Chapter 7 Linear Programming Continued file 7a

Here is a basic linear programming problem. We want to Maximize profit = 7x1 + 5x2, but we have constraints 4x1 + 3x2 <= 240 2x1 + 1x2 <= 100, Where x1 stands for tables and x2 stands for chairs and <= means less than or equal to. The point is we want to make as much profit as we can by making tables and chairs, but we face constraints that give us road blocks to overcome.

Both constraints x2 x1 Feasible region We saw this feasible region for the product mix problem and saw in theory how the isoprofit line solution method could give us the answer. And as I suggested with the three examples, the solution will probably be at a corner. The corner solution method says just check profit at the corners - choose These are corners Corner with highest profit.

Corner point solution method The corners at the axes are relatively easy to check. Plug the x1 and x2 values into profit and see what number is obtained. The corner inside the graph is a little harder to find in math terms. A math systems of equations approach will never fail you. You are expected to have studied the method in your past and use it here. You can do so, but the method takes practice to get the tricks of the trade down. I want to show you a way to find the inside corner point with matrix algebra. Essentially we are finding where the two constraints cross.

Matrix algebra You are familiar with the basic algebra of equations like 2 x = 4. To solve for x you take 4/2 = 2. It turns out we could write x = , where the 2 raised to the power -1 really says divide. In general we have Ax = w and x = A -1 w. Now think about our two constraints in equality form 4x1 + 3x2 = x1 = 240 2x1 + 1x2 = x2 100 In this second form with brackets you can see we would have Ax = w, where A, x and w now are tables with columns and rows (sometimes only one column). We need to learn a few rules here, but it is easy. What does A -1 really mean now?

Matrix algebra Here I want you to see how the matrix is formed 4x1 + 3x2 = x1 = 240 2x1 + 1x2 = x2 100 This first row is The 4 and x1 split The 3 and x2 split Similar process for the second line.

Matrix algebra In general say A = ab cd. Then A -1 = 1/(ad - cb) d -b -c a. Example A = Then A -1 = 1/(4- 6) 1 -3 =

Matrix algebra x = A -1 w or x1 = = (-.5)(240) + 1.5(100) = 30 x (1)(240) + (-2)(100) 40 Nothing to it! The other methods require the building up of some skill. This is just a BRUTE FORCE method, it will never fail. Note: Dividing by zero would mean the solution is undefined. Here that would mean the curves do not cross.

Corner points Our three corner points are (50, 0), (0, 80) and (30, 40). Profit is 7x1 + 5x2. Profit at each point (50, 0) 350 (0, 80) 400 (30, 40) 410. Corner (0, 0) should also be checked - profit is 0 there. 410 is highest profit. Make 30 tables and 40 chairs.