Solving 2 × 2 Matrix Games Using Geometric Methods

Question

I think my text book may be wrong. I am trying to learn how to solve matrix games using a geometric method. So I am given the matrix $$\begin{pmatrix}-2 & 4 \\1 & -3\end{pmatrix}$$ So the first step is to get rid of the negatives, so I make k=4 and add k to each number resulting in $$\begin{pmatrix}2 & 8 \\5 & 1\end{pmatrix}$$ Now the next step is to SET UP TWO CORRESPONDING LINEAR PROGRAMMING PROBLEMS. This is where I need help. The text book says the two problems will be \begin{align} \text{Minimize } \ y=x_1 +x_2\\ \text{subject to }\ 2x_1 +8x_2 \ge 1\\ 5x_1+x_2\ge1\\ x_1,x_2\ge0\end{align} and \begin{align} \text{Minimize }\ y=z_1+z_2\\ \text{subject to }\ 2z_1+8z_2\ge1\\ 5z_1+z_2\ge1\\ z_1,z_2\ge0 \end{align}

The next step shows two different graphs which confuses me because the numbers of the two problems are the same. This leads me to believe something is wrong in this example. Should one be a minimization and the other be a maximization? If someone could explain this that would be great. Thanks!

Answer 1

I have that same textbook and although the instructions are good, the attention to detail is terrible. When they form the dual of the problem (the second linear programming problem) they erroneously use the same coefficients. The linear programming systems should be:

Minimize    y=x1+x2
subject to  2x1+5x2 ≥ 1
            8x1+x2 ≥ 1
            x1, x2 ≥ 0

Maximize    y=z1+z2
subject to  2z1+8z2 <= 1
            5z1+z2 <=  1
            z1, z2 ≥ 0

This should produce the results you see in example graphs.