Chapter 4, Problem 36RE

In Chapter 2 we wrote a system of linear equations using matrix notation. Similarly, we can use matrix notation to write a system of linear inequalities.

1. (a) Find matrices A, B, C, and X such that the maximization problem in Example 1 of Section 4.1 can be written as

$\begin{array}{ll}\text{Maximize}\hfill & CX\hfill \\ \text{subject}\text{to:}\hfill & AX\le B\hfill \\ \text{with}\hfill & X\ge O.\hfill \end{array}$

(Hint: Let B and X be column matrices, and C a row matrix.)

1. (b) Show that the dual of the problem in part (a) can be written as

$\begin{array}{ll}\text{Minimize}\hfill & YB\hfill \\ \text{subject}\text{to:}\hfill & YA\ge C\hfill \\ \text{with}\hfill & Y\ge O,\hfill \end{array}$

  where Y is a row matrix.

1. (c) Show that for any feasible solutions X and Y to the original and dual problems, respectively, CX ≤ YB. (Hint: Multiply both sides of AX ≤ B by Y on the left. Then substitute for YA.)

EXAMPLE 1 Slack Variables

Restate the following linear programming problem by introducing slack variables.

$\begin{array}{ll}\text{Maximize}\hfill & z=3x_1+2x_2+x_3\hfill \\ \text{subject}\text{to:}\hfill & 2x_1+x_2+x_3\le 150\hfill \\ \hfill & 2x_1+2x_2+8x_3\le 200\hfill \\ \hfill & 2x_1+3x_2+x_3\le 320\hfill \\ \text{with}\hfill & x_1\ge 0,x_2\ge 0,x_3\ge 0.\hfill \end{array}$