Question

A farm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. In view of the need to ensure certain nutrient constituents (call them XY and Z). it is necessary to buy two additional products, say A and B. One unit of product A contains 36 units of X, 3 units of Y, and 20 units of Z. One unit of product B contains 6 units of X, 12 units of Y and 10 units of Z. The minimum requirement of XY and Z is 108 units, 36 units and 100 units respectively. Product A costs Rs 20 per unit and product B costs Rs 40 per unit. Formulate the above as a linear programming problem to minimize the total cost, and solve the problem by using graphical method. 

Solution

Correct option is

x = 4, y = 2, Z = 160 & 2 units of product A and 4 units of product B are sufficient to fulfill the minimum requirement at a minimum cost of Rs 160 

 

The data given in the problem can be summarized in the following tabular form:

Product

Nutrient constituent

X                          Y                         Z

Cost in Rs

A

B

36                         3                        20

6                          12                       10

20

40

Minimum Requirement

108                      36                     200

 

Let x units of product A and y units of product B are bought to fulfill the minimum requirement of XY and Z and to minimize the cost.   

The mathematical formulation of the above problem is as follows:

               Minimize Z = 20x + 40y

Subject to   

                36x + 6y ≥ 108   

                3x + 12y ≥ 36

              20x + 10y ≥ 100 

and,      xy, z ≥ 0.

The set of all feasible solutions of the above LPP is represented by the feasible region shaded darkly in fig. The coordinates of the corner points of the feasible region are A2 (12, 0), P1 (4, 2), P2 (2, 6) and B1 (0, 18). 

                                                                             

 

Now, we have to find a point or points in the feasible region which give the minimum value of the objective function. For this, let us give some value toZ, say 20, and draw a dotted line 20 = 20x + 40y. Now, draw lines parallel to this line which have to least one point common to the feasible region and locate a line which is nearest to the origin and has at least one point common to the feasible region. Clearly, such a line is Z1 = 20x + 40y and it has a point P1 (4, 2) common with the feasible region. Thus, Z1 = 20x + 40y is the minimum value of Z, and the feasible solution which gives this value of Z is the corner P1 (4, 2) of the shaded region. The values of the variables for the optimal solution are x = 4, y = 2. Substituting these values in Z = 20x + 40y, we get Z = 160 as the optimal value of Z.   

Hence, 2 units of product A and 4 units of product B are sufficient to fulfill the minimum requirement at a minimum cost of Rs 160.   

SIMILAR QUESTIONS

Q1

 

Solve the following LPP graphically:  

Maximize   Z = 5x + 7y  

Subject to

              x + y ≤ 4

             3x + 8y ≤ 24  

            10x + 7y ≤ 35

            xy ≥ 0  

Q2

Solve the following LPP graphically:  

Minimize Z = 3x + 5y     

Subject to  

         – 2x + y ≤ 4  

            x + y ≥ 3

           x – 2y ≤ 2   

           xy ≥ 0

Q3

 

A house wife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A,12 units of vitamin B and 8 units of vitamin C.

The vitamin contents of one kg of food is given below:   

 

Vitamin A

Vitamin B

Vitamin C

Food X:

1

2

3

Food Y:

2

2

1

One kg of food X costs Rs 6 and one kg of food Y costs Rs 10. Find the least cost of the mixture which will produce the diet.

Q4

A dietician wishes 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/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 5.00 per kg to purchase food ‘I’ and Rs 7.00 per kg to produce food ‘II’. Determine the minimum cost to such a mixture. formulate the above as a LPP and solve it. 

Q5

Every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates. The corresponding values of rice are 0.05 gm and 0.5 gm respectively. Wheat costs Rs. 4 per kg and rice Rs. 6. The minimum daily requirements of proteins and carbohydrates for an average child are 50 gms and 200 gms respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost.    

Q6

A manufacturer produces nuts and bolts for industrial machinery. It takes 1 hour or work on machine A and 3 hours on machine B to produce a package of nuts while 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 2.50 per package of nuts and Re 1.00 per package of bolts. How many packages or each should he produce each day so as to maximize hit profit, if he operates his machines for at most 12 hours a day? Formulate this mathematically and then solve it.  

Q7

An oil company requires 12,000, 20,000 and 15,000 barrels of high-grade, medium grade and low grade oil, respectively. Refinery A produces 100, 300 and 200 barrels per day of high-grade, medium-grade and low-grade oil, respectively, while refinery B produces 200, 400 and 100 barrels per day of high-grade, medium-grade and low-grade oil, respectively. If refinery A costs Rs 400 per day and refinery B  costs Rs 300 per day to operate, how many days should each be run to minimize costs while satisfying requirements.

Q8

A company produces soft drinks that has a contract which requires that a minimum of 80 units of the chemical A and 60 units of the chemical B to go into each bottle of the drink. The chemicals are available in a prepared mix from two different suppliers. Supplier S has a mix of 4 units of A and 2 units of B that costs Rs 10, the supplier T has a mix of 1 unit of A and 1 unit of B that costs Rs 4. How many mixes from S and T should the company purchase to honour contract requirement and yet minimize cost?

Q9

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5760.00 to invest and has space for at most 20 items. A fan costs him Rs 360.00 and a sewing machine Rs 240.00. His expectation is that he can sell a fan at a profit of Rs 22.00 and a sewing machine at a profit of Rs 18.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Translate this problem mathematically and then solve it. 

Q10

A toy manufacturer produces two types of dolls; a basic version doll Aand a deluxe version doll B. Each doll of type B takes twice as long to produce as one doll of type A. The company have time to make a maximum of 2000 dolls of type A per day, the supply of plastic is sufficient to produce 1500 dolls per day and each type requires equal amount of it. The deluxe version, i.e. type B requires a fancy dress of which there are only 600 per day available. If the company makes a profit of Rs 3 and Rs 5 per doll, respectively, on doll A and B; how many of each should be produced per day in order to maximize profit? Solve it by graphicl method.