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

 

Comments 2

  • Alan,

    Your QRS II is a god-send for us professors who want to teach students principles with a simple project. This is the 6th time I am using QRS II Demo in my class. I noticed that you have updated to version 9 and 8.3.

    Where can I download the manuals for the latest demo versions?

    Thanks.
    -Mohan

  • Full reference manuals to QRS II and GNE are contained in the demo setup packages. After setup there should be a link to each reference manual on the start menu. I do not anticipate updating the reference manuals for QRS II 9.1 and GNE 8.4 when ready. You will need to refer to supplemental documentation when these new releases become available. Also, please look at the readme file for QRS II 9, since it mentions some important actions that might be needed when upgrading networks from version 8.

    Alan