Question

Suppose now that the daily demand for the t-shirts is not deterministic. It is random with mean 2 and standard deviation 0.5. In addition, the lead time (the time it takes for an order to arrive at the store) is also random. Specifically, the mean lead time is six days and the standard deviation for the lead time is 1 day. The store uses an (s, Q) type inventory control policy. It sets the order quantity using the EOQ formula assuming as though demand is deterministic, i.e., it uses the order quantity computed in Question 1. As for s, the reorder point, it chooses s so that the probability of stockout during a replenishment period is only one percent. What should the value of s be assuming that the demand during lead time can be assumed to have a normal distribution? (Question 1 Suppose that the demand for a particular t-shirt the UNC Student Stores sells is deterministic with 2 units per day. Each t-shirt costs $10 and the monthly charge of carrrying one t-shirt is 50 cents. If the fixed cost of placing an order (e.g. transportation cost etc.) regardless of the order size is $200 and the order arrives instantaneously, what is the optimal number of t-shirts the UNC Student Stores should order every time it places an order and how frequently should the orders be placed? )

Answer 1

The reorder quantity is given as the prooduct of daily demand and lead time. That is, $R=D\times T$

Here both D and T are normally distributed with and Hence we are essentially required to evaluate the probability distribution of the product of two random variables. This is a rather complex analysis, and a good reference is available at

http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1146/c/dmps.1146.pdf

Going by the result presented on page 7, the distribution of R is $R\(2\times 6,0.25\times 36+1\times 4)=R(12,13)$

So we now need this reoder point to have a 0.01 significance level. Hence, that reorder point corresponds to a one sided critical Z value of 2.33. Hence,

$\frac{s-12}{\sqrt{13}}=2.33\Rightarrow s=20.4$

We must round up, hence the reorder point is 21 units