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418 CHAPTER 10 INVENTORY MODELS

production situation the ordering cost is more correctly referred to as the produc-
tion setup cost. This cost, which includes labour, material and lost production costs
incurred while preparing the production system for operation, is a fixed cost that
occurs for every production run regardless of the production lot size. EnviroHealth
estimates the setup cost to be E135 largely made up of cleaning, preparation and
disinfecting. This cost is incurred every time a batch of the product is produced. In
order to schedule the setup for this product, the company must have at least six
working days notification that production is to be scheduled. The company also
knows that it costs E4.50 per litre to produce the product and the annual holding
cost is 24 per cent of the inventory value.

Total Cost Model
Let us begin building the production lot size model by showing the holding cost in
terms of the production lot size Q. Again, the approach is to develop an expression for
average inventory and then establish the holding costs associated with the average
inventory. We use a one-year time period and an annual cost for the model.
In the EOQ model the average inventory is one-half the maximum inventory or
½ Q. Figure 10.5 shows that for a production lot size model a constant inventory
buildup rate occurs during the production run and a constant inventory depletion
rate occurs during the nonproduction period; thus, the average inventory will be
one-half the maximum inventory. However, in this inventory system the production
lot size Q does not go into inventory at one point in time, and so the inventory never
reaches a level of Q units.
To show how we can calculate the maximum inventory, let:
d ¼ daily demand rate
p ¼ daily production rate
t ¼ number of days for a production run
At this point, the logic of Because we are assuming that p will be larger than d, the daily inventory buildup rate
the production lot size during the production phase is p d. If we run production for t days and place
model is easier to follow
using a daily demand rate p d units in inventory each day, the inventory at the end of the production run will
D and a daily production be (p d)t. From Figure 10.5 we can see that the inventory at the end of the
rate P. However, when the production run is also the maximum inventory. Thus:
total annual cost model is
eventually developed, we
recommend that inputs Maximum inventory ¼ðp dÞt (10:8)
to the model be
expressed in terms of the
annual demand rate D
and the annual If we know we are producing a production lot size of Q units at a daily production
production rate P. rate of p units, then Q ¼ pt, and the length of the production run t must be:

Q
t ¼ days (10:9)
p

So

Q
Maximum inventory ¼ðp dÞt ¼ðp dÞ
p
(10:10)
d
¼ 1 Q
p

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