$A$ firm has to transport $1200$ packages using large vans which can carry $200$ packages each and small vans which can take $80$ packages each. The cost for engaging each large van is $Rs. 400$ and each small van is $Rs. 200$. Not more than $Rs. 3000$ is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as an $LPP$ given that the objective is to minimize cost.
(A) Let $x$ be the number of large vans and $y$ be the number of small vans.
The objective is to minimize the total cost $Z$. The cost of one large van is $Rs. 400$ and one small van is $Rs. 200$. Thus,the objective function is $Z = 400x + 200y$.
Constraints:
$1$. Total packages to be transported is at least $1200$: $200x + 80y \geq 1200$,which simplifies to $5x + 2y \geq 30$.
$2$. Total cost cannot exceed $Rs. 3000$: $400x + 200y \leq 3000$,which simplifies to $2x + y \leq 15$.
$3$. Number of large vans cannot exceed the number of small vans: $x \leq y$.
$4$. Non-negativity constraints: $x \geq 0, y \geq 0$.
Thus,the $LPP$ is:
Minimize $Z = 400x + 200y$
Subject to:
$5x + 2y \geq 30$
$2x + y \leq 15$
$x \leq y$
$x, y \geq 0$
The objective is to minimize the total cost $Z$. The cost of one large van is $Rs. 400$ and one small van is $Rs. 200$. Thus,the objective function is $Z = 400x + 200y$.
Constraints:
$1$. Total packages to be transported is at least $1200$: $200x + 80y \geq 1200$,which simplifies to $5x + 2y \geq 30$.
$2$. Total cost cannot exceed $Rs. 3000$: $400x + 200y \leq 3000$,which simplifies to $2x + y \leq 15$.
$3$. Number of large vans cannot exceed the number of small vans: $x \leq y$.
$4$. Non-negativity constraints: $x \geq 0, y \geq 0$.
Thus,the $LPP$ is:
Minimize $Z = 400x + 200y$
Subject to:
$5x + 2y \geq 30$
$2x + y \leq 15$
$x \leq y$
$x, y \geq 0$
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