## 1 Introduction  
Conveyors are used when material is to be moved frequently between specific points [1]. When there is sufficient throughput between fixed points, the use of conveyors is suggested [2] for high-efficiency transport over short and medium distances [3, 4]. Conveyors are also used in assembly line systems, generally in two different configurations. In the first, items are transported between machines by conveyors. Once the items arrive at a machine, the operators move them off the conveyor for assembling and then put them back on the conveyor to move them to subsequent machines. In this case, the positions of operators are fixed when assembling items. In the second case, the items may be large or not easily handled, and are not easily moved. The items are fixed to the moving conveyor, and operators execute their assembly tasks moving beside the conveyor as they work. After finishing all their assembly tasks, the operators walk back to their original locations and execute the assembly tasks for the next item. The present study aims to solve an assembly line balancing problem in which the conveyor belongs to the second case. Line balancing involves allocating an equal amount of work to each workstation along the line [5]. The fundamental line balancing problem is to assign a set of tasks to an ordered set of workstations, so that the precedence relationships are satisfied and some measure of performance is optimized [6]. For the second case of conveyor assembly lines, operators have to move along the conveyors while executing assembly tasks and walk back to their original locations after finishing the assembly tasks assigned to them. Walking time between assembly tasks should be taken into consideration when balancing such assembly lines. In traditional straight assembly lines, the walking times are proportional to the corresponding assembly times and can be integrated into assembly time, because the operatorswalk back to their original locations beside the conveyor and the times depend on the distance that the conveyor moves while the operators execute the assembly tasks. A U-shaped conveyor assembly line, however, is divided into two sub-lines: the entrance sub-line and exit sub-line. An operator may be able to execute assembly tasks on both sub-lines within the same work cycle [7], and the estimation of walking times between assembly tasks for each operator is more complicated. For example, in Figure 1, there are 7 assembly tasks. The assembly sequence is 1, 2, 3, 4, 5, 6, and 7, so that no precedence relationships are violated. Tasks 1 and 7 are assigned to Operator 1, Tasks 2 and 6 are assigned to Operator 2, and Tasks 3, 4 and 5 are assigned to Operator 3. According to the sequence and the corresponding assembly times, \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\) indicate the starting and completing points of tasks. The dotted lines with arrows are the walking paths between assembly tasks. For example, Operator 1, executes assembly Task 1 from point \(a\) to point \(b\), taking the assembly time of Task 1, and then walks to point \(g\) without executing assembly tasks. He/she then executes assembly Task 7 from point \(g\) to point \(h\), taking the time of Task 7, and then walks back to point \(a\) without executing assembly tasks. According to the example shown in Figure 1, the time spent walking between assembly tasks depends on the sequence of tasks, the assembly time of tasks, the allocation of tasks to workstations and the configuration of the U-shaped conveyor assembly line. Therefore, it is not easy to take walking times into consideration when balancing the second case of U-shaped conveyor assembly lines where the items are fixed to the conveyor. There are four types of assembly line balancing problem [8]. Type 1 aims to minimize the number of stations for a given cycle time. Type 2 aims to minimize the cycle time for a given number of workstations. Type E aims to maximize the efficiency with a specified cycle time or number of workstations.