Answer:
Let number of executive class tickets = x And number of economy class tickets = y Now, required linear programming problem is given by Maximise \[Z=1000x+600y\] Subject to the constraints \[x+y\le 200,\] \[x\ge 20,\] \[y\ge 4x\] \[\Rightarrow \] \[4x-y\le 0\,\,\text{and}\,\,\,x,\,\,y\ge 0\] On considering the constraints as equation, we get \[x+y=200\] ?(i) x = 20 ?(ii) and \[4x-y=0\] ?(iii) Table for x + y = 200 is
So, line x + y = 200 passes through the points (0, 200) and (200, 0). Put (0, 0) in the inequality \[x+y\le 200,\] we get \[0+0\le 200\] [true] \[\therefore \] Shaded region is towards the origin. table for\[4x-y=0\] is x 0 200 y 200 0
So, line \[4x-y=0\] passes through the points (40, 160) and (20, 80). Put (1, 0) in the inequality \[4x-y\le 0\] we get \[4(1)-0\le 0\] \[\Rightarrow \] \[4\le 0\] [false] \[\therefore \] Required shaded region is the region not containing (1, 0). Also, the line x = 20 is parallel to Y-axis, so it passes through the point (20, 0). Put (0, 0) in the inequality \[x\ge 20,\] we get \[0\ge 20\] [false] \[\therefore \] Shaded region is away from the origin. On plotting the above points, we get the required feasible region. The intersection point of lines (i) and (ii) is (20, 180), intersection point of lines (i) and (iii) is (40, 160) and intersection point of lines (ii) and (iii) is (20, 80). Thus, corner points of the region are A(20, 180), B (40, 160) and C (20, 80). Now, consider value of Z at corner points which are given below: x 40 20 y 160 80
Hence, maximum profit is Rs. 136000 and it is achieved when 40 tickets of executive class and 160 tickets of economy class are sold. Value Yes, more passengers would prefer to travel by such an airline because some amount of profit is invested for welfare fund. Corner points \[\mathbf{Z=1000x+600y}\] A(20, 180) \[1000(20)+600(180)=20000\] \[+108000=Rs.\text{ }128000\] B(40, 160) \[1000(40)+600(160)=40000\] \[+\text{ }96000=\text{ }Rs.136000\,(\text{maximum})\] C(20, 80) \[1000(20)+600(80)=20000\] \[+\text{ }48000=Rs.\text{ }68000\]
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