Two-machine Job Shop Scheduling Research Articles

Two-machine job-shop scheduling with one joint job

In this research, we consider a two-machine job-shop scheduling problem with one joint job where a joint job is defined as a job whose operations are to be processed by different machines. The machine repetition and transportation times between machines are also considered. This problem is associated with real-world applications in the fields of production planning and supply chains. We first demonstrate that this problem is NP-hard in the strong sense. We then propose polynomial-time algorithms based on dynamic programming for cases with a fixed number of jobs. We also propose lemmas to reduce the number of states in dynamic programming without loss of optimality, so that the time complexity is improved. Other methodologies, including preprocessing and the two-pointers method, are also embedded to improve the algorithms. The proposed algorithms have better time complexities than those of variants of a well-known algorithm when applied to the cases of our problem.

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.

Decentralization cost in two-machine job-shop scheduling with minimum flow-time objective

A decentralized two-machine job-shop system is considered, where each machine minimizes its own flow-time objective. Analyzing the system as a non-cooperative game, we investigate the Decentralization Cost (DC), the ratio in terms of the system flow-time between the best Nash equilibrium and the centralized solution. Settings generating significant inefficiency are identified and discussed. We provide bounds on the maximal DC, and prove they are tight for two-job problems. For larger problems, we use a cross entropy meta-heuristic that searches for DC maximizing job durations. This supports the tightness of the proposed bounds for a flow-shop. Additionally, for a flow-shop, a simple, scheduling-based mechanism is proposed, which always generates efficiency.

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.

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.

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.

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.

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.

SEQUENCE-DEPENDENT SETUP AND CLEAN-UP TIMES IN A TWO-MACHINE JOB-SHOP WITH MINIMIZING MAKESPAN

This article addresses the job-shop scheduling problem of minimizing the length of a schedule (makespan) for processing n jobs by one or two machines with sequence-dependent setup times and clean-up times. The processing of each job includes at most two operations that have to be non-preemptive. Machine routes may differ from job to job. If all setup and clean-up times are equal to zero, this problem is polynomially solvable via Jackson's pair of job permutations, otherwise it is NP-hard even if each of n jobs consists of one operation on the same machine. We present sufficient conditions when Jackson's pair of job permutations may be used for solving the two-machine job-shop scheduling problem with sequence-dependent setup times and clean-up times. For the general case of the latter problem, the results obtained provide polynomial lower and upper bounds for the objective function which may be used in an implicit enumeration technique, e.g., in a branch-and-bound algorithm.

On-line and off-line preemptive two-machine job shop scheduling

On-line and off-line preemptive two-machine job shop scheduling

Optimal Scheduling for a Two-Machine Robotic Cell of Jobshop Type

In this paper we consider a scheduling problem for a two-machine robotic cell of jobshop type. The problem is an extension of the classical two-machine jobshop scheduling addressed by Jackson 6), but is NP-hard. We propose a greedy heuristic algorithm and test for its performance with numerical experiments.