Types of Linear Programing Problems

Q. A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires $9$ labour hours of fabricating and $1$ labour hour for finishing. Each type of B requires $12$ labour hours for fabricating and $3$ labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are $180$ and $30$ respectively. The company makes a profit of $80$ on each piece of type A and $120$ on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

Q. Solve the following problem:
Maximize $Z=-P{x}_1+4x_2$
Subject to: $-3x_1+x_2\le 6$
$x_1+2x_2\le 4$
$x_2\le -3$
No lower bound constraint for $x_1$

Q. The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs $\square$20 per kg and sand costs of $\square$6 per kg. Strength consideration dictates that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Form the LPP for the cost to be minimum.

Q. A company manufactures two articles $A$ and $B$. There are two department through which these article are processed : (i) assembly and (ii) finishing department. The maximum capacity of the first department is $60$ hours a week and that of other department. $48$ hours, per week. The product of each unit of article $A$ requires $4$ hours in assembly and $2$ hours in finishing and that of each unit of $B$ requires $2$ hours in assembly and $4$ hours in finishing. If the profit is Rs. $6$ for each unit $A$ and Rs. $8$ for each unit of $A$ and Rs. $8$ for each unit of $B$, find the number of units of $A$ and $B$ to be produced per week in order to have maximum profit.

Q. A firm manufactures two products $A$ and $B$. Each product is processed on two machines $M_1$ and $M_2$. Product A requires $4$ minutes of processing time on $M_1$ and $8$ min. on $M_2$; product $B$ requires $4$ minutes on $M_1$ and $4$ min. on $M_2$. The machine $M_1$ is available for not more than $8$ hrs $20$ min. while machine $M_2$ is available for $10$ hrs during any working day. The products $A$ and $B$ are sold at a profit of Rs. $3$ and Rs. $4$ respectively.
Formulate the problem as a linear programming problem and find how many product of each type should be produced by the firm each day in order to get maximum profit.

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

Q. 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, while each packet of the same quality of food $Q$ contains $3$ units of calcium, $20$ units of vitamin A. The diet requires atleast $240$ units of calcium, atleast $460$ units of iron and almost $300$ 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?

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. A firm has the cost function $C=\frac{x^3}{3}-7x^2+111x+50$ and demand function $x=100-p$.
Write the total revenue function in terms of $x$

Q. Minimize $Z=x+2y$
Subject to the constraints
$2x+y\ge 3$
$x+2y\ge 6$
and $x\ge 0,y\ge 0$