Q7 of 31 Page 525

An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distances (in km) between the depots and the petrol pumps is given in the following table:


Assuming that the transportation cost of 10 liters of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?

let x and y liters of oil be supplied by A to shop D and E respectively. Then 7000 - x - y will be supplied to F. Obviously x ≥ 0, y ≥ 0.


Since requirement at D is 4500L but supplied x quintal hence, 4500 - x is sent form B to D.


Similarly requirement at E is 3000L but supplied y quintal hence, 3000 - y is sent form B to E.


Similarly requirement at E is 3500 quintal but supplied 7000 - x - y quintal hence, 3500 - (7000 - x - y) i.e. x + y - 3500 is sent form B to E.


Diagrammatically it can be explained as:


2nd.JPG


As x ≥ 0, y ≥ 0 and 7000 - x - y ≥ 0 ⇒ x + y ≤ 7000


4500 - x ≥ 0, 3000 - y ≥ 0 and x + y - 3500 ≥ 0


⇒ x ≤ 4500, y ≤ 3000 and x + y ≥ 3500


Cost of delivering 10L of petrol = Re.1


Then Cost of delivering 1L of petrol = Rs.1/10


Total transportation cost Z can be calculated as:


Z =


Z = 0.3x + 0.1y + 3950


Mathematical formulation of given problem is as follows:


Minimize Z = 0.3x + 0.1y + 3950 ….(1)


Subject to the constraints,


x + y ≤ 7000 …..(2)


x ≤ 4500 …..(3)


y ≤ 3000 ….(4)


x + y ≥ 3500 …(5)


x ≥ 0, y ≥ 0 ….(6)


Now let us graph the feasible region of the system of inequalities (2) to (6). The feasible region (shaded) is shown in the fig. Here, we can observe that the feasible region is bounded.



The coordinates of the corner points A(3500,0),B(4500,0),C(4500,2500), D(4000,3000) and E(500,3000) .



Now, we find the minimum value of Z. According to table the minimum value of Z = 4400 at point E (500,3000).


Hence, the amount of oil delivered from A to D, E and F is 500L, 3000L and 3500L quintals respectively and from B to D,E and F is 4000L, 0 L and 0 L respectively.


More from this chapter

All 31 →
5

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?

 6

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:


How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

 8

A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?


 9

Refer to Question 8. If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?