Question 921432: Pearl's company has just received and urgent order for its bathroom cabinets,which it makes in two styles,standard and de luxe.the order is for 'at least 100 bathroom cabinets of either variety,including at least thirty of the de luxe style.
The standard model takes two hours of assembly time and has variable costs of P40,whereas de luxe takes 5 hours of assembly time and has variable costs of P60.there are 400 hours in total available for assembly.the equipment can be used to assemble either style of cabinet in any combination.other reasons dictate that atleast as many standard cabinets as de luxe cabinets must be made.the company wishes to minimize its variable costs of production on this special order.
A.formulate this problem as a linear programming problem
b.graph the constraints,shading the unwanted region
c.recommend the best product mix for the company.what is the variable cost incurred
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = number of standard cabinets.
let y = number of deluxe cabinets.
at least 100 cabinets have to be made.
constraint:
x + y >= 100.
solve for y to get:
y >= 100 - x
region to shade that does not meet this requirement:
y <= 100 - x
have to make at least 30 of the deluxe model.
constraint:
y >= 30
region to shade that does not meet this requirement:
y <= 30
standard model takes 2 hours of assembly time and the deluxe model takes 5 hours of assembly time and the available assembly time is less than or equal to 400 hours.
constraint:
2x + 5y <= 400
solve for y to get:
y <= (400 - 2x) / 5
simplify to get:
y <= 80 - (2/5)x
region to shade that does not meet this requirement:
y >= 80 - (2/5)x
at least as many standard models as deluxe models need to be made.
x >= y
solve for y to get:
y <= x
region to shade that does not meet this requirement:
y >= x
standard model has variable costs of p40 per unit and deluxe model has variable costs of p60 per unit.
you want to minimize the cost.
objective function:
cost = 40x + 60y
you need to graph and shade the region that does not meet the requirements.
the region that is not shaded is the feasible region.
you need to find the corner points of this region.
your minimum cost will be at one of these corner points.
you need to evaluate the objective function at each of these corner points to find the minimum cost for the objective function.
note that, theoretically, if the region that meets the requirements is y >= x, then the region that does not satisfy those requirements would need to be y < x, and not y <= x.
i used y <= x to make the graphing software create a solid line that i wanted to show rather than a dashed line.
i was technically not correct but graphically correct because, even though i shaded the region that did not meet the requirements, the region that did meet the requirements need to be a solid line because the requirements was y >= x rather than y > x.
this logic applied to all the equations used, not just to y >= x.
your graph looks like this:
the equations you are graphing are the equations that do not meet the requirements.
the regions for those equations will be shaded, the result being that you have shaded the regions that do not meet the requirements.
the regions that are not shaded are therefore the feasible regions.
the equations that are graphed are:
y <= 100 - x
y <= 30
y >= 80 - (2/5)x
y >= x
since x has to be greater than or equal to 0, you would also graph x <= 0 to shade the region that does not meet this requirement.
since y cannot be less than or equal to 0, you would also graph y <= 0 to shade the region that does not meet this requirement.
the objective function to minimize is 40x + 60y
the corner points of the feasible region are at:
(50,50)
(57.14,57.14)
(125,30)
(70,30)
the objective function is evaluated at each of these corner points.
(50,50) yields 50 * 40 + 50 * 60 = P5000
(57.14,57.14) yields 57.14*40 + 57.14 * 60 = P5714
(125,30) yields 125*40 + 30*60 = P6800
(70,30) yields 70*40 + 30*60 = P4600
looks like the cost will be minimized when 70 standard cabinets and 30 deluxe cabinets are built.
the constraints also need to be met.
y >= 100 - x is satisfied because 30 is greater than or equal to 30.
y >= 30 is satisfied because 30 is greater than or equal to 30.
y <= 80 - 2/5x is satisfied because 30 is smaller than or equal to 52.
y <= x is satisfied because 30 is smaller than or equal to 70.
all constraints are satisfied at the minimum cost solution of x = 70 and y = 30.
note that the company had the resources to make more than 100 cabinets but then the variable cost would not have been minimized.
the calculator i used is at http://www.desmos.com/calculator
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