Suppose you are the operations manager for a peanut processing company, and you need to supply peanuts to two stores: S$1$ and S$2.$ S$1$ requires $250$ pounds of peanuts, and S$2$ requires $100$ pounds. There are two plants processing peanuts, P$1$ and P$2,$ and the shipping costs vary: it costs cij to send one pound of peanuts from Pi to Sj$.$ Additionally, P$2$ can only produce up to $250$ pounds of peanuts. Formulate an LP to determine the most cost$-$efficient way to supply the two stores with peanuts. Then, find the dual of this LP$.$