Question 1201219: A company wishes to produce two types of souvenirs: Type A and Type B. Each Type A souvenir will result in a profit of $1, and each Type B souvenir will result in a profit of $1.20. To manufacture a Type A souvenir requires 2 minutes on Machine I and 1 minute on Machine II. A Type B souvenir requires 1 minute on Machine I and 3 minutes on Machine II. There are 3 hours available on Machine I and 5 hours available on Machine II.
(a) The optimal solution holds if the contribution to the profit of a Type B souvenir lies between $____ and $____ . (Enter your answers from smallest to largest.)
(b) Find the contribution to the profit of a Type A souvenir (with the contribution to the profit of a Type B souvenir held at $1.20), given that the optimal profit of the company will be $172.80.
$______
(c) What will be the optimal profit of the company if the contribution to the profit of a Type B souvenir is $2.50 (with the contribution to the profit of a Type A souvenir held at $1.00)?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the optimal solution, when profit of x = 1 and profit of y = 1.20 is at (x,y) = (48,84), as shown in the following graph.
the constraints were:
2x + y <= 180
x + 3y <= 300
the objective function was:
1 * x + 1.2y = max
the opposite of the contraints was graphed.
the area not shaded is the feaible solution area.
the optimal solution is at the corner points of the feasible solution.
those corner points are:
(0,100)
(48,84)
(90,0)
x is the number of type A souveniers and y is the number of type B souveniers.
when profit of type A souvenier is 1 dollar per unit and profit of type B souvenier is 1.20 dollars per unit becomes:
at (0,100, profit = 0 + 120 = 120.
at (48,84), profit = 48 + 100.8 = 148.8
at (90,0), profit = 90 + 0 = 90
(a) The optimal solution holds if the contribution to the profit of a Type B souvenir lies between $____ and $____ . (Enter your answers from smallest to largest.)
when the profit of souvenier type a is held at 1 dollar per unit, then the 3 optimal solutions becomew:
0 * 1 + 100 * y at (0,100) = 100 * y
48 * 1 + 84 * y at (48,84) = 48 + 84 * y ***** optimal solution
90 * 1 + 0 * y at (90,0) = 90
the optimul soluton will hold when:
48 + 84y > 100y and 48 + 84y > 90
subtract 84y from both sides of the first equation to get 48 > 16y
solve for y to get 3 > y
this is the same as y < 3.
subtract 48 from both sides of the second equation to get 84y > 42
solve for y to get y > .5
your optimal solution will be at (48,84) when y > .5 and y < 3.
that can be rewritten as .5 < y < 3.
to confirm, evaluate the optimal solution points when x = 1 and y is allow to fluctuate between .5 and 3, as shown in the following excel printout.
you can see that 48 + 84y is max when y is greater than .5 and smaller than 3.0.
in the spreadsheet, x was set at 1 and y was free to roam in .5 increments.
when y = .5 or y = 3.0, then 48 * x + 84 * y is still the max solution but it has to share it with 90 * x + 0 * y at y = .5 and 0 * x + 100 * y at y = 3.
it is not the max solution when y < .5 and y > 3.0.
(b) Find the contribution to the profit of a Type A souvenir (with the contribution to the profit of a Type B souvenir held at $1.20), given that the optimal profit of the company will be $172.80.
$______
when the optimal profit is 172.8 and profit for type B souvenier is held at 1.20, then your optimal profit equation becomes:
48 * x + 84 * 1.20 = 172.8
simplify to get:
48 * x + 100.8 = 172.8
subtract 100.8 from both sides of the equation to get:
48 * x = 72
solve for x to get:
x = 1.5.
the maximum profit wil be 172.8 when the profit of x is 1.5 and the profit of y is 1.2.
you get 48 * 1.5 + 84 * 1.2 = 172.8.
when profit for x = 1.5, .....
(0,100) becomes 0 * 1.5 + 100 * 1.2 = 120
(48,84) becomes 48 * 1.5 + 84 * 1.2 = 172.8
(90,0) becomes 90 * 1.5 = 135
the maximum solution remains at (48,84).
(c) What will be the optimal profit of the company if the contribution to the profit of a Type B souvenir is $2.50 (with the contribution to the profit of a Type A souvenir held at $1.00)?
when y = 2.50 and x = 1.00, the maximum profit equation becomes:
1.00 * x + 2.5 * y
at (0,100, profit is 0 * 1.00 + 100 * 2.5 = 250
at (48,84), profit is 48 * 1.00 + 84 * 2.5 = 258
at (90,0), profit is 90 * 1.00 + 0 * 2.5 = 90.
the maximum solution remains at (48,84).
as a reminder.
x is the number of type A souveniers.
y is the number of type B souveniers.
here's the excel printout again with the additional rows for the last two questions.
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