27 : Linear Programming in Two Dimensions CS 473

Consider a production facility for a manufacturing company. The facility is capable of producing a variety of products that we shall enumerate as 1, 2, · · · , n. These products are constructed/manufactured/produced out of certain raw materials. Let us assume that there are m different raw materials, which again we shall simply enumerate as 1, 2, · · · , m. Furthermore, the ith raw material has at this moment in time a known unit market value ρ i. In addition, each product is made from known amounts of the various raw materials. That is, producing one unit of product j requires a certain known amount, say a ij units, of raw material i. Also, the jth final product can be sold at the known market price of σ j dollars per unit. The company production manager has to decide how to use the materials at hand. Let us assume that she decides to produce x j units of the jth product, j = 1, 2,. .. , n. The revenue associated with the production of one unit of product j is σ j. But there is also a cost of raw materials that must be considered. The cost of producing one unit of product j is m i=1 ρ i a ij. Therefore, the net revenue associated with the production of one unit is the difference between the revenue and the cost. Namely, c j = σ j − m i=1 ρ i a ij , Note that c j is the profit of manufacturing one using of product j, and we can compute it from the given input. As such c 1 ,. .. , c n are just constants. Now, the net revenue corresponding to the production of x j units of product j is simply c j x j , and the total net revenue is n j=1 c j x j. The production planner goal is to maximize this quantity. However, there are constraints on the production levels that she can assign. For example, each production quantity x j must be nonnegative, and so she has the constraint x j ≥ 0 for j = 1,. .. , n.