Managing Inventory of Perishable or Seasonal Items

Many of the inventory items that we need to manage are short-lived (e.g., newspapers, flowers, available cabins on a boat cruise) or very seasonal (e.g., greeting cards, toys, etc. during the holiday season).   Typically, these items have a relatively low salvage value, and we incur losses on any items that we do not manage to sell.   On the other hand, we make profits on the items that we sell. We will assume that we realize marginal profit MP on any unit that we sell and marginal loss ML on each unit that we do not sell.

Now we can try to determine the order quantity that will maximize our expected profits.   This problem is amenable to solution through the classic economic approach of marginal analysis.   The optimal order size Q occurs at the point where the expected benefits derived from ordering an additional unit are less than the expected costs.   In other words, we keep increasing our order size as long as an additional unit is expected to increase our profits.

The expected profit from ordering a unit is:

Exp(profit) = P(selling the unit)∙MP – P(not selling the unit)∙ML

Note that because the probability of selling additional units goes down (we are more likely to sell the first unit than the second unit, etc), the expected profit from ordering additional units decreases.   We simply stop, that is, we have reached optimal order size Q, once the expected profit from the last unit is zero.

If the demand D is an approximately normally distributed random variable with mean  and standard deviation , and our order size is Q, we can write the expected profit for the last unit as:
Exp(profit) = P(D≥Q)∙MP – P(D
(*Why is P(D≥Q) the probability that we will sell the last unit and P(D
We also know that P(D≥Q) = 1 – P(D (*Why do probabilities of selling and not selling an item sum up to 1?*)
We can now substitute... [continues]

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