Question

If the objective is to minimize cost \( Z = 16x + 20y \), find the values of \( x \) and \( y \) at which cost is minimum. Also, find the minimum cost assuming that minimum cost is possible for the given unbounded region.

Hint:

In linear programming, always evaluate the objective function at the vertices of the feasible region to determine the minimum or maximum value. Verify the result if the feasible region is unbounded.

Answer 1

Step 1: Calculate \( Z \) at each vertex. At \( A(10, 0) \): 
\[ Z = 16(10) + 20(0) = 160. \] At \( B(2, 4) \): 
\[ Z = 16(2) + 20(4) = 32 + 80 = 112. \] At \( C(1, 5) \): 
\[ Z = 16(1) + 20(5) = 16 + 100 = 116. \] At \( D(0, 8) \): 
\[ Z = 16(0) + 20(8) = 0 + 160 = 160. \] 
Step 2: Identify the minimum cost. 
The minimum value of \( Z \) is \( 112 \) at \( B(2, 4) \). 
Conclusion: 
The values of \( x \) and \( y \) that minimize the cost are: \[ \boxed{x = 2, \, y = 4, \, \text{Minimum cost} = \text{₹} 112.} \]
Step 3: Verify for unbounded region. 
Since the feasible region is unbounded, it is necessary to verify the validity of the minimum cost. The objective function \( Z = 16x + 20y \) increases as \( x \) or \( y \) increases. Hence, the minimum cost of \( 112 \) at \( B(2, 4) \) is valid.