Question

 

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.              

Solution

Correct option is

 

Minimize Z = x – 7y + 190  

Subject to  

            x + y ≤ 8

            x + y ≥ 4   

            x ≤ 5  

            y ≤ 5  

and,     x ≥ 0, y ≥ 0

The given information can be exhibited diagrammatically as follows:

                                        

Let the factory at P transports x units of commodity to depot at A and yunits to depot at B. Since the factory at P has the capacity of 8 units of the commodity. Therefore, the left out (8 – x – y) units will be transported to depot at C.

Since the requirements are always non-negative quantities. Therefore,   

         x ≥ 0, y ≥ 0 and 8 – x – y ≥ 0 ⇒ x ≥ 0, y ≥ 0 and x + y ≤ 8

Since the weekly requirement of the depot at A is 5 units of the commodity and x units are transported from the factory at P. Therefore, the remaining (5 – x) units are to be transported from the factory at Q. Similarly, 5 – yunits of the commodity will be transported from the factory at Q to the depot at B. But the factory at Q has the capacity of 6 units only, therefore the remaining 6 – (5 – x + 5 – y) = x + y – 4 units will be transported to the depot at C. As the requirements at the depots at AB and C are always non-negative.

∴               5 – x ≥ 0, 5 – y ≥ 0 and x + y – 4 ≥ 0   

⇒              x ≤ 5, y ≤ 5 and x + y ≥ 4.   

The transportation cost from the factory at P to the depots at AB and Care respectively Rs 16x, 10y and 15(8 – x – y). Similarly, the transportation cost from the factory at Q to the depots at AB and C are respectively Rs 10(5 – x), 12(5 – y) and 10(x + y – 4). Therefore, the total transportation cost Z is given by  

Z = 16x + 10y + 15(8 – x – y) + 10(5 – x) + 12(5 – y) + 10(x + y – 4)

         

Hence, the above LPP can be stated mathematically as follows:

Find x and y which

             Minimize Z = x – 7y + 190   

Subject to   

            x + y ≤ 8

            x + y ≥ 4    

            x ≤ 5   

            y ≤ 5   

and,     x ≥ 0, y ≥ 0

SIMILAR QUESTIONS

Q1

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?

Q2

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.

Q3

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.

Q4

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.

Q5

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.

Q6

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.     

Q7

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.   

Q8

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.

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

 

A brick manufacturer has two depots, A and B, with stocks of 30,000 and 20,000 bricks respectively. He receives orders from three builders PQand R for 15,000, 20,000 and 15,000 bricks respectively. The cost in Rs of transporting 1000 bricks to the builders from the depots are given below:

              From

To

P

Q

R

A

B

40

20

20

60

30

40

How should the manufacturer fulfil the orders so as to keep the cost of transportation minimum?

Formulate the above linear programming problem.