Two-machine Job Shop Research Articles

Two-machine job shop scheduling with optional job rejection

Two-machine job shop scheduling with optional job rejection

A complexity analysis and algorithms for two-machine shop scheduling problems under linear constraints

We study several two-machine shop scheduling problems, namely flow shop, job shop and open shop scheduling problems under linear constraints. In these problems, the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal of each problem is to determine the processing time of each job, and to schedule the jobs to the shop machine such that the makespan, i.e., the completion time of all jobs, is minimized. These problems can find application in various areas, such as industrial production, advertising and automotive maintenance. We study the computational complexity and propose polynomial-time optimal or approximation algorithms for them. In particular, we show that although a two-machine flow shop scheduling problem and a two-machine job shop scheduling problem without recirculation can be solved in polynomial time, the problems where processing times satisfy linear constraints are generally NP-hard in the strong sense. Then, we design algorithms for various settings of these problems. We design polynomial-time algorithms for them when there are a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial-time approximation schemes. In contrast to the previous two problems, we show that the two-machine open shop scheduling problem under linear constraints can be solved in polynomial time.

Schedule Execution for Two-Machine Job-Shop to Minimize Makespan with Uncertain Processing Times

This study addresses a two-machine job-shop scheduling problem with fixed lower and upper bounds on the job processing times. An exact value of the job duration remains unknown until completing the job. The objective is to minimize a schedule length (makespan). It is investigated how to best execute a schedule, if the job processing time may be equal to any real number from the given (closed) interval. Scheduling decisions consist of the off-line phase and the on-line phase of scheduling. Using the fixed lower and upper bounds on the job processing times available at the off-line phase, a scheduler may determine a minimal dominant set of schedules (minimal DS), which is based on the proven sufficient conditions for a schedule dominance. The DS optimally covers all possible realizations of the uncertain (interval) processing times, i.e., for each feasible scenario, there exists at least one optimal schedule in the minimal DS. The DS enables a scheduler to make the on-line scheduling decision, if a local information on completing some jobs becomes known. The stability approach enables a scheduler to choose optimal schedules for most feasible scenarios. The on-line scheduling algorithms have been developed with the asymptotic complexity O(n2) for n given jobs. The computational experiment shows the effectiveness of these algorithms.

Two-Machine Job-Shop Scheduling Problem to Minimize the Makespan with Uncertain Job Durations

We study two-machine shop-scheduling problems provided that lower and upper bounds on durations of n jobs are given before scheduling. An exact value of the job duration remains unknown until completing the job. The objective is to minimize the makespan (schedule length). We address the issue of how to best execute a schedule if the job duration may take any real value from the given segment. Scheduling decisions may consist of two phases: an off-line phase and an on-line phase. Using information on the lower and upper bounds for each job duration available at the off-line phase, a scheduler can determine a minimal dominant set of schedules (DS) based on sufficient conditions for schedule domination. The DS optimally covers all possible realizations (scenarios) of the uncertain job durations in the sense that, for each possible scenario, there exists at least one schedule in the DS which is optimal. The DS enables a scheduler to quickly make an on-line scheduling decision whenever additional information on completing jobs is available. A scheduler can choose a schedule which is optimal for the most possible scenarios. We developed algorithms for testing a set of conditions for a schedule dominance. These algorithms are polynomial in the number of jobs. Their time complexity does not exceed O ( n 2 ) . Computational experiments have shown the effectiveness of the developed algorithms. If there were no more than 600 jobs, then all 1000 instances in each tested series were solved in one second at most. An instance with 10,000 jobs was solved in 0.4 s on average. The most instances from nine tested classes were optimally solved. If the maximum relative error of the job duration was not greater than 20 % , then more than 80 % of the tested instances were optimally solved. If the maximum relative error was equal to 50 % , then 45 % of the tested instances from the nine classes were optimally solved.

Some Results on Shop Scheduling with S-Precedence Constraints among Job Tasks

We address some special cases of job shop and flow shop scheduling problems with s-precedence constraints. Unlike the classical setting, in which precedence constraints among the tasks of a job are finish–start, here the task of a job cannot start before the task preceding it has started. We give polynomial exact algorithms for the following problems: a two-machine job shop with two jobs when recirculation is allowed (i.e., jobs can visit the same machine many times), a two-machine flow shop, and an m-machine flow shop with two jobs. We also point out some special cases whose complexity status is open.

A column generation-based approach for proportionate flexible two-stage no-wait job shop scheduling

Job shop scheduling, as one of the classical scheduling problems, has been widely studied in literatures, and proved to be mostly NP-hard. Although it is extremely difficult to solve job shop scheduling with no-wait constraint to optimality, the two-machine no-wait job shop scheduling to minimise makespan could be solvable in polynomial time when each job has exactly two equal length operations (proportionate job shop). In the present paper, an extension is attempted by considering a proportionate flexible two-stage no-wait job shop scheduling problem with minimum makespan, and a set-covering formulation is put forward which contains a master problem and a pricing problem. To solve this problem, a column generation (CG)-based approach is implemented. In comparison, a mixed integer programming model is constructed and optimised by Cplex. A series of randomly generated numerical instances are calculated. And the testing result shows that the mixed integer model handled by Cplex can only solve small scale cases, while the proposed CG-based method can conquer larger size problems in acceptable time.

Two-Machine Job-Shop Scheduling with Equal Processing Times on Each Machine

In this paper, we consider a two-machine job-shop scheduling problem of minimizing total completion time subject to n jobs with two operations and equal processing times on each machine. This problem occurs e.g., as a single-track railway scheduling problem with three stations and constant travel times between any two adjacent stations. We present a polynomial dynamic programming algorithm of the complexity O ( n 5 ) and a heuristic procedure of the complexity O ( n 3 ) . This settles the complexity status of the problem under consideration which was open before and extends earlier work for the two-station single-track railway scheduling problem. We also present computational results of the comparison of both algorithms. For the 30,000 instances with up to 30 jobs considered, the average relative error of the heuristic is less than 1 % . In our tests, the practical running time of the dynamic programming algorithm was even bounded by O ( n 4 ) .

Minimising the makespan in the two-machine job shop problem under availability constraints

Classical scheduling problem assumes that machines are available during the scheduling horizon. This assumption may be justified in some situations but it does not apply if maintenance requirements, machine breakdowns or other availability constraints have to be considered. In this paper, we treat a two-machine job shop scheduling problem with one availability constraint on each machine to minimise the maximum completion time (makespan). The unavailability periods are known in advance and the processing of an operation cannot be interrupted by an unavailability period (non-preemptive case). We present in our approach properties dealing with permutation dominance and the optimality of Jackson's rule under availability constraints. In order to evaluate the effectiveness of the proposed approach, we develop two mixed integer linear programming models and two schemes for a branch and bound method to solve the tackled problem. Computational results validate the proposed approach and prove the efficiency of the developed methods.

No-idle, no-wait: when shop scheduling meets dominoes, Eulerian paths and Hamiltonian paths

In shop scheduling, several applications require that some components perform consecutively. We refer to “no-idle schedules” if machines are required to operate with no inserted idle time and to “no-wait schedules” if tasks cannot wait between the end of an operation and the start of the following one. We consider here no-idle/no-wait shop scheduling problems with makespan as the performance measure and determine related complexity results. We first analyse the two-machine no-idle/no-wait flow shop problem and show that it is equivalent to a special version of the game of dominoes which is polynomially solvable by tackling an Eulerian path problem on a directed graph. We present for this problem an O(n) exact algorithm. As a by-product, we show that the Hamiltonian path problem on a digraph G(V, A) with a special structure (where any two vertices i and j either have all successors in common or have no common successors) reduces to the two-machine no-idle/no-wait flow shop problem. Correspondingly, we provide a new polynomially solvable special case of the Hamiltonian path problem. Then, we show that also the m-machine no-idle/no-wait flow shop problem is polynomially solvable and provide an $$O(mn \log n)$$ exact algorithm. Finally, we prove that the decision versions of the two-machine job shop problem and the two-machine open shop problem are NP-complete in the strong sense.

Two-machine job shop problem under availability constraints on one machine: Makespan minimization

Abstract This paper considers a two-machine job shop scheduling problem with availability constraints on one machine in order to minimize makespan. We consider the problem when unavailability periods are known in advance and operations are non-preemptive. First, two mixed integer linear programming models MILP1, MILP2 are presented. Secondly, we introduce some properties concerning the optimality of Jackson’s algorithm under availability constraints. Consequently, new lower bounds are provided and an upper bound is obtained using heuristics based on Jackson’s rule. Then, a branch and bound algorithm (B&B) incorporating these bounds is proposed to solve the problem. The performances of the proposed approaches are evaluated by comparing their solutions through well known benchmarks. Computational results prove the efficiency of the proposed B&B.

Heuristic algorithms for two-machine job shop problem under availability constraints on one machine: makespan minimization

We discuss the two-machine job shop scheduling problem with availability constraints on one machine for maximum completion time (makespan) minimization. We consider the problem when unavailability periods are planned in advance and operations are non-preemptive. Firstly, some propositions considering Jackson’s rule under availability constraints are introduced. Then, we provide polynomial-time algorithm based on Jackson’s rule and an iterated local search tabu based algorithm exploiting the optimality of Jackson’s rule between each two consecutive availability periods to solve the problem. Finally, we present some experimental results.

Approximation results for the two-machine job shop under limited machine availability

This paper tackles the two-machine job shop scheduling problem with limited machine availability under the resumable scenario. We provide a number of characterizations for the optimal solution. We also develop polynomial-time approximation schemes for two particular configurations and identify a polynomially solvable case.

Two-machine job shop problem for makespan minimization under availability constraint

Abstract: We treat a two-machine job shop scheduling problem with availability constraint on one machine to minimize the makespan. We consider the deterministic case where the unavailability period, corresponding to preventive maintenance tasks, is known in advance and fixed. We assume that jobs are non-preemptive. First, two mixed-integer programming (MIP) models are first presented. Some propositions concerning the optimality of Jackson’s algorithm when availability constraint exists is provided. Then a branch and bound (B&B) method is developed to solve the problem. The obtained results prove the efficiency of the proposed B&B.

Heuristic and Exact Algorithms for the Two-Machine Just in Time Job Shop Scheduling Problem

The problem addressed in this paper is the two-machine job shop scheduling problem when the objective is to minimize the total earliness and tardiness from a common due date (CDD) for a set of jobs when their weights equal 1 (unweighted problem). This objective became very significant after the introduction of the Just in Time manufacturing approach. A procedure to determine whether the CDD is restricted or unrestricted is developed and a semirestricted CDD is defined. Algorithms are introduced to find the optimal solution when the CDD is unrestricted and semirestricted. When the CDD is restricted, which is a much harder problem, a heuristic algorithm is proposed to find approximate solutions. Through computational experiments, the heuristic algorithms’ performance is evaluated with problems up to 500 jobs.

Job selection in two-stage shops with ordered machines

We consider job selection problems in two-stage flow shops, open shops and job shops. The objective is to select the best job subset with a given cardinality to minimize either the total job completion time or the maximum job completion time (makespan). An O(n2) algorithm is available for the two-stage flow shop with ordered machines and the minimum makespan objective; we utilize this algorithm to solve the same problem for the total job completion time objective. We then propose an O(n1) algorithm for the two-machine open shop job selection problem with ordered machines and the minimum makespan objective. We also consider a two-machine job shop in which the first operation of each job is no longer than the second one. We show that the job selection problem in this job shop with the minimum makespan objective is ordinary NP-hard and that the problem becomes solvable in O(nlogn) time with the additional assumption of ordered jobs.

Two-machine interval shop scheduling with time lags

We introduce and analyze the interval shop scheduling problem, where the objective is to maximize the weighted number of jobs that can be processed in a two-stage flow shop, job shop, or open shop, where each job has a fixed start and end time and requires a given transportation time, i.e., a time lag, for moving from one stage to the other. This problem is a natural extension of the parallel-machine scheduling problem with fixed start and end times, which is relatively well understood. We prove that the two-machine interval flow shop problem is $$\mathcal{NP }$$NP-hard in the strong sense for general time lags, even in the case of equal processing times and equal weights. The problem is solvable in polynomial time if all time lags are equal. The two-machine interval job shop problem is solvable in $$O(n^3)$$O(n3) time if the time lags are equal and relatively small. For the two-machine interval open shop problem, this is true as well.

Batch scheduling on a two-machine jobshop with machine-dependent setup times

The problem of minimum makespan on an m machine jobshop with unit execution time (UET) jobs (m ≥ 3) is known to be strongly NP-hard even with no setup times. We focus in this article on the two-machine case. We assume UET jobs and consider batching with batch availability and machine-dependent setup times. We introduce an efficient \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document} \begin{align*}(O(\sqrt{n}))\end{align*} \end{document} algorithm, where n is the number of jobs. We then introduce a heuristic for the multimachine case and demonstrate its efficiency for two interesting instances. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

Maximizing the weighted number of just-in-time jobs in several two-machine scheduling systems

The problem of maximizing the weighted number of just-in-time jobs in a two-machine flow shop scheduling system is known to be $\mathcal {NP}$ -hard (Choi and Yoon in J. Shed. 10:237---243, 2007). However, the question of whether this problem is strongly or ordinarily $\mathcal{NP}$ -hard remains an open question. We provide a pseudo-polynomial time algorithm to solve this problem, proving that it is $\mathcal{NP}$ -hard in the ordinary sense. Moreover, we show how the pseudo-polynomial algorithm can be converted to a fully polynomial time approximation scheme (FPTAS). In addition, we prove that the same problem is strongly $\mathcal{NP}$ -hard for both a two-machine job shop scheduling system and a two-machine open shop scheduling system.

On-line two-machine job shop scheduling with time lags

We consider the on-line two-machine job shop scheduling problem with time lags so as to minimize the makespan. Each job consists of no more than two operations and time lags exist between the completion time of the first and the start time of the second operation of any two-operation job. We prove that any greedy algorithm is 2-competitive. For the non-clairvoyant variant of the problem, no on-line algorithm can do better. For the clairvoyant variant, no on-line delay algorithm has a competitive ratio better than 5 + 1 2 ≈ 1.618 , and a greedy algorithm is still the best on-line non-delay algorithm. We also show that the same results hold for the two-machine flow shop problem with time lags.

On the connection between a cyclic job shop and a reentrant flow shop scheduling problem

Hall et al. (J. Sched. 2002; 5:307---327) investigated the cycle time minimization problem in a two-machine job shop, where each job consists of at most three operations. In this note, we reduce the problem to a two-machine reentrant flow shop problem and briefly discuss some consequences.