Embibe Experts Solutions for Chapter: Linear Programming, Exercise 1: Exercise

If the objective function is $Z =6x-7y$. Identify the decision variables

Solve the linear programming problem graphically:

Maximize $Z=4x+y,$ subject to the constraints $x+y\le 50, 3x+y\le 90, x\ge 0, y\ge 0.$

State advantages of linear programming problems.

State advantages of linear programming problems.

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 $Z=px+qy,$ where $p,q>0.$
Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $\left(3,4\right)$ and $\left(0,5\right)$ is

A dealer wishes to purchase a number of fans and sewing machines. He has only $₹5760$ to invest and space for at most $20$ items. A fan costs him $₹360$ and a sewing machine,$₹240$. He expects to gain $₹22$ on a fan and $₹18$ on a sewing machine. Assuming that he can sell all the items he can buy, how should he invest the money in order to maximise the profit?

A carpenter has $90, 80$ and $50$ running feet respectively of teak wood, plywood and rosewood which is used to produce product $A$ and product $B$. Each unit of product $A$ requires $2,1$ and $1$ running feet and each unit of product $B$ requires $1, 2$ and $1$ running feet of teak wood, plywood and rosewood respectively. If product $A$ is sold for $₹48$ per unit and product $B$ is sold for $₹40$ per unit, how many units of product $A$ and product $B$ should be produced and sold by the carpenter, in order to obtain the maximum gross income?

Formulate the above as a Linear Programming Program and solve it, indicating clearly the feasible region in the graph.

A company produces two types of goods, $A$ and $B$, that require gold and silver. Each unit of type $A$ requires $3 \mathrm{g}$ of silver and $1 \mathrm{g}$ of gold, while that of type $B$ requires $1 \mathrm{g}$ of silver and $2 \mathrm{g}$ of gold. The company can use at the most $9 \mathrm{g}$ of silver and $8 \mathrm{g}$ of gold. If each unit of type $A$ brings a profit of $₹ 120$  and that of type $B$ $₹ 150$, then find the number of units of each type that the company should produce to maximise profit.

Formulate the above LPP and solve it graphically. Also, find the maximum profit.

A furniture trader deals in only two items - chairs and tables. He has $50,000$ rupees to invest and a space to store at most $35$ items. A chair costs him $1000$ rupees and a table costs him $2000$ rupees . The trader earns a profit of $150$ rupees and $250$ rupees on a chair and table, respectively. Formulate the above problem as an LPP to maximise the profit and solve it graphically.