Chapter 4, Problem 9RP

Explanation of Solution

Optimal solution:

• Consider the following linear programming problem
• min z= $-3x_1+x_2$
• Subject to constraints:
  • $x_1-2x_2\ge 2$
  • $-x_1+x_2\ge 3$
  • $x_1,x_2\ge 0$
• Surplus variables and artificial variables are added, $e_1,a_1,e_2,a_2$
• Therefore, the standard form of the linear programming problem is,
• min z=$-3x_1+x_2+M{a}_1+M{a}_2$
• Subject to constraints:
  • $a_2=3$
  • $-x_1+x_2-e_2+a_2=3$
  • $x_1,x_2,e_1,a_1,e_2,a_2\ge 0$
• The basic feasible solution is
  • $a_1=2$
  • $a_2=3$
• Since basic feasible solution contains artificial variable, we need to eliminate the artificial variable.
• We apply,
  • row 0+M(row 1)=0+M(2)+M(3)
  • $2x_3+x_2+sp5-sm5=93$
  • $+M(x_1-2x_2-e_1+a_1)+M(-x_1+x_2-e_2+a_2)=$