# Class 12 RD Sharma Solutions – Chapter 30 Linear Programming – Exercise 30.5

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

Solution:

Let us assume godown A supply x and y quintals of grain to the shops D and E.

Afterwards, (100 â€“ x â€“ y) will be supplied to shop F.

The requirement at shop D is 60 quintals

Since, x quintals are transported from godown A.

Therefore, the remaining (60 âˆ’ x) quintals will be transported from godown B.

Similarly,

(50 âˆ’ y) quintals and 40 âˆ’ (100 âˆ’ x âˆ’ y) i.e. (x + y âˆ’ 60) quintals will be transported from godown B to shop E and F respectively.

The diagrammatic representation of the given problem:

x â‰¥ 0 , y â‰¥ 0 and 100 â€“ x â€“ y â‰¥ 0

â‡’ x â‰¥ 0 , y â‰¥ 0 , and x + y â‰¤ 100

60 â€“ x â‰¥ 0 , 50 â€“ y â‰¥ 0 , and x + y â€“ 60 â‰¥ 0

â‡’ x â‰¤ 60 , y â‰¤ 50 , and x + y â‰¥ 60

Total transportation cost Z is given by,

Z = 6x + 3y + 2.5(100 â€“ x â€“ y) + 4(60 â€“ x) + 2(50 â€“ y) + 3( x + y â€“ 60)

= 6x + 3y + 250 â€“ 2.5x â€“ 2.5y + 240 â€“ 4x + 100 â€“ 2y + 3x + 3y â€“ 180

= 2.5x + 1.5y + 410

Thus, the required mathematical formulation of linear programming is:

Minimize Z = 2.5x + 1.5y + 410

subject to the constraints,

x + y â‰¤ 100

x â‰¤ 60

y â‰¤ 50

x + y â‰¥ 60

x, y â‰¥ 0

The feasible region obtained by the system of constraints is:

The corner points are A(60, 0), B(60, 40), C(50, 50), and D(10, 50).

The values of Z at these corner points are given below.

Therefore,

The minimum value of Z is 510 at D (10, 50).

Hence, the amount of grain transported from A to D, E, and F is 10 quintals, 50 quintals, and 40 quintals respectively.

From B to D, E, and F is 50 quintals, 0 quintals, and 0 quintals respectively.

The minimum cost is Rs 510.

### How many packets from each factory be transported to each agency so that the cost of transportation is minimum? Also, find the minimum cost?

Solution:

The diagrammatic representation of the given problem:

Assume x and y packets be transported from factory A to the agencies P and Q respectively.

After that,

[60 âˆ’ (x + y)] packets be transported to the agency R.

First constraint â‡¢  x, y â‰¥ 0 and

Second constraint â‡¢ 60 âˆ’ (x + y) â‰¥ 0

(x + y) â‰¤ 60

The requirement at agency P is 40 packets.

Since, x packets are transported from factory A,

Therefore, the remaining (40 âˆ’ x) packets are transported from factory B.

Similarly,

(40 âˆ’ y) packets are transported by B to Q and 50âˆ’ [60 âˆ’ (x + y)]

i.e. (x + y âˆ’ 10) packets will be transported from factory B to agency R respectively.

Number of packets cannot be negative.

Hence,

Third constraint â‡¢ 40 â€“ x â‰¥ 0

=> x â‰¤ 40

Fourth constraint â‡¢ 40 â€“ y â‰¥ 0

=> y â‰¤ 40

Fifth constraint â‡¢ x + y â€“ 10 â‰¥ 0

=> x + y â‰¥ 10

Thus,

Costs of transportation of each packet from factory A to agency P, Q, R are Rs 5, 4, 3.

Costs of transportation of each packet from factory B to agency P, Q, R are Rs 4, 2, 5.

Let total cost of transportation be Z.

Z = 5x + 4y + 3[60 â€“ x + y] + 4(40 â€“ x) + 2(40 â€“ y) + 5(x + y â€“ 10]

= 3x + 4y + 10

Hence,

The required mathematical formulation of linear programming is:

Minimize Z = 3x + 4y + 370

subject to constraints,

x + y â‰¤ 60

x â‰¤ 40

y â‰¤ 40

x + y â‰¥ 10

where, x, y â‰¥ 0

Let us convert inequations into equations as follows:

x + y = 60, x = 40, y = 40, x + y = 10, x = 0 and y = 0

Region represented by x + y â‰¤ 60:

The line x + y = 60 meets the coordinate axes at A1 (60, 0) and B1 (0, 60) respectively.

Region containing origin represents x + y â‰¤ 60 as (0, 0) satisfies x + y â‰¤ 60.

Region represented by x â‰¤ 40:

The line x = 40 is parallel to y-axis, meets x-axis at A2 (40, 0). Region containing origin represents x â‰¤ 40 as (0, 0) satisfies x â‰¤ 40.

Region represented by y â‰¤ 40:

The line y = 40 is parallel to x-axis, meets y-axis at B2 (0, 40).

Region containing origin represents y â‰¤ 40 as (0, 0) satisfies y â‰¤ 40.

Region represented by x + y â‰¥ 10:

The line x + y = 10 meets the coordinate axes at A2 (10, 0) and B3 (0, 10) respectively.

Region not containing origin represents x + y â‰¥ 10 as (0, 0) does not satisfy x + y â‰¥ 10.

Shaded region A3 A2 P Q B2 B3 represents feasible region.

Point P(40, 20) is obtained by solving x = 40 and x + y = 60

Point Q(20, 40) is obtained by solving y = 40 and x + y = 60

The value of Z = 3x + 4y + 370 at

A3(10, 0) = 3(10) + 4(0) + 370 = 400

A2(40, 0) = 3(40) + 4(0) + 370 = 490

P(40, 20) = 3(40) + 4(20) + 370 = 570

Q(20, 40) = 3(20) + 4(40) + 370 = 590

B2(0, 40) = 3(0) + 4(40) + 370 = 530

B3(0, 10) = 3(0) + 4(10) + 370 = 410

Hence, minimum value of Z = 400 at x = 10, y = 0

So,

From A -> P = 10 packets

From A -> Q = 0 packets

From A -> R = 50 packets

From B -> P = 30 packets

From B -> Q = 40 packets

From B -> R = 0 packets

Therefore, minimum cost = Rs 400

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