Linear Programming in relationship to the Profit Maximization of the Business - Math Problem Example

Dx=y C A2x+3y=30 B x + y = 10500000The feasibility area would be the region with boundaries ray BC, ray AD and segment AB. The co-ordinates of A and B are (5250000, 5250000) and (6, 6) respectively. The value of the objective function at these points is 0.45 X 5250000 = 2362500 and 2.7 respectively. The value of the objective function at the points of ray AD beyond point A would be 0.2x + 0.25(10500000 - x) i.e. 2625000 - 0.05x and this value will be maximum when 0.05x is minimum i.e. when x=0 as we cannot take x as negative since x is the value of new houses and this maximum value of 2625000 will be attained at point D.

Similarly the value of objective function on ray BC beyond points B is 0.2x + 0.25(30-2x)/3 i.e. 2.5 +0.03x and this will be maximum when x is maximum i.e. at point B itself. Thus the maximum value of profit in this case is at point D i.e. 2625000 and it is more than that in the earlier case. Therefore there would be increase in the profit of 2625000-2624999.8=0.2 million.b)would it be worthwhile increasing the skilled workforce The cost of taking an another skilled laborer is 15000.Suppose there are 181 laborers instead of 180.

then the constraint line BC on page two will be shifted right. The co-ordinates of B and C will be (4, 7.38) and (9.083, 4) and the values of the objective function at B and C will be 2.645 and 2.8166 respectively. This means at point C there will be increase in profit of 16000 which would cover up the overhead of additional laborer of 15000. So, I think it is worthwhile increasing the skilled workforce.c)would the optimal solution change if the profit contributions. 2625000 - 0.05x and this value will be maximum when 0.

05x is minimum i.e. when x=0 as we cannot take x as negative since x is the value of new houses and this maximum value of 2625000 will be attained at point D. Similarly the value of objective function on ray BC beyond points B is 0.2x + 0.25(30-2x)/3 i.e. 2.5 +0.03x and this will be maximum when x is maximum i.e. at point B itself. Thus the maximum value of profit in this case is at point D i.e. 2625000 and it is more than that in the earlier case. Therefore there would be increase in the profit of 2625000-2624999.8=0.2 million.

Suppose there are 181 laborers instead of 180. then the constraint line BC on page two will be shifted right. The co-ordinates of B and C will be (4, 7.38) and (9.083, 4) and the values of the objective function at B and C will be 2.645 and 2.8166 respectively. This means at point C there will be increase in profit of 16000 which would cover up the overhead of additional laborer of 15000. So, I think it is worthwhile increasing the skilled workforce. Suppose the profit contributions are 19% and 26% respectively and that the objective function is 0.19x + 0.26y and the value of objective function at point A on page 2 will be 2729999.72 i.e.

there will be increase. If we just interchange the profit contributions i.e.