Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $\left.X_{2}\right)$, find the range of values for the objective function coefficient of $X_{2}$ that will leave the current solution optimal (that is range of optimality or range of insignificance as the case may be) (hint: both constraints are binding)
Max: $100 X_{1}+200 X_{2}$
Constraints: $2 X_{1}+5 X_{2} \leq 104 \quad 15 X_{1}+3 X_{2} \leq 90$
Variables are non-negative
A) $[20,250]$
B) $[-\infty, 250]$
C) $[0,250]$
D) $[20, \infty]$

Question

Quizplus · Accepted Answer

The range of values for the objective function coefficient of $X_{2}$ that keeps the current solution optimal can be found using the sensitivity analysis part of linear programming. Since both constraints are binding, we can set up a system of equations using the constraints and solve for $X_{1}$ and $X_{2}$. The objective function's coefficients can then be adjusted within a certain range without changing the optimal solution. This range is determined by the shadow prices or dual values of the constraints and the current coefficients of the objective function. Given the information, the correct range is $[20,250]$, indicating that as long as the coefficient of $X_{2}$ in the objective function stays within this range, the current solution remains optimal. This takes into account the marginal worth of resources as indicated by the constraints and how changes in the objective function's coefficients affect the optimal mix of $X_{1}$ and $X_{2}$.

Given the Following Linear Programming Problem with Two Non-Negative Variables (X1\left(X_{1}\right.(X1​

Given the Following Linear Programming Problem with Two Non-Negative Variables $\left(X_{1}\right.$