize the total cost with release dates

This paper considers a two-agent scheduling problem with arbitrary release dates on a single machine. The cost of the first agent is the maximum weighted completion time of its jobs while the cost of the second agent is the total weighted completion time of its jobs. The goal is to schedule the jobs such that the total cost of the two agents is minimized. The problem is known to be strongly NP-hard. Thus, as an alternative, a branch-and-bound algorithm incorporating several dominance properties and a lower bound is provided to derive the optimal solution and a largest- order-value method combined with proposed three initials is developed to derive the near-optimal solutions for the problem. Computational results are also presented to evaluate the performance of the proposed algorithms.

Speed Scaling for Maximum Lateness

We consider the power-aware problem of scheduling non-preemptively a set of jobs on a single speed-scalable processor so as to minimize the maximum lateness, under a given budget of energy. In the offline setting, our main contribution is a combinatorial polynomial time algorithm for the case in which the jobs have common release dates. In the presence of arbitrary release dates, we show that the problem becomes strongly NP$\mathcal {N}\mathcal {P}$-hard. Moreover, we show that there is no O(1)-competitive deterministic algorithm for the online setting in which the jobs arrive over time. Then, we turn our attention to an aggregated variant of the problem, where the objective is to find a schedule minimizing a linear combination of maximum lateness and energy. As we show, our results for the budget variant can be adapted to derive a similar polynomial time algorithm and an NP$\mathcal {N}\mathcal {P}$-hardness proof for the aggregated variant in the offline setting, with common and arbitrary release dates respectively. More interestingly, for the online case, we propose a 2-competitive algorithm.

Online heuristic for the preemptive single machine scheduling problem of minimizing the total weighted completion time

The preemptive single machine scheduling problem of minimizing the total weighted completion time with arbitrary processing times and release dates is an important NP-hard problem in scheduling theory. In this paper we present an efficient high-quality heuristic for this problem based on the Weighted Shortest Remaining Processing Time (WSRPT) rule. The running time of the suggested algorithm increases only as a square of the number of jobs. Our computational study shows that very large size instances might be treated within extremely small CPU times and the average error is always less than 0.1%.

A branch-and-bound algorithm for a single machine sequencing to minimize the total tardiness with arbitrary release dates and position-dependent learning effects

This study considers an NP-hard problem of minimizing the total tardiness on a single machine with arbitrary release dates and position-dependent learning effects. A mixed-integer programming (MIP ) model is first formulated to find the optimal solution for small-size problem instances. Some new dominance rules are then presented which provide a sufficient condition for finding local optimality. The branch-and-bound (B& B) strategy incorporating with several dominance properties and a lower bound is proposed to derive the optimal solution for medium- to-large-size problem instances, and four marriage-in-honey-bees optimization algorithms (MBO) are developed to derive near-optimal solutions for the problem. To show the effectiveness of the proposed algorithms, 3600 situations with 20 and 25 jobs, are randomly generated for experiments.

An investigation on a two-agent single-machine scheduling problem with unequal release dates

This paper addresses a two-agent scheduling problem on a single machine with arbitrary release dates, where the objective is to minimize the tardiness of one agent, while keeping the lateness of the other agent below or at a fixed level Q. A mixed integer programming model is first presented for its optimal solution, admittedly not to be practical or useful in the most cases, but theoretically interesting since it models the problem. Thus, as an alternative, a branch-and-bound algorithm incorporating with several dominance properties and a lower bound is provided to derive the optimal solution and a marriage in honey-bees optimization algorithm (MBO) is developed to derive the near-optimal solutions for the problem. Computational results are also presented to evaluate the performance of the proposed algorithms.

Tight complexity analysis of the relocation problem with arbitrary release dates

The paper considers makespan minimization on a single machine subject to release dates in the relocation problem, originated from a resource-constrained redevelopment project in Boston. Any job consumes a certain amount of resource from a common pool at the start of its processing and returns to the pool another amount of resource at its completion. In this sense, the type of our resource constraints extends the well-known constraints on resumable resources, where the above two amounts of resource are equal for each job. In this paper, we undertake the first complexity analysis of this problem in the case of arbitrary release dates. We develop an algorithm, based on a multi-parametric dynamic programming technique (when the number of parameters that undergo enumeration of their values in the DP-procedure can be arbitrarily large). It is shown that the algorithm runs in pseudo-polynomial time when the number m of distinct release dates is bounded by a constant. This result is shown to be tight: (1) it cannot be extended to the case when m is part of the input, since in this case the problem becomes strongly NP-hard, and (2) it cannot be strengthened up to designing a polynomial time algorithm for any constant m > 1 , since the problem remains NP-hard for m = 2 . A polynomial-time algorithm is designed for the special case where the overall contribution of each job to the resource pool is nonnegative. As a counterpart of this result, the case where the contributions of all jobs are negative is shown to be strongly NP-hard.

A branch and bound algorithm for scheduling jobs with controllable processing times on a single machine to meet due dates

In most deterministic scheduling problems, job-processing times are regarded as constant and known in advance. However, in many realistic environments, job-processing times can be controlled by the allocation of a common resource to jobs. In this paper, we consider the problem of scheduling jobs with arbitrary release dates and due dates on a single machine, where job-processing times are controllable and are modeled by a non-linear convex resource consumption function. The objective is to determine simultaneously an optimal processing permutation as well as an optimal resource allocation, such that no job is completed later than its due date, and the total resource consumption is minimized. The problem is strongly \(\mathcal{NP}\)-hard. A branch and bound algorithm is presented to solve the problem. The computational experiments show that the algorithm can provide optimal solution for small-sized problems, and near-optimal solution for medium-sized problems in acceptable computing time.

A decomposition scheme for single stage scheduling problems

This paper introduces a general decomposition scheme for single stage scheduling problems with jobs that have arbitrary release dates. We assume that the objective function is monotone in the completion time of each job. The decomposition scheme has significant theoretical and practical relevance. When assuming equal processing times, we can reduce the number of steps required to solve several well-known nonpreemptive single machine scheduling problems by \(\mathcal{O}(n^{3})\), provided the processing time p is constant. Specifically, we develop new approaches that solve the problems 1|r i ,p i =p|∑f i (C i ) and 1|r i ,p i =p|∑w i U i in \(\mathcal{O}(n^{4})\) time; the algorithms that have been described in the literature for these problems operate in \(\mathcal{O}(n^{7})\). Moreover, solution approaches for \(\mathcal{NP}\)-hard problems with unequal processing times may also benefit from our decomposition rule. This is particularly true if p max/p min is close to 1. Using the decomposition rule, either the problem size is reduced or additional information about the maximal schedule length is obtained.

Solving a Multiprocessor Problem by Column Generation and Branch-and-price

This work presents an algorithm for solving exactly a scheduling problem with identical parallel machines and malleable tasks, subject to arbitrary release dates and due dates. The objective is to minimize a function of late work and setup costs. A task is malleable if we can freely change the set of machines assigned to its processing over the time horizon. We present an integer programming model, a Dantzig-Wolfe decomposition reformulation and its solution by column generation. We also developed an equivalent network flow model, used for the branching phase. Finally, we carried out extensive computational tests to verify the algorithm's efficiency and to determine the model's sensitivity to instance size parameters: the number of machines, the number of tasks and the size of the planning horizon.

Minimizing the number of machines for scheduling jobs with equal processing times

In this paper, we consider a parallel machine environment when all jobs have the same processing time and arbitrary release dates and deadlines of the jobs are given. We suppose that the available number of machines, which can be used simultaneously, may vary over time. The aim is to construct a feasible schedule in such a way that the maximal number of simultaneously used machines is minimal. We give a polynomial algorithm for this problem.