Lesson 5: Optimization — Practice Questions

1. 1. What is the maximum value of the objective function $f(x,y) = 4x + y$ for a feasible region with vertices at $(1,2)$, $(5,1)$, $(6,4)$, and $(2,5)$? The maximum value is ___.

2. 2. Find the minimum value of the objective function $P = 2x - 3y$ over a feasible region with vertices at $(0,5)$, $(4,0)$, and $(6,8)$. The minimum value is ___.

3. 3. At which vertex does the maximum value of the objective function $C = 5x + 2y$ occur for a feasible region with vertices $(0,0)$, $(7,0)$, $(5,4)$, and $(0,9)$?

   • A. $(0,0)$
   • B. $(7,0)$
   • C. $(5,4)$
   • D. $(0,9)$

4. 4. According to the Vertex Theorem for linear programming, where must the optimal value of a linear objective function be found for a bounded feasible region?

   • A. At the origin $(0,0)$
   • B. At one of the vertices of the feasible region
   • C. Along an edge of the feasible region
   • D. At the center of the feasible region

5. 5. A feasible region is defined by the vertices $(1,1)$, $(6,2)$, $(8,5)$, and $(3,7)$. What is the minimum value of the objective function $Z = x - 5y$ on this region? The minimum value is ___.

6. 6. Consider maximizing $P = x + y$ subject to $x > 0$, $y > 0$, and $x + y < 10$. Why does this linear programming problem have no maximum value?

   • A. The feasible region is unbounded.
   • B. The optimal value occurs on a boundary that is excluded from the feasible region.
   • C. The objective function is not linear.
   • D. The constraints are contradictory and define an empty region.

7. 7. Consider minimizing the objective function $C = 4x + 5y$ subject to the constraints $x > 3$ and $y > 1$. The minimum value is approached at the excluded vertex $(3, 1)$. The value of $C$ at this point is ___.

8. 8. True or False: If a feasible region is defined entirely by strict inequalities (e.g., $x > a, y > b, ...$), it is impossible to find a maximum or minimum value for a linear objective function.

   • A. True
   • B. False

9. 9. Consider maximizing $Z = 3x + y$ subject to $x > 0$, $y > 0$, and $3x + y < 15$. The maximum value is approached but never reached. This limiting value, or supremum, of $Z$ is ___.

10. 10. Consider maximizing $P = 5x - y$ subject to $x \leq 6$, $y \geq 1$, and $x + y < 10$. Does an optimal maximum value exist for $P$?

    • A. Yes, because the optimal vertex (6, 1) is included in the feasible region.
    • B. No, because the inequality $x + y < 10$ is strict.
    • C. No, because the feasible region is unbounded.
    • D. Yes, but the optimal value occurs on the line $x + y = 10$.