Question

The corner points of the feasible region determined by $x + y \leq 8$, $2x + y \geq 8$, $x \geq 0$, $y \geq 0$ are $A(0, 8)$, $B(4, 0)$, and $C(8, 0)$. If the objective function $Z = ax + by$ has its maximum value on the line segment $AB$, then the relation between $a$ and $b$ is:

• A. $8a + 4 = b$
• B. $a = 2b$
• C. $b = 2a$
• D. $8b + 4 = a$

Hint:

When dealing with linear optimization problems, the slope of the constraint line often provides a critical relationship between the coefficients of the objective function. Make sure to relate the slope of the constraint line to the ratio of coefficients for finding optimal values.

Answer 1

The Correct Option is B

To determine the relationship between coefficients $a$ and $b$ for the objective function $Z = ax + by$ to attain its maximum value along the line segment $AB$, we examine the vertices $A(0, 8)$, $B(4, 0)$, and $C(8, 0)$. The equation of the line segment $AB$ is derived as follows. The slope of line $AB$ is calculated as:

$$\text{slope of } AB = \frac{0 - 8}{4 - 0} = -2.$$This yields the line equation $y = -2x + 8$. Substituting this into the objective function, we get $Z = ax + b(-2x + 8)$, which can be rewritten as $Z = x(a - 2b) + 8b$.
The objective function $Z$ will exhibit a constant value (either maximum or minimum) along segment $AB$ if the coefficient of $x$ is zero, i.e., $a - 2b = 0$.

Thus, the condition $a = 2b$ ensures that the objective function $Z$ remains constant throughout the line segment $AB$.