Example

Consider again Example $1($manufacturing television speakers$)$ from Section $18\mathrm{.}1.$ Remember that the preparation cost to produce the speakers is K $=$ $$12,000,$ the unit maintenance cost is h $=$ $$0\mathrm{.}30$ per month per speaker, and the unit cost for shortages is p $=$ $$1\mathrm{.}10$ per speaker per month.

The original fixed demand rate was $8,000$ speakers per month to be assembled into televisions manufactured on the production line at this fixed rate. However, sales have been highly variable, so the inventory level of finished televisions fluctuates significantly. To reduce the costs of holding inventory of finished products, management has decided to adjust the daily production rate to better fit the orders received.

Consequently, the current demand for speakers is variable. There is a $1$ month lead time between ordering a production run of speakers and having them ready for assembly. The demand for speakers during this delivery time is a random variable D that has a normal distribution with a mean of $8,000$ and a standard deviation of $2,000.$ To minimize the risk of interrupting the television production line, management has decided that The safety inventory of speakers must be sufficient to avoid shortages $95\%$ of the time during this delivery period.