Explaining P-A matrix and O-D matrix in a class

>>>> In the above example, let's separate only HBW trips. Then P-A matrix is: {0,55,0; 0,0,0; 0,0,0} and O-D matrix is: {0,30,0; 25,0,0; 0,0,0} Even now the PA-to-OD method does not work.

In your HBW example, the PA-to-OD method works as

OD = 6/11 * PA + 5/11 * PA_transpose

One might think of it as a weighted average of the PA matrix and its transpose, or as temporal factors.
>>>> Then I realize - for the method to work, there should not be any tours.

Tours have historically been incorporated in trip-based models through the Non-Home-Based purpose, including subsets such as Work-Based-Other, School-Based-Other, and Other-Based-Other.

Scott Ramming, PhD PE | Senior Travel Modeler | Transportation Planning & Operations
Direct: 303-480-6711 | Fax: 303-480-6790 | Email: sramming@drcog.org

I usually explain P/A versus O/D in terms of how the concept emerged historically.

The original unit of data collection for trip generation (still "splattered" across the graphs in the ITE Trip Generation Manual) consisted of tube counts, with each click considered to be the "end" of some trip. More important, the counts did not include an estimate of directionality - a tube stretched across  a driveway can't tell if the trip is "coming" or "going".

To resolve the directionality, the basic trip-based model needs to assemble the separate ends into a pair of ends that define a physical trip.  But we do that in multiple stages.  The analytic approach is to understand first of all why those trips are taking place - the trip purpose.  The purpose associated with the trip ends actually has two components: first, the classic trip purpose (HBW, HBO, NHB, etc.) but second, the more basic relationship of transportation (which is that we travel in order to satisfy a need that we cannot meet where we are).

Without knowing any more than that we got a bunch of "clicks" on our counter, we can still say something about why the click was there - for example, at the mouth of a subdivision, we can presume (based on household surveys for instance) that people are traveling because something they needed was not available at home (e.g. ice cream, or a paycheck).  So we'll see one click when they left in search of that thing, and another click when they came back.  But in the world of trip generation, because this is the 1950s and we're using stupid tube counters, we don't know which trip is which (outbound or return). What we do know is that some trips are "produced" (i.e. generated because this zone has a "need" that is unmet at that location) and some are "attracted" (i.e. because this zone offers a certain amount of "satisfaction" for needs located elsewhere).

When we get to trip distribution (using P's and A's), the job is to make the connections between the trip ends by matching productions (need) to attractions (satisfaction).  There is no physical directionality implied in that connection: it's about allocation of demand, and not (yet) about making an actual trip.  For each trip end with a certain purpose in zone X, we'll find matching ends in zones P, Q, R, S etc.  We can further divide those up by mode (if we have a four-step model, rather than a three-step).

We don't worry about the direction of the trip until we start getting ready for trip assignment (mapping the demand onto physical trips on highway and transit networks).  That's where we'll roll out factors like vehicle occupancy and directionality by time of day.  In the simplest case (divide P-A matrix by two, transpose, and add) we just assume that half the demand is an "outbound" trip (P traveling to A) and half is "inbound" (A travelling to P). In more sophisticated models with daily time periods, we'll assign directionality using different factors in different time periods before we transpose and add.  That will probably yield more HBW P-A direction trips in the morning and more A-P direction trips in the afternoon - though there are still some A-P trips in the morning and some P-A trips in the afternoon. For NHB trips, we'll almost always presume a 50/50 split on the assumption that half the trip ends happen because someone has a need at that location (production), and half happen because a need is being satisfied there (attraction).

Finally, I'll observe that while we usually presume that within each overarching purpose (HBW, NHB, etc.) every trip "out" is matched by a trip "in", that is absolutely not a requirement of the four-step conceptual framework.  The bookkeeping requirement is that when you add up all trips by all purposes, each zone "gives" as well as it "takes" (i.e. over all purposes, trips out balances with trips in).  So for example, we might set up the factors so a majority of the HBW trips end up going from P to A (and mostly in the morning), with most of the HBOther trips going from A to P in the afternoon.  In practice, however, that's almost impossible to work out consistently (and to justify effectively), which is part of the reason why (a) almost nobody does it; and (b) people gravitate toward tour-based models where the structure of the tours takes care of generating (and explaining) the asymmetrical trip factors.

I hope that's helpful!

Jeremy Raw
FHWA Office of Planning

The original question (shown in the email below) was:
“And what is the best way to inform the students about this issue without confusing them so that they don't lose interest.”

I think it may be useful to help students understand that use of a production-attraction format for trip generation, distribution and mode choice is indeed a simplification for representation of real-world person travel that will sometimes have two, three or even more “different places” that are reached between when a traveler leaves and gets back home. But just because a PA-format trip-based model is “simpler” than other model constructs does not automatically mean that it will therefore be “less useful” in the preparation of plausible forecasts to support a local decision-making process, particularly if the “simpler” model has a clear connection to “data” (ideally “data” that goes beyond just a model calibration year) and is being used by a “forecaster” (as opposed to the “just run the model and summarize the outputs” person) who understands the limitations of any particular model application.

My intent is not to “defend” PA-format trip-based models over tour-based models, but rather to point out the importance of both “meaningful data” and “knowledgeable forecasters,” regardless of model structure. For ridership forecasting, an example of “meaningful data” these days is still a “good” transit rider survey that can be used to perform district-to-district transit passenger flow checks. An example of a “good forecaster” would be one who realizes that instead of simply preparing and summarizing a 30-year forecast, and proclaiming “here is what the model output says,” to find the “coherent story” that goes back to the model calibration year.

From: bushan515=yahoo.com@mg.tmip.org [mailto:bushan515=yahoo.com@mg.tmip.org] On Behalf Of bushan515@yahoo.com
Sent: Friday, June 14, 2019 10:15 PM
To: TMIP
Subject: [TMIP] Explaining P-A matrix and O-D matrix in a class

I am teaching four step travel demand modelling to my undergraduate students as elective.

I always wonder what would be the best way to teach the topic on P-A matrix and O-D matrix. Currently I teach the difference between production and attraction, and origin and destination while I introducing trip generation. I give an example where 2 trips travel from home to work and return. Then I will show P-A matrix and O-D matrix for this case. 0,4;0,0 vs 0,2;2,0. Then I can say that O-D matrix is relevant to real world because everyone understands start and end of trip. But in our modelling we start with Productions and attractions and so conceptually we develop P-A matrix. There is a need for transformation from P-A to O-D. Then I give them one easiest way of transforming: Average of P-A matrix and its transpose. But I tell them that this is only one method and in reality it could be more complicated. To tell you the truth, I know that this is not a correct method most of the time but I have never used any other method.

In fact I have once faced difficulty in class when I gave a three zone example: Zone 1- all homes; Zone 2 - all offices; Zone 3 - all shops. Then say, morning 30 trips started from home to work; evening 25 trips returned from office to home; 5 trips went to shop and returned to home after that. P-A matrix is: {0,55,5; 0,0,5; 0,0,0} O-D Matrix is: {0,30,0; 25,0,5; 5,0,0} My method does not work here at all. This is a realistic case.

In the above example, let's separate only HBW trips. Then P-A matrix is: {0,55,0; 0,0,0; 0,0,0} and O-D matrix is: {0,30,0; 25,0,0; 0,0,0} Even now the PA-to-OD method does not work.

Then I realize - for the method to work, there should not be any tours. OR, we should separate only those trips which start from home and return to home and apply the method. In a small example like above, we can do it. But in a full citywide model, how is it done?

What is the correct procedure?

And what is the best way to inform the students about this issue without confusing them so that they don't lose interest.
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