Question
A diet is to contain at least 400 units of carbohydrate, 500 units of fat, and 300 units of protein. Two foods are available: F_{1}_{’} which costs Rs 2 per unit, and F_{2}_{’} which costs Rs 4 per unit. A unit of food F_{1} contains 10 units of carbohydrate, 20 units of fat, and 15 units of protein; a unit of food F_{2} contains 25 units of carbohydrate, 10 units of fat, and 20 unit of protein. Find the minimum cost for a diet consists of a mixture of these two foods and also meets the minimum nutrition requirements. Formulate the problem as a linear programming problem.
Solution
Minimize Z = 2x + 4y
Subject to
10x + 25y ≥ 400
20x + 10y ≥ 500
15x + 20y ≥ 300
x, y ≥ 0
The given data may be put in the following tabular form:
Food 
Carbohydrate 
Fat 
Protein 
Cost per unit 
F_{1} F_{2} 
10 25 
20 10 
15 20 
Rs 2 Rs 4 
Minimum requirement 
400 
500 
300 

Suppose the diet contains x units of food F_{1} and y units of food F_{2}.
Since one unit of food F_{1} costs Rs 2 and one unit of food F_{2} costs Rs 4. Therefore, total cost of x units of food F_{1} and y units of food F_{2} is Rs (2x+ 4y).
Let Z denote the total cost. Then,
Z = 2x + 4y.
Since each unit of food F_{1} contains 10 units of carbohydrate. Therefore, xunits of food F_{1} contain 10x units of carbohydrate. A unit of food F_{2}contains 25 units of carbohydrate. So, y units of food F_{2} contain 25y units of carbohydrate.
Thus, x units of food F_{1} and y units of food F_{2} contain 10x + 25y units of carbohydrate.
But, the minimum requirement of carbohydrate is 400 units.
Similarly, the total amount of fat supplied by x units of Food F_{1} and yunits of food F_{2} is 20x + 10y and the minimum requirement is of 500 units.
∴ 20x + 10y ≥ 500.
Finally, the total amount of protein supplied by x units of food F_{1} and yunits of food F_{2} is 15x + 20y. But the minimum requirement of protein is of 300 units.
∴ 15x + 20y ≥ 300.
Clearly, x ≥ 0 and y ≥ 0.
Since we have to minimize the total cost Z = 2x + 4y.
Thus, the mathematical form of the given LPP is as follows:
Minimize Z = 2x + 4y
Subject to
10x + 25y ≥ 400
20x + 10y ≥ 500
15x + 20y ≥ 300
x, y ≥ 0.
SIMILAR QUESTIONS
A factory produces two products P_{}_{1} and P_{2}. Each of the product P_{1}requires 2 hrs for moulding, 3 hrs for grinding and 4 hrs for polishing, and each of the product P_{2} requires 4 hrs moulding, 2 hrs for grinding and 2 hrs for polishing. The factory has moulding machine available for 20 hrs, grinding machine for 24 hrs and polishing machine available for 13 hrs. The profit is Rs. 5 per unit of P_{1} and Rs. 3 per unit of P_{2} and the factory can sell all that it produces. Formulate the problem as a linear programming problem to maximize the profit.
A toy company manufactures two types of doll; a basic version doll A and a deluxe version doll B. Each doll of type B takes twice as long to produce as one of type A, and the company would have time to make a maximum of 2000 per day if it produces only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. If the company makes profit of Rs 3 and Rs 5 per doll respectively on doll A and doll B; how many of each should be produced per day in order to maximize profit?
A firm can produce three types of cloth, say C_{1}, C_{2}, C_{3}. Three kinds of wool are required for it, say red wool, green wool and blue wool. One unit of length C_{1} needs 2 metres of red wool, 3 metres of blue wool; one unit of cloth C_{2} needs 3 metres of red wool, 2 metres of green wool and 2 metres of blue wool; and one unit of cloth C_{3} needs 5 metres of green wool and 4 metres of blue wool. The firm has only a stock of 16 metres of red wool, 20 metres of green wool and 30 metres of blue wool. It is assumed that the income obtained from one unit of length of cloth C_{1} is Rs. 6, of cloth C_{2} is Rs. 10 and of cloth C_{3} is Rs. 8. Formulate the problem as a linear programming problem to maximize the income.
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 A, 1 hour on machine B, and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit realized by selling one chair is Rs 30 while for a table the figure 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? Develop a mathematical formulation.
A manufacturer of a line of patent medicines is preparing a production plan on medicines A and B. There are sufficient ingredients available to make 20,000 bottles of A and 40,000 bottles of B but there are only 45,000 bottles into which either of the medicines can be put Further more, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes one hour to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit is Rs 8 per bottle for A and Rs 7 per bottle for B. Formulate this problem as a linear programming problem.
A resourceful home decorator manufactures two types of lamps say A andB. Both lamps go through two technicians, first a cutter, second a finisher. Lamp A requires 2 hours of the cutter’s time and 1 hour of the finisher’s time. Lamp B requires 1 hour of cutter’s and 2 hours of finisher’s time. The cutter has 104 hours and finisher has 76 hours of time available each month. Profit on one lamp A is Rs. 6.00 and on one lamp B is Rs 11.00. Assuming that he can sell all that he produces, how many of each type of lamps should he manufacture to obtain the best return.
A company makes two kinds of leather belts, A and B. Belt A is high quality belt, and B is of lower quality. The respective profits are Rs 4 and Rs 3 per belt. Each belt of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (bothA and B combined). Belt A requires a fancy buckle, and only 400 buckles per day are available. There are only 700 buckles available for belt B. What should be the daily production of each type of belt? Formulate the problem as a LPP.
A dietician whishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of Vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C while food ‘II’ contains 1 unit per kg of vitamin A and 2 units per kg of vitamin C. It costs Rs 5.00 per kg to purchase food ‘I’ and Rs 7.00 per kg to produce food ‘II’. Formulate the above linear programming problem to minimize the cost of such a mixture.
The objective of a diet problem is to ascertain the quantities of certain foods that should be eaten to meet certain nutritional requirement at minimum cost. The consideration is limited to milk, beaf and eggs, and to vitamins A, B, C. The number of milligrams of each of these vitamins contained within a unit of each food is given below:
Vitamin 
Litre of milk 
Kg of beaf 
Dozen of eggs 
Minimum daily requirements 
A B C 
1 100 10 
1 10 100 
10 10 10 
1 mg 50 mg 10 mg 
Cost 
Rs 1.00 
Rs 1.10 
Re 0.50 

What is the linear programming formulation for this problem?
There is a factory located at each of the two places P and Q. From these locations, a certain commodity is delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are 8 and 6 units respectively. The cost of transportation per unit is given below.
To From 
Cost (in Rs) 


A 
B 
C 

P Q 
16 10 
10 12 
15 10 
How many units should be transported from each factory to each in order that the transportation cost is minimum. Formulate the above as a linear programming problem.