A contribution to the stochastic flow shop scheduling problem

This paper deals with performance evaluation and scheduling problems in m machine stochastic flow shop with unlimited buffers. The processing time of each job on each machine is a random variable exponentially distributed with a known rate. We consider permutation flow shop. The objective is to find a job schedule which minimizes the expected makespan. A classification of works about stochastic flow shop with random processing times is first given. In order to solve the performance evaluation problem, we propose a recursive algorithm based on a Markov chain to compute the expected makespan and a discrete event simulation model to evaluate the expected makespan. The recursive algorithm is a generalization of a method proposed in the literature for the two machine flow shop problem to the m machine flow shop problem with unlimited buffers. In deterministic context, heuristics (like CDS [Management Science 16 (10) (1970) B630] and Rapid Access [Management Science 23 (11) (1977) 1174]) and metaheuristics (like simulated annealing) provide good results. We propose to adapt and to test this kind of methods for the stochastic scheduling problem. Combinations between heuristics or metaheuristics and the performance evaluation models are proposed. One of the objectives of this paper is to compare the methods together. Our methods are tested on problems from the OR-Library and give good results: for the two machine problems, we obtain the optimal solution and for the m machine problems, the methods are mutually validated.     

A New Heuristic for a Single Machine Scheduling Problem with Set-up Times

This paper examines the problem of scheduling jobs on a single machine with set-up times. The jobs are divided into mutually exclusive classes and a set-up task is required when processing switches from a job of one class to a job of another class. The set-up times are assumed to be sequence independent. A number of necessary conditions for a schedule to minimize mean flow time have previously been stated, but do not uniquely define the optimal solution, and the problem is apparently NP-complete. We propose a new polynomial-time heuristic, based on these conditions, and compare its performance with some existing heuristics.     

A graph-theoretic approach to interval scheduling on dedicated unrelated parallel machines

We study the problem of scheduling n non-preemptive jobs on m unrelated parallel machines. Each machine can process a specified subset of the jobs. If a job is assigned to a machine, then it occupies a specified time interval on the machine. Each assignment of a job to a machine yields a value. The objective is to find a subset of the jobs and their feasible assignments to the machines such that the total value is maximized. The problem is NP-hard in the strong sense. We reduce the problem to finding a maximum weight clique in a graph and survey available solution methods. Furthermore, based on the peculiar properties of graphs, we propose an exact solution algorithm and five heuristics. We conduct computer experiments to assess the performance of our and other existing heuristics. The computational results show that our heuristics outperform the existing heuristics.     

Scheduling coupled-operation jobs with exact time-lags

Scheduling coupled-operation jobs with exact time-lags on a single machine has a wide range of applications. In that problem, each job consists of two operations with given processing times, which should be scheduled on a single machine observing a given time-lag. The general case of the problem with arbitrary processing times of operations and arbitrary time lags is known to be NP-hard in the strong sense and the problem remains NP-hard for many special cases. In order to develop a local search algorithm for the problem, we first explore two possible approaches for representing feasible solutions and their neighborhoods based on maintaining a permutation of first operations of the jobs or maintaining a full permutation of all operations. The first representation appears to be unpromising since, as we prove, the problem of finding an optimal sequence of second operations for a fixed sequence of first operations is NP-hard in the strong sense even in the case of unit processing times. We elaborate the second approach by developing a tabu search heuristic based on efficient job re-insertion. Empirical evaluation demonstrates superiority of the developed algorithm in comparison with the earlier published algorithms.     

Sequencing and due-date determination in the stochastic single machine problem with earliness and tardiness costs

This paper studies the problem of simultaneous due-date determination and sequencing of a set of n jobs on a single machine where processing times are random variables and job earliness and tardiness costs are distinct. The objective is to determine the optimal sequence and the optimal due-dates which jointly minimize the expected total earliness and tardiness cost. We present an analytical approach to determine optimal due-dates, and propose two efficient heuristics of order O(n log n) to find candidates for the optimal sequence. It is demonstrated that variations in processing times increase cost and affect sequencing and due-date determination decisions. Our illustrative examples as well as computational results show that the proposed model produces optimal sequences and optimal due-dates that are significantly different from those provided by the classical deterministic single machine models. Furthermore, our computational experiments reveal that the proposed heuristics perform well in providing either optimal sequences or good candidates with low overcosts.     

Single machine earliness and tardiness scheduling

We examine the problem of scheduling a given set of jobs on a single machine to minimize total early and tardy costs without considering machine idle time. We decompose the problem into two subproblems with a simpler structure. Then the lower bound of the problem is the sum of the lower bounds of two subproblems. A lower bound of each subproblem is obtained by Lagrangian relaxation. Rather than using the well-known subgradient optimization approach, we develop two efficient multiplier adjustment procedures with complexity O(nlog n) to solve two Lagrangian dual subproblems. A branch-and-bound algorithm based on the two efficient procedures is presented, and is used to solve problems with up to 50 jobs, hence doubling the size of problems that can be solved by existing branch-and-bound algorithms. We also propose a heuristic procedure based on the neighborhood search approach. The computational results for problems with up to 3 000 jobs show that the heuristic procedure performs much better than known heuristics for this problem in terms of both solution efficiency and quality. In addition, the results establish the effectiveness of the heuristic procedure in solving realistic problems to optimality or near optimality.     

工件具有子工件工期的排序问题

研究了工件具有子工件工期的排序问题.需要在一台单机上加工若干个给定的工件.每个工件由若干个子工件组成,每个子工件都有各自的工期.只有当工件的每个子工件都按时完成,才能称该工件是按时完工工件,否则,称该工件产生延误.目标是最大化按时完工的工件个数.证明当每个工件都被分成两个子工件时,该问题是NP-难的,而且不存在完全多项式时间近似方案(fully polynomial time approximation scheme,简记为FPTAS).提出两个启发式算法,利用数值模拟比较它们的性能,并且将这两个启发式算法的解与最优解的上界进行比较.     

Fast approximation algorithms for bi-criteria scheduling with machine assignment costs

We consider parallel machine scheduling problems where the processing of the jobs on the machines involves two types of objectives. The first type is one of two classical objective functions in scheduling theory: either the total completion time or the makespan. The second type involves an actual cost associated with the processing of a specific job on a given machine; each job-machine combination may have a different cost. Two bi-criteria scheduling problems are considered: (1) minimize the maximum machine cost subject to the total completion time being at its minimum, and (2) minimize the total machine cost subject to the makespan being at its minimum. Since both problems are strongly NP-hard, we propose fast heuristics and establish their worst-case performance bounds.     

极小化最大提前完工费用的可中断排序问题

讨论了带截止期限的$n$个工件在单机上加工,工件间存在优先约束,在允许机器空闲的条件下,确定一个工件的可中断排序,极小化最大提前完工费用.首先考虑两种特殊情形:(1)截止期限相同,存在优先约束;(2)截止期限任意,不存在优先约束.针对两种情形分别给出了时间复杂度为$O(n^2)$的算法.在此基础上,考虑普遍情形,即截止期限任意,存在优先约束,也给出了一个时间复杂度为$O(n^2)$的算法.由于工件不允许延迟,问题可能会无可行排序,需先对问题的可行性进行讨论.     

Minimizing the sum of squares lateness on a single machine

This paper proposes a new procedure that considers the use of inserted idle time to optimally solve the single machine scheduling problem where the objective is to minimize the sum of squares lateness. A heuristic procedure is also proposed for this problem. These procedures and an existing procedure developed by Gupta and Sen are tested using problems of various sizes in terms of the number of jobs to be scheduled and for different degrees of due date tightness. The performance measures are CPU time and sum of squares lateness of the solutions generated. The results show that a simple descent procedure yields very good results and is efficient even on large problems.     

Parallel machine scheduling problems with a single server

In this paper, we consider the problem of scheduling jobs on parallel machines with setup times. The setup has to be performed by a single server. The objective is to minimize the schedule length (makespan), as well as the forced idle time. The makespan problem is known to be NP-hard even for the case of two identical parallel machines. This paper presents a pseudopolynomial algorithm for the case of two machines when all setup times are equal to one. We also show that the more general problem with an arbitrary number of machines is unary NP-hard and analyze some list scheduling heuristics for this problem. The problem of minimizing the forced idle time is known to be unary NP-hard for the case of two machines and arbitrary setup and processing times. We prove unary NP-hardness of this problem even for the case of constant setup times. Moreover, some polynomially solvable cases are given.     

Johnson’s rule,composite jobs and the relocation problem

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.     

Single Machine Scheduling with Learning Effect Considerations

In this paper we study a single machine scheduling problem in which the job processing times will decrease as a result of learning. A volume-dependent piecewise linear processing time function is used to model the learning effects. The objective is to minimize the maximum lateness. We first show that the problem is NP-hard in the strong sense and then identify two special cases which are polynomially solvable. We also propose two heuristics and analyse their worst-case performance.     

带强制工期的可中断单机排序问题

工件带强制工期,指工件必须在已给定的工期内完工,不得延迟.这种环境在实际应用中随处可见.如果工件过早提前完工,意味着工件还需要保管,将会产生额外费用.本文讨论了在单机上,加工带准备时间与强制工期的n个可中断工件,在机器可空闲条件下,确定一个工件排序,使得提前完工时间和最小.先考虑了问题的复杂性,通过奇偶划分问题归约,证明了其是NP-complete的.而后,讨论了加工时间相等的特殊情形,由于工件不允许延迟,问题可能会无可行排序,因此提出了—个多项式时间算法,既能判定可行性,又能针对可行问题获得最优排序.     

Single-machine scheduling problems with an aging effect

This paper considers single machine scheduling problems where the processing time of a job increases as a function of its position in the sequence. In this model, the later a given job is scheduled in the sequence, the longer its processing time. It is shown that the optimal schedule may be very different from that of the classical version of the problem. We introduce polynomial solutions for the makespan minimization problem, the sum of completion times minimization problem and the sum of earliness penalties minimization problem. For two resource constrained problems, based on the analysis of the problems, the optimal resource allocation methods are presented, respectively.     

Scheduling problems with position dependent job processing times: computational complexity results

In this paper, we analyse single machine scheduling problems with learning and aging effects to minimize one of the following objectives: the makespan with release dates, the maximum lateness and the number of late jobs. The phenomena of learning and aging are modeled by job processing times described by non-increasing (learning) or non-decreasing (aging) functions dependent on the number of previously processed jobs, i.e., a job position in a sequence. We prove that the considered problems are strongly NP-hard even if job processing times are described by simple linear functions dependent on a number of processed jobs. Additionally, we show a property of equivalence between problems with learning and aging models. We also prove that if the function describing decrease/increase of a job processing time is the same for each job then the problems with the considered objectives are polynomially solvable even if the function is arbitrary. Therefore, we determine the boundary between polynomially solvable and strongly NP-hard cases.     

Maximization of solution flexibility for robust shop scheduling

We consider the problem of introducing flexibility in the schedule determination phase, for shop scheduling problems with release dates and deadlines. The flexibility is provided by generating a family of schedules, instead of a unique one. We represent a family of schedules by an ordered group assignment defining for each machine a sequence of groups where the operations within a group are totally permutable. We propose a polynomial time algorithm to evaluate the worst case completion time of operations in an ordered group assignment. We then consider the single machine problem with heads and deadlines associated to operations, as a sub-problem of the job shop problem. We propose polynomial time dynamic programming algorithms for minimizing the number of groups and for maximizing the number of characterized sequences, under specific constraints. Finally, computational experiences on job shop benchmarks, show the impact of grouping operations on the solution makespan value.     

Search heuristics for a flowshop scheduling problem in a printed circuit board assembly process

We consider a scheduling problem in a factory producing printed circuit boards (PCBs). The PCB assembly process in this factory can be regarded as a flowshop which has two special characteristics: jobs have sequence dependent setup times and each job consists of a lot (batch) of identical PCBs. Because of the latter characteristic, it is possible to start a job on a following machine before the job is entirely completed on a previous machine, that is, there is time-lag between machines. In this paper, we propose several heuristics, including taboo search (TS) and simulated annealing (SA) methods, for this generalized flowshop scheduling problem with the objective of minimizing mean tardiness. We compare suggested heuristics after series of tests to find appropriate values for parameters needed for the two search algorithms, TS and SA. Results of computational tests on randomly generated test problems are reported.     

A heuristic for single machine scheduling with early and tardy costs

The problem of scheduling jobs with distinct ready times and due dates in a single machine to minimise the total earliness and tardiness penalties is considered. A constructive heuristic, which determines the sequence of jobs and simultaneously inserts idle times, is proposed. Adjacent pairwise interchange is then applied to the schedule obtained. For problems involving at most 12 jobs the heuristic solutions are compared to optimal solutions. For larger problems with up to 80 jobs the heuristic is tested against a local search based on pairwise interchanges and four dispatching rules presented in the literature. In each case, idle times are optimally inserted.     

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.