Evolutionary search for difficult problem instances to support the design of job shop dispatching rules

Dispatching rules are simple scheduling heuristics that are widely applied in industrial practice. Their popularity can be attributed to their ability to flexibly react to shop floor disruptions that are prevalent in many real-world manufacturing environments. However, it is a challenging and time-consuming task to design local, decentralised dispatching rules that result in a good global performance of a complex shop.An evolutionary algorithm is developed to generate job shop problem instances for which an examined dispatching rule fails to achieve a good solution due to a single suboptimal decision. These instances can be easily analysed to reveal limitations of that rule which helps with the design of better rules. The method is applied to a job shop problem from the literature, resulting in new best dispatching rules for the mean flow time measure.     

Production scheduling with applications in the printing industry

This paper presents a technique for the solution of scheduling problems encountered in a printing company that can be applied to a range of practical industrial problems. The aim of the research was to develop a due date scheduling algorithm within the framework of the scheduling system already in use by the company. The objective was to enable jobs to be scheduled as close to their due dates as possible, while ensuring that the resultant schedule was feasible with respect to work centre capacities and earliest start date constraints.     

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.     

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.     

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.     

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.     

Approximating a two-machine flow shop scheduling under discrete scenario uncertainty

This paper deals with the two machine permutation flow shop problem with uncertain data, whose deterministic counterpart is known to be polynomially solvable. In this paper, it is assumed that job processing times are uncertain and they are specified as a discrete scenario set. For this uncertainty representation, the min-max and min-max regret criteria are adopted. The min-max regret version of the problem is known to be weakly NP-hard even for two processing time scenarios. In this paper, it is shown that the min-max and min-max regret versions of the problem are strongly NP-hard even for two scenarios. Furthermore, the min-max version admits a polynomial time approximation scheme if the number of scenarios is constant and it is approximable with performance ratio of 2 and not (4/3 − ?)-approximable for any ? > 0 unless P = NP if the number of scenarios is a part of the input. On the other hand, the min-max regret version is not at all approximable even for two scenarios.     

Mixed integer formulation to minimize makespan in a flow shop with batch processing machines

Batch processing machines are commonly used in wafer fabrication, kilns, and chambers used for environmental stress screening (ESS). This paper proposes two models to schedule batches of jobs on two machines in a flow shop. A set of jobs with known processing times and sizes has to be grouped, to form batches, in order to be processed on the batch processing machines. The jobs are nonidentical in size. The processing time of a batch is the longest processing time of all the jobs in that batch. Mixed integer formulations are proposed for the flow shop problem when the buffer capacity is unlimited or zero. Numerical examples are presented to demonstrate the application of our model.     

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.     

Bounding strategies for the hybrid flow shop scheduling problem

In this paper, we investigate new lower and upper bounds for the multiple-center hybrid flow shop scheduling problem. We propose a family of center-based lower bounds as well as a destructive lower bound that is based on the concept of revised energetic reasoning. Also, we describe an optimization-based heuristic that requires iteratively solving a sequence of parallel machine problems with heads and tails. We present the results of extensive computational experiments that provide evidence that the proposed bounding procedures consistently improve the best existing ones.     

Two-machine flow shop scheduling problem with an outsourcing option

We consider a two-machine flow shop problem in which each job is processed through an in-house system or outsourced to a subcontractor. A schedule is established for the in-house jobs, and 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 total outsourcing costs. We show that the problem is NP-hard in the ordinary sense. We consider a special case in which each job has a processing requirement, and each machine a characteristic value. In this case, the time a job occupies a machine is equal to the job’s processing requirement plus a setup time equal to the characteristic value of that machine. We introduce some optimality conditions and present a polynomial-time algorithm to solve the special case.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.