Solution of Linear Programming Problems

For an LPP the objective function is $Z=4x+3y$ and the feasible region determined by set of constraints(linear inequations) is shown in the graph.

Which of the following statement is true?

What are conflicting constraints. Show that the LPP in which the objective function $z=6x+4y$ is to be minimized subject to the constraints $3x+2y\ge 18$ and $2x+y\le 16 \left(x\ge 0, y\ge 0\right)$ has infinitely many optimal solutions.

What are conflicting constraints. 

What are conflicting constraints. Show that if each of the infinitely many optimal solutions of an LPP with objective function $z=ax+by$, lies on the line $15 x+25 y=32$ with $5a=3b$.

What are conflicting constraints. Find optimal solution of the following LPP, Maximize $z=2x+3y$ subject to $5x+4y\le 20$, where $x\ge 0, y\ge 0$.

What are conflicting constraints. 

Show that the optimal solution of the following LPP

Maximize $z=5x+3y$

Subject to, $x+2y\le 16$,

$0\le y\le 3$,

$x\ge 0$

lies on the straight line $2x+5y=32$.

Solve $z=2x+3y$, subject to $x+y\le 6$, $2x+y\ge 16$, $x\ge 0$, $y\ge 0$ graphically. Check whether it has feasible or infeasible solution.

Solve $z=3x+2y$, subject to $x+y\le 5$, $2x+y\ge 20$, $x\ge 0$, $y\ge 0$ graphically. Check whether it has feasible or infeasible solution.

Solve $z=2x+2y$, subject to $x+y\le 5$, $x+2y\ge 14$, $x\ge 0$, $y\ge 0$ graphically. Check whether it has feasible or infeasible solution.

Solve $z=x+2y$, subject to $x+y\le 5$, $3x+y\ge 21$, $x\ge 0$, $y\ge 0$ graphically. Check whether it has feasible or infeasible solution.

Solve $z=4x+3y$, subject to $x+y\le 6$, $2x+y\ge 20$, $x\ge 0$, $y\ge 0$ graphically. Check whether it has feasible or infeasible solution.

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 pumps, $D, E, F$, whose requirements are $4500 L, 3000 L,$ and $3500 L$ respectively. The distances $\left(\mathrm{in} \mathrm{km}\right)$ between the depots and the petrol pumps are given in the following table:

Assuming that the transportation cost per $\mathrm{km}$ is $\mathrm{Rs} 1$ per litre, the delivery is scheduled in order that the transportation cost is minimum. What is the minimum transportation cost?

The feasible region of an $\mathrm{LPP}$ is shown in the figure. If $z=3x+9y$, then the minimum value of $z$ occurs at 

The corner points of the feasible region are $\left(800, 400\right), \left(1050, 150\right), \left(600, 0\right)$. The objective function is $P=12x+6y$. The maximum value of $P$ is

Reshma wishes to mix two types of food $P$ and $Q$ in such a way that the vitamin contents of the mixture contain at least $8$ units of vitamin $A$ and $11$ units of vitamin $B.$ Food $P$ costs $\mathrm{R}\mathrm{s}.60/\mathrm{k}\mathrm{g}$ and Food $Q$ costs $\mathrm{R}\mathrm{s}.80/\mathrm{k}\mathrm{g}.$ Food $P$ contains $3 \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{k}\mathrm{g}$ of vitamin $A$ and $5 \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{k}\mathrm{g}$ of vitamin $B$ while food $Q$ contains $4 \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{k}\mathrm{g}$ of vitamin $A$ and $2 \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{k}\mathrm{g}$ of vitamin $B.$ Determine the minimum cost of the mixture?

Find the maximum value of $Z=7x+7y$ subject to the constraints.

$x \ge 0, y \ge 0, x + y \ge 2$ and $2x + 3y \le 6.$

A dietician wishes to mix two types of food, $X$ and $Y$, in such a way that the vitamin contents of the mixture contains at least $8$ units of vitamin $A$ and $10$ units of vitamin $C$. Food $X$ contains $2 \mathrm{units}/\mathrm{kg}$ of vitamin $A$ and $1 \mathrm{unit} /\mathrm{kg}$  of vitamin $C$, while food $Y$ contains $1 \mathrm{unit}/\mathrm{kg}$ of vitamin $A$ and $2 \mathrm{units}/\mathrm{kg}$ of vitamin $C$. It costs $₹5 \mathrm{per} \mathrm{kg}$ to purchase the food $X$ and $₹7 \mathrm{per} \mathrm{kg}$ to purchase the food $Y$. Determine the minimum cost of such a mixture.

The objective function $Z=4x_1+5x_2,$ subject to $2x_1+x_2\ge 7, 2x_1+3x_2\le 15, x_2\le 3 \& x_1,x_2\ge 0$ has minimum value at the point

Solution of the LPP Min. $\mathrm{Z}=5x+10y$ subject to: $x+2y\le 120,$ $x+y\ge 60,$ $x-2y\ge 0,$ $x, y\ge 0$ is

A small firm manufactures necklace and bracelets. The total number of necklace and bracelet that it can handle per day is at most $24$. It takes $1$hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is $16$. If the profit on a necklace is $₹100$ and that on a bracelet is $₹300$, how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.