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Actually, this is a very common type of problem for all products that are perishable or have very low shelf lives. This includes both goods as well as services. A simple way to think about this is to consider how much risk we are willing to take for running out of inventory.

The classical case illustrated in most texts is the ‘newspaper seller’s dilemma’. Let’s take the example where the newspaper vendor has collected data over a few months that show that each Sunday, on an average, 100 papers were sold with a standard deviation of 10 papers. With this data, it is possible for our newspaper vendor to state a service rate that he feels is acceptable to him. For example, the newspaper vendor might want to be 90 per cent sure of not running out of newspapers each Sunday.

In the lesson on forecasting, we described a normal distribution. If we assume that the distribution is normal and the newspaper vendor stocked exactly 100 papers each Sunday morning, the risk of stock running out would be 50 per cent. The demand would be expected to be less than 100 newspapers 50 per cent of the time, and greater than 100 the other 50 per cent. To be 90 per cent sure of not stocking out, he needs to carry a few more papers. From the “standard normal distribution”, we know that we need to have additional papers to cover 1.282 standard deviations, in order to ensure that the newspaper vendor is 90 per cent sure of not stocking out.

A quick way to find the exact number of standard deviations needed for a given probability of stocking out is provided by Microsoft Excel. Press ‘insert’ and you will find ‘functions’. Click on ‘function’ and select the category ‘statistical’. You can then use the NORMSINV (probability) function to get the answer. NORMSINV returns the inverse of the standard normal cumulative

distribution. In this case, (NORMSINV (.90) = 1.281552. This means that the number of extra newspapers required by the vendor would be 1.281552 × 10 = 12.81552, or 13 papers. This result is more accurate than what we can get from the tables and is sometimes very useful.

If we know the potential profit and loss associated with stocking either too many or too few papers on the stand, we can calculate the optimal stocking level using marginal analysis. The optimal stocking level occurs at the point where the expected benefits derived from carrying the next unit are less than the expected costs for that unit. This can be mathematically expressed as follows:

If Co = Cost per unit of demand overestimated, and Cu = Cost per unit of demand overestimated and the probability that the unit will be sold is ‘P’; the expected marginal cost equation can be represented as:

P (Co) < (1–P)Cu

Here (1–P) is the probability of the newspaper not being sold. Solving for P, we obtain

P < [Cu/(Co + Cu)]

This equation states that we should continue to increase the size of the order so long as the probability of selling what we order is equal to or less than the Ratio Cu/(Co+Cu).

Single-period inventory models are useful for a wide variety of service and manufacturing applications. This mode is very useful and is used for many service sector problems, such as in yield analysis.

The basic difference between the two systems is that the fixed-order quantity models are “event triggered” and fixed time period models are “time triggered”. In other words, at an identified level of the stock the fixed-order quantity model initiates an order. This event may take place at any time, depending on the demand for the items considered. In contrast, the fixed time period models review the stocks at time intervals that are fixed and orders are placed at the end of predetermined time periods. In these models, only the passage of time triggers action.