Mathematical Formulation of a Linear Programming Problem

The feasible region for an LPP is always a _____ polygon.

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 $Z=-3x+4y$.

Any point in the _____ region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.

For the following linear programming problem, find the minimum value of $z=8000x+12000y$,  where constraints are 

$3x+4y\le 60$

$x+3y\le 30$

$x⩾0, Y⩾0$

The region of feasible solution under the constraints $2x+y\le 6, x\ge 0, y\ge 0$ is:

Solve the following problem graphically :

Minimise and Maximise

$\mathrm{Z}=3x+9y$

Subject to the constraints :

$x+3y\le 60$

$x+y\ge 10$

$x\le y$

$x\ge 0, y\ge 0$.

For the following linear programming problem:

Objective function: $z=150x+250y$

Subject to: $4x+y\le 40$

$3x+2y\le 60$

$x\ge 0$

$y\ge 0$

The maximum value of $z$ is

For the following L.P.P. Problem, the minimum value of $z$ is

Minimize $z=8x+10y$, subject to $2x+y\ge 7, 2x+3y\ge 15, y\ge 2, x\ge 0, y\ge 0$.

Maximise $z=4x+y$

Subject to constraints:

$x+y\le 50$

$3x+y\le 90$

$x\ge 0$

$y\ge 0$

by graphical method.

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

Maximize $z=x+2y$ subject to $x+2y\ge 50$, $2x-y\le 0$, $2x+y\le 100$, $x, y>0$.

Prizes are to be distributed among the students of class $XI$ and class $XII$. It is decided that at least $5$ students from class $XI$ and at least $4$ students from class $XII$ should get the prizes.The prize amount for class $XI$ students is $Rs 300$ and that for the class $XII$ students is $Rs 400$. The total number of prize holders should not be less than $10$ and more than $15$. How many students from each standard be selected to maximise the amount of prize money?

Prizes are to be distributed among the students of class $XI$ and class $XII$. It is decided that at least $5$ students from class $XI$ and at least $4$ students from class $XII$ should get the prizes.The prize amount for class $XI$ students is $Rs 300$ and that for the class $XII$ students is $Rs 400$. The total number of prize holders should not be less than $10$ and more than $15$. How many students from each standard be selected to minimise the amount of prize money?

If a motorcyclist rides his motorcycle at $30$ km per hour, he has to spend $Rs 3$ per km on petrol. If he rides at a faster speed of $50$ km per hour, the petrol cost increases to $Rs 5$ per km. He has $Rs 150$ to spend on petrol and wishes to travel the maximum distance within an hour. Express this as $LPP$ and solve it graphically.

A farmer mixes two brands $A$ and $B$ of cattle feed. Brand $A$, costing $500\text{ ₹ }$ per bag, contains $6$ units of nutritional element $P$, $5$ units of element $Q$ and $4$ units of element $R$. Brand $B$ costing  $400\text{ ₹ }$per bag contains $3$ units of nutritional element $P$, $2.5$ units of element $Q$ and $6$ units of element $R$. The minimum requirements of $P,Q \& R$ are $36,25 \& 48$ units respectively. Determine the number of bags of each brand that should be mixed in order to produce a mixture having a minimum cost per bag. What is the minimum cost of the mixture per bag?

A company manufactures two types of hats, $A$ and $B$. Market survey has indicated that the combined production level should not exceed $1200$ hats per week and the demand for hats of type $B$ is at most half of that for hats of type $A$. Further, the production level of hats of type $A$ can exceed three times the production of hats of type $B$ by at most $600$ units. The company makes the profit of $24\text{ ₹ }$ and $32\text{ ₹ }$ per hat of type $A\&B$ respectively. How many hats of each type should be produced weekly in order to maximize the profit?