Question

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.

Solution

Correct option is

x = 1000, y = 500, Z = 3 × 1000 + 5 × 500 = 5500 & 1000 dolls of type A and 500 dolls of type B should be produced to maximize the profit and the maximum profit is Rs 5500

 

Let x dolls of type A and y dolls of type B be produced per day to maximize the profit.   

The mathematical form of the given LPP is as follows:

Maximize Z = 3x + 5y

Subject to x + 2y ≤ 2000 

                 x + y ≤ 1500   

                       y ≤ 600  

and,             xy ≥ 0.   

The set of all feasible solutions of the given LPP is represented by the feasible region shaded darkly in fig. The coordinates of the corner points of the feasible region are O (0, 0), A2 (1500, 500), Q (800, 600) and R (0, 600).    

Now, to find a point or points in the feasible region which give the maximum value of the objective function Z = 3x + 5y, let us given some value to Z, say 1500 and draw the dotted line 3x + 5y = 1500 as shown in fig. 

                                                                                       

 

Now, draw lines parallel to the line 3x + 5y = 1500 and obtain a line which is farthest from the origin and have to least one point common to the feasible region. Clearly, Z1 = 3x + 5y is such a line. This line has only one point P (1000, 500) common to the feasible region. Thus, Z = 3 × 1000 + 5 × 500 = 5500 is the maximum value of Z and the optimal solution is x = 1000, y = 500.

Hence, 1000 dolls of type A and 500 dolls of type B should be produced to maximize the profit and the maximum profit is Rs 5500.  

SIMILAR QUESTIONS

Q1

Solve the following LPP graphically:  

Minimize Z = 3x + 5y     

Subject to  

         – 2x + y ≤ 4  

            x + y ≥ 3

           x – 2y ≤ 2   

           xy ≥ 0

Q2

 

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.

Q3

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. 

Q4

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.    

Q5

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.  

Q6

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.

Q7

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?

Q8

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. 

Q9

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. 

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 these 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 respectively 8 and 6 units. The cost of transportation per unit is given below:  

                   To

From

Cost (in Rs)

A

B

C

P

16

10

15

Q

10

12

10

         

How many units should be transported from each factory to each depot in order that the transportation cost in minimum. Formulate the above LPP mathematically and then solve it.