Method of Solving LPP

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 $2x+5y=32$.

State the Convex polygon theorem.

Find convex region from the following figures.

A gardener has a supply of fertilizers of the type one which consist of $10\%$ nitrogen and $6\%$ phosphoric acid, and of the type two which consist of $5\%$ nitrogen and $10\%$ phosphoric acid. After testing the soil condition, he finds that he needs at least $14 \mathrm{g}$ of nitrogen and $14 \mathrm{g}$ of phosphoric acid for his crop. If the type one fertilizer costs $60$ paise per $\mathrm{g}$ and the type two fertilizer costs $40$ paise per $\mathrm{g}$, determine how many grams of each type of fertilizer should be used so that the nutrient requirement are met at a minimum cost. What is the minimum cost?

State the Convex polygon theorem.

Find convex region from the following figures.

An oil company has two depots, $a$ and $b$, with capacities of $7000 \mathrm{ml}$ and $4000 \mathrm{ml}$ respectively. The company is to supply oil to three pumps, $d,e,f$, whose requirements are $4500 \mathrm{ml}, 3000 \mathrm{ml},$ and $3500 \mathrm{ml}$ respectively. The distances (in $\mathrm{m}$) between the depots and the petrol pumps are given in the following table:

Assuming that the transportation cost per $\mathrm{m}$ is $1$ paise per $\mathrm{ml}$, how should the delivery be scheduled in order that the transportation cost is minimum?

State the Convex polygon theorem.

Find convex region from the following figures.

The corner points of the feasible region determined by the following system of linear inequalities:
$2x+y\le 10,x+3y\le 15,x,y\ge 0$ are $\left(0,0\right),\left(5,0\right),\left(3,4\right)$ and $\left(0,5\right)$
Let $Z=sx+ty,$ where $s,t>0.$
Condition on $s$ and $t$ so that the maximum of $Z$ occurs at both $\left(3,4\right)$ and $\left(0,5\right)$ is

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.

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

A company makes two products ($X$ and $Y$) using two machines ($A$ and $B$). Each unit of $X$ that is produced requires $50$ minutes processing time on machine $A$ and $30$ minutes processing time on machine $B$. Each unit of $Y$ that is produced requires $24$ minutes processing time on machine $A$ and $33$ minutes processing time on machine $B$. At the start of the current week there are $30$ units of $X$ and $90$ units of $Y$ in stock. Available processing time on machine $A$ is forecast to be $40$ hours and on machine $B$ is forecast to be $35$ hours. The demand for $X$ in the current week is forecast to be $75$ units and for $Y$ is forecast to be $95$ units. Company policy is to maximize the combined sum of the units of $X$ and the units of $Y$ in stock at the end of the week. Formulate the problem of deciding how much of each product to make in the current week as a linear program. Solve this linear program graphically

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.

Mr.Dass wants to invest $₹12000$ in public provident fund ($\mathrm{PPF}$) and in national bonds. He has to invest at least $₹1000$ in $\mathrm{PPF}$ and at least $₹2000$ in bonds. If the rate of interest on $\mathrm{PPF}$ is $12\%$ per annum and that on bonds is $15\%$ per annum, how should he invest the money to earn maximum annual income? Also find the maximum annual income.

For the following Linear Programming problems with given constraints $4x+6y\le 60, 2x+y\le 20 \text{and} x\ge 0, y\ge 0$. The maximum value of $z=2x+3y$ is

For the following linear Programming problems, subject to the constraints $x+y\le 4$ and $x\ge 0$ , $y\ge 0$. Find the maximum value of $Z=3x+4y$

For the following linear Programming problem, subject to the constraints $x+2y\le 8$, $3x+2y\le 12$ and $x\ge 0$ , $y\ge 0$. Find the minimum value of $Z=-3x+4y$.