Chapter 6.4, Problem 12P

Explanation of Solution

Optimal solution of the diet problem:

• The case 1 result for right- hand side states that “All constraints whose right-hand sides are being modified and non-binding constraints”.
• Now it can be seen that is a constraint is non-binding in the optimal solution, then its slack or excess variable is in the optimal basis and the corresponding column of $B^{-1}$ will have a single 1 and all other elements equal to 0.
• Suppose that the right –hand side of constraint 1 is changed then, b₁ is changed to $b_1{}^{\text{'}}=b_1+\Delta$ then the right-hand side of the optimal table is given by,
  • $B^{-1}{b}^{\text{'}}=B^{-1}b+\Delta B_1{}^{-1}$
• In this case $B_1{}^{-1}$ stands for first column of $B^{-1}$ matrix