A Two-Stage Three-Machine Flow Shop Assembly Problem Mixed with a Controllable Number and Sum-of-Processing Times-Based Learning Effect by Simulated Annealing Algorithms

The two-stage assembly scheduling problem is widely used in industrial and service industries. This study focuses on the two-stage three-machine flow shop assembly problem mixed with a controllable number and sum-of-processing times-based learning effect, in which the job processing time is considered to be a function of the control of the truncation parameters and learning based on the sum of the processing time. However, the truncation function is very limited in the two-stage flow shop assembly scheduling settings. Thus, this study explores a two-stage three-machine flow shop assembly problem with truncated learning to minimize the makespan criterion. To solve the proposed model, we derive several dominance rules, lemmas, and lower bounds applied in the branch-and-bound method. On the other hand, three simulated annealing algorithms are proposed for finding approximate solutions. In both the small and large size number of job situations, the SA algorithm is better than the JS algorithm in this study. All the experimental results of the proposed algorithm are presented on small and large job sizes, respectively.


Introduction
For decades, scheduling models usually assumed that the processing time of the job is known and fixed [1]. But, in a real production system, both the efficiency of the machine and the skill of the worker can increase with the working time, thereby reducing the actual work processing time with the development of production. Many researchers indicated certain types of learning effects, such as [2] in the positionbased learning effect and [3] in the time-dependent learning effect. is topic continues to attract a lot of research interest [4][5][6].
On the other hand, Biskup, Kuo and Yang [2,3], and Kuolamas and Kyparisis [7] introduced models that involve a learning effect on two-machine scheduling or flow shop scheduling settings following the same or different learning ideas. ere are a multitude of studies related to two-machine scheduling or flow shop scheduling settings with the learning effect consideration, including [8][9][10][11] and [12]. Besides, there are some two-stage models also with the learning effect consideration. Wu et al. [13] adopted the learning model developed by Biskup [2] to solve a two-stage flow shop scheduling problem with three machines to minimize the makespan. ey devised three simulated annealing (SA) algorithms and three cloud theory-based SA algorithms. Adopting the learning model in [3,14], a branch-and-bound (B&B) algorithm is devised incorporating six (hybrid) particle swam optimization (PSO) methods to solve the two-stage flow shop assembly scheduling problem to minimize the total job completion time.
Most studies using the learning model are applied in the single-machine or flow shop setting. However, the two-stage assembly scheduling problem is relatively unexplored. Recently, Wu et al. and Zou et al. [15,16] considered positionbased learning in connection with a two-stage assembly scheduling problem. ey proposed different versions of simulated annealing and cloud theory-based simulated annealing to solve this problem. Moreover, Wu et al. [17] considered this problem with general learning effects. ey developed and evaluated the performances of six hybrid particle swarm optimizations (PSO). Wu et al. [18] have conducted the research about a combined approach for twoagent scheduling with a sum-of-processing-times-based learning effect. To the best of our knowledge, no other research considers the two-stage assembly scheduling problem with learning effects. Table 1 presents a comparative study of existing models to solve the considered scheduling problem. Many studies claim that most of the productive items in manufacturing systems may be formulated in a two-stage assembly scheduling model. However, the literature neglects accumulated learning experience in solving a two-stage assembly scheduling problem. In fact, the sum-of-processing times-based learning model is pertinent to process manufacturing in which an initial setup is often followed by a lengthy uninterrupted production process. Motivated by this observation, this study introduces the 2-stage 3-machine assembly problem with a sum-of-processing times of already processed jobs learning to minimize the makespan criterion.
is model assumes that there are two machines in the first stage and an assembly machine in the second stage. is study first provides some dominances and a lower bound applied in the branch-and-bound method for an optimal solution. ree heuristics modified from Johnson's rule with and without improving by an interchange pairwise method are, then, separately applied in three simulated annealing algorithms for finding near-optimal solutions. Finally, the statistical results of the three proposed algorithms are evaluated and reported.

Problem Statement
e considered two-stage assembly scheduling problem is formally described in this section. We assume a series of n given jobs to be processed on three machines. e main idea of the studied problem is executing n given jobs on three machines in two stages. Each job has strictly more than two operations. For the operations of the n job, the first stage is performed in a parallel way in two machines. e operation processing of the second stage only begins when the operations of the first stage are completed. All jobs are available at time zero. Each job j is decomposed on three tasks as follows: the ordinary processing times (without learning effect) of job j are a j , b j , and c j on machines M 1 , M 2 , and M 3 , respectively. Other assumptions are no machine can process more than one job at a time; no idle time is allowed on machines M 1 and M 2 ; and once a job has begun processing, it cannot be interrupted.
Following the works of Kuo and Yang [3] and Liu et al. [19], the real processing times of J j , if J j scheduling on the position r of a job sequence S, are considered as a j max α, ) a 3 on machines M 1 , M 2 , and M 3 , respectively, where α denotes the controllable parameter with α < 0, and a 1 , a 2 , a 3 denote the learning indices for M 1 , M 2 , and M 3 . For a schedule S, the finishing time of a job, say J j , to be scheduled on the rth position (in S) on machine M 3 is defined, and denoted, as C 3[r] (S): where t 1 , t 2 , and t 3 are denoted as the beginning times of J j on machines M 1 , M 2 , and M 3 in S, respectively. e optimal criterion of this study is to find a scheduled S to minimize the makespan (or C 3[n] (S)).

Lower Bounds and Some Lemmas
is section proposes some useful lower bounds used in a branch-and-bound method. Before deriving the lower bounds, we define some notations. Suppose t 1 , t 2 , and t 3 denote the starting time of the first job in s US on all three machines.
en, we assume that s PS are the partially scheduled k jobs and s US are the remaining unscheduled and c 1 , c 2 , . . . , c n− k in σ US , respectively. Notably, a i , b i , c i are not necessarily from the same job, and s spt1 , s spt2 , and s spt3 are three subsequences of n jobs n increasing sequences of a 1 , a 2 , . . . , a n− k ; b 1 , b 2 , . . . , b n− k ; and c 1 , c 2 , . . . , c n− k in s US , respectively.
Additionally, by using the ideas of Kuo and Yang [3] and Liu et al. [19], a lower bound for a subsequent can be yielded as follows: Parameter Discrete Dynamics in Nature and Society 2 . (2) In a similar way, lower bd � max lower bd 1 , lower bd, lower bd 3 .

(3)
In what follows, we will propose four properties used in a branch-and-bound to improve its search power. Before deriving our properties, we first give four useful lemmas which will be used in the proof of Property 1.
Lemmas 1 and 2 have been proved by Kuo and Yang and Liu et al. [3,19], respectively.
So, the inequality holds.
e inequality holds.
Discrete Dynamics in Nature and Society which is less than 1 − (1 + λt) e . From Lemma 2, the equality holds.
In summary, the inequality always holds for the given conditions.
Proof. For a job sequence S, if the job J j is assigned to the r th position of S, then according to the definition of the completion time of a job, To simplify the proof, let 3 . en, Using the given conditions a i < a j , b i < b j , and 4 Discrete Dynamics in Nature and Society Applying the given condition, and then, follows and, thus, completes the claim.
Finally, one needs to show that C 3i (S 2 ) − C 3j (S 1 ) ≥ 0 to fulfill the proof. From equations (15) and (16), one obtains, applying e details of the proofs of Property 2-4 are omitted because they can be obtained in a similar way to Property 1.
Discrete Dynamics in Nature and Society

Heuristics and Metaheuristics
is section validates the performance levels of nine algorithms, namely, JS_max, JS_min, JS_mean, JS_maxpi, JS_minpi, JS_meanpi, SA_max, SA_min, and SA_mean. Note that all algorithms were developed in FORTRAN and executed on a computer with a 3.00-GHz Intel ® Core ™ i9-10980XE CPU and 16-GB RAM. Our experiments are divided into two parts. e first part is dedicated to the small size instances, and the second is devoted to the large size instances. For the two cases, we randomly generated the normal processing times from uniform distribution U (1, 100) on M1, M2, and M3 by fixing an individual seed for each case. e factor β would be set at 0.2, 0.4, and 0.6. e learning effect would be set at − 0.1, − 0.1, and − 0.2, and the computational results for each small and large size instances are shown. is section describes the considered two-stage assembly scheduling problem.

Small Size Number of Jobs.
e parameters of the small size number of jobs are selected as the number of work pieces (n) at 10, number of machines (m) of 3, and the final temperature (Tf ) is fixed to 2. e selections of other parameters are shown in Table 1.
As a result of parameter selection, the starting temperature (T i ) is 10 − 3 , cooling coefficient is 0.95, and the number of tests (N r ) of Johnson's algorithm is 10.
Next, the parameter selection process is explained in detail.
We select the parameter starting temperature (T i ). e range is from 10 − 4 to 10 4 at 10 times the interval, and the cooling coefficient (c f ) is fixed at 0.95. e number of tests of Johnson's algorithm (N r ) is 5, and there are also generated data and parameters defined before selecting parameters.
By observing Figure 1, it is obvious that only the initial temperature (T i ) is changed. In this range, it can be observed that the AEP is relatively low at 10 − 3 , so the initial tem- After fixing the starting temperature (T i ) to 10 − 3 , we select the parameter cooling coefficient (c f ), ranging from 0.05 to 0.95 with 0.05 intervals, and fix the number of tests (N r ) of the Johnson's algorithm as 5. ere are also generated data and parameters defined before selecting the parameters.
By selecting the parameter of the cooling coefficient (c f ), the overall decline in its value is obvious. It becomes flat after reaching 0.80, so we choose a lower value in this part. e cooling coefficient (c f ) is fixed at 0.95. After fixing the starting temperature (T i ) to 10 − 3 and the cooling coefficient (Cf ) to 0.95 (please see Figure 2), we select the parameter Johnson's algorithm test times (N r ) with a range of 1 to 20 and the interval of 1 and see the difference. e generated data and parameters are fixed and defined before selecting the parameters.
When selecting the parameter Johnson's algorithm test times (N r ), we can see from Figure 3 that although it fluctuates up and down, its AEP value is actually very low, so we choose a lower value in this part. Johnson's algorithm test times (N r ) is 10. It shows when the number of work pieces of the B&B algorithm is 12. e results of finding the best solution in a few cases cannot be found, so only a few values (in the Fs column) are recorded. From the viewpoint of the number of nodes and CPU time, the difference in the number of jobs can also be clearly seen. Table 2 uses the β values of 0.2, 0.4, and 0.6 to distinguish the table. It can be seen regardless of whether the number of work pieces is 8, 10, or 12, the smaller the β value, the greater the number of nodes. Also, there is no best solution for the CPU time. Table 3 is a table distinguished by three different situations of α 1 , α 2 , α 3 . It can be observed that regardless of whether the number of work pieces is 8, 10, or 12, the  Discrete Dynamics in Nature and Society number of nodes and CPU time for finding the best solution for the second case are all more than the third case, and the third case is more than the first case. e simulation result for the small size number of jobs from the total mean at the bottom shows that the SA-related algorithm performs better in the small size number of jobs. Table 4 has β values of 0.2, 0.4, and 0.6, and no obvious features are noted. Table 5 is divided into three different situations of α 1 , α 2 , α 3 . Clearly, in the case of various numbers of jobs, the AEP of the second case is better than the AEP of the first case. e AEP for each case is better than the AEP for the third case. Table 6 compares nine algorithms in total. ere are three changes in the number of jobs (n) and β and 2 changes in α 1 , α 2 , α 3 .
Before comparing algorithms, normality verification is carried out on the data. If the data is normal, you can use the average of the algorithm to compare pairwise. e residuals are not in line with the normal state, so they cannot be compared using means. We instead use the Kruskal-Wallis test to observe whether the medians of the data are the same (please see Table 7). If they are not the same, we can use the median of the algorithm to make pairwise comparisons and observe the calculations for the difference in the median between methods. e null hypothesis is that the median of each group is the same, where the P value <0.05, so the null hypothesis is rejected, meaning the median of each group is not the same and multiple comparative analysis can be performed to show the difference between each group (please see Table 8). Figure 4 is a box diagram of the nine algorithms scored by nonparametric statistics. Clearly, the results of the SArelated algorithms under the small size number of jobs are superior. Table 9 shows the results of the pairwise comparison. We take the significance level of 0.05 as the standard. e JS_max algorithm and the JS_mean algorithm are grouped into one group, the JS_mean algorithm is divided into a group with the JS_maxpi algorithm, and the JS_maxpi algorithm is divided into a group with the JS_meanpi algorithm. Finally, the SA-related algorithms are all divided into a group. From this, it can be seen that the SA algorithm is a relatively better group for the small size number of jobs, followed by the JS_meanpi Algorithm, JS_maxpi algorithm, JS_mean algorithm, JS_max algorithm, and finally, the JS_minpi algorithm and JS_min algorithm. ere are two key points clearly observed in the grouping. First, the JS algorithm with pi is better. Second, the result obtained by mean is better than max, and max is better than min.

Large Size Number of Jobs.
e parameters of the large size number of jobs are selected when the number of work pieces (n) is 10, the number of machines (m) is 3, and the final temperature (T f ) is fixed to 10 − 8 , β is 0.2, α 1 is − 0.1, α 2 is − 0.1, and α 3 is − 0.2. e selections of other parameters are shown in Table 10.
From the summary of Tables 11 and 12, the SA algorithm is the best in calculating the RPD. However, the CPU_time shows that the SA algorithm takes a relatively long time. With pi or without pi, the JS algorithm clearly shows that the solution with pi to the run out of RPD is slightly inferior to the solution without pi. e JS algorithm with pi for the CPU_time is the fastest. Table 13 shows the residual error in the large size number of jobs is not in line with the normal state, so we also Discrete Dynamics in Nature and Society use the same method as the small size number of jobs to compare the relationship between the algorithms. In the Kruskal-Wallis test, the null hypothesis is rejected as with the small size number of jobs, meaning the median of each group is not the same and multiple comparison analysis      Table 14). Figure 5 is a box diagram of nine algorithms scored by nonparametric statistics. e figure shows clear differences among SAs and others under the large size number of jobs. Table 15 shows the results of the pairwise comparison. If the P value value is greater than 0.05, it means the effect between the pair is not significant, but they are correlative. In    Parameter         e results of the SA algorithm and JS algorithms are also significant, so they would not be divided into the same group. e RPD summary of the large size number of jobs shows the solution of the SA algorithm is comparatively better. e SA algorithm is better than the JS algorithm in this case.

Conclusions
is study focuses on two-stage three-machine flow shop assembly problems mixed with a controllable number and sum-of-processing times-based learning effect, in which job processing time is considered to be a function of the control of the truncation parameter and learning based on the sum of the processing time. We derive several dominance rules, lemmas, and lower bounds applied in the branch-and-bound method. On the other hand, three simulated annealing algorithms are proposed for finding approximate solutions. Computational results show that the SA algorithm is relatively better for the small size number of jobs. In the large size number of jobs, the solution of the SA algorithm is comparatively better. Both in the small and large size number of jobs situation, the SA algorithm is better than the JS algorithm in this study. For future research, it would be interesting to develop more powerful dominance rules and a sharper lower bound on the optimal solution for mediumsize instances.

Data Availability
All data are available on request.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.