Types of Linear Programing Problems

Q.

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol. How Many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

Q.

Maximize Z = 3x + 2y, subject to constraints are $x+2y\le 10,3x+y\le 15andx,y\ge 0$.

Q.

Minimize Z = 3x + 5y, subject to constraints are x + 3y $\ge$ 3, $x+y\ge 2andx,y\ge 0$.

Q.

Minimize Z = -3x + 4y, subject to constraints are $x+2y\le 8,3x+2y\le 12,x\ge 0andy\ge 0$.

Q. Solve the following linear programming problem graphically:

Maximise $Z=34x+45y$

under the following constraints

$x+y\le 3002x+3y\le 70x\ge ,y\ge 0$

Q. A cooperrative society of farmers has $50$ hectare of land to grow two crops $X$ and $Y$. The profit from crop $X$ and $Y$ per hectare are estimated as Rs $10,500$ and Rs $9,000$ respectively. To control weeds, a liquid herbicide has to be used for crops $X$ and $Y$ at rates of $20$ litres and $10$ litres per hectare. Further, no more than $800$ litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?

Q.

Determine the maximum value of Z =11x +7y subject to the constraints $2x+y\ge 6,x\ge 2,x\ge 0,y\ge 0$

Q.

What is maximization in linear programming?

Q.

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 a LPP given that the objective is to minimise cost.

Q. The coordinates of the corner points of the bounded feasible region are $(10,0),(2,4),(1,5)$ and $(0,8)$. The maximum of objective function $z=60x+10y$ is

1. $700$
2. $800$
3. $600$
4. $110$

Q.

Maximize Z = 3x + 4y, subject to the constraints are x + y $\le$ 4, x $\ge$ 0 and y $\ge$ 0.

Q.

A manufacturer produces two models of bikes -, model X and model Y. Model X takes a 6 man-hours to make per unit, while model Y takes 10 man hours per unit. There is a total of 450 man-hour available per week. Handling and marketing costs are Rs 2000 and Rs 1000 per units for models X and Y, respectively. The total funds available for these purposes are Rs 80000 per week. Profits per units for models X and Y are Rs 1000 and Rs 500, respectively. How many bikes of each model should the manufacturer produce, so as to yield a maximum profit ? Find the maximum profit.

Q.

Refer to question 13. Solve the linear programming problem and detrmine the maximum profit to the manufacturer.

Q.

Fill in the blanks:

Sum borrowed is called $\left(\_\_\_\_\right)$

Q. how to solve binomial series through integration?

Q.

A company makes 3 model of calculators; A, B and C at factory I and factory II. The company has orders for atleast 6400 calculators of model A, 4000 calculators of model B and 4800 calculators of model C. At factory I, 50 calculators of model A, 50 of model 8 and 30 of model C are made everyday; at factory II, 40 calculators of model A, 20 of model B and 40 of model C are made everyday. It costs ! 12000 and Z 15000 each day to operate factory I and II, respectively. Find the number of days each factory should operate to minimise the operating costs and still meet the demand.

Q.

Refer to question 11. How many of circuits of type A and of type B, should be produced by the manufacturer, so as to maximise his profit? Derermine the maximum profit.

Q.

There are two types of fertilizers $F_1$ and $F_2$. $F_1$ consists of 10 % nitrogen and 6 % phosphoric acid and $F_2$ consists of 5 % nitrogen and 10 % of phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If $F_1$ costs Rs. 6 kg and $F_2$ costs Rs. 5 kg, determine how much of each at a minimum cost. What is the minimum cost ?

Q. The coordinates of four angular points of a tetrahedron are $(0,0,0),(0,0,2),(0,4,0)$ and $(6,0,0)$. A point $P$ inside the tetrahedron is at the same distance $r$ from the four plane faces of tetrahedron. Which of the following CANNOT be the value of $r$?

1. $\frac{2}{3}$
2. $\frac{2}{5}$
3. $2$
4. $3$

Q. Corner points of the bounded feasible region for an LP problem are $A(0,5)B(0,3)C(1,0)D(6,0)$. Let $z=-50x+20y$ be the objective function. Minimum value of z occurs at ______ center point.

1. $(0,5)$
2. $(6,0)$
3. $(0,3)$
4. $(1,0)$

Q. Use the graphical method to solve each of the following LP problems.
A factory manufactures two products, each requiring the use of three machines. The first machine can be used at most $70$ hours; the second machine at most $40$ hours; and the third machine at most $90$ hours. The first product requires $2$ hours on Machine 1, $1$ hour on Machine 2, and $1$ hour on Machine 3; the second product requires $1$ hour each on machine 1 and 2 and $3$ hours on Machine 3. If the profit in $E40$ per unit for the first product and $E60$ per unit for the second product, how many units of each product should be manufactured to maximize profit?

Q. Solve the following linear programming problem graphically:
Maximize $Z=7x+10y$
subject to the constraints
$4x+6y\le 240$
$6x+3y\le 240$
$x\ge 10$
$x\ge 0,y\ge 0$

Q. (Diet problem) A dietician has to develop a special diet using two foods $P$ and $Q$. Each packet (containing $30g$) of food $P$ contains $12$ units of calcium, $4$ units of iron, $6$ units of cholesterol and $6$ units of vitamin A. Each packet of the same quantity of food $Q$ contains $3$ units of calcium, $20$ units of iron, $4$ units of cholesterol and $3$ units of vitamin A. The diet requires atleast $240$ units of calcium, atleast $460$ units of iron and at most $300$ units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

Q.

Maximise the function Z =11x +7y, subject to the constraints $x\le 3,y\le 2,x\le 0$ and $y\le 0$

Q. Solve the following linear programming problem using graphical method
Minimize: $Z=60x+30y$
Subject to:
$2x+3y\ge 120$
$2x+y\ge 80$
$x\ge 0,y\ge 0$

Q. Consider two families $A$ and $B.$ Suppose there are $4$ men, $4$ women and $4$ children in family $A$ and $2$ men, $2$ women and $2$ children in family $B.$ The recommended daily amount of calories is $2400$ for a man, $1900$ for a woman, $1800$ for a child and $45$ grams of proteins for a man, $55$ grams for a woman and $33$ grams for a child.
The requirement of calories and proteins for each person is given by matrix $R$ and the number of family members in each family is given by matrix $F.$

Matrix $R$ is

1. $\begin{array}{ccc}\text{Calories}& \text{Proteins}& \end{array}\left[\begin{array}{cc}2400& 45\\ 1900& 55\\ 1800& 33\end{array}\right]\begin{array}{c}\text{Men}\\ \text{Women}\\ \text{Children}\end{array}$
2. $\begin{array}{ccc}\text{Calories}& \text{Proteins}& \end{array}\left[\begin{array}{cc}1900& 55\\ 2400& 45\\ 1800& 33\end{array}\right]\begin{array}{c}\text{Men}\\ \text{Women}\\ \text{Children}\end{array}$
3. $\begin{array}{ccc}\text{Calories}& \text{Proteins}& \end{array}\left[\begin{array}{cc}1800& 33\\ 1900& 55\\ 2400& 45\end{array}\right]\begin{array}{c}\text{Men}\\ \text{Women}\\ \text{Children}\end{array}$
4. $\begin{array}{ccc}\text{Calories}& \text{Proteins}& \end{array}\left[\begin{array}{cc}2400& 33\\ 1900& 55\\ 1800& 45\end{array}\right]\begin{array}{c}\text{Men}\\ \text{Women}\\ \text{Children}\end{array}$

Q. Minimise $Z=3x+2y$
subject to constraints:
$x+y\ge 8$
$3x+5y\le 15$
$x\ge 0,y\ge 0$

Q. Maximize z = 2x + 4y subject to the constraints

x ≥ 0, y ≥ 0, x + y ≥ 1, 3x + 2y <= 6

Q.

The value of c for which Lagrange's theorem f(x) = |x| in the interval [-1, 1] is

1. 1

2. non-existent in the interval

3. -1/2

4. 1/2

Q. A retired person wants to invest an amount of Rs. $50,000.$ His broker recommends investing in two type of bonds A and B yielding $10\%$ and $9\%$ return respectively on the invested amount. He decides to invest at least $Rs.20,000$ in bond A and at least $Rs.10,000$ in bond B. He also wants to invest at least as much in bond A as in bond B. Solve this linear programming problem graphically to maximize his returns.

1. $4900$
2. $2900$
3. $5400$
4. $4000$