Solving Multi-Objective Two-Sided Assembly Line Balancing Problems by Harmony Search Algorithm Based on Pareto Entropy

Two-sided assembly lines are designed to produce large and complex products, where workers can perform on both sides at the same time. This paper establishes a mathematical model for the multi-objective two-sided assembly line balancing problems with additional constraints (MOATALBP). The model considers both workers skills and the balance of the assembly line, aiming to maximize efficiency and minimize workers cost and smoothness index. A harmony search algorithm (HS) based on Pareto entropy (PE-MHS) is proposed to solve MOATALBP. The difference entropy of Pareto solutions is employed to adjust the algorithm parameters to enhance the optimization ability of PE-MHS. Moreover, a fine-tuning operation combining insertion and inverse sequence is utilized to avoid the algorithm from falling into local optima. Ultimately, non-dominated sorting ensures a set of well-distributed Pareto solutions. The experimental results of different problems indicate that the proposed algorithm can achieve better solutions than three classical algorithms (NSGAII, SPEA2 and HS) for the MOATALBP.


I. INTRODUCTION
Assembly lines play an important role in modern enterprises. They can reduce the operating scope of workers and increase their repeatability, thus improving workers' skill and the efficiency of the assembly line. Hence, assembly lines are widely used in the automotive, consumer electronics industries and other industries which produce high-quality and standardized products [1], [2]. The assembly line problems are usually categoried by shape, such as one-sided assembly line problems, U-shaped assembly line problems, parallel assembly line problems, two-sided assembly line problems, and etc. Two-sided assembly lines are better than one-sided assembly line, because they can shorten the length of the assembly line, reduce output time, decrease the cost of tools and fixtures, and reduce material handling and workers' movement [3]. Like the assembly line balancing The associate editor coordinating the review of this manuscript and approving it for publication was Diego Oliva . problem (ALBP), The two-sided assembly line balancing problem (TALBP) is also an NP-hard problem [4]. Its complexity increases rapidly with the problem scale. This research focuses on the problem of two-sided assembly lines, which are usually applied to produce large-size products such as trucks, armored vehicles, and locomotives.
As shown in Figure 1, the mated-stations (e.g. station 1 and station 2) are important components of the twosided assembly lines. The two-sided assembly line balancing problem is constrained by direction and precedence, thus some tasks can only be performed at a specific side of the assembly line, while others can be assigned to either side. Because certain sequence has to be followed, there is a certain amount of idle time between the two tasks. For example, in the mated-station 2, task m and task q are allocated to the left line and task n is allocated to the right line. Assuming that task n is an immediate predecessor of task q, that is, task q cannot be performed until task n is completed. Then, station3 will have some sequence-depended idle time.   For TALBP, tasks, operations directions and the priority relationship can be expressed by the precedence diagram, as shown in Figure 2. The number in the circle indicates the task number. Each task is associated with a label. The label shows the time required for the task to be executed and the direction in which the task is located. L indicates that the task is on the left side of the assembly line, R indicates that the task is on the right side of the assembly line, and E indicates that the task can be on either side of the assembly line. The arrow indicates the priority relationship between tasks. Depending on the objective function, TALBP can be classified into three types: TALBP-I aims to minimize the number of stations with the predetermined cycle time, TALBP-II aims to minimize the cycle time with a given number of stations, and TALBP-III aims to minimize the smoothness index [5], [6].
The precedence constraints and cycle time constraints are prior consideration in the ALBP [7]. However, in the actual two-sided assembly lines, there are several additional constraints, including positioning constraints, zoning constraints and synchronizing constraints, distance constraints and resource constraints. Moreover, in the view of real-world situations, TALBP with several objectives is sufficient for practical application. Therefore, the multiobjective TABLP with additional constraints is closer to actual production situation, and the study of this problem has more significance for assembly line of the enterprise. In this paper, positioning constraints, zoning constraints and synchronizing constraints are all considered. Two objectives, minimization of ES and minimization of workers' skill, are developed.
Researcher suggests [8] that the most effectiveness optimization methods in tackle large scale problems utilized are meta-heuristic methods, such as genetic algorithm (GA), ant colony optimization (ACO), simulated algorithm (SA). In this paper, a new PE-MHS algorithm is presents to solve the MOATALBP. Some benchmark instances are tested.
The rest of the paper is organized as follows: Section 2 describes the literature review about TALBP. Section 3 presents the mathematical model of MOATALBP. Section 4 illustrates the harmony search (HS) algorithm based on Pareto entropy for MOATALBP. Section 5 shows the experimental study and data analysis. Section 6 describes conclusion and future research. Experimental comparisons of different repair strategies prove the effectiveness and stability of the proposed model of moving the task violating to any position after the maximum immediate predecessor and the proposed model helps to improve the assembly line efficiency, ensures the balance of the station load, and reduces the workers cost.

II. LITERATURE REVIEW
The study of TALBP has become a more active field in recent years [9]. Many optimization methods, objective functions, and constraints on TALBP have been studied to addresses the ALBP. However, there are relatively few studies on the TALBP considering workers' skills [10]. Hence, the following reviews the works on TALBP, and later presents the reports works on multi-objective two-sided assembly line balancing problems.
Bartholdi [3] firstly proposed and solved the TALBP by conducting a computer program with a balancing algorithm, but his main achievement is to start the research of two-sided assembly line problem, the algorithm is relatively backward for the moment. Then, a large number of researchers have utilized various optimization algorithms to explore the TALBP. Hu et al. [11] proposed a branch and bound algorithm to solve the large-size TALBP, and the majority of the solution sets are optimal solutions, although the branch and bound method he proposed can obtain most of the optimal solutions, for some special cases, the optimal solution obtained by the branch and bound method cannot meet the requirements. Yang et al. [12] implemented a multi-neighborhood-based path relinking algorithm to address TALBP. For TALBP-I, the following representative algorithms are utilized: Genetic algorithm [13], ant colony algorithm, late acceptance hill-climbing algorithm [14]. For TALBP-II, the following representative algorithms has been reported: simulated annealing algorithm [15], iterated greedy algorithm [16].
Özcan [17] addressed the stochastic TALBP by simulated annealing algorithm. Mosadegh et al, but they did not consider some realistic constraints, and only considered the normal distribution of task time. [18] implemented a simple SA algorithm to solve the balancing and sequencing problems in the mixed-model assembly line (MMAL), But he did not conduct research on the uncertain assembly time. Pinar Tapkan et al. [19] employed bees algorithm and artificial bee colony algorithm for parallel two-sided assembly line balancing problem with walking times. Li et al. [20] developed co-evolutionary PSO algorithm to address TALBP, VOLUME 9, 2021 however, the problem of the assembly line on both sides that he solved did not consider constraints such as partition, location, and synchronization tasks. Ibrahim [21] used nondominated sorting ant colony optimization (NSACO shortly) to address the problem of balancing mixed-model two-sided assembly lines. A paced mixed-model two-sided assembly lines considering balancing and sequencing problems were established by Li et al. [15]. They developed a restarted SA algorithm for this model. Kim et al. [22] employed the hybrid genetic algorithm to balance a mixed-model assembly line with unskilled temporary workers, but he did not consider multiple constraints and random extended operation time.
Recently, such economic factors as production cost and efficiency have attracted increasing attention. Janardhanan et al. [23] proposed an improved migrating birds optimization algorithm with restart mechanism to deal with the worker assignment and line balancing problems. Purnomo et al. [24] applied Harmony Search to balance the efficiency and workload of the two-sided assembly lines. They obtained the Pareto optimal solutions by an non-dominated sorting method. Ritt et al. [25] solve assembly line worker assignment and balancing problem considering uncertain worker availability. Moreira et al. [26] established a mathematical model, which used Miltenburg's regularity criterion and cycle time as the evaluation metrics of workers and productivity, respectively.
Besides TALBP with the precedence and cycle time as constraints, TALBP with other constraints have also been addressed. Wang et al. [27] proposed a hybrid imperialist competitive algorithm, which considered the positioning constraints, the zoning constraints and the synchronizing constraints. Li et al. [28] established a MOATALBP, which can maximize production efficiency, and minimize the smoothness index and the total cost per unit of product. Some other researches on TALBP with multiple constraints are conducted by Baykasoglu et al. [29], Chiang et al. [30], Sepahi et al. [31] and Li et al. [16]. Tang et al. [32] developed a hybrid teaching-learning-based optimization (HTLBO) approach. They adopted a priority-based decoding to ensure satisfaction of the constraints. However, few researchers consider these constraints and economic factors con-currently.
Recently, more and more researchers take into account the multi-objective TALBP. At first, Volling et al. [33] and Wang et al. [34] earlier converted multi-objective into single objective problem by linear combination, however, the adaptability of this method is not strong, and the adaptability to different problems of the assembly lines on both sides is not strong. Subsequently, a form of solutions called Pareto solutions is presented for multi-objective problems. M Mokhtarzadeh et al. [35]developed a two-stage framework to balance a mixed-model parallel U-shaped assembly line by considering ergonomic risks. Chutima and chimklai [36] applied the particle swarm optimization algorithm with negative knowledge to solve the multi-objective two-sided mixed-model assembly line balancing problem. The relative position knowledge of different particles was employed to generate new solutions, and the pareto strategy was used to solve the conflict of the objective function. A Yadav et al. [37] proposed objective function is to maximize the workload at each station such that the number of stations is minimized. Z Li et al. [38] presents two simple local search methods, the iterated greedy algorithm and iterated local search algorithm, to deal with type I mixed-model twosided assembly line balancing problems. Z Li et al [39] investigates the impact of various structural parameters on the performance of exact methods, including branching methods, search direction, method to achieve upper bounds, utilized lower bounds, utilized dominance rules and search strategy. Zhang et al. [40] proposed a mathematical model to minimize cycle time and rebalancing costs, however, he did not consider many more meaningful situations and practical constraints in the problems of the assembly line on both sides. They applied MNSGA-II to solve the multi-objective two-sided assembly line rebalancing problem with special constraints. Multiobjective optimization problems can be towards realistic situation, and there have been many discussions in the literature [41]- [43].
The harmony search algorithm is a superior heuristic global search algorithm, which has been successfully applied to many combinatorial optimization problems and has the advantages of simple concept, easy programming and few parameters, It has been successfully applied in many combinatorial optimization problems. On related issues, it shows better performance than genetic algorithm, simulated annealing algorithm and tabu search. Results show that it has better performance than GA, SA and tabu search [44]. Wang et al. [45] applied an improved harmony search algorithm to optimize the multi-objective vulcanization workshop scheduling. After studying a large number of papers on the assembly line on both sides, the author found that there are few researches on the application of HS algorithm in TALBP. In recent years, Literature [46] applied the HSA to various scenarios of automatic production lines, but did not focus the main research on the problem of bilateral assembly lines. Literature [47] applies the HSA to the robot assembly line problem(RALB), and has achieved good results, but the research focus of the RALB problem and the TALBP problem is still different. In summary, it is innovative to apply the HSA to the TALBP.
Moreover, few studies focus on the solution repair strategy of MOATALBP. Hence, this study concentrates on MOATALBP with the harmony search algorithm based on Pareto entropy (PE-MHS), and develops a strategy to deal with infeasible solutions violating constraints. The Pareto entropy is used to determine the diversity and evolutionary status of the population in order to adjust adaptively the harmony memory consideration rate (HMCR) and the pitch adjusting rate (PAR).

III. THE MODLE OF MOATALBP
This section presents the mathematical model for the MOATALBP which can maximize assembly line efficiency, and minimize worker costs and the smoothness index. In this paper, the zoning constraints, positioning constraints and synchronizing constraints are considered, not only that, we also consider the constraints of the distribution of workers and the constraints of cost in these constraints. In our proposed TALBP, every task in the station must be completed within a given cycle time. The current task cannot operate until the previous task has completed. Tasks with direction constraints must be assigned to the station of specified direction. Each pair of tasks with the positive zoning constraints must be assigned to the same station. While the tasks with the negative zoning constraints must be operated on different stations. Synchronizing constraints mean that the tasks on both sides of a mated-station must be performed at the same time. Further more, the operational difficulty level of the station task must match the skill level of the worker in the station. The mathematical model is established, and its parameters are introduced.

A. PROBLEM ASSUMPTIONS
The assumptions of the MOATALBP are introduced as follows: 1) The assembly line produces only one type of product, and the assembly process has been determined.
2) The operation time of all tasks are determined as constant. 3) Each task can be assigned to any station on the twosided assembly line. 4) A single-side station can process only one task at a time. 5) Once a task starts, it cannot be stopped until the task is completed. 6) No buffer is set on the assembly line. 7) Special circumstances are not taken into account, such as machine failures worker rest, etc.

B. NOTATIONS
The parameters and decision variables used in the mathematical model of the proposed problem are summarized as follows:

C. MATHEMATICAL MODEL
Based on the research by Kim et al. [48], Li et al. [20] and Moreira et al. [26], the mathematical model of MOATALBP presented in this paper is as follows. Objectives: Subject to: Objective (1) represents the optimization of the cost of the workers based on the task level. Objective (2) and (3) aim at optimizing the smoothness index and efficiency, respectively. Since these two objective functions are not mutually exclusive, the Pareto front cannot be obtained. Therefore, equation (4) can be established by combining these two objective functions, where LE inital and SI inital indicate the assembly line efficiency and smoothness index in the initial solutions.
Constraint (5) is a positive zoning constraint that requires two tasks to be assigned to the same station. Constraint (6) is a negative zoning constraint which ensures two tasks being at different mated-stations. Constraint (7) is a positioning constraint. Synchronization is considered by constraints (8) and (9), which ensure that task i and task e start simultaneously at different stations of the mated-station. Constraint (10) indicates that all tasks are performed in their precedence constraints. Constraint (11) means that every task only be assigned to one station. Operation direction constraints are obtained from constraints (12) indicating that tasks with direction L must be assigned to the left side of the line. The same is true for the tasks in the direction R, which are expressed in the constraint (13). Constraint (14) means that one worker can only be assigned to a station. Constraint (15) is the restrictive conditions of the cycle time. Constraint (16) deals with the sequence of tasks, if task i and task m are assigned to the same mated-station, and task m is an immediate precedence of task i, then the starting time of task i is not earlier than the finishing time of task m. Constraint (17) represents that task i and task g have no constraint and are allocated to the same station. If task i starts earlier than task g, task g starts before task i finishes. Otherwise, constraint (18) works.

IV. THE HS ALGORITHM BASED ON PARATO ENTROPY
In this section, the HS algorithm based on Pareto entropy is developed. Decoding based on the start time of the earliest task is proposed to solve MOATALBP. The insertion and reverse operation are applied to pitch adjusting to diversity solutions. At the same time, Pareto entropy and difference entropy are used to judge the evolution state of the population, and the parameters are adaptively adjusted to improve the diversity of the solutions and to speed up the convergence. The proposed algorithm is described in detail as follows.

A. INTRODUCTION TO THE HS ALGORITHM
The Harmony Search Algorithm is a non-derivative metaheuristic algorithm, which simulates the improvisation process of a musician [49]. Although the HS algorithm is relatively simple, it has been proved to be competitive with more complex optimization algorithms and has been applied to solve many practical problems. The basic HS algorithm is shown in Table 4.

B. ENCODING AND DECODING
Considering characteristics of the TALBP, we not only consider the operating sequence, but also constrain the operating direction, especially the E-type tasks. The E-type tasks indicates that this task can be operated on either side of the assembly line. Consequently, based on the contributions reported in Li et al. [9], an encoding method combining tasks sequence and operation direction is adopted. The chromosomes are composed of strings whose length is the size of the problem. The operation direction of the E-type tasks, such as task3, task6, task7 and task9, is randomly generated. In order to avoid infeasible solutions, the tasks sequence which violates the precedence constraints, if any, is repaired. All tasks that violate the constraints are moved to a position after their preceding tasks. Re-checking the newly generated tasks sequence to prevent generation of infeasible solutions.  For example, as shown in Figure 3 below, The constraint states that Task 7 must proceed after Task 4 and Task 5 have completed. Therefore, when Task 7 is selected to move, it can move to any position after Task 5 to get a new sequence combination.
Under the condition that all synchronizing constraints, and positive zoning and negative zoning constraints are satisfied, all tasks are assigned to the station and the specific task allocation scheme is obtained. Compared to the decoding proposed by Wang [50], the task assignment in this paper is different in several ways. If the operating task does not meet the negative zoning constraints, a new station is opened. Besides, if the operating task violates the synchronizing constraints or positive zoning constraints, this task is placed at the end of task sequence, and adjusting the priority of task sequence.

C. THE OPERATION OF PITCH ADJUSTING
The operation of pitch adjusting applying the insertion and reverse sequence is proposed to solve the TALBP. Then, the operation direction of the E-type tasks is selected randomly. The operation taking P9 as an example is depicted in Figure 4. The operation enhances the local search ability of the algorithm and avoids prematurity. The operation of pitch adjusting is shown as follows: Step 1: Selecting a solution X randomly; Step 2: The operation of combining insertion and reverse sequence is applied to the solution X, generating a solution X'; Step 3: For the E-type tasks of initial operation directions, a random direction is taken again. And generating a new solution X''. VOLUME 9, 2021

D. ADAPTIVE PARAMETER ADJUSTMENT BASED ON PARATO ENTROPY
In the actual production situation, there are two or more optimization objectives that need be considered simultaneously. However, the objectives are in conflict, and it is difficult to obtain a solution that optimizes multiple objectives. The multi-objective optimization problem is to obtain a Pareto-optimal solution that has at least one better objective without destroying another objective. The Paretooptimal set is consisted of these solutions. The curve (for two objectives) formed by the Pareto-optimal set is called Pareto front [51].
Entropy usually refers to the species diversity and the degree of confusion in the system. In this paper, entropy is used to determine the evolutionary state of population in the PE-MHS algorithm. The approximate Pareto front stored in external archives is mapped to a twodimensional planar grid in parallel coordinates to obtain the entropy of Pareto front, so as to judge the diversity of population in PE-MHS algorithm. The difference of an approximate Pareto front in a new objective space represents the entropy change. Therefore, we can estimate the population's ability to discover new solutions and its evolution state. The theoretical basis for developing and designing the evolution strategy is provided. Since the new solution generated by the algorithm causes a large change in the Pareto front, the difference entropy increases. Otherwise, the algorithm generates fewer good new solutions, which only causes the change of individual grid coordinate components.
The calculation formula of the Pareto entropy Entropy (t) in the t-th iteration of the approximate Pareto front stored in the external archive is as follows: where Cell k,m (t) denotes the number of cell coordinate components within the cell of the m-th column of the k-th row after the Pareto front is mapped to the PCCS. By comparing the difference entropy of two iterations with the critical value of evolutionary state [52], the evolutionary state of the current population can be inferred. and the HMCR and PAR can be adaptively adjusted. An adaptive parameter adjustment strategy is designed, as shown in Equations (20) and (21). Diversification where Step par and Step HMCR adjusting steps of PAR and HMCR respectively, the values of which are equal to the interval length divided by the maximum number of iterations.

E. THE PE-MHS ALGORITHM PROCEDURE
The HS algorithm is a heuristic global search algorithm, which is an intelligent optimization algorithm abstracted from the harmony created by musicians. It has been widely used in multi-dimensional optimization, pipeline optimization design and other combinatorial optimization problems. Geem et al. [53] first used HS algorithm to solve the multi-objective optimization problem of the satellite heat pipe design in 2006. In the MOATALBP, multiple objectives need to be considered to evaluate feasible solutions. Since these objectives usually are conflicting with each other, the improvement of a certain objective's performance may worsen the performance of other objective. Therefore, the optimal solution of the MOATALBP is no longer a certain value but a Pareto optimal solution set. Decision-makers can choose suitable solutions from a large number of Pareto solutions based on actual needs. For intelligent algorithms, the balance between development and exploitation is a significant factor in performance evaluation. In this study, several improvements are introduced to the HS algorithm. The detailed flowchart is shown in Figure 5. The PE-MHS algorithm adaptively adjusts the parameters HMCR and PAR according to the evolutionary state of the population, which ensures the diversity of the population and prevents premature convergence of the population. It is worth noting that though the insertion and reverse sequence operation are simple, they are applied to pitch adjusting for solutions. The utilization of two operations not only enhances the exploitation performance of the algorithm, but also prevents the algorithm from being trapped into local optimum.
After combining the harmony memory generated by the PE-MHS with the initial harmony memory, the nondominated sorting and crowding distance operations are performed to form a new harmony memory. That is because the NSGA-II applies the non-dominated sorting and crowding distance, and the performance of convergence and diversity are excellent.

V. EXPERIMENTAL STUDY A. EXPERIMENTAL CASE
In order to demonstrate the effectiveness of PE-MHS algorithm for the MOATALBP, four TALBPs are solved in different cycle times. The benchmark problem of P24 can be 121734 VOLUME 9, 2021  obtained from Kim et al. [13]. P16, P65 and P205 are taken from Lee et al. [54]. P148 is taken from Bartholdi [48]. The additional constraints are depicted in Table 5. The skill level of tasks and the wage of workers' skill level are detailed in Appendix A. All problems are programmed in Java. The tests are run on a personal computer with on Intel Core 7 Duo 3.60GHz CPU and 16GB memory.
Since the ES and workers cost defined in Equation (1) and (4) are two objective functions of different dimensions, the dimensions need to be normalized, that is, the ''min-max normalization'' is adopted, whose minimum and maximum values are respectively the minimum and maximum values of each set of objective function.
The performance of the repair strategy is evaluated by the distribution of Pareto solutions and the range of Pareto front. The distribution of Pareto solutions can be obtained by calculating the average spacing deviation function (ASDF) [55]. ASDF can be obtained by Equation (22). It can represent the distribution of solutions of Pareto frontier and the number of solutions. The range of Pareto front can be calculated using Equation (23) which indicates the value of spacing metric (SP).
where, d i,j represents the distance from solution i to solution j on the Pareto front.
where, Z i a represents the value of objective. O is the number of objectives. K is the number of Pareto solutions.

B. UNCERTAIN PARAMETER ANALYSIS
Under the condition of PAR = 0.05, the harmony memory consideration rate (HMCR) are analyzed with 5 sets of data (0.5, 0.6, 0.65, 0.7, 0.8) using the proposed model. The experimental results are shown in Figure 6. As the different value of HMCR, the values of ASDF (the Average Spacing Deviation Function) and SP (Spacing metric) will also fluctuate with the same trends. Therefore, the setting of the HMCR value will affect the solution distribution of the MOATALBP. According to the experimental results, when the value of PMCR is 0.60∼0.70, the values of ASDF and SP are relatively small. It indicates that the solution points included in the Pareto front are distributed, and there are a large number of solution points.
When PMCR = 0.65, the influence of pitch adjusting rate on proposed model performance is analysed. The experimental results are shown in Figure 7. In the proposed model, pitch adjusting rate (PAR) can also have a great impact on the distribution of the Pareto solution. When PAR is 0.05, the values of ASDF and SP are the smallest. It indicates that the deviation of the distance between the Pareto points and the VOLUME 9, 2021   average distance of the Pareto points in a front is the smallest, and the solution points are evenly distributed.

C. COMPARISON OF THE SOLUTION REPAIR STRATEGY
To test the effectiveness of the proposed repair strategy (strategy 1), two strategies are used for comparison: the tasks that violate precedence constraints are put after the last gene of the chromosome (strategy 2) or the maximum immediate predecessor (strategy 3). These repair strategies are embedded into the proposed PE-MHS, the parameters of whose algorithm are shown in Table 6. And the values of ASDF and SP, with P65, are utilized for comparison in different cycle times. The test run 10 times with each cycle time to obtain the ASDF and SP. Figure 8 and Figure 9 show the values of ASDF and SP based on different strategies. It can be seen that the values of ASDF and SP obtained from strategy 1 at different cycle times are less than the values of other strategies. The interquartile range (IQR) of strategy 1 is smaller than those of other strategies. These indicate that the results obtained from the proposed strategy would be relatively stable. Compared with other strategies, it can give more even distribution of Pareto solutions at the front. However, there are some error   points, which are caused by uncertain solutions obtained from strategy 1 and strategy 2.

D. CALCULATION RESULTS OF MOATALBP
To test the performance of the proposed PE-MHS, the results obtained from NSGAII and SPEA2 for the MOATALBP are compared. The current problem has multi-objectives to be optimized simultaneously and the non-dominated solutions exist in the form of Pareto set of solutions. All algorithms are encoded with the proposed solution repair strategy. And the parameter values selected for the test are illustrated in Table 7. Other parameters of SPEA2 are the same as NSGAII.
After all parameters are determined, the experiment is run for 10 times with five test problems of different algorithms to get Pareto solutions. The description of two small-scale problems (P16 and P24) and their corresponding computational results of the four algorithms are shown in Table 8. where, f 1 represents the objective value ES, and f 2 is workers cost. It can be seen from Table 8 that PE-MHS is significantly better than NSGAII, SPEA2 and HS in solving multi-objective problems of small-scale. The PE-MHS has obtained at least one better objective value for each case, which proves that it is able to achieve greater assembly line efficiency and smaller smoothness index at the same workers cost. For P16 with cycle time 21, it is observed that PE-MHS The MOATALBP has three objectives to be optimized simultaneously and the solution exists in the form of Pareto set of solutions, In this article, the solution will be represented by (f 1 , f 2 ), Among them, f 1 a represents the target value ES, f 2 represents the worker cost. the average number of whose solutions is taken as the evaluation criterion. The average number of the non-dominated solutions is listed in Table 9. It can be seen from Table 9 that in most cases, by average, PE-MHS obtains more solutions than other algorithms. These results indicate that the proposed PE-MHS algorithm may give more assembly line balancing schemes as compared with the schemes obtained from other algorithms.
The ASDF and SP show the effect of distribution of Pareto solutions and the extent of the obtained non-dominated front, respectively. Consequently, the proposed PE-MHS algorithm results are compared with the results of NSGAII, SPEA2 and HS based on ASDF and SP. The results are presented in Table 10 where the average ASDF values and the average SP values by different algorithms are exhibited. Each case contains the average value of ASDF and SP for 10 repetitions. It can be observed from Table 10.
In Table, the average values of ASDF and SP obtained from PE-MHS algorithm are smaller than the others in most cases. Under condition of guaranteeing the preferred values of objectives, the proposed algorithm has more evenly distribution of solutions on Pareto front as compared with the others. For the problem of P16 and P24, the difference VOLUME 9, 2021  between the ASDF and SP values for each case is very small. And as the problem scale increases, the difference gradually increases. For the problem of P59, P65 P148 and P205, the proposed algorithm can obtain the smallest value in most cases. Therefore, the PE-MHS algorithm has the best performance among the four tested algorithms for different problems. However, due to the complexity of the problem studied, the part values obtained by the PE-MHS algorithm are worse than the others.
The above computational results show that the PE-MHS algorithm and the proposed approach for solving MOATALBP are very beneficial for the balance optimization of twosided assembly lines during production. For small-scale problems, the proposed algorithm is able to obtain smaller objective values relative to others. The values of ASDF and SP indicate that the proposed PE-MHS algorithm can give Pareto solutions of more even distribution on front as compared with the solutions obtained from other algorithms. In addition, the proposed algorithm can get more average number of solutions than others. As the number of solutions increases, more alternatives are available to decision makers.

VI. CONCLUSION AND FUTURE RESEARCH
The MOATALBP is addressed in this paper. Workers with different skills are assigned to corresponding stations, with the different skills required for each task taken into account. Assembly line efficiency, smoothing index and workers cost are used as objective functions. In addition, positioning constraints, zoning constraints, and synchronizing constraints are considered in the MOATALBP. The PE-MHS algorithm is constructed for the MOATALBP. In this algorithm, the concept of Pareto entropy is introduced to determine the evolution state of the population and to adjust the parameters adaptively. It can effectively balance the exploitation and development ability of the algorithm, and improve the accuracy of finding the optimal solution. The operation of insertion and reverse sequence enhances the local search capability of the algorithm. If the task violates the precedence  constraints, the repair strategy moves it after the maximum immediate predecessor. It ensures that a reasonable task allocation scheme is obtained. Therefore, the above improvements enhance the optimization performance of the proposed algorithm.
A series of experiments prove the outstanding performance of the proposed PE-MHS algorithm. Experimental comparisons of different repair strategies prove the effectiveness and stability of the proposed strategy of moving the task violating to any position after the maximum immediate predecessor. The solutions of different problems based on PE-MHS, NSGAII and SPEA2 algorithms are compared. Results indicate that the Pareto solutions obtained from the proposed PE-MHS algorithm are better than the solutions obtained from others in most cases. The distribution of Pareto solutions based on three large-size problems (P59, P65, P148) of PE-MHS, NSGAII, SPEA2 and HS algorithms is compared by the values of ASDF and SP. Results indicate that the proposed algorithm has excellent quality of Pareto solutions as compared with other three algorithms.
The proposed model helps to improve the assembly line efficiency, ensures the balance of the station load, and reduces the workers cost. Therefore, it can increase the economic efficiency of the line. In the future, the PE-MHS algorithm can be applied to other problems, such as the U-shaped assembly line and the parallel assembly line. Moreover, some more realistic constraints can be considered for TLABP to enhance significance of the research.