Chapter 6.5, Problem 3P

Explanation of Solution

Duals of LP:

• Consider the linear programming problem given below:
• max z= $4x_1-x_2+2x_3$
• Subject to constraints
  • $x_1+x_2\le 5$
  • $2x_1+x_2\le 7$
  • $2x_2+x_3\ge 6$
  • $x_1+x_3=4$
  • $x_1\le 0,x_2,x_3$ (unrestricted in sign(urs))
• Dual of the above LPP can be found with the use of three steps:
• Step1:
  • Construct the following table according to the given LP,

	Max Z
Min W	$(x_1\ge 0)$	$(x_{2}urs)$	$(x_{3}urs)$
	$x_1$	$x_2$	$x_3$
$(y_1\ge 0)$	$y_1$	1	1	0	$\le 5$
$(y_2\ge 0)$	$y_2$	2	1	0	$\le 7$
	$y_3$	0	2	1	$\ge 6^{*}$
	$y_4$	1	0	1	$=4^{*}$
	$\ge 4$	$=-1$	$=2$

• Step 2:
  • From the above given table, it can be said that the third primal constraint is $\ge$ constraint