TV Circuit has 30 large-screen televisions in a warehouse in Erie and 60 large-s
ID: 2083611 • Letter: T
Question
TV Circuit has 30 large-screen televisions in a warehouse in Erie and 60 large-screen televisions in a warehouse in Pittsburgh. Thirty-five are needed in a store in Blairsville, and 40 are needed in a store in Youngstown. It costs $19 to ship from Pittsburgh to Blairsville and $25 to ship from Pittsburgh to Youngstown, whereas it costs $20 to ship from Erie to Blairsville and $27 to ship from Erie to Youngstown. How many televisions should be shipped from each warehouse to each store to minimize the shipping cost? Hint: If the number shipped from Pittsburgh to Blairsville is represented by x, then the number shipped from Erie to Blairsville is represented by
35 x.
from Pittsburgh to Blairsville ? televisions from Pittsburgh to Youngstown ? televisions from Erie to Blairsville ? televisions from Erie to Youngstown ? telsevisonsExplanation / Answer
#detailed step by step explanation of answer
Taking Contraints:
1. x+y<= 60
2. (35-x)+(40-7) <=30
x <=35
y<= 40
x>=0
y>=o
From Pittsburg: 20 to Blairsville, 40 to Youngstown
From Erie: 15 to Blairsville, 0 to Youngstown
Minimum cost: $1540
So x is the number shipped from Pittsburgh to Blairsville
and y is the number shipped from Pittsburgh to Youngstown,
which means
35-x are shipped from Erie to Blairsville
and 40-y are shipped from Erie to Youngstown.
the problem constraint is best cleaned up using algebra:
(35-x) + (40-y) <= 30
35 + 40 - x - y <= 30
35 + 40 - 30 <= x + y
45 <= x + y
which makes sense: there are only 30 televisions in Erie and we need to ship 75, so at least 45 must come from Pittsburgh.
So the graph starts with a rectangle defined by
0 <= x <= 35 and
0 <= y <= 40
and then is crossed by two parallel lines slanting downward to the right, delimiting the constraints
45 <= x + y <= 60
The lower bound
45 <= x + y
cuts through the rectangle from (5,40) to (35,10)
[eliminating from consideration the corners at (0,40), (0,0), and (35,0)].
The upper bound
x + y <= 60
cuts through the rectangle from (20,40) to (35,25)
[eliminating from consideration the corner at (35,40)].
This leaves us with a trapezoidal feasible area with corners at
(5,40), (35,10), (20,40) and (35,25).
The objective function is the total shipping costs, and it's the sum of the costs for all four shipping paths:
C(x,y) = 18x + 22y + 20(35-x) + 25(40-y)
= 18x + 22y + 700 - 20x + 1000 - 25y
= 1700 - 2x - 3y
Now we just have to compute
C(5,40) = 1700 - 10 - 120 =$1570
C(35,10) = 1700 - 70 - 30 = $1600
C(20,40) = 1700 - 40 - 120 = $1540
C(35,25) = 1700 - 70 - 75 = $1555
which confirms answer.
Notice that the final version of the cost function
C(x,y) = 1700 - 2x - 3y
basically says that shipping cost is $1700 minus whatever can be saved by shipping from Pittsburgh in preference to Erie, with greater savings for sets shipped from Pittsburgh to Youngstown than for those shipped to Blairsville.