Preemptive Scheduling Problem Research Articles

On preemptive scheduling on unrelated machines using linear programming

<abstract><p>We consider a basic preemptive scheduling problem where $ n $ non-simultaneously released jobs are to be processed by $ m $ unrelated parallel machines so as to minimize maximum job completion time. An optimal LP-solution has been used to construct an optimal preemptive schedule for simultaneously released jobs in time $ O(n^3) $. We propose fast $ O(m) $ time algorithm that finds an optimal schedule in case the above LP-solution possesses "small enough" number of non-zero elements. We propose another linear program for non-simultaneously released jobs and show how an optimal schedule can be constructed also in time $ O(m) $ from the optimal solution to that linear program. Based on another stronger linear program formulation, we extend the earlier known schedule construction procedure for non-simultaneously released jobs. The procedure is important, in particular, because there may exist no optimal schedule that agrees with an optimal LP-solution. An optimal LP-solution imposes a number of preemptions, and additional preemptions may occur during the schedule construction process, a job might be forced to be split on the same machine. We show that if no job split is allowed, even a restricted version of the problem on three unrelated machines is NP-hard. As a result, we obtain that, given an optimal LP-solution, it is NP-hard to find an optimal schedule that agrees with that LP-solution. As another side result, we obtain that it is NP-hard to find an optimal schedule with at most $ m-1 $ preemptions.</p></abstract>

Robust algorithms for preemptive scheduling on uniform machines of non-increasing job sizes

Preemptive scheduling problems of minimizing the makespan on parallel machines are basic problems. Motivated by sensitivity analysis and online algorithms, we investigate the problem of designing robust algorithms that are faced with the input one job at a time, but unlike online algorithms they are allowed to modify a small portion of the solution whenever a new job is released. For the settings of uniformly related machines, it is known that such algorithms cannot maintain an optimal solution (for general input sequences). Thus, we consider the semi-online case where the job sequence has non-increasing job sizes and design an optimal algorithm with migration factor 1. Previously such result was known only for the case of identical machines where it holds for an arbitrary input sequence.

Unary NP-hardness of preemptive scheduling to minimize total completion time with release times and deadlines

We revisit the single-machine preemptive scheduling problem to minimize total completion time with release times and deadlines. Du and Leung (1993) showed that the problem is binary NP-hard. But the exact complexity (unary NP-hardness or pseudo-polynomial-time solvability) of the problem is a long standing open problem. We show in this paper that the problem is unary NP-hard.

An exact solution approach using a novel concept for single machine preemptive scheduling problem in the just-in-time production system

ABSTRACT This paper addresses single machine scheduling with preemption, machine idle time and release date to minimize the total weighted Earliness/Tardiness penalties and Work-In-Process (WIP) costs. This research brings a novel concept in the solution approach by introducing a new Mixed Integer Linear Programming (MILP) model which provides a new insight of the studied problem. The proposed MILP model is benchmarked with an existing model in four divisions under 144 randomly numerical problems. In addition, a discussion is made on several features of the introduced approach. The results demonstrate that the new model can easily find optimum solution up to six jobs in small computational times where values of parameters can be discrete or continuous within a wide range of values. While the existing model can find optimum solution up to four jobs in larger computational times where parameters can only take certain values within a small range of values.

Preemptive scheduling with transportation delays between machines

PurposeThis paper studies the preemptive scheduling problem of independent jobs on identical machines. The purpose of this paper is to minimize the makespan under the imposed constraints, namely, the ones that relate the transportation delays which are required to transport a preempted job from one machine to another. This study considers the case when the transportation delays are variable.Design/methodology/approachThe contribution is twofold. First, this study proposes a new linear programming formulation in real and binary decision variables. Then, this study proposes and implements a solution strategy, which consists of two stages. The goal of the first stage is to obtain the best machines order using a local search strategy. For the second stage, the objective is to determine the best possible sequence of jobs. To solve the preemptive scheduling problem with transportation delays, this study proposes a heuristic and two metaheuristics (simulated annealing and variable neighborhood search), each with two modes of evaluation.FindingsComputational experiments are presented and discussed on randomly generated instances.Practical implicationsThe study has implications in various industrial environments when the preemption of jobs is allowed.Originality/valueThis study proposes a new linear programming formulation for the problem with variable transportation delays as well as a corresponding heuristic and metaheuristics.

Modeling single machine preemptive scheduling problems for computational efficiency

We propose two modeling approaches to solve single machine preemptive scheduling problems with tardiness related objectives. Employing the conventional time-indexed formulation, we first build a model that explicitly identifies completion times of jobs with varying release times, due dates, and processing times. The second model adopts the aggregate planning view and eliminates binary constraints. Under this approach, each job is seen as a unit demand while its due date is mapped to a period where this unit is demanded. With this mapping, the periodic job allocation decisions are transformed into periodic production decisions that are measured by fraction of demand. Consequently, instead of explicit representation of the job completion times, this model tracks the amounts of production completed and backlogged via inventory and shortage variables and conservation of units constraints. We establish that the latter model provides tighter bounds and demonstrate that it provides a more efficient platform for optimization via computational analysis employing four commonly used tardiness related criteria and a case study from a real life application. Numerical computations reveal that aggregate planning view becomes more dominant in terms of computational performance as the problem size grows.

Scheduling preemptable position-dependent jobs on two parallel identical machines

A problem of preemptive scheduling of jobs with position-dependent processing times on two parallel identical machines and the schedule length criterion is addressed. Jobs may be preempted but only restricted preemption of a single job is allowed. Several properties of this problem and two exact algorithms for its solution are presented. Computational experiments show that the exact algorithms find optimal schedules for medium size instances in a relatively short time, and that the largest average percentage of nodes discarded by the proposed branch-and-bound algorithm is close to 65%.

������ ������� ������� �� ���I�� ���������� ��� �I�I�I���I� ���������� ��������� ���� ���������� � �����I ���������� � ������������� �I ���������� ���������I ��������� ������� ���������� ��'���

The scheduling theory studying, in particular, minimization of the total weighted completion time, which refers to planning, organizing, and executing complex or multistep processes of assembling, manufacturing, building, dispatching, computing, etc., possesses both exact and heuristic approaches to the schedule computation. The computation time of the exact schedule approach grows immensely when the number of jobs is increased off just a few jobs (roughly, off 6 to 9, depending also on how jobs are divided into job parts). Therefore, a lot of heuristics are used to find the approximate schedule but to obtain it much faster. The heuristics’ approximate schedule is not always executed in the exactly minimal total weighted completion time, but the loss is commonly not so great. Moreover, when the number of jobs is of order of hundreds, the scheduling problems become intractable by any exact schedule approaches, and so the heuristics remain the single way to find a schedule. Considering the preemptive scheduling problem by subsequent length-equal job importance growth, there are two ways to input the job release dates and the respective priority weights. On the one hand, the release dates can be given in ascending order; then the respective priority weights will be a set of, generally speaking, non-decreasing values. On the other hand, the release dates can be given in descending order; and then the respective priority weights will be a set of, generally speaking, non-increasing values. Having estimated the averaged time of obtaining the approximate schedule by both ascending and descending orders of inputting the job release dates, the heuristic’s job order is revealed to be very significant. Its significance grows as the number of jobs increases. The influence of the heuristic’s job order also grows as the number of job parts increases. The descending job order has the growing advantage for scheduling about 300 jobs and more. In particular, the descending job order’s advantage in scheduling 100000 jobs divided in two parts each is almost 42 %. So, for total weighted completion time minimization in the preemptive scheduling problem by subsequent length-equal job importance growth, the job release dates are to be input in the descending order. However, the heuristic’s job order gain in scheduling a lesser number of jobs (a few tens and up to 100, 200, 300) remains uncertain due to considerable fluctuations of the much shorter computation time.

An energy-aware multiobjective ant colony algorithm to minimize total completion time and energy cost on a single-machine preemptive scheduling

Energy-aware scheduling in manufacturing operations with time-of-use electricity tariffs is a challenging problem. Sustainable manufacturing is gaining significant momentum: companies are not only improving their product quality, but also optimizing the production processes to improve energy consumption (i.e., minimizing energy cost) in order to manage environmental challenges which contribute to global climate change. The purpose of this paper is to study a preemptive scheduling problem on a single-machine to minimize the total completion time and total energy cost under time-of-use electricity tariffs, which is a mixed-integer multiobjective mathematical programming model. To solve these objectives, we develop several new holistic ant colony optimization algorithms. The proposed model is solved via several methods including weighted sum method (WSM) using CPLEX, and multiobjective ant colony optimization based on a dominance ranking (ACO-DR) or based on a dominance ranking procedure and crowding distance comparison (ACO-DRC) or based on a heuristic approach to obtain an approximate Pareto-front and also provide information on when to start and resume each job for any solution on the Pareto-front. We provide detailed experimental results evaluating the performance of the proposed algorithms. In a case study, we demonstrate how the results of the multiobjective model could be utilized in decision making using the multiobjective optimization on the basis of ratio analysis (MOORA) method. This proposed model and heuristics allows decision makers to operate in challenging-data enabled environments in industrial internet of things ecosystem, and assists in optimizing production planning to improve energy cost.

Scheduling in a two-stage flowshop with parallel unrelated machines at each stage and shared resources

Abstract The paper deals with the problem of preemptive scheduling in the two-stage flowhop with parallel unrelated machines at both the stages and renewable resources shared among the stages. A novel heuristic is proposed which minimizes the flowshop makespan taking into account both the stages simultaneously. This heuristic introduces a new concept of the sets of jobs which are allowed to be processed at stage 1 and stage 2 in successive time intervals. The definition of these sets is based on a priority rule. The constraints resulting from this definition are included into the optimization problem solved by a column generation (CG) algorithm. The CG algorithm creates the schedule which is composed of partial schedules assigning jobs to machines for simultaneous processing at the first and second stages during some periods of time, so that the resource constraints are satisfied at any moment. We developed four heuristic algorithms which start either from a straightforward initial solution or from a solution provided by a linear programming based procedure, and use a simulated annealing or tabu search procedure to create partial schedules in each successive iteration. The results of the extensive computational experiment indicate that these algorithms provide good quality solutions with reasonable computational effort, even for the difficult problems with strong resource constraints.

Scheduling in Multiprocessor Systems with Additional Restrictions

An admissible multiprocessor preemptive scheduling problem is solved for the given execution intervals. In addition, a number of generalizations are considered—interprocessor communications are arbitrary and may vary in time; costs for processing interruptions and switches from one processor to another are taken into account; and besides the processors, additional resources are used. Algorithms based on reducing the original problem to finding paths of a specific length in a graph, a flow problem, and an integer system of linear restrictions are developed.

The minimization of exact total weighted completion time in the preemptive scheduling problem by subsequent length-equal job importance growth

For the preemptive scheduling problem in case of subsequent job importance growth, it is studied whether the optimal schedule might be found faster within an exact model. It is ascertained that when the number of jobs up to six (except for the case of four jobs) and there is no randomness in problem forming, a little advantage of weight-descending job order exists only on average. As the number of jobs increases, the advantage of either weight-descending or weight-ascending job order becomes more certain. When priority weights are formed randomly, weight-descending job order is expected to be faster than weight-ascending.

Scheduling Position-Dependent Maintenance Operations

This paper addresses one-machine scheduling with maintenance restrictions. A maintenance operation is position dependent in a sequence of normal jobs if the maintenance has to be performed after at most some defined number of job changes on the machine. We show that several problems with objective functions Cmax and Lmax are still solvable in polynomial time if position-dependent maintenance is considered. We then consider the problem of preemptive scheduling with ready times and due dates on one machine with the Lmax criterion. We show that this problem is computationally hard and present the characteristics of this problem—for example, the fact that optimum schedules may be nonactive. After determining a set of dominance properties, branch-and-bound and local search algorithms are proposed. The performance of the algorithms is evaluated using a series of computational experiments.

Computation scheduling in multiprocessor systems with several types of additional resources and arbitrary processors

The problem of feasible preemptive scheduling in a multiprocessor system is considered for when scheduled intervals are assigned, processor performance can be arbitrary, there are several types of additional resources, and the time for executing tasks depends linearly on the amount of additional resources allocated to them. Polynomial algorithms based on reducing the original problem to a flow problem and a linear programming problem are developed.

Scheduling under a non-reversible energy source: An application of piecewise linear bounding of non-linear demand/cost functions

In this paper, we address a preemptive scheduling problem involving a non-reversible energy source. To the classical scheduling issue, additional information is added regarding the characteristics of the energy source used to satisfy the total power demand of tasks processed at each instant. Different non-reversible energy sources can have very different characteristics in terms of power range and energy demand/cost function also known as efficiency function. The objective is to identify the best combination between scheduling and energy resource utilization that minimizes the total energy cost of the project. The non-linear efficiency function used to compute energy costs is bounded from above and below by two piecewise-linear curves, yielding two instances of a scheduling problem with a piecewise-linear objective that can be solved separately. For the piecewise-linear scheduling problem, we show that the problem involving multiple non-reversible sources is equivalent to a single-source problem, the particular case of a linear function is polynomially solvable, and the case with a piecewise-linear function with two pieces is NP-hard. A pseudo-polynomial size time-indexed mixed-integer linear formulation of the problem and its Dantzig–Wolfe decomposition yielding an extended formulation are presented. A branch-and-price procedure is proposed to solve the extended formulation. The formulations are compared on a set of scheduling instances, considering artificial and real-world efficiency functions.

Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

We revisit the classic problem of preemptive scheduling on m uniformly related machines. In this problem, jobs can be arbitrarily split into parts, under the constraint that every job is processed completely, and that the parts of a job are not assigned to run in parallel on different machines. We study a new objective which is motivated by fairness, where the goal is to minimize the sum of the two maximal job completion times. We design a polynomial time algorithm for computing an optimal solution. The algorithm can act on any set of machine speeds and any set of input jobs. The algorithm has several cases, many of which are very different from algorithms for makespan minimization (algorithms that minimize the maximum completion time of any job), and from algorithms that minimize the total completion time of all jobs.

Solving preemptive job-shop scheduling problems using a true concurrency model

A true concurrency model called stopwatch durational action timed automata (S-DATA) is proposed to model preemptive timed concurrent systems. We demonstrate its usefulness in tackling the preemptive job-shop scheduling problem (PJSSP). This yields a compact reachability tree representing the possible schedules such that the durations and preemptions of the scheduled actions are dynamically managed. Different techniques are demonstrated to reduce the space search over such reachability tree, preserving the optimal schedules of the problem. The efficiency of approach is tested on a number of well-known benchmark problems and compared with the best method of scheduling in common use.

Two-agent scheduling in open shops subject to machine availability and eligibility constraints

Purpose: The aims of this article are to develop a new mathematical formulation and a new heuristic for the problem of preemptive two-agent scheduling in open shops subject to machine maintenance and eligibility constraints. Design/methodology: Using the ideas of minimum cost flow network and constraint programming, a heuristic and a network based linear programming are proposed to solve the problem. Findings: Computational experiments show that the heuristic generates a good quality schedule with a deviation of 0.25% on average from the optimum and the network based linear programming model can solve problems up to 110 jobs combined with 10 machines without considering the constraint that each operation can be processed on at most one machine at a time. In order to satisfy this constraint, a time consuming Constraint Programming is proposed. For n = 80 and m = 10, the average execution time for the combined models (linear programming model combined with Constraint programming) exceeds two hours. Therefore, the heuristic algorithm we developed is very efficient and is in need. Practical implications: Its practical implication occurs in TFT-LCD and E-paper manufacturing wherein units go through a series of diagnostic tests that do not have to be performed in any specified order. Originality/value: The main contribution of the article is to split the time horizon into many time intervals and use the dispatching rule for each time interval in the heuristic algorithm, and also to combine the minimum cost flow network with the Constraint Programming to solve the problem optimally.

Optimal correction of execution intervals for multiprocessor scheduling with additional resource

A problem of admissible multiprocessor preemptive scheduling is considered when execution intervals are specified, processors have arbitrary performance, and the execution time of tasks linearly depends on the amount of the additional resource allocated to their execution. If there is no admissible schedule for a given amount of the additional resource, we consider the problem of optimal correction of execution intervals. The solution is based on reducing the original problem to the problem of minimum cost flow.

Preemptive scheduling in a two-stage supply chain to minimize the makespan

This paper deals with the problem of preemptive scheduling in a two-stage supply chain framework. The supply chain environment contains two stages: production and transportation. In the production stage jobs are processed on a manufacturer's bounded serial batching machine, preemptions are allowed, and set-up time is required before a new batch is processed. In the transportation stage each batch is delivered to a customer by a single vehicle. The objective is to minimize the makespan by making decisions for both mutually coordinated stages. Specifically, two versions are studied. The first one is that all jobs are available to be processed at time zero, and the second one is that jobs have different release times. An time algorithm is developed for the first version, and we show that it can produce the optimal schedule for the entire problem. For the second version, based on some useful properties we have designed an time heuristic and a novel lower bound. The worst-case performance ratio of our algorithm is bounded by 2. Our computational study with random instances of different scales shows high-quality solutions for either small-scale or large-scale instances returned in a reasonable PC times.