7.5 THE CASE OF NON-IDENTICAL ITEMS WITH DIFFERENT PROCESSING TIMES Up until now, we always assumed that the printing times would always be the same for all items, even though they might be different SKUs. We will now explain what happens when we have a multi-item setup, where processing/printing times are different, depending on what SKU they are. We will denote the deterministic print time of item đ?‘– ∈ đ??ź by đ?‘Ąđ?‘–đ??´đ?‘€ (> 0). A means to calculate the expected lead time (waiting time in the queue + processing time) will be given, followed by a numerical example. Queueing theory At the printer, orders for all items in đ??ź arrive, where item đ?‘– arrives according to a Poisson process with rate đ?‘šđ?‘– , requiring a print that takes đ?‘Ąđ?‘–đ??´đ?‘€ time units. The queue is serviced by a single server, and has infinite waiting room. Such a queue is called an M|G|1 queue (Kulkarni, 1999). The mean service time đ??¸[đ?‘Ą đ??´đ?‘€ ] equals đ??¸[đ?‘Ą đ??´đ?‘€ ] = ∑ đ?‘Ąđ?‘–đ??´đ?‘€ ( đ?‘–∈đ??ź
đ?‘šđ?‘– ) đ?‘€
The variance of the service times equals 2
đ?œŽ 2 = ∑(đ?‘Ąđ?‘–đ??´đ?‘€ − đ??¸[đ?‘Ą đ??´đ?‘€ ]) ( đ?‘–∈đ??ź
đ?‘šđ?‘– ) đ?‘€
The utilization is calculated as follows đ?œŒ = ∑ đ?‘šđ?‘– ∙ đ?‘Ąđ?‘–đ??´đ?‘€ đ?‘–∈đ??ź
The second moment is đ?‘ 2 = đ??¸[đ?‘Ą đ??´đ?‘€ ]2 + đ?œŽ 2 Once we know the total order arrival rate đ?‘€ and the utilization of the server (in our case a 3D printer) đ?œŒ, we can calculate the expected time in the queue as đ??¸[đ?‘Š] =
đ?‘€đ?‘ 2 2(1 − đ?œŒ)
The total time spent in the system đ?œ?đ?‘– , also known as the lead time for additive manufacturing then becomes đ?œ?đ?‘– = đ?‘Ąđ?‘–đ??´đ?‘€ + đ??¸[đ?‘Š] We can now calculate the expected inventory and backorder levels with the known formulas from inventory theory, as provided in section 2.1.3, which we will do in Example 2. We have to emphasize that this approximation ignores dependence of successive lead times, as assumed in Section 7.1.
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