A furniture firm manufactures chairs and tables, each requiring the use of three machines 'A', 'B' and 1 hour on machine 'C'. Each table requires 1 hour each on machines 'A' and 'B' and 3 hours on machine 'C'. The profit realized by selling one chair is RS. 30 while for a table is Rs. 60. The total time available per week o machine 'A' is 70 hours, on machine 'B' is 40 hours and on machine 'C' is 90 hours. Find the mathematical formulation so as to find the number of chairs and tables that should be made per week so as to maximize the profit. 

Solve the following Linear Programming Problems graphically:
Maximise Z = 5x + 3y
subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0

If a man rides his motor cycle at 25 km/hr., he has to spend Rs. 2 per km on petrol, if he rides at a faster speed of 40 km/hr., the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find maximum distance he can travel within one hour. Express this as a linear programming problem and then solve it graphically.

A furniture firm manufactures chairs and tables, each requiring the use of three machines -A, B and C. Production of one chair requires 2 hours on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as LLP and solve it graphically.

A merchant plans to sell two types of personal computers-a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchantshould stock to get maximum profit if he does not want to invest more than Rs.70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. Make an LPP and solve it graphically.

A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops 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?

(Transportation Problem) A catering agency has two kitchens to prepare food at two places A and B. From these places 'Mid-day Meal' is to be supplied to three different schools situated at P,Q,R. The monthly requirements of the schools are respectively 40, 40 and 50 food packets. A packet contains lunch for 1000 students. Preparing capacity of kitchen A and B are 60 and 70 packets per month respectively. The transportation cost per packet for the kitchen to schools is given below:

 How many packets from each kitchen should be transported to school so that the cost of transportation is minimum? Also find the minimum cost.

A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 unit/ kg of vitamin A and 1 unit/kg of vitamin C while food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs. 5 per kg to produce Food I and Rs. 7 per kg to produce food II. Determine the minimum cost of such a mixture. Formulate the above as a LLP and solve 3 it graphically.

If a 20 year old girl drives her car at 25 km/h, she has to spend Rs. 4/km on petrol. If she drives her car at 40 km/h, the petrol cost increases to Rs. 5/km. She has Rs. 200 to spend on petrol and wishes to find the maximum distance she can travel within one hour. Express the above problem as a Linear Programming Problem. Write anyone value reflected in the problem.

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below: 2x + 4y \(\le \) 8 \(\Rightarrow\) x + 2y \(\le \) 4
3x + y \(\le \) 6
x + y \(\le \) 4
x \(\\ \ge \) 0, y  \(\\ \ge \) 0

A cottage industry manufacturers pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs. 25 and that from a shade is Rs. 15.Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit? Formulate an LPP and solve it graphically.

A manufacturer produces nuts and bolts. It takes 1 hours of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package of nuts and Rs. 7.00 per package of bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machine for at most 12 hours a day? Form the above as a linear programming problem and solve it graphically.