Choosing a station loading rule in assembly line design.

ABSTRACT

Paced assembly lines remain the dominant means of efficiently mass producing products such as major appliances and automobiles. Innovations in operating assembly plants make it important to be able to quickly and efficiently balance assembly lines in a practical setting. For example, manufacturers rebalance lines to adjust for demand changes, accommodate specialized equipment, and innovations in work force management. Previous research in assembly line balancing described methods for minimizing the cost of line design under conditions of variable task times. However, there has not been an adequate study of the relative effectiveness of various station loading rules. This paper compares the performance of two commonly used station loading rules in a variety of contexts. The performance of the rules are examined in cases of high and low task time variability, high and low remedial costs when the cycle time is exceeded, 45 and 70 task problems and two actions that can be taken when the cycle time is exceeded. In most of the contexts studied, the probability rule outperformed the percent rule. Significantly, the optimal percentage and probability threshold are not stable and must be determined for each situation.

1. INTRODUCTION

This paper compares the effectiveness of two station loading rules when balancing paced assembly lines with random task times. In the case of deterministic task times, the objective of the assembly line designer is to load each station as closely as possible to the cycle time so as to minimize the number of work stations. If task times are random variables, there is the possibility of workers at a work station exceeding the cycle time, thus incurring a cost of completing the work. By building in slack when making task assignments to stations, the cost of remedial action can be controlled. Two ways of introducing slack are to load stations to a threshold defined in terms of the percent of cycle time or allowed probability of exceeding the cycle time.

2. LOADING RULES

The percent rule disregards task time variance in determining tasks eligible to enter a station. A task is eligible if the station's cumulative mean (the sum of the means for assigned tasks) would be no more than some percentage (often 90%) of the cycle time [Ignall (1965)]. Thus, if two tasks with the same means are being considered for a station, the choice will be made randomly, even if one has a much larger variance than the other. A line design produced by this rule could well have many stations which have a high probability of exceeding the cycle time, even if the station means are well balanced.

The probability rule utilizes both the task means and variances in determining which tasks are eligible to be included in a station [Moodie and Young (1965), Reeve and Thomas (1973) and others]. A task is eligible if the probability of the station completing all tasks--including the one under consideration within the cycle time is greater than a specified value. Thus, high variance tasks would be less likely to be eligible than lower variance tasks with the same mean. Line designs with high probabilities of exceeding the cycle time are precluded. Therefore, more designs of a "higher quality" can be generated within a given number of trials. The consequence of evaluating variance in addition to mean times is to force a more relevant allocation of slack capacity. The probabilities of stations completing their assigned tasks are balanced rather than mean times being balanced. Such lines will be perceived by workers as being more equitable.

3. PURPOSE OF THE STUDY

The purpose of the study is to compare the effectiveness of the two loading rules under a variety of conditions. The conditions for the study are:

The assembly line is stopped to allow the station to complete its assigned work (intermittent line stoppage);

a). There is a high overtime cost of making up for lost production when the line is stopped;

b). There is a low overtime cost of making up for lost production when the line is stopped;

2. The item being produced is allowed to continue to the end of the line, where it is completed (offline repair);

a). There is a high repair rate for completing the work at the end of the line;

b). There is a low repair rate for completing the work at the end of the line;

3. Task times are highly variable;

4. Task times are slightly variable.

In this study, we use the standard assumptions that there is a single, paced assembly line, units produced are identical, workers receive equal pay, tasks can be completed in any order that meets precedent constraints and there is no paralleling. Task times are assumed to be normally distributed independent random variables with known means and variances in place of the standard assumption of constant task times.

4. REMEDIAL ACTIONS

The two remedial actions described above are both used in manufacturing. Intermittent line stoppage is appropriate for the production of heavy, complex products for which an off-line repair would require major disassembly. Such a system is used by the Toyota Corporation in automobile manufacturing. Off-line repair is preferable when products may be easily repaired at the end of the line. Identification of deficiencies could be accomplished either by inspection or by encouraging line workers to attach a report to the production unit, indicating to the end-of-line repair crews what still needs to be done. To be successful, the latter approach depends on providing positive incentives for the worker to supply data which show that he or she was unable to get the job done.

5. LITERATURE REVIEW

The assembly-line balancing problem has received considerable attention in the literature since its formulation by Bryton (1954). A variety of approaches have been proposed to solve the problem, including linear and dynamic programming as well as heuristic techniques. The objective in most papers is to minimize the number of stations for a given cycle time. For example, Carraway (1989) presents two computationally efficient dynamic programming models for solving the problem subject to a lower bound on the probability of the work at any station being completed within the cycle time. However, computational complexity grows geometrically with the number of work elements, so that heuristic techniques have been most popular for problems of real-world complexity.

Relatively few authors have attempted to formally take incompletion costs into account in the process of designing an assembly line with stochastic task times. For lines in which repairs are made off-line, Kottas and Lau (1973), developed a balancing heuristic which selects tasks from an updated "marginal desirability list" containing tasks for which expected labor cost equals or exceeds expected incompletion cost. Being an incremental approach, the procedure builds each station in turn without regard to the potential system's benefit of assigning a currently desirable task to a different station. In a second paper, the same authors presented a total cost model for evaluating any proposed design [Kottas and Lau (1976)]. The model is based on identifying all incompletion combinations, their probabilities and expected total costs. In their third paper, Kottas and Lau (1981) use a probabilistic procedure to generate many premising designs and select the best of these by applying the cost model developed in their second paper. Satin, Erel and Dar-El (1999) use a dynamic programming approach assuming a single parameter completion time and off-line repair. Silverman and Carter (1984, 1986) use an Arcus-based algorithm, coupled with a comprehensive cost function, to obtain approximately lowest (expected) cost balances when using the probability loading rule for the two remedial actions. Again in this paper, the total costs of operating the assembly line are calculated for each candidate line design generated, so that a balance can be selected which directly minimizes the sum of normal and incompletion costs. Gokcen and Baykoc (1999) propose a remedial policy of substituting a good item from buffer storage between stations for the item that is not completed.

6. COST FUNCTIONS

The cost functions are used to evaluate each line design as it is generated. The functions used for the two remedial actions are not comparable. In the off-line repair model, the cost of exceeding the cycle time is simply a function of the repair rate, the total work content and the probability of exceeding the cycle time. It is not directly affected by the number of work stations.

In the intermittent line stoppage model, all stations are idled while the line is stopped to allow a station to complete its assigned work. Thus the overtime production required to compensate is affected by the number of stations. The larger the number of stations, the more costly it will be to stop the line. Details of the two cost functions are presented in Appendix I.

7. THE ALGORITHM

Work done by Mastor in evaluating heuristic line balancing algorithms led to the modification of the Arcus technique by the authors for use in this research [Arcus (1963), Arcus (1966), Mastor (1970), Silverman and Carter (1984)]. The modified algorithm works by first generating a list of tasks whose precedence constraints have been satisfied (the precedence list). From this list, a fit list is generated to include all tasks that will fit into the station such that the conditions of the loading rule are not violated. Tasks are selected at random from the fit list for assignment to the station. After each task is assigned, the precedence list and fit list are updated. When the fit list is empty, a new station is started. After all tasks have been assigned, the total cost of the balance is calculated using the stochastic cost function. The procedure is repeated a predesignated number of times with various threshold levels used for calculation of the fit list. The lowest cost balance is retained as the "best."

Lowering the percent and probability thresholds increases the constraint on the allocation process. It becomes harder to find tasks to fit into a station as this constraint increases. At some point it becomes necessary to add a new station to accommodate all tasks. At that point there is a large increase in total cost. Prior to reaching that point, however, the balance of tasks across stations is more uniform, thus tending to reduce the cost of exceeding the cycle time.

8. HYPOTHESES

We hypothesize that the probability rule should be superior to the percentage rule. It explicitly takes task variability into consideration and precludes loading patterns with high variance and therefore high cost of exceeding the cycle time. More designs of a higher "quality" can be generated in a finite number of trials.

The above argument gives rise to the following hypotheses:

1. The probability rule will find a lower cost balance than the percent rule.

2. The probability rule will require fewer iterations than the percent rule to locate its lowest cost balance.

3. The greater the remedial cost, the greater will be the advantage provided by the probability rule.

4. The greater the task variability, the greater will be the advantage provided by the probability rule.

9. EXPERIMENTAL DESIGN

To test the above hypotheses, the 45 task problem of Kilbridge and Wester (1961) and the 70 task problem of Tonge (1960) were used. Since the task times presented in the original problems are deterministic, they had to be transformed into normal random variates for this line balancing study. This was accomplished by using a task's original time as the mean and applying a uniformly distributed coefficient of variation to produce a standard deviation.

The experimental design can be structured as a comparison of the two loading rules in each of sixteen contexts. The average coefficients of variation used were 0.05 and 0.25 for low and high variability, respectively. For the intermittent line stoppage remedial action, overtime rates of 1 and 3 were used to compute the cost of making up lost production. For the off-line repair action, repair rates were calculated to make the total cost of balancing the 45 task assembly line using off-line repair approximately equal to the cost using intermittent line stoppages. The repair rates used were 0.146 and 0.438. In the latter case, for each hour of work content, it takes an average of 0.438 of an hour to repair an item, given that it needs repairing.

An automated procedure was devised to perform a search of the solution domain. For the percent rule, the search was made in decrements of two percent from 100 to 78 percent of the original problem's cycle time. For the probability rule, the search was made from an allowable probability of any station exceeding the original problem's cycle time from 0.5 to 0.0001 using decrements of 0.03.

For each of the 16 contexts, 500 balances were generated at each threshold level, using the algorithm described earlier. For example, one of the 16 contexts was as follows:

Problem                    45 task
Loading rule               Percent of cycle time
Remedial action            Intermittent line stoppage
Overtime rate              3 times regular rate
Coefficient of variation   0.25 ("high")

For this context, the problem was balanced using percentage thresholds of 100, 98, ... ,78. For each threshold, 500 balances were obtained and the lowest cost balance retained as the "best." Because tasks are allocated to stations randomly and because the threshold values are discrete, there is no expectation that an optimum will be found for any finite number of balances. As with any heuristic procedure, as the number of trials increases, there is an increased likelihood of finding an optimal balance.

10. RESULTS

The results of the analyses are summarized in Tables 1 and 2. (Results for the OLR and ILS remedial actions are recorded in the two tables for succinctness. The cost results for the two actions are not comparable.) The first hypothesis was that the probability rule would be able to find a lower cost balance than the percentage rule. A finite number of trials is an assumption implicit in this hypothesis; with unlimited trials, a given "best" balance could be found by both loading rules. Tables 1 and 2 show that, in 500 trials, the probability rule produced a lower cost balance in 10 of the 16 contexts, an equal cost balance in 4 cases and a higher cost balance in 2 cases. Many of the cost differences are very small, so the support for the hypothesis comes from the 63 percent of many different contexts in which the probability rule prevailed.

The second hypothesis concerns the efficiency with which the lowest cost balance can be found by the two loading rules. The assignment number for the best balance in a run is a rough indicator of efficiency. In neither problem did the probability rule demonstrate an ability to find its best balance in fewer assignments than the percent rule, although it was outperformed only once in 8 opportunities in the 70 task problem. This suggests that the probability rule may have an efficiency advantage with larger scale problems. The CPU requirement also tended to be higher for the probability rule in the 45 task problem, but that situation was reversed for the 70 task problem.

An important part of the study was to see whether the level of remedial cost and task time variability affected the choice of loading rule. The effects of these two variables were similar. In the 45 task problem, the probability rule gave lower cost balances in six of the eight comparisons of high and low remedial costs. It also gave lower cost balances in six of the eight comparisons of high and low variability. In the 70 task problem, the probability rule did better than the percent rule in half the cases and equaled the percent rule in the other half. The probability rule's advantage occurred more frequently for high remedial costs and low variability, yet the single largest advantage came with both high remedial costs and high variability.

In conclusion, the experiment conducted for this research has demonstrated quite a robust advantage of the probability rule over the percent loading rule. Although some of the differences between the results for the two rules are small, they tend to be more pronounced for the 70 task problem. This research makes it clear, however, that there is no universally applicable percentage or probability which will produce an economically sound assembly line layout. Best percent levels ranged from 82 to 100 and best allowed probabilities of exceeding the cycle time ranged from .02 to .5. If costs, task time variability and level of remedial cost are taken into account, a simulation of various possible thresholds is necessary to achieve a good balance.

The number of trials used to find an approximately lowest cost balance for each threshold level within each context was set at 500. This provided balances which were close to optimal and sufficient for the comparative purposes of the study. When the number of trials was increased to 2000 and 10000, lower cost balances could be found. But the process clearly demonstrated rapidly diminishing marginal returns; the average number of sequences between successive assignments increased rapidly and the improvement in total cost was often very small.

1. Off-line repair cost function

In their 1984 paper, the authors developed the following cost function:

TC = normal operating cost (including idle time) the

expected cost of exceeding the cycle time;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

TC = expected total cost to assemble one unit,

[t.sub.c] = cycle time,

n = number of work stations,

c = (constant) station operating cost per unit time,

1-[[PI].sub.j] F([z.sub.j]) = probability that one or more stations will exceed the cycle time,

r[[SIGMA].sub.j] [M.sub.j] = cost of ill-will and off-line repair, which is assumed to be the product of a fixed dollar rate (r) and the expected total

2. Intermittent line stoppage cost function

The derivation of this cost function is presented in the authors' 1986 paper.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

A = penalty rate for operating the line overtime, which is a multiple of the standard rate (nc),

P = probability that at least one station will exceed the cycle time,

G(w) = probability that all stations complete their tasks within time w,

[[integral].sup.[infinity]. sub.t.sub.c]/P = expected time in excess of the cycle time to complete all tasks when the cycle time is exceeded at one station.

TABLE 1
Results for the 45 Task Problem

                           Task Time Variation
                           OLR               ILS
                           High     Low      High     Low

High     Perc.   Percent   84       94       84       92
Repair   Rule    Cost      831.83   622.07   818.45   621.15
Cost             Seq       132      18       288      315
                 Assmt     6        4        9        6
                 CPU       9.13     1.58     19.86    27.94

         Prob.   Prob      .17      .02      .11      .02
         Rule    Cost      830.06   622.71   802.18   621.11
                 Seq       40       494      393      494
                 Assmt     4        10       13       10
                 CPU       3.05     45.9     28.95    45.92

Low      Perc.   Percent   94       94       94       92
Repair   Rule    Cost      692.74   621.59   701.34   621.05
Cost             Seq       348      18       318      315
                 Assmt     7        4        7        6
                 CPU       22.96    1.59     20.87    27.81

         Prob.   Prob      .32      .02      .29      .02
         Rule    Cost      693.27   621.57   699.51   621.04
                 Seq       141      494      460      494
                 Assmt     5        10       7        10
                 CPU       9.64     45.88    33.22    46.27

TABLE 2 Results for the 70 Task Problem

                           Task Time Variation
                           OLR                 ILS
                           High      Low       High      Low

High     Perc.   Percent   100       92        82        96
Repair   Rule    Cost      5431.06   4262.81   7119.36   4116.05
Cost             Seq       454       321       228       312
                 Assmt     11        6         11        6
                 CPU       43.53     55.98     21.77     52.97

         Prob.   Prob      .50       .05       .11       .05
         Rule    Cost      5431.06   4223.04   6851.8    4072.76
                 Seq       454       26        310       26
                 Assmt     11        6         11        4
                 CPU       44.67     5.35      44.12     5.38

Low      Perc.   Percent   100       96        100       100
Repair   Rule    Cost      4391.69   4173.95   5046.09   3988.63
Cost             Seq       454       312       489       489
                 Assmt     10        6         9         9
                 CPU       43.92     52.94     47.06     97.41

         Prob.   Prob      .50       0.05      .50       .50
         Rule    Cost      4391.69   4106.34   5046.09   3988.63
                 Seq       454       26        489       489
                 Assmt     10        6         9         9
                 CPU       43.59     5.38      46.86     97.32

REFERENCES

Arcus, A., "An Analysis of a Computer Method of Sequencing Line Operations," Ph.D. Thesis, University of California, Berkeley, 1963.

Arcus, A. "COMSOAL: A Computer Method of Sequencing Operations for Assembly Lines" in E. Buffa (Ed.), Readings in Production and Operations Management, Wiley, 1966.

Bryton, B., "Balancing of a Continuous Production Line," M.S. Thesis, Northwestern University, 1954.

Carraway, R. L., "A Dynamic Programming Approach to Stochastic Assembly Line Balancing," Management Science, Vol. 35, No. 4, 1989, pp. 459-471.

Gokcen, H and B. Baykoc, "A New Line Remedial Policy for the Paced Lines with Stochastic Task Times," International Journal of Production Economics, Vol. 58, 1999, pp. 191-197.

Ignall, E., "A Review of Assembly Line Balancing," Journal of Industrial Engineering, Fourth Quarter, 1965.

Kilbridge, M. and L. Wester, "A Heuristic Method of Assembly Line Balancing," Journal of Industrial Engineering, Fourth Quarter, 1961.

Kottas J. K. and H. S. Lau, "A Cost Oriented Approach to Stochastic Line Balancing," AIIE Transactions, Second Quarter, 1973.

Kottas J. K. and H. S. Lau, "A Total Operating Cost Model for Paced Lines with Stochastic Task Times," AIIE Transactions, Second Quarter, 1976.

Kottas J. K. and H. S. Lau, "A Stochastic Line Balancing Procedure," International Journal of Production Research, February, 1981.

Mastor, A., "An Experimental Investigation and Comparative Evaluation of Assembly Line Balancing Techniques," Management Science, July, 1970.

Moodie C. and H. Young, "A Heuristic Method of Assembly Line Balancing for Assumptions of Constant or Variable Work Element Times," Journal of Industrial Engineering, First Quarter, 1965.

Reeve, N. R. and W. H. Thomas, "Balancing Stochastic Assembly Lines," AIIE Transactions, Third Quarter, 1973.

Sarin, C, E. Erel and E. Dar-El, "A Methodology for Solving Single-model, Stochastic Assembly line Balancing Problem," Omega, The International Journal of Management Science, Vol 27, 1999 pp. 525-535.

Silverman, F. and J. Carter, "A Cost Effective Approach to Stochastic Line Balancing with Off-Line Repairs," Journal of Operations Management, February, 1984.

Silverman, F. and J. Carter, "A Cost-based Methodology for Stochastic Line Balancing with Intermittent Line Stoppages," Management Science, April, 1986.

Tonge, F., "Summary of a Heuristic Line Balancing Procedure," Management Science, October, 1960.

Fred Silverman, Pace University, White Plains, NY

John Carter, Pace University, White Plains, NY

Author Profiles

Dr. Fred Silverman earned his Ph.D. at Columbia University in 1974. He is a professor of management and management science at the Lubin School of Business, Pace University.

Dr. John Carter earned his Ph.D. at Columbia University in 1975. He is a professor of management and management science at the Lubin School of Business, Pace University.