A$507$ Project $3$

Pump$-$It$-$Yourself is a new, independently owned and operated service station. As such, it will not receive regular deliveries from the local wholesale gasoline distributorship, which is controlled by a major oil company. Instead, the station will have the opportunity to purchase any or all of the excess gasoline remaining on the distributor$$s truck after all regular deliveries have been made each day. There is no delivery charge if all the gasoline remaining in the truck is taken, but if not, there is a back$-$shipping charge. The station will take all of the gasoline that it can from the truck. The delivery truck will stop each day after the station has closed for business. The amount of gasoline remaining on the truck is a random variable with the following distribution:

Gallons Remaining Probability

$0\mathrm{.}20$

$1000\mathrm{.}15$

$2000\mathrm{.}25$

$3000\mathrm{.}18$

$4000\mathrm{.}12$

$5000\mathrm{.}10$

The station is planning a $3,000-$gallon storage capacity, and the demand for gasoline is expected to be normally distributed with a mean of $2,000$ gallons and a standard deviation of $800$ gallons. Assume that unsatisfied demand is lost $($customer goes to a competitor$).$ Simulate the activities involving the supply and demand of gasoline for $30$ days. Start with an inventory on hand of $500$ gallons. Given:

Profit per gallon sold $($before deducting the following costs$)$ $$\mathrm{.}20$

Cost of ill will per gallon of demand not met $$\mathrm{.}05$

Cost of shipping back per gallon $$\mathrm{.}02$

$*$Cost of maintaining a gallon of capacity for $30$ days $$\mathrm{.}30$

$*$For a $3,000-$gallon storage capacity, the cost for maintaining that capacity for $30$ days is $3000*0\mathrm{.}30)$

The simulation model should be designed to report the expected total profit for a $30-$day period, the percentage of days for which the storage capacity is not sufficient to take all of the gasoline from the truck, and the percentage of days on which demand for the day cannot be met. Use at least $1000$ iterations. $($Note that a $30-$day period represents one iteration.$)$ Construct a histogram of the total profit.

b$)$ Suppose that the station can consider more or less storage capacity in increments of $1000$ gallons. What storage capacity would you recommend and why? $($Show the results for the storage capacities considered.$)$ Please offer additional relevant observations about the performance of the business. Using the recommended storage capacity, report the expected total profit for a $30-$day period, the percentage of days for which the storage capacity is not sufficient to take all of the gasoline from the truck, and the percentage of days on which demand for the day cannot be met.