Chapter 6, Problem 6RP

a.

Explanation of Solution

Dual linear problem and its optimal solution

• The dual of a given linear program is another linear program that is derived from the original linear program in following ways:
  • Each variable in the original linear program becomes a constraint in the dual linear program.
  • Each constraint in the original linear program becomes a variable in the dual linear program.
  • The objective direction is inversed which becomes maximum in the original and minimum in the dual.
• Hence the dual of the linear problem is

  ${\text{Min w = 8y}}_{\text{1}}{\text{+ 10y}}_{\text{2}}$

  Subject to

  $\begin{array}{l}{\text{2y}}_{\text{1}}{\text{+ 4y}}_{\text{2}}\text{>= 3}\hfill \\ {\text{y}}_{\text{1}}{\text{+ y}}_{\text{2}}\text{>= 1}\hfill \end{array}$

• The formulas for computing the optimal table include:
  • ${\text{x}}_{\text{j}}$ column in optimal table’s constraints = ${\text{B}}^{\text{-1}}$

b.

Explanation of Solution

New Optimal solution

• For a maximization problem, the new optimal value = (old optimal value) + (Constraint i’s shadow price).
• For a minimization problem, the new optimal value = (old optimal value) – (Constraint i’s shadow price)...