question_answer

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is almost half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by almost 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, then how many of each should be produced weekly in order to maximize the profit? Why are small scale industries important in India? What values are being promoted by establishing small scale industries?

Answer:

Let the company manufactures x dolls of type A and j y dolls of type B. Then, objective function is maximum profit,\[Z=12x+16y.\]                           Subject to the constraints               \[x+y\le 1200,\]\[y\le \frac{x}{2}\] \[\Rightarrow \]   \[x-2y\ge 0\] and \[x\le 3y+600\] \[\Rightarrow \]        \[x-3y\le 600\]          Consider the given constraints as equations, we get \[x+y=1200\]                              ... (i) \[x-2y=0\]                                   ? (ii) and       \[x-3y=600\]                              ? (iv) Table for \[x+y=1200\]is

x	0	1200
y	1200	0

So, the line \[x+y=1200\] passes through the point (0, 1200) and (1200, 0).                         On putting (0, 0) in the inequality \[x+y\le 1200,\] we get \[0+0\le 1200\] \[\Rightarrow \,\,\,\,0\le 1200\]        [true] So, the shaded region is towards the origin.  Table for \[x-2y=0\] or x = 2y is

x	400	800
y	200	400

So, the line x = 2y passes through the points (0, 0) (400, 200) and (800, 400). On putting (3, 0) in the inequality \[x-2y\ge 0,\] we get \[3-0\ge 0\] \[\Rightarrow \] \[3\ge 0\]                      [true] So, the half plane is towards the X-axis.        Table for \[x-3y=600\] is 

x	0	600
y	\[-\,200\]	0

So, the tine \[x-3y=600\] passes through the points \[(0,-\,200)\] and (600, 0).                           On putting (0, 0) in the inequality\[0-3(0)\le 600\] we get \[0-3(0)\le 600\] \[\Rightarrow \] \[0\le 600\]            [true) So, the half plane is towards the origin.           Now, intersection point of Eqs. (i) and (ii) is C (800, 400) and intersection point of Eqs. (iii) and (i) is B (1050, 150), Now, plotting the graph of equations, the shaded portion OABC represents the feasible region which is bounded and coordinates of the corner points are 0(0, 0), A (600, 0), 8 (1050, 150) and C (800, 400).     Now, the value of Z at each corner point is given below

Corner points	\[\mathbf{Z=12x+16y}\]
O(0, 0)	\[Z=12(0)\,\,16(0)=0+0=0\]
A(600, 0)	\[Z=12(600)+16(0)=7200\]
B(1050, 150)	\[Z=12(1050)+12(150)\] \[=12600+2400=15000\]
C(800, 400)	\[Z=12(800)+16(400)=9600+\]\[6400=16000\] (maximum)

\[\therefore \] Maximum value of Z is 16000 at the point C (800, 400). Hence, maximum profit is Rs. 16000 when 800 dolls of type A and 400 dolls of type 8 are produced, Small scale mousses generate more employment with low investment. Hence, resulting into equitable distribution of money and progress of the country. Values promoted are generally employment and helping to removing poverty.