Two-sided assembly line balancing using teaching–learning based optimization algorithm

Accepted Manuscript Two-sided assembly line balancing using teaching-learning based optimization algorithm Gonca Tuncel, Dilek Aydin PII: DOI: Reference:

S0360-8352(14)00186-7 http://dx.doi.org/10.1016/j.cie.2014.06.006 CAIE 3732

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

17 August 2013 21 March 2014 8 June 2014

Please cite this article as: Tuncel, G., Aydin, D., Two-sided assembly line balancing using teaching-learning based optimization algorithm, Computers & Industrial Engineering (2014), doi: http://dx.doi.org/10.1016/j.cie. 2014.06.006

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Two-sided assembly line balancing using teaching-learning based optimization algorithm

Gonca Tuncel* and Dilek Aydin

Dokuz Eylul University, Dept. of Industrial Engineering, Buca-Izmir, Turkey

* Corresponding Author; E-mail: [email protected] (Gonca Tuncel) Postal Address: Dokuz Eylul University, Department of Industrial Engineering, Tinaztepe Campus, Buca - Izmir, 35160 Turkey Phone: 00902323017617, Fax: 00902323017608

Assembly line balancing plays a crucial role in modern manufacturing companies in terms of the growth in productivity and reduction in costs. The problem of assigning tasks to consecutive stations in such a way that one or more objectives are optimized subject to the required tasks, processing times and some specific constraints is called the Assembly Line Balancing Problem (ALBP). Depending on production tactics and distinguishing working conditions in practice, assembly line systems show a large diversity. Although, a growing number of researchers addressed ALBP over the past fifty years, real-world assembly systems which require practical extensions to be considered simultaneously have not been adequately handled. This study deals with an industrial assembly system belonging to the class of two-

sided line with additional assignment restrictions which are often encountered in practice. Teaching-Learning Based Optimization (TLBO), which is a recently developed natureinspired search method, is employed to solve the line balancing problem. Computational results are compared with the current situation in terms of the line efficiency, and the solution structure with workload assigned to the stations is presented. Keywords: Two-sided assembly lines; line balancing; teaching-learning based optimization; application, manufacturing

1. Introduction Assembly lines are the mostly used technique in mass production, as they enable the assembly of complicated products by operators with restricted training and devoted robots and/or machines. Thus, production of standardized similar products is performed in cost efficient flow-line systems. A classic assembly line is composed of serial stages, in which workpieces (jobs) are flowed down the line and transferred from one workstation to the other through workforce or material handling equipment. At each stage, definite assembly operations are completed repeatedly in order to obtain finished products. The tasks are allocated to workstations considering some restrictions including precedence constraints, number of workstations, cycle time and incompatibility relations between tasks. The problem of assigning jobs to consecutive workstations that one or more goals are optimized based on the required tasks, processing times and some particular constraints are named the Assembly Line Balancing Problem (ALBP). The process of balancing is a crucial task in designing highly efficient and cost effective assembly lines. The establishment or re-arrangement of a line is quite an expensive investment so effective regulations of lines are essential at the beginning of process. Lines need to be balanced in the design stage; otherwise unbalanced lines cause inefficiency in production, increased cost, and a lot of casualties such as waste of labor or equipment. Since the classical ALB problem was first described in 1955 by Salveson, many studies have been done with regard to assembly line design problems. Researchers have focused on improving qualified and fast solution approaches for solving the line balancing problem in assembly systems. In the first researches, the authors studied on mostly minimizing number of workstations and used mathematical modeling methods, e.g. integer programming and goal programming. Then, they head towards heuristic approaches to handle large size problems.

Based on the restrictions on operation directions, assembly lines can be classified as onesided and two-sided assembly lines. Two-sided assembly lines are usually designed to produce high-volume large-sized standardized products, such as automobiles, trucks, buses and home appliances, in which some tasks must be performed at a specific side (left-side or right-side) of the product. Although a large number of methods for solving one-sided assembly line balancing problem have been studied in literature, little attention has been paid to balancing of two-sided assembly lines (Simaria & Vilarinho, 2009). In the last decade, the researchers started to study on two-sided assembly lines that are recognized to be of crucial importance in real life (Lee et al., 2001; Wu et al., 2008; Ozcan & Toklu, 2009a; Ozcan & Toklu, 2009b; Kim et al., 2009; Simaria & Vilarinho, 2009; Ozcan, 2010; Ozbakır & Tapkan, 2011; Chutima & Chimklai, 2012; Tapkan et al., 2012). However, problems considered in these studies were generally test problems from the literature. Real-world assembly systems which require practical extensions to be considered simultaneously have not been adequately handled by the authors. The challenge lies in putting theory into practice, which involves simultaneously handling efficiency, practical assignment restrictions and competitiveness. This study deals with an industrial assembly system belonging to the class of two-sided line with additional assignment restrictions which are often encountered in practice. A novel optimization method, Teaching-Learning Based Optimization (TLBO) Algorithm, is employed for solving two-sided assembly line balancing problem. TLBO algorithm was proposed by Rao et al. (2011) for the optimization of mechanical design problems, and then it has been applied to various engineering problems including some unconstrained and constrained non-linear programming problems by its developers and several other researchers (Rao & Kalyankar, 2011; Satapathy & Naik, 2011; Amiri, 2012; Rao & Patel, 2012; Naik et al., 2012; Togan, 2012; Roy& Bhui, 2013; Singh et al., 2013). Recently, the performance of the TLBO algorithm on some combinatorial optimization problems, namely flow shop and job shop scheduling problems is tested by Baykasoglu et al. (2014). The authors concluded that the performance of the TLBO algorithm is comparable with the best known solutions from the literature. Assembly line balancing is one of the most important problems in the field of production management and takes part in the NP-hard class of combinatorial optimization problems. This study is the first to apply the TLBO algorithm to the assembly line balancing problem. The remainder of this paper is organized as follows. A brief introduction to basic characteristics of two-sided assembly lines is given in the next section. In Section 3, we described an industrial assembly system, which can be characterized as a two-sided assembly

line. In order to improve the line balance implemented by the company for a given cycle time, the assembly line balancing problem is solved by using a population based optimization algorithm, TLBO, which simulates the teaching–learning process within a classroom environment. Computational results are compared with current state in terms of the line efficiency, and the solution structure with workload assigned to the stations was presented in Section 4. Finally, the summary and concluding remarks are presented in Section 5.

2. Two-sided Assembly Lines

Depending on production tactics and different conditions in practice, assembly line systems show a large diversity; therefore they can be classified in various ways. One of those is to categorize ALs based on the restrictions on operation directions. If only one side (left or right) is used in an assembly line, then it is called as one-sided assembly line. Most of the studies in the literature dealt with balancing of one-sided assembly lines. A two-sided assembly line is a type of production line in which different assembly tasks are performed in parallel at both sides of the line as shown in Figure 1. In this case, some of the assembly operations should be performed at strictly one side of the line (right or left side) and the others can be assigned to either side of the line. Thereby, tasks are classified into three types according to the restrictions on the operation directions: L (left), R (right) and E (either)-type tasks. Two-sided assembly lines are usually designed to produce large-sized high-volume products such as automobiles, buses, trucks, and some home appliances. These lines have some advantages over one-sided assembly lines: (i) shorter line length (ii) reduced throughput time, worker movements, and setup time (iii) lower cost of tools and fixtures (iv) less material handling (Bartholdi, 1993).

Insert Figure 1 here

Besides the cycle time and precedence restrictions, the following constraints may appear in a two-sided assembly line balancing problem (TALBP) in practice:

Synchronization constraints: The essential difference between the assignment of tasks in one-sided lines and in two-sided lines is mainly related with the sequence in which the tasks are performed. In one-sided lines, the sequence of the tasks within a workstation is not

important considering providing precedence relations. But in two-sided lines, sequencing is a critical job for an efficient assignment of tasks. If a task has synchronization constraint, it has to be assigned to a workstation at the opposite side of the line where its mated-task was started in parallel. Otherwise, tasks performed opposite sides of the line can conflict with each other through precedence constraints which might cause idle time if a workstation needs to wait for a predecessor task to be completed at the opposite side of the line (Gunasekaran & Sandhu, 2010). Task zoning constraints: Some zoning restrictions constrain the assignment of various operations to a specific station which is named positive zoning constraints and others forbid the assignment of operations to the same station which is named negative zoning constraints. Positive zoning constraints are mostly related with the usage of common equipment or tooling. Hence, some of the operations are needed to assign to the same workstation. Negative zoning constraints are usually related with the technological issues. It may not be possible to perform some tasks in the same workstation because of safety reasons or any other causes. Workstation related constraints: Some operations need particular equipment or material that is only available at a certain workstation so these tasks should be assigned to that workstation. Position related constraints: In producing of the large and heavy workpieces, they have a fixed position and cannot be turned. In this case, we come up position related constraints which are commonly faced in real-world problems (Tuncel and Topaloglu, 2012). Then, tasks are grouped according to the position in which they are performed. Operator related constraints: Some tasks need different levels of skill depending on the operation complexity. Assigning a qualified operator to a determined task is better to combine more monotonous tasks and more variable tasks in the same workstation in order to induce higher levels of job satisfaction and motivation. Although there are several studies in literature that consider TALBP with additional assignment restrictions such as synchronization constraints and position related constraints, none of them takes into consideration more than one constraints including positive and negative zoning constraints.

3. System Description In this study, we considered a TALBP in an international home appliances company. The company is one of the leading manufacturers and distributors of major domestic appliances in

Europe. In the plant under consideration, cooling products are produced which can be ranged with several main models. Production process involves 7 assembly lines, and each assembly line is composed of two stages. In the first stage, products are processed on pre-assembly lines, where tasks are performed with inner liner of refrigerator and side-and-top panels. These tasks include drilling, banding, siliconizing, fixing, and placing some components on inner liner so that two opposite workers can execute various operations on the same individual item in parallel. After pre-assembly stage, products are transferred to assembly line where doors, internal-external accessories, and thermodynamic components are installed. Data of the assembly line is presented in Table 1. There are 70 tasks; each of them has a specific operation direction (e.g., left, right or either side). The second column lists operation times of each task in terms of cts. Letters written in the third column indicate the direction of related operation. L shows that task should be assigned to the left side of the line, R shows that task should be assigned to the right side of the line, E represents that task can be assigned either side of the line. Figure 1 below illustrates the precedence diagram that indicates the order of performing each task. Red circles represent dummy nodes with

.

In addition to the precedence relations among the tasks, the following assignment restrictions should also be taken into account while balancing the two-sided ALBP considered in this study: 1. Total task times in the first station has to be less or equal to 137 cts because of the machine capacity existed in this station. 2. Tasks 1 and 2 should be assigned to station 1 because of the equipment available only at this workstation (workstation related constraint). 3. Tasks 19 and 25 should be performed together and therefore assigned to the same station. (inclusion/positive zoning constraint) 4. Tasks 20 and 27 should also be performed together and therefore assigned to the same station. (inclusion/positive zoning constraint) 5. Tasks 69 and 70 represent quality control and fixture operations, respectively. Thus, their operation times are changeable in some cases. In order to reduce the cycle time variation, these tasks should be executed on the different stations with no other tasks assigned. (exclusion/negative zoning constraint) 6. The pair of tasks (3-4), (28-33), (30-40), (43-45), and the set of task groups (18-19-21) and (20-23-24), (61-62-63-64) and (65-66-67-68) should be performed at the opposite sides of a workstation (synchronization constraint).

7. Right tasks have to be assigned to a station at the right side of the line, and left tasks have to be assigned to a left station. Tasks, which can be done at any side of the line, should be assigned to a right or left station considering sum of the task times at each station. (side restrictions) 8. Sum of the task times assigned to a station should not exceed the cycle time (cycle time constraint).

Insert Figure 2 here

Insert Table 1 here

Insert Table 1 (continued) here

Initial workloads of the workstations according to the current layout of the assembly line are depicted in Figure 3. Cycle time (239cts) is marked with red line in the graphic. It can be easily noticed that workstations are not fulfilled and there are quite idle times with respect to the cycle time of 239 cts. Main goal is to improve the line balance implemented by the company for the given cycle time and also considering the additional aim of smoothing the workloads between workstations (i.e., to decrease the workload difference between stations).

Insert Figure 3 here

4. Balancing of the TALBP using Teaching-learning Based Optimization Algorithm

Teaching-Learning Based Optimization (TLBO) algorithm is a nature-inspired search method that has been recently introduced for solving large scale optimization problems with less computational effort and high consistency. TLBO, as a population based algorithm, is inspired by passing on knowledge within a classroom environment. The essential advantages of the TLBO algorithm over other evolutionary methods such as genetic algorithms, ant colony optimization, harmony search, and particle swarm optimization are its simplified numerical structure (Togan, 2012). The basic philosophy of the method is based on the two factors: (i) influence of a teacher on the output of learners in terms of results or grades, and

(ii) interaction between learners, which also helps in their results. Thus, the learning process consists of two fundamental parts: “teacher phase” and “learner phase” (Rao et al., 2011; 2012): Teacher phase represents the learning mode where the students acquire knowledge from the teacher. The teacher tries to increase the mean of class from any value (Ma) to his/her level (Ta). But in real life, it is not completely possible and a teacher just can raise the mean of class from Ma to Mb which is higher than Ma depending on many factors (e.g. capability of the class etc). In this aspect, teaching role is assigned to the person who has the best level in the class (Xteacher). The algorithm tries to increase other persons’ level (Xi) to teacher’s level considering the current mean (Xmean). Let Mj be the mean and Ti be the teacher at any iteration i. Now Ti will try to improve existing mean Mj towards it so the new mean will be Ti designated as Mnew. It is obvious that the quality of a teacher also affects the outcome of the learners. The difference between the existing mean and new mean is given by Rao et al, 2011. Difference_Meani = ri (Mnew − TFMi)

(1)

where TF is a teaching factor that decides the value of mean to be changed, and ri is a random number in the range [0, 1]. The value of TF can be either 1 or 2, which is a heuristic step, and it is decided randomly with equal probability as: TF = round[1 + rand(0, 1) {2 − 1}].

(2)

The existing solution is modified according to the following expression,

Xnew,i = Xold,i + Difference_Meani.

(3)

Learner phase of the algorithm simulates the learning of the students (i.e. learners) through interaction among themselves, where a solution is randomly interacted to learn something new with other solutions in the population. Thus, Xi tries to improve his/her knowledge by learning from Xii (i.e. a classmate) which is assumed to have better knowledge than Xi. In this case, Xi is moved to Xii. Otherwise, it means that Xii is not better than Xi, then Xi is moved away from Xii. Learner modification of this phase is expressed below.

Xnew, i = Xold,i + ri* (Xii - Xi)

if

f(Xi) < f (Xii)

(4)

Xnew, i = Xold,i + ri* (Xi - Xii)

if

f(Xii) < f (Xi)

(5)

The TLBO requires only controlling parameters such as population size and number of generations which are common in running any population based optimization algorithms. The population and the candidate solution in the TLBO’s representation correspond to a class and a student in that class, respectively. Each subject taught to the students describes the design variable (e.g. assignment of tasks to workstations). The candidate solution composes of design variables (e.g. a line balance) and is qualified according to its fitness. The procedure for implementing the TLBO algorithm is briefly described with the following steps.

Step 1: Define the optimization problem and initialize the optimization parameters such as the population size, number of generations, and number of design variables. Step 2: Generate a random population according to the number of students (i.e. learners’) in the class and design variables which indicate the subjects (i.e. courses) offered to the learners. Evaluate the mean grade of each subject offered in the class. Step 3: Based on the overall grade (objective value) of each individual learner, select the best learner as a teacher and calculate mean result of learners in each subject. Step 4: Evaluate the difference between current mean result and best mean result according to Eq. (1) by utilizing the teaching factor (TF) (Eq. (2)). Step 5: Update the learners’ knowledge (i.e. the grade point of each subject) with the help of teacher’s knowledge according to Eq. (3). Step 6: Update the learners’ knowledge through the mutual interaction with the other learners according to Eqs. (4) and (5). Step 7: Repeat the procedure from step 3 to 6 until the termination criterion is met.

For further reading see in Rao & Patel, 2012 and Rao et al., 2012.

4.1 Implementation of the TLBO algorithm

This section is devoted to the explanation of important features and the details of the proposed TLBO algorithm for two sided assembly line balancing problem (TALBP). In TALBP a learner is constructed as a priority list (PL).

PL={xi(1), xi(2), .., xi(n)}, is

represented as a n-dimensional real-number vector, where xi denotes the priority of task i. The priorities of the tasks (priority vector) are generated by using the following equation. xi= 1 + rand[0,1]×(UB − 1)

(6)

where; rand[0, 1] represents a uniformly distributed random value that ranges from zero to one and UB represents the maximum number of tasks. It is necessary to convert priority vector to a task permutation for evaluating the objective function value. The tasks are assigned to the stations sequentially by the priority value of tasks. The position of a PL represents a task i, and the value of the position represents the priority value of task i (PRi). The length of a PL is equal to the number of tasks (n). For obtaining a feasible line balance, the tasks are assigned to stations as follows: The first assignable task which has the highest priority value is assigned to the first mated station according to its preferred operation direction. Then the set of assignable tasks is updated and this process continues until all tasks are assigned (see decoding algorithm (Ozcan & Toklu (2009b)). At any iteration of the solution process, a feasible solution is obtained by using the decoding algorithm. The terms and variables used in the decoding algorithm are given as follows.

C is the cycle, AT(.) is the set of assignable tasks, IPi(.) is the list of immediate precedence(s) of task i, STi is starting time of task i, FTi is finishing time of task i, d is the direction of task i, r is the index number of right-side station, Sr is the station times of right-side station, l is the index number of left-side station, Sl is the station times of left-side station. Step 1. Set r=1, l=1, Sr=0, Sl=0 and go to Step 2. Step 2. Determine AT(.). If AT(.) is not include any task go to step 5. Otherwise go to Step 3. Step 3. Sort the tasks in AT(.) in decreasing order of PL(.) and go to Step 4.

Step 4. Remove the first task i from AT(.) and go to Step 5. Step 5. Assign the first task h in AT(.) for which: a. If dh=R, th+Sr≤C, and th+FTa≤C (FTa=max{FTp|p∈IPh have already been assigned to the companion of this station}), then assign task h to station r and set to STh=max{[Sr], [FTa]}, FTh=th+STh, Sr=FTh, and go to step 2. Otherwise, set l=l+1, r=r+1, Sr=0 and Sl=0 and go to step 2. b. If dh=L, th+Sl≤C, and th+FTa≤C (FTa=max{FTp|p∈IPh have already been assigned to the companion of this station}), then assign task h to station l and set to STh=max{[Sl], [FTa]}, FTh=th+STh, Sl=FTh, and go to step 2. Otherwise, set l=l+1, r=r+1, Sr=0 and Sl=0 and go to step 2. c. If dh=E, then generate a random number rn for task h between 0 and 1 using uniform distribution. If rn<0.5, then go to step 5a. Otherwise, go to step 5b. Step 6. Calculate the objective function value.

Once a task assignment is done according to the decoding algorithm, positive and negative zoning constraints are also checked. If the zoning constraints are not fulfilled, then a high penalty cost is added to the objective function (z =1000), which is infeasible because of the assignment restrictions. The objective of TALBP under consideration is to minimize the number of workstations as a primary goal and to smooth the workload balance of the workstations (i.e., to distribute the workload evenly as possible to the workstations) as a secondary goal given in Eq. (7) (Vilarinho & Simaria, 2006).

min z = K + Cb

(7)

where, K

total number of workstations utilized on the line

T

total idle time of all workstations

Cb

line smoothness index

Ik

idle time at workstation k

C

cycle time

Wk

total workload assigned to workstation k

Eq. (8) calculates the total idle time of all workstations, and Eq. (9) calculates the smoothness index among workstations within the value range [0, 1). When the smoothness index is closer to 0, the idle time is more evenly distributed.

Insert Figure 4 here

The usage of the proposed TLBO algorithm is demonstrated using an example of the twosided ALB problem, which is obtained from Lee et al., 2001. The problem is represented by the precedence diagram that is shown in Figure 4. A circle indicates a task. Each task is associated with a label (ti, d), where ti is the ith task processing time and d=(L, R, or E) denotes the preferred operation direction of task i. The arrows linking two tasks represent the precedence relation between tasks. Suppose that the cycle time is fixed to 22. The iterations of the procedure for task assignments to the first two stations are given in Appendix A. A random key based approach is used for obtaining task priorities. In Appendix B, an assignment of tasks to workstations obtained from ith learner at any time is shown in a Gantt chart, and the objective function value of the example is turned out to be 6.049.

4.2. Computational Results

In the preliminary parameter adjustment experiments, the following levels were selected for the operation parameters of the TLBO: population size (i.e. number of learners) = 40, maximum number of generations = 100, and number of replications = 10. The TLBO algorithm is coded in Matlab 7.10.0. The implementation results are presented in Table 2. As it can be seen from the results, the best solution is obtained on the first replication with 9 workstations and the smoothness index of 0.131.The average number of workstations over 10 replications of the algorithm is equal to 9.70 and its variance is 0.233. It is noted that the proposed TLBO algorithm provided solutions with low variance in short computational time. The workload balance between workstations and new layout achieved by using TLBO algorithm is depicted in Figure 5 and 6, respectively.

Insert Table 2 here

Insert Figure 5 here

Insert Figure 6 here

In Table 3, the solution obtained by using TLBO algorithm is compared with the current line balance in the production system in terms of the line efficiency, number of workers, and workload balance of the workstations.

Insert Table 3 here

It is observed from the results of Table 3 that, the TLBO algorithm provided a line balance with 9 operators and less idle time at each workstation compared to the initial line configuration. Additionally, the smoothness index (Cb) in new layout is also lower. We can conclude that we have an improvement rate about 35% in terms of workload balance between workstations. Moreover, line efficiency increased from 63% to 85% with TLBO approach. On the other hand, if we had excluded the scynchronization restrictions and positive/negative zoning constraints, we would have reached higher line efficiency with fewer workers. However, these practical constraints should be also taken into consideration in order to get an applicable solution to the considered problem.

5. Conclusion

In this study, we considered a real life two-sided ALBP with additional assignment restrictions. Thus, we take into account operating sides of tasks in addition to precedence and cycle time constraints, when the allocation of the tasks to an ordered sequence of workstations is determined. Moreover, the problem considered in this paper involves several compatible and incompatible zoning constraints. Accordingly, some groups of tasks must be executed together on the same station (compatible tasks) and other tasks were prevented from being assigned to the same station (incompatible tasks). Finally, the problem is also composed of several tasks which should be assigned to the different stations with no other tasks assigned

(negative zoning constraint). Objective function was to minimize number of workstations and to ensure a smooth distribution of workload between workstations. We used teaching-learning based optimization algorithm to obtain high quality solutions in a very short computational time. In this solution, current line was balanced with fewer workstations in the new configuration with lower smoothness index. We achieved a considerably improvement in the distribution of workloads between the workstations, and significant reduction in total number of operators for the assembly system. Two-sided assembly lines have more complicated structures than the other types of assembly lines. Thus, we can face additional conditions in real world such as symmetric tasks, task separation, station layout, parallel stations, setup times, multi or mixed model, capacity of machines in stations (local cycle times) and ergonomic constraints. Beyond benchmark/test problems, if real-world problems are addressed by including above conditions, gap between real world and academic world can be shortened. Moreover, teaching-learning based optimization algorithm is a quite new method within the other meta-heuristics methods. We can see from the relevant literature that this algorithm give promising results. As a further research area, its phases can be improved using different teachers in teacher phase and using tutorials or self-learning capability in learner phase.

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LIST OF CAPTIONS:

Figure 1. Configuration of a two-sided assembly line Figure 2. Precedence diagram Figure 3. Initial workloads of the workstations Figure 4. Example of two-sided ALB problem Figure 5. The line balance using the TLBO algorithm Figure 6. The new layout of the assembly line

Table 1. Data of the problem Table 1. (continued) Data of the problem Table 2. Implementation results Table 3. Summary of the computational results

Appendix A. The iterations of the TLBO for the example problem Appendix B. Representation of a balancing solution

APPENDİX A. The iterations of the TLBO for the example problem Suppose that ith learner obtained from the TLBO at any time is as follows; Task index Learner Priority

1 3.12 9

2 -1.45 15

3 0.13 13

4 9.65 4

5 4.37 8

6 6.33 5

7 1.34 12

8 15.81 1

9 2.65 10

10 4.95 7

11 -0.12 14

12 -3.45 16

13 1.78 11

14 10.38 3

15 5.89 6

16 11.35 2

Step 1. r=1, l=1, Sr=0, and Sl=0. Step 2. AT={1,2} Step 3. PL={15, 9}, AT={2, 1} Step 4. Remove the task 2 from AT(.). Step 5c. rn=0.38 If rn<0.5, then go to step 5a. Step 5a. dh=R, 5+0≤22, and 5+0≤22, then assign task 2 to station r=1 and set to STh=0, FTh=5+0, Sr=5, and go to step 2. Step 2. AT={1,5} Step 3. PL={9, 8}, AT={1, 5} Step 4. Remove the task 1 from AT(.). Step 5c. rn=0.63 If rn>0.5, then go to step 5b. Step 5b. dh=L, 6+0≤22, and 6+0≤22, then assign task 1 to station l=1 and set to STh=0, FTh=6, Sl=6, and go to step 2. Step 2. AT={3,4,5} Step 3. PL={13, 8, 4}, AT={3, 5, 4} Step 4. Remove the task 3 from AT(.). Step 5b. dh=L, 2+6≤22, and 2+6≤22, then assign task 3 to station l=1 and set to STh=6, FTh=2+6, Sl=8, and go to step 2. Step 2. AT={4,5,6} Step 3. PL={8, 5, 4}, AT={5, 6, 4}

Step 4. Remove the task 5 from AT(.). Step 5a. dh=R, 8+5≤22, and 8+5≤22, then assign task 5 to station r=1 and set to STh=5, FTh=13, Sr=13, and go to step 2. Step 2. AT={4,6} Step 3. PL={5, 4}, AT={6, 4} Step 4. Remove the task 6 from AT(.). Step 5b. dh=L, 4+8≤22, and 4+8≤22, then assign task 6 to station l=1 and set to STh=8, FTh=12, Sl=12, and go to step 2. Step 2. AT={4} Step 3. PL={4}, AT={4} Step 4. Remove the task 4 from AT(.). Step 5c. rn=0.12 If rn<0.5, then go to step 5a. Step 5a. dh=R, 9+13≤22, and 9+6≤22, then assign task 4 to station r=1 and set to STh=13, FTh=22, Sr=22, and go to step 2. Step 2. AT={7} Step 3. PL={12}, AT={7} Step 4. Remove the task 7 from AT(.). Step 5c. rn=0.41 If rn<0.5, then go to step 5a. Step 5a. dh=R, 7+22>22, and 7+22>22 then set r=2, l=2, Sr=0, Sl=0 and go to step 2. Task assignments to the first station are completed. Step 2. AT={7} Step 3. PL={12}, AT={7} Step 5c. rn=0.88 If rn>0.5, then go to step 5b. Step 5b. dh=L, 7+0≤22, and 7+0≤22, then assign task 7 to station l=2 and set to STh=0, FTh=7, Sl=7, and go to step 2. Step 2. AT={8,9,10} Step 3. PL={10,7,1}, AT={9,10,8} Step 4. Remove the task 9 from AT(.). Step 5a. dh=R, 5+0≤22, and 5+7≤22, then assign task 9 to station r=2 and set to STh=7, FTh=12, Sr=12, and go to step 2. Step 2. AT={8,10,12} Step 3. PL={16,7,1}, AT={12,10,8} Step 4. Remove the task 12 from AT(.). Step 5b. dh=L, 5+7≤22, and 5+12≤22, then assign task 12 to station l=2 and set to STh=12, FTh=17, Sl=17, and go to step 2. Step 2. AT={8,10 } Step 3. PL={7,1}, AT={10,8} Step 4. Remove the task 10 from AT(.). Step 5a. dh=R, 4+12≤22, and 4+7≤22, then assign task 10 to station r=2 and set to STh=12, FTh=16, Sr=16, and go to step 2. Step 2. AT={8,13 }

Step 3. PL={11,1}, AT={13,8} Step 4. Remove the task 13 from AT(.). Step 5c. rn=0.33 If rn<0.5, then go to step 5a. Step 5a. dh=R, 6+16≤22, and 6+16≤22, then assign task 13 to station r=2 and set to STh=16, FTh=22, Sr=22, and go to step 2. Step 2. AT={8,16 } Step 3. PL={2,1}, AT={16,8} Step 4. Remove the task 16 from AT(.). Step 5c. rn=0.70 If rn>0.5, then go to step 5b. Step 5b. dh=L, 4+17≤22, and 4+22>22, then then set r=3, l=3, Sr=0, Sl=0 and go to step 2. Task assignments to the second station are completed. … The procedure is repeated until all of the tasks are assigned to stations.

Appendix B. Representation of a balancing solution Station 1

R

L

Task2 5

Task1 6

Task5 8

Task3 2

Task6 4

Station 2

Station 3 Task4 9

Task9 5

Station 5

Task10 4

Task7 7

Task12 5

Station 4

Task13 6

Task8 4

Task16 4

Task11 6

Task14 4

Task15 3

Station 6

Table 1. Data of the problem Task No.

Processing Times- cts

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

82 34 8 11 22 11 9 30 13 38 24 24 20 10 18 10 16 37 34 23 32 16 29 47 7 9 36 10 17 22 9 38 8 11 18 31 10 10 15 29 34 26 16 12 13

Operation directions E E E E E E E E E E E L E L L L E R R L R R E L R R L R R R R R L E R R R L L L E L L L E

Immediate precedence(s) 1 2 3 2 2 2 4,5,6,7 8 8 9,10 11 11 12 11 15 11 13,14,16,17 18 18 19 20,21 13,14,16,17 18,23 19,24 25 20 22,26,27 22,26,27 22,26,27 22,26,27 29 22,26,27 22,26,27 22,26,27 22,26,27 28,30,31,32,33,34,35,36 22,26,27 22,26,27 22,26,27 22,26,27 22,26,27 22,26,27 22,26,27 22,26,27

46 47 48 49

10 19 10 51

L L L L

38,39,40,41,42,43,45,44 47 48

Table 1.(continued) Task No.

Processing Times- cts

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

9 69 53 6 25 15 15 26 25 10 20 6 14 14 28 38 24 15 8 156 162

Operation directions E R R R R R E R R R R L E L E L L L L E E

Table 2. Implementation results Objective function Replication no

value

CPU-time (s.)

1

9.131

20.6987

2

10.143

20.7376

3

10.541

20.5989

4

10.296

20.5804

5

10.145

20.9565

6

10.129

21.2433

7

9.258

20.6703

Immediate precedence(s) 49 51 37,44,50,52 53 53 53 54,55,56,62,63,64 57 58 59 37,44,50,52 61 61 61 54,55,56,62,63,64 65 65 66,67 60,68 69

8

10.210

20.791

9

9.216

20.7008

10

10.340

20.6737

Table 3. Summary of the computational results Number of operators

Line efficiency

Smoothness index (Cb)

Initial Situation

12

63%

0.20

TLBO Algorithm

9

85%

0.13

A real-life two-sided assembly line balancing problem (TALBP) is considered. The problem includes additional assignment restrictions. Teaching-Learning Based Optimization (TLBO) Algorithm is employed. A high quality solution is obtained in a short computational time.

Figure1

Figure2

Figure3

Figure4

(6,E)

(2,L)

(4,L)

(4,E)

(6,E)

(4,E)

1

3

6

8

11

14

(7,E)

(5,R)

(5,L)

(3,E)

7

9

12

15

(8,R)

(4,R)

(6,E)

(4,E)

5

10

13

16

(5,E) 2

(9,E) 4

Figure5

Figure6