# Trade-offs between travel time and monetary cost

#### 2025-03-08

Abstract

This vignette shows how to use the `pareto_frontier()` function to examine the trade-offs between travel time and monetary cost in travel time matrices in r5r.

# 1. Introduction

In most cases, transport routing models find either the fastest or the lowest-cost routes that connect places in a given transport network. Sometimes, though, we might want a more sophisticated analysis that considers both the time and monetary costs that public transport passengers have to face. The problem here is that simultaneously accounting for both time and monetary costs is a major challenge for routing models because of the trade-offs between the objectives of minimizing trip duration and cost (Conway and Stewart 2019).

To address this problem, `r5r` has a function called `pareto_frontier()`, which calculates the most efficient route possibilities between origin destination pairs considering multiple combinations of travel time and monetary costs. This vignette uses a reproducible example to demonstrate how to use `pareto_frontier()` and interpret its results.

## 2. What the `pareto_frontier` means.

Imagine a hypothetical journey from A to B. There are multiple route alternatives between this origin and destination with varying combinations of travel time and cost (figure below).

- Walking from A to B would be the cheapest option but it would take 50 minutes.
- The fastest option would be to take a bus to a subway station and transfer to the subway. This option would only take 15 minutes, but it would cost $8.
- There are some intermediary alternatives, such as taking:
  - a single bus, $3 for 35 min.
  - two buses with one transfer, $5 for 29 min.
  - taking the subway after walking to the station , $6 for 20 min.

This figure illustrates the Pareto frontier of alternative routes from A to B. In other words, it shows the most optimal set of route alternatives between A and B. There are certainly other route options, but there is no other option that is both faster and cheaper at the same time.

This kind of abstraction allows us to have a better grasp of the trade-offs between travel time and monetary cost passengers face when using public transport. It also allows us to calculate cumulative-opportunity accessibility metrics with cutoffs for both time and cost (e.g. the number of jobs reachable from a given origin with limits of 40 minutes and $5) (ref paper by Matt and Anson).

Let’s see a couple concrete examples showing how `r5r` can calculate the Pareto frontier for multiple origins.

## 3. Demonstration of `pareto_frontier()`.

### 3.1 Build routable transport network with `setup_r5()`

First, let’s build the network and create the routing inputs. In this example we’ll be using the a sample data set for the city of Porto Alegre (Brazil) included in `r5r`.

    # increase Java memory
    options(java.parameters = "-Xmx2G")

    # load libraries
    library(r5r)
    library(data.table)
    library(ggplot2)
    library(dplyr)

    # build a routable transport network with r5r
    data_path <- system.file("extdata/poa", package = "r5r")
    r5r_core <- setup_r5(data_path)

    # routing inputs
    mode <- c('walk', 'transit')
    max_trip_duration <- 90 # minutes

    # load origin/destination points of interest
    points <- fread(file.path(data_path, "poa_points_of_interest.csv"))

### 3.2 Set up the fare structure

Now we need to set what are the fare rules of our public transport system. These rules will be used by `R5` to calculate the monetary cost of alternative routes. In the case of Porto Alegre, the fare rules are as follows: * Each bus ticket costs R$ 4.80. * Riding a second bus adds `$` 2.40 to the total cost. Subsequent bus rides cost the full ticket price of $ 4.80. * Each train ticket costs $ 4.50. Once a passenger enters a train station, she can take an unlimited amount of train trips as long as she doesn’t leave a station. * The integrated fare between bus and train has a 10% discount, which totals $ 8.37.

We can create `list` object with these fare rules with the support of the `setup_fare_structure()` function as shown in the code below. A detailed explanation of how to use the fare structure of `5r5` can be found in (this other vignette).

    # create basic fare structure
    fare_structure <- setup_fare_structure(r5r_core,
                                           base_fare = 4.8,
                                           by = "MODE")

    # update the cost of bus and train fares
    fare_structure$fares_per_type[, fare := fcase(type == "BUS", 4.80,
                                                 type == "RAIL", 4.50)]

    # update the cost of tranfers
    fare_structure$fares_per_transfer[, fare := fcase(first_leg == "BUS" & second_leg == "BUS", 7.2,
                                                     first_leg != second_leg, 8.37)]

    # update transfer_time_allowance to 60 minutes
    fare_structure$transfer_time_allowance <- 60

    fare_structure$fares_per_type[type == "RAIL", unlimited_transfers := TRUE]
    fare_structure$fares_per_type[type == "RAIL", allow_same_route_transfer := TRUE]

For convenience, we can save these fare rules as a `zip` file and load again for a future application.

    # save fare rules to temp file
    temp_fares <- tempfile(pattern = "fares_poa", fileext = ".zip")
    r5r::write_fare_structure(fare_structure, file_path = temp_fares)

    fare_structure <- r5r::read_fare_structure(file.path(data_path, "fares/fares_poa.zip"))

### 3.3 Calculating a `pareto_frontier()`.

In this example, we calculate the Pareto frontier from all origins to all destinations considering multiple cutoffs of monetary costs: - $1, which would only allow for walking trips - $4.5, which would only allow for rail trips - $4.8, which would allow for a single bus trip - $7.20, which would allow for bus + bus - $8.37, which would allow for walking walking + bus + rail

    departure_datetime <- as.POSIXct("13-05-2019 14:00:00",
                                     format = "%d-%m-%Y %H:%M:%S")

    prtf <- pareto_frontier(r5r_core,
                          origins = points,
                          destinations = points,
                          mode = c("WALK", "TRANSIT"),
                          departure_datetime = departure_datetime,
                          fare_structure = fare_structure,
                          fare_cutoffs = c(1, 4.5, 4.8, 7.20, 8.37),
                          progress = TRUE
                          )
    head(prtf)

For the sake of illustration, let’s check the optimum route alternatives from the Farrapos train station to (a) the Praia de Belas shopping mall and (b) the Moinhos hospital. An optimum route alternative means that one cannot make a faster trip without increasing costs, and one cannot make a cheaper trip without increasing travel time.

### Cleaning up after usage

`r5r` objects are still allocated to any amount of memory previously set after they are done with their calculations. In order to remove an existing `r5r` object and reallocate the memory it had been using, we use the `stop_r5` function followed by a call to Java’s garbage collector, as follows:

    r5r::stop_r5(r5r_core)
    rJava::.jgc(R.gc = TRUE)

If you have any suggestions or want to report an error, please visit the package GitHub page.

## References

Conway, Matthew Wigginton, and Anson F. Stewart. 2019. “Getting Charlie Off the MTA: A Multiobjective Optimization Method to Account for Cost Constraints in Public Transit Accessibility Metrics.” International Journal of Geographical Information Science 33 (9): 1759–87. https://doi.org/10.1080/13658816.2019.1605075.