Approximation algorithms for two-machine flow shop scheduling with batch setup times

In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

The flow shop problem with no-idle constraints: A review and approximation

The makespan minimization problem in flow shops with no-idle constraints on machines is considered. The latter means that each machine, once started, must process all its operations without intermediate idle time until all those operations are completed. The problem is known to be strongly NP-hard already for three machines. While being based on a geometrical approach, we propose several polynomial time heuristics (for the general case and for special cases of 3 and 4 machines) which provide asymptotically optimal solutions for the increasing number of jobs. A comprehensive review of relevant results is also presented.     

Routing open shop and flow shop scheduling problems

We consider a generalization of the classical open shop and flow shop scheduling problems where the jobs are located at the vertices of an undirected graph and the machines, initially located at the same vertex, have to travel along the graph to process the jobs. The objective is to minimize the makespan. In the tour-version the makespan means the time by which each machine has processed all jobs and returned to the initial location. While in the path-version the makespan represents the maximum completion time of the jobs. We present improved approximation algorithms for various cases of the open shop problem on a general graph, and the tour-version of the two-machine flow shop problem on a tree. Also, we prove that both versions of the latter problem are NP-hard, which answers an open question posed in the literature.     

Outsourcing and scheduling for two-machine ordered flow shop scheduling problems

This paper considers a two-machine ordered flow shop problem, where each job is processed through the in-house system or outsourced to a subcontractor. For in-house jobs, a schedule is constructed and its performance is measured by the makespan. Jobs processed by subcontractors require paying an outsourcing cost. The objective is to minimize the sum of the makespan and the total outsourcing cost. Since this problem is NP-hard, we present an approximation algorithm. Furthermore, we consider three special cases in which job j has a processing time requirement p_j, and machine i a characteristic q_i. The first case assumes the time job j occupies machine i is equal to the processing requirement divided by a characteristic value of machine i, that is, p_j/q_i. The second (third) case assumes that the time job j occupies machine i is equal to the maximum (minimum) of its processing requirement and a characteristic value of the machine, that is, max{p_j, q_i} (min{p_j, q_i}). We show that the first and the second cases are NP-hard and the third case is polynomially solvable.     

Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration

In this paper, we address a two-machine flow shop scheduling problem under simple linear deterioration. By a simple linear deterioration function, we mean that the processing time of a job is a simple linear function of its execution start time. The objective is to find a sequence that minimizes total weighted completion time. Optimal schedules are obtained for some special cases. For the general case, several dominance properties and two 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 analysis on randomly generated problems is conducted to evaluate the branch-and-bound algorithm and heuristic algorithm.     

Approximability of flow shop scheduling

Shop scheduling problems are notorious for their intractability, both in theory and practice. In this paper, we construct a polynomial approximation scheme for the flow shop scheduling problem with an arbitrary fixed number of machines. For the three common shop models (open, flow, and job), this result is the only known approximation scheme. Since none of the three models can be approximated arbitrarily closely in the general case (unless P = NP), the result demonstrates the approximability gap between the models in which the number of machines is fixed, and those in which it is part of the input of the instance. The result can be extended to flow shops with job release dates and delivery times and to flow shops with a fixed number of stages, where the number of machines at any stage is fixed. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.A preliminary version of this paper appeared in theProceedings of the 36th Annual IEEE Symposium on the Foundations of Computer Science, 1995.Research supported by NSF grant DMI-9496153.     

Two-machine flow shop scheduling problems with no-wait jobs

In this paper we provide a fairly complete complexity classification of various versions of the two-machine permutation flow shop scheduling problem to minimize the makespan in which some of the jobs have to be processed with no-wait in process. For some version, we offer a fully polynomial-time approximation scheme and a -approximation algorithm.     

A note on the two-stage assembly flow shop scheduling problem with uniform parallel machines

We study the problem of minimizing the makespan in a two-stage assembly flow shop scheduling problem with uniform parallel machines. This problem is a generalization of the assembly flow shop problem with concurrent operations in the first stage and a single assembly operation in the second stage. We propose a heuristic with an absolute performance bound which becomes asymptotically optimal as the number of jobs becomes very large. We show that our results slightly improve earlier results for the simpler assembly flow shop problem (without uniform machines) and for the two-stage hybrid flow shop problem with uniform machines.     

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.     

The two-machine open shop problem: To fit or not to fit, that is the question

We consider the open shop scheduling problem with two machines. Each job consists of two operations, and it is prescribed that the first (second) operation has to be executed by the first (second) machine. The order in which the two operations are scheduled is not fixed, but their execution intervals cannot overlap. We are interested in the question whether, for two given values D₁ and D₂, there exists a feasible schedule such that the first and second machine process all jobs during the intervals [0,D₁] and [0,D₂], respectively.We formulate four simple conditions on D₁ and D₂, which can be verified in linear time. These conditions are necessary and sufficient for the existence of a feasible schedule. The proof of sufficiency is algorithmical, and yields a feasible schedule in linear time. Furthermore, we show that there are at most two non-dominated points (D₁,D₂) for which there exists a feasible schedule.     

The hybrid flow shop scheduling problem

The scheduling of flow shops with multiple parallel machines per stage, usually referred to as the hybrid flow shop (HFS), is a complex combinatorial problem encountered in many real world applications. Given its importance and complexity, the HFS problem has been intensively studied. This paper presents a literature review on exact, heuristic and metaheuristic methods that have been proposed for its solution. The paper briefly discusses and reviews several variants of the HFS problem, each in turn considering different assumptions, constraints and objective functions. Research opportunities in HFS are also discussed.     

Multi-family scheduling in a two-machine reentrant flow shop with setups

This paper considers a two-machine multi-family scheduling problem with reentrant production flows. The problem consists of two machines, M₁ and M₂, and each job has the processing route (M₁, M₂, M₁, M₂). There are identical jobs in the same family and the jobs in the same family are processed in succession. Each machine needs a setup time before the first job in a family is processed. The objective is to minimize the maximum completion time. Examples of such a problem occur in the bridge construction, semiconductor industry and job processing on numerical controlled machines, where they usually require that the jobs are reprocessed once and there are identical jobs in the same family. This problem is shown to be NP-hard. A branch-and-bound algorithm is proposed, and computational experiments are provided.     

A note on flow shop scheduling problems with deteriorating jobs on no-idle dominant machines

The main results in a recent paper [M. Cheng, S. Sun, L. He, Flow shop scheduling problems with deteriorating jobs on no-idle dominant machines, European Journal of Operational Research 183 (2007) 115–124] are incorrect because job processing times are variable due to deteriorating effect, which is not taken into account by the authors. In this note, we show first by counter-examples that the published results are incorrect, and then we provide corrected results.     

Approximative procedures for no-wait job shop scheduling

In this article we consider the no-wait job shop problem with makespan objective. Based on a decomposition of the problem into a sequencing and a timetabling problem, we propose two local search algorithms. Extensive computational tests in which the algorithms compare favorably to the best existing strategies are reported. Although not specifically designed for that purpose, our algorithms also outperform one of the best no-wait flow shop algorithms in literature.     

An approximation algorithm for the m-machine permutation flow shop scheduling problem with controllable processing times

The paper deals with the m-machine permutation flow shop scheduling problem in which job processing times, along with a processing order, are decision variables. It is assumed that the cost of processing a job on each machine is a linear function of its processing time and the overall schedule cost to be minimized is the total processing cost plus maximum completion time cost. A algorithm for the problem with m = 2 is provided; the best approximation algorithm until now has a worst-case performance ratio equal to . An extension to the m-machine (m ≥2) permutation flow shop problem yields an approximation algorithm with a worst-case bound equal to

Full-size image

, where is the worst-case performance ratio of a procedure used, in the proposed algorithm, for solving the (pure) sequencing problem. Moreover, examples which achieve this bound for = 1 are also presented.     

Two-machine flow shop total tardiness scheduling problem with deteriorating jobs

This paper studies a two-machine scheduling problem with deteriorating jobs which their processing times depend on their waiting time. We develop a branch and bound algorithm to minimize the total tardiness criteria. A lower bound, several dominance properties and an initial upper bound derived from a heuristic algorithm are used to increase the speed of branch and bound algorithm and decrease its required memory space. Computational results are presented to evaluate effectiveness and efficiency of the algorithms.     

An O(n) algorithm for the variable common due date,minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines

We consider the ordinary NP- hard two-machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs. We present an O(n²) algorithm for the problem when the machines are ordered, that is, when each job has its smaller processing time on the first (second) machine. We also discuss the applicability of the proposed algorithm to the corresponding single-objective problem in which the common due date is given.     

Bicriteria hierarchical optimization of two-machine flow shop scheduling problem with time-dependent deteriorating jobs

We study a two-machine flowshop scheduling problem with time-dependent deteriorating jobs, i.e. the processing times of jobs are an increasing function of their starting time. The objective is to minimize the total completion time subject to minimum makespan. We propose a mixed integer programming model, and develop two pairwise interchange algorithms and a branch-and-bound procedure to solve the problem while using several dominance conditions to limit the size of the search tree. Several polynomial-time solvable special cases are discussed. Finally, numerical studies are performed to examine the effectiveness and the efficiency of the proposed algorithms.     

A note on the proportionate flow shop with a bottleneck machine

Choi, B.-C., Yoon, S.-H., Chung, S.-J., 2007. Minimizing maximum completion time in a proportionate flow shop with one machine of different speed. European Journal of Operational Research 176, 964–974 consider the proportionate flow shop with a slow bottleneck machine and propose the SLDR heuristic for it. Choi et al. (2007) derive a data-dependent worst-case ratio bound for the SLDR heuristic which is then bounded by two. In this note, we show that the tight worst-case ratio bound of the SLDR heuristic is 3/2.     

Flow shop scheduling with heterogeneous workers

We propose an extension to the flow shop scheduling problem named Heterogeneous Flow Shop Scheduling Problem (Het-FSSP), where two simultaneous issues have to be resolved: finding the best worker assignment to the workstations, and solving the corresponding scheduling problem. This problem is motivated by Sheltered Work centers for Disabled, whose main objective is the labor integration of persons with disabilities, an important aim not only for these centers but for any company desiring to overcome the traditional standardized vision of the workforce. In such a scenario the goal is to maintain high productivity levels by minimizing the maximum completion time, while respecting the diverse capabilities and paces of the heterogeneous workers, which increases the complexity of finding an optimal schedule. We present a mathematical model that extends a flow shop model to admit a heterogeneous worker assignment, and propose a heuristic based on scatter search and path relinking to solve the problem. Computational results show that this approach finds good solutions within a short time, providing the production managers with practical approaches for this combined assignment and scheduling problem.