Quantitative Analysis for Management, 13/e

Solved Problems

1. Solved Problem M7-1 Convert the following constraints and objective function into the proper form for use in the simplex method:

   $$\begin{array}{lll}\text{Minimize cost}\hfill & =\hfill & \text{4}{X}_{\text{1}}\text{ + 1}{X}_{\text{2}}\hfill \\ \text{subject to}\hfill & \text{3}{X}_{\text{1}}\text{ + }{X}_{\text{2}}\text{ = }\text{3}\hfill \\ \text{4}{X}_{\text{1}}\text{ + 3}{X}_{\text{2}}\ge 6\hfill \\ X_{\text{1}}\text{ + 2}{X}_{\text{2}}\le \text{3}\hfill \end{array}$$

   Solution

   $$\begin{array}{lllll}\text{Minimize cost =}\hfill & \text{4}{X}_{\text{1}}\text{ + 1}{X}_{\text{2}}\text{ + 0}{S}_{\text{1}}\hfill & \text{+ 0}{S}_{\text{2}}\hfill & \text{+ }M{A}_{\text{1}}\hfill & \text{+ }M{A}_{\text{2}}\hfill \\ \text{subject to}\hfill & \text{3}{X}_{\text{1}}\text{ + 1}{X}_{\text{2}}\hfill & \text{+ 1}{A}_{\text{1}}\hfill & \text{= 3}\hfill \\ \text{4}{X}_{\text{1}}\text{ + 3}{X}_{\text{2}}\text{ − 1}{S}_{\text{1}}\hfill & \text{+ 1}{A}_{\text{2}}\hfill & \text{= 6}\hfill \\ \text{1}{X}_{\text{1}}\text{ + 2}{X}_{\text{2}}\hfill & \text{+ 1}{S}_{\text{2}}\hfill & \text{= 3}\hfill \end{array}$$

2. Solved Problem M7-2 Solve the following LP problem:

   $$\begin{array}{ll}\text{Maximize profit}\text{=}\hfill & \text{\$9}{X}_{\text{1}}\text{ + \$7}{X}_{\text{2}}\hfill \\ \text{subject to}\hfill & \text{2}{X}_{\text{1}}{\text{ + 1X}}_{\text{2}}\le \text{ 40}\hfill \\ {\text{X}}_{\text{1}}{\text{ + 3X}}_{\text{2}}\le \text{ 30}\hfill \end{array}$$

   Solution

   We begin by adding slack variables and converting inequalities into equalities.

   $$\begin{array}{ll}\text{Maximize profit}\text{=}\hfill & \text{9}{X}_{\text{1}}\text{ + 7}{X}_{\text{2}}{\text{+ 0S}}_{\text{1}}{\text{ + 0S}}_{\text{2}}\hfill \\ \text{subject to}\hfill & \text{2}{X}_{\text{1}}\text{+ 1}{X}_{\text{2}}\text{+}\text{1}{S}_{\text{1}}\text{ + 0}{S}_{\text{2}}\text{ = 40}\hfill \\ \text{1}{X}_{\text{1}}\text{+ 3}{X}_{\text{2}}\text{+ 0}{S}_{\text{1}}\text{+ 1}{S}_{\text{2}}\text{ = 30}\hfill \end{array}$$

   The initial tableau is then as follows:

   7.1-28 Full Alternative Text ...