A Well Choreographed OD Table Dance, Part 2: Example

This blog post provides a numerical example for the previous blog post.  Recall that we are given a before OD table, a before travel time matrix and an after travel time matrix.  We want to find the after OD table.

A further goal of this problem is to avoid fancy software, so I have attempted to illustrate this technique solely in Excel.  However, I later demonstrated the strength and weaknesses of the technique by comparing its results to travel demand forecasting software.  I used the very familiar UTown test network in QRS II, starting with the outputted OD table and travel time matrix that comes from a “before” alternative of the model.  The modification for the “after” alternative was a 15% increment of free speeds on the UTown freeway.

You have likely seen UTown in years past, but here is what it looks like in QRS II.

There are 5 zones, and there are many congested links. Should you want to replicate the complete example and the comparison, you can do so easily with the free demo edition of QRS II, downloadable from ajhassoc.com.  The UTown network is included.  Note:  QRS II rounds travel times to the nearest whole minute for trip distribution calculations, so we will too in the spreadsheet.

The Data

QRS II was used for creating the “true” before case and the after travel time matrix.  QRS II’s default trip distribution step is a doubly-constrained gravity model.  All parameters were QRS II defaults, except I substituted 0.07 for the HBW coefficient of the gravity model, as suggested in Part 1 of this blog.  I also ran 30 iterations of MSA with feedback.  The UTown network was originally set up for a PM peak hour (much preferred), but it was changed to a 24-hour time period for this example.  UTown has three trip purposes – HBW, HBNW and NHB.  The before OD table, rounded, is shown just below.

	3    Northern	4    Northwestern	2    Central	1    Eastern	5    Southwestern
3    Northern	45035	4927	9033	16753	8723
4    Northwestern	4929	11101	3110	6489	4441
2    Central	9032	3107	28849	20468	5261
1    Eastern	16649	6449	20344	49062	14750
5    Southwestern	8750	4451	5279	14849	21046

Note that this table is almost symmetrical, as expected. All spreadsheet calculations were performed using unrounded trips so that I could get a clean evaluation of the technique.

The rounded, before travel time matrix is shown next.

	3    Northern	4    Northwestern	2    Central	1    Eastern	5    Southwestern
3    Northern	6	23	21	30	19
4    Northwestern	22	6	24	33	17
2    Central	20	25	5	24	22
1    Eastern	27	31	22	6	28
5    Southwestern	24	24	27	36	7

 And finally, the after travel time matrix is shown, also rounded.

	3    Northern	4    Northwestern	2    Central	1    Eastern	5    Southwestern
3    Northern	6	22	21	30	18
4    Northwestern	22	6	24	34	17
2    Central	20	25	5	24	21
1    Eastern	27	32	22	6	28
5    Southwestern	23	24	26	36	7

 Only a few cells are affected by the speed change, given the rounding.

The Model

I factored the before OD table into 4 component tables: HBW, P-A; HBW, A-P; HBNW+NHB, P-A; and HBNW+NHB, A-P.  Simple constant factors of 0.1, 0.1, 0.4, and 0.4, respectively, were used.  Singly-constrained gravity models (exponential form) were fitted to each table, conventionally.  The process for the A-P tables were almost exactly the transpose of the P-A tables.  That is, for the P-A tables row totals were held constant and for the A-P tables column totals were held constant. For example, the computed HBW, P-A table is shown below.

	3    Northern	4    Northwestern	2    Central	1    Eastern	5    Southwestern
3    Northern	4491	486	1241	1067	1161
4    Northwestern	703	767	482	415	641
2    Central	1327	333	2993	1279	741
1    Eastern	1257	338	1408	6970	753
5    Southwestern	1167	415	747	643	2465

Observe that symmetry was no longer preserved, but symmetry is never expected in a P to A table or an A to P table, anyway.

The after case was calculated by changing the travel time matrix.  The total before and total after OD matrices were then found by summing across purposes and by summing across P-A and A-P.

The Adjustments

Below is the total calculated before OD table.

	3    Northern	4    Northwestern	2    Central	1    Eastern	5    Southwestern
3    Northern	53049	5237	10427	7219	10910
4    Northwestern	5343	10828	3513	2434	5739
2    Central	11237	3402	37247	9816	6558
1    Eastern	9211	3036	11838	82654	5796
5    Southwestern	8066	3628	4873	3401	27426

Comparing the calculated and “true” before OD matrices revealed some large errors, particularly for the Eastern Zone, but the calculations seemed good enough to give reasonable sensitivities, so I continued.

Next I found the ratio of the calculated after table to the calculated before table. Numbers greater than 1 represent forecasted increases.

	3    Northern	4    Northwestern	2    Central	1    Eastern	5    Southwestern
3    Northern	0.987109	1.037067	0.988521	1.000678	1.057502
4    Northwestern	1.037	1.006882	1.000194	0.976218	0.977942
2    Central	0.988516	1.000166	0.991816	1.002809	1.055211
1    Eastern	1.000671	0.97618	1.002801	1.001599	0.999468
5    Southwestern	1.057384	0.97781	1.055104	0.999466	0.966808

Notice that the largest fractional increases corresponded to those cells that experienced significant travel time decreases.  The remaining cells changed seemingly at random, but nothing is truly random in this problem.

The Forecast

The ratios were then applied to the original “true” OD table, recalling that the only relevant output of this exercise was the set of cell adjustment factors. Here, finally, is the computed after OD table.

	3    Northern	4    Northwestern	2    Central	1    Eastern	5    Southwestern
3    Northern	44455	5110	8929	16764	9225
4    Northwestern	5112	11177	3111	6335	4343
2    Central	8928	3108	28613	20525	5551
1    Eastern	16660	6295	20401	49141	14742
5    Southwestern	9253	4352	5570	14841	20347

 Since, the total of all trips closely matches the “true” OD table, further scaling was not needed.

The Evaluation

When the QRS II after OD table (being the “true” result) was compared with the spreadsheet calculated OD table, the largest deviation in any cell value was less the 0.5 trip (a negligible amount).  Thus, one could reasonably conclude that this techniques is a decent substitute for a full travel demand forecasting model run, with some provisos.  Particularly important is the need for a good set of friction factors.  In this experiment the good quality of the friction factors was assured, so it is likely that truly borrowed friction factors will not perform quite as well.

Perhaps this technique should be added to a revised NCHRP Report 765 or the Hawaii Guidelines.

Alan Horowitz, Whitefish Bay, March 29, 2017