Juan Ortuzar’s book indeed provides a pivot point formula that allows two or more choice alternatives to be varied at the same time. So a valid question may be raised: which method works best, pivot point or the gravity model, for OD table forecasts? This blog article extends the example problem to pivot point and compares the answers. As we will see, the selection of the method to be adopted hinges on the amount of error* in the original OD table and how well the gravity assumption matches the reality of the forecast.
[*I am using “error” to refer to any inaccuracies in measurement. I am using “mistakes” to refer to any failures or misapplications of theory.]
Recall the original question posed by Jim Bunch. He has before and after travel time matrices and an observed OD table. He wants to forecast the after OD table without using travel demand modeling software.
The pivot point formula, attributed to Kumar in 1980 and adapted to this application is:
P_{j} = p_{j}^{o}exp(bΔt_{j})/Σ_{k}p_{k}^{o}exp(bΔt_{k}) for choices across rows, and
P_{i} = p_{i}^{o}exp(bΔt_{i})/Σ_{m}p_{m}^{o}exp(bΔt_{m}) for choices down columns.
Here Δt is the difference between the after and before travel times, p is the probability of choosing the the ith or jth OD pair, p^{o} is the observed proportion of trips using the ith or jth OD pair, and b is a coefficient on travel time. The subscripts j and k are row indices and i and m are column indices.
Notice that the pivot point formula is somewhat simpler than the gravity model presented in the first article in this series.
The setup of this example is identical to the earlier one. QRS II is used to establish “true” before and after OD tables, considering a big speed limit increase on the UTown freeway. The analysis is broken out by trip purpose (HBW and HBNW+NHB) and by direction of travel (PtoA and AtoP) exactly as before.
In this instance the forecasted OD table calculates to these numbers.
3 Northern 
4 Northwestern 
2 Central 
1 Eastern 
5 Southwestern 

3 Northern 
44454 
5379 
8927 
16667 
9389 
4 Northwestern 
4953 
11228 
3129 
5982 
4429 
2 Central 
8951 
3100 
28626 
20444 
5686 
1 Eastern 
16610 
5896 
20320 
49329 
14607 
5 Southwestern 
9445 
4403 
5705 
14705 
20524 
And the sizes of the mistakes of this forecast are as so.
3 Northern 
4 Northwestern 
2 Central 
1 Eastern 
5 Southwestern 

3 Northern 
1 
269 
2 
97 
164 
4 Northwestern 
159 
51 
18 
353 
86 
2 Central 
23 
8 
13 
81 
135 
1 Eastern 
50 
399 
81 
188 
135 
5 Southwestern 
192 
51 
135 
136 
177 
Which compares with the earlier calculated mistakes from the gravity model, rounded.
3 Northern 
4 Northwestern 
2 Central 
1 Eastern 
5 Southwestern 

3 Northern 
0 
0 
0 
0 
0 
4 Northwestern 
0 
0 
0 
0 
0 
2 Central 
0 
0 
0 
0 
0 
1 Eastern 
0 
0 
0 
0 
0 
5 Southwestern 
0 
0 
0 
0 
0 
All zeros! The mistakes from the application of the gravity model were, as stated earlier, inconsequential.
Although the pivot point method was OK, the mistakes appear to be far worse than seen earlier with the gravity model. I am dismissing the possibility of a theoretical error in the pivot point equation derivation (which I have not checked) or in the spreadsheet calculations (which I have checked), so what gives?
To some extent the comparison of the two methods is biased in favor of the gravity model, but I would not have expected the pivot point technique to have done this much worse. A major reason for the differing qualities of the results relates to how each technique treats the errors in the given, before OD table. Remember that splitting the full OD table into four smaller tables introduces a lot of error in each of those smaller tables. The pivot point method thinks these errors are useful, while the gravity model pretty much steamrolls over anything it doesn’t like.
And the pivot point technique uses the OD proportions from each cell multiple ways. Hence any errors inherent in the OD table are amplified. The effect of any error on the gravity model technique is more muted, since it only utilizes the row and column totals. This by itself is not a reason to abandon the pivot point technique, but it is a good reason for caution. Most empirical OD tables have a lot of error. Do we want a technique that accentuates the error or do we want a technique that quiets the error?
Notwithstanding the better performance of the gravity model, there are tradeoffs. I am sure this is not the last word on this subject. Give me a few weeks to concoct a better test that more fairly demonstrates the effect of errors on these techniques.
Alan Horowitz, Whitefish Bay, April 26, 2017