﻿ A diet is to contain at least 400 units of carbohydrate, 500 units of fat, and 300 units of protein. Two foods are available: F1’ which costs Rs 2 per unit, and F2’ which costs Rs 4 per unit. A unit of food F1 contains 10 units of carbohydrate, 20 units of fat, and 15 units of protein; a unit of food F2 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. : Kaysons Education

# A Diet Is To Contain At Least 400 Units Of Carbohydrate, 500 Units Of Fat, And 300 Units Of Protein. Two Foods Are Available: F1’ which Costs Rs 2 Per Unit, And F2’ which Costs Rs 4 Per Unit. A Unit Of Food F1 contains 10 Units Of Carbohydrate, 20 Units Of Fat, And 15 Units Of Protein; A Unit Of Food F2 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.

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## Question

### Solution

Correct option is

Minimize Z = 2x + 4y

Subject to

10x + 25y ≥ 400

20x + 10y ≥ 500

15x + 20y ≥ 300

xy ≥ 0

The given data may be put in the following tabular form:

 Food Carbohydrate Fat Protein Cost per unit F1 F2 10 25 20 10 15 20 Rs 2 Rs 4 Minimum requirement 400 500 300

Suppose the diet contains x units of food F1 and y units of food F2.

Since one unit of food F1 costs Rs 2 and one unit of food F2 costs Rs 4. Therefore, total cost of x units of food F1 and y units of food F2 is Rs (2x+ 4y).

Let Z denote the total cost. Then,

Z = 2x + 4y.

Since each unit of food F1 contains 10 units of carbohydrate. Therefore, xunits of food F1 contain 10x units of carbohydrate. A unit of food F2contains 25 units of carbohydrate. So, y units of food F2 contain 25y units of carbohydrate.

Thus, x units of food F1 and y units of food F2 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 F1 and yunits of food F2 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 F1 and yunits of food F2 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

xy ≥ 0.

#### SIMILAR QUESTIONS

Q1

A factory produces two products P­1 and P2. Each of the product P1requires 2 hrs for moulding, 3 hrs for grinding and 4 hrs for polishing, and each of the product P2 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 P1 and Rs. 3 per unit of P2 and the factory can sell all that it produces. Formulate the problem as a linear programming problem to maximize the profit.

Q2

A toy company manufactures two types of doll; a basic version doll and a deluxe version doll B. Each doll of type 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 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 and doll B; how many of each should be produced per day in order to maximize profit?

Q3

A firm can produce three types of cloth, say C1, C2, C3. Three kinds of wool are required for it, say red wool, green wool and blue wool. One unit of length C1 needs 2 metres of red wool, 3 metres of blue wool; one unit of cloth C2 needs 3 metres of red wool, 2 metres of green wool and 2 metres of blue wool; and one unit of cloth C3 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 C1 is Rs. 6, of cloth C2 is Rs. 10 and of cloth C3 is Rs. 8. Formulate the problem as a linear programming problem to maximize the income.

Q4

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.

Q5

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.

Q6

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.

Q7

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.

Q8

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.

Q9

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 ABC. 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?

Q10

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 AB 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.