Page 262 - DMGT501_OPERATIONS_MANAGEMENT

Operations Management

Notes 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 percent sure of not running out of
newspapers each Sunday.
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 percent. The demand would be expected to be less than 100 newspapers 50 percent
of the time, and greater than 100 the other 50 percent. To be 90 percent 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 percent 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 C = Cost per unit of demand overestimated, and C = Cost per unit of demand overestimated
o u
and the probability that the unit will be sold is ‘P’; the expected marginal cost equation can be
represented as:
P (C ) < (1 – P)C
o u
Here (1-P) is the probability of the newspaper not being sold. Solving for P, we obtain
P < [C /(C + C )]
u o u
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 C /(C + C ).
u o u
Single-period inventory models are useful for a wide variety of service and manufacturing
applications.

11.2 Multiple Period Inventory Models

Multi-period inventory systems are designed to ensure that an item will be available on an
ongoing basis throughout the year. There are two general types of systems and these inventory
systems can be distinguished on the basis of the ordering criteria. The models of these two
systems are; (a) Fixed-Order Quantity Models (also called the Economic Order Quantity models)
and (b) Fixed-Time Period Models (also referred to as the Periodic System or P-models).

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.
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