Optimizing batch operations with batch-position-dependent learning effect and aging effect

https://doi.org/10.1016/j.cie.2021.107325Get rights and content

Highlights

  • We consider the problem of optimizing batch operations.

  • Both batch-position-dependent learning effect and aging effect are considered.

  • Three different models are studied and fast algorithms are proposed.

  • An optimal algorithm is proposed for the tractable model.

  • Approximation algorithms are proposed for two NP-hard models respectively.

Abstract

Motivated by applications in the metal pickling process, we study the optimization scheduling problems of batch operations with batch-position-dependent learning effect and aging effect. In production, a set of metal work-pieces are pickled on a fixed capacity single batch facility. If the pickling solution is regarded as the machine, the processing time of the metal work-pieces will be longer and increase in a power function because of the aging effect. At the same time, workers need to heat the solution. A worker can finish the task more and more quickly because of the learning effect, resulting in shorter processing time which decreases in a power function. Then we consider three models of batch operations. In the first model, the work-pieces have identical sizes and an optimal algorithm is proposed with time complexity of Onlogn. In the second model, the work-pieces have identical processing time and it is shown that the model is NP-hard in the strong sense. Then, we propose an approximation algorithm. The absolute and asymptotic worst-case ratios of the algorithm are 2 and 1.494. In the third model, the work-pieces have arbitrary processing time which is proportional to their sizes and the model is also NP-hard in the strong sense. Finally, an approximation algorithm is proposed with an absolute and asymptotic worst-case ratio is less than 2.

Introduction

In general scheduling problems, it is usually assumed that the processing time of a job is a constant. However, the actual processing time of the jobs will be affected by factors in practice, such as machine facility, jobs themselves and processing sequence. In production, when workers handle the same task repeatedly, he or she can finish the task more and more quickly. At the same time, the processing speed of the machine will gradually decrease. These two conditions are called learning effect and aging effect respectively. Obviously, it is necessary to find optimization methods to organize workers, machines and processing tasks under the condition of considering both learning effect and aging effect in practical work.

A typical application is metal pickling, which is a very common process in large-scale manufacturing industry and one of the important stages in metal finishing. It is to put a batch of metal work-pieces into the pickling tank for cleaning, and the oxidation layer and rust on the metal surface are removed by acid solution to improve the corrosion resistance of metal. Fig. 1 shows the process of metal pickling, in which the cleaning of metal work-pieces will reduce the concentration of pickling solution, resulting in the decrease of pickling tank efficiency. For instance, the amount of ferrous salt increases, but the concentration of active substance decreases due to chemical reactions. As a result, the activity of pickling solution is reduced. As more contaminants dissolve in the pickling bath, the time required for pickling a single batch metal work-pieces increases in a power function. At the same time, workers heat the pickling solution, and the temperature rise can improve the pickling efficiency. For example, the pickling speed can be increased by about 70% by heating it at 10°C in a pickling bath based sulfuric acid. The same batch of cleaning work-pieces should not exceed the capacity limit of pickling tank. In order to avoid chemical reaction, work-pieces of different metal types cannot be cleaned together. Aging effect and learning effect are very significant in pickling process. First, put the same metal type of work-pieces into the pickling bath. The aging effect takes place in the deterioration process of pickle solution, and the learning effect takes place in the treatment process of pickle solution by workers. Second, the deterioration of the machine is inevitable, but working experience is important because the acidification process will produce harmful gases, i.e., new workers need more time to increase the concentration, and experienced workers are familiar to controlling system of equipment which need less time. After enough practice, workers can perform the pickling operations skillfully.

Sequencing and scheduling problems originated from manufacturing, and then widely used in engineering technology, industrial engineering and supply chain. In addition to metal pickling process, similar applications also occur in food making, semiconductor manufacturing industries in practice. In recent years, the combination of learning effect and optimization has aroused heated discussions. Biskup (1999) introduced position-based learning effect into scheduling problem for the first time. Learning effect refers to the steady decline of processing time, which is usually achieved by repeating the same task. And he modeled the actual processing time of piwhich is scheduled in position ris pir=pira. Jiang, Chen, and Kang (2013) introduced a time-dependent and job-dependent learning effect into single-machine scheduling problems and considered a new learning effect model. Bai and Zhao (2020) studied the single-machine scheduling problem with DeJong’s effect and machine availability constraints. Wang, Gao, Wang, Liu, and He (2020) addressed how to minimize total completion time on a single machine with a position-weighted learning effect. Ji, Yao, Yang, and Cheng (2015) proposed a fully polynomial-time approximation scheme for minimizing makespan in parallel-machine case based on DeJong’s learning effect. Meanwhile, minimizing completion time was polynomially solvable. Xu et al. (2016) introduced a multiple-machine order scheduling problem with position-dependent learning effect, and heuristic algorithms were proposed to minimize the total tardiness. Przybylski (2018) proposed a new parallel-machine scheduling model, in which the processing time of a job was described by Riemann integral. Qin, Zhang, and Bai (2016) studied the flow shop scheduling problems with position-dependent learning effect and group effect. The processing time of each job on any machine depends not only on its working position, but also on its group position. On this basis, Bai, Tang, Zhang, and Santibanez-Gonzalez (2018) added release dates of jobs into flow shop scheduling problems. A branch and bound algorithm was proposed to obtain the optimal solutions for small-scale problems, and a heuristic algorithm was proposed to obtain the approximate solutions for large-scale problems. Besides, part of the research on learning effect has evolved into truncated learning effect. Cheng, Wu, Chen, Wu, and Cheng (2013), Wang, Zhou, Zhang, Ji, and Wang (2013) and Wu, Yin, and Cheng (2013) studied two-machine flowshop scheduling with truncated position-based learning effect. And a new model was proposed, pjr=pjmaxra,ρ. The permutation flow shop problems with truncated exponential sum of logarithm processing times based and position-based learning effects are explored by Wang, Liu, and Wang (2019). Cheng et al. (2013) introduced a branch-and-bound and genetic algorithms to find the optimal and approximate solutions respectively. Cheng, Zhu, Li, and Li (2019) considered batch optimization problems with truncated batch-position-dependent learning effect, and obtained the approximate solutions by heuristic algorithms.

Aging effect is also studied in the optimization problem, and most of them consider the maintenance activities at the same time. Yang and Yang (2010a) investigated three basic types of aging or deterioration effects and applied them to single-machine scheduling. They introduced that aging effect refers to the degradation of production equipment performance caused by machine fatigue. Yang and Yang (2010b) introduced the position-based aging effect and maintenance activities in single-machine scheduling problems, and the optimal job sequences were gotten by polynomial time algorithms. Sun and Geng (2019) proposed efficient solution algorithms (O(n2logn)) to solve single-machine scheduling problems with deteriorating effects and maintenance activities. Yang, Cheng, Yang, and Hsu (2012) studied the aging effects and the multiple maintenance activities in parallel-machine scheduling problems. Cheng, Xiao, Luo, and Lian (2017) introduced the batch-based aging effect and variable maintenance activated between batches in single-machine scheduling problems. Yang, Lai, and Yang (2014) explored multiple due window assignment in a single-machine scheduling problem with aging effect, assuming that the actual processing time of a job can be controlled by introducing additional resources. Kozik and Rudek (2018) proposed a fast innovative neighborhood search method to solve the general scheduling problem, i.e., aging effect (deterioration) of machines. Pei, Wei, Liao, Liu, and Pardalos (2019) considered the aging effect of job-position-dependent in two-agent parallel-machine scheduling problems and designed a heuristic algorithm. Zhang, Liu, Lin, and Wu (2020) studied deteriorating jobs and maintenance activities in the potential disrupted parallel-machine scheduling problems.

Other researchers considered both learning and aging effect (or deterioration) in the scheduling problems. (Wang, 2006) and Wang (2007) considered job processing times were defined by functions of their starting time and positions in the sequence of single machine. Huang, Wang, Wang, Gao, and Wang (2010) introduced time-dependent deterioration and exponential learning effect in the single-machine scheduling problems. Rudek (2017a) proved that the single-machine scheduling problems based on learning or aging effects to minimizing the weighted completion time both were NP-hard. Wang and Wang (2014) considered position-dependent learning effect and time-dependent aging effect in the unrelated parallel-machine scheduling problems. Ji, Tang, Zhang, and Cheng (2016) studied parallel machines scheduling with DeJong’s learning effect and deteriorating jobs, and a fully polynomial-time approximation scheme was provided. Lu (2016) explored time-based learning effect and deterioration in no-idle permutation flowshop scheduling problems. By constructing a pseudo-polynomial dynamic programming algorithm to simulate the learning effect or aging effect of the parallel processor, Rudek (2017b) obtained a fully polynomial time approximation scheme. Mehdizadeh, Niaki, and Hemati (2018) developed a bi-objective optimization model to solve an aggregate production planning (APP) with worker learning effect and machine deterioration. Pei, Song, Liao, Liu, and Pardalos (2020) studied a parallel-machine serial-batching scheduling problem for jobs with arbitrary release times under the effects of job-position-based learning and sum-of-normal-processing-time-based deterioration and a hybrid Society and Civilization-Variable Neighborhood Search (SC-VNS) algorithm was proposed.

However, Litter research has be done on batch operation optimization scheduling problems with position-dependent learning and aging effect. Optimizing batch machines with effects is more complex than traditional scheduling problems. This problem exists widely in practice. In this paper, the optimization problems of single batch equipment with batch-position-dependent learning effect and aging effect are explored. The specific models are given for different problems, and the effective optimization algorithms are provided respectively.

The key contributions of our research are as follows:

First, we propose an accurate optimization model for optimizing batch machines which includes batch-position-dependent learning effect and aging effect. We consider the metal pickling industry manufacturers who have batch-processing equipments to process metal work-pieces with different processing requirements. The objective is to minimize the makespan.

Second, we prove that the difference of processing time does not complicate our target. The optimal solution of the batch-optimization scheduling problem can be obtained in polynomial time when the processing time of the work-pieces is different but the size is the same. Therefore, manufacturers can increase product diversity to improve customer satisfaction properly.

Third, we show that the difference in sizes of metal work-pieces make the target problem intractable. When customers have different requirements for metal work-pieces’ sizes, the problem is NP-hard in the strong sense. We propose effective approximation algorithms to solve the problems and obtain good worst-case performance ratios. It can be obtained that the problem of aging effect is more difficult to solve than the problem of learning effect when customers have same demands on processing time. The results show that, in general problems, the diversity of metal work-pieces often brings difficulties to production and processing, resulting in more cost losses.

The rest of the paper is arranged as follows. In Section 2, we introduce the required models and the meanings of the different notations. In Section 3, we propose an optimization algorithm for the first model (sj=1). In Section 4, we propose an approximate algorithm for the second model (pj=1) and calculate the absolute and asymptotic worst case ratios. In Section 5, we consider the general model of the problem and propose an approximate algorithm. Finally, in Section 6, we conclude this paper and provide managerial insights and directions for future research.

Section snippets

The model and preliminaries

The problem under investigation can be described as follows. A set of metal jobs J={1,2,n}needs to be processed: job j has a size sjand a processing time pj. We define η=pminpmaxto denote the difference degree of metal jobs, where pmin=minpj:jJand pmax=maxpj:jJ. Obviously, we have 0η1, where η=odenotes that there is a great difference between the processing time of metal jobs and η=1means that all the metal jobs have identical processing time. The pickling bath has a single batch facility

Optimal algorithm A1for problem Γ1

In Γ1:1|p-batch,P[i]=Pirα+β,sj=1,pj|Cmax, the jobs have arbitrary processing time but the same sizes. Since sj=1for all jobs, we can put D jobs together in a batch.

Algorithm A1

When -1<α+β<0, we have following algorithm.

Step 1. Sort the jobs in non-increasing order of their processing time.

Step 2. Assign the jobs into batches using the First Fit Decreasing rule and generate Zbatches.

Step 3. Sort the batches in non-decreasing order of their processing time.

When 0<α+β<1, we have following

Approximation algorithm A2for problem Γ2

In Γ2:1|p-batch,LEr,AEr,sj,pj=1|Cmax, the jobs have arbitrary sizes but the same processing time.

Proposition 2. Γ2is NP-hard in the strong sense.

Proof. In Γ2, since the jobs have the same processing time p, the batches also have the same processing time p. To minimize the makespan, it is sufficient to minimize the number of batches. ThereforeCmax=i=1ZP[i]=r=1Zrα+β.This problem is equivalent to the Bin Packing Problem (BPP) without considering learning effect and aging effect. That isCmax=i=1Z

Approximation algorithm A3for problem Γ3

In this section, we consider the general problem Γ3:1|p-batch,LEr,AEr,pjsj|Cmax. In Γ3, the jobs have arbitrary size and the processing time is proportional to its size. Γ3is more difficult than Γ2, so Γ3is also NP-hard in the strong sense. We have following proposition.

Proposition 3. Γ3is NP-hard in the strong sense.

Algorithm A3

We propose Algorithm A3 to solve Γ3.

When α+β<0, it means that learning effect plays a greater role than aging effect.

Step 1. Sort the jobs in non-increasing order of

Conclusions

In this paper, we study the optimization scheduling problems of batch operations with the learning effect of workers and the aging effect of machines. The objective is to minimize the makespan when all jobs with arbitrary sizes and processing time are to be processed on a fixed capacity batch facility. We consider three models. First, we propose an optimal algorithm for the jobs with the identical sizes but arbitrary processing time. Second, we propose an approximate algorithm for jobs with the

Acknowledgements

This work is partly supported by the National Natural Science Foundation of China under Grants 71671055, 72071056, 71971075, 71531008 and 71690230. This work is also partly supported by the National Key Research and Development Program of China 2019YFE0110300.

References (34)

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