## 1 Introduction
Therefore, if a location in the entrance sub-line (or exit sub-line) is close to the entrance (or exit), then the corresponding tasks cannot be assigned to the workstation that is further from the entrance (or exit). This is called \"location sequence\" in the present study. A task sequence and a location sequence are illustrated in Figure 5. Figure 5 illustrates a situation where the number of tasks is even, and Figure 5 illustrates a situation where the number of tasks is odd. Moreover, the distances that the operators need to walk and the required travel times to stations are also important criteria. In the case of U-shaped assembly systems with workers covering several stations, these values may be lower than those in the case of straight-line systems. However, the necessary working and walking space must be available [14]. In particular, the minimization of the walking distance problem could be a significant issue when work interference occurs. In this instance, it may have a negative impact on operator productivity. Therefore, walking times should be taken into account to derive the exact cycle time [15]. For example, in Figure 6, although all tasks assigned to locations follow the  
Figure 4: An example of interruption between operators. Figure 5: An illustration of task and location sequences for U-shaped production line. (**a**): Number of tasks is even; (**b**): Number of tasks is odd. task sequence and all locations assigned to workstations follow the location sequence, the tasks in location 1 and 5 are assigned to workstation 2. Locations 1 and 5 are the furthest apart of all pairs of locations, and the walking time between tasks usually depends on the locations of the tasks. Therefore, the design illustrated in Figure 6 would increase the walking time of the operator in workstation 2 and reduce the performance of the entire U-shaped production line. The assembly line balancing problem was first proposed by [PERSON] [16]. When only one single product is considered, the type of problem is classified as a simple assembly line balancing problem (SALBP), the main characteristics of which can be found in [17; 18]. [PERSON] and [PERSON] [19] summarized four types of SALBPs. SALBP-1 aims to minimize the number of stations for a given cycle time. SALBP-2 aims to minimize the cycle time for a given number of workstations. SALBP-E aims to maximize efficiency, thereby simultaneously minimizing cycle time and the number of workstations. Finally, Type-F aims to find a feasible design for a given cycle time and number of workstations. This research deals with SALBP-2, which is encountered frequently when the production rhythm must be adjusted in the manufacturing process. If only one operator is assigned to a workstation in a U-shaped production line, SALBP-2 involves balancing the design of a product type for a given number of operators. This means that, for every product, the cycle times can be determined when given a different number of operators. According to the balancing results, when there is demand for the corresponding products, managers can make appropriate operator assignment decisions based on the demand and the number of available operators. This research aims to balance a U-shaped production line by minimizing the cycle time for a given number of workstations. The task sequence has to satisfy all precedence relationships, and the location sequence has to satisfy the relationship between workstations. Moreover, the walking times between all locations in the U-shaped production line are also considered. LINGO is a simple tool that can utilize the power of linear and nonlinear optimization to concisely formulate large problems and solve them [20]. It employs branch-and-bound methods to break a model down into many convex subregions. It also has a multistart feature that restarts the nonlinear solver from a number of intelligently generated points. This allows the solver to find a number of locally optimal points and report the best one found.