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420 CHAPTER 10 INVENTORY MODELS

Therefore, we can write the total annual cost model as follows:

1 D D
TC ¼ 1 QC h þ C o (10:15)
2 P Q

Equations (10.14) and (10.15) are equivalent. However, Equation (10.15) may be
used more frequently because an annual cost model tends to make the analyst think
in terms of collecting annual demand data (D) and annual production data (P)
rather than daily data.
Economic Production Lot Size

Given estimates of the holding cost (C h ), setup cost (C o ), annual demand rate (D)
and annual production rate (P), we could use a trial-and-error approach to
compute the total annual cost for various production lot sizes (Q). However, trial
and error is not necessary; we can use the minimum cost formula for Q* that has
been developed using differential calculus (see Appendix 10.2). The equation is as
As the production rate p follows:
approaches infinity, D/P
approaches zero. In this s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
case, Equation (10.16) is

equivalent to the EOQ Q ¼ 2DC o (10:16)
model in Equation (10.5). ð1 D=PÞC h

We are now in a position to advise EnviroHealth. Recollect that its annual
production capacity was 60 000 litres with annual demand at 26 000 and a
constant rate of demand through the year. Setup costs were E135 whilst
holding cost was 24 per cent of inventory value which, given the product cost
of E4.50 per litre, gives an annual holding cost of 0.24 E4.50 or E1.08 per
litre.
EXCEL file Using equation (10.16), we have:
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
LOT SIZE 2ð26 000Þð135Þ

Q ¼ ¼ 3387
ð1 26 000=60 000Þð1:08Þ
So, we have an optimum production lot size of 3387 litres. That is we
should produce 3387 litres during each production run. This lot size minimizes
annual inventory costs at E2073 (using Equation (10.15)). We also know that
it takes six working days to schedule and set up a production run. Let us
also assume that the company works 300 days a year. The lead time demand
is then:

ð26; 000=300Þð6Þ¼ 520 litres
That is, during the six days it takes to schedule and set up a production run,
inventory will need to be 520 litres to cope with demand. 520 litres is then the
reorder point for EnviroHealth. The cycle time is the time between each production
run. Using Equation (10.7) we can calculate the cycle time as:

T ¼ 300Q =D ¼½ð300Þð3387ފ=ð26000Þ¼ 39 days
So, EnviroHealth should plan for production runs of 3387 litres every 39 working
days. For practical convenience, the company might schedule this as 3400 litres every
six-and-a-half working weeks.

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