$$
For our purposes today, it is assumed that the first hub is going directly to the second hub, without an intermediary hubs along the way. So:
\[ T_{h_1 \rightarrow h_2} = T_{N(h_1 \rightarrow h_2)} \]
We’re also going to assume for today that no one is interested in making more than 2 transfers. I think this is a reasonable assumption to make. There’s some fraction of the populous who at least doesn’t mind transfers. But you’d be pretty hard pressed to find someone who likes them. This may seem like a count against my proposal, but:
a: for most people, I imagine, shorter travel time is worth the trouble of transferring.
b: for those of which it isn’t, local routes would necessarily still be available.
(I also don’t subscribe to the idea that bus users are poor they therefore aren’t going to abandon the system if they need to transfer more. For a litany of reasons; chief among them is the fact that you’re setting up the transit system to fail if/when prosperity is gained)
we can start by putting in some arbitrary values for our variables.
dwell_time <- 15
stopping_delay <- 0.5#writing a function for the distance in local route
local_route_time <- function(
time_between_stops= 1,
n=length(time_between_stops),
stopping_delay=0.5,
dwell_time=15,
direct_time=n*time_between_stops
) {
return((dwell_time / 2) # averaging the dwell time
+ sum(time_between_stops)
+ n*stopping_delay) # and the amount of additional time each stop creates
}ex_time_vector <- c(3, 4, 5, 2, 4, 5, 4)route_time <- function(
stops_to_first_hub=0,
stops_to_second_hub=0,
stopping_delay=0.5,
dwell_time=15,
time_between_stops= 1,
hub_transit_time
) {
r_time <- local_route_time(
n=stops_to_first_hub,
stopping_delay=stopping_delay,
dwell_time=dwell_time,
time_between_stops=time_between_stops) +
hub_transit_time +
local_route_time(
n=stops_to_second_hub,
stopping_delay=stopping_delay,
dwell_time=dwell_time,
time_between_stops=time_between_stops)
return(r_time)
}lets start by considering the simplest example that we can muster. Imagine a straight line, with equally spaced stops along it. Where do you put the two hubs? Or more specifically, how many stops between the hubs
local_stops <- 700
to_hub_vector <- 2
from_hub_vector <- 4
time_between_stops <- 1
hub_to_hub_vector <- local_stops + to_hub_vector + from_hub_vector
hub_transit_time <- hub_to_hub_vector*time_between_stops
route_time(to_hub_vector, from_hub_vector, time_between_stops=c(1), hub_transit_time=hub_transit_time) %>% print()## [1] 726local_route_time(n=local_stops)## [1] 358.5# saving my default values
#local_stops=0
# stops_to_first_hub=0
# stops_to_second_hub=0
# stopping_delay=0.5
# dwell_time=15
# time_between_stops= 1
# to_hub_vector=0
# from_hub_vector=0
# hub_transit_time=local_stops + to_hub_vector + from_hub_vectorhub_advantage <- function(v) {
# returns the advantage of using a hub route over a route with multiple stops along the way
print("hello function")
local_stops=v[1]
stops_to_first_hub=v[2]
stops_to_second_hub=v[3]
stopping_delay=v[4]
dwell_time=v[5]
time_between_stops=v[6]
to_hub_vector=v[7]
from_hub_vector=v[8]
tryCatch(
{hub_transit_time=v[9]},
error = function(e) {
hub_transit_time=local_stops + to_hub_vector + from_hub_vector
}
)
print(v)
hub_route_time <- route_time(
stops_to_first_hub=stops_to_first_hub,
stops_to_second_hub=stops_to_second_hub,
stopping_delay=stopping_delay,
dwell_time=dwell_time,
time_between_stops=time_between_stops,
hub_transit_time=hub_transit_time
)
local_route_time <- local_route_time(
n=local_stops,
stopping_delay=stopping_delay,
time_between_stops=time_between_stops,
dwell_time=dwell_time,
)
# takes the time it would take to stop along the way and finds the difference
return(local_route_time - hub_route_time)
}hub_advantage(c(1,2,3,4,5,6,7,8,9)) %>% print()## [1] "hello function"
## [1] 1 2 3 4 5 6 7 8 9
## [1] -33.5hub_advantage.tester <- function(v) {
hub_route_time <- 9
# takes the time it would take to stop along the way and finds the difference
return(local_route_time() - hub_route_time)
}
print(hub_advantage.tester())## [1] 0number_of_stops = 1:50
df <- data_frame(
number_of_stops = number_of_stops,
hub_advantage = number_of_stops %>% hub_advantage()
)## Warning: `data_frame()` was deprecated in tibble 1.1.0.
## ℹ Please use `tibble()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## [1] "hello function"
## [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## [26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50df## # A tibble: 50 × 2
## number_of_stops hub_advantage
## <int> <dbl>
## 1 1 -33.5
## 2 2 -33.5
## 3 3 -33.5
## 4 4 -33.5
## 5 5 -33.5
## 6 6 -33.5
## 7 7 -33.5
## 8 8 -33.5
## 9 9 -33.5
## 10 10 -33.5
## # ℹ 40 more rows# TODO: understand why the hub_advantage keeps coming up with NA#uniroot(hub_advantage, interval = c(1, 50))$rootnumber_of_stops <- 1:50
dwell_time <- 15:60
#the_hub_advantage = sapply(number_of_stops, function(x) hub_advantage(local_stops = x))
#the_hub_advantage
testing_dwell_time <- function(dwell_time = 15:60) {
sapply(dwell_time, uniroot(hub_advantage(dwell_time = dwell_time), interval = dwell_time)) %>% return()
}
#testing_dwell_time() %>% print()
#sapply(dwell_time, uniroot(hub_advantage, interval = c(1, 50))$root)set_param_hub_adv <- function(first_value) {
if(first_value == T) {
}
lapply()
#hub_advantage(v)
}
#set_param_hub_adv()## [1] 4 8 12 5 10 15