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How to pick the optimal batch size?


If the batch size is so small, that the process step with the set-up time (assuming, that there is only one) becomes the bottleneck of the process, the process looses on overall efficiency. Thus, the batch size needs to be chosen in a way that assures, that it will not generate a new bottleneck.

If, however, the batch size is too big, any increase in capacity at the station with the set-up time (assuming, again, that there is only one) is ultimately of no use, because it will only lead to inventory piling up somewhere else in the process – wherever the bottleneck may be.

This goes to show, that the ideal batch size is one, in which the station with the set-up time has a processing time which is just identical to the process bottleneck. Only then will the batch size not lead to the creation of a new bottleneck or additional inventory pile-up. This is calculated as:

capacity determined by the batch size = capacity of the bottleneck

b / (s + b * p) = m / p

with:

b = batch size
s = set-up time
p = processing time
m = number of resources

These lecture notes were taken during 2013 installment of the MOOC “An Introduction to Operations Management” taught by Prof. Dr. Christian Terwiesch of the Wharton Business School of the University of Pennsylvania at Coursera.org.
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Re-definition of the batch size in accordance with demand


The batch size was previously defined as the number of flow units that are produced between two set-ups. While this definition is correct, it does not take into account the actual demand for the flow units. If a process is able to produce multiple flow units (e.g. cheeseburgers and veggie sandwiches) with one set-up time in between, a batch in a mixed-model production is re-defined as a number of mixed flow units produced during a certain amount of time (before the used pattern of production is repeated). The additional set-up times for switching between the flow units during the production of the batch have, of course, to be recognized.

This brings us to the following formula:

target flow = batch size / (set-up time + batch size * processing rate)

Here, the target flow is defined as the number of flow units needed per time frame in order to stay on top of the demand (e.g. 100 units per hour). The processing rate is determined by the bottleneck of the process or by the demand while set-up time and batch size have previously been defined.

If the goal is determining the ideal batch size, the formula can be resolved for the batch size. The result has to be set in ratio to the demand for the various flow units within the batch in order to find out, how many flow units of each type are produced within the ideal batch size. Note, that the set-up time needed to start the production pattern at the beginning is part of the overall set-up time and thus needs to be included in the total sum of set-up times needed for this calculation.

Obviously, the batches will become larger and the inventory will become bigger the more set-ups are necessary as long as the overall demand does not change (but is simply spread out over more offered product choices). Variety thus leads to more set-ups and thus to more inventory, which is one of the biggest problems associated with offering more variety.

These lecture notes were taken during 2013 installment of the MOOC “An Introduction to Operations Management” taught by Prof. Dr. Christian Terwiesch of the Wharton Business School of the University of Pennsylvania at Coursera.org.

The effect of set-up times


Set-up times often have a significant effect on the performance of a process and can even determine a process bottleneck. The most important definition here is that of the batch: A batch is the number of flow units, which are produced between two set-ups. To calculate the capacity of a process with respect to the batch size, the following formula is needed:

capacity = (batch size) / (set-up time + batch size * time per unit)

Note, that this is the capacity of the process for producing a batch. For example, if the batch size happens to be 10 flow unites per minute, this calculation answers the question of how long it will take to produce one complete batch. This is a deviation from the previous definition of capacity, which did not take the batch size into account and was simply calculated as:

capacity = number of resources / processing time

The capacity can by this basic definition be calculated for every station in a process. It is always m / processing time with m being the number of resources (e.g. workers) being devoted to this process step. If, for example, one worker needs 40 seconds to put together a sandwich, the capacity of this station is 1/40 per second or 1,5 sandwiches per minute. If there are two workers on the same station, the capacity increases to 2/40 per second or 3 sandwiches per minute.

Usually, the larger the batch, the more efficient the production process becomes (economics of scale). Companies with custom-made batches are therefore trying to get their customers to order large batches (sometimes even forcing them). The bigger a batch grows, the more irrelevant the set-up time becomes with the process capacity getting closer and closer to the original definition of m / processing time. This is because the processing time is less and less determined by the set-up time with larger batch sizes. Thus, set-ups reduce capacity – and therefore, companies have an incentive to aim for such large batches. However, large batches also increase inventory – with all of the negative consequences (e.g. storage costs, ageing of shelved products etc.).

These lecture notes were taken during 2013 installment of the MOOC “An Introduction to Operations Management” taught by Prof. Dr. Christian Terwiesch of the Wharton Business School of the University of Pennsylvania at Coursera.org.
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