Equal Processing Times Research Articles

Scheduling a single machine with multiple due dates per job

AbstractIn this paper, we consider single-machine scheduling with multiple due dates per job. This is motivated by several industrial applications, where it is not important by how much we miss a due date. Instead the relevant objective is to minimize the number of missed due dates. Typically, this situation emerges whenever fixed delivery appointments are chosen in advance, such as in the production of individualized pharmaceuticals or when customers can only receive goods at certain days in the week, due to constraints in their warehouse operation. We compare this previously unexplored problem with classical due date scheduling, for which it is a generalization. We show that single-machine scheduling with multiple due dates is NP-hard in the strong sense if processing times are job dependent. If processing times are equal for all jobs, then single-machine scheduling with multiple due dates is at least as hard as the long-standing open problem of weighted tardiness with equal processing times and release dates $$1 \mid r_j, p_j = p \mid \sum w_j T_j$$ 1 ∣ r j , p j = p ∣ ∑ w j T j . Finally, we focus on the case of equal processing times and provide several polynomially solvable special cases as well as an exact branch-and-bound algorithm and heuristics for the general case. Experiments show that our branch-and-bound algorithm compares well to modern exact methods to solve problem $$1 \mid r_j, p_j = p \mid \sum w_{j} T_{j}$$ 1 ∣ r j , p j = p ∣ ∑ w j T j .

An asymptotically optimal solution for the minimization of the variation of job completion times in two-stage proportionate no-wait flow shops

We propose an asymptotically optimal solution for the two-stage no-wait proportionate flow shop scheduling problem with the objective of minimizing the total absolute deviation of job completion times. The proposed solution is within 1% of the optimum for problems with more than 20 jobs. We also show that the problem is solved optimally in constant time for the special case where the two smallest jobs have equal processing times. Our results provide answers to most of the outstanding research questions for this problem.

Bounded scheduling two-component jobs simultaneously or hierarchically

This paper studies the multi-objective problem of scheduling n two-component jobs on a fabrication machine with bounded batch capacity in terms of the number of jobs. Each job consists of two components, one is common among all jobs (called standard component) and the other is unique to itself (called specific component). The machine processes all components so that it might become a capacitated bottleneck machine. Standard components have equal processing times. They are fabricated in batches sequentially on the machine, and a setup time is required to fabricate a batch. Each batch contains the standard components of at most b(b < n) jobs. Specific components are manufactured individually, therefore their associated setup times could be absorbed into their processing times. Standard components in a batch are not available until all these components are completed (meaning that the processing time of a batch is equal to the total processing time of the standard components in it), whereas a specific component is available on completion of its processing. A job is completed when both its two components are completed and available. For simultaneous optimization of makespan and maximum lateness, an O(n3)-time algorithm is obtained. Through a careful use of the heap data structure, the time bound can be improved to O(n2 log n). For hierarchical optimization of two maximum lateness objectives and makespan, an O(n5)-time algorithm is obtained, which can be improved to run in time O(n4).

Solving Preemptive Single Machine Scheduling with Lovebird Algorithm and Constructive Heuristics for Equal-Length Jobs

Scheduling is essential in manufacturing to maintain machinery and increase revenue. Machines that are routinely maintained will boost production, whereas scheduling based on total weighted completion time (TWCT) can improve the performance of production equipment. The Lovebird Algorithm (LBA), Constructive Heuristics (CH), and combination of LBA-CH are used in this study to schedule a single machine with equal processing times. LBA, a population-based algorithm, represents job parts as birds for jobs that can be preempted. The experiments, which used ten jobs divided into dozens of job parts, show that LBA-CH can effectively reduce TWCT compared to CH by 18.16%.

Bicriteria multi-machine scheduling with equal processing times subject to release dates

<abstract><p>This paper addresses the problem of scheduling $ n $ equal-processing-time jobs with release dates non-preemptively on identical machines to optimize two criteria simultaneously or hierarchically. For simultaneous optimization of total completion time (and makespan) and maximum cost, an algorithm is presented which can produce all Pareto-optimal points together with the corresponding schedules. The algorithm is then adapted to solve the hierarchical optimization of two min-max criteria, and the final schedule has a minimum total completion time and minimum makespan among the hierarchical optimal schedules. The two algorithms provided in this paper run in $ O(n^3) $ time.</p></abstract>

Single machine Pareto scheduling with positional deadlines and agreeable release and processing times

<abstract><p>This paper studies the problem of scheduling $ n $ jobs on a single machine to minimize total completion time and maximum cost, simultaneously. Each job is associated with a positional deadline that indicates the largest ordinal number of this job in any feasible schedule. The jobs have agreeable release and processing times, meaning that jobs with larger release times also have larger processing times. The agreeability assumption is reasonable since both the single-criterion problems (without positional deadline constraints) of minimizing total completion time and maximum lateness on a single machine with arbitrary release and processing times are strongly NP-hard. An $ O(n^3) $-time Pareto optimal algorithm is presented. The previously known algorithms only solve two special cases of the agreeability assumption: either the case of equal release times in $ O(n^4) $ time, or the case of equal processing times (without positional deadline constraints) in $ O(n^3) $ time.</p></abstract>

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.

Two-agent bounded parallel-batching scheduling for minimizing maximum cost and makespan

This paper considers the bounded parallel-batching scheduling with two agents to minimize maximum cost of agent A and makespan of agent B simultaneously, in which all jobs of agent A have equal processing time, the jobs from different agents can be processed in a common batch and the cost function of each agent is only determined by its own jobs. In the paper, we present a polynomial-time algorithm to generate all Pareto optimal points for the problem and determine a corresponding Pareto optimal schedule for each Pareto optimal point.

Pareto‐scheduling with double‐weighted jobs to minimize the weighted number of tardy jobs and total weighted late work

AbstractWe consider the single‐machine Pareto‐scheduling problem to minimize the weighted number of tardy jobs and total weighted late work simultaneously. The problem is to find the set of all the Pareto‐optimal points, that is, the Pareto frontier, and their corresponding Pareto‐optimal schedules. We consider the corresponding weighted‐sum scheduling problem and primary‐secondary scheduling problems, being subproblems of the general Pareto‐scheduling problem. The NP‐hardness of the general problem follows directly from the NP‐hardness of the two constituent single‐criterion problems. We present a pseudo‐polynomial algorithm and a fully polynomial‐time approximation scheme (FPTAS) running in weakly polynomial time to deal with the general problem. When all the jobs have a common due date, we further provide an FPTAS running in strongly polynomial time. We also study some special cases of the general problem where the jobs have equal processing times, a common due date, or a common weight, and analyze their computational complexity status.

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.

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.

Iterated Local Search and Other Algorithms for Buffered Two-Machine Permutation Flow Shops with Constant Processing Times on One Machine.

The two-machine permutation flow shop scheduling problem with buffer is studied for the special case that all processing times on one of the two machines are equal to a constant c. This case is interesting because it occurs in various applications, for example, when one machine is a packing machine or when materials have to be transported. Different types of buffers and buffer usage are considered. It is shown that all considered buffer flow shop problems remain NP-hard for the makespan criterion even with the restriction to equal processing times on one machine. However, the special case where the constant c is larger or smaller than all processing times on the other machine is shown to be polynomially solvable by presenting an algorithm (2BF-OPT) that calculates optimal schedules in O(nlogn) steps. Two heuristics for solving the NP-hard flow shop problems are proposed: (i) a modification of the commonly used NEH heuristic (mNEH) and (ii) an Iterated Local Search heuristic (2BF-ILS) that uses the mNEH heuristic for computing its initial solution. It is shown experimentally that the proposed 2BF-ILS heuristic obtains better results than two state-of-the-art algorithms for buffered flow shop problems from the literature and an Ant Colony Optimization algorithm. In addition, it is shown experimentally that 2BF-ILS obtains the same solution quality as the standard NEH heuristic, however, with a smaller number of function evaluations.

Transcending Conventional Biometry Frontiers: Diffusive Dynamics PPG Biometry.

This paper presents the first photoplethysmographic (PPG) signal dynamic-based biometric authentication system with a Siamese convolutional neural network (CNN). Our method extracts the PPG signal’s biometric characteristics from its diffusive dynamics, characterized by geometric patterns in the -planes specific to the 0–1 test. PPG signal diffusive dynamics are strongly dependent on the vascular bed’s biostructure, unique to each individual. The dynamic characteristics of the PPG signal are more stable over time than its morphological features, particularly in the presence of psychosomatic conditions. Besides its robustness, our biometric method is anti-spoofing, given the complex nature of the blood network. Our proposal trains using a national research study database with 40 real-world PPG signals measured with commercial equipment. Biometric system results for input data, raw and preprocessed, are studied and compared with eight primary biometric methods related to PPG, achieving the best equal error rate (ERR) and processing times with a single attempt, among all of them.

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness Lmax for integer due dates to the scheduling problem, where along with precedence constraints given on the set V={v1,v2, …,vn} of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set V. We prove that the extended scheduling problem with integer release times ri≥0 of the jobs V to minimize schedule length Cmax may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) {1,2, …,t} to the vertices {v1,v2, …,vn}=V of the mixed graph G=(V,A, E) such that, if two vertices vp and vq are joined by the edge [vp,vq]∈E, their colors have to be different. Further, if two vertices vi and vj are joined by the arc (vi,vj)∈A, the color of vertex vi has to be no greater than the color of vertex vj. We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs G=(V,A, E), have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.

Mixed graph colouring as scheduling multi-processor tasks with equal processing times

A problem of scheduling partially ordered unit-time tasks processed on dedicated machines is formulated as a mixed graph colouring problem, i. e., as an assignment of integers (colours) {1, 2, …, t} to the vertices (tasks) V {ν1, ν2, …, νn}, of the mixed graph G = (V, A, E) such that if vertices vp and vq are joined by an edge [νp, νq] ∈ E their colours have to be different. Further, if two vertices νp and νq are joined by an arc (νi, νj) ∈ A the colour of vertex νi has to be no greater than the colour of vertex νj. We prove that an optimal colouring of a mixed graph G = (V, A, E) is equivalent to the scheduling problem GcMPT|pi = 1|Cmax of finding an optimal schedule for partially ordered multi-processor tasks with unit (equal) processing times. Contrary to classical shop-scheduling problems, several dedicated machines are required to process an individual task in the scheduling problem GcMPT|pi = 1|Cmax. Moreover, along with precedence constraints given on the set V {ν1, ν2, …, νn}, it is required that a subset of tasks must be processed simultaneously. Due to the theorems proved in this article, most analytical results that have been proved for the scheduling problems GcMPT |pi = 1|Cmax so far, have analogous results for optimal colourings of the mixed graphs G = (V, A, E), and vice versa.

Mixed graph colouring as scheduling multi-processor tasks with equal processing times

Mixed graph colouring as scheduling multi-processor tasks with equal processing times

Arc-flow approach for single batch-processing machine scheduling

We address the problem of scheduling jobs with non-identical sizes and distinct processing times on a single batch-processing machine, aiming at minimizing the makespan. The extensive literature on this NP-hard problem mostly focuses on heuristics. Using an arc-flow based optimization approach, we construct a novel formulation that represents it as a problem of determining flows in graphs. The size of the formulation increases with the machine capacity and with the number of distinct sizes and processing times among the jobs, but it does not increase with the number of jobs, which makes it very effective to solve large instances to optimality, especially when multiple jobs have equal size and processing time. We compare our model to other models from the literature, showing its clear superiority on benchmark instances and proving optimality of random instances with up to 100 million jobs.

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.

Scheduling jobs with sizes and delivery times on identical parallel batch machines

We consider the problem of scheduling jobs with sizes, release times and delivery times on identical parallel batch machines. All machines have the same capacity. Each machine can simultaneously process several jobs as a batch as long as the total size of these jobs does not exceed the capacity. The release time, processing time and delivery time of a batch are defined to be the latest release time, longest processing time and largest delivery time of all the jobs in the batch, respectively. The objective is to minimize the maximum delivery completion time, i.e., the time by which all jobs are delivered. For this strongly NP-hard problem, there does not exist any polynomial time algorithm with approximation ratio better than 2 unless P=NP, even if all jobs have equal processing times and all release times and delivery times are zero. We present an algorithm with approximation ratio arbitrarily close to 2.