question_answer
A manufacturer considers that men and women workers are equally efficient, so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B, To produce 1 unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce 1 unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, then how should he use his resources to maximise the total revenue? Form the above as an LPP and solve it graphically. Also, write what quality of manufacturer reflects here?
Answer:
Let the manufacturer use x workers and y units of capital to maximize the total revenue. Given, information can be written in tabular form as given below | | A | B | Available capacity |
| Workers | 2 | 3 | 30 |
| Capital | 3 | 1 | 17 |
| Profit function (Z) | 100 | 120 | |
Then, the required LPP is Maximise \[(Z)=100x+120y\] Subject to constraints are \[2x+3y\le 30\] ?(i) \[3x+y\le 17\] ?(ii) and \[x,\,\,y\ge 0\] Consider the inequalities as equation, we get \[2x+3y=30\] ?(iii) and \[3x+y=17\] ?(iv) Table for line \[2x+3y=30\] is \[\therefore \] Eq. (iii) passes through points (0, 10) and (15, 0). On putting (0, 0) in the inequality \[2x+3y\le 30,\] we get \[2(0)+3(0)\le 30\] \[\Rightarrow \] \[0\le 30\] [true] \[\therefore \] The shaded portion is towards the origin. Table for line \[3x+y=17\] is \[\therefore \] Eq. (iv) passes through the points (0, 17) and (17/3, 0). On putting (0, 0) in the inequality \[3x+y\le 17,\] we get \[3(0)+0\le 17\] \[\Rightarrow \] \[0\le 17\] [true] \[\therefore \] The shaded portion is towards the origin. The intersection point of both lines is P(3, 8). Now, on plotting these points on graph paper, we get the feasible, region OAPCO, which is bounded and its corner points are O(0, 0), \[A\left( \frac{17}{3},\,\,0 \right),\] P(3, 8) and C(0, 10). 
The values of Z corner points are given below | Corner points | \[\mathbf{Z=100x+120y}\] |
| O(0, 0) | \[100\times 0+120\times 0=0+0=0\] |
| \[A\left( \frac{17}{3},\,\,0 \right)\] | \[100\times \frac{17}{3}+120\times 0=\frac{1700}{3}=566.66\] |
| C(0, 10) | \[100\times 0+120\times 10=0+1200=1200\] |
| P((3, 8) | \[100\times 3+120\times 8=300+960=1260\] [maximum] |
Here, the maximum value of Z is 1260 at point P(3, 8). Hence, the manufacturer use 3 workers and 8 units of capital to minimize the total revenue and the maximize revenue is Rs. 1260. Value To develop equality in society and professionalism.
