5. Linear Programming

So far we have looked at modeling problems that involve quantities that change with time. Time, however, is not always part of the picture. In a modeling scenario that arises very often in economics , as well as in other sciences, one has to describe relationships between different quantities that are additionally subject to constraints. For instance, these quantities might be prices of raw materials and of finished products: how do the latter depend on the former? More to the point, modeling problems of this type often are set up in order to address optimization questions. For example, how can one set the prices of an array of finished products so as to maximize profit, given the costs of the raw materials and possibly constraints on their availability? The following simple example motivates the technique of linear programming, arguably one of the most widely used optimization algorithms. The example is from the Wikipedia entry for linear programming, Suppose that a farmer has a piece of farm land, say A square kilometers large, to be planted with either wheat or barley or some combination of the two. The farmer has a limited permissible amount F of fertilizer and P of insecticide which can be used, each of which is required in different amounts per unit area for wheat (F 1 , P 1) and barley (F 2 , P 2). Assume that the crop from one square kilometer of wheat can be sold at a price S 1 , and that from one square kilometer of barley can be sold at a price S 2 .The problem is to find the areas x 1 and x 2 to be planted with wheat and barley respectively. In this problem, the target function is profit, which depends on the two variables x 1 and x 2 that are under the farmer's control. If we assume for simplicity that fertilizer and insecticide were purchased with earlier income, profit is simply the overall selling price: so the question is what combination of x 1 and x 2 yields the greatest value of f (x 1 , x 2). In the absence of any constraints, this function has obviously no maximum. However, the circumstances of the problem impose quite a few constraints on x 1 and x 2. First, these two areas cannot add up to more than the total available area A: x 1 + …