linear programming 5
the fence post company
Train A can has 4 carriages for large posts and 1 carriage for small posts per trip.
Train B has 2 carriages for large posts and 2 carriages for small posts per trip.
Each day the trains need to deliver at least 20 carriages large posts and 14 carriages of small posts.
The maintenance crew has told management that the trains cannot make any more than 8 trips per day (each train)
Cost, Train A cost $360 per trip and Train B, $240 per trip
Train B has 2 carriages for large posts and 2 carriages for small posts per trip.
Each day the trains need to deliver at least 20 carriages large posts and 14 carriages of small posts.
The maintenance crew has told management that the trains cannot make any more than 8 trips per day (each train)
Cost, Train A cost $360 per trip and Train B, $240 per trip
Cost = 360x + 240y
A(1,8) = 360x1+240x8=$2,280
B(8,8) = 360x8+240x8=$4,800
C(8,3)= 360x8+240x3=$3,600
D(2,6)= 360x2+240x6 = $2,160
The point that minimises fuel costs is D(2,6)
To minimise the fuel costs, The Fence Post Company should run 2 trips a day for Train A and 6 trips a day for Train B. The daily fuel cost is at a minimum of $2160
A new cost equation is provided
Cost = 500x + 250y
Cost = 500x + 250y
Cost = 500x + 250y
A(1,8) = 500x1+250x8=2,500
B(8,8) = 500x8+250x8=6,000
C(8,3)= 500x8+250x3=4,750
D(2,6)= 500x2+250x6 = 2,500
A(1,8) = 500x1+250x8=2,500
B(8,8) = 500x8+250x8=6,000
C(8,3)= 500x8+250x3=4,750
D(2,6)= 500x2+250x6 = 2,500
The company now has two options that give the lowest fuel cost. It can either run 1 trip with train A and 8 trips with train B each day or 2 trips with Train A and 6 trips with Train B each day. The minimum cost is $2500, an increase of $___ over the last option. That takes it to high Merit.
excellence
For Excellence you need to notice the there are two minimum values achieved, at points A and D.
(It isn't always A and D but that happens in most of our activities). The equation for that line is
4x + 2y ≥ 20, now find its gradient by rearranging it.
2y ≥ -4x + 20
y ≥ -2x + 10 so the gradient = -2
The new cost function 500x + 250y has the same gradient as 2y ≥ -4x + 20 above as follows,
500x + 250y = cost - don't worry about c
y = -500/250x + c
y = -2x + c so the gradient is -2 as above.
What does this mean?
Any integer (whole) coordinates along the line 4x + 2y ≥ 20 for 1≤ x ≤ 2, will minimise the cost. This means the company can run the trains in different combinations. However, in this example, there are still only 2 options available so there are no more combinations possible than that were achieved earlier. When Train A's trip number is increased by 1, Train B's trip number will decrease by 2 (see desmos graph). shown as (1,8), (2,6)
The daily fuel cost will be at minimum of $ 2500.