Netw Spat Econ (2010) 10:241–259 DOI 10.1007/s11067-008-9066-7
Commercial Vehicle Empty Trip Models With Variable Zero Order Empty Trip Probabilities José Holguín-Veras & Ellen Thorson & Juan C. Zorrilla
Published online: 1 July 2008 # Springer Science + Business Media, LLC 2008
Abstract This paper considers enhanced formulations to model commercial vehicle empty trips. These formulations relax a limitation of the original trip chain models, i.e., the assumption of a constant probability of a zero order trip chain, p, for the entire range of trip types. The formulations considered in the paper assumed that p follows a logistic function of the opposing commodity flow, and trip distance. The consideration of a variable p significantly improves the relative performance of the models. The relative error for the models with constant p ranged from 4.2% to 19.3% higher than that for the models with a variable p as a function of either commodity flow or distance. Keywords Freight demand modeling . Empty trip modeling
1 Introduction One of the most important activities in transportation planning is the estimation of future transportation demand which, together with network (supply) analyses, provide the basis for decisions pertaining to transportation investment needs. In order to make sound decisions, a comprehensive set of transportation demand
J. Holguín-Veras (*) Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, JEC 4030, 110 8th Street, Troy, NY 12180, USA e-mail:
[email protected] E. Thorson University Transportation Research Center, The City College of New York, Marshak Hall Suite 910A 138th Street & Convent Avenue, New York, NY 10031, USA e-mail:
[email protected] J. C. Zorrilla 100 Cambridge Park Drive, Suite 400, Cambridge, MA 02140, USA e-mail:
[email protected]
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models should be developed to represent the behavior of the different user segments (e.g., passengers, freight) as a function of social, economic and policy variables. Once such models have been estimated and calibrated to ensure they reflect the local conditions, they are used for forecasting purposes. Unfortunately, the lack of adequate freight transportation demand models remains a formidable obstacle to taking full advantage of the process described above. There are several reasons that may help explain why freight transportation modeling is lagging behind passenger transportation modeling. A first reason may be related to the fact that transportation modeling started as an exercise that focused almost exclusively on passenger transportation, with freight being no more than an afterthought. This point of view may have had its origins in the fact that commercial truck traffic usually represents a relatively small proportion of total traffic (typically between 6–10%). However, in spite of the relatively small numerical significance of commercial truck traffic, its economic value and its contributions to network congestion are considerable. With travel time values that routinely reach $50–70/h, compared to $10–15/h for passenger demand, the economic value of commercial traffic could easily represent 20–35% of total economic value of the trips made in a region. (The same line of reasoning applies to network congestion.) A second set of reasons pertains to the chronic lack of understanding and appreciation of the importance of freight activity to the economy, and the history of confrontations between the government and the freight industry (for discussions of the American case, see Holguín-Veras and JaraDíaz 1999; Holguín-Veras et al. 2005a) that have shaped the minds of countless transportation professionals and community leaders into believing that freight activity, and particularly truck traffic, is something to get rid of; not a topic that deserves to be understood and researched. A third set of factors is associated with the staggering complexity of the underlying dynamics of freight activity. There are multiple players that dynamically interact in a highly confidential and commercially sensitive environment (e.g., shippers, freight forwarders, third party logistic systems, freight carriers, receivers, government regulatory agencies). There are literally hundreds of thousands of different commodities with a very wide range of opportunity costs and handling needs. More important for the purposes of this paper, freight demand can be measured and defined according to different dimensions: cargo value, weight and number of vehicle-trips. The multidimensional nature of freight demand has given rise to two major modeling platforms: vehicle-trip based and commodity based (cargo value is only used in Input–Output models). Vehicle-based models focus on modeling the actual number of vehicle trips, which has several advantages. Among them are the relative ease and high-quality with which traffic data can be obtained; and, since the model focuses on vehicle trips, no distinction is made between empty and loaded trips. A key limitation of vehicle-trip modes is that they cannot be applied to multimodal systems because the vehicle-trip is the result of a mode choice that already took place (Holguín-Veras and Thorson 2003a). Furthermore since the models assume that the vehicle-trip is the unit of demand, as opposed to the commodity being transported, there is no way to consider the economic characteristics of the shipments. This is a rather serious limitation because the commodity type has been found to be a very important explanatory variable of a number of choice processes involving freight (Holguín-Veras 2002). Commodity based models, as the name
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points out, focus on modeling the flow of goods between zones (measured in a unit of weight). Since the cargo’s weight is the unit of demand, the consideration of cargoes’ attributes (e.g., value, weight, type) is straightforward. In this platform, the loaded trips are estimated by dividing the total flow from one region to the other by a suitable payload for all loaded trucks. The problem with commodity-based models is that they are unable to model empty trips, which can make up about 30 to 40 percent of the total trips in a region (Holguín-Veras and Thorson 2003a). This occurs because the commodity flow in one direction determines the corresponding loaded trips, but does not bear a direct relationship with the empty trips. To resolve this, complementary empty trip models have been developed, such as Noortman and van Es’ (in Hautzinger 1984), Hautzinger’s (1984), and Holguín-Veras and Thorson (2003a). In this context, the empty trip models use the commodity flows estimated by a freight demand model as an input for the estimation of the corresponding empty trips. Having done that, the empty trips are added to the loaded trips to obtain the total vehicle trips that would be used in the traffic assignment process. The main advantage of this sequential estimation is that it enables the researcher to take advantage of existing commodity flow models. It is also clear that if comprehensive freight models—which are able to estimate both commodity flows and vehicle trips— are used, there is no need to use separate empty trip models (e.g., Holguín-Veras 2000). Far from being of purely academic interest, the correct estimation of commercial vehicle empty trips is very important for transportation planning purposes because not doing correctly will lead to severe directional errors in the estimation of commercial vehicle traffic, as shown in Holguín-Veras and Thorson (2003b). This, in turn, may have important implications in terms of determining road capacity improvement needs. The main objective of this paper is to contribute to freight transportation modeling by enhancing the methodologies used to estimate empty trips from previously estimated commodity flow matrices. The paper builds on the developments of Noortman and van Es (in Hautzinger 1984), Hautzinger (1984) and Holguín-Veras and Thorson (2003a). The paper considers enhanced formulations of Holguín-Veras and Thorson’s that relax a key assumption of their original formulations, i.e., that the probability of a trip chain with only one stop (zero order trip chain) is constant. As shall be seen later in the paper, the formulations considered in this paper, originally suggested in Holguín-Veras et al. (2005b), shed light on the trip chaining behavior of the different vehicle types. The paper has four major sections in addition to this introduction. Previous Developments provides the reader with an idea about the alternative approaches to model (or to avoid modeling) commercial vehicle empty trips. Further Improvements discusses the new formulations developed in this paper. Brief description of the data, as the name implies, provides the reader with a succinct idea about the data used in the paper. Results presents the findings from a case study. Conclusions summarizes the key findings from this research.
2 Previous developments This section discusses the empty trip models reported in the literature. In order to ensure consistency with these works, the authors follow the notation used by
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Hautzinger (1984) and Holguín-Veras and Thorson (2003a). Because of the obvious linkages between the Holguín-Veras and Thorson (2003a) and the models proposed in this paper, the latter models are discussed in more detail than the other previous developments. This should enable the reader to fully understand the differences between these models and the models introduced in this paper. Notation: mij dij aij m xij ¼ aijij yij zij =xij +yij
commodity flow between origin i and destination j distance (or any measure of trip impedance) between origin i and destination j average payload (tons/trip) for loaded trips between i and j estimated number of loaded trips between i and j estimated number of empty trips between i and j estimated total number of trips (loaded+empty) between i and j
2.1 Naïve proportionality model This model attempts to approximate the total number of trips from i to j as a function of the commodity flow in the same direction, as shown in Eq. (1). The constant M, to be determined empirically, is multiplied by the average payload in order to obtain a “load factor” that takes into account empty trips as well. Although the estimates produced by this model could be made to match the total number of trips in a region, it leads to significant errors in the directional flows. This is because of its mathematical construction that produces overestimation of the vehicle-trips flows in the predominant direction of the commodity flows, and underestimation of the vehicle-trips in the opposite direction (see Holguín-Veras and Thorson 2003b). For example, if the loaded trips from j to i increase, it is likely that the number of trips from i to j will also increase (because of the return trips), but this model will not account for this since the commodity from i to j remains unchanged (assuming return trips are empty). Other variations of naïve formulations exist, however they all have the same limitations, as shown in Holguín-Veras and Thorson (2003b). mij zij ¼ ð1Þ Maij 2.2 Modeling empty trips as a commodity Another approach that has been used entails modeling the empty trips as a (rather unique) commodity (e.g., Tamin and Willumsen 1988; Fernández et al. 2003). In this type of application, distribution models of commercial vehicle empty trips are calibrated as a function of socio-economic variables, typically using an observed matrix of empty trips as the input. There are, at least, two major problems with this approach. First, there is no way to ensure consistency between the forecasts of loaded and empty trips. This is important because the percentage of empty trips in a given area has been found to be very stable between 30–50% of total trips and it is not likely that independently estimated models maintain such consistency. Second, since the empirical evidence demonstrates the existence of a significant correlation
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between the empty trips and the opposing commodity flows, the distribution models of empty trips are likely to end up using variables traditionally associated with trip production as trip attraction variables, and vice versa. Needless to say, this is of questionable conceptual validity. 2.3 Noortman and van Es Noortman and van Es (in Hautzinger 1984) modeled the empty trips as a function of the commodity flow in the opposite direction, which undoubtedly represents a step forward. A fraction, p, of the loaded trips in the opposite direction are expected to return empty to their origin. Therefore the number of empty trips in direction i-j can be estimated as the total number of loaded trips in the opposite direction multiplied by this constant p, see Eq. (2) below. Note that this model assumes that the total number of trips from i to j is given only by the commodity flow between the two areas therefore not considering trip chains, which occur more often than not. Nevertheless, it still represents an improvement over the naïve proportionality model. zij ¼ xij þ yij ¼
mij mji þp aij aji
ð2Þ
2.4 Hautzinger Hautzinger pointed out that Noortman and van Es’ model implies that the difference in vehicle flows (between two zones) is directly proportional to the difference in commodity flows. He claims that empirical evidence shows that even in the cases of extreme differences between commodity flows, zij and zji tend to be approximately equal. To correct this he developed the model shown in Eq. (3). Here pi represents the probability of a trip going from i to j returning empty to i. It can clearly be seen that with this formulation zij is equal to zji. Hautzinger also specified a formulation for the probabilities, assuming that. pi mij þ pj mji zij ¼ ð3Þ a 1 ð1 pi Þ 1 pj However, the analyses of the dataset used in this paper highlighted some limitations of the Hautzinger model. First, attempts to compute pi proved unsuccessful because of the numerical impossibility of computing exponentials of large ratios. Second, the authors also found that the difference in vehicle flows is in fact directly proportional to the difference in commodity flows, as originally assumed by Noortman and van Es (the Pearson correlation coefficient between the two variables for the dataset in study was relatively high, i.e., 0.78). This finding was corroborated by the analyses of two different data sets (see Holguín-Veras et al. 2005a, b). There was also an overall considerable difference in directional vehicle flows in the dataset, reaching up to 32% of the flows. In any case, Hautzinger’s model still presents the same limitation as Noortman and van Es’, i.e., it assumes
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that the vehicle flow between two zones is only a function of the commodity flow between these two zones. Although it is true that a large portion of the commercial traffic is a function of the commodity flow between the two zones in consideration, trip chains are the norm in commercial vehicle operations. As a result, there is an increased need for formulations that explicitly consider trip chains. In order to overcome this limitation, Holguín-Veras and Thorson (2003a) introduced a formulation based on trip chains to model commercial vehicle empty trips. 2.5 Holguín-Veras and Thorson The next stage in the development of empty trip models involved the use of probability and spatial interaction concepts to model empty trips as trip chains (Holguín-Veras and Thorson 2003a). These formulations attempt to model the trip chain formation process, as opposed to Noortman and van Es’ and Hautzinger’s that only focus on modeling empty trips back to the home base. Holguín-Veras and Thorson (2003a) defined the order of a trip chain as the number of destinations—in addition to the primary trip—that a vehicle visits in a tour. In this context, the models proposed by Noortman and van Es, and Hautzinger effectively model zero order trip chains. The models developed by Holguín-Veras and Thorson (2003a) are based on a simplified depiction of commercial vehicle trip chains. These formulations are able to consider the flow of commercial vehicles as part of a first order trip chain, i.e., in addition to modeling the vehicle flow from i to j, the model considers a third zone. Although in real life trip chains are far more complicated than presented in this model, it still represents a step forward. An example of a 1st order trip chain is shown in Fig. 1 (after Holguín-Veras and Thorson 2003a). Figure 1(a) presents the flow of vehicles from i to j, and a set of possible destinations after j. Figure 1(b) presents the flow in the opposite direction. Holguín-Veras and Thorson (2003a) defined the vehicle flow from zone i to zone j as the summation of the expected number of loaded trips and the expected number of empty trips as shown in Eq. (4). As defined earlier, the expected number of loaded
xji=mji/aji
xij=mij/aij
mji
j mij i (a) Loaded trips between i and j
i
(b) Loaded trips between j and i
Legend: Alternative destinations after completion of primary trip Fig. 1 Schematic of commercial vehicle trip chains
j
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trips is equal to the commodity flow, mij, divided by the average payload, aij. Therefore, after substitution, Eq. (4) reduces to the model shown in Eq. (5). E zij ¼ E xij þ E yij
ð4Þ
mij E zij ¼ þ E yij aij
ð5Þ
Holguín-Veras and Thorson (2003a) decomposed the second term into a summation of the vehicle-trips associated with the trip chains of different orders. They defined xnij as the number of loaded trips from i to j as part of an nth order trip chain. Similarly, ynij is the number of empty trips from i to j as part of an nth order trip chain. Then, Eq. (5) can be expressed as the summation of the loaded and empty trips from i to j associated with all the trip chains of different orders. This is shown in Eq. (6) below. N N X X E zij ¼ E xnij þ E ynij n¼0
ð6Þ
n¼0
Substituting for the first term from Eq. (5) results in: mij E zij ¼ þ E y0ij þ E y1ij aij
P n 3 E yij X mij 6 n>1 7 þ E ynij ¼ þ E y0ij þ E y1ij 41 þ 5 aij E y1ij n>1 2
ð7Þ
Then, by assuming that the term within brackets is equal to a constant γ*, Holguín-Veras and Thorson arrived to the following model: mij E zij ¼ þ E y0ij þ g E y1ij aij
ð8Þ
As shown in Eq. (9), Holguín-Veras and Thorson assumed (in a manner similar to the model of Noortman and van Es) that zero order empty trips are a function of the opposing commodity flow and the probability, p, of a of a zero order trip chain. Empty trips of first order were modeled using the total number of vehicles arriving at zone i multiplied by the probability that these will not be involved in a zero order chain (1-p). Furthermore these vehicles would then have to choose zone j as the next destination and be empty, i.e., P(j)P(E/j). X mhi mij mji E zij ¼ þp þ ð1 pÞg Ph ð jÞPh ðE=jÞ aij aji a h6¼j hi
ð9Þ
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If the probability p is constant, Eq. (9) can be rewritten as: X mij mji E zij ¼ þp þg xhi Ph ð jÞPh ðE=jÞ aij aji h6¼j
ð10Þ
Where: P γ=(1-p)γ* xhi Ph(j) Ph(E/j)
probability of a zero order trip chain parameter to be determined empirically number of loaded trips from h to i probability that a vehicle that came from h to i chooses j as the next destination probability that a vehicle following the tour h-i-j does not get cargo to j
Holguín-Veras and Thorson (2003a) proposed a number of different specifications of the destination choice probability function. In general terms, they assumed that the probability that a vehicle that came from h to i chooses j as the destination, Ph(j), is a function of the commodity flow in the same direction, mij, and the trip impedance between i and j. Some formulations expressed the choice probabilities as a function of the previous origin h, while others only consider the interaction between zones i and j. These formulations are shown in Table 1. The resulting empty trip models are shown in Table 2. The first Eq. (16) only considers the commodity flow between the two zones, while Eqs. (17) and (18) also take into account the impedance between zones i and j. These formulations are more effective given that they consider the fact that trip chains with short trips are more likely than with long trips. Finally, Eq. (19) also takes into account the amount of travel already done by the vehicle starting at node h. For that reason, Eq. (19) is said to have “memory;” while Eq. (16) thru (18) are considered to be “memoryless.”
3 Further improvements As shown in Eq. (9), Holguín-Veras and Thorson originally assumed that the probability of a zero order trip chain, p, is constant which enables Eq. (10) to be obtained. However, there is compelling evidence and theoretical support to argue that p should be a function of a number of variables including trip distance and the magnitude of the opposing commodity flow mji. A situation that may lead to a Table 1 Destination choice probability functions
Pð jÞ ¼ P ijm (16) m
il
l
P ð jÞ ¼
bdij Pmij e bd mil e il l
(17)
b
mij ðdij Þ Pð jÞ ¼ P m ðd Þb (18) il
l
il
b
mij ðdij þdhi Þ
P ð jÞ ¼ P m h
l
il ðdil þdhi Þ
b
(19)
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Table 2 Holguín-Veras and Thorson’s trip chain formulations m P m m HV TModel1 :E zij ¼ aijij þ p aijij þ g xhi P ijm PðE=jÞ (20) il
h6¼j
l m P m m ebðdij Þ HV TModel2 : E zij ¼ aijij þ p ajiji þ g xhi P ij bðdil Þ PðE=jÞ (21)
h6¼j
mil e
b m P mij ðdij Þ m HV TModel3 : E zij ¼ aijij þ p ajiji þ g xhi P m ðd Þb PðE=jÞ (22)
h6¼j
l
il
il
l
b m P mij ðdij þdhi Þ m HV TModel4 : E zij ¼ aijij þ p ajiji þ g xhi P m ðd þd Þb PðE=jÞ (23)
h6¼j
il
il
hi
l
The initials HV-T are used when discussing the models presented by Holguín-Veras and Thorson
variable p involves cases in which large values of the opposing commodity flow mji may increase the probability of undertaking a zero order trip chain. In this context, a large value of mji could make it more attractive to undertake a zero order trip chain because it increases the likelihood of getting a load for the return trip. Trip distance could also be a factor as well because longer trips may have lower probabilities of zero order trip chains. It is interesting to note that both Noortman and van Es (in Hautzinger 1984) and Hautzinger (1984) hypothesized that distance is an important variable, though neither suggested formulae. Earlier versions of the models discussed here considered simpler formulations (i.e., a simple exponential of the opposing commodity flow, and a linear function of distance). Although important because they highlighted the potential benefits of using variable p functions, these simpler models do not ensure the values of p stay within the range expected for probabilities (0, 1). In order to overcome this limitation, the paper considers logistic functions of the kind shown in Eqs. (11), (12), and (13). Pm ¼
ep0 þp1 mji 1 þ ep0 þp1 mji
ð11Þ
Pd ¼
ep0 þp1 dij 1 þ ep0 þp1 dij
ð12Þ
Pmþd ¼
ep0 þp1 mji þp2 dij 1 þ ep0 þp1 mji þp2 dij
ð13Þ
Where pm and pd represent, respectively, p as a function of the opposing commodity flow and trip distance, pm+d represents p as a function of both, and p0, p1, and p2 are parameters to be determined empirically. As shown in Fig. 2, a direct relationship between the independent variable used in the p function would lead to increasing values of the probability of a zero order trip chain; while an inverse relationship would lead to the opposite. In this context, the model introduces heterogeneity in the estimation by adjusting the value of the parameters to the specifics of the trip characteristics. These formulations were applied, together with the formulations with constant p, to three datasets corresponding to different vehicle types (i.e., small trucks and pick-
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Fig. 2 Idealized depictions of the p functions
1
Direct Inverse
0.8
p
0.6 0.4 0.2 0 0
50
100
150
Independent variable
ups, large 2 and 3 axle trucks, and semi-trailers). The results shed some light into the trip chaining behaviors exhibited by these different types of vehicles. 3.1 Parameter search procedure The estimation of the parameters of the models considered in this paper was conducted by a parameter optimization search that consists of finding the parameters that best fit a given data set. The parameters were determined by finding the values which minimized the error: 2 X "¼ zaij zij ð14Þ i;j
zaij
Where is the actual number of trips from i to j, and zij is the estimated value. The minimization procedure for the models with more than one parameter consists of a multidimensional downhill simplex method (Nelder and Mead 1965; Press et al. 1992). A simplex is a geometrical figure in N dimensions with N+1 vertices. The initial simplex is defined by a starting point P0 which is a vector with N dimensions and a set of N points defined by: Pi ¼ P0 þ lei
ð15Þ
Where ei the are N unit vectors and l is a constant. The method then takes a series of moves such as moving the point of the simplex where the function is largest through the opposite simplex face to a lower point until the function value at all points are within a specified tolerance of a minimum value. For the one dimensional model (NVE with constant p), a golden search was conducted (Press et al. 1992).
4 Brief description of the data The models were applied to an origin–destination (OD) sample corresponding to a major modeling project conducted by the first author in Guatemala City. In that project, roadside origin–destination interviews were conducted and complemented
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by classified traffic counts to expand the sample according to time of day and type of vehicle. The origin–destination questionnaire included questions about: time of the interview, vehicle type, origin, destination, commodity type, load factor, number of units, type of trucking company, shipment size, economic sectors at both origin and destination, and activities performed at both origin and destination. The sample, comprised of 5,276 observations, was expanded by time of day and type of vehicle, and processed to eliminate double counting of trips. The overall expansion factor was 6.476. There were 17 survey stations, five within 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 (2000a). Although the survey indicated the type of commodity being transported, the models were applied to flows which were aggregated across commodity types. The number of observations and expanded trips are shown for each type of truck in Table 3.
5 Results Tables 4, 5, and 6 show a comparison of the different models for three data subsets taken from the Guatemala City origin–destination dataset. These data sets consisted of the following three groups of vehicle types: (1) small trucks—pickups and small 2 axle trucks; (2) large trucks—large 2 axle and 3 axle single unit trucks; and (3) semi trailers. Five different models are considered, the model proposed by Noortman and van Es and models 1–4 by Holguín-Veras and Thorson (Eqs. (20), (21), (22), and (23)). The first row for each model shows the results obtained when maintaining the parameter p as a constant, the second row shows the results of assuming p is a function of the commodity flow (Eq. (11)), the third row shows the results of assuming p is a function of the trip length (Eq. (12)), while the fourth row shows the results for the model that combines both distance and the opposing commodity flow (Eq. (13)). The parameter values are given in columns 2 through 6 followed by the Sum of Squared Differences (SSD), that is, the square of the difference between the actual number of trips and the estimated ones. Although in previous papers, the authors used the Root Mean Squared Error (RMSE) as a performance measure, it was found that the RMSE does not always produce a consistent evaluation of performance. This is because different models produce a different number of estimates, n, which may translate into an artificially low value of the RMSE. In this context, to avoid this problem, the authors decided to focus on a measure of total error (SSD) instead. The relative performance of the models is quantified using two different metrics. The first one, i.e., % Diff-W, represents the percent difference between the SSD of a given model and the SSD corresponding to best model from Table 3 Number of observations and expanded trips
Vehicle set
Number of observations
Expanded total trips
Small Large Semitrailer
1,138 3,288 848
21,572 9,769 3,645
12.5295 12.5295 14.2109
24.7726 14.4708 1.2400 0.0000 1.6089 0.5712 495.2543 0.0000 12.3353 0.6200 41.4669 0.0000 1.6708 21.2785
1.0955 27.2058 27.2058 43.0869
1.0046 40.1161 48.5678 0.5237
1.0154 0.1659 0.1652 −28.9949
1.1844 −0.2911 0.3439 −1.4196
1.1424 0.5390 −1.1725
0.0865
0.2023
0.1638
23.3046
p2
−5.6007 −0.3408 −2.3415 −0.4030 −15.8898 −2.2441 3.1792
−18.0839 20.3912 11.5682 19.1212 −17.0385 18.9054 41.5939
1,669,428 1,399,450 1,809,224
1,651,897 1,540,627 1,430,473 1,739,378
1,553,544 1,416,014 1,457,557 1,763,558
−47.8341 −8.2950 −6.2537 −55.1018
−244.7926 14.6481 11.7225 5.9396
2,218,882 2,226,649 2,226,649 2,226,649
Error
1,631,792 2,226,649 2,226,649 2,345,547
Beta
−951.4684 −924.8251 −925.0837 0.3135
Gamma
19.292 0.000 29.281
15.479 7.701 0.000 21.595
9.712 0.000 2.934 24.544
0.000 36.454 36.454 43.741
0.000 0.350 0.350 0.350
% Diff-W
Performance measures
19.292 0.000 29.281
18.039 10.088 2.217 24.290
11.011 1.184 4.152 26.018
16.602 59.109 59.109 67.605
58.554 59.109 59.109 59.109
% Diff-B
The italicized results did not fully converge. % Diff-W is the percentage difference in error calculated relative to the best model within each group and % Diff-B is the percentage difference in error calculated relative to the best overall model
NVE dest. choice: none p function Constant Function of (p0+p1*mji) Function of (p0+p1*dij) Function of (p0+p1*mji+p2*dij) HVT 1 dest. choice: mij p function Constant Function of (p0+p1*mji) Function of (p0+p1*dij) Function of (p0+p1*mji+p2*dij) HVT 2 dest. choice: mij*exp(b*dij) p function Constant Function of (p0+p1*mji) Function of (p0+p1*dij) Function of (p0+p1*mji+p2*dij) HVT 3 dest. choice: mij*dij^b p function Constant Function of (p0+p1*mji) Function of (p0+p1*dij) Function of (p0+p1*mji+p2*dij) HVT 4 dest. choice: mij*(dij+dhi)^b p function Constant Function of (p0+p1*mji) Function of (p0+p1*dij) Function of (p0+p1*mji+p2*dij)
p0
Model characteristics
p1
Model parameters
Small trucks
Table 4 Results for small trucks
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Table 5 Results for (two/three axle) large trucks Large trucks
Model parameters
Model characteristics
p0
NVE dest. choice: none p function Constant 0.6802 Function of (p0+p1*mji) 2.5548 Function of (p0+p1*dij) 1.7119 Function of (p0+p1*mji+ 3.5985 p2*dij) HVT 1 dest. choice: mij p function Constant 0.4164 Function of (p0+p1*mji) 0.7791 Function of (p0+p1*dij) 0.6931 Function of (p0+p1*mji+ 0.9045 p2*dij) HVT 2 dest. choice: mij*exp(b*dij) p function Constant 0.4109 Function of (p0+p1*mji) 0.7532 Function of (p0+p1*dij) 0.2216 Function of (p0+p1*mji+ 0.6731 p2*dij) HVT 3 dest. choice: mij*dij^b p function Constant 0.3996 Function of (p0+p1*mji) 0.3681 Function of (p0+p1*dij) 0.6889 Function of (p0+p1*mji+ 0.3506 p2*dij) HVT 4 dest. choice: mij*(dij+dhi)^b p function Constant 0.4079 Function of (p0+p1*mji) 0.3394 Function of (p0+p1*dij) 0.5275 Function of (p0+p1*mji+ 0.3224 p2*dij)
p1
Performance measures p2
Gamma Beta
Error
% Diff-W
% Diff-B
−3.3502 −3.6270 −3.4148 −3.7079
74,391 66,802 67,021 60,419
23.125 10.564 10.928 0.000
33.185 19.598 19.991 8.170
2.2053 −3.0091 2.5699 −4.3319 1.8871 −0.7775 −4.2773 1.9425
60,733 57,152 56,484 55,890
8.666 2.258 1.063 0.000
8.733 2.322 1.126 0.062
60,324 56,994 57,097 56,727
6.341 0.472 0.653 0.000
8.000 2.039 2.223 1.560
2.2408 −0.2536 59,484 −1.7725 2.3577 −0.3665 56,109 −3.4518 1.8290 −0.0118 56,729 −1.9332 −0.0658 2.3551 −0.2979 55,855
6.496 0.454 1.564 0.000
6.496 0.454 1.564 0.000
5.954 0.156 1.014 0.000
6.419 0.596 1.457 0.439
0.0000 −2.6498 −2.4303 −2.6557 −0.0985
0.0000 −1.6631 −3.3117 −1.6910 −0.0384
2.2093 2.3439 2.0569 2.4301
2.2441 2.4142 1.9522 2.4027
−0.7244 −0.4459 −0.3867 −0.5240
−0.5175 −0.4769 −0.0103 −0.4885
59,441 56,188 56,669 56,100
% Diff-W is the percentage difference in error calculated relative to the best model within each group and % Diff-B is the percentage difference in error calculated relative to the best overall model
the group. A value of % Diff-W equal to zero identifies the best model from the group. The second metric, i.e., % Diff-B, is the percent difference between the SSDs of a given model and the best overall model. A value of % Diff-B equal to zero identifies the best overall model. When interpreting the results, the reader is advised to keep in mind that the results for small trucks must be used in caution. This is because the original data included a significant portion of pick-up trips used for passenger related purposes. These passenger trips have an impact on the performance of the models that, in essence, only consider commodity related flows. This is most obvious in the case of the Noortman and van Es’ model. As the reader can see, the parameter p is larger than
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Table 6 Results for semi-trailers Semi-trailers
Model parameters
Model characteristics
p0
NVE dest. choice: none p function Constant 0.4550 Function of (p0+ 0.7944 p1*mji) Function of (p0+p1*dij) −0.8341 Function of (p0+ 0.1011 p1*mji+p2*dij) HVT 1 dest. choice: mij p function Constant 0.3374 Function of (p0+ 0.1918 p1*mji) Function of (p0+p1*dij) −1.8645 Function of (p0+ −1.7911 p1*mji+p2*dij) HVT 2 dest. choice: mij*exp(b*dij) p function Constant 0.3216 Function of (p0+ 0.2826 p1*mji) Function of (p0+p1*dij) −0.7600 Function of (p0+ 0.2248 p1*mji+p2*dij) HVT 3 dest. choice: mij*dij^b p function Constant 0.3327 Function of (p0+ 0.2084 p1*mji) Function of (p0+p1*dij) −0.7064 Function of (p0+ 0.0725 p1*mji+p2*dij) HVT 4 dest. choice: mij*(dij+dhi)^b p function Constant 0.3296 Function of (p0+ 0.1976 p1*mji) Function of (p0+p1*dij) −0.6847 Function of (p0+ 0.16962 p1*mji+p2*dij)
p1
Performance measures p2
Gamma Beta
8.295 2.534
29.283 22.405
28,049 26,710
5.014 0.000
25.365 19.380
1.4446 1.6953
24,099 23,183
7.711 3.615
7.711 3.615
1.4106 1.4195
22,583 22,374
0.935 0.000
0.935 0.000
1.2841
−1.5433 2.7542 −0.4422
2.8642
−1.9481 0.1181 −1.8838
0.1157
0.0000 −1.6744 0.6769 −1.4095
0.0000 −1.6540
0.1440
% % Diff-W Diff-B
28,926 27,387
−1.6138 1.5617 −1.3897
Error
1.5370 1.9317
1.4121 1.7395
23,730 22,594
5.413 0.367
6.063 0.986
1.5052 1.9034
1.3675 1.7233
23,626 22,512
4.949 0.000
5.597 0.617
1.4439 1.7597
−0.0084 0.4494
24,100 23,056
4.610 0.078
7.714 3.048
1.1481 1.6997
−0.3315 0.0073
23,679 23,038
2.784 0.000
5.835 2.968
1.4640 1.7616
0.3566 0.4711
24,015 23,056
4.561 0.383
7.337 3.048
0.4065 23,972 0.50084 22,968
4.370 0.000
7.141 2.655
0.0609 1.3993 −1.63250 0.10122 1.74738
% Diff-W is the percentage difference in error calculated relative to the best model within each group and % Diff-B is the percentage difference in error calculated relative to the best overall model
one indicating that a unit of loaded trip in the direction j to i is going to generate more than one unit of empty trip in the direction j to i, which is not conceptually correct (because it would lead to more empty trips than loaded trips). In spite of this problem, the authors decided to leave the results for small trucks for the sake of completeness.
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The modeling results clearly confirm the superiority of the models with variable p functions. As shown in Tables 4 thru 6, these models outperform the variants with constant p by margins ranging from 4% to 23% (see the column labeled “Diff-W” that shows the within group differences between each model and the group’s best). The obvious exceptions are the first two models for the Small Trucks dataset that, as discussed before, is of suspect quality. Figure 3 shows the histogram of the % difference with respect to the best model in its group (% Diff-W) for the models with constant p. As shown, the mean difference is 9.66% with a standard deviation of 5.90. Figure 4 shows the histogram of the errors for the % difference with respect to the best overall model (% Diff-B) (again, for the models with constant p). In this case, the mean error is 13.02 with a standard deviation of 9.16. As shown, the models with variable p simply outperform the models with constant p. It is also noteworthy to highlight that the formulations that consider a trip chain formation model (HVT 1, 2, 3, and 4) outperform the Noortman and van Es’ model. As shown, the errors for the NVE model are between 28% to 58% higher than the best model (see the results in the column “Diff-B” that shows the differences with respect to the best model overall). As expected, the formulation with p as a function of both commodity flow and distance outperforms the other variable p formulations for each data set, except small trucks, with errors ranging from 0.156% to almost 11% lower than the respective models with p as a function of either commodity flow or distance. These models were included for completeness’ sake and the subsequent discussion will focus on the other two variable p models. As shown in the tables, the best models with p as a function of commodity flow or distance are: HVT 4b for Small trucks; HVT 1d/3d for Large trucks; and HVT 1d/2b for Semi-trailers (these two models are virtually tied). The common feature among most of these models (with the exception of HVT 1c) is that they take into account the spatial separation in the destination choice model. Again, this confirms the original assumptions that these models provide a better depiction of the key dynamics of the underlying decision making process. The results show that in all the cases the p function of both trip distance and the opposing commodity flow outperforms the other formulations (with the obvious exception of the suspicious small trucks dataset). However, the differences with respect to the second best models (which consider only one variable and have one less parameter) are relatively small. This may indicate that the observed improvement
60.00% Frequency
50.00%
% of cases
Fig. 3 Distribution of errors for models with constant p with respect to best model within group
40.00% 30.00% 20.00% 10.00% 0.00% 4
8
12
16
20
24
28
32
% with respect to best model within group
More
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Fig. 4 Distribution of errors for models with constant p with respect to best overall model
60.00% Frequency
% of cases
50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 4
8
12
16
20
24
28
32
More
% with respect to best overall model
is due to the extra parameter. Unfortunately, without a formal econometric specification of the model, there is no way to assess how statistically significant the individual parameters are. As shown in Table 4, the best overall model for small trucks is HVT 4b, which uses a p function of the opposing commodity flow. Since both parameters of the p function are positive, it indicates that the probability of a zero order trip chain increases with the opposing commodity flow. The best model with a p function of distance (HVT 3c) suggests a similar behavior with respect to trip distance. Table 5 shows that the behavior of Large Trucks stands in contrast to the case of Small Trucks. As shown, the parameters of the independent variables in the best models with variable p functions are negative, indicating that the probabilities of a zero order trip chain decreases with both the magnitude of the opposing commodity flow and trip distance. This suggests a predominance of trip chains in cases where the trip distances or the opposing commodity flow are high. This result seems to make sense because this type of truck is frequently used to make deliveries to/from regional distribution centers. The results for Semi-Trailers, shown in Table 6, indicate that the best models for each of the variants with variable p functions of commodity flow or distance are HVT 1c (overall best) and HVT 2b. The parameters of these models suggest that for long trips, semi-trailers are mostly used for trips going from the home base to the destination and back to the home base (zero order trip chains). To see why this is the case, the reader is referred to the idealized functions of Fig. 2, as well as Fig. 6. This makes sense because this is the behavior that has been observed in the field. On the other hand, the coefficients of the opposing commodity flow suggest that increasing values of the opposing commodity flow lead to increasing amounts of trip chains. It is possible that both models are capturing the same process because of the fact that commodity flows tend to decrease with trip distance (see Holguín-Veras and Thorson 2000b). However, without a formal econometric formulation of the empty trip models disentangling the individual contributions of the independent variables is not possible. In order to analyze the role of the independent variables considered in the p functions, the authors decided to select the best models for each of the p functions of commodity flow or distance used in each of the three data sets. Figure 5, shows the p functions for the best models with a p function of the opposing commodity flow (i.e., models HVT 4b, HVT 3b and HVT 2b for small trucks, large trucks and semi-
Commercial Vehicle Empty Trip Models With Variable Zero Order… Fig. 5 p functions of the opposing commodity flow
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1 0.9
Small trucks
0.8
Large trucks
0.7
Semi-trailers
p
0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10
mji (tons)
trailers respectively); while Figure 6 shows the p functions of trip distance for the best models (i.e., models HVT 3c, HVT 1c and HVT 1c for small, large trucks and semi-trailers respectively). As shown, all the functions tend to approach the extreme values of either zero or one vary rapidly. This implies that the p functions act as a switching function that turns off the second term of Eq. (9) when p tends to zero, or the third term when p tends to one. This is a rather interesting finding because it indicates that some trip characteristics play an influential role on the observed behavior of commercial vehicles.
6 Conclusions This paper considers enhanced formulations to model commercial vehicle empty trips. These formulations relax a limitation of the original formulations developed by Holguín-Veras and Thorson (2003a), i.e., the assumption of a constant probability of a zero order trip chain for the entire range of trip types. The formulations considered in the paper assumed that p was a function of the opposing commodity flow, or a function of the trip distance. The consideration of a variable p significantly improved the relative performance of the models. For each data subset, the relative error for the models with constant p was higher than that for the models with p as a function of
Fig. 6 p functions of trip distance
1 0.9
Small trucks
0.8
Large trucks
0.7
Semi-trailers
p
0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
dij (kilometers)
8
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either commodity flow or distance. As discussed in the paper, the models with variable p outperform the equivalent models with constant p. In average, the variable p functions have errors 9.66% lower, with a standard deviation of 5.90. When compared to the best overall models, the difference is even more pronounced reaching 13.02 in average, with a standard deviation equal to 9.16. The models suggest contrasting behaviors that depend on the type of vehicle. The negative values of the coefficients of trip distance indicate that the probability of a zero order trip chain decreases with distance, implying more trip chains as distance increases. In contrast, the positive values of the same coefficient in the case of semitrailers indicate exactly the opposite, i.e., that the longer the trip the more likely a semitrailer is used for a trip from the home based to the destination and back to the home base. These results capture the observed behavior of the trucking industry in Latin America, where semitrailers tend to do long haul movements, e.g., connecting a port to distribution centers; while large 2–3 axle trucks focus on the deliveries to end users. (The results for small trucks were deemed not conceptually correct because of the passenger traffic in pickups.) The results show that the p functions that combine trip distance and the opposing commodity flow perform better than the one with either only the trip distance or the opposing commodity flow. However, the relative small difference between the best overall model (with three parameters) and the second best model (with two parameters) may indicate that the better performance is the result of the extra parameter. Unfortunately, the lack of a formal econometric specification prevents the assessment of the statistical significance of individual parameters. In spite of the acknowledged limitations of the work, it is clear that considering variable p functions holds the potential to significantly improve the performance of empty trips models, which would hopefully facilitate the development of new paradigms of freight transportation modeling. Acknowledgements The research reported in this paper was supported by the National Science Foundation’s grant CMS-0324380. This support is both acknowledged and appreciated.
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