Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $X_{2}$ ), find the range of values for the objective function coefficient of $X_{1}$ 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) $[0,1000]$
B) $[-\infty, 1000]$
C) $[80,900]$
D) $[80,1000]$

Question

Quizplus · Accepted Answer

The range of values for the objective function coefficient of $X_{1}$ that keeps the current solution optimal can be found using sensitivity analysis in linear programming. Given that both constraints are binding, we can set up a system of equations based on these constraints and solve for $X_{1}$ and $X_{2}$. The optimal solution is found at the intersection of the two constraints. To find the range of optimality for $X_{1}$'s coefficient, we adjust the objective function coefficient of $X_{1}$ and observe the effect on the optimal solution, ensuring that the solution remains feasible and optimal. The correct range is $[80,1000]$, indicating that as long as the coefficient of $X_{1}$ in the objective function is within this range, the current solution remains optimal.

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 $\left(X_{1}\right.$