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Quick Response Freight Manual: Final Report

4.0 Incorporating Commercial Vehicles into the Travel Forecasting Process

4.1 Introduction

This chapter describes a simplified quick-response procedure for incorporating commercial vehicles into the travel forecasting processes used by Metropolitan Planning Organizations, State Departments of Transportation, and other planning agencies. This chapter also provides alternative approaches that might be used if more data are available (or can be collected) and more accuracy is desired.

The procedure produces trip tables that can be assigned to highway networks for three classes of commercial vehicles:

Figure 4.1 shows the simplified quick-response procedure as consisting of the following steps:

  1. Obtain data on economic activity for traffic analysis zones (including employment by type and the number of households),
  2. Apply trip generation rates to estimate the number of commercial vehicle trip destinations for each traffic analysis zone,
  3. Estimate commercial vehicle volumes at external stations,
  4. Estimate the number of commercial vehicle trips between pairs of traffic analysis zones or external stations,
  5. Develop a preliminary estimate of commercial vehicle VMT by assigning trips to a network (or using a table of zone-to-zone distances),
  6. Develop control totals for commercial VMT based upon (1) estimates of total VMT in the region for each functional class, and (2) vehicle classification data indicating the percentage of total VMT associated with commercial vehicles
  7. Compare the results of Step 5 and Step 6, and, if necessary, develop adjustment factors to trip generation rates or trip distribution factors.

Figure 4.1 Simplified Quick Response Freight Forecasting Procedure

Steps 2 through 7 are repeated until the estimates of commercial vehicle VMT developed in Step 5 is reasonably close to the control totals developed in Step 6. The following sections describe each of these steps. A hypothetical example is included to illustrate the procedures.

Finally, a section on time-of-day characteristics discusses the temporal distribution of travel by commercial vehicles.

4.2 Trip Generation

In the quick-response procedure, the number of commercial vehicle destinations per day in each traffic analysis zone is calculated by:

The trip generation rates shown in Table 4.1 are for trip destinations (which, on an average day, are equal to trip origins). These rates were taken from a Phoenix, Arizona study.1,2 The Phoenix study results are used as the basis for default values because they provide an internally-consistent set of trip generation rates and trip times, compared with potentially inaccurate rates derived from mixing results from a large number of studies in which the exact trip generations, vehicle definitions and employment categories used are mostly unclear or unknown. Appendix D, however, contains site-specific trip generation rates and regression equations that a user may find more suitable for a particular state or region being analyzed.

Table 4.1 Trip Generation Rates

Generator Commercial Vehicle Trip Destinations (or Origins) per Unit per Day
Four -Tire Vehicles Single Unit Trucks (6+ Tires) Combinations TOTAL
Employment:*
•
Agriculture, Mining and Construction
1.110 0.289 0.174 1.573
•
Manufacturing, Transportation, Communications, Utilities and Wholesale Trade
0.938 0.242 0.104 1.284
•
Retail Trade
0.888 0.253 0.065 1.206
•
Office and Services
0.437 0.068 0.009 0.514
Households 0.251 0.099 0.038 0.388
*If employment data is available only in terms of retail and non-retail employment, the trip generation rates shown above for non-retail employment should be weighted by the following national employment average percentages: (1) Agriculture, Mining and Construction - 10.9%; (2) Manufacturing, Transportation, Communications, Utilities and Wholesale Trade - 29.5%, (3) Office and Services - 59.6%.

Information on the number of households and employees by traffic analysis zone may be available to States or metropolitan/regional planning agencies through local data used for passenger transportation planning. If not, other sources and methodologies may be used including allocation of business-specific county or zip code data to census tracts.

For example, County Business Patterns presents county-level data on establishments (total and by employment size class) as well as total employment by SIC code (see Appendix C for listing and description of SIC codes). This data is tabulated by industry as defined in the 1987 Standard Industrial Classification (SIC) Manual. This tabulation is consistent with the classification methods used to create the default values in Table 4.1.

The same categories may also be obtained at the zip code level by special order from the Bureau of Census.3

County Business Patterns examines activity by specific sites or establishments. An establishment is a single physical location at which business is conducted or services or industrial operations are performed. It is not necessarily identical with a company or enterprise, which may consist of one or more establishments. When two or more activities are carried on at a single location under a single ownership, all activities generally are grouped together as a single establishment. For example, the administrative and shipping personnel of a manufacturing facility will be classified as manufacturing. However, administrative and auxiliary establishments that primarily manage or support the activities of other establishments of the same company (such as the headquarters of a multi-establishment conglomerate) are shown separately by industry division.

County Business Pattern data by major SIC code and by zip code may be allocated to associated census tracts where there is reasonable correspondence, based on total employment in each census tract from the Census Transportation Planning Package (CTPP). Comparisons of total employment between the two data sources should be performed, recognizing that County Business Patterns excludes self-employed and government workers, and that CTPP includes 1990 Census data. Local knowledge should also be employed to fine-tune allocations. For example, if County Business Patterns identifies 2,000 persons engaged in manufacturing employment in a zip code area, and the regional planning organization knows that the manufacturing site in a single census tract employs approximately that many people, it is appropriate to allocate the zip code manufacturing employment to that single census tract, rather than distribute it among all census tracts in the zip code area.

A general caution in using Bureau of Economic Analysis (BEA) or Census data for employment is that in some cases a headquarters office or central administrative facility is used as the address for dispersed activities such as construction, transportation, or electric and gas utilities. Data are excluded for self-employed persons, domestic service workers, railroad employees, agricultural production workers, most government employees, and employees on ocean-borne vessels or in foreign countries.

If the planning agency cannot secure employment by business type at the regional, county or zip code level, analysis can be performed using total employment by census tract. Under this method (not recommended for quick-response freight planning) total employment by census tract is multiplied by the average trip generation rate per employee. Employment by census tract is available from the CTPP. It includes tabulations by small areas of work, which are traffic analysis zones (TAZ's) in areas where the MPO supplied block-to-zone correspondence (Part 2 in MPO tabulations) and tabulations by census tract of work (Part 7 in MPO tabulations). Tabulations for CTPP do not include business type.

To achieve a higher level of accuracy, it is necessary to identify employment by type and by census tract or TAZ, rather than allocate from zip code or county level. If even greater accuracy is desired, commercial data sources can provide detailed information in various formats, including employment by census tract and by SIC code (see Chapter 6 and Appendix L for details).

4.2.1 Example

Figure 4.2 shows a hypothetical study area consisting of three traffic analysis zones (TAZ's) and a number of major radial and circumferential highways. Four external stations have also been designated for the study area as shown.

Data on employment and number of households for each zone in the study area are given in the table below.

Number of Households and Employees in Each Zone

Land Use Type
Zone
Z1 Z2 Z3
No. of Households
 3,120 4,364 5,985
No. of Employees:
 
Agriculture, Mining and Construction
0 0 0
 
Manufacturing, Transportation/ Communications/Utilities and Wholesale Trade
6,241 9,362 20,209
 
Retail Trade
8,916 17,831 7,430
 
Office and Services
23,775 8,916 5,944
Total Employment
38,932 36,109 33,583

Multiplying the numbers in this table with the trip generation rates in Table 4.1 gives the estimated number of commercial vehicle destinations per day for each vehicle type in each zone. For example, the estimated number of 4-tire commercial vehicle destinations generated by office/services employees for Zone Z1 is:

= [23,775] * [0.437] = 10,390 destinations/day.

Figure 4.2 Map of Hypothetical Study Area for the Example

Figure 4.2

The estimated total daily commercial vehicle destinations generated for the vehicle types and land use classification in the three zones are shown in the tables below.

Estimated Total Daily 4-Tire Commercial Vehicle Destinations Generated

Land Use Type
Zone
Z1 Z2 Z3
Households
 783 1,095 1,502
Employees:
 
Agriculture, Mining and Construction
0 0 0
 
Manufacturing, Transportation/ Communications/Utilities and Wholesale Trade
5,854 8,782 18,956
 
Retail Trade
7,917 15,834 6,598
 
Office and Services
10,390 3,896 2,598
TOTAL
24,944 29,607 29,654

Estimated Total Daily Single Unit (6+ tire) Commercial Vehicle Destinations Generated in Each Zone

Land Use Type
Zone
Z1 Z2 Z3
Households
309 432 593
Employees:
 
Agriculture, Mining and Construction
0 0 0
 
Manufacturing, Transportation/ Communications/Utilities and Wholesale Trade
1,510 2,266 4,891
 
Retail Trade
2,256 4,511 1,880
 
Office and Services
1,617 606 404
TOTAL
5,692 7,815 7,767

Estimated Total Daily Combination Vehicle Destinations Generated in Each Zone

Land Use Type
Zone
Z1 Z2 Z3
Households
119 166 227
Employees:
 
Agriculture, Mining and Construction
0 0 0
 
Manufacturing, Transportation/ Communications/Utilities and Wholesale Trade
649 974 2,102
 
Retail Trade
580 1,159 483
 
Office and Services
214 80 53
TOTAL
1,561 2,379 2,866

The total estimated daily commercial vehicle destinations generated for each land use type in each zone and the total trips for all zones are shown in the table below.

Estimated Total Daily Commercial Vehicle Destinations Generated for Each Vehicle Type and Zone

Vehicle Type Zone TOTAL
Z1 Z2 Z3
4-Tire Trucks
24,944 29,607 29,654 84,205
Single Unit (6+ Tire) Trucks
 5,692 7,815 7,767 21,274
Combination Vehicles
1,561 2,379 2,866 6,806
All Commercial Vehicles
 32,197 39,801 40,287 112,285

4.2.2 Alternative Approaches

As stated above, the trip generation rates proposed in the quick response method were derived from Phoenix, Arizona data. In many situations one would want to use site-specific trip generation rates particularly if the site characteristics are very much different from the Phoenix area. In addition, the trip generation rates presented in Table 4.1 are for four groups of land use or industrial classification. Each group pertains to several specific land use and employment characteristics (see Appendix C for Standard Industrial Classification (SIC) codes). More accurate estimates of commercial vehicle trips can be obtained using trip generation rates that correspond to specific land use or industrial classification, if the employment data as well as trip generation rates exist for the specific employment category.

Tables D-1a through D-1d in Appendix D contain trip generation rates (per employee) gathered from a large number of locations throughout the United States and Australia.4 The tables are arranged according to the four groups of SIC codes pertaining to the land use classification in Table 4.1. Specific SIC codes for some trip generation rates in many locations have been identified (e.g. SIC 42 for Truck Transportation). This information can be very useful in detailed site analysis and planning for specific types of establishments, and for more accurate estimation of commercial vehicle trip generation in a traffic analysis zone. Land use types that could not be classified under any one of the SIC codes are shown in Table D-1e.

Chapter 6 also presents other data collection methods and data sources pertaining to truck trip generation. Chapter 5 also describes procedures for estimating trip generation rates for major intermodal facilities and other special trip generators.

If employment data are not available for estimating commercial vehicle trips, other measures of economic activity such as total floor space or total land area devoted to specific employment categories can be used. Trip generation rates per one thousand square feet (TSF) and per acre of various employment (SIC) categories are shown in Table D-2b through D-2e, and Table D-3a through D-3e, respectively. These tables are arranged according to SIC codes (similar to Table D-1). Tables D-2e and D-3e contain trip generation rates for sites whose land use category cannot be classified under any one of the SIC codes.

More elaborate procedures (compared to the one-variable, fixed-rate approach in the quick response method) for predicting commercial vehicle trips involve various forms of equations as well as more than one independent variable. These equations have been developed and calibrated using a variety of estimation techniques, most commonly the ordinary least squares regression. Table D-4a through D-4e summarize some of the regression equations developed from various site studies and which can be used for predicting number of commercial vehicle trips as a function of one or more variables. If the required information exists, these equations can produce more accurate trip generation estimates compared with the simple fixed-rate approach.

4.3 External Stations

Most metropolitan area and regional travel forecasting networks include external stations through which trips with one or both ends outside the study area are loaded onto the network. Trips through external stations include:

These trips are usually classified into one of the following four categories:

  1. Passenger vehicles (which may be subdivided into Light vehicles and Buses),
  2. Four-tire commercial vehicles,
  3. Single unit trucks with six or more tires, and
  4. Combination vehicles.

In the quick-response procedure, commercial vehicle volumes at external stations may be estimated by applying percentages to estimate volumes for each of the three commercial vehicle classes based on the functional classification of the highway.

In some cases, however, a comprehensive data gathering effort to determine actual volumes and vehicle classifications at external stations (possibly including an origin-destination survey, see Chapter 6) may be warranted as a means for estimating volumes for each of the three commercial vehicle classes. Such effort will be particularly useful for a small study area and/or an area with significant volumes of through trips. Sources of actual data include traffic counts using field surveys, weigh-in-motion equipment or pneumatic tubes. Field counts will generally include truck counts by truck class (axles/weight or both), site, roadway type and time of day, usually accumulated as parts of other studies from one or combinations of the following:

It may be necessary to perform new counts on major external stations with old, suspect or missing data. Select data from sites near the border of the region in question, preferably without intervening roads to add or divert traffic. If data are available for a broad representation of lane and highway classifications, it is possible to expand the data to lanes and highways that were not sampled.

Agencies should be alert to a number of cautions and potential definitional problems in all counts related to freight movement, both internal and external. Research suggests wide variances in truck counts or classifications based on tube counts due to equipment calibrations, vehicle speed and traffic density. Therefore caution should be exercised in applying tube counts for vehicle classification to the model. Weigh-in-motion data, and even some visual classification schemes, may not clearly identify small commercial vehicles from other four-tired vehicles such as autos and vans. Further, truck classifications such as pickups, mini-vans and panel or full-sized vans are used for personal transportation as well as business applications. A key parameter of freight forecasting is to identify and model vehicle trips which are not typically captured in a household survey. If the survey or classification method does not clearly distinguish between personal and commercial vehicle use, then counts of pickups, mini-vans and panel or full vans should be discounted by the number of vehicles used for personal transportation. The 1992 Truck Inventory and Use Survey (TIUS) identifies the national average commercial use percentages for these vehicle types:

As illustrated below, these percentages can be applied to counts of four-tire trucks to estimate commercial vehicle traffic:

Type Total Count Percent Commercial 4-Tire Commercial Vehicle
Pickups
1,200 32.2% 386
Mini-vans
500 25.0% 125
Panels or Vans
400 45.7% 183
TOTAL
2,100   694

If it is not practical to conduct traffic count and classification studies at external stations, the default percentages shown in Table 4.2 may be used to obtain estimates of volumes at external stations for each of the three commercial vehicle classes. The percentages are based on: (1) vehicle classification data collected by States and compiled by the Federal Highway Administration, and (2) information from the Bureau of the Census' Truck Inventory and Use Survey (TIUS) on the use of light trucks.5

Table 4.2 Percent Distribution of Traffic by Vehicle Class

Functional Class Non-Commercial Vehicles Commercial Vehicles Total
Four-Tire Single Unit Combination
RURAL
Interstate
81.6% 3.3% 2.9% 12.2% 100%
Other Principal Arterials
87.2% 4.7% 3.2% 4.9% 100%
Minor Arterial, Collector and Local
88.5% 5.3% 3.6% 2.6% 100%
Average - Rural
86.6% 4.7% 3.4% 5.3% 100%
URBAN
Interstate
 88.2% 5.5% 1.8% 4.5% 100%
Other Freeways and Expressways
90.5% 5.5% 1.7% 2.3% 100%
Other Principal Arterials
89.5% 6.6% 1.7% 2.2% 100%
Minor Arterials
90.4% 6.4% 1.7% 1.5% 100%
Collectors
90.3% 6.4% 1.8% 1.5% 100%
Local
91.0% 6.4% 1.8% 0.8% 100%
Average - Urban
89.8% 6.2% 1.7% 2.3% 100%
Source: Vehicle Classification Data of FHWA and Census' Truck Inventory User Survey.

If data on average daily traffic for all vehicles at one or more external stations are not available, data on annual average daily traffic per lane from the Highway Performance Monitoring System (HPMS) database can be used to estimate AADT (see Table 4.3 below). The minimum information needed to accomplish this method is an inventory of the number of lanes by functional class, classified as urban or rural, for all highways where the stations are located. However, it should be noted that the variability in AADT per lane across facilities and geographic areas is huge, and estimates of AADT based on the default values given in Table 4.3 could be off by an order of magnitude, as indicated by the 10th percentile and 90th percentile values of traffic volumes in the HPMS database as shown in the table.

Table 4.3 Annual Average Daily Traffic (AADT) per Lane

Functional Class 2-Lanes 4-Lanes 6-Lanes 8-Lanes 10-Lanes
RURAL
Interstate
Average
2,581 4,251 8,500 9,004 ----
10th %-ile
304 1,493 4,613 5,888 ----
90th %-ile
20,355 7,325 13,299 15,788 ----
Other Principal Arterial
Average
2,268 3,159 7,100 ---- ----
10th %-ile
671 975 3,416 ---- ----
90th %-ile
4,432 6,425 9,546 ---- ----
Minor Arterial
Average
1,758 2,752 7,878 ---- ----
10th %-ile
335 712 5,047 ---- ----
90th %-ile
3,900 5,518 16,533 ---- ----
Major Collector
Average
1,062 2,774 4,970 ---- ----
10th %-ile
84 650 2,183 ---- ----
90th %-ile
2,665 5,909 8,167 ---- ----
Minor Collector
Average
407 926 ---- ---- ----
10th %-ile
24 79 ---- ---- ----
90th %-ile
1,035 2,500 ---- ---- ----
URBAN
Interstate
Average
8,321 8,649 12,940 15,700 16,654
10th %-ile
3,115 3,020 6,249 8,160 10,579
90th %-ile
15,300 15,063 21,000 23,865 23,420
Other Freeways/Expressways
Average
6,887 7,448 11,932 17,084 19,145
10th %-ile
2,420 2,495 4,140 7,000 14,330
90th %-ile
13,475 14,000 21,500 26,638 25,965
Other Principal Arterials
Average
4,823 4,924 6,075 6,936 ----
10th %-ile
1,500 1,833 2,650 2,743 ----
90th %-ile
9,000 8,550 9,779 10,918 ----
Minor Arterials
Average
3,242 3,993 4,747 5,004 ----
10th %-ile
705 1,335 2,200 1,500 ----
90th %-ile
6,748 7,065 8,200 10,594 ----
Collectors
Average
1,737 2,696 3,243 ---- ----
10th %-ile
285 528 1,286 ---- ----
90th %-ile
4,025 5,407 5,793 ---- ----
Source: Highway Performance Monitoring System (HPMS) Database, Federal Highway Administration

4.3.1 Example

To illustrate the methods discussed above, the characteristics of the four external stations in the hypothetical study area shown in Figure 4.2 are given in the table below:

Characteristics of External Stations in Hypothetical Study Area

Characteristic External Station
S1 S2 S3 S4
Functional Class
Urban Interstate Rural Interstate Urban Principal Arterial Urban Interstate
No. of Lanes
8 6 4 8
AADT per Lane
13,400 9,100 not available 11,500

No vehicle classification data are available, hence the composition of traffic at these external stations is unknown. Using Table 4.2 and Table 4.3, the total commercial vehicle trips at each station can be estimated as shown in the table below:

Estimated Daily Vehicle Trips at External Stations

Characteristics
 External Station
S1 S2 S3 S4 Total
Functional Class
 Urban Interstate Rural Interstate Urban Princ. Arterial Urban Interstate  
AADT per Lane
 13,400 9,100 4,924
(Table 4.3)		 11,500  
No. of Lanes
8 6 4 8  
AADT
107,200 54,600 19,696 92,000  
% Distribution:
(from Table 4.2) (from Table 4.2) (from Table 4.2) (from Table 4.2)  
 
Four-Tire
5.50% 3.30% 6.60% 5.50%  
 
Single Unit
1.80% 2.90% 1.70% 1.80%  
 
Combination
4.50% 12.20% 2.20% 4.50%  
Truck AADT: (2-Way)
 
Four-Tire
5,896 1,802 1,300 5,060 14,058
 
Single Unit
1,930 1,583 335 1,656 5,504
 
Combination
4,824 6,661 433 4,140 16,059
Total
12,650 10,046 2,068 10,856 35,620
Truck AADT: (1-Way)
 
Four-Tire
2,948 901 650 2,530 7,029
 
Single Unit
965 792 167 828 2,752
 
Combination
2,412 3,331 217 2,070 8,029
Total
6,325 5,023 1,034 5,428 17,810

4.3.2 Alternative Approaches

Chapter 6 provides additional detail on how to obtain more accurate data on external stations by conducting vehicle classification counts and origin-destination surveys at major external stations to develop external-external, external-internal and internal-external trip tables. Surveys at external stations, in which individual vehicles are stopped and asked about their origin and destination, are the preferred method for analyzing traffic at these stations. However, as stated earlier, budget limitations or other constraints may prevent the agency from conducting such surveys at all external stations. If such surveys are not possible, the agency should consider the possibility of at least conducting traffic classification counts at external stations, either manually or by using automatic classification equipment.

4.4 Trip Distribution

Trip distribution is the process by which trips between traffic analysis zones, or between external stations, are connected. The output of trip distribution is a trip table in which the origins and destinations of individual trips are identified.

The quick-response procedure uses the following standard gravity model for trip distribution:

Equation

where

Vij= trips (volume) originating at analysis area i and destined to analysis area j;
Oi = total trip originating at i;
Dj= total trip destined at j;
Fij= friction factor for trip interchange ij,
i = origin analysis area number, i = 1, 2, 3 . . . n;
j= destination analysis area number,j = 1, 2, 3 . . . n; and
n = number of analysis areas.

Applying the above equation for each zone pair can result in a trip table in which the total number of trips ending in a given zone differs significantly from the desired number of destinations (Dj). To address this problem, the Gravity Model can be applied in an iterative manner. After each iteration, the adjusted destination total to be used for the next iteration is calculated by the following equation:

Equation

where

Djq = adjusted destination factor for destination analysis area (column) j, iteration q;
Djq-1 = Dj when q = 1;
Cjq-1 = destination (column) total for analysis area j, resulting from the previous iteration of the gravity model;
Dj= original and desired destination total for destination analysis area (column) j, developed from trip generation;
j= destination analysis area, j = 1, 2, . . . n;
n= number of analysis areas; and
q= iteration number.

For the quick response procedure involving manual calculations, it would be tedious to perform more than three iterations using the above equation, especially if the study area consists of numerous zones. The option to iterate more depends upon the level of accuracy required which is the percentage difference between the destination totals at the end of each iteration and that originally input for each analysis area. According to the NCHRP Report No. 187 - Quick-Response Urban Travel Estimation Techniques and Transferable Parameters Users Guide,6 a 5- to 10-percent difference is generally acceptable. However, these levels of accuracy may not be attained within three iterations. A computer program may be utilized if the planning agency wishes to achieve a certain accuracy level without manually going through multiple iterations.

Friction factors (Fij ) for use with the gravity model can be based on travel time or distance between analysis areas. Most state or regional planning agencies have well-developed databases describing road networks that include distances and travel times. If an assignment network is available it should be modified to represent available truck facilities and operational characteristics and used to develop the necessary input to trip distribution. For example, the planning agency should annotate roads on the network that may restrict certain classes of trucks due to height or weight, or conversely may be designated as truck routes. The agency may also add a time-value to particular segments to represent the effect of large tolls.

If a network is not available, the agency may develop trip distribution estimates based on map distances. The minimum data needed for the quick response method are distances in miles between zones. These may be derived from actual miles on the existing network or using methods identified in NCHRP Report No. 187. In addition, either map tracings using a map wheel or driven surveys with odometer may be used for distances. Distances should be calculated to and from zone centroids, using appropriate routes.

The following data sources for networks may supplement local data. (Data sources for rail and other modal networks are included in Chapter 6).

In the quick response method, for the different types of commercial vehicles, the following friction factors based on travel time (tij) in minutes between analysis areas are recommended:

Four-tire commercial vehicles:

Fij=e-0.08*tij

Single unit trucks (6+tires):

Fij=e-0.1*tij

Combinations:

Fij=e-0.03*tij

These friction factors are based on average trip times from Phoenix, with a judgmental adjustment to account for the fact that the Phoenix survey did not cover trips beginning or ending outside the MPO region.7

If information on external-to-external trips can be obtained from other sources (e.g., from a special survey or statewide network analysis), these trips should be treated separately from other trips in the trip distribution step. Section 8.2 of this report demonstrates how trip distribution might be carried out for an external-to-external trip table.

If separate information on external-to-external trips is not available, then it will be necessary to apply the trip distribution model to both external and internal trips. In this case, we recommend that the analyst review a map showing the location of external stations and identify all pairs of external stations that are unlikely to share trips. Usually these will be pairs of stations that are adjacent to one another or on the same side of the metropolitan area. Examples include external stations on two highways that intersect outside the metropolitan area or serve the same nearby city. The analyst should then put a very small number or zero in the friction factor matrix to greatly reduce or eliminate trips between such pairs of external stations.

In applying the gravity model to external stations, it is necessary to estimate: (1) the travel time from origin to external station for trips that begin outside the study area, and (2) the travel time from external station to destination for trips that end outside the study area. The following default values can be used if no other information is available:

These default values are based on an analysis of data about the primary range of operations for trucks from the Bureau of the Census' Truck Inventory and Use Survey. While these default values may be reasonable on average, their use could considerably understate or overstate travel times for a given external station. Accordingly, the analyst is urged to examine state or regional highway maps and make judgmental adjustments if necessary.

4.4.1 Example

Assume that the origin-destination travel times for the three commercial vehicle types in the hypothetical study area are as shown in the tables below (Note: Zi = Origin Zone i , Sj = External Station destination j, the entry Zi - Sj in the table corresponds to the average travel time from Zone i to anywhere outside the study area through Sj, and the entry Si - Zj in the table corresponds to the average travel time from outside the study area to destination Zone j through external station Si):

Travel Time (tij) Matrix for Four-Tire Trucks, in Minutes

>
  Destination Zone (j)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 10 18 24 54 60 70 75
Z2 18 12 18 60 55 65 68
Z3 24 18 12 70 62 55 55
S1 54 60 70 --- 98 115 115
S2 60 55 62 98 --- 105 108
S3 70 65 55 115 105 --- 90
S4 75 68 55 115 108 90 ---

Travel Time (tij) Matrix for Single-Unit Trucks, in Minutes

  Destination Zone (j)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 12 22 28 54 50 60 65
Z2 22 14 20 50 45 55 58
Z3 28 20 14 60 52 45 45
S1 54 50 60 --- 78 95 95
S2 50 45 52 78 --- 85 88
S3 60 55 45 95 85 --- 70
S4 65 58 45 95 88 70 ---

Travel Time (tij) Matrix for Combination Trucks, in Minutes

  Destination Zone (j)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 14 25 30 214 220 230 235
Z2 25 16 22 220 215 225 228
Z3 30 22 16 230 222 215 215
S1 214 220 230 --- 422 438 435
S2 220 215 222 422 --- 428 428
S3 230 225 215 438 428 --- 413
S4 235 228 215 435 428 413 ---

Using the formulae for friction factor given earlier, the matrix of friction factors to be used in the gravity model have been calculated as shown in the following tables:

Friction Factors (Fij) Matrix for Four-Tire Trucks

  Destination Zone (j)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 0.4493 0.2369 0.1466 0.0133 0.0082 0.0037 0.0025
Z2 0.2369 0.3829 0.2369 0.0082 0.0123 0.0055 0.0043
Z3 0.1466 0.2369 0.3829 0.0037 0.0070 0.0123 0.0123
S1 0.0133 0.0082 0.0037 --- 0.0004 0.0001 0.0001
S2 0.0082 0.0123 0.0070 0.0004 --- 0.0002 0.0002
S3 0.0037 0.0055 0.0123 0.0001 0.0002 --- 0.0007
S4 0.0025 0.0043 0.0123 0.0001 0.0002 0.0007 ---

Friction Factors (Fij) Matrix for Single-Unit Trucks

  Destination Zone (j)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 0.3012 0.1108 0.0608 0.0045 0.0067 0.0025 0.0015
Z2 0.1108 0.2466 0.1353 0.0067 0.0111 0.0041 0.0030
Z3 0.0608 0.1353 0.2466 0.0025 0.0055 0.0111 0.0111
S1 0.0045 0.0067 0.0025 --- 0.0004 0.0001 0.0001
S2 0.0067 0.0111 0.0055

0.0004

 --- 0.0002 0.0002
S3 0.0025 0.0041 0.0111 0.0001 0.0002 --- 0.0009
S4 0.0015 0.0030 0.0111 0.0001 0.0002 0.0009 ---

Friction Factors (Fij) Matrix for Combination Trucks

  Destination Zone (j)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 0.6570 0.4724 0.4066 0.0016 0.0014 0.0010 0.0009
Z2 0.4724 0.6188 0.5169 0.0014 0.0016 0.0012 0.0011
Z3 0.4066 0.5169 0.6188 0.0010 0.0013 0.0016 0.0016
S1 0.0016 0.0014 0.0010 --- 0.0000 0.0000 0.0000
S2 0.0014 0.0016 0.0013 0.0000 --- 0.0000 0.0000
S3 0.0010 0.0012 0.0016 0.0000 0.0000 --- 0.0000
S4 0.0009 0.0011 0.0016 0.0000 0.0000 0.0000 ---

For the four-tire truck, the gravity model is applied as follows:

1. Create an origin-destination matrix (trip table) which shows the row (origin) and column (destination) totals from the results of trip generation step for the internal zones and traffic estimates at external stations. The distribution of these trips (e.g. Vij) are still unknown. (See matrix below)

2. For each origin-destination pair, multiply the column (destination) total by the friction factor for origin-destination pair (e.g. Dj*Fij). Calculate the total for each row. For example, for Z1, INSERT SUM SYMBOL (Dj*Fij) = (24,944*0.4493) + (29,607*0.2369) + (29,654* 0.1446) + (901*0.0082) + (650*0.0037) + (2530*0.0025) ~= 22,626. The results are shown below:

Four-Tire Truck Trip Table
Iteration = 0

  Destination Zone (j) Total (Oi) Sum(Dj*Fij)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 ? ? ? ? ? ? ? 24,944 22,625.56
Z2 ? ? ? ? ? ? ? 29,607 24,321.98
Z3 ? ? ? ? ? ? ? 29,654 22,082.25
S1 ? ? ? 0 ? ? ? 2,948 685.74
S2 ? ? ? ? 0 ? ? 901 778.49
S3 ? ? ? ? ? 0 ? 650 622.03
S4 ? ? ? ? ? ? 0 2,530 555.32
  Total (Dj) 24,944 29,607 29,654 2,948 901 650 2,530 91,234  

3. First Iteration: Distribute the row totals to each cell in the trip table by using the trip distribution formula for Vij given earlier. For example, V12 = (24,944 * 29,607 * 0.2369)/22625.56 ~= 7,734. Calculate the total for each column (destination). Determine the percentage difference (% Diff.) between the column totals and the original column total. For example, the % Difference for Column 2 (Destination Z2) is equal to [(33,177-29,607)/29,607] * 100% ~= 12.1%. If one or more of the % Differences exceed the threshold value (say ± 5%), adjust the destination totals using the adjustment formula discussed earlier (i.e. Djq ).

Four-Tire Truck Trip Table
Iteration = 1

  Destination Zone (j) Oi Sum(Dj*Fij)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 12,357 7,734 4,793 43 8 3 7 24,944 22,723.50
Z2 7,194 13,800 8,553 30 13 4 13 29,607 23,960.30
Z3 4,911 9,420 15,248 15 8 11 42 29,654 22,426.45
S1 1,426 1,047 471 0 2 0 1 2,948 655.17
S2 238 421 241 1 0 0 1 901 774.10
S3 96 171 380 0 0 0 2 650 665.19
S4 282 585 1,659 1 1 2 0 2,530 546.87
  Total Dj 26,503 33,177 31,344 90 33 20 66  
% Diff. 6.3% 12.1% 5.7% -96.9% -96.4% -96.9% -97.4%
Adj. Dj 23,476 26,421 28,055 96,126 24,908 20,716 97,603

Note that for the first iteration (q=1), all the column totals are above the 5% threshold limits. Therefore we need to adjust the column totals. For example, for Column 2, the adjusted column total is:

Equation

4. Second Iteration: Repeat Step 3 above, but using the adjusted column total in the trip distribution formula. Again, for V12, the new value is (24,944 * 26,421 * 0.2369)/22,724 @ 6,872. The results of the calculation are shown below. Note that the % Differences are all below the threshold value, which means that the no further iteration is necessary.

Four-Tire Truck Trip Table
Iteration = 2

  Destination Zone (j) Oi Sum(Dj*Fij)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 11,579 6,872 4,515 1,403 225 84 266 24,944 22,704.55
Z2 6,873 12,501 8,213 978 378 141 523 29,607 23,950.49
Z3 4,551 8,277 14,204 470 231 336 1,584 29,654 22,450.34
S1 1,405 978 467 0 44 9 44 2,948 654.51
S2 225 378 229 44 0 5 20 901 773.76
S3 85 142 337 9 5 0 71 650 666.39
S4 269 530 1,593 45 20 72 0 2,530 547.64
  Total Dj 24,987 29,678 29,558 2,949 904 648 2,509  
% Diff. 0.2% 0.2% -0.3% 0.0% 0.3% -0.3% -0.8%

As stated earlier, more than two iterations may be needed in other cases to meet the level of accuracy criterion. A computer program may have to be implemented if a very accurate trip table is desired, especially if it involves many iterations. The table below shows that five iterations are needed to balance the trip table for the above example.

Four-Tire Truck Trip Table
Iteration = 5

  Destination Zone (j) Oi Sum(Dj*Fij)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 11,566 6,861 4,536 1,404 224 84 268 24,944 22,701.13
Z2 6,861 12,475 8,247 977 377 142 528 29,607 23,950.08
Z3 4,536 8,247 14,238 469 230 337 1,597 29,654 22,453.43
S1 1,404 977 469 0 44 9 45 2,948 654.41
S2 224 377 230 44 0 5 20 901 773.74
S3 84 142 337 9 5 0 72 650 666.52
S4 268 528 1,597 45 20 72 0 2,530 547.75
  Total Dj 24,944 29,607 29,654 2,948 901 650 2,530  
% Diff. 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

For the single-unit trucks and combination trucks, the same procedures will be followed to distribute the trips to various origin-destination pairs. The following tables show the results of each iteration:

Single Unit Truck Trip Table
Iteration = 0

  Destination Zone (j) Total (Oi) Sum(Dj*Fij)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 ? ? ? ? ? ? ? 5,692 3,063.99
Z2 ? ? ? ? ? ? ? 7,815 3,627.48
Z3 ? ? ? ? ? ? ? 7,767 3,336.91
S1 ? ? ? 0 ? ? ? 965 98.02
S2 ? ? ? ? 0 ? ? 792 168.57
S3 ? ? ? ? ? 0 ? 167 133.32
S4 ? ? ? ? ? ? 0 828 118.85
  Total (Dj) 5,692 7,815 7,767 965 792 167 828 24,026  


 
Single Unit Truck Trip Table
Iteration = 1
  Destination Zone (j) Oi Sum(Dj*Fij)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 3,185 1,609 877 8 10 1 2 5,692 3,089.45
Z2 1,359 4,152 2,265 14 19 1 5 7,815 3,611.32
Z3 806 2,462 4,458 6 10 4 21 7,767 3,369.23
S1 253 518 190 0 3 0 1 965 94.00
S2 180 408 201 2 0 0 1 792 164.73
S3 18 40 108 0 0 0 1 167 142.67
S1 60 165 601 1 1 1 0 828 112.83
  Total Dj 5,860 9,353 8,700 30 43 8 31  
% Diff. 2.9% 19.7% 12.0% -96.9% -94.5% -95.3% -96.2%
Adj. Dj 5,529 6,530 6,934 30,907 14,492 3,539 21,926


 
Single Unit Truck Trip Table
Iteration = 2
  Destination Zone (j) Oi Sum(Dj*Fij)
Z1 Z2 Z3 S1 S2 S3 S4
Origin Zone (i) Z1 3,068 1,333 777 257 180 16 61 5,692 3,091.16
Z2 1,326 3,485 2,031 451 348 31 144 7,815 3,609.45
Z3 775 2,037 3,942 177 184 91 562 7,767 3,369.91
S1 256 452 176 0 61 3 17 965 93.96
S2 179 349 184 61 0 3 16 792 164.68
S3 16 31 90 3 3 0 23 167 142.80
S4 61 145 565 17 16 24 0 828 112.81
  Total Dj 5,681