Question

The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let $z = px + qy$, where $p, q>0$. The relation between $p$ and $q$, so that the maximum $z$ occurs at both points (15, 15) and (0, 20) is

• A. $p = q$
• B. $p = 2q$
• C. $q = 2p$
• D. $q = 3p$

Hint:

Whenever a problem states that an objective function attains its optimal value at multiple specific points, simply evaluate the function at those points and equate the results. This is a very common and straightforward question type in linear programming.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In linear programming, the fundamental theorem states that if an optimal value (maximum or minimum) exists, it must occur at one or more corner points of the feasible region. If the maximum value occurs at two different corner points, it means the value of the objective function evaluated at these two points must be exactly the same.

Step 2: Key Formula or Approach:
1. Evaluate the objective function $z = px + qy$ at the first given point $(15, 15)$.
2. Evaluate the objective function $z = px + qy$ at the second given point $(0, 20)$.
3. Set these two expressions equal to each other because they both represent the same maximum value.
4. Solve the resulting equation to find the relationship between $p$ and $q$.

Step 3: Detailed Explanation:
The objective function is given as $z = px + qy$. Let's find the value of $z$ at the corner point $(15, 15)$: \[ z_1 = p(15) + q(15) = 15p + 15q \] Next, find the value of $z$ at the corner point $(0, 20)$: \[ z_2 = p(0) + q(20) = 0 + 20q = 20q \] The problem states that the maximum value of $z$ occurs at both of these points. Therefore, the value of $z$ must be equal at these two points: \[ z_1 = z_2 \] Substituting our expressions: \[ 15p + 15q = 20q \] Now, solve for the relation between $p$ and $q$. Subtract $15q$ from both sides: \[ 15p = 20q - 15q \] \[ 15p = 5q \] Divide both sides by 5: \[ 3p = q \] This can be written as $q = 3p$.

Step 4: Final Answer:
The required relation between $p$ and $q$ is $q = 3p$.