Chapter 6.9, Problem 3P

Explanation of Solution

To determine whether the current basis remains optimal:

• The primal of 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 problem, the optimal solution is ,

  $\begin{array}{l}\text{z =}-\text{280, }\\ {\text{x}}_1\text{= 2, }\\ {\text{x}}_2\text{= 0, }\\ {\text{x}}_3\text{= 8}\\ {\text{s}}_1\text{=24}\\ {\text{s}}_2\text{ = 0}\\ {\text{s}}_3\text{ = 0}\end{array}$

• The dual of the Dakota problem is,

${\text{min w=48y}}_{\text{1}}{\text{+ 20y}}_{\text{2}}{\text{+ 8y}}_{\text{3}}$

• The above function subject to the following conditions,
  • “${\text{8y}}_{\text{1}}{\text{ +4y}}_{\text{2}}{\text{ +2y}}_{\text{3}}\ge \text{60}$” denotes the desk condition.
  • “${\text{6y}}_{\text{1}}{\text{ +2y}}_{\text{2}}\text{ +1}{\text{.5y}}_{\text{3}}\ge \text{ 30}$” denotes the table condition.
  • “${\text{y}}_{\text{1}}\text{ +1}{}_{}$

Explanation of Solution

To determine the mistake in the given reasoning:

The interpretation is tabulated is as follows,

Resource	Shadow price	Quantity of resource used	Value of resource used
Lumber	0	“8” board feet	8(0) =$0
Finishing	10	“4” hours	4(10)=$40
Carpentry	10	“15” hours	15(10)=$150