The feasible region for a $LPP$ is shown in the figure. Find the minimum value of $Z=11x+7y$.
(A) The lines $x+y=5$ and $x+3y=9$ intersect at $(3,2)$. From the figure,the feasible region is bounded with corner points $C(0,3)$,$A(3,2)$,and $B(0,5)$.
We evaluate the objective function $Z=11x+7y$ at each corner point:
Comparing the values of $Z$,the minimum value is $21$ at the point $(0,3)$.
We evaluate the objective function $Z=11x+7y$ at each corner point:
| Corner Point | Value of $Z=11x+7y$ |
| $C(0,3)$ | $11(0)+7(3) = 21$ |
| $A(3,2)$ | $11(3)+7(2) = 33+14 = 47$ |
| $B(0,5)$ | $11(0)+7(5) = 35$ |
Comparing the values of $Z$,the minimum value is $21$ at the point $(0,3)$.
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The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:
An oil company has two depots $A$ and $B$ with capacities of $7000 \, L$ and $4000 \, L$ respectively. The company is to supply oil to three petrol pumps,$D, E$ and $F$ whose requirements are $4500 \, L, 3000 \, L$ and $3500 \, L$ respectively. The distances (in $km$) between the depots and the petrol pumps are given in the following table:
From/To $A$ $B$ $D$ $7$ $3$ $E$ $6$ $4$ $F$ $3$ $2$
Assuming that the transportation cost of $10 \, L$ of oil is $Rs. \, 1$ per $km$,how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?
| From/To | $A$ | $B$ |
| $D$ | $7$ | $3$ |
| $E$ | $6$ | $4$ |
| $F$ | $3$ | $2$ |
Assuming that the transportation cost of $10 \, L$ of oil is $Rs. \, 1$ per $km$,how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?
Difficult
View SolutionMaximize the function $Z = 11x + 7y$,subject to the constraints:
$x \leq 3, y \leq 2, x \geq 0, y \geq 0$
$x \leq 3, y \leq 2, x \geq 0, y \geq 0$
Medium
View SolutionDetermine graphically the minimum value of the objective function
$Z = -50x + 20y$ .....$(1)$
subject to the constraints:
${2x - y \geqslant -5}$ .....$(2)$
${3x + y \geqslant 3}$ .....$(3)$
${2x - 3y \leqslant 12}$ .....$(4)$
${x \geqslant 0, y \geqslant 0}$ .....$(5)$
$Z = -50x + 20y$ .....$(1)$
subject to the constraints:
${2x - y \geqslant -5}$ .....$(2)$
${3x + y \geqslant 3}$ .....$(3)$
${2x - 3y \leqslant 12}$ .....$(4)$
${x \geqslant 0, y \geqslant 0}$ .....$(5)$
Medium
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