In the LP with constraints X1 + X2 ≤ 4 and 2X1 + 5X2 ≤ 10, which constraint is binding at the optimal solution?

• A. Both constraints binding
• B. X1+X2 ≤ 4
• C. 2X1+X2 ≤ 10
• D. None binding

Answer: (B)

A constraint is binding when the optimal solution lies on its boundary, meaning the constraint holds with equality there.

If the goal is to maximize X1 + X2, the first constraint sets the highest possible sum: X1 + X2 ≤ 4, so the maximum value of the objective is 4. Any optimal solution must satisfy X1 + X2 = 4, placing you on that boundary. Along that edge, the second constraint 2X1 + 5X2 ≤ 10 is not required to reach the maximum; it’s only tight at the single intersection point (3.333, 0.667). Therefore, the constraint X1 + X2 ≤ 4 is the one binding at the optimum.