Question

The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure: 
(i) If \( Z = 3x - 4y \) be the objective function, then find the maximum value of \( Z \). 
(ii) If \( Z = px + qy \) where \( p, q>0 \) be the objective function, find the condition on \( p \) and \( q \) so that maximum value of \( Z \) occurs at \( B(4, 10) \) and \( C(6, 8) \). 


 

Hint:

When working with objective functions and constraints, always evaluate the function at the corner points of the feasible region to determine the maximum or minimum value. For multiple objective functions, set them equal to each other to find the relationship between the coefficients.

Answer 1

The corner points are given as: \[ A(0,8), \quad B(4,10), \quad C(6,8), \quad D(6,5), \quad E(4,0), \quad O(0,0) \] 
(i) Objective function: 
\( Z = 3x - 4y \) Now, substitute the coordinates of the corner points into the objective function: For \( A(0,8) \): \[ Z = 3(0) - 4(8) = -32 \] For \( B(4,10) \): \[ Z = 3(4) - 4(10) = -28 \] For \( C(6,8) \): \[ Z = 3(6) - 4(8) = -14 \] For \( D(6,5) \): \[ Z = 3(6) - 4(5) = -2 \] 
For \( E(4,0) \): \[ Z = 3(4) - 4(0) = 12 \quad {(Maximum value)} \] For \( O(0,0) \): \[ Z = 3(0) - 4(0) = 0 \] 
Thus, the maximum value of \( Z \) is \( 12 \) at point \( E(4,0) \). 
(ii) Objective function: 
\( Z = px + qy \) where \( p, q>0 \) For \( Z_B = Z_C \), we have the condition: 
\[ 4p + 10q = 6p + 8q \] Simplifying this: \[ 4p + 10q - 6p - 8q = 0 \] \[ -2p + 2q = 0 \] \[ p = q \] Thus, the condition on \( p \) and \( q \) is \( p = q \).