Chapter 6.5, Problem 4P

Explanation of Solution

Duals of LP:

• Consider the linear programming problem given below:
• max z= $4y_1+2y_2-y_3$
• Subject to constraints
  • $y_1+2y_2\le 6$
  • $y_1-y_2+2y_3=8$
  • $y_1,y_2\ge 0,y_3$
• • $x_1+x_2\le 5$
  • $2x_1+x_2\le 7$
  • $2x_2+x_3\ge 6$
  • $x_1+x_3=4$
  • $x_1\le 0,x_2,x_3$ (unrestricted in sign(urs))
• As the above linear programming problem is a normal min problem, and therefore, perform the following steps to convert it into a normal form.
• Step1:
  • Multiply each $\le$ constraint by -1 as follows,
  • $y_1+2y_2\le 6$
  • $-\left(y_1+2y_2\right)\ge -6$
  • $-y_1-2y_2\ge -6$
• Step 2:
  • Replace the equality constraint $y_1-y_2+2y_3=8$ by two inequality constraints as,
    • $y_1-y_2+2y_3\ge 8$
    • $y_1-y_2+2y_3\le 8$
      • Now convert $\le$ constraint $\ge$ constraint as,
      • $-y_1+y_2-2y_3\ge 8$
• Step 3:
  • Now replace each urs variable $y_1$ by $y_i=y_i^{\text{'}}-y_i^{\text{'}\text{'}}$ as,
    • $y_3=y_3^{\text{'}}-y_3^{\text{'}\text{'}}$
      • Here,
        • $y_3^{\text{'}},y_3^{\text{'}\text{'}}\ge 0$
• Now the standard min LP is,
• min w= $4y_1+2y_2-y_3^{\text{'}}+y_3^{\text{'}\text{'}}$
• Subject to constraints,
  • $-y_1-2y_2\ge -6$
  • $y_1-y_2+2y_3^{\text{'}}-2y_3^{\text{'}\text{'}}\ge 8$
  • $-y_1+y_2-2y_3^{\text{'}}+2y_3^{\text{'}\text{'}}\ge -8$
  • $y_1,y_2,y_3^{\text{'}},y_3^{\text{'}\text{'}}\ge 0$
• As the above linear programming problem is a normal min problem, it can be solved with the tabular approach as follows