question_answer

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs. 5760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at a profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18 Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise his profit? Make it as an LPP and solve it graphically. Keeping the rular background in mind to justify the values to be promoted for the selection of the manually operated machine.

Answer:

Let the dealer purchased x electronic sewing machines and y manually operated sewing machines. Now, we can construct the following table

Type of sewing machine	Number	Investment (in Rs.)	Profit (in Rs.)
Electronic	\[x\]	\[360\,x\]	\[22x\]
Manually	\[y\]	\[240\,y\]	\[18y\]
Total	\[x+y\]	\[360x+240y\]	\[22x+18y\]
Availability	20	\[5760\]

Then, given LPP is Maximise           \[Z=22x+18y\]             ?(i) Subject to constraints, \[x+y\le 20\]                ?(ii) \[360x+240y\le 5760\] or                     \[3x+2y\le 48\]            ?(iii) and                   \[x\ge 0,\,\,y\ge 0\]                   ?(iv) Firstly, draw the graph of the line,             \[x+y=20\]                              ?(v)            

X	0	20
y	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. Secondly, draw the graph of the line, \[3x+2y=48\]                         ?(vi)            

X	0	16
y	24	0

On putting (0, 0) in the inequality\[3x+2y\le 48\], we get \[0\le 48,\]which is true. So, the half plane is towards the origin. On solving Eqs. (v) and (vi), we get \[x=8,\,\,y=12\] Since, \[x\ge 0\]and\[y\ge 0\], so the feasible region lies in the first quadrant. Graphical representation of the Eqs. is given below From the graph, feasible region is OABCO. The corner points of the feasible region are O (0, 0), A (16, 0), B (8, 12) and C (0, 20). (1) The value of Z at these points are as follow:

Corner points	\[Z=22x+18y\]
O (0, 0)	\[Z=22\,(0)+18\,(0)=0\]
A (16, 0)	\[Z=22\times 16+0=352\]
B (8, 12)	\[Z=22\times 8+18\times 10=392\] (maximum)
C (0, 20)	\[Z=22\times 0+18\times 20=360\]

From the table, maximum value of Z = Rs. 392 at point B (8, 12). Hence, dealer should purchased 8 electronic and 12 manually operated sewing machines to get maximum profit. Value keeping the ?Save environment? and ?Conservation of exhaustive resources? in mind the manually operated machine should be promoted to energy could be saved.