Prologue

This document is one in a series being developed by ODOT Urban Engineering and Project Analysis Sectionsrelating to travel demand forecasting procedures. Other documents in this series will include, HighwayNetwork Coding Procedures (1), Transit Network Coding Procedures (2) and Trip Table SynthesisProcedures(3). Taken together these documents represent a complete set of guidelines for using four steptravel demand forecasting models in Ohio. Other procedures dealing with pre and post processing to the fourstep process have been deliberately omitted from this series and will be dealt with individually as necessary.

Throughout this document, Ohio procedural guidelines are underlined to make them easily distinguished fromthe accompanying descriptive information. These guidelines are summarized in Appendix B.

2I. Introduction

Traffic assignment is defined as the process of allocating a set of trip interchanges to a specific transportationsystem based on specific criteria as to choice of route. The choice of route criteria is that the travelimpedance through the transportation system be minimized for a given origin-destination pair. The output oftraffic assignment is "an estimate of user volumes on each segment of a transportation network as well as theturning movements at each intersection of the transportation network".(4)

Purpose

The purposes of performing a traffic assignment have been variously given in a number of sources (5), (6) as:1. To determine deficiencies in the existing system2. To assist in the development of the future transportation system3. To develop construction priorities4. To provide highway designers with design-hour traffic volumes

In addition to these, many other uses of traffic assignments have developed such as air quality analysis,congestion management analysis and major investment studies. The number of uses for traffic assignmentsare many and can be summed up simply as: traffic assignments are used whenever a system wide predictionof transportation segment volumes (highway vehicle volumes or transit ridership) are needed.

Input

There are two basic inputs to traffic assignment: a matrix of trip interchanges and a computerized descriptionof the transportation system.

Creation of the trip interchange matrix or trip table is covered in the aforementioned document; Trip TableSyntheses Procedures. The trip table is a matrix of volumes of vehicles or persons that desire to move fromone area to another. Each row of the matrix represents those trips wishing to go from one area to all otherareas while each column represents those trips wishing to come to one area from all others. The matrix as awhole allows for the interaction of trips between all areas.

These areas are known as traffic analysis zones. Traffic analysis zones are merely geographic areas used toaggregate travel behavior into manageable units. Travel actually occurs between discrete traffic generatorssuch as homes and businesses. To model the interactions between all homes and businesses in an area wouldbe nearly impossible, so travel desires are aggregated into traffic analysis zones. The use of traffic analysiszones to aggregate travel behavior is a fundamental concept of travel demand forecasting. The delineation ofthese zones will be discussed in more detail later as well as in the Trip Table Synthesis Procedures Document.

A trip table can be obtained in one of two ways, through an origin-destination survey or through synthesiswith trip generation, trip distribution and mode split models. These synthesis models are calibrated using atrip table obtained from an origin-destination survey. Regardless how it is obtained, any trip table can beassigned to a transportation network. Which trip tables are used when and why will be discussed later in thisdocument.

The second input to traffic assignment is the computerized transportation network. There are two types oftransportation network used in Ohio, the highway network and the transit network. Development of thehighway network is discussed in Highway Network Coding Procedures while development of the transitnetwork is discussed in Transit Network Coding Procedures. A transportation network is represented by aseries of links and nodes. A link represents a segment of street and contains data on the characteristics of thatstreet, a node represents an intersection or other point where conditions on the street change. Additionally,transit networks also contain lines which are lists of nodes through which a given transit route passes. Atransit line is a path in its own right, which all transit minimum time paths are restricted to follow. A giventransit passenger may transfer between transit lines to reach his destination, but he must always follow the

3prescribed lines. Special nodes called centroid nodes or just centroids, are coded into a highway network foreach traffic analysis zones. These centroids are connected to the highway network through special linkscalled centroid connectors. A centroid connector is a fictitious street segment which may represent a numberof local streets and access points which are not otherwise part of the transportation network. Centroids areused to load trips from the trip table onto the transportation network and are thus analogous to sources andsinks in say a pipe flow network.

The most basic information about a transportation network that is needed for traffic assignment purposes isthe connectivity between links and a measure of impedance on that link. Connectivity is indicated on a linkby listing the nodes that it connects. All links that meet at a given node are said to be connected. A link is, infact, identified by this node to node listing. Thus a link that connects node 52 to node 121 is referred to aslink 52-121. Travel impedance is indicated by the time to traverse the link. This time includes all delays dueto intersections and other conditions on the link. This can be indicated in one of two ways, either byindicating the travel time directly or the average speed that can be maintained across the link as well as thedistance of the link. Impedance coding and other network parameters will be discussed later.

Overview of Process

Traffic assignment is basically a two step procedure. First minimum impedance paths are determinedbetween all traffic analysis zones using the network impedance discussed above. Paths are not built one at atime for each zonal interchange but instead are built for each origin zone in such a way that all overlappingportions of the zone to zone paths from that origin only need be built once. For example, if a given centroidhad 4 centroid connectors and the network as a whole had 500 zones, at most only 4 calculations would berequired in finding the first link of the path instead of 500 if each path were calculated individually.

The second step is to load the trip table to the network by utilizing the minimum impedance paths. Each tripinterchange in the trip table is individually loaded to the path between the two zones it represents. Thevolume of the trip interchange is accumulated on each link that belongs to that path. Each zonal interchangewhose path crosses a given link adds its volume to the link in turn until a final traffic assignment volume isaccumulated after the entire trip table has been loaded. Trip loading occurs backwards from the destinationzone to the origin zone. This is because the path set from the origin (often called a tree or vine) emanates outfrom the origin and goes to each other node in the network exactly once. Loading backwards ensures thatyou eventually reach the origin node. An analogy can be drawn from a watershed where the origin is themouth of the primary river and the destinations are at the headwaters of the various tributaries. Goingdownstream from the destinations will always eventually get you to the origin while going upstream from theorigin might get you to any of the destinations. Paths are built upstream to ensure all destinations are reachedand loaded downstream to ensure the origin is reached.

Assignment Types

In Ohio there are four basic types of traffic assignment used. These assignments are all based upon a 24 hourmodeling period and are:

1. Base year calibration assignment2. Existing year assignment (when different from base year)3. Future year no-build assignment4. Future year build assignment

All other assignment types can be classified as one of these three. The base year calibration (or validation)assignment is intended to be made every 10 years to correspond to the availability of census data. Unlikeother assignments this assignment involves collection and coding of ground counts into the network to checkthe accuracy of the modeling process. Much additional work goes into the creation of trip tables for the baseyear validation as discussed in Trip Table Synthesis Procedures. The model checks discussed in Chapter IV.of this document are then used to validate and adjust the network model until it adequately represents groundcounts. One very important point that should be made is that the final base year validation modeling chain

4must be reproduced exactly for forecast year assignments. This condition prohibits the use of countrestraining and requires the use of capacity restraining in the assignment process. Count and capacityrestraint will be discussed more later in this document.

Because the base year assignment is only made once every 10 years, it is sometimes necessary to have anexisting year assignment in the intermediate years. This assignment can be used to check the forecastassumptions and to serve as the basis of further analysis.

The future year no-build and build assignments utilize forecasted trip tables to evaluate the system effects oftransportation improvements. The build and no-build scenarios are each assigned a forecasted trip table andthe differences in various measures of congestion are analyzed. For Long Range Planning purposes, the no-build network is the existing + committed network (E+C). This network represents currently existingconditions plus those projects already committed for building where in the past committed variously meantprojects which are programmed, projects that have gone into the environmental process or projects appearingin the TIP. ODOT recommends that in the future, E+C networks be equated to the TIP network. The TIPnetwork which is currently used primarily for air quality analysis contains all projects in the TIP plussignificant local projects not in the TIP. Equating these networks would provide a measure of consistence inthe process and result in less networks to be maintained. The build scenario involves the E+C plus allprojects that will appear in the Long Range Plan. Several possible alternative LRP's should be analyzed andthe best selected for implementation. This of course assumes that the alternatives are all feasible given fiscalconstraint, the political climate and other factors.

Build/no-build analysis is also used for various other purposes including air quality analysis, congestionmanagement analysis and MIS analysis. For analysis, the year is determined and networks representingproposed conditions that year (both with and without the subject projects) are coded. A trip table for thatyear is created using the trip table synthesis procedures and assigned. Plots of the assigned volumes andvolume to capacity ratio are then analyzed to determine project benefits which can then be used in benefitcost analysis. Other analyses use only build scenarios such as air quality budget analysis and design yeartraffic analysis.

History

The beginnings of traffic assignment can be traced back to the introduction of origin-destination surveys.These surveys produce trip tables which show the trips people make independent of route choice. Toproperly analyze the effects of this trip table a method was needed to determine which routes the trips in thetrip table would use to move from origin to destination. Early methods used manual techniques to determinethe routes people would use between each origin and destination, computers were only used to accumulatetrips from the trip table on the manually determined routes. Other work was in the area of diversion analysisin which various methods were proposed for determining how many trips would divert from current routes toa proposed route. These methods utilized diversion curves which expressed the number of trips divertingbetween routes as a function of travel time ratio. These methods were only conducted at the corridor leveland often gave unpredictable results.

The most critical aspect of a traffic assignment program is that of route selection or path building. Attemptsto develop a computerized program for system wide traffic assignment had been blocked for some time by aneed to find an efficient route selection algorithm. According to USDOT's Traffic Assignment Manual (4),"The breakthrough in the network path determination came from work undertaken to solve the route selectionproblem of the telephone systems for direct dialing of long distance telephone calls...Two papers published in1957 provided the impetus for computerized route selection process for traffic assignment: The Shortest PathThrough A Maze, by E.F. Moore(7); The Shortest Route Problem, by G. Dantzig.(8)" Transportationagencies were quick to exploit these methods and by 1960 work by Chicago, Washington D.C. Detroit,California and Minnesota had resulted in the first Bureau of Public Roads Traffic Assignment Battery.

One early development came when researchers realized that there is a relationship between volume and traveltime (known as a speed (time) - volume curve, the most commonly used is know as the BPR curve) as shown

5in the Highway Capacity Manual (9-11). In an attempt to reflect this relationship in assignments and toproduce a "multi-path" diversion type effect between zones capacity restraint algorithms were incorporated,first by the staff of the Chicago area study. Capacity restraint is an iterative process whereby an assignmentis made and the link impedances are adjusted based on a speed-volume curve. The assignment is then redoneand impedance readjusted. The final capacity restrained assignment is not used alone but rather weightedwith previous iterations to produce the final weighted capacity restrained assignment. The TrafficAssignment Manual has an excellent discussion of Capacity Restraint which states:

"There is a relationship between speed and volume on all types of facilities, both for interrupted and uninterrupted flow.On facilities such as freeways, there is a constant decrease in speed with increase in traffic volume up to a point of criticaldensity. Beyond this point, however, both volume and speed decrease with an increase in density. The situation issimilar at interrupted flow type facilities. Here, however, speed is influenced by external influences such as signalprogression, speed limits, and the conditions on adjacent sections...The traffic assignment process assigns trips inaccordance with impedances coded on each network segment. These impedances are usually travel time or somederivative of time. The assignment process results in the traffic load on each network segment. Since there is a verydirect relationship between travel time (or speed) on a section and the volume on the section, a process is necessary toallow consideration of this relationship. This process is referred to as capacity restraint. Specifically, the capacityrestraint process attempts to bring the assigned volume, the capacity of a facility, and the related speed into the properbalance. There are several problems in the application of speed-flow relations in the assignment process. Most importantis the assignment of trips for some extended period of time, such as a day...Most critical flow problems actually occurover shorter time spans. Secondly, the assignment process may load a facility far in excess of capacity based upon someoriginally coded speed. Observed conditions are limited to some maximum capacity. Because of this, capacity restraintfunctions are theoretical extensions beyond some critical capacity point."

The last two points should be emphasized. First, capacity restraining conditions actually only occur over ashort time period so their use in 24 hour assignments is not straight forward and rigorous but requiresjudgment and experimentation to produce good results. Second, capacity restraint functions are somewhattheoretical in nature because they must extend beyond the realm of observable values.

Early research (12) showed that capacity restraint reduced the overall error of assignments and often obviatesthe need for manual adjustment of assignment networks due to over-loaded conditions. Capacity restraint isdiscussed in more detail later in this document.

The next development occurred in the mid 1960's with the introduction of a new minimum path buildingalgorithm often referred to as the vine algorithm (13). Early in the development of trip assignment methodsthe capability to model turning movement penalties and prohibitions at intersections was added to themethodology. This sometimes resulted in unrealistic paths using the traditional tree building or Moorealgorithm. The tree algorithm built paths from node to node while the new vine algorithm built paths to eachlink at a node thus allowing turn penalties and prohibitions to be accounted for correctly. The distinctionbetween these two methods will be discussed in more detail later.

The introduction of the "Equilibrium Model" (14-16) in the 1970's brought traffic assignment to its currentlyused form. Equilibrium assignment arose when it was realized that traditional iterative capacity restraintprocedures do not converge to a unique solution. Equilibrium assignments use optimization techniques tocreate a linear combination of assignment runs which produce a user optimum assignment (i.e. one in whichno trip can reduce its travel time by switching paths). This is accomplished by minimizing the area under thespeed (time)-volume relationship curve from zero to the assigned volume for all links in the network (see (16)for a discussion of why this is so). This process is similar to traditional capacity restraint in that it can utilizethe same speed-volume relationships to modify link impedances and bring assigned volume and impedance inbalance, however, instead of the user choosing the weights to be applied to each assignment iteration, they arechosen automatically by the computer. Also while capacity restraint only weights each assignment togetherafter all assignments have been made, equilibrium performs the weighting after each iteration and uses theweighted volume to determine the travel impedance for the next iteration. At each iteration of theequilibrium assignment, link impedance is adjusted based on the previously weighted assignment using thespeed-volume curve for that link. The use of equilibrium assignments is discussed in detail later.

6Traffic assignment as employed for region wide systems analysis has not changed much since theintroduction of equilibrium methods. This is not to say that the field has remained stagnant, new trafficassignment techniques focusing on dynamic micro assignment of vehicles such as that found in the NETSIMproducts have been developed which give much more realistic traffic assignments. To date these methodshave not been employed region wide simply because they are too computationally intensive for computers touse at this scale. For example the NETSIM program has a limit of 600 links and 200 nodes which is far toosmall for the typical urban area model. The future of travel demand forecasting appears to be in the area ofactivity based modeling. Activity based modeling is a micro-simulation of individual households andtravelers. New traffic assignment techniques are being developed to operate within this paradigm and mayone day replace many of the techniques discussed here.

7II. Transportation Network Considerations

This section discusses aspects of one of the two inputs to traffic assignment, the transportation network,relative to traffic assignment.

Traffic Analysis Zones

As discussed previously, a set of traffic analysis zones are needed for traffic assignment. These zones areused to aggregate all trip making into discrete locations which can then be assigned to the transportationnetwork via zone centroids. Traffic analysis zones are generally pre-existing, however, the following criteriashould be considered when creating or modifying traffic analysis zones:

1. Zones should generally be of regular geometric shape (i.e. it should be generally square notgerrymandered)

2. Zones should follow geographic boundaries where possible including roadways, streams, railroads andpolitical boundaries.

3. Zones should follow census boundaries to allow socio-economic data collection utilizing the census.4. Zones should have relatively equal generating power (i.e. the total number of trips generated by each

zone should be similar).5. The total trips generated by a zone should be less than 10-15 thousand to avoid large discontinuities in

traffic assignments near the zone.6. The density of development in a zone should be relatively even across the zone.7. The land use in a zone should be somewhat homogenous.8. Most importantly, and this point is sometimes forgotten, zones should be created so that they give good

traffic assignment results!

It is impossible to meet all of these criteria all the time. Judgment and criteria number 8 are the final decidingfactors in creating traffic analysis zones. The size of the zones to be used is dependent on the final purpose ofthe traffic assignment. For region-wide studies the best guide is to make zones similar in size to the existingzones. Finer zones can be used for subarea analysis and to obtain more accurate results if the extra codingand processing time are not important. It must always be remembered, however, that the size of the zones isdirectly related to the number of roadways included in the transportation network.

Zone-Network Relationship

There is a relationship between the number of traffic analysis zones and the number of links in thetransportation network. According to the Traffic Assignment Manual (4): "Experience in traffic assignmentapplication indicates that this zone-network compatibility helps insure that assignment results will be asaccurate as possible regardless of the level of detail and the objectives of the study." In general it isrecommended that there be one traffic analysis zone for each area delimited by the transportation networklinks. This requirement implies that it is undesirable to have network links which bisect traffic analysiszones. Using this criteria strictly in a grided street system with 4 centroid connectors per zone would result in8 links per traffic analysis zones. According to (4): "it would appear that 10 links per zone is the[nationwide] average". In Ohio the average for regional models is about 8 links per zones. There is somevariance, however, from a low of 5.3 links per zone in the Steubenville area to a high of 10.8 links per zonein the Cleveland area. The zone-network relationship should only be changed in existing models if a problemwith the final output is discovered, as mentioned in criteria 8 of the previous section, the most important thingis that they give good traffic assignment results.

What is the reason for this relationship and what are its implications? First, recall that traffic analysis zonesare used to aggregate trip making behavior into manageable geographic units. Those trips whose origin anddestination are within the same zone are called intrazonal trips. These trips will not load onto thetransportation network. Larger zones produce more intrazonal trips and thus less trips on the network.Second, it should be noted that not all facilities (roads) are coded into the network. For a given number of

8trips being loaded to the network, fewer facilities will have to carry larger volumes because trips that wouldhave used the roads not coded into the network must use those that were. Thus very small zones loading acoarse network will result in over-assignment because there are few intrazonal trips and the few roads codedin the network must carry all of the traffic. On the other hand, large zones loading a fine network will resultin under-assignment because many of the trips are now intrazonal and the few trips that are left mustdistribute to many roadways. The key is to find the proper balance to produce reasonable assignments whichis about 8 links per zones. Given this discussion it might be expected that the Steubenville model with 5.3links per zone would be over-assigned while the Cleveland model with 10.8 links per zone might be under-assigned.

Knowledge of this relationship provides a useful calibration tool. If a particular area of a network is over orunder assigned a check on the zone-network relationship in that area will indicate if this is the problem. If itis, addition of links representing certain minor facilities can alleviate over-assignment while removal of linksrepresenting lower classed facilities can alleviate under-assignment. Network calibration will be discussedmore later, it should be noted that there can be many reasons for poor assignment results in an area such asincorrect trip generation, network impedances etc., a check of the zone-network relationship should be madeto determine if this is truly the problem.

Transportation Network Layout

Given that there is a relationship between the number of network links and the number of traffic analysiszones, the next question is how many links should be included in the network? It was mentioned previouslythat not all facilities are necessarily coded into a transportation network. The number of links to include isbased on the use of the traffic assignment. For highway networks this is usually done using functionalclassifications. The general rule of thumb is that roadways with a functional class one lower than that whichwill be studied should be coded as links. In other words, if I want to analyze arterial streets and above then Ishould code collectors and above. In addition, network connectivity must be taken into consideration bycoding lower classification streets where they provide important network connections. Another criteria thatcan be used is that all streets with signalized control at intersections should be included in an urban areanetwork. For region-wide models in Ohio we have traditionally coded collectors and better, however inrecent years the number of local streets in the regional models has increased. Generally centroid connectionsare made to the lowest class of roadway. Those roadways closest to the centroid connectors and with lowervolumes (usually locals and collectors) will have poorer assignments while those further away and withhigher volumes (usually arterials and freeways) will have better assignments. The implication here is that thenetwork model should not be used to analyze traffic on the lowest classification of street (usually collectors)unless a lower classification (usually locals) is coded as well. This point is often forgotten by users of themodel who wonder why every link in the network doesn't match ground counts well. Thus the number oflinks is governed by the number needed to code all streets of a certain functional class and higher (usuallycollectors), the total number of zones is then related to this number and the size of individual zones based onthe criteria given in the Traffic Analysis Zones section above.

Details on coding the network are given in (1) and (2) with some additional comments related to trafficassignment given for specific data items in the sections to follow. Appendix A. contains the network formatsfor both Planpac and Tranplan for highway networks.

Centroid Coding

All travel activity (trips) is loaded to the transportation network through traffic analysis zones. This isaccomplished by coding into the transportation network a centroid for each zone. This centroid is a fictitiousnode from which all trips begin and end. The centroid node is located at the center of activity of the zone itrepresents (not necessarily the geographic center). Thus a zone with all of its activity at one end of the zonewill have its centroid node at that end. Determination of the center of activity is a judgment call based onmaps and aerial photographs of the area as well as personnel knowledge when appropriate. Centroid nodesmust be the first nodes in a network (starting at 1 and continuing consecutively without gaps until one iscoded for each zone.) Thus if there are 500 zones there will be 500 centroid nodes numbered from 1 to 500.

9Each centroid node is connected to the network by centroid connectors. These are fictitious linksrepresenting all local streets and access points to the network within the zone. A centroid can be connectedby as many as 4 centroid connectors. It is best to use as many centroid connectors as possible to reducediscontinuities on the network in the vicinity of the centroid connector (a single centroid connectorconnecting a zone with many trips may produce a large jump in volume between the two network linksconnecting it). At times less than 4 are used when there are less than 4 access points from a zone, or less than4 roadways surrounding the zone. Only one centroid connector should connect a centroid to thetransportation system between any two intersection nodes. Centroid connectors should not connect directlyto intersections but rather should connect at the mid block area, thus splitting an intersection to intersectionlink in two. Centroid connectors are assumed to have unlimited capacity and thus do not affect capacityrestraint calculations (leave the capacity field blank or code 999999 in Tranplan).

Impedance Coding

As previously stated, the only link data absolutely necessary for traffic assignment is the connectivity of thelink (which nodes it connects) and a measure of impedance. In Ohio travel impedance is measured strictly bythe time necessary to traverse the link. It is possible, however, to use various combinations of time, distance,tolls, operating expenses, intersection stop time etc. for travel impedance, it is simply our convention to useonly travel time. The use of travel time has been found to produce good traffic assignment results not only inOhio but elsewhere as well. Travel impedance on a link can be coded as time directly (in hundredths ofminutes) or speed (in mph). If speed is coded, then the length of the link (distance) must also be coded sothat the speed can be converted to time. Usually speed is coded. This value gives a better "feel" for theoperating conditions of a link since the dependence on distance has been removed. It must be re-emphasized,however, that the speed coded on a link is a measure of total travel impedance not necessarily the mid-blockspeed or the posted speed limit. Thus the speed must reflect time spent sitting at traffic lights or slowing toallow someone to turn. In addition, since the speed (or time) is the only value which affects path building, itis sometimes necessary to incorporate extraneous route choice impedances into the speed. For example,many people prefer not to use the freeway when going across town, possibly due to the stress of high speedtravel. This is an impedance to using a freeway link and thus must be reflected in the speed. The speeds usedfor traffic assignment are thus not necessarily realistic speeds and thus should not be interpreted as such.They are a measure of travel impedance and nothing more.

Traditionally link speeds have represented travel times under actual traffic conditions. These speeds bythemselves produce good all or nothing type assignments and are needed as the basis of the capacity restraintassignment. Equilibrium assignments, on the other hand, use free flow speed for their calculations. Thereason for this distinction is primarily associated with the way the speed-density relationship (BPR curve) isset up and used by the two methods. It must be remembered, however, that all or nothing assignments willnot produce satisfactory correlation with ground counts when using free flow speeds, thus free flow speedsshould only be used for equilibrium assignments. Free flow speeds can be used with all or nothingassignments to produce a travel desires assignment for planning analysis purposes only.

Travel times (speed) are obtained for highway networks by field work utilizing the floating car technique.This methodology is discussed in Chapter 11 of the Highway Capacity Manual (11) and in the proceduremanual "Determining Travel Time" (17). A number of travel time runs must be made on the same highwaysegment to obtain an acceptable confidence level. The Traffic Assignment Manual suggests no more than10% standard deviation on high speed roadways (50-70 mph) and 20% standard deviation on low speedroadways (10-30 mph). Travel time runs must also be made for both peak and off-peak conditions. Traveltime studies can be made for all highways in the study area (note that a travel time run would be made for onestreet comprising many links at once) if possible, however if not, a sample of roadways can be studied withthe objective of producing a speed table.

Speed tables give average travel speeds for different combinations of area type and facility type. For exampleone might use the following 8 facility types, 2 lane arterials, collectors, locals, multi-lane arterials, collectors,locals, freeways, ramps. Area type might include 5 categories such as, CBD, Urban, Outlying Business

10

District, Residential and Rural. Using this breakdown there would be 40 cells in the speed table for both peakand off-peak speed tables (80 total). In addition it is desirable to make speed runs for several facilities withdifferent posted speed limits within each cell. The more roadways sampled for a given cell, the less runs needto be made on each roadway to obtain the desired level of confidence. A regression equation relatingnetwork speed to posted speed can then be formulated for each cell. Posted speed can then be coded on thenetwork (outside the range of link information used for traffic assignment) and converted to network speedby the speed table. The network speed and not the posted speed are then written in the speed fields of the linkfile and are subject to further modification for network calibration purposes. This methodology for handlingspeeds has been in use since the inception of traffic assignment. The details and origins of the networkspeeds have been forgotten by many because they have not been updated in a long time. It is therecommendation of this manual that consideration be given to updating these speeds. Regardless ofmethodology, if 24 hour assignments are to be made using all or nothing or capacity restraint assignment,then the peak and off-peak speeds must be factored together to obtain a 24 hour speed in the ration 1/3 peakspeed plus 2/3 off-peak speed. If equilibrium assignments are to be used then the free flow speed can berepresented using data from off-peak speed runs. In this case, peak period travel time data is not needed fortraffic assignment purposes.

It may also be possible to use posted speed limit to represent free flow speed for equilibrium assignments.This may only be done if the resulting assignment can be shown to adequately reproduce ground counts in thebase year per the assignment checks discussed later.

When adding links to an existing network, speeds can be coded from the speed table if one has been created.If not, speed should be taken from links with similar area type, facility type in the network. This secondprocedure is the one currently used by ODOT and is obviously the same as the speed table look up procedurein a less formal form.

The speeds coded in the link file are used for both base year and future year assignments. Speeds producedutilizing capacity restraint functions are not saved for future use, their sole purpose is to produce capacityrestrained traffic volumes during a given assignment.

Distance Coding

Link distances represent the actual node to node distance of the link. These distances can often be obtainedfrom road-inventory data bases or from field measurement (distance will be measured in a travel time run).Distance can also be scaled from a map or obtained from digital mapping files. In this case the distanceshould be measured along the actual roadway segment, not the node to node straight line distance. Whencoding roadways that do not yet exist distance coding must be the analysts best guess, distance can becalculated using the node coordinates of the link in this case.

Turn Penalties/Prohibitors

Turn penalties and prohibitors are used to add time to certain turning movements (usually left turns) orprohibit them altogether. Turn penalties may be used if it is felt they yield a better assignment. During modelvalidation, tests with and without turn penalties should be made to ensure the penalties produce betterassignment results. Turn penalties are often used in the assignment process to prevent "stair stepping". Stairstepping sometimes occurs in grided street systems when a minimum path follows a pattern of alternate leftand right turns instead of traveling straight down one road and turning once onto another. Printing the pathsfrom a network with and without turn penalties will show any stair stepping and is another useful check of theneed and accuracy of turn penalties. As discussed later, it is ODOT policy that minimum time paths be builtwith the vine building algorithms not trees. It should be noted that if this policy is not followed, turnpenalties and prohibitors in a network will yield unpredictable results.

Capacities and Other Network Information

11

Hourly capacities are coded in the network for the purpose of capacity restraint of the assigned volumes.These capacities are no longer hard coded to the links. Instead, capacity is calculated at run time by theprogram CAP94 which uses the 1994 Highway Capacity Manual procedures. Use and documentation of thisprogram is contained in Capacity Calculator Program Documentation (18). This program requires variousinformation including area type, functional class, number of lanes and total street width to be coded on thelinks. Additional optional link information including % trucks, terrain type, intersection turn bays, throughlanes, and signal coordination refines the capacity calculation.

ODOT policy is that capacity coding calculations will be made using three levels of detail. The same level ofdetail must be applied to all links in a study area as described below. Level one detail involves havingupdated functional class, area type, total street width and number of mid-link lanes on all links in thenetwork. All study areas should at least be at level of detail one. A time frame has been set for all areas to beat or beyond level one by the end of c.y. 1996. ODOT has agreed to update all network functional classes aswell as the number of lanes and width on the state system. The MPO's have agreed to update all area typesand the number of lanes and width off of the state system. Level of detail two involves coding theintersection turning and through lanes, median left turn lanes, % trucks and terrain type for all links on thestate system (i.e. interstate, U.S. and state routes.) Level of detail three extends this coding to all links in thenetwork.

Other link data including, district and administrative class are coded and will be maintained for the purpose ofgenerating reports and classifying assignment results for calibration purposes as will be discussed later.

External Stations

Traffic entering/exiting and passing through the study area must be accounted for in the travel demandforecasting model. IE and EE trip tables will be created based upon road-side interview surveys as discussedin Trip Table Synthesis Procedures. These trip tables are loaded to the network through centroids andcentroid connectors which tie into the highway network at the point of the road-side survey on the cordonline. External stations will typically have only one centroid connector (except multi-lane divided freewayswhich are represented by 2 one way links and thus have 2 centroid connectors at the cordon line) since theyrepresent a single access point. Centroids representing external stations are generally the last zones in thezone numbering scheme and must follow the restrictions on zone numbering mentioned under the TrafficAnalysis Zones section. For example an area with 500 internal zones and 30 external stations would have530 centroids with the internals numbered 1-500 and the externals numbered 501-530. It is wise to includesome dummy centroids between the last internal zone and first external zone to allow for latter expansion ofthe internal zones. These dummy centroids are simply tied to the network at a convenient place and given 0trips. The use of large external zones to generate and distribute external traffic is generally not used in Ohio.Their use is not strictly prohibited, however. For those who would like to undertake this type of analysisexternal zones provide a much better tool for modeling near cordon events such as new roadways which leadout of the study area. The cost is bigger networks, data collection outside the MPO region and reducedcorrespondence to ground counts at the cordon line.

12

III. Assignment Techniques

The following section describes the techniques to be used for traffic assignment in Ohio. These techniquesare limited to 3 methodologies namely, all or nothing assignments, capacity restraint assignments andequilibrium assignments. There are other types of traffic assignment such as incremental and stochasticassignment. As these are not used in Ohio they will not be discussed.

Hourly vs. 24 Hour Assignment

Standard traffic assignments are done on a 24 hour basis. This is because the model calibration is madeversus 24 hour ground counts which are easier to collect than hourly counts and because one 24 hour modelis easier to maintain than multiple hourly models. There are several problems with the 24 hour assignment.First, applying capacity restraint which is a short time period phenomenon to a 24 hour representation is notentirely correct. Second, current post processing analysis for congestion management and air quality requirehourly volumes which must be approximated from the 24 hour values. The use of hourly assignments arerecommended to alleviate these difficulties. Generally, 3 different representative hours are analyzed, an AMpeak hour, PM peak hour and off peak hour. Each hour requires its own network with speeds, capacities,parking etc. tailored to the given period. In addition hourly ground counts are needed for calibrationpurposes. Hourly trip tables are also needed. Creation of hourly trip tables requires origin-destinationinformation. Because of this, hourly models should not be attempted without an updated origin-destinationsurvey. A method has been developed (19) for creating hourly trip tables from 24 hour trip tables by trippurpose and might be employed to create daily distributions by trip purpose from the 1995-1996 Road-SideInterview Surveys. This method may be particularly useful in smaller urban areas where the cordon linepatterns are more likely to hold over the area as a whole. The applicability and validity of this method whencompared to hourly counts is yet to be proven, however. While hourly assignments are expected to givebetter results in the post processing analysis it must be remembered that since three hourly periods areanalyzed (and possibly the 24 hour assignment as well) there will be 3 or 4 times the modeling effort toproduce this type of assignment. Also each hourly period model must be calibrated separately. As modelcalibration is very time consuming this requirement may be prohibitive.

All or Nothing Assignment

The simplest type of traffic assignment is the all or nothing (AON) assignment. In this type all tripinterchanges between a given origin-destination pair are assigned to the minimum time path between thezones. Because of this AON assignments contain no diversion or multi-path effects. These terms refer to thefact that in reality not all trips between two places will use the same route. This is due to the psychology ofhuman choice. In reality people do not have perfect information and are biased by past experience and otherfactors extraneous to rapid movement through the transportation network. In addition, route choice is basedon congestion levels which are not reflected in an AON assignment where a particularly speedy route mayreceive traffic far in excess of its capacity which in reality would produce a traffic jam. To reflect this,various procedures have been developed to produce a diversion effect primarily based on congestion levels.These methods will be discussed later.

In Ohio it has been found that AON assignments give reasonable results (with some manual modelcalibrations) for relatively uncongested networks. It is ODOT policy, however, that some form of capacityrestraint (including equilibrium) be used for all forecasts. This requirement taken in conjunction with therequirement that the base year model chain be duplicated in forecasts implies that base year runs must alsouse a capacity restraining method. AON assignments have been found to be most useful during thecalibration process. Because the AON assignment is more straight forward it is easier to trace modelproblems in an AON assignment. Once manual calibration is complete, a capacity restrained assignment ismade and the results compared to the AON assignment to ensure that the results are still valid.

All of the assignment techniques described here actually use AON assignment. The capacity restrainingtechniques iteratively calculate AON assignments and weight fractions of each assignment together to

13

produce a final assignment. As the paths may be different in each of these iterations, a multi-path effect iscreated with as many possible paths between zones as the number of iterations run.

The basic methodology of the AON assignment has already been described as building paths followed byloading the paths with the trip interchanges. The loading process is relatively straight forward. As wasdescribed previously, each trip interchange in the trip table is loaded to its respective path backwards (fromthe destination to the origin). All trip interchanges that use paths traversing a given link are accumulated onthat link to produce a loaded volume. Creation of the paths is that part of the assignment process which wasmore difficult to develop and an example will be given below.

Recall that path building does not occur for each possible zonal interchange but rather paths are built fromone origin zone to all other nodes in the network at once. This process creates what it called a tree of paths.The analogy used previously is a water shed. Each point in a water shed (node) is served by exactly one pathover the ground to a stream to a river and eventually the sea (minimum time path). These paths are allinterconnected and lead to the mouth of the river (origin zone). There are two primary methods forconstructing minimum time paths. The original method sometimes called the Moore method (7) or the treealgorithm will be described first. The newer vine algorithm is a modification of this method and will bedescribed second. ODOT policy is to use the vine algorithm for traffic assignments in Ohio.The tree building algorithm will be demonstrated with an example. For a more detailed explanation of thealgorithm see (4). Figure 1 shows a simple network with 6 nodes (A-F) and 7 links. Listed on each link isthe impedance of that link (say in minutes). The origin node is node A. The dashed lines show the minimumpath tree for this network. Path building begins at node A in Step 1. Minimum time paths are built to nodesB and C from A. In Step 2 path building proceeds to the node closest to the origin which in this case is nodeC. From C minimum paths are built to all nodes connected to it (D and E). Note that a path was not builtback to A. One of the constraints of the path building algorithm is that it not cross the same link twice. InStep 3, paths are again built from the next closest node to the origin which is now B (4 minutes compared to 5minutes for D and 8 minutes for E). The sole path built from B to D results in a total time to node D of 6minutes. As this is longer than the existing path to D the new path is discarded. In Step 4, path buildingproceeds from node D to nodes B and F. The path to B is 7 minutes long so it is discarded since the existingpath is 4 minutes. The closest node is now node E and a path is constructed to F. This path, however, is 14minutes long while the existing path to F is 9 minutes so the new path is discarded. Finally, in Step 6, a pathis constructed from node F to E, however its total length is 15 minutes compared to the existing 8 minutes soit is discarded.

Origin

4

2

3

2

6

6

4

A B

C D

E F

Origin

4

2

3

2

6

6

4

A B

C D

E F

Origin

4

2

3

2

6

6

4

A B

C D

E F

Step 1 Step 2 Step 3

Origin

4

2

3

2

6

6

4

A B

C D

E F

Step 4

Origin

4

2

3

2

6

6

4

A B

C D

E F

Step 5

Origin

4

2

3

2

6

6

4

A B

C D

E F

Step 6

Figure 1.

14

This simple example may not make the great advantage of calculating all paths from a given origin at onceimmediately clear. Note, however, that with only 9 computations, this algorithm has created the minimumtime path to all 5 nodes in the network. Note also that in this example, the existing path was always selectedin preference to the new path. This is not necessarily the case. For example if the time from E to F were 0.5minutes then path ACEF (8.5 minutes) would have been retained to node F in Step 5 and the path from D to Fdropped (ACDF = 9 minutes, ACD would still be the minimum path to D). Notice in the final tree there isalways exactly one link on a minimum time path entering every node in the network (except the origin). Iftrip interchange A-F were being loaded to this network it would be loaded starting at F, first to link DF sincethis is the only link entering F, then to link CD, the only link entering D, lastly to link AC, the only linkentering C.

The tree building method demonstrated builds paths to each node in the network. This method is fine unlessturn penalties or prohibitors are introduced into the network. Minimum time paths are not always found in anetwork with turn penalties/prohibitors by the tree algorithm. To solve this problem the vine algorithm wasdeveloped. The vine algorithm builds minimum time paths to each link at each node in the network. Thus inStep 2 of the above example, when a minimum time path was built to node D from C, two additional pathswere actually created, that from link CD to link DB and that from link CD to link DF. Because more pathsare created the computation time and computer storage requirements are larger than those of the tree method.The difference will be pointed out in an example. For a more rigorous formulation of the vine algorithm see(13).

Figure 2. shows the same network used in the previous example. In this example a turn penalty may exist atnode D as shown. Three cases of turn penalties are listed below the diagram with the path selected by thevine and tree algorithms shown beneath each. Note that with no turn penalty both methods give the samepath that was found in the previous example.

Origin

Destination

4

2

3

2

6

6

4

T

Path 1

Path 3

Path 2

A B

C D

E F

T=0 (No Turn Penalty)Tree VinePath 2 Path 29 minutes 9 minutes

T=2 minutesTree VinePath 2 Path 111 minutes 10 minutes

T=Infinite (Prohibitor)Tree VinePath 3 Path 114 minutes 10 minutes

Figure 2.

15

Without the turn penalty both algorithms choose the correct path as would be expected. With a turn penaltyof 2, the tree algorithm continues to use Path 2 even though the time penalty has made this path longer thanPath 1. When the right turn is prohibited altogether the tree algorithm selects Path 3 which is much longerthan Path 1. Why is this? Recall in the first example in Step 2 the tree algorithm built a 5 minute path tonode D from C. In Step 3 a 6 minute path to node D via node B was constructed but is discarded because it islonger. When paths are built from node D in Step 4 it results in an 11 minute path to node F if there is a turnpenalty of 2 minutes and no path to node F if there is a prohibitor. For 2 minute turn penalty the 11 minutepath ends up being shortest because the path via link BD is no longer available having been discarded in Step3. For the case of the turn prohibitor a viable path is not found until Step 5 when the 14 minute path via E isconstructed. Another thing to note, if link EF did not exist, the tree algorithm would not find any path fromA to F if the turn prohibitor were used. Thus trips from A to F would not be able to load to the network andan error message would be received. Even in this extraordinary case the tree algorithm will not use Path 1.Thus it is important to be careful with turn prohibitors when using a tree building algorithm.

The vine algorithm on the other hand builds minimum time paths to the links connected to a node. Thus inStep 1 instead of building 2 paths to B and C, three paths are constructed, one from link AB to link BD (4minutes), one from link AC to link CD (2 minutes) and one from link AC to CE (2 minutes). Note that thetime to CD and CE are the same because there are no turn penalties. If there is a 2 minute turn penalty at Dthen in Step 2, the path from CD to DF will be 7 minutes instead of 5 because the turn penalty is accountedfor in order to turn onto link DF from CD. When the path from link BD to DF is constructed in Step 3 it isnot discarded as before because this 6 minute path is now being compared to the 7 minute path via node Cinstead of the 5 minute path constructed with the tree algorithm.

A further point made by example 2 has to do with stair stepping. It was mentioned previously that turnpenalties are often used to alleviate stair stepping. This is a condition whereby minimum time paths makealternating left and right turns through a grided network instead of remaining on one route as long as possiblebefore turning. It is known that actual drivers tend not to stair step through a network. Notice that Path 2(selected with no turn penalty) is stair stepped, while Path 1 (selected when the turn penalty is added) is not.This is a good illustration of how turn penalties can be applied to eliminate stair stepping. The rationale isthat drivers do not stair step because there is an impedance associated with making a turn, this impedancemay be due to the extra time to turn (especially in the case of left turns) or it may be partly psychological.Whatever the case, the turn penalty is used to represent this impedance.

The AON assignment with the vine building algorithm is guaranteed to yield minimum time paths through anetwork with or without turn penalties/prohibitors. Unfortunately, as mentioned previously, the AONassignment does not take into account the fact that not all trips between two place use the same route. Mostsignificantly, it does not account for the effects of congestion on the choice of route. Volume of traffic on aroadway is directly related to the speed (travel time) that can be maintained on that road which in turn affectsthe selection of minimum paths which affects the volume of traffic. This circular relationship must be solvediteratively until speeds and volumes are brought into balance. Before this can be done a relationship isneeded between speed (or travel time) and volume on a roadway. The relationship that has traditionally beenused in Ohio and the rest of the country is known as the BPR (Bureau of Public Roads) equation.

The BPR Equation

The BPR equation is the relationship used in Ohio to relate network link volumes to the travel time on thatlink. This relationship was developed by the Bureau of Public Roads, however its sources are obscure. Itsapplicability to actual traffic flow conditions has been questioned. Despite this, it continues to be the mostused volume-travel time relationship in the country. The reason is that this equation does produce a trafficdiversion effect due to over-capacity conditions and can be made to give results which compare well withground counts. Despite its heuristic success, the primary reason for the BPR equations continued use is thatso many practitioner use it that it is the accepted standard. The BPR equation is:

T = To[1 + 0.15(V/C)4]

16

whereT = Balance Travel Time (travel time adjusted based on assigned volume)To = Free Flow Time (0.87 * time at practical capacity for capacity restraint)V = Assigned VolumeC = Practical Capacity of Link

The Free Flow Time To is listed as 0.87 times the time at practical capacity for capacity restraintmethodologies. This time is that coded on the link (or that resulting from the speed on the link) during thefirst iteration and the time from the previous iteration thereafter. (Planpac (6) uses a different definition ofTo. It is calculated from the BPR curve based on the ground count, capacity and coded speed unless there isno ground count in which case it uses the coded speed for To. In Planpac To does not change with eachiteration.) Thus another way of stating this formula when using capacity restraint is:

Tn = 0.87Tn-1[1 + 0.15(V/C)4]

For equilibrium assignments, this definition of free flow speed is not used. Free flow speed is held constantthroughout the various iterations and is generally set equal to the link speed which should be coded as freeflow speed instead of congested travel speed.

The BPR curve can be equivalently stated in terms of speed by inverting it to:

S = So/{[1 + 0.15(V/C)4]}whereS = Balance Speed

This is the form most often seen graphically as seen below.

B P R C U R V E

V /C

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1

Figure 3.The capacity in the BPR equation has been defined as the practical capacity of the link. Unfortunately, manypractitioners have forgotten what practical capacity means. Practical capacity is a term that was last used inthe 1950 Highway Capacity Manual. It is defined there as: "the maximum number of vehicles that can pass agiven point on a roadway or in a designated lane during one hour without the traffic density being so great asto cause unreasonable delay, hazard, or restriction to the driver's freedom to maneuver under the prevailing

17

roadway and traffic condition." Practical capacity is later equated to design capacity in that manual. Designcapacity has typically meant LOS C in Ohio, thus LOS C capacity is used in the BPR equation and is thuscoded as such in transportation networks. This equation of LOS C capacity to practical capacity is reinforcedin the UTPS (20) UROAD documentation which performs an automatic conversion from LOS E to LOS Ccapacity when using the BPR curve.

The implications of this definition of capacity are many. Notice that the BPR equation bears no relationshipto the possible (LOS E ) capacity. This is intentional. Recall that the BPR equation extends theoretically toany V/C ratio even those that would be impossible in reality. This is necessary because the assignmentprocess may create volumes far in excess of capacity. Because of this there is no special significance to theLOS E capacity. The relationship to LOS C capacity is important because it was assumed by the developer ofthe curve that the speeds (or time) coded in a network from current travel time runs representedapproximately those at practical (LOS C) capacity and that any more congested situations would only occurin the future (an assumption that may have been true when the equation was created in the 1950's). Thisassumption is easily seen in the equation itself where at practical capacity (V/C=1) the equation yields:

Tn = 0.87Tn-1[1 + 0.15(1)4] = Tn-1

Thus at practical capacity no adjustment is made to link times because the volume and time are alreadyconsidered to be in balance. This also points out that the link times are practical capacity times not free flowtimes. The implications of changing the definition of capacity from practical (LOS C) to possible (LOS E) orof changing the definition of link speeds from actual (practical capacity) speeds to free flow speeds as hasbeen suggested are thus complicated and inter-related. If free flow speeds are coded to a network as isrequired for equilibrium assignment, then the factor 0.87 (sometimes called the level of service factor) shouldbe set to 1.0. Note that in this case at V/C=1 there is a travel time adjustment of:

Tn = Tn-1[1 + 0.15(1)4] = 1.15Tn-1

or in terms or speedSn = Sn-1/[1 + 0.15(1)

4] = 0.87Sn-1

If the capacity definition is changed, then the .15 factor needs to be changed so that the speed at V/C= 1.0represents the proper fraction of free flow speed. Notice that the assumption in the .15 factor is that atpractical capacity the travel time is 1.15 times that at free flow conditions and the speed is .87 times the freeflow speed. Current research (11) on the speed-density relationships indicate that speed is between 0.98 and1.00 times free flow speed at LOS C and between 0.85 and 0.90 free flow speed at LOS E. This would seemto indicate that the given coefficient of 0.15 is actually more suited to LOS E capacities instead of LOS Ccapacities.

The problem with changing the capacity definition is that to do so without changing the BPR curve will makethe model relatively less sensitive to capacity and there will be less diversion effect due to the capacityrestraining process. The question that must be answered is: which capacities give better assignment results?This is a question that can easily be answered with the capacity calculator program CAP94 which canproduce LOS C or LOS E capacities. At this time it is still ODOT policy to use LOS C capacities in themodel. If it can be shown that reasonable results (compared to ground counts) are obtained using LOS Ecapacities, then this policy may be waived.

The last part of the BPR equation that deserves mention is the exponent. This exponent governs the shape ofthe curve shown in Figure 3. Larger exponents result in flatter curves with a sharper drop in theneighborhood of V/C=1.0. Smaller exponents result in smoother curves with more speed decline (comparedto the standard exponent of 4) when V/C < 1.0 and less speed decline when V/C > 1.0. A plot of BPR stylecurves with various exponents is given below:

18

BPR Exponent Variation

V/C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

EXP=0.5

EXP=1.0

EXP=3.0

EXP=4.0

EXP=6.0

Figure 4.

The only justification for changing this exponent would have to come from empirical data available for agiven area, possibly for different classifications of roadway. It is entirely possible to use different BPRcurves for different types of roadways in the assignment process.

Capacity Restrained Assignment

Capacity Restraint is the traditional method whereby the relationship between travel speeds (or times) andtraffic density (assigned volume) is taken into account. The capacity restraint procedure is an iterativeprocess in which volumes are assigned, speeds adjusted, volumes reassigned, etc. until a sufficient number ofiterations have been performed whence the speed and volume are said to be balanced. In Ohio, 2 iterations ofspeed adjustment using the BPR curve with 3 AON assignments are used in the capacity restraint process.The general sequence of operations is:

1. Perform AON assignment using actual travel time speeds coded on network (This is called iteration 0).2. Adjust individual link speeds based on iteration 0 assignment and the BPR curve. (Note that To in the

BPR curve is 87% of actual travel time from the network thus So is 115% of the actual speed coded onthe network).

3. Calculate a weighted speed equal to 75% of the original speed plus 25% of the adjusted speed.4. Perform AON assignment using this weighted speed (iteration 1).5. Adjust individual link speeds based on iteration 1 assignment and the BPR curve. (Note that So in the

BPR curve is now 115% of the weighted speed calculated in step 3.6. Calculate a weighted speed equal to 75% of the previous weighted speed plus 25% of the adjusted speed.7. Perform AON assignment using this weighted speed (iteration 2).8. Weight the three iterations together in the ratio 40% of iteration 0 plus 40% of iteration 1 plus 20% of

iteration 2 to produce a final weighted assignment.

19

The weighting that occurs in steps 3 and 6 is standard practice (the given weights are defaults supplied withboth Planpac and Tranplan) to reduce the oscillations that occur in the link speeds due to the capacity restraintprocess. These weights are heuristic values that need to be employed because of the fact that this capacityrestraint procedure is not a mathematically convergent solution. Rather it has been found through trial anderror that the weights and factors used in the capacity restraint process tend to yield good correlation withground counts. The same is true of the weighting percentages used for the three AON assignments. Thismethod cannot be justified mathematically, however, it appears that the method in a very rough waysomehow simulates human behavioral patterns on a 24 hour basis. Perhaps for example people (who makedecisions based on limited information) select their travel paths from a set of only 3 alternatives and onaverage 40% are likely to take the AON minimum path, 40% the path we calculate in iteration 1 and 20% thelast path.

It is important to point out that this method needs actual travel speeds coded on the links, not free flowspeeds. This is because the method is very dependent on the first AON assignment being close to the correctsolution. As will be seen later, the equilibrium solution which is mathematically convergent has no suchrestriction. Another point that should be made is that the speed adjustment in step 5 bases its free flow speedon the speed from the previous iteration. For links that were highly over assigned in iteration 0 this speedwill be unrealistically low and the link will not be able to recover much volume in subsequent iterations.Take for instance the following example:

Assignment Speed V/C BPR Speed Weighted SpeedIteration 0 50 mph (link coded spd) 3.0 5 mph 38.75 mphIteration 1 38.75 mph 0.0 44.56 mph 40.20 mphIteration 2 40.20 mph

What this example shows is that it is easier for speed to decrease due to high V/C ratios than it is to increasedue to low ones. The BPR curve can reduce a speed to almost zero while the maximum speed increase isonly 15%. This example also points out how the 75/25 weighting step reduces speed oscillation, in iteration1 the assignment speed is 38.75 mph while it would have been 5 mph without the weighting.

Equilibrium Assignment

The equilibrium assignment process arose as a result of the recognized short comings of the capacity restraintprocess. Most notable, the capacity restraint process's failure to converge to the true solution. What is thetrue solution? It has been stated (16) that the true solution to the assignment of traffic to a congested networkis that which gives: "the assignment of vehicles to links such that no traveler can reduce his or her travel timefrom origin to destination by switching to another path." This is achieved through the use of the equilibriumassignment. Equilibrium is a mathematically sound and convergent algorithm utilizing the Frank-Wolfe (22)method to solve the following nonlinear programming problem:

Let L = The set of all linksp = The set of all pathso = The set of all originsd = The set of all destinationsvL = The number of vehicles on each link LF(vL) = a function relating travel time to volume (such as the BPR curve)tpod = The number of vehicles from origin o to destination d on path pLpod = 1 if a link L belongs to path p from origin o to destination d, 0 otherwiseTod = The trip table of all trip interchanges

then find min L 0vL F(x)dxsubject to vL = odp Lpod tpodp tpod = Tod

tpod > 0

20

What this says in English is: minimize the area under the speed density curve for all links in the networksubject to three conditions. First the volume on each link is determined by the paths in the network and thevolumes of the trip interchanges in the trip tables (this is simply what occurs when performing an AONassignment). Second, the number of vehicles between 2 zones on all paths must sum to the total vehiclesbetween those zones (this is automatically satisfied in an AON assignment because each trip interchange isloaded to only one path). Third, no trip interchange on a given path may be negative (again automaticallysatisfied in AON). For an explanation of why the objective function involves minimization of the area underthe speed-density (BPR) curve see (16) where a graphical example is presented which intuitively describesthis function.

Because the AON assignment process meets all of the above constraints, it seems logical that it should formthe basis of the equilibrium algorithm. The speed-density curve can be any suitable relationship, Tranplan(21) and UTPS use the BPR curve in the equilibrium algorithm. The algorithm for solving an equilibriumassignment then becomes:

1. Perform an AON assignment based on link speeds. (iteration 0)2. Recompute travel time using BPR curve based on last assignment. (Note the BPR curve is constant with

each iteration, To is not adjusted based on the previous assignment)3. Perform another AON assignment using the new travel times. (iteration 1a)4. Combine iterations 0 and 1a linearly using a value (such that it1b = (1-)it1a + it0) selected so as to

minimize the objective function. (iteration 1b)5. Check for convergence, if close enough stop, if not go to step 2.

In Tranplan it is possible to modify this algorithm by inserting a step 2a in which the travel time (speed)resulting from the BPR formula is weighted based on the previous speed (such as is done in steps 3 and 6 ofthe capacity restraint procedure.)

Equilibrium assignments have not previously been used in Ohio because it has been found that using existingnetworks, this method does not reproduce ground counts as well as the capacity restraint process. This is notsurprising given that the existing networks contain impedances that have been calibrated to make the capacityrestraint process yield acceptable results.

Additionally, it should be remembered that the equilibrium assignment process as currently formulated insoftware such as UTPS and Tranplan utilizes a free flow definition of speed in the BPR curve (i.e. link traveltimes are not multiplied by 0.87) while our current networks have actual travel speeds coded. The reasons forusing free flow speed are twofold. First, it has been argued that free flow speed should be used because itrepresents an upper bound to the range of possible speeds. When actual speeds are used, the BPR curve canincrease the speed due to a low V/C ration by as much as 15% at each iteration. Use of free flow speedprevents this. This is not possible in the capacity restraint process which depends on actual speeds in the firstiteration to produce good results. Second, the equilibrium method holds To constant with each iteration andthe use of free flow speed allows the congested speed to vary from To to zero at each iteration. If actualcongested speed were used with the level of service factor of 0.87 then the maximum speed obtainable wouldbe 115% of the congested speed, it is thus more rigorous and straight forward to use the free flow speedwhich then becomes the maximum possible speed. This definition of speed alleviates the problem withcapacity restraint, discussed previously, in which each iteration defines To based on the previous iteration andthus speeds tend to decrease with each iteration.

Finally, it should be remembered that the congestion constraints which the equilibrium algorithm modelsactually occur on a sub-hourly basis not 24 hours. Because of this, a rigorous mathematical formulationshould not be expected to yield good correspondence to 24 hour ground counts but should be expected toyield good results on an hourly basis. Recall that the 40-40-20 capacity restraint process was heuristic innature, meaning the various weighting factors were chosen because they gave the best results for 24 hourmodeling, therefore it may have an advantage in this realm (although it is impossible to say whether or notthis is always true given that the capacity restraint algorithm cannot be proven mathematically).

21

Because this method has not been previously used in Ohio there are currently no guidelines governing its use.Equilibrium assignments may be used, however, it is important that model checks be made whichdemonstrate that this methodology produces assignment results which match ground counts within thetolerances prescribed later in this document.

Feedback Loops

Feedback refers to taking the results of the traffic assignment, most specifically the congested travel timesand feeding them back into the trip table synthesis process. As discussed in (3), trip table synthesis dependsto some degree on the travel time between traffic analysis zones. These travel times are modified in thetraffic assignment process when using some type of capacity restraint so it would seem logical to use thesemodified travel times for trip table creation and then iteratively recalculate the traffic assignment. Feedbackfrom traffic assignment to trip table synthesis models is currently not used in Ohio for several reasons. First,the difference in final 24 hour assignment volumes is known to be minimal, therefore in the past theadditional effort was not deemed worthwhile. Second, trip table creation and traffic assignment havetraditionally been performed by separate groups at ODOT, thus ODOT was not organizationally prepared touse feedback. Finally, feedback to trip table creation is invalid when using 40-40-20 capacity restraintbecause this method is an heuristic approach in which only the final 24 hour volume outputs are known to beaccurate. The travel time data that results should not be used in trip table creation. Because of the currentdesire to use these models to analyze air quality, congestion management, TCM's etc. the first of the abovereasons is no longer relevant, particularly in light of recent law suits which have forced areas to implementfeedback regardless of its technical efficacy. The second reason disappears in an MPO environment and canbe over come by ODOT as well. The final reason is not valid if a study area converts its capacity restrainingprocess to the equilibrium assignment method. This convergent algorithm should give reasonable travel timedata for use in trip table synthesis. Thus implementation of feedback loops is only recommended when and ifan area has successfully implemented an equilibrium assignment process.

Count Restraint

Count restraining is a process whereby impedances are adjusted on links to make the assigned volume matchground counts automatically. This procedure is generally not used today primarily because of therequirement that the base year and future year assignment processes be the same and there are no groundcounts for the future year network.

22

IV. Assignment Limitations, Checks and Refinements

This section first summarizes some of the limitations inherent in the traffic assignment process describedherein. It then discusses checks that should be made to determine how well the traffic assignment reproducesground counts. These checks are obviously only necessary when validating the base year model and areskipped when creating forecasts. Finally, a set of refinements that can be used to improve the assignmentresults are described.

Assignment Limitations

The traffic assignment process described here is an approximate solution based upon aggregations of varioustravel behavior phenomenon. As such there are a number of errors built into the process. The first source oferror is due to aggregation of travel behavior into traffic analysis zones and the loading of the trips atcentroids. This process can create discontinuity in link volumes near the centroid connector and can result inunrealistic assignments in the neighborhood of the centroid connectors since real traffic loads onto thehighway system from many access points not just 1 to 4. In addition, the use of traffic analysis zones meansthat some vehicle trips will be intrazonal and therefore will not enter the transportation system. Anothersource of error is the level of detail of the network. Not all roadways are included in the traffic assignmentnetwork. It is possible that there will be over assignment if not enough links are included and underassignment if there are too many. The relationship between the number of zones and the number of links isimportant for diagnosing this problem. A balance must be struck such that the intrazonal trips, and thevolumes on the centroid connectors approximate the traffic on the unmodeled highway system.

The routing rules of AON and capacity restrained assignments are another source or error. These methodsassume that traffic takes the minimum time paths (with or without the effects of congestion) between originsand destination. This is not a completely accurate reflection of human choice behavior. While travel time isan important factor in route choice, it is not the only factor. Distance, tolls, vehicle operating costs, comfort,stress etc. all play a role in route selection. In addition, because people make decisions based upon limitedinformation, they do not always make the optimum choice when selecting routes.

Another error occurs when capacity restraining procedures are applied to 24 hour assignments. Congestionconditions which cause route diversion actually only occur over a short time period. Additionally, the trafficassignment process itself is an aggregate representation of static link volumes. It does not take into accountthe effects of individual vehicle interactions nor the dynamics behind vehicle movements and congestionevents. Dynamic traffic assignments have been designed to replicate these phenomena, however, they havenot been applied to regional models due to the large amount of computations required.

When forecasted traffic assignments are used an error can occur due to the basic assumption that conditionsin the current year will be present in the future. It is true that the future year network will represent proposedchanges to the system but it does not reflect more subtle changes. For instance, the definition of ultimatecapacity on a freeway has changed (11) from 2000 pcphpl to 2200 pcphpl between the last two printings ofthe Highway Capacity Manual. This change reflects differences in driver behavior as drivers become moreaccustomed to congestion. The change will have some effect on traffic assignment results though it wouldnot have been reflected in a past forecast assignment.

Finally, the errors occurring in a traffic assignment may be the result of errors in the input trip table. The triptable synthesis process has many of its own errors built in. These are described more fully in the Trip TableSynthesis Procedures document. Two of the most basic problems with trip table synthesis are that the wrongnumber of trips are created in a zone or the trips are sent to the wrong zone. These conditions can berecognized to a certain degree using the assignment checks discussed in this section. Other checks (4) shouldbe made, however, when problems with trip table synthesis are expected.

Assignment Checks

23

There are four basic checks that should be made on a base year assignment. These are the root mean squareerror check, vehicle miles traveled (VMT) check, screen line check and plot check of individual linkvolumes. In addition, a fifth check, the select link check, is made when necessary to diagnose assignmentproblems.

Plot CheckThis is the best check of a traffic assignment. A plot is made of the network with assignment volume andground count annotate on each link. Each link is inspected for assignment accuracy and routes are analyzedfor assignment consistency. This check involves individual judgment based on experience as to how goodthe traffic assignment is. It is the only convenient way to find spot problems with the transportation networksuch as problems with centroid connectors and incorrect impedances. GIS software is useful for makingthese plots. With this software, paper plots are not even necessary as the results can be viewed on the screen.Unfortunately, this check takes a considerable amount of time and provides no firm numbers as to the relativeaccuracy of various assignments. To make quicker checks to determine if one assignment is better thananother, the following aggregate checks are made.

Root Mean Square Error CheckThe root mean square error is a measure of the relative error of the assignment compared to ground counts.Root mean square error is a statistical formula given as such:

RMSE = SQRT(L(GC-VA)2/N-1)where GC = Ground Count on Link L

VA = Volume Assigned to Link LL = Set of all linksN = Total links

The percent root mean square error (as compared to the ground count) is used as the measure of relativeaccuracy in the assignment. %RMSE is applied to volume groups rather than all links as a whole. Volumegroups are used because the %RMSE can be higher for lower volume links. The reason is what has beencalled the "add a lane, drop a lane" criteria of traffic assignment accuracy. The idea is that traffic assignmentsshould be accurate enough such that a highway design resulting from it will have the right number of lanes.For high volume roads, smaller percentages will result in different numbers of lanes while at low volumeerrors well over 100% will not change the number of lanes. Take for example a road with a ground countAADT of 100 and an assigned volume of 500. Even though the assignment is 500% off, it still results in a 2lane road. The volume group ranges to be used for validation of traffic assignment models in Ohio is asfollows:

Required0-499, 500-1499, 1500-2499, 2500-3499, 3500-4499, 4500-5499, 5500-6999, 7000-8499, 8500-999910000-12499, 12500-14999, 15000-17499, 17500-19999Optional20000-24999,25000-34999,35000-54999,55000-74999,75000-120000

In some cases a study area will have very few if any counts in the higher volume groups. In these cases thehighest volume groups from the optional group may be dropped if they contain no data. In addition, if thehighest optional volume groups contain very few counts they may be grouped together into a larger volumegroup. Thus examples of an areas top two groups could be 17500-19999 and 20000-34999 or 20000-24999and 25000-74999 or 15000-17499 and 17500-19999 etc.

These volume groups are based upon the volume of the directional ground count. Note also that the statisticsare only accumulated for links with ground counts. Summaries of %RMSE for links without ground counts

24

should not be analyzed. The add a lane, drop a lane criteria results in a curve of maximum %RMSE (4)asfollows:

Allowable Percent Root M ean Square Error

Volum e G roup

0

10

20

30

40

50

60

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Figure 5.

This curve is shown in tabular format below:

Volume (100's) 25 28 40 47 57 70 80100120140170210262343468586700802

Allowable %RMSE 56 54 49 46 43 40 38 35 33 31 29 27 25 23 21 19 17 15

The %RMSE from each of the above volume groups should be plotted vs. this curve. If all volume groupsfall below the curve the assignment is said to have passed the %RMSE test or the add a lane, drop a lanecriteria. The %RMSE from two assignments can be compared and that with the lowest is said to be a betterrepresentation of ground counts overall. Note that this test is performed on an aggregation of links notindividual links. Therefore, individual links do not necessarily meet the add a lane, drop a lane criteria. It isalmost impossible to get all links to pass this test. Assignment plots as discussed previously should be usedto find those locations where this does not hold. Assignment results in these areas are usually adjusted asdiscussed later. This test is the single most important gauge in determining the validity of a base yearassignment.

Vehicle Miles Traveled CheckInstead of looking at the assigned volume directly, this check looks at the volume times the distance of thelink to produce vehicle miles. The check is made on aggregations of links cross tabularized in two ways.The first way cross tabulates VMT's by functional class and administrative class. This check should result inVMT's within about 10% of the ground count for each functional class and each administrative class (as a

25

whole not within each functional/administrative class combination). Certain classes may have only a fewlinks such as local roads and township roads. These classes may have slightly higher errors. This check isimportant in determining the relative travel on freeways vs. arterials (and other streets). The VMT splitsbetween these is important because it is the basis of air quality analysis. It has been found in the past thatfreeways tend to take too many of the network trips when using the traditional capacity restraint and all ornothing methods.

The second method of cross tabulating VMT is by rings and sectors. Rings and sector are aggregates ofzones which have been coded in most networks and should be maintained. The ring and sector are indicatedby the district number of a link. The first number of the district is the sector number and the second is thering number. Districts are the intersections of rings and sectors thus district 34 is the intersection of sector 3and ring 4. Rings are established circumferentially (concentric circles) from the CBD while sectors emanateradially (like pieces of a pie). Ring and sector boundaries are such that they encompass traffic analysis zones,therefore, a given district will be an aggregation of several zones. The use of ring/sector VMT's allows theassignment to be studied by area either in radial (sectors) or circumferential (rings) corridors or in smallersubareas (districts). This analysis can be used to diagnose two problems. First it points out problems withtrip generation, second it points out problems with zone/network compatibility. If a given district were underassigned the first thing to check is the zone/network relationship in this district. It may be that there are toomany links compared to the number of zones. If this is not the problem then the area may not be generatingenough trips. If over assigned then either there are two few links compared to the number of zones or toomany trips are being generated in the area. Generally the individual rings and sectors should be within about10% of the ground count VMT. This is often impossible to achieve, however, for the CBD ring and sectorwhich generally coincide and are thu