Publication: IIE Transactions
Publication Date: 01-APR-07
Delivery: Immediate Online Access
Author: Chiang, Wen-Chyuan ; Kouvelis, Panagiotis ; Urban, Timothy L.
Article Excerpt
1. Introduction
The Assembly Line Balancing (ALB) problem is a well-known problem in which a finite set of tasks or work elements are assigned to workstations subject to given precedence relationships. With Type I ALB problems, the objective is to minimize the number of workstations needed for a given production rate or cycle time. Type II ALB problems seek to maximize output (minimize the cycle time) for a given number of stations. Various economic criteria, using operating cost or profitability measures as the objective, have also been investigated.
Recently, however, many companies have established multiple lines, with the lines having longer cycle times and each station being responsible for more tasks than under the traditional model. One of Schonberger's principles of world-class, customer-driven performance (Schonberger, 1990) is "seek to have plural instead of singular ... flow lines for each product or customer family". Each cell consists of small, multi-functional work teams; Sekine (1992) goes so far as to suggest that "the production line should consist mainly of single-operator U-shaped cells".
The use of multiple lines also allows each cell to have a backup. Therefore, if a machine in a cell breaks down, this redundancy will allow the company to continue production (Harmon and Peterson, 1990). Harmon (1992) also notes that redundancy of processes can facilitate necessary adjustments to capacity, since changes in capacity are less costly with smaller-sized equipment and can be accomplished with lower levels of inventory. Merli (1990) states that dismantling extended, high-output flow lines with several low-output short lines can allow the production of extremely small economical lots and greater flexibility in terms of production volume.
Shtub (1984) recognizes the value of implementing multiple lines for assembling heavy products; when the cycle time and the length of each shift cause a synchronization problem, this can be solved by using parallel assembly lines at their "natural cycle time". Ghosh and Gagnon (1989) identify several suggestions for future research in assembly systems, one of which is to "allow some strategic design considerations (such as the ... number of lines ...) to be variables". Although numerous line-balancing procedures have been developed that incorporate parallel stations and tasks (see Becker and Scholl (2006) for a review), no one has yet investigated the line-balancing problem that incorporates the possibility of several, independent lines.
As an example, consider the 11-task problem from Jackson (1956). Using traditional line-balancing techniques requiring an output of 62 units per 480-minute day (a cycle time of 7.74 minutes), the optimal solution is to establish eight stations, each with one or two tasks. If we were to allow multiple stations, we can establish three lines, each with two stations and a cycle time of 23 minutes, resulting in a reduction of two stations. In addition to the benefits of establishing multiple lines discussed above, we can meet the required output with a 25% reduction in the numbers of stations required. The need for fewer stations, and the associated reduction in idle time, will result in lower labor costs, floorspace requirements, work-in-process inventory, etc.; although this would likely require additional fixed equipment costs (Daganzo and Blumenfeld, 1994).
Another consideration in the application of Just-In-Time (JIT) production principles is the utilization of U-shaped lines, such that the entrance and exit are at the same position. Monden (1983) notes that the most important advantage of this layout is the flexibility to change the number of workers as the demand changes. Furthermore, with the U-line entrance adjacent to the exit, the JIT "pull" production can easily be maintained, as one unit of material can be introduced into the system as one unit of output is completed. Other advantages of a U-shaped production line include a reduction in work-in-process inventory and wasted movement of operators (Hirano, 1988) as well as improved material handling (Sekine, 1992). On the other hand, there may be a need for a greater number of machines and for more versatile personnel (Merli, 1990) as well as reduced learning effects (as each worker performs fewer cycles per unit of time). It could, though, be argued that requiring versatile personnel is beneficial, as it is associated with job enrichment and job enlargement that may better suit modern workers.
The U-Line Balancing (ULB) problem was introduced and modeled by Miltenburg and Wijngaard (1994); they developed a dynamic programming procedure to solve problems with up to 11 tasks and a heuristic to solve larger problems. Urban (1998) presented an integer programming formulation of the problem, optimally solving problems up to 45 tasks with commercially available, general-purpose software. Scholl and Klein (1999) have since proposed a branch-and-bound procedure that was found to be effective in solving problems with up to 297 tasks. Recently, the N U-line balancing problem has been investigated (Miltenburg, 1998; Sparling, 1998; Sparling and Miltenburg, 1998); however, these studies consider lines in which stations may include tasks from adjacent lines, not multiple, independent lines.
Thus, the purpose of this paper is to introduce and characterize the Multiple U-Line Balancing (MULB) problem. The MULB problem is modeled, and mixed-integer linear programs are developed to solve the problem with multiple, identical lines as well as with multiple lines of varying sizes. An efficient procedure is proposed to solve large MULB problems, and is shown to optimally solve instances with over 100 tasks. This analysis is then extended to explicitly incorporate the effect of line configuration and task assignment on equipment requirements; a branch-and-bound algorithm is developed that can be used to identify optimal solutions for moderate-sized problems, or as a heuristic for large problems. Finally, computational results are presented concerning the effectiveness of the proposed procedures in solving the MULB problem.
2. The MULB problem
Consider the situation in which we have a number of tasks, N, each with given precedence relationships and completion times, [t.sub.i] (i = 1,... N), that are to be assigned to stations (j = 1,..., [H.sub.k]) on each line (k = 1,..., l) that is established. Let R = {1,..., r,..., |R|} represent a set of ordered pairs of tasks reflecting the precedence relationships; for example, r = (p, s) is the ordered pair indicating task p immediately precedes task s.
Let us first examine the Type I MULB problem, in which the objective under consideration is to minimize the total number of stations required to meet a given output per unit time, D = 1/C, where C is the required cycle time of the process. We also assume that the resulting number of lines must remain within a closed interval [l.sup.-], [l.sup.+]. Practically speaking, the lower limit will likely have a value of one; however, it may be desirable to place a limit on the maximum number of lines, due to the required duplication of equipment, span of control, etc. (this limit may be somewhat subjectively established based on experience; in a later section, we will develop a branch-and-bound technique that explicitly incorporates such costs). In addition to the benefits of using multiple U-lines, the solution to the MULB problem will always provide a configuration with no more stations than the solution to the traditional straight line or ULB problem.
To illustrate the effect of employing multiple U-lines, Table 1 presents the solution to the Jackson problem (Jackson, 1956)--with [l.sup.-] = 1 and [l.sub.+] = [??][SIGMA][t.sub.i]/C[??], where [??]x[??] is the smallest integer greater than or equal to x--over a range of cycle times (note that, unless limits are placed on the maximum number of lines, the upper bound for the number of lines will be the same as the lower bound for the number of stations since, at the extreme, we could have [??][SIGMA][t.sub.i]/C[??] lines with one station each). For large cycle times, using multiple lines provides little advantage over the traditional or single U-line configurations. However, as the cycle time decreases, the advantage of the multiple U-line becomes more apparent. For a cycle time less than nine, the use of multiple U-lines always outperforms the traditional line and usually outperforms a single U-line. As the cycle time is reduced to the point of the largest task time ([t.sub.max] = 7), traditional lines or U-lines become impractical without the use of parallel tasks or stations.
Another effect of employing multiple U-lines is that the flexibility provided by a multiple U-line configuration may enable a large number of alternate optimal solutions. Consider the Jackson problem with a system cycle time of 7.9. One solution is to have three identical lines, each with two stations and a cycle time of 23 (tasks A, B, I, J, and K assigned to the first station; C, D, E, F, G, and H assigned to the second). Another solution is to have two lines of varying size: one with four stations and a cycle time of 12 (tasks A, B, and K assigned to the first station; C, F, and J assigned to the second; D, G, and H to the third; E and I to the fourth), another with two stations and a cycle time of 23 (tasks A, B, I, J, and K assigned to the first station; C, D, E, F, G, and H to the second). Alternate optimal solutions can be identified for four, five, and six lines, as well. Furthermore, slight changes in the system cycle time, even within the ranges shown in Table 1, may result in different solutions. For example, the...
NOTE: All illustrations and photos have been removed from this article.
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