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Question 1118663: Each day , a factory makes three types of widget : basic , standard and luxury. The widget produced need three different components : type A , type B and type C.
Basic widget need 6 components of type A , 6 components type B and 12 components of type C.
Standard widgets need 4 components of type A ,3 compliance of type B , and 18 components of type C.
Luxury widgets need 2 components of type A , 9 components of type B and 6 components of type C.
Each day , there are 240 components of type A available , 300 of type B. 900 of type C.
Each day ,the factory must use at least twice as many components of type C as type B.
Each day , the factory makes X basic widgets , y standard widgets and z luxury widgets.
(a) In addition to x>\= 0 , y>\= 0 and z>\=0 , find four inequalities in x, y and z that model the above constraints , simplify each inequality.
(b) Each day ,the factory makes the maximum possible number of widget . On a particular day , the factory must make the same number of luxury widgets as basic widgets.
(i) show that your answers in part (a) become
2x + y <\= 60
5x + y <\= 100
X + y <\= 50
Y >\= x
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you would create a table as follows:
type widget basic standard luxury
number of widgets x y z
number of type A components 6 4 2
number of type B components 6 3 9
number of type C components 12 18 6
the number of type A components has to be less than or equal to 240.
the equation for that is:
6x + 4y + 2z <= 240
this can be simplified to:
3x + 2y + z <= 120
the nunmber of type B components has to be less than or equal to 300.
the equation for that is:
6x + 3y + 9z <= 300
this can be simplified to:
2x + y + 3z <= 100
the number of type C components has to be less than or equal to 900.
the equation for that is:
12x + 18y + 6z <= 900
this can be simplified to:
2x + 3y + z <= 150
the number of type C components has to be greater than or equal to 2 times the nmber of type B components.
the equation for that is:
12x + 18y + 6z >= 2 * (6x + 3y + 9z)
this can be simplified to:
12x + 18y + 6z >= 12x + 6y + 18z
this can be simplified to:
2x + 3y + z >= 2x + y + 3z
your simplified equations are:
3x + 2y + z <= 120
2x + y + 3z <= 100
2x + 3y + z <= 150
2x + 3y + z >= 2x + y + 3z
in part b, you are told that the factory must make the same number of luxury widgets as basic widgets.
this means that x must be equal to z.
since x must be equal to z, you can replace z with x in the equations.
you will get:
3x + 2y + x <= 120
2x + y + 3x <= 100
2x + 3y + x <= 150
2x + 3y + x >= 2x + y + 3x
combine like terms to get:
4x + 2y <= 120
5x + y <= 100
3x + 3y <= 150
3x + 3y >= 5x + y
simplify to get:
2x + y <= 60
5x + y <= 100
x + y <= 50
in 3x + 3y >= 5x + y, subtract 3x from both sides of the equation and subtract y from both sides of the equation to get:
3y - y >= 5x - 3x
combine like terms to get:
2y >= 2x
simplify to get:
y >= x
there you have it.
the 4 equations (they are really inequalities even though i called them equations) become:
2x + y <= 60
5x + y <= 100
x + y <= 50
y >= x
that matches what you showed above, so this solution appears to be correct.
i used excel to come up with a solution, just to see what it would be.
the solution that excel came up with was 13 basic, 34 standard, 13 luxury.
that makes the maximum number of widgets to be produced equal to 60.
all the constraints were satisfied.
i forced all the number of widgets to be integers.
excel claims the answer, in that case, would be within 1% of the optimal solution.
the total number of widgets was the same when i didn't force the number of widgets to be integers.
the non-integer solution was x and z = 13.33333 and y = 33.33333
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