$(25$ pts$.)$ Recall the Inventory Planning problem $($Example $2\mathrm{.}4)$ discussed in class:

A Californian water company is preparing for the summer when water may be harder to come

by$.$ The monthly demand remains at a constant $75$ thousand gallons over the five months,

but their aquafer produces less water each month. They own storage tanks capable of holding

$100$ thousand gallons water. The projected cost of buying water from out of state is given in

the table below.

Due to risk of contamination, the water company cannot use the water stored at the end of

a month after the next $2$ months. Therefore, the water available in storage at the beginning

of month $1,$ can only be used in month $1$ and $2.$ Similarly, water available in storage at the

beginning of month $2($or at the end of month $1)$ can be used in month $2$ and $3$ only, and

so on$.$ The per gallon cost of storing the water is $$0\mathrm{.}75.$ There is already $30,000$ gallons in

storage for use in the beginning of month $1.$

The company seek to minimize the cost incurred in supplying the water over the $5$ months.

Formulate the problem as a mathematical program.

$($Hint: Based on the new information identify the changes in the flow diagram discussed in

the class for each month.$)$