question_answer

A dealer wishes to purchase a number of fans and radios. He has only Rs. 5760 to invest and has a space for at most 20 items. A fan costs him Rs. 360 and a radio Rs. 240. His expectation is that he can sell a fan at a profit of Rs 18. Assuming that, he can sell all the items that he buys, how should he invest his money for maximum profit? Translate the problem as LPP and solve it graphically.

Answer:

Let the number of fans purchased be x and the number of radios purchased by y.             \[\therefore \] The objective function id maximize \[Z=22x+18y\] Subject to the constraints             \[360x+240\,y\le 5760\]             or  \[3x+2y\le 48,\] [on dividing both sides by 120]                         \[x+y\le 20\]             and       \[x,\,\,y\ge 0\]             On considering the constraints as equation, we get                         \[3x+2y=48\]                           ? (i)                         \[x+y=20\]                                ? (ii)                         x = 0, y = 0                              ? (iii)             Tablet for the line \[3x+2y=48\] or \[y=\frac{48-3x}{2}\] is          

x	0	16
y	24	0

            So, line  \[3x+2y=48\] passes through the points (0, 24) and (16, 0).             On putting (0, 0) in the inequality \[3x+2y\le 48,\] we get \[3(0)+2(0)\le 48\Rightarrow 0\le 48,\] which is true.             So, the half plane is towards the origin.             Table for the line \[x+y=20\] or \[y=20-x\] is 

x	0	20
y	20	0

            So, line \[x+y=20\] passes through the points (0, 20) and (20, 0).             On putting (0, 0) in the inequality \[x+y\le 20,\]we get                         \[0+0\le 20\]             \[\Rightarrow \]   \[0\le 20,\] Which is true.             So, the half plane is towards the origin.             The graphical representation of these lines is given below The point of intersection of lines (i) and (ii) is B (8, 12). The shaded region id the graph represents the feasible region and its corner points are O (0, 0), A (16, 0), B (8, 12) and C (0, 20). Now, the values of Z at corner points are given below

Corner points	\[\mathbf{Z = 22x + 18y}\]
O(0, 0)	\[Z=22(0)+18(0)=0\]
A(16, 0)	\[Z=22(16)+18(0)=352\]
B(8, 12)	\[Z=22(8)+18(12)=392\](maximum)
C(0, 20)	\[Z=22(0)+18(20)=360\]

So, Z is maximum at x = 8 and y = 12 and the maximum value of is Rs. 392. Hence, the dealer should invest \[8\times 360=\,\,Rs.\,2880\] and \[12\times 240=\,\,Rs.\,2880\] in fan and radio respectively, to have a maximum profit of Rs. 392.