Note: *This is part 2 in a series of posts about mathematical modelling of public transit systems. If you haven’t already, I suggest you take a look at the first post where I try to motivate the need for, and validity of mathematical models in public transit.*

Another Note: *This post will make use of some basic calculus, however results will be explained in a way that does not require you to know calculus.*

TL;DR: *Looking for trade-offs in public transit allows us to balance competing factors and find an “optimal solution”. An example is given which calculates the optimal headway of buses on a route.*

### The Compromise of Public Transit

When looking at the design of public transit networks, there are a number of opposing factors that one is faced with. Often one has to make a decision about what balance of these factors to strike according to the needs and wants of the community these decisions affect. In his book *Human Transit*^{1}, Jarrett Walker calls these trade-offs “Plumber’s Questions”, since they are usually asked and facilitated by an expert (a plumber or a transit planner), but should be answered collectively by the people affected by the decision (the homeowner or the community). A couple of the trade-offs he presents:

*Ridership or coverage*– You can plan a transit system to maximize the number of people who ride, but that is done at the expense of not serving everyone.*Connections or complexity*– Your network can avoid making people transfer, but this comes at the expense of network frequency and therefore ride time.

In reality, there is a spectrum between both extremes, and most transit systems fall somewhere in between those opposing ideas. In essence, transit systems are built on compromises.

It is said that a good compromise leaves nobody truly happy. In a way, all public transit systems provide a service that is a compromise across many individuals. As a single person, we would like to begin our trip somewhere, and be transported as quickly and comfortably as possible to our destination. This is why the personal automobile is such a powerhouse: it provides the freedom to do just that. The only way for some other method of transportation to compete is to provide a service with similar flexibility and freedom, (something along the lines of Car2Go), or to aggregate the needs of many individuals and transport them all at a lower cost.

### Compromise Can Be Optimal

While *Human Transit* endeavors to enable the communities it serves with the tools to answer these difficult questions as best they can, mathematical modelling of public transit aims to convert these compromises into equations and from there find the “ideal” balance. This balance will often depend on a certain number of parameters which reflect how people place value on the system. Essentially, it is assigning weight to one side of the compromise or the other. The nice thing is, with a mathematical solution, you can see how changing these values can affect the system.

These solutions are usually considered to be “optimal”, in the *mathematical sense*. This does not mean that it is the best situation for everyone, or that common sense cannot be applied to the solution afterwards to make it practical and usable. For those familiar with calculus, the optimal solution is found by maximizing or minimizing some function (of travel time, total cost, etc.).

So what does one of these optimal solutions look like?

### Example – Optimal Headway

Headway is one of those concepts that is important in transit, but non-existent from a car-oriented perspective. Headway, which is the time between two consecutive buses on a route, is directly related to how closely it approximates the freedom enabled by a car. If a bus comes very often, it is there when you need it, it gives you the freedom to change your plans (an important factor in increasing transit use), and it means you can *trust *it. Those are some of the “seven demands of transit” that Walker outlines.

As a result, Walker suggests that having short headway (or *high frequency*) is the key to having consistent, high ridership. The temptation, then, is to lower headways as much as possible, subject to some form of budget constraint. It is possible, however, that there are some inefficiencies that appear if you make transit *too* frequent. We will explore this idea with this example.

One way of balancing the needs of passengers with the costs of operating transit is to find some sort of *total cost* for a system, in which two aspects compete with respect to some adjustable factor. In our case, the adjustable factor will be the headway of buses on a route, which will be denoted by *h*. In this simple example, developed by Dr. Chan Wirasinghe^{2}, the waiting time of passengers at a bus stop due to varying headway is balanced with the costs of dispatching buses. If the headway is *h*, and passengers arrive at random, they will on average wait a duration of *h/2* for their bus. Let us say that there are *p* passengers who want to take the bus at any particular point along the route, and they all value their time equally, at a value of *γ* *dollars per hour per passenger*. Additionally, there is a cost of dispatching a bus, which we will denote with *λ*, and the number of buses that we will dispatch is inversely related to the headway we choose. In this way, both terms will provide us a total cost in *dollars per hour*, which we will denote as *z*:

By minimizing this cost, we can find an *optimal* headway, which *minimizes the sum of passenger waiting costs and dispatching costs*. This may sound specific, but we can use calculus to find an optimal headway for this scenario, namely

Even with such a simplified model, some insights can be gained:

- The optimal headway is sensitive only to the
*square root*of the other factors. This means, for example, that in order to double the optimal headway, the dispatching cost would have to*quadruple*. This tells us that we can have some uncertainty of our measurement of these parameters without it greatly affecting the optimal headway. - The ratio of dispatching cost and the value of passenger time is a key component. If we value passenger time more (that is one way to try and attract ridership), headways will decrease, and frequency will go up.
- If ridership goes up (demand, or
*p*increases), it is efficient to reduce headways, effectively reducing the time spent by all these extra people waiting for the bus.

But can we gain some numerical reality from this equation? Let’s try it. Often, people value their time at about $20/hr, and let’s say it costs $150 to dispatch a bus over a route (most of that cost is driver’s wages). Let’s also say that 200 people/hour would like to ride this bus. In that case,

Or about 16 minutes. This is a *resonable number*, considering we only accounted for people’s waiting time, and a rough stab at what a bus dispatching cost might be.

Generally, the demand is the most domineering factor in this set-up. Large demand can drive headways down to practical limits. It is also, as Walker argues, influenced by the headway, since a lower headway can increase ridership. This self-referential type of relationship can be iterated until a suitable balance is found.

So that’s how the cost-based approach to modelling works. Thanks for reading!

### References

- Walker, J. (2012). Human Transit. Washington, D.C.:
*Island Press*. - Wirasinghe, S. C. (2003). Initial planning for urban transit systems. In W. H. Lam & M. G. H. Bell (Eds.),
*Advanced Modeling for Transit Operations and Service Planning*(pp. 1–29). New York.

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