NEH-based heuristics for the permutation flowshop scheduling problem to minimise total tardiness
Introduction
The Permutation Flowshop Scheduling Problem (denoted as PFSP) is one of the most studied problems in the Operations Research literature (see the reviews by [4], [27]). This decision problem deals with scheduling n jobs that have to be processed on m machines in the same order and with the same job sequence on every machine. Different criteria have been considered in the literature for this decision problem (see e.g. the reviews by [6], [32], [19]), such as the maximum completion time among the jobs or makespan, the total flowtime (sum of completion times of all jobs), and the total tardiness (sum of the tardiness of each job). Makespan and total completion time are related to the fast processing of the products and to a balanced use of resources, both issues being of great importance in make-to-stock manufacturing scenarios. In contrast, total tardiness focuses on the satisfaction of customers and it is therefore better suited for make-to-order manufacturing scenarios as due dates play a key role [21], [15]. Thereby, among the objectives considered for the PFSP, the total tardiness highlights a critical concern for manufacturing systems (see e.g. [25], [20]), since delays may lead to an increase in costs such as penalty clauses in a contract, loss of customers and/or bad reputation for other customers [29]. The PFSP with total tardiness minimisation objective is denoted as (see e.g. [22]).
Since the problem is known to be NP-hard, most researchers have focused on developing solution procedures (i.e. heuristics) that do not guarantee the optimality of the solution, but that can provide a (hopefully) good solution in a reasonable time interval. More specifically, several heuristics and metaheuristics have been proposed in the literature for the problem, such as those by e.g. [8], [16], [24], [5], [31]. Among them, the NEHedd heuristic [14] stands out since, as we will discuss later, many works employ it to obtain an initial solution. The NEHedd is an adaptation for the tardiness objective of the well-known NEH heuristic by Nawaz et al. [18] for makespan minimisation. In the NEH heuristic, jobs are initially arranged in non-ascending order of their processing times. Then, a job sequence is constructed by evaluating the partial schedules originating from the initial order: assuming a sequence already set for the first k−1 jobs, k candidate (sub)sequences are obtained by inserting job k in the k possible slots of the current sequence. Out of these k (sub)sequences, the one yielding the minimum makespan is kept as the relative (sub)sequence for these first k jobs. Then, job k+1 from the initial order is considered analogously, and so on until all n jobs have been sequenced. In order to reduce the computational burden of the NEH heuristic, Taillard [30] proposed a mechanism (known as Taillard׳s acceleration) to reduce the complexity of the NEH heuristic from to . The excellent performance of the NEH heuristic and its easy adaptation to similar problems have led to its application to other scheduling decisions, such as the PFSP with total completion time minimisation (see e.g. [7]), denoted as , or the hybrid flowshop scheduling problem (see e.g. [1]). For these problems, Taillard׳s acceleration cannot be applied, but Li et al. [17] present a mechanism that saves between 30 and 50% of CPU time for the problem, however without reducing its complexity.
The NEHedd heuristic differs from the NEH heuristic in the starting order (jobs are arranged now according to the Earliest Due Date or EDD rule), and in the evaluation of the partial sequences (as the one with lowest total tardiness is selected). Taillard׳s acceleration cannot be applied to the NEHedd, and, although Vallada and Ruiz [31] propose a mechanism similar to that by Li et al. [17], the complexity of the NEHedd remains .
The extensive computational evaluation of heuristics for the problem carried out by Vallada et al. [32] shows that NEHedd is a key constructive heuristic for the problem since, aside to being very efficient, the rest of efficient heuristics in the literature with more average CPU time employ NEHedd as an initial solution. More specifically, more than half of the state-of-the-art improvement heuristics or metaheuristics for the problem use NEHedd as a starting solution. This fact can be also seen in more recent works, such as Vallada and Ruiz [31], or Schaller [28].
Despite the excellent performance of the NEHedd heuristic, we believe that additional improvements could be gained by further analysis of the problem under consideration. First, the tardiness minimisation problem could resemble different scheduling problems depending on the due dates of the jobs for each specific instance: intuitively, it is clear that, for an instance with due dates much greater than the sum of the processing times of its jobs, almost every schedule may yield zero total tardiness, thus turning the problem into a trivial one. Analogously, unachievable due dates for each job results in an instance for which almost every sequence yields tardiness for every job and therefore the problem resembles that of minimising flowtime. By conducting an analysis of these possible scenarios, further insights on the problem can be obtained, so the performance of the NEHedd procedure can be enhanced. More specifically, we will show in this paper that such an analysis reveals the importance of adequately addressing the high number of ties appearing in the constructive phase of the NEHedd. In order to handle these ties in an efficient way, we propose several tie-breaking mechanisms for the problem and conduct an extensive computational experiment to test their performance. The results show that one of these mechanisms (based on machine idle time) improves the original results obtained by NEHedd by roughly 25% while requiring the same CPU time. Another one (based on Taillard׳s acceleration for makespan) outperforms the NEHedd by 15% while employing less CPU time. Furthermore, when using the idle time-based version of the NEHedd as starting solution for the metaheuristic by Vallada and Ruiz [31] (which is the state-of-the-art metaheuristic for the problem), the metaheuristic improves its result for different stopping criteria.
The remainder of the paper is organized as follows: in Section 2 the problem under consideration is described and analysed to derive some properties of the problem that may serve to identify the different scenarios and related decision problems. As a result, it is shown that there is a high number of ties for the instances in the most tardiness-specific scenario (i.e. that one not leading to trivial problems or to flowtime minimisation). Therefore, in Section 3, eight tie-breaking mechanisms are proposed. An extensive comparison among them and with the original NEHedd procedure are performed in Section 4. Finally, conclusions and future research lines are discussed in Section 5.
Section snippets
Analysis of the problem
The problem under study can be stated as follows: n jobs have to be scheduled in a flowshop composed of m machines. Each job j has a processing time, tij, on each machine i. Considering a sequence , the processing time of the job in position j, i.e. job πj, is denoted as pij, where . Cij denotes the completion time of job πj on machine i. The completion time of the last job of the sequence in the last machine, , is defined as the maximum completion time or makespan of
Proposed tie-breaking mechanisms
As mentioned in the previous section, no specific tie-breaking mechanism is mentioned in the original NEH heuristic for makespan minimisation. Indeed, it is cited [18] that ‘… Next, the job with the third highest total process time is selected and the three partial sequences are tested in which this job is placed at the beginning, middle and end of the partial sequence…’, which seems to indicate that the first position where a tie is found is selected. In the following, we will denote this
Computational experience
Each proposed tie-breaking mechanism has been compared under the same computer conditions, which means the same computer and the same programming language (C#). Algorithms were tested using the set of instances of the benchmark of [32] described in Section 2. The different tie-breaking mechanisms were compared by means of the RDI (described in Section 2) as an indicator of the quality of the solution.
The results of the heuristics are shown in Table 1 in terms of their values of RDI. The best
Conclusions
In this paper, several tie-breaking mechanisms for the NEH heuristics have been proposed to solve the problem. It is clear that, depending on the due dates, the decision problem to be solved is different. Extremely tight due dates induce to a problem, whereas very loose due dates lead to a trivial problem. Thereby, the problem has been first analysed in detail, depicting the limits between the tardiness problem and other problems. As a conclusion, it was obtained that
Acknowledgements
The authors are sincerely grateful to the anonymous referees, who provide very valuable comments on the earlier version of the paper. This research has been funded by the Spanish Ministry of Science and Innovation, under projects “SCORE” with reference DPI2010-15573/DPI, and “ADDRESS” with reference DPI2013-44461-P/DPI.
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