Question 1172680: A Papyrus company owns two automated machines. Machine A costs ₱25,000 per day
to operate, and it can produce 40 rims of high-grade paper, 30 rims of medium-grade
paper, and 20 rims of low-grade paper each day. Machine B costs ₱35,000 per day to
operate, and it can produce 30 rims of high-grade paper, 40 rims of medium-grade
paper, and 50 rims of low-grade paper each day. The company has orders totaling
2,000 rims of high-grade paper, 2,000 rims of medium-grade paper, and 3,000 rims of
low-grade paper. How many days should it run each machine to minimize its costs but
have enough supply of paper for the orders?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! the graphical solution is shown below.
the simplex method tool solution is shown below:
both tell you that the optimum solution is when x = 7.143 and y = 57.143.
the graphical solution is rounded to 3 decimal places.
the simplex method tool solution rounds to 6 significant digits.
rounding to decimal digits only includes the fractional part of the answer.
rounding to significant digits includes the whole and fractional part of the answer.
the minimal cost is 2178580 from the graphical solution and 2178570 from the simplex method tool solution.
with the graphical solution, using the desmos.com calculator, you graph the opposite of the constraint inequalities.
the region of feasibility is the area on the graph that is not shaded.
the corner points of the region of feasibility are where the optimum solutions lie.
you evaluate the objective function at each of these corner points to find the minimal cost solution.
the graphical solution shows all the possible values of x and y, one of which will provide the optimal solution.
the simplex method tool only shows optimal solution.
all the constraints need to be satisfied at the optimal solution.
with (7.143, 57.143):
40x + 30y = 2000.01 which is >= 2000.
30x + 40y = 2500.01 which is >= 2000.
20x + 50y = 3000.01 which is >= 3000.
keep in mind that these are rounded numbers, so the answers won't necessarily be right on.
with the simplex method tool, you can set the rounding.
for example, it was set at 6 significant digits.
this will get you closer to the actual results.
it can also be set to greater than 6 significant digits.
i set it to 12 significant digits and this is what is said:
Optimal Solution: c = 2178571.42857; x = 7.14285714286, y = 57.1428571429
the desmos.com calculator can be found at https://www.desmos.com/calculator
the simplex method tool can be found at https://www.zweigmedia.com/RealWorld/simplex.html
your objective function was:
cost = 25000x + 35000y
this you want to minimize.
your constraint inequalities were:
40x + 30y >= 2000 (hi)
30x + 40y >= 2000 (med)
20x + 50y >= 3000 (lo)
x >= 0
y >= 0
i'll be available to answer any questions you might have regardng this.
theo
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