Markovian analysis for performance evaluation and scheduling in m machine stochastic flow-shop with buffers of any capacity

This paper deals with performance evaluation and scheduling in m machine static stochastic permutation flow-shop with buffers of any capacity (unlimited, limited or null). The processing time of a given job for a given machine is assumed to be exponentially distributed with a known rate. We propose a theorem which provides recursive scheme based on Markov chains and Chapman–Kolmogorov equations to compute the expected completion time of the last job for any sequence of jobs. This scheme is combined with metaheuristics based on simulated annealing for the scheduling problem. Computational results are given.     

Generalization of Johnson's and Talwar's scheduling rules in two-machine stochastic flow shops

The paper deals with the classical problem of minimizing the makespan in a two-machine flow shop. When the job processing times are deterministic, the optimal job sequence can be determined by applying Johnson's rule. When they are independent and exponential random variables, Talwar's rule yields a job sequence that minimizes the makespan stochastically. Assuming that the random job processing times are independent and Gompertz distributed, we propose a new scheduling rule that is a generalization of both Johnson's and Talwar's rules. We prove that our rule yields a job sequence that minimizes the makespan stochastically. Extensions to m-machine proportionate stochastic flow shops, two-machine stochastic job shops, and stochastic assembly systems are indicated.     

Minimizing maximum completion time in a proportionate flow shop with one machine of different speed

A flow shop with identical machines is called a proportionate flow shop. In this paper, we consider the variant of the n-job, m-machine proportionate flow shop scheduling problem in which only one machine is different and job processing times are inversely proportional to machine speeds. The objective is to minimize maximum completion time. We describe some optimality conditions and show that the problem is NP-complete. We provide two heuristic procedures whose worst-case performance ratio is less than two. Extensive experiments with various sizes are conducted to show the performance of the proposed heuristics.     

A branch and bound algorithm for the two-stage assembly scheduling problem

This paper develops a branch and bound algorithm for the two-stage assembly scheduling problem. In this problem, there are m machines at the first stage, each of which produces a component of a job. When all m components are available, a single assembly machine at the second stage completes the job. The objective is to schedule the jobs on the machines so that the maximum completion time, or makespan, is minimized. A lower bound based on solving an artificial two-machine flow shop problem is derived. Also, several dominance theorems are established and incorporated into the branch and bound algorithm. Computational experience with the algorithm is reported for problems with up to 8000 jobs and 10 first-stage machines.     

An empirical analysis of the optimality rate of flow shop heuristics

If a certain optimization problem is NP-hard or even harder, one could expect that the chances of solving it optimally should rather decrease with an increase of the problem size. We reveal, however, that the opposite occurs for a strongly NP-hard problem, which requires sequencing n jobs through an m machine flow shop so as to minimize the makespan. In particular, we empirically examine optimality rates (the probability of being optimal) of the famous NEH heuristic of Nawaz et al. [Nawaz, M., Enscore, Jr., E., Ham, I., 1983. A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega, The International Journal of Management Science 11, 91–95] and two improved versions of NEH. By using millions of simulation trials and a new effective lower bound on the shortest makespan, we observe relatively high optimality rates of the three heuristics for small values of m. Rather surprisingly, for larger values of n, the heuristics become more frequently optimal as n increases. Neither theoretical nor empirical studies of optimality rates of flow shop heuristics have been conducted so far, and – to the best of our knowledge – no similar studies are known in the field of operations research.     

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).     

NP-hard cases in scheduling deteriorating jobs on dedicated machines

In this paper problems of time-dependent scheduling on dedicated machines are considered. The processing time of each job is described by a function which is dependent on the starting time of the job. The objective is to minimise the maximum completion time (makespan). We prove that under linear deterioration the two-machine flow shop problem is strongly NP-hard and the two-machine open shop problem is ordinarily NP-hard. We show that for the three-machine flow shop and simple linear deterioration there does not exist a polynomial-time approximation algorithm with the worst case ratio bounded by a constant, unless P=NP. We also prove that the three-machine open shop problem with simple linear deterioration is ordinarily NP-hard, even if the jobs have got equal deterioration rates on the third machine.     

Parallel machine scheduling with machine availability and eligibility constraints

In this paper we consider the problem of scheduling n independent jobs on m identical machines incorporating machine availability and eligibility constraints while minimizing the makespan. Each machine is not continuously available at all times and each job can only be processed on specified machines. A network flow approach is used to formulate this scheduling problem into a series of maximum flow problems. We propose a polynomial time binary search algorithm to either verify the infeasibility of the problem or solve it optimally if a feasible schedule exists.     

Flow shop scheduling algorithms for minimizing the completion time variance and the sum of squares of completion time deviations from a common due date

We consider two problems of m-machine flow shop scheduling in this paper: one, with the objective of minimizing the variance of completion times of jobs, and the other with the objective of minimizing the sum of squares of deviations of job completion times from a common due date. Lower bounds on the sum of squares of deviations of job completion times from the mean completion time of jobs for a given partial sequence are first presented. Using these lower bounds, a branch and bound algorithm based on breadth-first search procedure for scheduling n jobs on m-machines with the objective of minimizing completion time variance (CTV) is developed to obtain the best permutation sequence. We also present two lower bounds and thereafter, a branch and bound algorithm with the objective of minimizing the sum of squares of deviations of job completion times from a given common due date (called the MSD problem). The computational experience with the working of the two proposed branch and bound algorithms is also reported. Two heuristics, one for each of the two problems, are developed. The computational experience on the evaluation of the heuristics is discussed.     

Heuristics for scheduling in a flow shop with multiple processors

This study investigates the performance of scheduling heuristics in a flow shop with multiple processors. We investigated five better performing flow shop heuristics for their performances of makespan and mean flow time criteria in a flow shop with multiple processors. The study examined the effects of problem characteristics (number of jobs, number of machine stages and number of parallel processors at each stage) and the performance of heuristics using regression analysis. We found that although structural characteristics explain most of the variation in performance, heuristics also had an effect. The experimental results showed that flow shop heuristics developed by Nawaz, Enscore, and Ham and that of Ho were comparable in performance in a flow shop with multiple processors. However, the former was slightly more consistent in results for both criteria.     

Minimizing makespan in a two-stage hybrid flow shop scheduling problem with open shop in one stage

This paper considers a scheduling problem in two-stage hybrid flow shop, where the first stage consists of two machines formed an open shop and the other stage has only one machine. The objective is to minimize the makespan, i.e., the maximum completion time of all jobs. We first show the problem is NP-hard in the strong sense, then we present two heuristics to solve the problem. Computational experiments show that the combined algorithm of the two heuristics performs well on randomly generated problem instances.     

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.     

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.     

On no-wait and no-idle flow shops with makespan criterion

The paper deals with the NP-hard problems of minimizing the makespan in m-machine no-wait and no-idle permutation flow shops. We identify networks whose longest path lengths represent the makespans. These networks reveal the duality between the two problems, and show graphical explanations of the fact that under no-wait and no-idle conditions the makespan can be a decreasing function of some job processing times. Moreover, they also lead to a natural reduction of the no-wait flow shop problem to the traveling salesman problem, some lower bounds on the shortest makespan, and new efficiently solvable special cases.     

The Two-stage Assembly Flow Shop Scheduling with an Availability Constraint: Worst Case Analysis

In this paper, we are interested in handling the limited availability of machines in the two-stage assembly flow shop scheduling problems. Emphasis is put on the semiresumable case with respect to the minimization of the makespan. We provide, when possible, heuristics with a tight worst-case ratio bound of 2.     

A genetic algorithm for flow shop scheduling problems

Many scheduling problems are NP-hard problems. For such NP-hard combinatorial optimization problems, heuristics play a major role in searching for near-optimal solutions. In this paper we develop a genetic algorithm-based heuristic for the flow shop scheduling problem with makespan as the criterion. The performance of the algorithm is compared with the established NEH algorithm. Computational experience indicates that genetic algorithms can be good techniques for flowshop scheduling problems.     

重入排序问题综述

重入排序问题打破传统假设：工件在加工过程中不止一次地访问某台机器,是一种新型的排序问题. 重入的特点源于半导体生产, 并广泛存在于其他领域. 对重入排序问题已有文献中的成果进行梳理和分析,按问题所处机器环境的不同, 对内容和方法进行分类介绍和总结：包括单机问题、流水作业问题、混合流水作业问题及其他机器环境下的重入排序问题. 最后展望未来的趋势和研究方向.     

Constraint-Based Job Shop Scheduling with IILOG SCHEDULER

We introduce constraint-based scheduling and discuss its main principles. An approximation algorithm based on tree search is developed for the job shop scheduling problem using ILOG SCHEDULER. A new way of calculating lower bounds on the makespan of the job shop scheduling problem is presented and we show how such results can be used within a constraint-based approach. An empirical performance analysis shows that the algorithm we developed performs well. Finally, taking the job shop scheduling problem as a start point, we discuss how constraint-based scheduling can be used to solve more general scheduling problems.     

Minimizing the makespan in a two-machine cross-docking flow shop problem

This paper studies a two-machine cross-docking flow shop scheduling problem in which a job at the second machine can be processed only after the processing of some jobs at the first machine has been completed. The objective is to minimize the makespan. We first show that the problem is strongly NP-hard. Some polynomially solvable special cases are provided. We then develop a polynomial approximation algorithm with an error-bound analysis. A branch-and-bound algorithm is also constructed. Computational results show that the branch-and-bound algorithm can optimally solve problems with up to 60 jobs within a reasonable amount of time.     

Complexity of flow shop scheduling problems with transportation constraints

In most manufacturing and distribution systems, semi-finished jobs are transferred from one processing facility to another by transporters such as Automated Guided Vehicles, robots and conveyors, and finished jobs are delivered to warehouses or customers by vehicles such as trucks.This paper investigates two-machine flow shop scheduling problems taking transportation into account. The finished jobs are transferred from the processing facility and delivered to customers by truck. Both transportation capacity and transportation times are explicitly taken into account in these models. We study the class of flow shop problems by analysing their complexity. For the makespan objective function, we prove that this problem is strongly NP-hard when the capacity of a truck is limited to two or three parts with an unlimited buffer at the output of the each machine. This problem with additional constraints, such as blocking, is also proven to be strongly NP-hard.