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.     

A New Heuristic Method for the Permutation Flow Shop Scheduling Problem

A new heuristic method for the permutation flow shop scheduling problem is presented and compared with two other heuristics named NEH and SPIRIT. The new heuristic method is based on a property of the scheduling problem that provides an upper bound on the idle time of the last machine between any two adjacent jobs regardless of their position in the sequence of jobs. The results from computational experience have shown that the new heuristic outperforms, in solution quality, all others for problems having up to 50 jobs and 30 machines.     

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

In this paper, we consider the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. For this strongly NP-hard problem, we develop and discuss different constructive heuristic algorithms. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is evaluated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimate of mean flow time. We observe that the recommendation of an appropriate constructive algorithm strongly depends on the ratio n/m.     

Complexity of shop-scheduling problems with fixed number of jobs: a survey

The paper surveys the complexity results for job shop, flow shop, open shop and mixed shop scheduling problems when the number n of jobs is fixed while the number r of operations per job is not restricted. In such cases, the asymptotical complexity of scheduling algorithms depends on the number m of machines for a flow shop and an open shop problem, and on the numbers m and r for a job shop problem. It is shown that almost all shop-scheduling problems with two jobs can be solved in polynomial time for any regular criterion, while those with three jobs are NP-hard. The only exceptions are the two-job, m-machine mixed shop problem without operation preemptions (which is NP-hard for any non-trivial regular criterion) and the n-job, m-machine open shop problem with allowed operation preemptions (which is polynomially solvable for minimizing makespan).     

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.     

Decomposition approaches in permutation scheduling problems with application to the M-machine flow shop scheduling problems

In this article, we consider three decomposition techniques for permutation scheduling problems. We introduce a general iterative decomposition algorithm for permutation scheduling problems and apply it to the permutation flow shop scheduling problem. We also develop bounds needed for this iterative decomposition approach and compare its computational requirements to that of the traditional branch and bound algorithms. Two heuristic algorithms based on the iterative decomposition approach are also developed. extensive numerical study indicates that the heuristic algorithms are practical alternatives to very costly exact algorithms for large flow shop scheduling problems.     

具有机器适用限制的分布式置换流水车间问题的模型与算法

考虑具有机器适用限制的多个不同置换流水车间的调度问题. 机器适用限制指的是每个工件只能分配到其可加工工厂集合. 所有置换流水车间拥有的机器数相同但是具有不同的加工能力. 首先, 针对该问题建立了基于位置的混合整数线性规划模型; 进而, 对一般情况和三种特殊情况给出了具有较小近似比的多项式时间算法. 其次, 基于NEH方法提出了启发式算法NEHg, 并给出了以NEHg为上界的分支定界算法. 最后, 通过例子说明了NEHg启发式算法和分支定界算法的计算过程, 并进行大量的实验将NEHg与NEH算法结果进行比较, 从而验证了NEHg算法的有效性.     

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 variable neighbourhood search for hybrid flow-shop scheduling problem with rework and set-up times

This paper deals with hybrid flow-shop scheduling problem with rework. In this problem, jobs are inspected at the last stage, and poorly processed jobs were returned and processed again. Thus, a job may visit a stage more than once, and we have a hybrid flow-shop with re-entrant flow. This kind of a shop may occur in many industries, such as final inspection system in automotive manufacturing. The criterion is to minimize the makespan of the system. We developed a 0–1 mixed-integer program of the problem. Since the hybrid flow-shop scheduling problem is NP-hard, an algorithm for finding an optimal solution in polynomial time does not exist. So we generalized some heuristic methods based on several basic dispatching rules and proposed a variable neighbourhood search (VNS) for the problem with sequence-dependent set-up times and unrelated parallel machines. The computational experiments show that VNS provides better solutions than heuristic methods.     

A high quality solution constructive heuristic for flow shop sequencing

This paper addresses the flow shop sequencing problem. Following an investigation of the problem characteristics, a property of this scheduling problem is presented, and is used for the development of a new constructive heuristic with the objective of minimizing the total time to complete the schedule (makespan). The new method, denoted by N&M, is compared with the best constructive heuristic reported in the literature, named NEH. Results from computational experience have shown that for problems having up to 10 machines and 100 jobs, the new heuristic outperforms, on average, the NEH heuristic. There is no significant difference regarding computation effort for both NEH and N&M heuristics.     

Approximation results in parallel machnies stochastic scheduling

We consider scheduling a batch of jobs with stochastic processing times on parallel machines. We derive various new formulae for the expected flowtime and weighted flowtime under general scheduling rules. Smith's Rule, which orders job starts by decreasing ratio of weight to expected processing time provides a natural heuristic for this problem. We obtain a bound on the worst case difference between the expected weighted flow time under Smith's Rule and under an optimal policy. For a wide class of processing time distributions, this bound is of oderO(1) and does not increase with the number of jobs.This research was supported in part by NSF Grant ECS-8712798.     

Realistic two-stage flowshop batch scheduling problems with transportation capacity and times

This paper investigates single-batch and batch-single flow shop scheduling problem taking transportation among machines into account. Both transportation capacity and transportation times are explicitly considered. While the single processing machine processes one job at a time, the batch processing machine processes a batch of jobs simultaneously. The batch processing time is the longest processing times of jobs assigned to that batch.Each problem is formulated as a mixed integer programming model to find optimal makespan. Lower bounds and heuristic algorithms are proposed and computational experiments are carried out to verify their effectiveness.     

Two-stage open shop scheduling with a bottleneck machine

It is known that for the open shop scheduling problem to minimize the makespan there exists no polynomial-time heuristic algorithm that guarantees a worst-case performance ratio better than 5/4, unless P≠NP. However, this result holds only if the instance of the problem contains jobs consisting of at least three operations. This paper considers the open shop scheduling problem, provided that each job consists of at most two operations, one of which is to be processed on one of the m⩾2 machines, while the other operation must be performed on the bottleneck machine, the same for all jobs. For this NP-hard problem we present a heuristic algorithm and show that its worst-case performance ratio is 5/4.     

The two-machine total completion time flow shop problem

In this paper we study the NP-hard scheduling problem of minimizing total completion time in a two-machine flow shop. Five known lower bounds are discussed and two new ones are presented. A new dominance criterion is also proposed. Several versions of a branch and bound method are derived by applying, both individually and combined, these lower bounds. A heuristic procedure is also presented that uses a constructive O(n²) time method, which computes a good starting solution, together with a neighborhood search based on pairwise interchanges. Computational results show that the exact method can handle problems of up to 30 jobs in size within a reasonable amount of time and that the heuristic procedure has an average error of less than 0.5% from the optimal value and less than 2.7% from the lower bound.     

一种求解延迟工件数最小的混合流水车间调度问题的模拟退火算法

针对延迟工件数最小的混合流水车间调度问题,给出了一种改进的模拟退火求解算法. 该算法首先给出一个启发式算法来获得初始解,然后用模拟退火算法对初始解改进. 通过交换工件在第一阶段的排序来获得一个新的解,采用最先空闲设备分配规则和先到先被加工规则,对工件在剩余各级的工序进行调度. 实验仿真表明算法是可行有效的.     

An improved heuristic for two-machine flow shop scheduling with an availability constraint and nonresumable jobs

In this paper, we deal with the two-machine flow shop scheduling problem having an unavailability interval on the first machine, and nonresumable jobs. We first present an enhancement procedure that, once applied to any arbitrary solution, produces a schedule that is at most equal 2 times the optimal makespan. We then develop an improved heuristic, with a relative worst-case error of 3/2.     

Scheduling flexible machining and assembly systems

Most scheduling papers consider flexible machining and assembly systems as being independent. In this paper, a heuristic two-level scheduling algorithm for a system consisting of a machining and an assembly subsystem is developed. It is shown that the upper level problem is equivalent to the two machine flow shop problem. The algorithm at the lower level schedules jobs according to the established product and part priorities. Related issues, such as batching, due dates, process planning and alternative routes, are discussed. The algorithm and associated concepts are illustrated on a number of numerical examples.     

Flexible flow shop scheduling with uniform parallel machines

We consider the multistage flexible flow shop scheduling problem with uniform parallel machines in each stage and the objective of minimizing makespan. We develop a general class of heuristics for this strongly NP-hard problem that extend several well-known heuristics for the corresponding embedded serial flow shop problem, and obtain absolute performance guarantees for heuristics in this class by building on similar absolute performance guarantees for the corresponding serial flow shop heuristics. Our approach is quite robust, since it can extend any heuristic for the serial flow shop problem (with an absolute performance guarantee) to a similar one for the flexible flow shop problem with uniform parallel machines.     

Two-machine open shop scheduling with special transportation times

The paper considers a problem of scheduling n jobs in a two-machine open shop to minimise the makespan, provided that preemption is not allowed and the interstage transportation times are involved. In general, this problem is known to be NP-hard. We present a linear time algorithm that finds an optimal schedule if no transportation time exceeds the smallest of the processing times. We also describe an algorithm that creates a heuristic solution to the problem with job-independent transportation times. Our algorithm provides a worst-case performance ratio of 8/5 if the transportation time of a job depends on the assigned processing route. The ratio reduces to 3/2 if all transportation times are equal.     

A penalty-shift-insertion-based algorithm to minimize total flow time in no-wait flow shops

This paper proposes a penalty-shift-insertion (PSI)-based algorithm for the no-wait flow shop scheduling problem to minimize total flow time. In the first phase, a penalty-based heuristic, derived from Vogel’s approximation method used for the classic transportation problem is used to generate an initial schedule. In the second phase, a known solution is improved using a forward shift heuristic. Then the third phase improves this solution using a job-pair and a single-job insertion heuristic. Results of the computational experiments with a large number of randomly generated problem instances show that the proposed PSI algorithm is relatively more effective and efficient in minimizing total flow time in a no-wait flow shop than the state-of-the-art procedures. Statistical significance of better results obtained by the proposed algorithm is also reported.