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Serial Batching Example

The following derives the flow time for a station with serial process batching as a function of batch size (see Hopp and Spearman, Factory Physics^®, Chapter 9 for further details):

• k = batch Size
• t₀ = time to process a single part (1)
• t_s = time to perform setup (5)
• c_e² = squared co-efficient of variation (c²) for batch (0.5)
• r_a = arrival rate for parts (0.4)
• c_a² = c² of batch arrival (1.0)

Then the total time to process a batch is: t_e = t_s+ k t₀

The arrival rate for the batches is : r_b = r_a/k

The utilization is (batch arrival rate * time for batch) = r_b t_e

Note that for stability we require that the utilization be less than 1 (can not have more than 100% utilization). This gives us

r_b t_e 1 = k (t_s* r_a)/(1- r_a t₀) = 3.33 for our example. This implies that we need the minimum batch size to be at least 4 for the system to be stable.

The average queue time for the station is given as:

The average flow time depends on whether we move the jobs in batches or we split the jobs. If we move batch, then the flow time is:

If we move only after a full batch is completed, flow time is:

The previous figure plots CT_move-batch and CT_split as a function of batch size, k.

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