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

A manufacturer has employed $5$ skilled men and $10$ semi-skilled men and makes two models $A$ and $B$ of an article.The making of one item of model $A$ requires $2$ hours work by a skilled man and $2$ hours work by a semi-skilled man. One item of model $B$ requires $1$ hour by a skilled man and $3$ hours by a semi-skilled man.No man is expected to work more than $8$ hours per day. The manufacturer's profit on an item of model $A$ is $₹15$ and on an item of model $B$ is $₹10$. How many of items of each model should be made per day in order to maximize daily profit $?$ Formulate the above $LPP$ and solve it graphically and find the maximum profit.

A factory manufactures two types of screws $\mathrm{A}$ and $\mathrm{B}$. Each type of screw requires the use of two machines automatic and a hand operated. It takes $4$ minutes on automatic and $6$ minutes on hand operated machines to manufacture a package of screws $\mathrm{A}$ while it takes $6$ minutes on automatic and $3$ minutes on hand operated machine to manufacture a package of screws $\mathrm{B}$. Each machine is available for at the most $4$ hours on any day. The manufacturer can sell a package of screws $\mathrm{A}$ at a profit to $70$ paise and screw $\mathrm{B}$ at a profit of $₹1$. Assuming that he sells all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

Minimize $Z=7x-5y$

subject to the constraints, $4x+5y\le 40$, $x, y\ge 0$.

Minimize $Z=10x-6y$

subject to the constraints, $6x+5y\le 30$, $x, y\ge 0$.

Maximize $Z=-3x+4y$

Subject to the constraints, $x+2y\le 8,3x+2y\le 12,x\ge 0,y\ge 0$.

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.

Solve the linear programming problem graphically:

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