A simheuristic algorithm for solving the permutation flow shop problem with stochastic processing times

Abstract

This paper describes a simulation–optimization algorithm for the Permutation Flow shop Problem with Stochastic processing Times (PFSPST). The proposed algorithm combines Monte Carlo simulation with an Iterated Local Search metaheuristic in order to deal with the stochastic behavior of the problem. Using the expected makespan as initial minimization criterion, our simheuristic approach is based on the assumption that high-quality solutions (permutations of jobs) for the deterministic version of the problem are likely to be high-quality solutions for the stochastic version – i.e., a correlation will exist between both sets of solutions, at least for moderate levels of variability in the stochastic processing times. No particular assumption is made on the probability distributions modeling each job-machine processing times. Our approach is able to solve, in just a few minutes or even less, PFSPST instances with hundreds of jobs and dozens of machines. Also, the paper proposes the use of reliability analysis techniques to analyze simulation outcomes or historical observations on the random variable representing the makespan associated with a given solution. This way, criteria other than the expected makespan can be considered by the decision maker when comparing different alternative solutions. A set of classical benchmarks for the deterministic version of the problem are adapted and tested under several scenarios, each of them characterized by a different level of uncertainty – variance level of job-machine processing times.

Introduction

The Flow Shop Scheduling Problem (FSP) is a well-known scheduling problem that can be described as follows: a set J of n jobs has to be processed by a set M of m machines. Each job i ∊ J is composed by an ordered set of m operations, O_ij, that must be sequentially performed by the m machines (one operation per machine). A special case of the FSP is the Permutation Flow Shop Scheduling Problem (PFSP), where the processing order of operations in machines is the same for all jobs – i.e., all jobs are processed by all machines in the same order. Each operation O_ij requires a processing time. In the PFSP, the goal is to find a sequence (permutation) of jobs so that a given criterion is optimized. The most commonly and studied criterion is the minimization of the completion time or makespan, i.e., the time it requires to process all the jobs throughout all the machines (Fig. 1). Although the usual goal is to minimize the makespan, other goals can also be considered, e.g., the minimization of the total processing time – i.e., the sum of the individual processing times of each job in each machine, or the total tardiness of jobs with respect to their scheduled deadlines. In Framinan et al. [22] and Ruiz and Maroto [50] surveys about the PFSP with the objective to minimize the makespan are provided. Similarly, Vallada et al. [58] contains a survey for the PFSP with the objective of minimizing total tardiness. In addition, it is usually assumed that: (i) there is unlimited storage buffer between machines, and preemption (task interruption) is not considered; (ii) machines are always available for processing jobs, but each machine can process only one job at a time; (iii) a job cannot be processed more than once for each machine; and (iv) job processing times are independent. In most of the existing literature, the processing time of each job i in each machine j is considered to be a known constant value, p_ij. This problem, which is NP-hard [24], has many practical applications in both economic and industrial fields as it exists in many real-life scenarios, including, among others: the manufacturing and production industry, the services sector, the biochemical and pharmaceutical industries, the computing and telecommunication sector, etc. [52].

The PFSP with Stochastic Times (PFSPST) can be seen as a generalization of the PFSP in which the processing time of each job i in each machine j is not a constant value, but instead it is a random variable, P_ij, following a non-negative probability distribution – e.g., Log-Normal, Exponential, Weibull, Gamma, etc. Since uncertainty is present in most real-life processes and systems, considering random processing times represents a more realistic scenario than simply considering deterministic times. In effect, unforeseen circumstances can lead to sudden changes in the processing time of certain jobs in certain machines, which is likely to have noticeable effects on the predicted makespan. Therefore, one goal that can be considered when dealing with the PFSPST is to determine a sequence (permutation) of jobs that minimizes the expected makespan or mean time to completion of all jobs.

The study of the PFSPST is within the practice of introducing randomness into combinatorial optimization problems as a way of describing more realistic problems in which most of the information and data cannot be known beforehand [59]. For these stochastic problems, efficient simulation–optimization methods have been developed during the last decades. Among these methods, we are particularly interested in the combination of simulation with metaheuristics (simheuristics), since this hybridization constitutes a promising approach yet to be explored in its full potential. Andradóttir [3] offers an overview of simulation–optimization using metaheuristics, while some recent articles combining simulation with metaheuristics can be found in Angelidis et al. [4] and Laroque et al. [40]. In the context of scheduling problems, several authors have discussed the role of simulation in solving real-life instances and have also proposed simulation-based approaches which are typically combined with optimization algorithms (including metaheuristics), among others: Azzaro-Pantel et al. [8], Deroussi et al. [19], Arakawa et al. [6], Fowler et al. [21], Klemmt et al. [39], Frantzén et al. [23], Hu and Zhang [29], Bassi et al. [13], or Kima and Choi [38].

Fig. 1 illustrates a simple PFSP with three jobs and three machines, where O_ij represents the operation of job i in machine j (1 ⩽ i ⩽ 3, 1 ⩽ j ⩽ 3). Notice that, for a given permutation of jobs, even a single change in the processing time of one job in one machine (O₁₂ and O₂₁ in the figure, respectively) can have a noticeable impact on the value of the final makespan.

As with other combinatorial optimization problems, a number of different approaches and methodologies have been developed to deal with the PFSP. These approaches range from exact optimization methods – such as linear and constraint programming, which can provide solutions to small-size problems, to approximate methods – such as heuristics and metaheuristics, which can provide near-optimal solutions for medium- and large-sized problems. Moreover, some of these methodologies are able to provide a set of alternative solutions among which the decision-maker can choose according to his/her expertise. However, the situation with the PFSPST is different. To the best of our knowledge, there is a lack of methods able to provide high-quality solutions to the stochastic version of the PFSP. As discussed in the literature review section, most of the existing approaches are quite theoretical and require many assumptions on the probability distributions that model job processing times, while other approaches seem to be valid only for small-size instances. In the articles by Dodin [20], Honkomp et al. [28], Gourgand et al. [27], and Baker and Altheimer [10], simulation-based techniques have been used to get results for the PFSPST. In some of these articles, however, simulation was mainly used as a backup method to validate the results generated by other analytical methods (e.g., Markov chains). Consequently, only Normal or Exponential probability distributions were employed to model processing times. Moreover, these papers made strong assumptions on the size of the instances being analyzed.

Accordingly, the main contributions of this paper to the existing PFSPST literature can be summarized as follows:

• To introduce a simulation–optimization algorithm able to solve large-size instances in short computing times – just a few minutes or even less – regardless the probability distributions that model the stochastic processing times at each job-machine pair. Our approach does not make any assumption on the size of the instances or on the specific distributions employed to model processing times. In fact, in a real-life scenario, the specific distributions to be used will have to be fitted from historical data (observations). The assumption of processing times following a Normal distribution – quite common in the existing literature – is quite unrealistic and restrictive, since distributions such as the Log-Normal or the Weibull are usually much better candidates to model processing times with positive values.

• To discuss how real-life observations associated with a given permutation of jobs – or, alternatively, simulation outputs provided by our approach – can be analyzed using reliability techniques [30] to provide more information on the stochastic makespan. As we discuss later in more detail, it can be very convenient to use survival functions (inverses of cumulative probability functions) to compare alternative PFSPST solutions with similar expected makespans, specially in those scenarios in which percentile information on the makespan variable associated with a given permutation of jobs can be useful. Reliability analysis is preferred here over classical statistical techniques due to the fact that censored or incomplete observations might be present when real-life data are collected.

• To introduce a set of well-defined and easily replicable PFSPST instances based on well-tested PFSP benchmarks.

The rest of the paper is organized as follows: Section 2 offers a complete literature review on the PFSPST. Section 3 provides an (upper-level) overview of our simulation-based approach. Section 4 gives the (lower-level) details, which might be necessary for implementing a software version of the proposed method. Section 5 presents some numerical experiments that illustrate our methodology. Section 6 discusses the use of reliability analysis techniques to compare alternative solutions. Finally, Section 7 summarizes the main contributions of the paper.