1. ## LP problem

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?

2. I'm not familiar with linear programming, but here is what I'd say: $\displaystyle X_1=x'_1+x_1-\frac{x_2}{3}$ (since $\displaystyle \frac{x_2}{3}$ is the number of dollars spent to buy francs) and $\displaystyle X_2=x'_2+x_2-\frac{x_1}{0.25}$. Maximize $\displaystyle X_1$ subject to the constraints $\displaystyle X_1\geq0,\ X_2\geq 0$.

(By the way, euros would be more up-to-date than francs)

Laurent.

3. Thats all there is too it right? No ambiguities?