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.     

The assembly of printed circuit boards: A case with multiple machines and multiple board types

In this paper a typical situation arising in the assembly of printed circuit boards is investigated. The planning problem we face is how to assemble boards of different types using a single line of placement machines. From a practical viewpoint, the multiplicity of board types adds significantly to the complexity of the problem, which is already very hard to solve in the case of a single board type. In addition, relatively few studies deal with the multiple board type case. We propose a solution procedure based on a hierarchical decomposition of the planning problem. An important subproblem in this decomposition is the so-called feeder rack assignment problem. By taking into account as much as possible the individual board type characteristics (as well as the machine characteristics) we heuristically solve this problem. The remaining subproblems are solved using constructive heuristics and local search methods. The solution procedure is tested on real-life instances. It turns out that, in terms of the makespan, we can substantially improve the current solutions.     

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.     

Routing heuristics for automated pick and place machines

We examine the problem of sequencing the placements of multiple part types for a computer controlled placement machine. The problem is modeled as a collection of interdependent traveling salesman problems. A heuristic based on a space filling curve is shown to be easy to compute and quite effective. Numerical experiments show that on problems of realistic size our heuristics show very little divergence from optimality. The probabilistic analysis of the proposed heuristic indicates that the proposed heuristics are asymptotically optimal. Finally we use our heuristic to perform the sequencing operation on real placement machines. The results of this experiment are in accord with our numerical simulations. The main ideas of this study have become part of the control system currently in use in a large electronic card assembly facility that produces approximately 12,000 boards per day. Improvements in throughput of between 4% and 9% have been reported.     

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.     

Resource leveling in a machine environment

We address resource leveling problems in a machine environment. Given a set of m machines, one or more renewable resources, and a set of n tasks, each assigned to exactly one of the machines. Each task has a processing time, an earliest start time, a deadline, and resource requirements. There are no precedence relations between the tasks. The tasks have to be sequenced on the machines while minimizing a function of the level of resource utilization from each resource over time. We provide various complexity results including a polynomial time algorithm for a one machine special case. We also propose an exact method using various techniques to find optimal or close-to-optimal solutions. The computational experiments show that our exact method significantly outperforms heuristics and a commercial MIP solver.     

The effect of material flow and workload on the performance of machine location heuristics

Depending on the problem structure and routing strategies a machine location problem plays an important role in controlling the material flow of work-in-process in discrete product manufacturing environment. In this paper we investigate the effect of material flow and workload on the performance of heuristics for solving an important design problem for job routing and material flow in a manufacturing system. In this research we first develop a model for workload or traffic intensity between machines in a shop floor and then identify different structures of the problems, especially the data. This measure is then used to evaluate the effect of workload on efficiency of the heuristics to solve machine location problems. Some concluding remarks are made on to the effect of the workload or the traffic intensity of materials within the machine cell on the performance of some known heuristics. Conclusions are also made on the performance measures such as makespan, transporter utilization and machine utilization, depending on the problem and data structures.     

Reducing work-in-process movement for multiple products in one-dimensional layout problems

This research describes a method to assign M machines, which are served by a material handling transporter, to M equidistant locations along a track, so that the distance traveled by a given set of jobs is minimized. Traditionally, this problem (commonly known as a machine location problem) has been modeled as a quadratic assignment problem (QAP), which is NP-hard, thus motivating the need for efficient procedures to solve instances with several machines. In this paper we develop a branching heuristic to obtain sub-optimum solutions to the problem; a lower bound on the optimum solution has also been presented. Results obtained from the heuristics are compared with results obtained from other heuristics with similar objectives. It is observed that the results are promising, and justify the usage of developed methods.     

Machine scheduling with an availability constraint

Most literature in scheduling assumes that machines are available simultaneously at all times. However, this availability may not be true in real industry settings. In this paper, we assume that the machine may not always be available. This happens often in the industry due to a machine breakdown (stochastic) or preventive maintenance (deterministic) during the scheduling period. We study the scheduling problem under this general situation and for the deterministic case.We discuss various performance measures and various machine environments. In each case, we either provide a polynomial optimal algorithm to solve the problem, or prove that the problem is NP-hard. In the latter case, we develop pseudo-polynomial dynamic programming models to solve the problem optimally and/or provide heuristics with an error bound analysis.This research was supported in part by NSF grant DDM 9201627     

Performance of scheduling algorithms for no-wait flowshops with parallel machines

We study the performance of scheduling algorithms for a manufacturing system, called the ‘no-wait flowshop’, which consists of a certain number of machine centers. Each center has one or more identical parallel machines. Each job is processed by at most one machine in each center. The problem of finding the minimum finish time schedule is considered here in a flowshop consisting of two machine centers. Heuristic algorithms are presented and are analyzed in the worst case performance context. For the case of two centers, one with a single machine and the other with m, two heuristics are presented with tight performance guarantees of 3 − (1/m) and 2. When both centers have m machines, a heuristic is presented with an upper bound performance guarantee of . It is also shown that this bound can be reduced to 2(1 + ε). For the flowshop with any number of machines in each center, we provide a heuristic algorithm with an upper bound performance guarantee that depends on the relative number of machines in the centers.     

Some effective heuristics for no-wait flowshops with setup times to minimize total completion time

In this work, we consider a complex flowshop scheduling problem in which both no-wait and separate setup times are considered. The optimisation criterion is the minimisation of the total completion time. We propose an effective dominance rule for the four machine case that can also be used for m machines. Five simple and fast heuristics are proposed along with two easy to code stochastic local search methods, one of them being based on Iterated Local Search (ILS). An extensive computational evaluation is carried out with two sets of 5,400 instances. All seven methods are compared to two recent algorithms. The results, confirmed by thorough statistical analyses, show that the proposed methods are more effective and efficient when compared to the best existing algorithms in the literature for the considered problem.     

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.     

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.     

A one-step tabu search algorithm for manufacturing cell design

As part of the cellular manufacturing design process, machines must be grouped in cells and the corresponding part families must be assigned. Limits on both the number of machines per cell and the number of parts per family can be considered. A weighted sum of intracell voids and intercellular moves is used to evaluate the quality of the solutions. We present a tabu search algorithm that systematically explores feasible machine cells configurations determining the corresponding part families using a linear network flow model. The performance of this tabu search is benchmarked against two simulated annealing approaches, another tabu search approach and three heuristics: (ZODIAC, GRAFICS and MST).     

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.     

Minimizing makespan in parallel flowshops

In this study, a new class of proportional parallel flow shop problems with the objective of minimizing the makespan has been addressed. A special case for this problem in which jobs are processed on only one machine as opposed to two or more machines in a flow shop, is the well-known multiple processor problem which is NP-complete. The parallel processor problem is a restricted version of the problems addressed in this paper and hence are NP-complete. We develop and test heuristic and simulation approaches to solve large scale problems, while using exact procedures for smaller problems. The performance of the heuristics relative to the LP lower bound as well as a comparison with the truncated integer programming solution are reported. The performance of the heuristics and the simulation results were encouraging.     

Scheduling job families on non-identical parallel machines with time constraints

This paper studies the scheduling of lots (jobs) of different product types (job family) on parallel machines, where not all machines are able to process all job families (non-identical machines). A special time constraint, associated to each job family, should be satisfied for a machine to remain qualified for processing a job family. This constraint imposes that the time between the executions of two consecutive jobs of the same family on a qualified machine must not exceed the time threshold of the family. Otherwise, the machine becomes disqualified. This problem comes from semiconductor manufacturing, when Advanced Process Control constraints are considered in scheduling problems. To solve this problem, two Mixed Integer Linear Programming models are proposed that use different types of variables. Numerical experiments show that the second model is much more effective, and that there is a trade-off between optimizing the scheduling objective and maximizing the number of machines that remain qualified for the job families. Two heuristics are also presented and studied in the numerical experiments.     

Analysis of Classes of Heuristics for Scheduling a Two-Stage Flow Shop with Parallel Machines at One Stage

This paper addresses two-stage flow shop scheduling with parallel machines at one stage. For finding a minimum makespan schedule, which is strongly NP-hard, some efficient heuristics have been proposed in the literature. In this paper we enrich the set of heuristics by introducing a few classes of heuristics, and show that the existing heuristics can be put into this classification scheme. Furthermore, we give a complete theoretical analysis of the worst-case performance of the classes. Some empirical evaluations and comparisons for the average-case performance of a few typical heuristics in the classes are also performed.     

On a methodology for discrete–continuous scheduling

Discrete–continuous problems of scheduling nonpreemptable jobs on parallel machines are considered. The problems arise e.g. when jobs are assigned to multiple parallel processors driven by a common electric, hydraulic or pneumatic power source. Existing models have assumed job processing rates as a function of the number of jobs currently being processed, or equivalently the number of machines currently in operation. In this paper a more general model is proposed in which processing rates of a job assigned to a machine depend on the amount of a continuous, i.e. continuously divisible resource (e.g. power) allotted to this job at a time. Thus the problem consists of two interrelated subproblems: (i) to sequence jobs on machines, and (ii) to allocate the continuous resource among jobs already sequenced. We provide a comprehensive analysis of the problem. This includes properties of optimal schedules, efficiently (in particular analytically) solvable cases, formulations of the possibly simplest mathematical programming problems for finding optimal schedules in the general case, heuristics and the worst-case analysis. Although our objective function in this paper is to minimize makespan of a set of independent jobs, the presented methodology can be applied to other criteria, precedence-related jobs, and many resource types (apart from, or instead of machines).     

Comparing the minimum completion times of two longest-first scheduling-heuristics

For the basic problem of non-preemptively scheduling n independent jobs on m identical parallel machines so that the minimum (or earliest) machine completion time is maximized, we compare the performance relationship between two well-known longest-first heuristics—the LPT- (longest processing time) and the RLPT-heuristic (restricted LPT). We provide insights into the solution structure of these two sequencing heuristics and prove that the minimum completion time of the LPT-schedule is at least as long as the minimum completion time of the RLPT-schedule. Furthermore, we show that the minimum completion time of the RLPT-heuristic always remains within a factor of 1/m of the optimal minimum completion time. The paper finishes with a comprehensive experimental study of the probabilistic behavior of RLPT-schedules compared to LPT-schedules in the two-machine case.