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.
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Solution and Explanation
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.
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