Leon Tusk is the President of Purchasing for the EV battery division of Panasonic $($PAN$).$ The company recently introduced a $500$ kWh battery for their biggest customer Nikola that has taken the market by storm. Although Leon is pleased with the strong demand for this product in the marketplace, it has been a challenge to keep up with Nikola$$s orders of this battery. His current challenge is how to meet the production requirements at the manufacturing plants in Detroit $(1),$ Freemont $(2),$ Fort Worth $(3),$ and Ames $(4)$ who have placed orders of $10,000,20,000,30,000,$ and $25,000$ units, respectively, for delivery in $2$ months $($there is a $1-$month manufacturing and one$-$month shipping lead time for this product$).$ Being the biggest EV battery manufacturer, PAN has contracts with companies in China $(5),$ Argentina $(6),$ and Australia $(7)$ that mine lithium and sell the metal exclusively to PAN. These contracts require PAN to order a specified minimum number of units each month at a guaranteed per$-$unit cost. The contracts also specify the maximum number of pounds of Lithium that may be ordered at this price. The following table summarizes these contracts: