Q. 25.0( 1 Vote )

A medical company

Answer :

Let factory A supply x and y packets of medicines to agency P and Q respectively. Remaining packets of factory A are 60 – x – y. So, now they will be supplied to agency R by factory A.

Now, remaining packets requirement of agency P is 40 – x which will be supplied by factory B.


Remaining packets requirement of agency Q is 40 – y which will be supplied by factory B.


Remaining packets requirement of agency R is 50 – (60 – x – y)


= x + y – 10 which will be supplied by factory B.


where,


x ≥ 0, y ≥ 0 and 60 – x – y ≥ 0


x 0, y 0 and x + y 60


40 – x ≥ 0, 40 – y ≥ 0 and x + y – 10 ≥ 0


x 40, y 40 and x + y 10


This can be illustrated by this:




Cost = Number of packets * Cost of transportation per packet


Total transportation cost z is given by,


z = 5x + 4y + 3(60 – x – y) + 4(40 – x) + 2(40 – y) + 5(x + y – 10)


z = 5x + 4y + 180 – 3x – 3y + 160 – 4x + 80 – 2y + 5x + 5y – 50


z = 3x + 4y + 370


We need to minimize the cost


Hence, mathematical formulation of LPP is


Minimize z = 3x + 4y + 370


subject to the constraints,


x+y≥10


y≤40


x≤40


x+y≤60


x,y≥0


The feasible region determined by the system of constraints is as follows:



The corner points of the enclosed region are A(10, 0) , B(40, 0), C(40, 20), D(20,40), E(0, 40) and F(0, 10)


The value of z at these corners points is as follows:


Case 1: A(10, 0)


z = 3x + 4y + 370


z = 3(10) + 4(0) + 370


z = 30 + 0 + 370


z = 400


Case 2: B(40, 0)


z = 3x + 4y + 370


z = 3(40) + 4(0) + 370


z = 120 + 0 + 370


z = 490


Case 3: C(40, 20)


z = 3x + 4y + 370


z = 3(40) + 4(20) + 370


z = 120 + 80 + 370


z = 570


Case 4: D(20, 40)


z = 3x + 4y + 370


z = 3(20) + 4(40) + 370


z = 60 + 160 + 370


z = 590


Case 5: E(0, 40)


z = 3x + 4y + 370


z = 3(0) + 4(40) + 370


z = 0 + 160 + 370


z = 530


Case 4: F(0, 10)


z = 3x + 4y + 370


z = 3(0) + 4(10) + 370


z = 0 + 40 + 370


z = 410


The value of z is minimum in first case at point A(10, 0)


As, x = 10, y = 0


Factory A supplies to:


Agency P = x = 10 packets


Agency Q = y = 0 packets


Agency R = 60 – x – y = 60 – 10 – 0 = 50 packets


Godown B supplies to:


Agency P = 40 – x = 40 – 10 = 30 packets


Agency Q = 40 – y = 40 – 0 = 40 packets


Agency R = x + y – 10 = 10 + 0 – 10 = 0 packets


Minimum cost for transportation of these packets to their respective agencies = Rs. 400



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