Chapter 6.7, Problem 4P

Program Plan Intro

Linear programming (LP):

• The linear programming (LP) is also known as linear optimization.
• Consider a mathematical model, and its requirements are used to represent by the linear relationships. The linear programming is the best method to achieve the best outcome of this mathematical model. The outcomes may be, maximum profit or lower cost.
• The linear optimization is also called as mathematical optimization because, it is a special case of mathematical programming.
• More formally, the LP is a technique for optimizing linear objective function subject to constraints of linear equality and linear inequality.

Dual of LP:

The dual of the given LP is another LP derived in the following schematic way from the original (the primal) LP.

• Every variable in the primal Linear Programming becomes a constraint in the dual LP.
• Every limit in the primal Linear Programming becomes a variable in the dual LP.
• In the dual LP, the objective direction is inversed. That is, the maximum in the primal becomes minimum in the dual and vice-versa.

Explanation of Solution

Optimal solution of the dual:

The optimal solution to the dual is,

$\begin{array}{c}{\text{x}}_{\text{1}}\text{=0,}\\ {\text{x}}_{\text{2}}\text{=1/2,}\\ {\text{x}}_{\text{3}}\text{=3/2}\end{array}$