Help me solve the Linear Programming problem?

A coal mining company owns 2 mines whose coal output differ in quality and accessibility.Suppose that for each day there are 12 available mining hours. Each mine produces high, middle and low quality coal.Because of storage restrictions, the company may mine not more than 60tons, 90tons and 80tons respectively of the high, middle and low quality coal in a day. An hour spent digging in mine I produces 2tons,4tons and 6tons respectively of the 3 kinds of coal, similarly an hour in mine II yields 3,1 and 5 tons respectively of the 3 kinds of coal. Profit per ton are #500,#400 and #300 per high,middle and low quality coal respectively. How many hours should be spent in each of the two mines in order to maximize total profit per day. Formulate the problem into a mathematical model.

Help me solve the Linear Programming problem?
Call x the hours in mine 1, and y the hours in mine 2. Since there are 12 hours in the day, x + y = 12, so we can write y = x - 12.





The maximum daily amounts are high enough that they do not affect the problem.





total profit = x(2*500 + 4*400 + 6*300) + y(3*500 + 1*400 + 5*300)


p = 4400x + 3400y


p = 4400x + 3400x - 40800


p = 7800x - 40800





x can be, at most, 12. Since this function is linear with no local maximum, the highest profit is found when x = 12.





p = 7800*12 - 40800


p = 52,800





Now to make sure that the daily maximums were not exceeded,


12*2 = 24 %26lt; 60


12*4 = 48 %26lt; 90


12*6 = 72 %26lt; 80





They should spend 12 hours in mine 1 and 0 hours in mine 2.
Reply:You have been told what the objective function is: maximise daily profit using two variables, the hours spent in each mine (I'll call them h1 %26amp; h2). Write an equation for profit as a function of h1 %26amp; h2.





Constraints:


You have been given limits on the total of h1 %26amp; h2.


For each grade of coal you'll have to write an inequality (using h1 %26amp; h2) limiting total output of the two mines to no more than the company can store.





And don't charge too much for the answer. At tonnages like that the CEO probably lives under a sheet of corrugated iron.
Reply:x = hours in mine 1


y = hours in mine 2


x%26gt;=0


y%26gt;=0


x + y %26lt;= 12





2x + 3y %26lt;= 60


4x + 1y %26lt;= 90


6x + 5y %26lt;= 80





Graph these and see the coordinates of the vertices of the resulting polynomial. The answer is at one of them. Figure out how much of each kind of coal is produced at each answer, then find the total value. Whichever is highest , take that.
Reply:What's linear programming?