A man deals with French currency (the franc) and American currency (the dollar). At $\displaystyle 12 $ midnight, he can buy francs by paying $\displaystyle .25 $ dollars per franc and dollars by paying $\displaystyle 3 $ francs per dollar. Let $\displaystyle x_{1} = \text{number of dollars bought (by paying francs)} $ and $\displaystyle x_{2} = \text{number of francs bought (by paying dollars)} $. Assume that both types of transactions take place simultaneously, and the only constraint it that at $\displaystyle 12:01 \ \text{A.M.} $ the man must have a nonnegative number of francs and dollars.

(a)Formulate an LP that enables the man to maximize the number of dollars he has after all transactions are completed.

Let $\displaystyle X_{1} = x_{1}+ x_{1}' $ where $\displaystyle x_{1}' $ is the number of dollars the man has before the transaction. Similarly, let $\displaystyle X_{2} = x_{2} + x_{2}' $ where $\displaystyle x_{2}' $ is the number of francs the man has before the transaction. So is this the correct LP formulation:

maximize $\displaystyle X_{1} $ subject to constraints that $\displaystyle X_{1} \geq 0, \ X_{2} \geq 0 $. We can solve this right? This formulation allows for the man to not enter any transaction, right?