Elsevier

Computers & Operations Research

Volume 100, December 2018, Pages 201-210
Computers & Operations Research

Bi-objective scheduling on a restricted batching machine

https://doi.org/10.1016/j.cor.2018.07.004Get rights and content

Highlights

  • We tackle the bi-objective single p-batching machine problem of minimizing maximum lateness and number of jobs with a restricted batch size.

  • We applied an epsilon-constraint method and develop new mathematical models that are enhanced with a family of valid inequalities and constraints that avoid symmetric solutions.

  • We find Pareto-optimal solutions in reasonable times for up to 50 jobs.

  • We develop a BRKGA for larger instances of up to 100 jobs with good results with smaller computational times than the exact methods.

Abstract

In this work, we consider a batching machine that can process several jobs at the same time. Batches have a restricted batch size, and the processing time of a batch is equal to the largest processing time among all jobs within the batch. We solve the bi-objective problem of minimizing the maximum lateness and number of batches. This function is relevant as we are interested in meeting due dates and minimizing the cost of handling each batch. Our aim is to find the Pareto-optimal solutions by using an epsilon-constraint method on a new mathematical model that is enhanced with a family of valid inequalities and constraints that avoid symmetric solutions. Additionally, we present a biased random-key genetic algorithm to approximate the optimal Pareto points of larger instances in reasonable time. Experimental results show the efficiency of our methodologies.

Introduction

A batching machine is one that can process a group of jobs simultaneously in the form of a batch. In this type of machines, jobs may not be added or removed from a batch before all jobs in the batch have been processed. As mention in Mathirajan and Sivakumar (2006) research on batching machines has been applied to the casting, furniture manufacturing, metal, aircraft, and shoe manufacturing industries, with an increasing volume of research focusing in the semiconductor manufacturing industry. Our research is partly motivated by the operations in semiconductor manufacturing.

A semiconductor chip is a highly miniaturized integrated electronic circuit whose process starts with raw wafers which are disks made of silicon or gallium arsenide. Mönch et al. (2011) characterizes several scheduling problems found in the different work areas in wafer fabrication facilities (wafer fabs). The literature considers two different ways of calculating the processing time of a batch: parallel batching (p-batch), where the processing time of a batch is the maximum processing time of all the jobs in the batch; and serial batching (s-batch), where the processing time of the batch is calculated as the sum of the processing times of the jobs inside the batch. Mönch et al. (2011) identifies a s-batch environment in the photolithography work area but states that the p-batch environment is more important in this industry since it appears in the oxidation, deposition and diffusion work areas, as well as in the burn-in ovens in the chip testing facilities. Effective burn-in operation scheduling is key, as it is frequently a bottleneck due to long processing time, as stated in Mathirajan et al. (2004), and as it occurs at the end of the manufacturing process has a strong influence on the shipping dates. Thus, due date related objective functions are relevant for these environments.

Single machines are not only interesting on their own but, as Mönch et al. (2011) states, they can also form subproblems in decomposition schemes of more complex scheduling problems. Batching machines can be divided into those that can process all n jobs within the same batch (unbounded batching machine), and those where batches have a limited size b (bounded batching machine) with b < n. In case the batch has a limited size, this can be restricted by the number of jobs that the batch can process or by the existence of a size capacity for the batch, as explained in Matin et al. (2017). This work relates to the problem of bounding the batches by the number of jobs as most articles do in the literature: Wang and Uzsoy (2002), He et al. (2015), Sabouni and Jolai (2010) and Aloulou et al. (2014) among others.

A burn-in operation consists of placing in an oven a batch of integrated circuits to stress them electrically and thermally to test their standing ability. The length of these operations can be as short as a few hours, or as long as 48  h, although some military applications require burn-in cycles with a total time of 240  h (Mathirajan et al., 2004). These long processing times result in negligible set-up time, as operators can be preparing the next batch while testing is taking place. Also, in earlier operations like oxidation, deposition and diffusion there is no need to consider set-up times, as stated in Mönch et al. (2011). These long processing times lead to the formation of long queues in front of non-batching machines, thus minimizing the number of batches becomes relevant.

In this article, we study a p-batch bounded machine limited by the number of jobs. We focus on the bi-objective problem of minimizing the maximum lateness Lmax and the number of batches #batch. The maximum lateness measures the maximum deviation between completion time and the due dates of all jobs, to ensure customer satisfaction and on-time delivery of the products. Following the standard three-field notation for scheduling problems (Graham et al., 1979), this problem can be denoted as 1|pbatch,b<n|(#batch,Lmax). To the best of our knowledge, this setting has not been studied in the literature before.

A batch, Bk, is a set of jobs, a schedule σ is a sequence of batches (B1,B2,,B#batch). Each job can only be processed on a single batch and all jobs must be executed in a batch, thus k=1#batch|Bk|=n. Let pj be the processing time of job j and dj its due date. Then, the processing time of batch Bk is p(Bk)=maxjBk{pj}, its completion time is C(Bk)=j=1kp(Bj), its due date is defined as d(Bk)=minjBk{dj} and the lateness is L(Bk)=C(Bk)d(Bk). Thus, the lateness of schedule σ can be calculated as Lmax=max1k#batch{C(Bk)d(Bk)}.

The single objective problem 1|pbatch,b<n|Lmax is shown to be unary (or strongly) NP-hard in Brucker et al. (1998). Thus, the bi-objective problem 1|pbatch,b<n|(#batch,Lmax) is also unary NP-hard.

A solution is called non-dominated, Pareto-optimal or Pareto-efficient if one of the objective functions cannot be improved without increasing the value of the other (see Hoogeveen, 2005). In Fig. 1, we show the complete Pareto frontier formed by optimal schedules obtained for an instance with n=50 jobs and batch size b=5 for the 1|pbatch,b<n|(#batch,Lmax) problem. The larger and darker points are the Pareto-optimal points while the smaller and lighter points are weak Pareto schedules.

Cabo et al. (2015) address the single-objective 1|pbatch,b<n|Lmax problem. The authors present a mathematical formulation and develop a split-merge neighborhood of exponential size that can be searched in polynomial time by dynamic programming. However, their dynamic programming approach is not easily generalized to the bi-objective problem we tackle. Dauzère-Pérès and Mönch (2013) minimize the weighted and unweighted number of tardy jobs on a single batch processing machine with incompatible job families. They propose two different mixed-integer linear programming formulations based on positional variables and a random-key genetic algorithm to solve this scheduling problem.

For bi-objective scheduling, a popular pair of objective functions is to minimize the makespan (completion time of the last batch, Cmax) which is a measure of the utilization of the machine, together with the maximum lateness Lmax. However, different pairs of objective functions in different environments have also been studied in the literature, as we summarize in Fig. 2 and explain in more detail below.

For the s-batch setting, Cmax is trivial to optimize for the unbounded machine case, as processing all jobs in the same batch minimizes Cmax. Hence, He et al. (2013) can obtain the Pareto-optimal schedules in polynomial time when studying Cmax together with a general cost function fmax. More recently, Aloulou et al. (2014) and He et al. (2015) consider bounded machines. Aloulou et al. (2014) present the problem of a 2-machine flow-shop to minimize the number of batches (#batch) and Cmax, and study its computational complexity, proposing approximation algorithms with guaranteed worst-case performance. He et al. (2015) consider a single batching machine with the objective of minimizing Cmax and Lmax. For this case, the makespan of a schedule is mainly dependent on the number of batches and hence to minimize both objectives it suffices to minimize Lmax for every fixed number of batches.

Concerning the p-batch setting, the unbounded case has been studied more widely. Fan et al. (2012) study two settings, both minimizing Cmax but one focusing on the total weighted completion time (∑wjCj), and the other on the total weighted number of tardy jobs (∑wjUj). They give NP-hard proofs for both problems and propose an approximation algorithm and a pseudo-polynomial time algorithm for each problem, respectively. Feng et al. (2013) present a 2-customer setting, where the first customer is interested in minimizing the makespan CmaxA, while the second requires minimizing the lateness LmaxB. Jobs from different customers cannot be processed in the same batch (incompatible jobs). They propose polynomial-time algorithms for finding all Pareto-optimal solutions for this problem. They also prove than the algorithm presented by Sabouni and Jolai (2010) is not optimal for this case. Geng and Yuan (2015) minimize makespan and maximum lateness when jobs are partitioned into families that cannot be processed together. Unlike the previous paper, this is not a two-agent problem. They present a dynamic programming algorithm for the constrained optimization problem of minimizing Cmax given an upper bound on Lmax. Improving this algorithm, they find all Pareto-optimal schedules, in polynomial-time when the number of families is constant.

There are only two papers dealing with the bounded case for the p-batch machine. In Sabouni and Jolai (2010), the authors present optimal algorithms for the 2-customer problem, with both compatible and incompatible jobs, to minimize makespan and lateness, when the processing times are the same for the customer whose objective is to minimize Lmax and propose this algorithm to be used as a heuristic for the non-constant processing time scenario. Unlike Sabouni and Jolai (2010), the processing times in our case do not have to be constant and there is no restriction on how to group the jobs into batches. Recently, in Shahvari and Logendran (2017) a bi-objective batch processing problem with dual-resources on unrelated-parallel machines is addressed. A mathematical programming model and particle swarm optimization algorithms are proposed for minimizing the production cost including total cost of tardy and early jobs (E-T) along with total batch processing cost as well as the makespan (Cmax) with dual-resources. This closely related problem differs from ours in the fact that an earliness-tardiness objective function induces natural idle times in the scheduling solution.

Our aim is to solve the 1|pbatch,b<n|(#batch,Lmax) problem. For this, we formulate a new integer linear programming that incorporates a family of valid inequalities and two sets of constraints that avoid symmetries in the solution space. This model is extremely efficient for medium size instances when solved by an ϵ-constraint methodology. To solve larger instances, we present a biased random-key genetic algorithm (BRKGA).

The remainder of the paper is structured as follows. In Section 2, we formally describe the 1|pbatch,b<n|(#batch,Lmax) problem and present some of its properties. In Section 3, we extend the model proposed by Cabo et al. (2015) for the 1|pbatch,b<n|Lmax problem but that can be easily generalized for our bi-objective function. The main contribution of this study is found in Section 3.2, where we present an integer linear programming with novel inequalities families that strengthen the convex hull of the solution space. In Section 4.1 we propose an ϵ-constraint methodology to find the Pareto-optimal solutions while in Section 4.2 we describe a BRKGA algorithm to approximate the Pareto points for instances of larger size. Finally, Section 5 analyzes the performance of the models and methods proposed, giving our final remarks in Section 6.

Section snippets

Properties of the problem

In this section, we state some relevant definitions and results that show interesting characteristics of the 1|pbatch,b<n|(#batch,Lmax) problem that are useful to strengthen the mathematical formulation of Section 3.1. Moreover, Theorem 1 shows that by minimizing the number of batches, one does not necessarily minimize the makespan. In other words, unlike the unbounded case, the 1|pbatch,b<n|(#batch,Lmax) and the 1|pbatch,b<n|(Cmax,Lmax) problems, are not equivalent.

To illustrate the

Mixed-integer linear formulations for the problem

In Section 3.1, we propose a mixed-integer linear programming (MILP) for our problem based on the model by Cabo et al. (2015). Their model only minimizes Lmax but is easy to generalize it to also consider the number of batches. Nevertheless, we show in Section 5 that this model is not efficient in time nor quality of the solutions. Thus, in Section 3.2, a new MILP is proposed with inequalities that strengthen the solution space polyhedron.

Notice that an upper bound on the number of batches in a

ϵ-constraint

As mentioned by Ehrgott (2006), besides the weighted sum approach, the ϵ-constraint method is probably the best-known technique to solve bi-objective optimization problems. There is no aggregation of objectives, instead only one of the original objectives is minimized, while the other is included as constraints in the formulation. An interesting property of the 1|pbatch,b<n|(#batch,Lmax) problem resides in its finite number of Pareto-optimal solutions since the number of batches is an integer

Experimental results

We show in Section 5.2 the efficiency of models CMILP and Bi-MILP solved by the ϵ-constraint method for the 1|pbatch,b<n|(#batch,Lmax) problem. In Section 5.3, the performance of our BRKGA algorithm is presented. All ϵ-constraint tests were executed using the integer linear solver of Gurobi C++ 6.05 with the default settings, except for the time limit of 1  h (3600  s) and precision set equal to 0 to obtain the optimal solution. We used a MacPro with 3.5 GHz 6-Core Intel Xeon E5 with 32GB

Conclusions

This paper tackles a new bi-objective scheduling problem to minimize Lmax and number of batches, on a restricted p-batching machine: 1|pbatch,b<n|(#batch,Lmax). We illustrate why this choice of functions is not as straightforward as minimizing (Lmax, Cmax) which has been a common choice in the literature before.

We adapt a MILP formulation to tackle the bi-objective problem and develop a new MILP model with different sets of inequalities to strengthen the formulation. We include restrictions to

Acknowledgments

M. Cabo and E. Possani would like to thank Asociación Mexicana de Cultura A. C. for their support. José Luis González-Velarde would like to thank Tecnológico de Monterrey Research Group in Industrial Engineering and Numerical Methods 0822B01006.

References (27)

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