# EOQ Model with Shortages

In this case, shortages are permitted which implies that shortage cost is finite or it is not large. The cost of a shortage is assumed to be directly proportional to the mean number of units short.

Further, all the assumptions of model I hold good here also.

The economic order quantity EOQ Model with Shortages is graphed in the following figure.

where
S = Back order quantity.
M = Maximum inventory level.
t1 = Time during which stock is available.
t2 = Time during which there is a shortage.
t = Time between receipt of orders.

The cycle time t is the sum of t1 and t2.

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 Q* = 2DCo --------- Ch X (Ch + Co) ------------ Cs M* = (2DCo) ----------- (Ch) X Cs ----------- (Ch + Cs) t* = 2Co -------- DCh X (Cs + Ch) ------------ Cs TC* = 2DCo Ch X Cs ----------- (Cs + Ch)

#### Example

The Wartsila Diesel Company has to supply diesel engines to a truck manufacturer at a rate of 10 engines per day. The ordering cost is Rs. 150 per order. The penalty in the contract is Rs. 90 per engine per day late for missing the scheduled delivery date. The cost of holding an engine in stock for one month is Rs. 140. His production process is such that each month (30 days) he starts procuring a batch of engines through the agencies and all are available for supply after the end of the month. Determine the maximum inventory level at the beginning of each month.

Solution.

Given
Demand (D) = 10 engines per day
Shortage cost (Cs) = Rs. 90 per day per engine
Carrying cost (Ch) = 140/30 = 14/3 per engine per day
Ordering cost (Co) = Rs. 150 per order

 M* = 2 X 10 X 150 ----------- 14/3 X 90 ----------- 14/3 + 90 X 30 = 741.65 = 742 engines (approx.)

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