Two-machine Flow Shop Scheduling Research Articles

Two-machine flow shop problem with effects of deterioration and learning

In this paper we consider a two-machine flow shop scheduling problem with effects of deterioration and learning. By the effects of deterioration and learning, we mean that the processing time of a job is a function of its execution starting time and its position in a sequence. The objective is to find a sequence that minimizes the total completion time. Optimal solutions are obtained for some special cases. For the general case, several dominance properties and some lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed, which is shown by computational experiments to perform effectively and efficiently in obtaining near-optimal solutions.

A branch-and-bound algorithm for solving a two-machine flow shop problem with deteriorating jobs

In this paper we consider a two-machine flow shop scheduling problem with deteriorating jobs. By a deteriorating job we mean that the job's processing time is an increasing function of its starting time. We model job deterioration as a function that is proportional to a linear function of time. The objective is to find a sequence that minimizes the total completion time of the jobs. For the general case, we derive several dominance properties, some lower bounds, and an initial upper bound by using a heuristic algorithm, and apply them to speed up the elimination process of a branch-and-bound algorithm developed to solve the problem.

Johnson’s rule, composite jobs and the relocation problem

Abstract Two-machine flowshop scheduling to minimize makespan is one of the most well-known classical scheduling problems. Johnson’s rule for solving this problem has been widely cited in the literature. We introduce in this paper the concept of composite job, which is an artificially constructed job with processing times such that it will incur the same amount of idle time on the second machine as that incurred by a chain of jobs in a given processing sequence. This concept due to Kurisu first appeared in 1976 to deal with the two-machine flowshop scheduling problem involving precedence constraints among the jobs. We show that this concept can be applied to reduce the computational time to solve some related scheduling problems. We also establish a link between solving the two-machine flowshop makespan minimization problem using Johnson’s rule and the relocation problem introduced by Kaplan. We present an intuitive interpretation of Johnson’s rule in the context of the relocation problem.

A 3/2-Approximation for the proportionate two-machine flow shop scheduling with minimum delays

We study the two-machine flow shop problem with minimum delays. The problem is known to be strongly NP-hard even in the case of unit processing times and to be approximable within a factor of 2 of the length of an optimal schedule in the general case. The question whether there exists a polynomial-time algorithm with a better approximation ratio has been posed by several researchers but still remains open. In this paper, we improve the above bound to 3/2 for the special case of the problem when both operations of each job have equal processing times (this case of flow shop is known as the proportionate flow shop). Our analysis of the algorithm relies upon a nontrivial generalization of the lower bound established by W. Yu for the case of unit processing times.

Minimizing makespan in a two-machine flowshop scheduling with batching and release time

This paper studies two-machine flowshop scheduling with batching and release time, whose objective is to minimize the makespan. We formulate the scheduling problem as a mixed integer programming model and show that it is a strongly NP-hard problem. We derive a lower bound and develop dynamic programming-based heuristic algorithms to solve the scheduling problem. Computational experiments are carried out to evaluate the performance of the heuristic algorithms. The numerical results show that some of the heuristic algorithms can indeed find effective solutions for the scheduling problem.

Two-machine flow shop scheduling with linear decreasing job deterioration

In this paper, we consider a two-machine flow shop scheduling problem with deteriorating jobs. By a deteriorating job, we mean that the processing time is a decreasing function of its execution start time. A proportional linear decreasing deterioration function is assumed. The objective is to find a sequence that minimizes total completion time. Optimal solutions are obtained for some special cases. For the general case, several dominance properties and some lower bounds are derived to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed to overcome the inefficiency of the branch-and-bound algorithm. Computational results for randomly generated problem instances are presented, which show that the heuristic algorithm effectively and efficiently in obtaining near-optimal solutions.

Flowshop scheduling problems with transportation or deterioration between the batching and single machines

In this paper, we consider two new types of the two-machine flowshop scheduling problems where a batching machine is followed by a single machine. The first type is that normal jobs with transportation between machines are scheduled on the batching and single machines. The second type is that normal jobs are processed on the batching machine while deteriorating jobs are scheduled on the single machine. For the first type, we formulate the problem to minimize the makespan as a mixed integer programming model and prove that it is strongly NP-hard. Furthermore, a heuristic algorithm along with a worst case error bound is derived and the computational experiments are also carried out to verify the effectiveness of the proposed heuristic algorithm. For the second type, the two objectives are considered. For the problem with minimizing the makespan, we find an optimal polynomial algorithm. For the problem with minimizing the sum of completion time, we show that it is strongly NP-hard and propose an optimal polynomial algorithm for its special case.

Schedule execution for two-machine flow-shop with interval processing times

This paper addresses the issue of how to best execute the schedule in a two-phase scheduling decision framework by considering a two-machine flow-shop scheduling problem in which each uncertain processing time of a job on a machine may take any value between a lower and upper bound. The scheduling objective is to minimize the makespan. There are two phases in the scheduling process: the off-line phase (the schedule planning phase) and the on-line phase (the schedule execution phase). The information of the lower and upper bound for each uncertain processing time is available at the beginning of the off-line phase while the local information on the realization (the actual value) of each uncertain processing time is available once the corresponding operation (of a job on a machine) is completed. In the off-line phase, a scheduler prepares a minimal set of dominant schedules, which is derived based on a set of sufficient conditions for schedule domination that we develop in this paper. This set of dominant schedules enables a scheduler to quickly make an on-line scheduling decision whenever additional local information on realization of an uncertain processing time is available. This set of dominant schedules can also optimally cover all feasible realizations of the uncertain processing times in the sense that for any feasible realizations of the uncertain processing times there exists at least one schedule in this dominant set which is optimal. Our approach enables a scheduler to best execute a schedule and may end up with executing the schedule optimally in many instances according to our extensive computational experiments which are based on randomly generated data up to 1000 jobs. The algorithm for testing the set of sufficient conditions of schedule domination is not only theoretically appealing (i.e., polynomial in the number of jobs) but also empirically fast, as our extensive computational experiments indicate.

Bicriteria scheduling of a two-machine flowshop with sequence-dependent setup times

A two-machine flowshop scheduling problem is addressed to minimize setups and makespan where each job is characterized by a pair of attributes that entail setups on each machine. The setup times are sequence-dependent on both machines. It is shown that these objectives conflict, so the Pareto optimization approach is considered. The scheduling problems considering either of these objectives are $$ \mathcal{N}{\wp } - {\text{hard}} $$ , so exact optimization techniques are impractical for large-sized problems. We propose two multi-objective metaheurisctics based on genetic algorithms (MOGA) and simulated annealing (MOSA) to find approximations of Pareto-optimal sets. The performances of these approaches are compared with lower bounds for small problems. In larger problems, performance of the proposed algorithms are compared with each other. Experimentations revealed that both algorithms perform very similar on small problems. Moreover, it was observed that MOGA outperforms MOSA in terms of the quality of solutions on larger problems.

Two-machine flowshop scheduling problems involving a batching machine with transportation or deterioration consideration

This paper considers two scheduling problems for a two-machine flowshop where a single machine is followed by a batching machine. The first problem is that there is a transporter to carry the jobs between machines. The second problem is that there are deteriorating jobs to be processed on the single machine. For the first problem with minimizing the makespan , we formulate it as a mixed integer programming model and then prove that it is strongly NP-hard. A heuristic algorithm is proposed for solving this problem and its worst case performance is analyzed. The computational experiments are carried out and the numerical results show that the heuristic algorithm is effective. For the second problem, we derive the optimal algorithms with polynomial time for minimizing the makespan , the total completion time and the maximum lateness, respectively.

Approximation results for flow shop scheduling problems with machine availability constraints

This paper considers two-machine flow shop scheduling problems with machine availability constraints. When the processing of a job is interrupted by an unavailability period of a machine, we consider both the resumable scenario in which the processing can be resumed when the machine next becomes available, and the semi-resumable scenario in which some portion of the processing is repeated but the job is otherwise resumable. For the problem with several non-availability intervals on the first machine under the resumable scenario, we present a fast ( 3 2 ) -approximation algorithm. For the problem with one non-availability interval under the semi-resumable scenario, a polynomial-time approximation scheme is developed.

A note on the complexity of flow shop scheduling with transportation constraints

This note investigates two-machine flow shop scheduling with transportation constraints to minimize makespan. Recently, Soukhal et al. [A. Soukhal, A. Oulamara, P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research 161 (2005) 32–41] proved that this problem is strongly NP-hard when the capacity of the truck is limited to two or three parts. The considered problem with blocking constraints is also proved to be strongly NP-hard by Soukhal et al. Unfortunately, their proofs contain mistakes. We point out their proofs’ invalidity and then show that, when the capacity of the truck is limited to two parts, the problem is binary NP-hard, and when the capacity of the truck is limited to three parts the problem is strongly NP-hard even if the jobs have a common processing time on machine one and all jobs have the same transportation time. We show also that the last result can be generalized to any fixed c ( c ⩾ 3) parts.

Single-machine and two-machine flowshop scheduling with general learning functions

We show that the O( n log n) (where n is the number of jobs) shortest processing time (SPT) sequence is optimal for the single-machine makespan and total completion time minimization problems when learning is expressed as a function of the sum of the processing times of the already processed jobs. We then show that the two-machine flowshop makespan and total completion time minimization problems are solvable by the SPT sequencing rule when the job processing times are ordered and job-position-based learning is in effect. Finally, we show that when the more specialized proportional job processing times are in place, then our flowshop results apply also in the more general sum-of-job-processing-times-based learning environment.

Single-machine scheduling problems with a learning effect

In many situations, the skills of workers continuously improve when repeating the same or similar tasks. This phenomenon is known as the “learning effect” in the literature. In most studies, the learning phenomenon is implemented by assuming the actual job processing time is a function of its scheduled position [D. Biskup, Single-machine scheduling with learning considerations, Eur. J. Oper. Res. 115 (1999) 173–178]. Recently, a new model is proposed where the actual job processing time depends on the sum of the processing times of jobs already processed [C. Koulamas, G.J. Kyparisis, Single-machine and two-machine flowshop scheduling with general learning functions, Eur. J. Oper. Res. 178 (2007) 402–407]. In this paper, we extend their models in which the actual job processing time not only depends on its scheduled position, but also depends on the sum of the processing times of jobs already processed. We then show that the single-machine makespan and the total completion time problems remain polynomially solvable under the proposed model. In addition, we show that the total weighted completion time has a polynomial optimal solution under certain agreeable solutions.

The no-wait two-machine flow shop scheduling problem with convex resource-dependent processing times

We extend the classical no-wait two-machine flow shop scheduling problem to the case where job-processing times are controllable through the allocation of a common, limited and nonrenewable resource. Our objective is to simultaneously determine the sequence of the jobs and the resource allocation for each job on both machines in order to minimize the makespan. By using the equivalent load method to obtain the optimal resource allocation on a series-parallel graph, we reduce the problem to a sequencing one and show that it is equivalent to a new special case of the Traveling Salesman Problem (TSP). We prove that although the reduced problem forms a subclass of the TSP on permuted Monge matrices, it is still strongly NP-hard. We provide an approximation result and present three special cases which are polynomially solvable. We have also tested two different subtour-patching heuristics in large-scale computational experiments on randomly generated instances of the problem. Both heuristics produced close-to-optimal solutions in most cases.

Approximation algorithms for UET scheduling problems with exact delays

In this paper we consider coupled-task single-machine and two-machine flow shop scheduling problems with exact delays, unit processing times, and the makespan as an objective function. The main results of the paper are fast 7 / 4 - and 3 / 2 -approximation algorithms for solving the single- and two-machine problems, respectively.

Planning Machine Maintenance in Two-Machine Shop Scheduling

We consider the two-machine open shop and two-machine flow shop scheduling problems in which each machine has to be maintained exactly once during the planning period, and the duration of each of these intervals depends on its start time. The objective is to minimize the maximum completion time of all activities to be scheduled. We resolve complexity and approximability issues of these problems. The open shop problem is shown to be polynomially solvable for quite general functions defining the length of the maintenance intervals. By contrast, the flow shop problem is proved binary NP-hard and pseudopolynomially solvable by dynamic programming. We also present a fully polynomial approximation scheme and a fast 3/2-approximation algorithm.

Two-machine flowshop scheduling to minimize mean flow time under simple linear deterioration

A real industrial production phenomenon, referred to as deteriorating jobs, has drawn increasing attention. However, most research on this issue considers only single-machine problems. Motivated by this limitation, this paper considers a simple linear deterioration model in a two-machine flowshop where the objective is to minimize the mean flow time. Several dominance rules and three lower bounds are proposed to speed up the search for an optimal solution, and several heuristic algorithms are provided to derive near-optimal solutions. In addition, a computational experiment is conducted to evaluate their performances. Results indicate that the algorithms perform well, and a combined heuristic algorithm is recommended for practitioners.

Two-machine flow-shop scheduling to minimize total late work

This article considers a two-machine flow-shop scheduling problem of minimizing total late work. Unlike tardiness, which is based upon the difference between the job completion time and the due date, the late work of a job is defined as the amount of work not completed by its due date. This article first shows that the problem remains non-deterministic polynomial time (NP) hard even if all jobs share a common due date. A lower bound and a dominance property are developed to design branch-and-bound algorithms. Computational experiments are conducted to assess the performance of the proposed algorithms. Numerical results demonstrate that the lower bound and dominance rule can help to reduce the computational efforts required by exploring the enumeration tree. The average deviation between the solution found by tabu search and the proposed lower bound is less than 3%, suggesting that the proposed lower bound is close to the optimal solution.

Heuristics for two-machine flowshop scheduling with setup times and an availability constraint

This paper studies the two-machine flowshop scheduling problem with anticipatory setup times and an availability constraint imposed on only one of the machines where interrupted jobs can resume their operations. We present two heuristics and show that their worst-case error bounds are no larger than 2 3 .