Base stock model

The base stock model is a statistical model in inventory theory. In this model inventory is refilled one unit at a time and demand is random. If there is only one replenishment, then the problem can be solved with the newsvendor model.
Fill rate, back-order level and inventory level.
In a base-stock system inventory position is given by on-hand inventory-backorders+orders and since inventory never goes negative, inventory position=r+1. Once an order is placed the base stock level is r+1 and if X≤r+1 there won't be a backorder. The probability that an order does not result in back-order is therefore:
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Since this holds for all orders, the fill rate is:
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If demand is normally distributed formula_15, the fill rate is given by:
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Where formula_17 is cumulative distribution function for the standard normal. At any point in time, there are orders placed that are equal to the demand X that has occurred, therefore on-hand inventory-backorders=inventory position-orders=r+1-X. In expectation this means:
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In general the number of outstanding orders is X=x and the number of back-orders is:
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The expected back order level is therefore given by:
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Again, if demand is normally distributed:
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Where formula_22 is the inverse distribution function of a standard normal distribution.
Total cost function and optimal reorder point.
The total cost is given by the sum of holdings costs and backorders costs:
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It can be proven that:
Where r* is the optimal reorder point. If demand is normal then r* can be obtained by:
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