Do any of you know how to come up with equations or mathematical way of representing these problems?
1) Finding probability of choosing number of routes
(i) If n of routes less than 4, consider all routes
(ii) else if n of routes is equal or more than 4, consider 40% of the n of
routes (round up decimal point)
2) Randomly choose which routes to consider out of all possible routes (this is for the case in which n of routes is more than or equal than 4)
3) Compare the chosen routes and take the route with lowest traveling time as the best route
Basically, I need to come up with a mathematical model to this problem and I would appreciate it if anyone could come up with a logical way to do so. Thank you :)
I have not understood completely what you want. From what I have understood, here is a way you can achieve the task.
Create a non uniform distribution such as below.
Now for the part I,
Let n be the number of routes
Then $P(n<3) = \sum_{i=0}^{3} P(Z_i)$
Then $P(n>=4) = \sum_{i=0}^{Roundup(.4*n)} P(rand(Z_i))$
Part II,
The following code shall enable you to choose at random any four routes.
r = rand()
If r lies between $A_{i}<=r<=A_{i+1}$
Choose $Z_{i+1}$
Repeat three more times for i's not chosen before.
Then do a bubblesort on the choosen routes based on the traveling time and choose the lowest. (Bubblesort will arrange the routes based on the traveling time in an ascending or descending order).
For i = 1 To 6
For j = i To 7
If (Time(i) <= Time(j)) Then
Temp = Time(i)
Time(i) = Time(j)
Time(j) = Temp
End If
Next
Next