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Question 1151913: Pulse Fuels produces two types of additive blends, A and B. Blend A sells at PhP 1200/liter while Blend B sells at PhP 2000/liter. Its manager would like to maximize revenue by producing the right
combination of the two additive blends.
Currently, the company outsources its labor due to the seasonal nature of its products. Its contract with its labor provider requires that a minimum of 5,400 hours be utilized every month. If not utilized, the company will still have to pay the equivalent of 5,400 hours. Each liter of Blend A requires 3 hours of labor while each liter of Blend B requires 1.8 hours. On the other hand, its equipment availability is limited to 9,600 hours per month. Each liter of Blend A requires 2 hours of equipment time while each liter of Blend B requires 3 hours of equipment time. Its Marketing Department has estimated that at least 1000 liters per month of Additive Blend A can be sold while the monthly market demand for Additive Blend B is at least 400 liters. Its Operations Manager has informed the GM that the existing configuration of the operating plant (with a 4000-liter/month capacity) will require that one liter of Additive Blend A is produced for every 2 liters of Additive Blend B. Furthermore, the 4000-liter/month plant must operate, at the least, at 67.5% of its monthly capacity to make it operationally viable.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! it took a few tries, but i think i might have it.
i had the biggest trouble with the statement that the existing configuration of the operating plan will require that one liter of additive blend A is produced for every 2 liters of additive blend B.
i finally resolved that this requirement needed to constraints.
the first was that x + y <= 4000 and the second was that x = 2y.
this forced the solution to give you 1 part of blend A for every 2 parts of blend B.
for example:
x + y <= 4000 (max capacity is 4000).
y = 2x.
at capacity, you get x = (1333 + 1/3) and y = (2666 + 2/3).
the sum of x + y is equal to 4000 with those values of x and y.
this represents 1 part of x for every 2 parts of y.
in other words, one liter of blend A is produced for every 2 liters of blend B, which i think is what is meant.
at any rate, that was my assumption and it's what i included in the list of constraints.
the objective function was r = 1200 * x + 2000 * y.
this is what you want to maximize.
r stands for revenue.
the constraint functions are:
3x + 1.8y >= 5400.
2x + 3y <= 9600.
x >= 1000
x >= 400
x + y <= 4000
x + y >= 2700 (equals .675 * 4000)
y = 2x
x >= 0
y >= 0
if you used the the desmos.com calculator, you will graph the opposite of the constraints and the area of the graph that is not shaded will be your region of feasibility.
in addition, your solution must be on the line of y = 2x.
if you did not have y = 2x as a constraint, the maximum revenue would be at x = 1000 and y = (2533 + 1/3).
this would give you a revenue of 6,266,666 and 2/3 dollars.
since you do have the restriction that, for every 1 liter of blend A, you need to produce 2 liters of blend B, then the possible solutions reduces to only 2 and the most revenue is generated when x = 1200 and y = 2400.
the maximum revenue at that point would be 6,240,000 dollars.
there are two graphs to show you.
the first graph is without the requirement that y = 2x.
the second graph is with the requirement that y = 2x.
here's the graphs.
the difference between the two is that your maximum revenue solution has to be on the line y = 2x as shown in the second graph.
that's my interpretation of this problem.
only when your solution is on the line y = 2x, will the processing capacity be able to support 1 liter of blend A for every 2 liters of blend B.
all the possible solutions are at the corner points of the feasible region.
i tried it different ways, but this was the only way it made sense, to my mind at least.
i also tried solving the problem using a simplex method tool.
it was able to solve without the requirement of y = 2x but was not capable of handling the additional constraint of y = 2x.
here's a display of the results from the simplex method tool.
the graphing software can be found at https://www.desmos.com/calculator
the simplex method tool can be found at https://www.zweigmedia.com/RealWorld/simplex.html
Answer by ikleyn(52943)  (Show Source):
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