You are an originator at a merchant energy trading company. It is August $13,2015.$ You are evaluating a potential deal that would allow you to purchase between $10,000$ MMBtu and $30,000$ MMBtu of natural gas per month for a unit price of $$2\mathrm{.}50$ for the next $24$ months, starting on September $1,2015.$ The choice of how much to purchase per month $($between these limits$)$ is at your discretion. However, you must purchase at least $150,000$ MMBtu and no more than $450,000$ MMBtu of natural gas during the contract term. You ask a structurer to obtain the NPV of this proposed contract. The structurer uses a constant risk$-$free rate equal to $1\%$ per year with continuous compounding normalizing the length of one year to $1,$ so that the current value of $$1$ received $15$ days from now is exp$(-0\mathrm{.}01*15/365)*$$$1$

The futures prices are as of the current date, August $132015$; e$.$g$.,$ F$($August $132015,$September $12015)=$ $$2\mathrm{.}787$

$1.$ Formulate a linear program to optimize the purchase contract NPV

$2.$ What is the optimal NPV of this contract?