Two factories manufacture 3 different grades of paper. The company that owns the factories has contracts to supply at least 16 tons of low grade, 5 tons of medium grade, and at least 20 tons of high grade paper. It costs $1000 per day to operate the first factory and $2000 per day to operate the second. Factory 1 produces 8 tons of low grade, 1 ton of medium grade, and 2 tons of high grade paper in one day's operation. Factory 2 produces 2 tons of low grade, 1 ton of medium grade, and 7 tons of high grade paper per day. How many days should each factory be in operation in order to fill the orders most economically?
this is very simple it is
you just add every number except for the the money so the answer 68o
To find the most economical way to fill the orders, we need to minimize the cost of operation while meeting the quantity requirements.
Let's assume that factory 1 operates for x days, and factory 2 operates for y days.
The objective function that we want to minimize is the total cost of operation, which can be calculated as follows:
Cost = (operating days for factory 1 * cost per day for factory 1) + (operating days for factory 2 * cost per day for factory 2).
Now, let's write equations to represent the constraints of the problem:
1) Quantity constraint for low grade paper:
Factory 1 produces 8 tons per day, so the total low grade paper produced by factory 1 would be 8 * x.
Factory 2 produces 2 tons per day, so the total low grade paper produced by factory 2 would be 2 * y.
To meet the requirement of at least 16 tons of low grade paper, we can write the constraint as:
8x + 2y >= 16.
2) Quantity constraint for medium grade paper:
Factory 1 produces 1 ton per day, so the total medium grade paper produced by factory 1 would be 1 * x.
Factory 2 produces 1 ton per day, so the total medium grade paper produced by factory 2 would be 1 * y.
To meet the requirement of 5 tons of medium grade paper, we can write the constraint as:
x + y >= 5.
3) Quantity constraint for high grade paper:
Factory 1 produces 2 tons per day, so the total high grade paper produced by factory 1 would be 2 * x.
Factory 2 produces 7 tons per day, so the total high grade paper produced by factory 2 would be 7 * y.
To meet the requirement of at least 20 tons of high grade paper, we can write the constraint as:
2x + 7y >= 20.
4) Non-negativity constraint:
Since we cannot have negative operating days, we need to include the non-negativity constraint:
x >= 0, y >= 0.
Now, we have the objective function and the set of constraints. We can solve this linear programming problem using techniques like the simplex method or graphical method to find the values of x and y that minimize the cost while satisfying all the constraints.