Freight Modelling

A Multicommodity Integrated FreightOrigin–destination Synthesis Model

José Holguín-Veras & Gopal R. Patil

Published online: 4 January 2008# Springer Science + Business Media, LLC 2007

Abstract This paper introduces a multi-commodity, single (generic) vehicleformulation of freight ODS model that combines a commodity-based model toestimate loaded truck trips and a complementary model of empty trips. Thisintegration is important because explicit modeling of empty trips—that account for30% to 40% of total truck trips—is required to avoid significant errors in theestimation of the directional traffic. The formulation is then applied to a case study.Two cases of the proposed model are studied. The first one uses total traffic in theestimation; while the second one is based on loaded and empty traffic. The resultsconclusively show that the models that consider an empty trip submodelsignificantly outperform the model that does not in their ability to replicate theobserved traffic counts. The comparison between the results from the multi-commodity ODS and the single commodity ODS previously developed by theauthors indicates that the multi-commodity formulation brings about substantialreductions in the error associated with the estimation of observed traffic counts.These reductions, in the order of 20% for empty traffic and 40% for loaded and totaltraffic, seem larger than the spurious improvement to be expected from the increasednumber of parameters, suggesting that the multi-commodity ODS formulationperforms better. The results also show some minor improvements in the ability of themulti-commodity ODS formulation to estimate the OD matrices. In terms of themodel's ability to correctly estimate the “true” value of the parameters of the modelsused, i.e., the parameter values estimated by calibrating the model directly from theOD data, it was found that the multicommodity ODS procedure is able to provide

Netw Spat Econ (2008) 8:309–326DOI 10.1007/s11067-007-9053-4

J. Holguín-Veras (*)Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute,JEC 4030, 110 8th Street, Troy, NY 12180, USAe-mail: [email protected]

G. R. PatilDepartment of Civil and Environmental Engineering, Rensselaer Polytechnic Institute,JEC 4002, 110 8th Street, Troy, NY 12180, USAe-mail: [email protected]

fairly good estimates Noortman and van Es's model parameters, though theparameters of the gravity models that came out to be quite different than the “true”values. The overall assessment of the formulation introduced here is that it representsa solid improvement with respect to comparable techniques.

Keywords Origin–destination synthesis . Matrix estimation . Freight demand model

1 Introduction

The transportation planning process usually entails the use of models to forecastdemand, in combination with network models to analyze supply. For the most part,the evolving transportation modeling paradigms have focused on the analysis ofpassenger transportation, while paying little or no attention to freight. An appropriateconsideration of freight issues is important because, notwithstanding its significantcontributions to the economy, truck transportation generates major externalities (e.g.,congestion, accidents, pollution, and discomfort to the population). As an example,suffices to mention that a research project found that the amount of particulate matter(PM2.5) and elemental carbon near truck routes at the Hunts Point terminal in NewYork City are up to three times higher than in a control site far away from the truckroutes (Lena et al. 2002).

However, the goal of improving freight transportation modeling techniques facessignificant challenges. One of the most significant ones is related to the complexityof freight decision making and operations. In this regard, it is widely acknowledgedthat freight is significantly more complex than passenger transportation (Ogden1992; Cambridge Systematics Inc. 1997). This complexity is the result of the uniquecharacteristics of freight transportation. While in passenger transportation, the keydecision maker is the passenger (that by virtue of being the unit of demand,simplifies things tremendously), in freight transportation there are multiple decisionmakers interacting dynamically and deciding on behalf of hundreds, thousands andeven millions of shipments. These interactions, that by force have to respond todynamic market conditions, take place outside the view of transportation plannersand modelers and almost always involve commercially sensitive data andinformation that the companies involved do not want to make public.

In this context, it is interesting to compare what happens to passenger and freightdemand models when the underlying behavioral rules are used in model derivation.In passenger demand modeling, the reasonable assumption of utility maximizationleads to fairly compact demand models that, in spite of their shortcomings, haveproven to be quite useful in practical applications. In freight transportation, however,the assumption of profit maximization and the recognition of the interactionsbetween the freight agents, lead to complex models based on game theory, spatialprice equilibrium, and the like, that are data hungry and difficult to use in practice.This has hampered the practical implementation of theoretically sound approaches todeal with freight transportation planning. As a result, the bulk of freighttransportation modeling has been characterized by a partial view of the problemwithout attempting to use first principles in model development. The implications ofthis practice deserve some discussion.

310 J. Holguín-Veras, G. R. Patil

Freight transportation involves activities and decisions taken at different layers(dimensions). In a simplistic fashion, there are three key dimensions: financial(value), user demands (commodities) and logistical (vehicle-trips). Profit maximiza-tion, which represents the objective function that each economic agent is trying tomaximize, belongs to the financial layer. The commodity flows are nothing morethan a reflection of the customer demands for goods. The vehicle-trips are simply theresult of the way in which the logistic industry organizes itself to transport thecommodities. It is also obvious that there are other layers, most notably the layer ofinformation flows and infrastructure, though for the purposes of this discussion theyare less relevant. The recognition of the need to explicitly take into account themultiple dimensions of the problem has lead to promising developments based ongame theory, spatial price equilibrium and other related concepts (Friesz and Harker1985; Nagurney and Siokos 1997; Holguín-Veras 2000a; Nagurney and Dong 2002;Friesz and Holguín-Veras 2005). Unfortunately, these techniques have not achievedan adequate level of penetration in practice.

The practical difficulties of integrating these dimensions into workable freighttransportation models that could be used in practice has, in turn, translated intomodeling platforms that, by focusing on only one dimension of the problem, offersonly a partial view of this complex multidimensional problem. For instance,focusing on vehicle trips enables the study of routing patterns and network impacts.However, vehicle-trip formulations cannot either consider the commodity type or beused in multi-modal systems because the vehicle-trip, in itself, is the result of modechoice process that already took place. The inability to consider the commodity typeis a major drawback because, literally, all the research conducted in freight behaviorhas concluded that the commodity type plays a fundamental role in all relevantdecision making processes. As shown in the literature, the commodity type has beenfound to play an influential role in: mode choice (McFadden et al. 1986;Abdelwahab and Sargious 1991); vehicle choice (Holguín-Veras 2002); technologyadoption (Holguín-Veras 2000b); response to pricing and other forms of financialincentives (Holguín-Veras et al. 2007a, b), and so on.

Focusing on the commodity flows enables the consideration of the commoditytype and the economics of production and consumption, and leads to models thatresemble the real life processes by which cargo is generated, transformed, distributedand consumed. However, it leads to other problems. First, there is no easy way totranslate commodity flows into vehicle-trips in the network, particularly in urbanareas where freight vehicles undertake complex routing patterns involving long tripchains that, in average, reach about five stops/tour (Holguín-Veras and Patil 2005).To deal with this, it has been routinely assumed that loaded vehicle trips (with cargo)are proportional to the commodity flows; and that the empty trips could be estimatedfrom the commodity matrices using empty trip models (Noortman and van Es 1978;Holguín-Veras and Thorson 2003a). In doing this, appropriate consideration ofempty trips is needed because it has been conclusively demonstrated that eitherdisregarding empty trips, or assuming that they could be directly estimated from thecommodity flow matrices, lead to significant errors in the estimation of thedirectional traffic (Holguín-Veras and Thorson 2003a, b).

Similarly, a focus on the financial transactions has its own set of advantages anddisadvantages. One of the advantages is that it enables the study of the economic

A Multicommodity Integrated Freight Origin–destination... 311

linkages among economic sectors across different geographic jurisdictions—whichcould be readily done with the assistance of Input–Output models. There are someissues, though. The first one is related to how to estimate the flow of commodityonce the model has estimated the flow of financial transactions. Traditionally, it hasbeen assumed that the flow of commodities run counter to the flow of money.However, due to the increasing economic globalization and complexity of themodern supply chains, this assumption is becoming more difficult to defend.Nowadays, it is typical for headquarters not to be co-located with the warehouses,distribution centers, or manufacturing sites, from where the cargo will ship out. Oncethe commodity flows have been estimated, the issue of how to estimate vehicle-trips(both loaded and empty) has to be dealt with, which ideally should be accomplishedusing the techniques outlined in the previous paragraph for commodity basedmodels.

However, in practice, the bulk of the modeling applications focus on either thecommodity flows, or the vehicle-trips, may be because both of them deal with aphysical unit bearing some resemblance to the passenger case. A similar situation isfound with respect to the modeling techniques used to estimate origin–destination(OD) matrices from secondary data—a process typically referred to as either ODsynthesis (ODS) or OD matrix estimation.

ODS represents an important class of simplified transportation models because ofits potential to reduce data collection costs. Since most of these techniques start witha demand model that is systematically re-calibrated so that its estimates are consistentwith observed secondary data, it is obvious that the shortcomings of the demandmodel assumed will cascade into the ODS process. As a result of this, the use ofvehicle-trip, commodity-based, or a cargo-value model will bring into the ODSprocess the shortcomings of these approaches, that were discussed before.

This paper is based on the assumption that, among the available unidimensionalmodeling platforms (i.e., vehicle-trips, commodity-based, cargo-value), a commodity-based approach is the best alternative from the conceptual point of view. This isbecause it enables to consider the role of the commodity type, and the development ofmodels that resemble the real-life process by which cargoes are produced, trans-formed, distributed and ultimately consumed. A second principle guiding this work isthat commodity based formulations need to be complemented with an empty tripmodel to produce the estimates of empty trips that the commodity based model is notable to produce. To overcome these problems, the paper considers an ODSformulation that combines a commodity based model for estimation of loaded trips,and an empty trip model to produce estimates of empty trips (that, as mentionedbefore, represent 30–40% of truck trips). This formulation is a more general form ofthe one previously developed by the authors, (Holguín-Veras and Patil 2008) that onlyconsidered a generic commodity.

The formulation introduced in this paper considers a multi-commodity, single(generic) vehicle ODS problem, in which the total amount of cargoes shipped outand received by the different transportation analysis zones (TAZ) are exogenous tothe model and known. The formulation is then tested using real-life OD datacollected as part of a modeling project conducted by the first author.

The paper has five sections in addition to the introduction. Section 2 provides asummary of the relevant literature. Section 3 introduces the ODS formulation and


discusses the key assumptions. Section 4 briefly describes the case study and thedata used in the analysis. Section 5 discusses the key results obtained from theapplication of the proposed model to the case study. The concluding sectiondescribes the key findings.

2 Literature review

As discussed before, the fundamental objective of OD synthesis (ODS) is toproduce an estimate of the OD matrix (or matrices) associated with secondarydata, typically link traffic counts or any other easily observable data. Among otherthings, ODS has the potential of minimizing data collection costs and, ultimately,speeding up the process of model calibration and updating. Because of thisimportance, a significant amount of research has been conducted in this importantsubject.

The possibility of reducing data collection costs is at the core of the interest inODS. Although most transportation modelers would agree that there is no substitutefor good and accurate data, collecting OD data is widely known to be a majorundertaking that has to contend with many methodological, sampling and financialissues. For instance: roadside interviews have been found to double count thenumber of trips, thus requiring special sample expansion procedures (Kuwahara andSullivan 1987); on board interviews if not properly expanded, lead to bias in theparameters of random utility models (Ben-Akiva et al. 1985); mail interviews areoften biased because the rate of response varies across the population; and homeinterviews, though able to provide statistically sound estimates of OD, require agreat deal of planning, time, effort and most significantly, money (Willumsen 1978;Ortúzar and Willumsen 2001).

ODS tries to overcome these limitations by bypassing the need for surveys. InODS, the traffic counts—linear combinations of the OD flows—are used to estimatethe OD matrices. Since the number of unknowns (OD pairs) exceeds the number ofindependent traffic counts, the estimation problem is under-specified. Two basicapproaches have been used to address this situation. Structured approaches impose amodel structure on the estimation problem, which reduces it to a parameterestimation problem, subject to traffic count constraints. Unstructured approaches usegeneral principles such as entropy maximization, information minimization, ormaximum likelihood to reduce the feasible space so that the problem has a uniquesolution (Willumsen 1978). In general terms, OD synthesis models can be classifieddepending upon the time-dimensionality of the estimation process and thecharacteristics of the underlying traffic assignment model. The former could besubdivided into: a) static estimation—in which the OD matrix is time invariant;and b) dynamic estimation—in which the resulting OD matrices are time-varying.The publications can be further classified, depending on the traffic assignmentprocess in: (1) not requiring route choice, i.e., problems in which the route choiceprocess can be disregarded (e.g., when estimating turning movements at inter-sections); (2) proportional route choice methods, i.e., problems in which theprobability of using a given route does not depend upon the OD flows—whichimplies separability between the route choice and the OD estimation problems; and


(3) non-proportional route choice methods, i.e., problems in which route choice andOD estimation are interdependent, thus requiring a joint estimation processinvolving equilibrium models.

Although there are dozens of publications in passenger OD estimation, only sixformulations of freight ODS were found (Tamin and Willumsen 1988; Gedeon et al.1993; List and Turnquist 1994; Tavasszy et al. 1994; Al-Battaineh and Kaysi 2005;Holguín-Veras and Patil 2008). Tamin and Willumsen (1988, reprinted in 1992),developed a formulation to obtain the parameters of the gravity-opportunity modelthat best reproduce a given set of traffic counts. Since the formulation developed byGedeon et al. (1993) is aimed at obtaining optimal multicommodity flows inmultimodal networks and since it does not model demand behavior, it will not befurther discussed. List and Turnquist (1994) developed an ODS formulation definingthe problem as a large-scale linear programming problem in which the decisionvariables are the OD flows, and the objective function is a weighted combination ofthe deviations of the estimated volumes with respect to the target values. Theirformulation was extended to estimate U.S.–Mexico travel patterns using the dollarvalues of each commodity group and port of entry as the control variables (Nozick etal. 1996). Tavasszy et al. (1994) estimated unobserved elements of the OD matrix ofan interregional problem using an input–output formulation to estimate productionsand attractions, and a Genetic Algorithm to estimate the OD matrix. The formulationof Al-Battaineh and Kaysi (2005) uses an input–output formulation to estimateproductions and attractions, and a Genetic Algorithm to compute the OD matrix.Holguín-Veras and Patil (2008) developed an integrated ODS model that explicitlytakes into account the empty trips, which it was found to significantly increase theaccuracy of the estimates produced. This formulation combines the ability ofconsidering the commodity type; with a formal estimation procedure to estimateempty trips. However, in spite of its usefulness, this formulation discussed above isonly able to consider a single generic commodity, which prevents the analysis ofpolicies that may have differential impacts on different market segments. In order toovercome this limitation, this paper expands the original formulation (Holguín-Verasand Patil 2008) to consider multiple commodities. This expanded formulationrepresents a more realistic and potentially useful formulation. Its basic mathematicalfeatures are discussed next.

3 Multicommodity OD Synthesis (ODS) model

This section describes the basic components of a multicommodity integrated freightODS model. It is worthwhile to mention that the formulation does not make adistinction between full-truckload and less-than-truckload, because it is assumed thatloaded truck-trips could be reasonably estimated from the estimated commodityflows using a suitable value of the payload. As in the previous formulation (Holguín-Veras and Patil 2008) the paper assumes that: (1) estimates of the productions andconsumptions of cargoes for each of the origin and destination zones, are available;(2) the underlying demand process that determines the commodity flows could beapproximated by a doubly constrained gravity model; and, (3) the flow of empty


trips could be approximated using a Noortman and Van Es's model (1978). In allcases, the (unknown) parameters of the models are determined during the estimationprocess.

Define:

mkij flow of commodity k from i to j (in tons)

akij average payload of commodity k from i to j (in tons per vehicle)yij empty trips from i to jxkij ¼

mkij

akijloaded trips of commodity k from i to j

xij ¼Pkxkij total loaded trips made from i to j

zij ¼ xij þ yij total number of trips from i to j

Assuming that mkij follows a doubly constrained gravity model, as in Eq. (1):

mkij ¼ Ok

i Dkj A

ki B

kj f

kij ð1Þ

Where:

Oki production of commodity k at origin i

Dkj consumption of commodity k at destination j

Aki ;B

kj balancing factors to ensure satisfaction of origin and attraction

constraintsf kij ¼ eb

k cij impedance function(s) for commodity kcij travel cost between from i to jbk impedance parameter for commodity k

The assumption of a gravity model is, in essence, a pragmatic one because itprovides a relatively easy, and to a certain extent flexible, way to estimate ODmatrices accounting for spatial interactions. Although the authors acknowledge theshortcomings of the gravity model, which were discussed elsewhere (Holguín-Verasand Patil 2008) its use as part of the proposed ODS seems justified because of itsrelative computational efficiency.

Again, practicality reasons suggests the use of the simplest, yet conceptually validempty trip model, i.e., the one postulated by Noortman and van Es (1978). Thismodel was selected because the more sophisticated formulations (Holguín-Veras andThorson 2003a) are computationally more expensive. In the Noortman and van Es'smodel, the total flow between an origin i and a destination j is the summation of thecorresponding loaded trips and the empty trips. The latter are estimated as aproportion p of the opposing loaded trip traffic, as shown in Eq. (2). For furtherdetails on the Noortman and van Es's model, its properties and basic limitations, thereader is referred elsewhere (Holguín-Veras and Thorson 2003a, b; Holguín-Veraset al. 2005). This assumption leads to:

zij ¼ xij þ pxij ð2Þ

Replacing the loaded trips by the commodity flows divided by the payloads:

zij ¼Xk

mkij

a kij

þ pXk

mkji

a kji

ð3Þ


If payloads are symmetric:

akij ¼ akji ¼ ak ð4Þ

zij ¼Xk

mkij

akþ p

mkji

ak

!ð5Þ

In matrix format:

Z ¼Xk

Xk þ pXkT� �

ð6Þ

As in most previous freight OD synthesis formulations, traffic counts are used toobtain estimates of the freight OD matrices. In this context, the problem reduces tothe estimation of the parameters of the demand model so that the resulting trafficflows resemble the observed traffic in the network. In terms of the traffic assignmentmodel needed in the ODS procedure, the authors decided to use techniques that arebased on proportional route choice, i.e., route choice that does not change withtraffic flows. Denoting:

plij fraction of traffic traveling from i to j using link lV el estimated truck traffic on link l

The value of fraction plij can be estimated using any route choice model includingall or nothing (AON), which is used in this study. We use distance in miles torepresent travel cost cij. In terms of the gravity and the empty trip models, theestimated traffic on link l becomes:

Vel ¼

Xi

Xj

zij plij ð7Þ

¼Xi;j

Xk

mkij

akþ p

mkji

ak

!plij ð8Þ

¼Xi;j

Xk

xkij þ pxkji

� �plij ð9Þ

Two different objective functions are used to compute the optimal parameters.The first one considers the summation of the squared differences in the observed andestimated total (loaded plus empty) truck traffic in the links, as shown in Eq. (10).

argmin β; pð Þ FV ¼Xl

V ol � Ve

l

� �2 ð10Þ

Where: Vol = observed total traffic volume on link l

The second objective function considers the case where there are data aboutempty and loaded truck traffic. This could represent the case in which data fromweight stations, that sometimes record empty and loaded traffic, is available. In thiscase, the objective function could be decomposed in two terms that represent the


total errors in the estimation of the loaded truck traffic and empty traffic respectively.The reader should be aware that, as discussed elsewhere (Holguín-Veras and Patil2008), the values of FV and FWU are not directly comparable because there is asystematic difference between them. In this case, Eq. (7) could be decomposed as(Holguín-Veras and Patil 2008):

Vel ¼

Xi

Xj

zeij plij ¼

Xi

Xj

xeij þ yeij

� �plij ¼

Xi

Xj

xeij plij þ

Xi

Xj

yeij plij

¼ Wel þ Ue

l ð11ÞWhere:

Wol , W

el observed and estimated loaded link volumes

Uol , U

el observed and estimated empty link volumes

The resulting objective function is:

arg min β; pð Þ FWU ¼Xl

Wol �We

l

� �2 þXl

Uol � Ue

l

� �2 ð12Þ

The paper assesses the accuracy of the formulations developed in the paper inreplicating the observed traffic counts and OD matrices. Three major cases areconsidered: (1) Minimization of total traffic error, i.e., Eq. (10) using a formulationwithout an empty trip submodel; (2) Minimization of total errors (Eq. (10)) with amodel that include an empty trip submodel (Eq. (2)); and (3) Minimization of errorsin loaded and empty traffic, Eq. (12), with an empty trip submodel, i.e., Eq. (2). Inthis study the estimation of β and p is performed iteratively using a golden searchprocedure that obtains the optimal value of a parameter, given the current values ofthe others. The procedure is systematically repeated for all parameters untilconvergence is reached.

4 Case study

The data used in this paper come from a major modeling project conducted by thefirst author in Guatemala City. In that project, roadside origin–destination interviewswere conducted and complemented by classified traffic counts to expand the sampleaccording to time of day and type of vehicle. The OD questionnaire includedquestions about: time of the interview, vehicle type, origin, destination, commoditytype, load factor, number of units, type of trucking company, shipment size,economic sectors at both origin and destination, and activities performed at bothorigin and destination. The sample, comprised of 5,276 observations, was expandedby time of day and type of vehicle, and processed to eliminate double counting oftrips. The overall expansion factor was 6.476. There were 17 survey stations, fivewithin the city itself and the remaining 12 located in the surrounding suburban areas.A full description of the sample can be found in Holguín-Veras and Thorson (2000).The number of observations and expanded trips are shown for each type of truck inTable 1. The test case considers only a single vehicle type (i.e., a generic vehiclerepresenting large two to three axles rigid trucks and semi-trailers). The data for


small trucks, mostly pick-ups, were not included because they were deemed ofquestionable validity for the purposes of this paper because they contain asignificant, though undetermined, number of passenger related trips. The basicnetwork is shown in Fig. 1.

The OD data were post-processed and regrouped into seven major commoditysuper-groups, that correspond to aggregations of two digits Standard Classificationof Transported Goods codes (SCTG) (Bureau of Transportation Statistics 1997).Table 2 shows the definitions of the super-groups and the corresponding averagepayloads from the sample. For the purpose of data collection the area was dividedinto 171 TAZs. These zones are aggregated to 24 TAZs for use in this paper.

5 Parameter estimation procedure

The procedure used for the estimation of the parameter of the gravity models foreach commodity super-group and the corresponding empty trip model is outlined inFig. 2. The estimation process for the version of the model that does not have anempty trip model is similar to the one in the figure. The only difference is that the

Table 1 Number of observations and expanded trips

Vehicle set Number of observations Expanded total trips

Small 1,138 21,572Large 3,288 9,769Semitrailer 848 3,645

Fig. 1 Location of survey stations and network


empty trips are not computed and, therefore, there is no search for the optimal valueof the parameter of the empty trip model. As outlined in the figure, the process startswith initializing the values of βk, k∈C and p. Using these initial values, the balancingfactors of the gravity model, Ak

i and Bkj , and OD matrices are estimated. The next

task is to find the optimal values of βk, k∈C and p, so as to minimize the error in theestimated link traffic (Eq. (10) or (12)). The estimation of Ak

i ;Bkj

� �and (βk, p) is

repeated until the convergence of βk and p. If the empty trip model is considered, thecommodity flow matrices are aggregated and then used as an input to thecomputation of the optimal value of the parameter of the Noortman and van Es'smodel.

Where:

βk,0 initial value of parameter β for commodity kn iteration counterC set of commoditiesκβ convergence limit for βκp convergence limit for pβ Vector of βAk Vector of parameters Ai

k

Bk Vector of parameters Bki

β−k, n Vector of β except for commodity k at iteration nβ−k (β1, β2,…, βk−1, βk+1,…)

6 Estimation results

The parameter estimation process outlined in was applied to the case study. Theprocedure was executed in a Dell Optiplex GX 620 with Pentium 4 CPU of 2.8 GHzand 1 GB of RAM, and took an average 2.5 min to complete the iterations. Forcomparative purposes, two alternative model formulations are used. In the first one,no empty trip model is considered because the gravity models are used to estimateboth empty and loaded trips. The second formulation uses gravity models to estimateloaded trips and an empty trip model to estimate the empty trips. This formulation isapplied to two different objective functions (minimizing the estimation error for total

Table 2 Super-groups of commodities and payloads (tons/truck)

Super-group Description Payload(tons/truck)

OD pairswith data

1 Milk products, perishables, grains, oil, fat, beverages,agricultural, poultry/cattle

19.09 214

2 Construction materials, raw material, and minerals 20.15 2373 Fuel 30.15 654 Chemicals 13.38 765 Textile, wood, lumber, plants 15.65 1056 Metals, machinery, manufacturer, industrial, and electrical 22.45 867 Household goods 17.31 173

All commodities 19.43 345

Metric tons are used


traffic, and minimizing the errors in the estimation of loaded and empty trips).Figure 3 shows the changes of the percent error in the estimation of the parametersas a function of the iteration number. As shown, there are isolations that tend todampen down with the number of iterations until convergence to the optimal values.

Fig. 2 Parameter estimation procedure


The performance of the alternative formulations was evaluated by assessing howwell they estimated: (1) the underlying OD matrices (commodities, vehicles); (2) theobserved traffic flows (total, loaded and empty); and (3) β and p parameter. Thecorresponding estimation errors are shown in Table 3. The values of correlationcoefficient between observed and estimated results are also given in the same table.The results conclusively show that the models that consider an empty trip submodel(Models II and III) significantly outperform the model that does not (Model I) intheir ability to replicate the observed traffic counts. Interestingly, Model I does quitewell in replicating the underlying OD matrices.

The comparison between Models II and III, that only differ in the objectivefunction used, indicates that both models perform relatively the same in thereplication of the calibration traffic counts (with Model II having a slight edge overModel III). However, Model III completely outperforms Model II in the estimationof the commodity flow matrices. This makes perfect sense because the ODSprocedure has more data with which to improve the estimates of the commodityflows produced by the gravity models. Although not very conclusive, this maysuggest that the main benefit of using an objective function such as FUW is toimprove the quality of the commodity flow matrices, as opposed to improving theestimates of traffic in the network.

It is also important to compare the results of the multi-commodity ODS proceduredeveloped in this paper, to the estimates produced by the ODS procedure developedbefore by the authors (Holguín-Veras and Patil 2008) that only considered a genericcommodity. This comparison must be done with some caution because of the factthat by adding additional parameters, the ability of the model to replicate the input

-10

0

10

20

30

40

50

60

1 3 5 7 9 11 13 15

Iterations

Per

cent

age

erro

r

Error for β

Error for p

Fig. 3 Percent error vs number of iterations


data is bound to increase. The comparison between the results from Table 3 (multi-commodity) and Table 4 (single commodity) indicates that the multi-commodityformulation brings about substantial reductions in the error associated with theestimation of observed traffic counts. These reductions, in the order of 20% forempty traffic and 40% for loaded and total traffic, seem much larger than thespurious improvement to be expected from the increased number of parameters,suggesting that the multi-commodity ODS formulation perform better. The resultsalso show minor improvements in the estimation of the OD matrices.

It is interesting to assess how well the ODS procedure used in this paper is ableto correctly estimate the “true” value of the parameters of the models used, i.e., theparameter values estimated by calibrating the model directly from the observed

Table 3 Estimation results from multicommodity ODS

Metric Model I Model II Model III

Characteristics of modelEmpty trip model No Yes, NVE Yes, NVEObjective function FV FV FUW

Number of sampled links 373 373 373Optimal value of p NA 0.4908 0.4549

Optimal value of βMilk products, perishables... 2.70089E−10 1.50515E−11 4.84655E−09Construction materials, raw … 2.28315E−02 1.01660E−09 1.39577E−02Fuel 2.89730E−10 5.74918E−12 1.85122E−09Chemicals 9.95389E−02 7.73304E−02 9.80853E−02Textile, wood, lumber, plants 8.21080E−10 1.46666E−11 1.61463E−08Metals, machinery, manufactures… 6.88586E−05 8.88640E−05 7.33137E−05Household goods 1.58400E−09 3.88041E−11 3.46124E−08

OD matricesCommodity flow matrices (tons)Total sum of squared errors 133,237,918 155,885,227 133,522,431Correlation coefficient 0.824 0.786 0.823

Empty vehicle-trip matrixTotal sum of squared errors NA 128,951 115,355Correlation coefficient NA 0.608 0.639

Loaded vehicle-trip matrixTotal sum of squared errors NA 333,627 295,702Correlation coefficient NA 0.745 0.776

Total vehicle-trip matrixTotal sum of squared errors 627,727 663,117 572,269Correlation coefficient 0.769 0.748 0.782

Traffic volumesEmpty link traffic errorTotal sum of squared errors NA 5,219,851 4,959,598Correlation coefficient NA 0.924 0.926

Loaded link traffic errorTotal sum of squared errors NA 10,859,660.98 11,254,761.63Correlation coefficient NA 0.9643 0.9634

Total link traffic errorTotal sum of squared errors 29,959,420 22,054,281 22,571,979Correlation coefficient 0.9545 0.9652 0.9647

True value of p=0.418


OD data. The true value of p can be found using (Holguín-Veras and Thorson2003b):

p* ¼

Pk

Pi; j

yijxkji

Pk

Pi; j

xkji

� �2

The true value of β is calculated using the standard gravity model calibrationprocedure, in which the difference in the observed and estimated OD values isminimized. This comparison reveals a number of interesting results. First, themulticommodity ODS procedure is able to provide fairly good estimates of theparameter of the Noortman and van Es's model. As shown in Table 3, the estimatesof 0.491 and 0.455 are fairly close to the “true” estimate of 0.418. Unfortunately, thesame cannot be said about the parameters of the gravity models. As shown inTable 5 the estimates from the ODS model are very different than the true values.

Table 4 Estimation results from single-commodity ODS

Metric Model I Model II Model III

Characteristics of modelEmpty trip model No Yes, NVE Yes, NVEObjective function FV FV FUW

Optimal value of p NA 0.554 0.433Optimal value of β 0.0107 1.54E-09908 0.0022Number of sampled links 373 373 373OD matricesCommodity flow matrices (tons)Total sum of squared errors 141,452,681 187,334,914 139,572,075Correlation coefficient 0.81 0.73 0.81Empty vehicle-trip matrixTotal sum of squared errors NA 151,076 114,780Correlation coefficient NA 0.58 0.65Loaded vehicle-trip matrixTotal sum of squared errors NA 408,092 335,487Correlation coefficient NA 0.69 0.76Total vehicle-trip matrixTotal sum of squared errors 700,660 811,951 614,047Correlation coefficient 0.76 0.70 0.77Traffic volumesEmpty link traffic errorTotal sum of squared errors NA 7,292,552 6,227,917Correlation coefficient NA 0.92 0.92

Loaded link traffic errorTotal sum of squared errors NA 17,544,689 19,013,319Correlation coefficient NA 0.95 0.95

Total link traffic errorTotal sum of squared errors 46,516,899 32,948,079 37,120,223Correlation coefficient 0.94 0.95 0.95

True value of p=0.418; true value of β=7.9×10−5 (source: Holguín-Veras and Patil 2008)


7 Conclusions

This paper has proposed and discussed a multicommodity ODS model that integratesa commodity-based demand model, based on a set of gravity models, and acomplementary model of empty trips. The performance of the new formulation wasassessed using data from an actual OD survey conducted by the first author inGuatemala. Two different formulations were tested: (1) the formulation discussed inthe paper; and (2) an alternative version that attempts to estimate total trips withoutexplicitly considering an empty trip model. The results conclusively show thesuperiority of the integrated ODS. The performance of the alternative formulationswas evaluated by assessing how well they estimated: (1) the underlying ODmatrices (commodities, vehicles); and (2) the observed traffic flows (total, loadedand empty). The results conclusively show that the models that consider an emptytrip submodel (Models II and III) significantly outperform the model that does not(Model I) in their ability to replicate the observed traffic counts. Interestingly,Model I (that does not have an empty trip submodel) does quite well in replicatingthe underlying OD matrices.

The comparison between Models II and III, that only differ in the objectivefunction used, indicates that both models perform relatively the same in thereplication of the calibration traffic counts (with Model II having a slight edge overModel III). However, Model III completely outperforms Model II in the estimationof the commodity flow matrices. This makes perfect sense because the ODSprocedure has more data with which to improve the estimates of the commodityflows produced by the gravity models. Although not very conclusive, this maysuggest that the main benefit of using an objective function such as FUW is toimprove the quality of the commodity flow matrices, as opposed to improving theestimates of traffic in the network.

The comparison between the results from the multi-commodity ODS and thesingle commodity ODS previously developed by the authors indicates that the multi-commodity formulation brings about substantial reductions in the error associatedwith the estimation of observed traffic counts. These reductions, in the order of 20%for empty traffic and 40% for loaded and total traffic, seem much larger than thespurious improvement to be expected from the increased number of parameters,suggesting that the multi-commodity ODS formulation perform better. The results

Table 5 “True” and ODS estimated parameters of gravity models

Super-group Description Trueparameter b

Parameter bfrom model III

1 Milk products, perishables, grains, oil, fat, beverages,agricultural, poultry/cattle

0.000101317 4.846549E-09

2 Construction materials, raw material, and minerals 0.250087769 1.395772E−023 Fuel 0.000125235 1.851217E−094 Chemicals 9.36344E−05 9.808529E−025 Textile, wood, lumber, plants 0.200862556 1.614634E−086 Metals, machinery, manufacturer, industrial, and electrical 9.56709E−05 7.331375E−057 Household goods 0.999997951 3.461241E−08


also show some minor improvements in the ability of the multi-commodity ODSformulation to estimate the OD matrices.

In terms of the model's ability to correctly estimate the “true” value of theparameters of the models used, i.e., the parameter values estimated by calibrating themodel directly from the OD data, it was found that the multicommodity ODSprocedure is able to provide fairly good estimates of the parameter of the Noortmanand van Es's model. Unfortunately, the same cannot be said about the parameters ofthe gravity models that came out to be quite different than the “true” values.

The paper shows that the formulation introduced here shows considerableimprovements with respect to previous ones reported in the literature. It seems clearthat the integration of a commodity based model with an empty trip model increasesthe level of realism of the formulation, though a considerable amount of research isstill needed on this important subject.

Acknowledgements The research reported in this paper was supported by the National ScienceFoundation's grant CAREER-0245165. This support is both acknowledged and appreciated.

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