Chapter 6.11, Problem 4P

Explanation of Solution

New optimal solution to the Dakota problem:

• The primal to the Dakota problem is,

  ${\text{max z =60x}}_1{\text{+30x}}_2{\text{+20x}}_3$

• The above function subject to the following conditions,
  • “${\text{8x}}_1{\text{+6x}}_2{\text{+x}}_3\le \text{ 48}$” denotes lumber condition.
  • “${\text{4x}}_1{\text{+2x}}_2\text{+ 1}{\text{.5x}}_3\le \text{ 20}$” denotes the finishing condition.
  • “${\text{2x}}_1\text{+1}{\text{.5x}}_2\text{+ 0}{\text{.5x}}_3\le \text{8}$” denotes the carpentry condition.
  • All the variables should be non-negative.
• For the above Linear Programming (LP), the system of linear equations can be defined in the table as follows,

z	$x_1$	$x_2$	$x_3$	$s_1$	$s_2$	$s_3$	RHS
1	-60	-30	-20	0	0	0	0
0	8	6	1	1	0	0	48
0	4	2	1.5	0	1	0	20
0	2	1.5	0.5	0	0	1	8

• The new RHS of the optimal table is as follows,

$\left[\begin{array}{ccc}1& 2& -8\\ 0& 2& -4\\ 0& -0.5& 1.5\end{array}\right]\left[\begin{array}{c}48\\ 20\\ 15\end{array}\right]=[\begin{array}{c}-32\\ -20\\ 12\end{array}$