A multi-modal supply–demand equilibrium model for predicting intercity freight flows

Transportation Research Part B 37 (2003) 615–640 www.elsevier.com/locate/trb

A multi-modal supply–demand equilibrium model for predicting intercity freight flows J. Enrique Fern
andez L.

a,*

, Joaqu
ın de Cea Ch. a, Alexandra Soto O.

b

a

b

Transport Engineering Department, Universidad Cat
olica de Chile, Vicu~ na Mackenna 4860, Casilla 306, Macul, Santiago 22, Chile Modelos Computacionales de Transporte Ltda. (MCT), Lota 2257, Of. 403, Providencia, Santiago, Chile Received 2 December 1997; received in revised form 17 April 2002; accepted 22 April 2002

Abstract In this paper a new approach to intercity freight transportation system modeling is developed. Modeling formulation considers supply–demand equilibrium, where the demand side represents the behavior of shippers (cargo owners) and the supply side represents the behavior of carriers (transportation operators). Shippers decisions considered include choice of destination, mode, carrier for pure modes and transfer point for combined modes. Carriers take routing decisions over a multi-modal, multi-product and multioperator network. A new mathematical formulation, not known before, is proposed to find consistent equilibrium solutions for modal O–D shipments, network flows and levels of service. Necessary conditions are deduced to show that the solutions obtained, from the mathematical formulations proposed, satisfy the behavioral principles assumed in each case. It is shown that special rationality conditions are required, with respect to fares charged and network routing decisions, to obtain consistent supply–demand equilibrium solutions. Sufficient conditions for the existence and uniqueness of solutions to diagonalized versions of the mathematical problem formulated are deduced. Finally, a general solution approach is proposed and an application example is developed to illustrate the characteristics of the model and solution algorithm.
2003 Elsevier Science Ltd. All rights reserved. Keywords: Transportation planning; Multi-modal network modeling; Demand–supply equilibrium; Interurban freight flows

*

Corresponding author. Tel.: +56-2-686-4270; fax: +56-2-553-0281/552-4054. E-mail address: [email protected] (J.E. Fern
andez L.).

0191-2615/03/$ - see front matter
2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0191-2615(02)00042-5

616

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

1. Introduction Modeling of intercity freight flows has received far less attention than passenger travel demand modeling and, in general, the decision process governing intercity freight markets are less well understood. Many of the studies and models presented in the literature in the past (CACI network model; Bronzini, 1980c; PetersenÕs model, Petersen and Fullerton, 1975; LansdowneÕs model; Lansdowne, 1981; PrincetonÕs model; Kornhaunser et al., 1979) adopt a partial approach or introduce important simplifications in their formulations. However, system wide views have also been developed, starting with the Harvard–Brookings model (Kresge and Roberts, 1971), later followed by the freight network equilibrium model (FNEM) (Friesz et al., 1986) and the multimode multi-product network assignment model for strategic planning of freight flows (STAN) (Guelat et al., 1990). FNEM introduced the explicit consideration of two agents sharing the decisions that determine how cargo moves on the transportation system: shippers and carriers. A specific sub-model with its own network represents each of them and the interactions are taken into account with a sequential (or two lebel) shipper–carrier model formulation (Friesz et al., 1986; op. cit., Friesz and Harker, 1985). In order to overcome the limitations of the sequential approach Harker and Friesz (1982) proposed SFNEM, which considers simultaneous decisions of shippers and carriers while maintaining two different networks. However, the formulation corresponds to a mathematical problem with nonlinear objective function and nonlinear constraints that requires path enumeration. Friesz et al. (1985) proposed a different simultaneous formulation using a mathematical problem with nonlinear objective function and linear constraints. This model keeps the assumption of two different networks, and adds the condition that the transportation (rail 1) market is perfectly competitive, with prices equal to marginal costs. However, the solution still requires path enumeration. STAN avoids the carriers–shipper consistency problem by modeling only carrier decisions (routing over a multi-modal network). A system optimal behavior is assumed. Shipper decisions, associated with demand modeling, are externally provided as trip matrices. In this paper a different modeling approach, based on a simultaneous demand–supply network equilibrium formulation, is proposed. Well-known demand and network models are used within an equilibrium framework, to represent shipper and carrier decisions. The model proposed has many important differences with previous models. Only one network is used, over which flows are assigned and transportation level of service is calculated. Supply (carriers) and demand (shippers) models are integrated in a simultaneous mathematical formulation whose solution does not require path enumeration. It is assumed that demand (or shipper) decisions are determined on the basis of level of services corresponding to the transportation alternatives considered. These alternatives are explicitly included in the demand models for which a hierarchical structure is assumed. Levels of service are determined by the assignment of flows to the network. An explicit mathematical formulation not known before is developed to find consistent equilibrium solutions for modal O–D trips, network flows and levels of service. It is explicitly shown that solutions obtained correspond to the supply–demand equilibrium conditions assumed. Sufficient conditions

1

Rail is the only mode explicitly considered.

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

617

for the existence and uniqueness of diagonalized versions of the mathematical problems formulated are given. In the final section, a general and efficient solution approach is proposed and a sample application is solved. Additionally it is shown that in a simultaneous equilibrium formulation of a transportation freight system, two conditions are required in order to obtain a consistent equilibrium solution, (i) fares charged by carriers must be consistent with operating costs. If fares are independent of operating costs, no consistent equilibrium solution can be obtained. However, we show that some rather general rational fare structure allows obtaining consistent equilibrium solutions, (ii) carrier network routing decisions, cannot be taken without consideration of the effect that they have on the costs experienced by shippers. In other words, carriers cannot have minimizing their own operating costs as only objective; they must also care about providing a good level of service for their customers. In todayÕs world both conditions seem to be reasonable for the sustainable operation of a freight transportation system based on market and commercial principles.

2. Modeling assumptions We take an equilibrium supply–demand modeling approach in order to simultaneously represent in a consistent way the decisions of shippers and carriers. Shippers decisions are simulated by appropriate distribution and modal split demand models and carriers behavior is simulated by an assignment sub-model over a multi-modal, multi-product network with asymmetric costs. 2.1. Shippers modeling We assume that shippers can represent: • Producers that are distributing their products and therefore sending them to distribution centers (warehouses or commercial centers), transfer points (ports for exporting) or specific customers. • Producers acquiring intermediate production inputs. • Wholesale distributors or retailers purchasing final products. Shippers decide either where to purchase or where to sell the corresponding products, and in general also how to ship the product (which carrier or combination of carriers to use). 2 With respect to the general modeling approach, we assume that shippers decisions of destination, mode and carrier, are better modeled as demand decisions, using distribution and modal split models and that the routing decisions over the network are taken by each individual carrier.

2

For relatively small shipments, specially for general cargo, this decision is in general transferred to a professional shipper who decides the best way to send the shipment.

618

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

Therefore, both the choice of mode and carrier type are modeled by the modal split submodel. We assume that the use of each mode or carrier reports a utility (equal to a negative generalized cost in this case) to the shipper. This corresponds to the usual approach of random utility theory, where each alternative i is represented by a utility Ui that takes the form: Ui ¼ Vi þ ei , with Vi equal to a function of the observed attributes of alternative i and (in our case) the observed characteristics of the product transported and ei is the random component derived from unobserved variables that influence the real utility. The formulation of Vi should consider variables that define the level of service of transportation alternatives for the product transported. When a combination of modes and carriers is used, we assume that the shipper decides the sequence of modes and the main transfer point where the shipment is transferred between modes, specially if it is a port. We consider that the choice of carrier type is relevant only for road transport (trucks) and train. In general, for a given combination of modes, i.e. train and ship, it is assumed that only a few transfer points (i.e., ports) are relevant and therefore can be identified and enumerated in advance. Therefore, the demand model includes a transfer choice sub-model for the case of combined modes; for an example of a similar approach to modeling of transfer choice in the urban case see: Fern
andez et al. (1994). This modeling approach is consistent with assuming that in the choice of transfer point between modes and carriers, the shippers consider, in addition to the observable operational characteristics and transfer costs, some non-observable factors that only can be taken into account through an appropriate calibration of a demand choice model. The shipments (and therefore the shippers) considered are classified according to the following categories: ii(i) Product transported. We assume that the characteristics of the product transported strongly influence the transportation decisions taken by the shipper. i(ii) Commercial position. Different criteria will be used to take decisions related to spatial product distribution if the shipper is a buyer (at destination) or a seller (at the origin) of the product to be distributed (see Section 3.2). (iii) Trade type. We assume that the influence of factors affecting transportation services consumption will be different if the product is being exported, imported or internally traded. (iv) Shipment size. Negotiation power and conditions obtained for transporting the products will be in general influenced by the size of the shipments. i(v) Fleet ownership. The fact that a shipper has his own private fleet will also importantly influence the transportation decisions; this requires treating separately to shippers that are in such position.

2.2. Carriers modeling Transportation operators are called ‘‘carriers’’. They will in general operate a fleet of vehicles corresponding to only one mode; nevertheless, on each mode different carriers can operate, providing different levels of service (different service quality and fare). Carriers receive requests to transport shipments between O–D pairs, ði; jÞ that are within the reach of the network that they

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

619

operate. These origin and destinations could be the primary origin and final destination of the shipment or intermediate points, where this is transferred between two different carriers. In general, carriers can offer: (i) scheduled services with known routes and frequencies and, (ii) services on request by a shipper (for-hire) to satisfy his special transportation needs; this will be performed under an special contract for a given period of time, to transport a predetermined number of product units, between prespecified centroids on the network, (iii) in addition we must also consider as a different type the private carrier that transport his own shipments. Carriers are identified by the characteristics of the service offered. For carriers with scheduled services, freight will be assigned to the corresponding network of services. Alternatively, to carry shipments between a given O–D pair w, carriers operating nonscheduled services, will choose network routes that minimize their operating costs; therefore in this case vehicles must be assigned to the corresponding modal infrastructure network. 2.3. Network and vehicles modeling A set of modes M, is considered; it contains two type of elements: (i) pure modes m, with ^ , made of pure modes joined by a transfer point operators r 2 Rm and, (ii) combined modes m d 2 Dm^ . Both, combined modes and transfer points (in addition to pure modes) are predefined as choice alternatives in the demand models. The set of paths Pw 2 P joining a given origin–destination pair w 2 W is made of: the subsets of paths Pwmr , corresponding to each operator r considered for each pure mode m, and the subsets Pwm^d , corresponding to the paths going through ^ . If any of these subsets contains only one path each transfer point d for each combined mode m this could be denominated by pwmr or pwm^d . The total flow on each arc a of the multi-modal network is equal to the sum of flows corresponding to all operators of pure modes, plus the flows corresponding to combined modes using the arc: XX X X damr famr þ dam^d fam^d ð1Þ fa ¼ m2M r2Rm

where



damr ¼  dam^d ¼

^ 2M d2Dm^ m

1 0

if arc a is used by carrier r of pure mode m otherwise

1 0

^ if arc a is used by paths going through transfer point d of combined mode m otherwise

ð2aÞ

ð2bÞ It is important noticing that transfer points on combined modes are choices equivalent to operators on pure modes. We assume that one or more types of vehicles v are identified with each of the products p to be transported. Empty vehicles are modeled as a special product (that is carried by all types of vehicles) and therefore, a special demand model is used to generate and distribute empty vehicles. Fleet constraints are considered by using a nonlinear increasing waiting time function, in the initial access and transfer arcs, with a capacity in TONs defined by the amount of empty vehicles available, to transport each product (Fern
andez et al., 2001).

620

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

3. Demand modeling 3.1. Structure of the demand decision process According with the approach defined in Section 2, the shippers decision process with respect to the choice of mode, carrier type, and transfer point in the case of combined modes, is represented through the use of a hierarchical choice model (Williams, 1977). This model has two levels: i(i) Bottom level contains shipper type alternatives conditional to a pure mode chosen (truck, train or ship) or transfer point alternatives conditional to a combined mode. (ii) Upper level contains mode alternatives (pure or combined). For combined modes, only important transfer points are considered, like ports between train and ship or truck and ship, or main transfer yards between truck and train. Therefore, only combined modes, corresponding to chains containing two stages separated by a transfer will be considered in the modal split model. If the chain contains more than two modes and therefore more than one transfer, only the choice of the most important transfer will be considered in the demand model; it is assumed that the choice of the other less important transfers will be made by the carriers in charge of delivering the shipment and therefore will be modeled as routing decisions over the appropriate combined network. Therefore, networks definition for the assignment step should be made consistently with these considerations. The demand model structure corresponds to a disaggregate hierarchical logit model (Williams, 1977) as shown in Fig. 1. 3.2. Trip generation The model proposed corresponds to a short run equilibrium model, therefore we assume that both prices of commodities and total supply and demands at different points in the territory are pts given. Then, total origins Opts i and destinations Dj corresponding to each product p, trade type t and shipment size s, at each relevant centroid in the network are known inputs to the model. Nevertheless some special care must be taken to consistently specify this data. Many important products normally considered, in freight transport models are interrelated because they belong to a same production chain. Thus, we can start with some row materials that are extracted at a given location and transported to other location, where they are inputted to a production process that transforms them in a different product; this in turn can be transported to a new production unit,

Fig. 1. General structure of the mode, carrier and transfer point choice model.

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

621

located in a different place and inputted in a new production process and thus several times until obtaining a final product, that is transported to distribution centers, for their final commercialization through wholesalers or retailers. Therefore, the Oip and Djp for different products, belonging to the same production chain, must satisfy technical input–output constraints. In developed countries these production chains can be long and complex making the analysis more difficult than in developing countries where the production chains are generally short and simple (i.e., production chains of the sugar, the cement, the iron, the pulp, etc.). In the case of row materials that are exported this chains can be formed by only one or at most two stages. Some care must also be taken for the case of seasonal products, depending on the modeling period defined. Thus, a different input–output coefficient should be considered for the relation between the input to the production of sugar (beet for example) if the modeling period is a year or corresponds only to the months in which such agricultural input is harvested. In this last case it must be taken into account that during the period of few months considered, the amount of beet transported will correspond to the total required for the yearly production of sugar, however the amount of sugar produced and transported out of the plant will be proportional only to the months considered. 3.3. Trip distribution modeling Both simple and double constrained aggregate gravity expressions must be used for distribution, and different models should be calibrated depending on the specific product p, trade type t, and shipment size s. Also, different formulations should be used for shippers buying products at destinations (or importers) that for shippers selling products at origins (or exporters). For double constrained gravity models the following general expression is used: pts pts pts pts pts pts p Twpts ¼ Apts i Oi Bj Dj expðb Lw þ q Pi Þ

ð3Þ

where, bpts is the distribution parameter for product p, trade type t and shipment size s, Pi p is the price of product p at origin i; by construction of the hierarchical choice model described in Fig. 1, bpts corresponds also to the parameter ‘‘phi’’ associated to the highest choice level (trip distribution). q is a conversion parameter expressed in utility units per dollar. 3 Lwpts is the ‘‘expected maximum utility’’ (EMU), for shipments between O–D pair w ¼ ði; jÞ. The formulation of the pts introduced in Section 2.1 EMU is given by the ‘‘logsum’’ of the corresponding modal utilities V~wk (Williams, 1977): X pts ~ pts Lpts expðbpts ð4Þ k Vwk þ ak Þ w ¼ ln k2M

where bpts k is the parameter of the EMU variable which represents the composite utility of the nest pts under mode k, in the modal choice level of the choice hierarchy (see, Fig. 1). Apts i and Bj are the balancing factors whose expressions are:

3

The value of q must be calibrated together with b. This includes the sign of the parameters for each case.

622

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

Apts i ¼ P

1  pts pts  exp b Lw þ qpts Pip

ð5Þ

Bpts j ¼ P

1  pts pts  exp b Lw þ qpts Pip

ð6Þ

pts pts j B j Dj

pts pts i A i Oi

Gravity model (3) represents the origin–destination distribution decisions made by shippers. It pts (or Lpts assumes that the influence of generalized transportation costs V~wk w ) on these decisions depends on the product considered, the trade type and the shipment size; therefore, different models, with different bpts values, must be calibrated for each of these cases. Expressions (3), (5) and (6) of the distribution model, implicitly assume that shippers are importers and therefore they are interested in minimizing the generalized delivered price pts p bpts Lpts w þ q Pi of product p at destination j. Therefore, given that L must be negative (see Eqs. (4), (7a), (7b), (11a) and (11b)), we should expect that b be positive and q be negative. Alternatively for shippers selling products at origins (or exporters) the objective will be to maximize the net price obtained at origin i. This is obtained subtracting from the price paid at the exporting destination j, the equivalent transportation cost from i to j. Therefore we should expect that both b and q be positive. 3.4. Modal, carrier and transfer choices We assume the following expressions for generalized costs experienced by shippers that use ^ , to send a shipment between carrier r of pure mode m or transfer point d for combined mode m O–D pair w: pts pts pts pts pts pts ¼ apts Vwmr mr þ hR Rwmr þ ht twmr þ cl lwmr þ cr rwmr

ð7aÞ

Vwpts ^d m

ð7bÞ

¼

apts ^d m

þ

pts hpts ^d R Rwm

þ

hpts ^d t twm

þ

cpts ^d l lwm

þ

cpts ^d r rwm

where a, h and c are parameters to be calibrated on real data in order to adjust the model to observed behavior with respect to carrier and transfer point choices. The variables included are: (i) transport fare R, between pair w by carrier r, or transfer d. Notice that for transfer choice (9), the fares corresponding to each mode segment of the combined mode (from the origin to the transfer point d and from the transfer to the final destination), plus the transfer fare, must be included, (ii) travel time t, for carrier r between O–D pair w, if a pure mode m is used, or for transfer point d in ^ , considering the travel times on each segment of the trip, plus the the case of combined mode m transfer process, (iii) the percentage of product loses l, experienced in each option, and (iv) the variability of travel times r, for each option, which can be calculated as the variance of the travel time values observed. 4 Then, given pure mode m (truck or train), the following model gives the proportion of shipments type (pts), traveling between O–D pair w, by carrier type r:

4

Given the formulation of functions (7a)–(9), negative values should be expected for calibration parameters, a, h and c in (7a) and (7b).

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640 pts ~pts Þ ¼ P exp Vwmr pts Gpts ð V wmr wm k2Rm exp Vwmk

623

ð8Þ

pts is the observed (or systematic) utility (or negative generalized cost) of using carrier type where Vwmr pts is the r of mode m between O–D pair w, for product type p, trade type t and shipment size s; V~wm vector of observed utilities corresponding to all carriers available for sending shipments by mode m between O–D pair w. ^ the following model determines the proportion of shipments type (pts) For combined modes m going through transfer point d to travel between O–D pair w:

exp Vwpts pts ^d m ~ P ð V Þ ¼ Gpts ^d ^ wm wm exp Vwpts m ^k k2Dm^

ð9Þ

In this case Vwpts ^d , represents shipperÕs generalized travel cost perception when using combined m ^ via transfer node d, to travel between O–D pair w. This is made up of the costs incurred mode m over each modal section plus the transfer cost. The proportion of shipments (pts) using each mode m (pure or combined) when traveling between O–D pair w is given by   pts ~ pts pts  pts  exp bm Vwm þ am ~ ~ ð10Þ ¼P Gpts wm V w pts ~ pts pts k2M expðbk Vwk þ ak Þ pts where, V~wm is the EMU, for the utilities corresponding to all relevant carriers available in mode m, to transport shipments (pts) between O–D pair w, r 2 Rpts m . For combined modes, it is the ‘‘EMU’’ for the utilities corresponding to all relevant transfer points between O–D pair w, to carry ^ , d 2 Dpts shipments (pts) by mode m ^ : m X pts pts V~wm ¼ ln exp Vwmr ð11aÞ r2Rm

V~wpts ^ m

¼ ln

X

exp Vwpts ^d m

ð11bÞ

d2Dm^

The above models (8)–(11), should be calibrated using observations from individual shippers mode choice, transfer point choice and carrier type choice. In principle, as many different models as the number of products defined, times the number of trade types and times the number of shipments sizes considered, should be calibrated. However in order to make more efficient the use of information, the calibration of different models could be pooled together, by using dummy variables to distinguish among products and shipment sizes, treating the rest of variables as generic. The appropriate approach and the final number of models calibrated depends on practical considerations with respect to the data available and the experimental results obtained with alternative formulations. The demand model considering both distribution and modal split (included, mode, carrier and transfer point) will then have the general form: pts pts ¼ Twpts Gpts Twmr wm Gwmr

ð12aÞ

Twptsm^d

ð12bÞ

¼

pts Twpts Gpts ^ Gwm ^d wm

624

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

4. Supply modeling 4.1. General considerations Carriers take operational decisions, including vehicle scheduling and detailed network routing, given the demand structure defined by shippers decisions. To be consistent with the behavioral principles assumed and the formulation of the demand models described in Section 3, a different network should be defined for each pure mode and for the combinations with transfers not resolved by the demand model. We assume that arc costs are function of the flows corresponding to all the carriers that share the use of the same arc. Therefore, because different carriers use different type of vehicles, with different congestion impacts, we will have a network assignment problem with asymmetric costs. All vehicles operating on scheduled services should be preassigned to the infrastructure used by them and the corresponding cost functions accordingly redefined (see, Fern
andez and De Cea, 1992). An appropriate service network based on the concept of route sections Fern
andez and De Cea, 1992) should be constructed in order to assign the matrix of shipments (physical units of the corresponding products that use this type of services), obtained from the modal split model. For non-scheduled services, O–D matrices by product, obtained in tons from the demand models, should be aggregated over shipment size and trade type and transformed to the equivalent number of vehicles type v, that should be assigned to the corresponding network. Thus, consolidated vehicle matrices, by vehicle type v, will be assigned to the multi-modal network, with the exception of matrices that contain demands for scheduled services, 5 that must be kept in tons, and assigned to the corresponding network of services. In a simultaneous equilibrium formulation, like the proposed in this paper, the complexity of the model and the possibility of obtaining consistent unique solutions and developing convergent algorithms are difficulted by the existence of interactions among the different sub-models included in the formulation. For the class of models considered, sub-models interactions appear as a consequence of the existence of congestion effects. Levels of service, that influence demand decisions, depend on network flows, which are obtained as a result of routing decisions simulated by the assignment sub-model. Therefore a circular process of influences is produced between demand and supply sub-models. 4.2. Network equilibrium conditions The equilibrium principle used to obtain freight network flows, depends on the type of carrier considered: i(i) For carriers operating scheduled services, O–D matrices are assigned assuming that routing decisions are taken according to WardropÕs system optimization principle; therefore, at equi-

5

However, different modes have different treatments, for instance for rail, the tons of products must be transformed to railcars before assignment.

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

625

librium marginal costs will have the same value on all used routes, and they will be lower than those that would be experienced on the alternative non-used routes. (ii) For carriers operating non-scheduled services that must share the use of infrastructure with other carriers (roads, rails, ports, etc.), O–D matrices are assigned using a Wardrop type equilibrium principle, but based on the consideration of the private marginal cost experienced by the carrier. Notice that in this case private and social marginal costs are different, because the first considers the congestion effects only over the own vehicles operating costs and the second takes into account the effect over all the vehicles operating on the same infrastructure (see, Harker, 1988). The following are considered as possible components of the cost function used by the carriers for routing decisions: cp is the average vehicle operating cost over route p, that is assumed to be independent of flow, tp , is the average travel time over route p, which depends on flows magnitude, tM p , is the private marginal travel time; we also define the total marginal private cost perceived by the operator as CM p ¼ cp þ /tM p , and the average (private) total operating cost as Cp ¼ cp þ /tp , where / represents the carrierÕs value of time which in general will be related to vehicle capital costs. For combined mode networks that consider modes operating according to different principles (private vs. system optimum) the operating costs are calculated over the combined network, but using private costs (average or private marginal) for the arcs on the network operating according to user optimum, and marginal costs for the arcs on the network operating according to system optimum. In all cases the costs are calculated considering the total flow of vehicles sharing the same infrastructure. As a way of simplifying notation, for the case of combined modes, the values of cp , tp , tM p , Cp , CM p , include the costs and/or travel times over the route segments corresponding to both component modes and the transfer point. Network equilibrium conditions corresponding to this model are the following: cp

cp

þ

 /tM p

þ

 /tM p

( þ

hpts tp

þ

hpts tp

(





pts ¼ Uwmr if hpts >0 p  pts pts P Uwmr if hp ¼ 0





¼ Uwptsm^d if hpts >0 p pts pts P Uwm^d if hp ¼ 0

w2W r 2 Rm p 2 Pwmr

ð13Þ

w2W d 2 Dm^ p 2 Pwm^d

ð14Þ

Notice that carriers perceive private marginal times tM p but shippers perceive average private times tp . Also notice that carriers take into account the shipments travel time, for their routing network decisions, in addition to their own operating costs; 6 in other words, they are sensitive to the shippers demand characteristics in order to improve the chances that their transportation services be used by them. In other words, in a competitive market, carriers must care about providing a good level of service for their customers.

6

Notice that parameter h in (13) and (14) is the same as parameter ht expressions (7a) and (7b).

626

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

In Section 6.4 we show that this is required to obtain consistent equilibrium solutions, in the context of a simultaneous supply–demand equilibrium model. Carriers consideration of shipments travel time can only be eliminated if travel time cost is also eliminated from the definition of the pts , Vwpts shippers generalized travel cost, Vwmr ^ d (see, (7a) and (7b)). m 4.3. Fare structure Given that we are using a simultaneous equilibrium formulation, it is required that fares charged by carriers be consistent with operating costs. As we will see in Section 6.4, if total independence would be assumed between fares and operating costs no consistent equilibrium solution could be obtained. This is an important result for freight transportation modeling. Therefore, we assume that the fare agreed between carrier and shipper, is in general equal to a value by unit of freight transported and is different for each product p and O–D pair w 2 W . Following Hurley and Petersen (1994a,b) the fare values are made of two parts: cost and benefit. The cost corresponds to the marginal cost perceived by the carrier and the benefit is equal to a constant value e, by unit transported. According with Hurley and Petersen, this type of fare structure represents most real cases, when two agents interact vertically, as is the case of shippers and carriers. Therefore, fares charged by carriers will be expressed as pts pts Rpts wmr ¼ Cwmr þ ewmr

ð15aÞ

pts pts Rpts ^ d ¼ Cwm ^ d þ e wm ^d wm

ð15bÞ

pts represents the equilibrium marginal private cost perceived by carrier r of mode m, where Cwmr between O–D pair w. This cost is equal to the marginal cost perceived over all paths with positive flow at an equilibrium solution: Cwmr ¼ CM p ¼ cp þ /tM p 8p 2 w; hp > 0, where tM p is the corresponding equilibrium value.

5. Mathematical formulation In order to simplify notation, the mathematical formulation will consider only one product p, one shipment size s and one type of commercial interchange t. In addition, two generic modes, one ^ , are considered; combined mode m ^ is made of pure pure mode m and one combined mode m  joined by several transfer points d 2 Dm^ . modes m and m Given the asymmetric interactions between flows and delays for different carriers, the model described in previous sections does not have an equivalent optimization problem. However, equilibrium conditions (3), (7a), (7b), (8), (10), (18) and (19) can be obtained from the following variational inequality network equilibrium formulation, as is shown in Section 6: Fa ; ~ Ca such that: VINE: Find the values of ~ Tw ; ~ X a2ðAm [Am Þ

~ Ca ð~ Fa ÞT ð~ Fa ~ Fa Þ

X w2W

~ gw ð~ Tw ÞT ð~ Tw ~ Tw Þ P 0

8~ F;~ T 2X

ð16Þ

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

627

where h iT T ~ Ca ð~ Fa Þ ¼ . . . ; CM akl ð~ Fa Þ þ htakl ð~ Fa Þ; . . .  1 1 T ~ ~ gw ðTw Þ ¼ ð ln Tw qPi Þ þ ln Tw ; . . . ; ð ln Twk ak Þ þ ln Twk ; . . . ; . . . ; b bk T ln Twkl akl Rwkl cl lwkl cr rwkl ; . . .

ð17Þ

ð18Þ

X: Tw ¼ Twm þ Twm^ 8w 2 W X Twmr 8w 2 W Twm ¼

ð19Þ ð20Þ

r

X

Twm^ ¼

Twm^d

8w 2 W

ð21Þ

d

X

Twmr ¼

hp

8w 2 W ;

8r 2 Rm

hp

8w 2 W ;

8d 2 Dm^

ðUwmr Þ

ð22Þ

p2Pwmr

X

Twm^d ¼

ðUwm^d Þ

ð23Þ

p2Pwm^d

Oi ¼

X

Tw

8i 2 Nm

ðki Þ

Tw

 8j 2 Nm [ Nm

ð24Þ

j

Dj ¼

X

ðkj Þ

ð25Þ

8r 2 Rm

ð26Þ

i

famr ¼

X X w

fam^d ¼

X X

 dap ¼

w

1 0

dap hp

8a 2 Am ;

dap hp

 8a 2 Am [ Am ;

p2Pwmr

ð27Þ

8d 2 Dm^

p2Pwm^d

if a 2 p if a 2 6 p

8a 2 fAm [ Am g;

Tw P 0; Twm P 0; Twm^ P 0; Twmr P 0; Twm^d P 0

8p 2 Pw ; 8w 2 W ;

8w 2 W 8r 2 Rm ;

ð28Þ 8d 2 Dm^

ð29Þ

6. Equilibrium conditions In this section we show that equilibrium conditions (3), (8), (9), (10), (13) and (14) are obtained by applying the necessary conditions for optimality, to the equivalent Lagrangean corresponding to the diagonalized formulation of problem VINE. Notice that if we assume known the equilibrium solution and we diagonalize problem VINE for that solution, solving such diagonalized problem will produce the same equilibrium solution. This because all flows considered for the diagonalization are by definition equilibrium flows. Therefore, we will use such diagonalized expression of VINE to apply the necessary conditions.

628

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

6.1. Equivalent Lagrangean We apply the necessary conditions for optimality to the Lagrangean function obtained by adjoining Eqs. (22)–(25) to a diagonalized version of variational inequality (16). Notice that such diagonalization allows formulating an equivalent optimization problem, which objective function corresponds to a BeckmanÕs type transformation included in the expression of Lagrangean (30). Eqs. (26) and (27) are used indirectly to apply the chain derivation rule and Eqs. (19)–(21) are used as definitions, in the derivation of the equilibrium conditions. In the following expression, t represents the diagonalized average travel time, perceived by each carrier, given the other flows corresponding to other carriers using the same infrastructure. min L ¼ ðh;T Þ

X X a2Am r2Rm

X

þ/

X

camr famr þ

X

cam^d fam^d þ /

X XZ   tam^d fam^d fam^d þ h a2Am r2Rm

a2ðAm [Am Þ d2Dm^

Z

fam^d

tam^d ð xÞdx þ 0

þ

XX w

þ

w

famr

tamr ð xÞdx þ h 0

X

X

a2ðAm [Am Þ d2Dm^

X 1 X Twm ðln Twm 1 am Þ Twm ð ln Twm 1Þ bm w w   X   1 X Twm^ ln Twm^ 1 am^ Twm^ ln Twm^ 1 bm^ w w  XX Twmr ðamr þ ewmr þ cl lwmr þ cr rwmr Þ 1 þ

Twmr ð ln Twmr 1Þ þ

r

XX

tamr ðfamr Þfamr

a2Am r2Rm

a2ðAm [Am Þ d2Dm^

X

X X

 Twm^d ln Twm^d

w

d

r

  1X Twm^d am^d þ ewm^d þ cl lwm^d þ cr rwm^d þ Tw ð ln Tw 1 qPi Þ b w w d 0 1 ! X X X X X Tw ð ln Tw 1Þ þ Uwmr Twmr hp þ Uwm^d @Twm^d hp A

þ

XX

w

þ

X

w

ki Oi

i

X

! Tw

j

þ

p2Pwmr

X

k j Dj

j

X

w

p2Pwm^d

! Tw

ð30Þ

i

6.2. Partial derivatives We first calculate the partial derivatives of the Lagrangean (30) with respect to the problem variables (flow variables: hp 8p 2 Pw , 8w 2 W , and demand variables: Tw , Twm , Twm^ , Twmr , Twm^d 8w 2 W , 8r 2 Rm , 8d 2 Dm^ ). CM p

oL zfflfflfflfflfflffl}|fflfflfflfflfflffl{ ¼ cp þ /tM p þhtp Uwmr ohp

8p 2 Pwmr

ð31Þ

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

629

CM p

oL zfflfflfflfflfflffl}|fflfflfflfflfflffl{ ¼ cp þ /tM p þhtp Uwm^d ohp

8p 2 Pwm^d

ð32Þ

oL 1 ¼ ð ln Tw qPi Þ ln Tw ki kj oTw b oL 1 ð ln Twm am Þ ln Twm ¼ oTwm bm  oL 1  ¼ ln Twm^ am^ ln Twm^ oTwm^ bm^ oL ¼ ln Twmr þ amr þ ewmr þ cl lwmr þ cr rwmr þ Uwmr oTwmr oL ¼ ln Twm^d þ am^d þ ewm^d þ cl lwm^d þ cr rwm^d þ Uwm^d oTwm^d

ð33Þ ð34Þ ð35Þ ð36Þ ð37Þ

6.3. Equilibrium of flows Making derivatives (31) and (32), equal to zero, the network equilibrium conditions (13) and (14) are directly obtained:  w2W  ¼ Uwmr if hp > 0   CM p þ htp ð38Þ r 2 Rm  if hp ¼ 0 P Uwmr p 2 Pwmr  w2W ¼ U if h > 0   CM p þ htp P Uwm^d if hp ¼ 0 ð39Þ d 2 Dm^ ^d wm p p 2 Pwm^d 6.4. Equilibrium of trips by carrier and transfer point From derivatives (34) and (36) we obtain that:  Xwm

 if Twm

 Ywm zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflffl}|fflffl{  1    >0) ln Twm am ln Twm ¼0 bm

ð40Þ

   > 0 ) ln Twmr þ amr þ ewmr þ cl lwmr þ cr rwmr þ Uwmr ¼0 if Twmr   Adding (40) and (41), isolating Twmr and replacing the value of Uwmr from (38):            Twmr ¼ exp Xwm Ywm exp CM p þ htp þ ewmr þ cl lwmr þ cr rwmr þ amr

Replacing the fare definition form R and using (7a) we get:           Ywm exp Rwmr þ htp þ cl lwmr þ cr rwmr þ amr ¼ exp Xwm Twmr |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}  Vwmr

ð41Þ

ð42Þ

ð43Þ

630

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

P   Then dividing (43) by the sum, over all carrier r, of Twmr , we can form: Gwmr ¼ Twmr = k2Rm wmk  , obtaining the equilibrium condition (8):  expðVwmr Þ  k2Rm expðVwmk Þ

Gwmr ¼ P

ð44Þ

Making a similar development we obtain condition (9). Eqs. (41)–(43) show that, for the derivation of consistent equilibrium conditions (see Section 4.3) it is necessary that: (i) fares charged by carriers be consistent with operating costs, (ii) carrier network routing decisions, must consider shippers travel time cost. As we can see, this is the only  , can be consistently replaced in (43). way in which Vwmr  in (7a) and (7b), such that htp is not considered, Obviously if we change the definition of Vwmr then it is neither necessary to include such term in (38) and (39). 7 Something similar happens with  ) if it is directly related to the the transport fare R; it can only be replaced in (43) (to obtain Vwmr operating cost cp . The meaning is that, the fare charged to the shipper must at least recapture the cost incurred by the shipper in the provision of the transportation service. Then the only case in which the value of the fare could be arbitrary, is when the operating cost is zero. 6.5. Equilibrium of trips by mode Replacing (43) in (20) we obtain: X      Twm ¼ exp ðXwm Ywm Þ expðVwmr Þ

ð45Þ

r   Then, simplifying, reordering and replacing the Xwm and Ywm values we get: X 1   expðVwmr Þ¼ ðln Twm am Þ ln b m r

ð46Þ

Applying the definition of the generalized utility associated to each mode, and reordering terms, (46) can be written as   ¼ ln Twm am bm V~wm

ð47Þ

In a similar way we can obtain that: bm^ V~wm^ ¼ ln Twm^ am^

ð48Þ

Subtracting (48) from (47) and using (19), we obtain equilibrium condition (10): G 1 wm

7

 ¼ ln

 Tw 1 þ am^ am  Twm

Actually, if the term is included an inconsistency appears.

ð49Þ

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

631

6.6. Equilibrium of total trips From the expression of partial derivative (33), we obtain:  1 if Tw > 0 ) ln Tw qPi ln Tw ki kj b

ð50Þ

Reordering and rising to the power e we get: Tw ¼ expðbki Þ expðbkj Þ expðb ln Tw þ qPi Þ

ð51Þ

Then replacing (51) in (24) and (25) we obtain: X expðbkj Þ expðb ln Tw þ qPi Þ Oi ¼ expðbki Þ

ð52Þ

j

Dj ¼ expðbkj Þ

X

expðbki Þ expðb ln Tw þ qPi Þ

ð53Þ

i

Now, let us define: expðbki Þ ) expðbki Þ ¼ Ai Oi Oi expðbkj Þ  Bi ¼ ) expðbkj Þ ¼ Bj Dj Dj

Ai ¼

ð54Þ ð55Þ

 and Twm^ from (47) and (48), we have: And taking the expressions of Twm   Twm ¼ expðbm V~wm þ am Þ   T ^ ¼ expðbm^ V~ ^ þ am^ Þ wm

ð56Þ ð57Þ

wm

Adding up both O–D demands and replacing in (19), then taking natural log, reordering and using the definition of generalized utility (logsum associated to each)-D pair, we get:  þ am Þ þ expðbm^ V~wm^ þ am^ ÞÞ ln Tw ¼ lnðexpðbm V~wm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

ð58Þ

Lw

Finally, replacing (54), (55) and (58) in (51), we obtain the gravity model that describes the demand for O–D trips: Tw ¼ Ai Oi Bj Dj expðbLw þ qPi Þ

ð59Þ

6.7. Sufficient conditions The Kuhn–Tucker conditions (Zangwill, 1969), derived above are necessary and sufficient for the optimal solution of problem VINE provided that the objective function of the diagonalized version of the objective function is convex, given that all the constraints are linear. By inspecting the expression of the Lagrangean (30), obtained from the diagonalized version of problem VINE, it is easy to notice that only terms involving variables T may be non-convex.

632

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

Therefore we must examine the second derivatives of L with respect to those variables. If we derive expression (33) with respect to Tw , we obtain the second derivative of L with respect to Tw : o2 L 1 b 1 ¼ oTw2 b Tw

ð60Þ

Now, deriving expressions (34) and (35) with respect to variables Twm and Twm^ we get, the following second derivatives: o2 L 1 bm 1 ¼ 2 oTwm bm Twm

ð61Þ

o2 L 1 bm^ 1 ¼ 2 oTwm^ bm^ Twm^

ð62Þ

Finally, deriving (36) with respect to Twmr and (37) with respect to Twm^d we obtain: o2 L 1 ¼ 2 oTwmr Twmr

ð63Þ

o2 L 1 ¼ 2 oTwm^d Twm^d

ð64Þ

Therefore, from (60)–(64) we have that given that all the variables T are always positive, VINE will be convex if: b < 1, bm < 1 and bm^ < 1. Notice first that, according to the definition of the demand models, all parameters must be positive. In addition, the conditions requiring that b < 1, bm < 1 and bm^ < 1, are consistent with the normal requirements for the validity of the hierarchical demand model assumed and represented in Fig. 1 (see, Williams, 1977). Therefore, we will assume that such conditions are satisfied and therefore, the necessary conditions derived above (Sections 6.3–6.6) for VINE, are also sufficient. 7. Solution approach and sample application 7.1. Solution approach Assuming that second-order conditions (60)–(64) are satisfied, the diagonalized version of problem VINE is a linearly constrained convex program. Therefore, similarly to many other network equilibrium models, it may be solved by the adaptation of the linear approximation algorithm of Frank and Wolfe (1956). This approach is the most frequently used to solve transportation network problems and in particular has been successfully applied to obtain numerical solutions from performance–demand models similar to those proposed here (see Fern
andez et al., 1994). The algorithm proposed has in general two main steps: (i) the linear approximation subproblem which results from the linearization of the convex objective function and yields the descent direction and (ii) the solution of a one-dimensional minimization problem (line search), which determines the optimal step size to minimize the objective function, given the current solution and the descent direction.

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

633

An specially useful and intuitively appealing variant of the algorithm proposed is the ‘‘partial linearization algorithm’’ (see Evans, 1976), where only the terms involving the link flows are linearized and those involving demand variables are not. Given the relative large number of demand terms on the objective function of problem VINE, it can be specially advantageous to use also the Horowitz approximation for solving the line search step of the algorithm (Horowitz, 1989). The necessary steps to solve a diagonalized problem VINE will be the following: ii(i) The minimum cost paths (based on the current feasible solution) would be computed for each carrier r on each network, corresponding to pure and combined modes. Transfers, d 2 Dm^ , are considered as additional origin and destination points for the corresponding modes: then combined mode minimum cost paths are computed through each transfer point for each O–D pair. i(ii) Using (7a) and (7b), generalized costs perceived by shippers are calculated and the corresponding EMUs, Vwm and Vwm^ , and Logsums, Lw are also computed. Next, new demands T are obtained applying demand models (3), (8), (9), (10), (12a) and (12b). (iii) Finally, the resulting O–D modal trips are assigned to each of the networks, according to the minimum cost paths calculated before. (iv) Once an auxiliary solution has been obtained by the procedure described in (ii) and (iii), a convex combination of this with the current solution is computed. For this, a one-dimensional search must be performed in the direction defined by the current and auxiliary solutions; this task can be importantly simplified, if only the network related terms of the objective function are considered, according to the use of the Horowitz approximation. We omit here the technical details of the adaptations proposed, since they are tedious and generally well known. 7.2. Application example In this section we apply the above described solution approach to a test network. It is shown in Fig. 2 and considers all the special elements of model VINE. It includes five geographical

Fig. 2. Test network.

634

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

Table 1 Shipments generations and attractions Zone

Demand level 1 Generations

1 2 3 4 5 Total

Demand level 2 Attractions

Generations

Attractions

68,500 15,000 30,000 72,000 40,000

22,000 70,000 73,500 10,000 50,000

137,000 30,000 60,000 144,000 80,000

44,000 140,000 147,000 20,000 100,000

225,500

225,500

451,000

451,000

zones, 18 nodes, two pure modes, one combined mode (with two transfer nodes: 6 and 7) and 54 unidirectional links (links shown in Fig. 2 are bi-directional). Nodes 1–5 correspond to centroids that generate and attract freight shipments; nodes 6 and 7 represent transfer nodes and nodes 8–18 represent network nodes. The seven bi-directional links depicted by segmented lines correspond to centroid connectors, representing local accessibility conditions within the five geographical zones considered. The 11 bi-directional links in the lower part of the network correspond to pure mode A (road network) and the nine red links (bi-directional) in the upper part correspond to pure mode B (rail network). All links belonging to pure modes. A and B can be also used by a combined mode C (road-rail). The combined mode can use any route going through transfer nodes 6 and 7 to provide transport services between any O–D pair. Three different carriers are assumed to operate mode A, and one mode B. Two carriers operate combined mode C, one through each transfer node: 6 and 7. We assume that there is no possibility of using only pure mode B. For instance, to send a shipment from zone 4 to zone 3 shippers can chose one of the three carriers of mode A or one of the two carriers of combined mode C. All carriers operate non-schedule transport services and therefore they select the minimum cost route (given the corresponding equilibrium flows) between each O–D pair. Only one homogeneous product to be transported is considered and the corresponding shipments generation and attraction data is shown in Table 1. In this case two different sets of values were assumed, to analyze the effect of different levels of network congestion. 7.2.1. Demand functions We assume that shippers are buyers that purchase the product at a given origin and transport it to the destination were it is consumed; therefore, they are interested in minimizing the delivered price, of the product transported, at the destination node. The values assumed for the parameters of the demand functions are the following (see Sections 3.3 and 3.4): (i) distribution parameter Eq. (3): b ¼ 0:1, (ii) conversion parameter for the price in the origin of the product transported Eq. (3): q ¼ 0:1, (iii) modal split parameters Eq. (10): bm ¼ bm^ ¼ 0:005. 7.2.2. Carriers operating costs and transportation generalized costs Carriers operating costs cra , for carrier r over link a, are shown in Table 2 in ($/km). These costs are constant and proportional to the length of the arcs and paths used. Arc lengths are also given in the same table.

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

635

Table 2 Carriers operating costs Node A

Node B

Length

Carrier 1 mode A

Carrier 2 mode A

Carrier 3 mode A

Carrier 1 mode C

14 14 115 15 16 16 17 17 7 18 1 8 8 8 6 9 10 10 10 2 11 12 12 7 3 4 5 6 15 6 16 6 17 7 18 18 3 8 6 10 9 10 10 2 12 11 12

6 15 6 16 6 17 7 18 18 3 8 6 10 9 10 10 2 12 11 12 12 7 13 13 13 10 13 14 14 115 15 16 16 17 17 7 18 1 8 8 8 6 9 10 10 10 2

5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 10.0 5.0 5.0 5.0 5.0 5.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 10.0 5.0 5.0 5.0 5.0 5.0 10.0 10.0 10.0 10.0

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 10.0 0.5 10.0 0.5 2.0 1.0 10.5 1.0 24.0 38.65 0.5 11.0 11.0 0.5 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 0.0 5.0 0.0 0.0 0.0 5.0 5.0 5.0 8.0 5.0 8.0 15.0 7.0 2.0 2.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 8.0 4.0 2.0 6.0 2.0 7.0 4.0 4.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 8.0 4.0 4.0 6.0

5.0 5.0 5.0 2.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 2.0 5.0 4.5 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

Carrier 2 mode C

5.0 5.0 5.0 5.0 5.0 5.0 5.0 2.0 4.15 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 8.0 7.0 5.0 4.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 2.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 (continued on next page)

636

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

Table 2 (continued) Node A 12 7 13 13 13 10 13

Node B 11 12 12 7 3 4 5

Length 10.0 10.0 10.0 10.0 10.0 10.0 10.0

Carrier 1 mode A 5.0 8.0 5.0 5.0 5.0 5.0 5.0

Carrier 2 mode A

Carrier 3 mode A

Carrier 1 mode C

Carrier 2 mode C

5.0 8.0 15.0 7.0 5.0 5.0 5.0

4.0 7.0 4.0 4.0 4.0 4.0 4.0

5.0 5.0 5.0 5.0 5.0 5.0 5.0

5.0 5.0 5.0 5.0 5.0 5.0 5.0

Travel times per km are obtained using the following BPR type function:  4 !  1 fvakl þ fvakl h 1þ ; takl ðfva Þ ¼ 90 km 1800

ð65Þ

To obtain the travel time for any particular link, the value of ta must be multiplied by the length value shown in Table 2. The parameters of formula (65) are derived from the following assumptions: (i) a speed of 90 km/h was assumed for the operation of vehicles in free flow conditions, (ii) a link capacity of 1800 veh/h was assumed for all one way links in the network, (iii) a speed of 45 km/h was assumed for the vehicles when the flow of the link is equal to its capacity, (iv) a congestion power of 4 was assumed for all links in the network. The generalized costs perceived by shippers, Vwmr (see Eqs. (7a) and (7b)), was assumed to depend only on the transport fare, Rwmr (charged by carrier r, of mode m between O–D pair w) and the travel time experienced, twmr : Vwmr ¼ hR Rwmr þ ht twmr ð66Þ Competitive conditions were assumed for the carriers market; therefore, the fare value, R, charged by carriers is equal to the its marginal private cost, Rwmr ¼ Cwmr , (e ¼ 0, see Eqs. (15a) and (15b)). It was assumed that hR ¼ 0:01 and ht ¼ 0:1:  Cwmr ¼ cp þ /tM p ð67Þ The travel time value for all carriers, / (see Section 4.3), was assumed equal to 2 $/h. 7.2.3. Model performance and outputs A computer implementation of the solution procedure described in Section 7.1 was used to obtain equilibrium flows, using the different values of shipment generations and attractions assumed in Table 1. The number of diagonalization iterations necessary to obtain an equilibrium solution was between 2 and 12 depending on the level of network congestion. The precision of the final solutions was between 0.000% and 0.002% for O–D trips and between 0.009% and 0.015% for link equilibrium flows. Convergence characteristics are graphically shown in Fig. 3 for demand level 2. As we can see, a monotonic reduction in the percent difference between the solutions obtained from successive diagonalizations was obtained, both in terms of O–D trips and link flows. We can also observe that O–D trips are significantly more stable than link flows; this is in general the case in network equilibrium models.

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

637

Fig. 3. Convergence of O–D trips and link flows for demand level 2 (high congestion).

In Table 3, O–D trips (total and per carrier shipments, expressed in tons) are shown, for the high congestion case, considering both modes A and C. Notice that the magnitudes of flows transferred at nodes 6 a and 7, correspond to demands for carriers 1 and 2 of mode C. Table 3 O–D shipments (tons) for demand level 2 (high congestion) O

D

Total trips

Carrier 1 mode A

Carrier 2 mode A

Carrier 3 mode A

Carrier 1 mode C

Carrier 2 mode C

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

13522.33 42551.41 44540.63 6074.65 30310.58 2912.87 9398.11 9739.57 1321.70 6627.93 5802.43 18534.76 19694.45 2632.83 13335.53 14030.33 44817.20 46912.34 6462.42 31777.07 7732.04 24698.54 26113.00 3508.39 17948.90

2483.92 24514.02 3567.48 2588.77 5259.99 1018.23 1726.34 364.42 468.90 553.93 1142.95 4473.50 3617.68 534.84 2289.49 2527.10 13126.17 1299.22 1187.09 1893.40 3453.87 12683.51 7356.07 1601.29 3297.04

2483.92 10996.13 4019.57 1800.95 5897.39 1018.23 1726.34 870.51 468.90 1315.44 1049.82 4108.97 3617.68 491.26 5584.03 5756.38 19233.80 4802.13 1187.09 6964.13 823.03 3035.62 4716.05 383.34 3297.04

2483.92 7041.26 13259.22 1684.93 19153.19 876.40 1726.34 3192.39 383.90 4758.55 3609.66 9952.29 3617.68 1606.74 5462.01 5746.85 12457.23 16046.87 1187.09 22919.54 3455.15 8979.41 5699.74 1523.75 3297.04

3035.28 0.00 17200.88 0.00 0.00 0.00 2109.54 2344.16 0.00 0.00 0.00 0.00 4420.70 0.00 0.00 0.00 0.00 16539.81 1450.58 0.00 0.00 0.00 1783.31 0.00 4028.89

3035.28 0.00 6493.48 0.00 0.00 0.00 2109.54 2968.10 0.00 0.00 0.00 0.00 4420.70 0.00 0.00 0.00 0.00 8224.32 1450.58 0.00 0.00 0.00 6557.83 0.00 4028.89

638

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

In order to analyze the characteristics of the solution shown in Table 3, a set of selected path flows and their corresponding transportation costs, experienced by shippers and carriers, are shown in Table 4. Total shipment flows transported between O–D pair (4–3) by each of the three carriers that operate mode A and carrier 2 of mode C (who operates through transfer node 7) are shown in the third column of the table. Column four shows all the paths available between O–D pair (4–3) and column five indicates which of them are used in each case. Column six shows the generalized transportation cost, Cwmr , that is perceived by the carrier when using a specific path to transport shipments between O–D pair (4–3). It can be observed that all used paths are at equilibrium: carrier marginal private cost is the same for all used paths and non-used paths present a higher cost. It can be also observed that carriers with lower transportation costs (and fares in this case), carry more shipments (tons), than the more expensive ones. The differences observed depend on the values taken by the demand calibration parameters, that represent the sensitivity of shippers to transportation variables (represented by parameters hR and ht ; see Eqs. (8), (9) and (66)). If hR increase, shipments will tend to be assigned in higher proportion to least cost carriers. Finally, the last column shows the fare charged to shippers, when shipping product between the O–D pair (4–3) by each of the carriers considered. Notice that in this case the carrier charges a fare equal to his marginal cost: Rwmr ¼ Cwmr , (e ¼ 0, see Eqs. (15a) and (15b)). Table 4 Path flows and transportation costs for demand level 2 O–D pair

Carrier number

4–3

1 (mode A)

4–3

Total shipments carried (tons)

Alternative paths available

Paths used Carrier path cost Cwmr

Fare charged Rwmr

1299.22

4-10-11-12-7-13-3 4-10-11-12-13-3 4-10-12-7-13-3 4-10-12-13-3 4-10-2-12-7-13-3 4-10-2-12-13-3

YES NO YES NO NO NO

535.79 679.11 535.94 679.26 643.77 787.09

536

2 (mode A)

4802.13

4-10-11-12-7-13-3 74-10-11-12-13-3 4-10-12-7-13-3 4-10-12-13-3 4-10-2-12-7-13-3 4-10-2-12-13-3

NO NO YES YES NO NO

438.79 440.24 390.20 391.65 486.19 487.64

391

4–3

3 (mode A)

16046.90

4-10-11-12-7-13-3 4-10-11-12-13-3 4-10-12-7-13-3 4-10-12-13-3 4-10-2-12-7-13-3 4-10-2-12-13-3

NO YES NO YES NO NO

344.92 281.48 344.90 281.47 459.03 395.59

281

4–3

2 (mode C)

8224.32

4-10-12-7-18-3 4-10-12-7-17-18-3 4-10-11-12-7-18-3 4-10-11-12-7-17-18-3 4-10-2-12-7-18-3 4-10-2-12-7-17-18-3

YES YES NO NO NO NO

325.60 325.61 354.65 354.66 358.15 358.16

326

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

639

8. Conclusions We have presented a demand–supply equilibrium formulation for the modeling of interurban, multi-modal, freight transportation systems. The formulation considers the most important factors present in the real decision process. It is specially interesting to notice that full consideration of congestion effects in the simultaneous equilibrium formulation proposed, is only possible under some important consistency requirements: (i) Transportation fares paid by shippers must be related to the operating costs experienced by carriers. In the formulation proposed, fares are equal to marginal costs plus profit. (ii) Routing decisions, taken by carriers, which determine the level of service offered by them, must take into account shippers preferences with respect to travel times. In other words, carriers must be rational in a microeconomic sense. The model VINE proposed corresponds to an entirely new, more consistent general network formulation for the simulation of interurban freight transportation systems; it allows to calibrate different parameters for the choices of mode, carrier and transfer point and considers that they are based on level of service values that are generated by the carriers network operations. A solution algorithm to obtain equilibrium flows has been proposed and a sample application solved. It clearly shows the feasibility of application and illustrates the characteristics of the outputs obtained and their relation with the model formulation.

Acknowledgements This research was financed by a grant from the Chilean National Research Fund (FONDECYT) and by the Catholic University of Chile.

References Bronzini, M.S., 1980c. Evolution of a multi-modal freight transportation network model. Proc. Transp. Res. Forum 21 (1), 475–485. Evans, S.P., 1976. Derivation and analysis of some models for combining trip distribution and assignment. Transp. Res. 10, 37–57. Fern
andez, J.E., De Cea, J., 1992. Characteristics of composite modes modeling in the context of network equilibrium. In: Proceedings of the World Conference on Transport Research, Lyon, France. Fern
andez, J.E., De Cea, J., Florian, M., Cabrera, E., 1994. Network equilibrium models with combined modes. Transp. Sc. 28 (3), 182–192. Fern
andez, J.E., De Cea, J., Giesen, R., 2001. A strategic model for freight operations on rail transportation systems. Transp. Planning Tech. (submitted for publication). Frank, M., Wolfe, P., 1956. An algorithm for quadratic programming. Naval Res. Logistics Quart. 3, 95–110. Friesz, T.L., Gottfried, J.A., Morlok, E.K., 1986. A sequential shipper–carrier network model for predicting freight flows. Transp. Sci. 20 (2), 80–91. Friesz, T.L., Harker P.T, 1985. Freight network equilibrium: a review of the state of the art. In: Daughty, A.F. (Ed.), Analytical Studies in Transportation Economics. Cambridge University Press, Cambridge (Chapter 7). Friesz, T.L., Viton, P.A., Tobin, R.L., 1985. Economic and computational aspects of freight network equilibrium models: a synthesis. J. Reg. Sci. 25, 29–49.

640

J.E. Fern
andez L. et al. / Transportation Research Part B 37 (2003) 615–640

Guelat, J., Florian, M., Crainic, B.C., 1990. A multimode multiproduct network assignment model for strategic planning of freight flows. Transp. Sci. 24 (1), 25–39. Harker, P.T., 1988. Multiple equilibrium behaviors on networks. Transp. Sci. 22 (1), 39–46. Harker, P.T., Friesz, T.L., 1982. A simultaneous freight network equilibrium model. Proc. Cong. Numer. 36, 365–402. Horowitz, A.J., 1989. Tests of an ad hoc algorithm of elastic demand equilibrium traffic assignment. Transp. Res. B 23 (4), 309–313. Hurley, W.J., Petersen, E.R., 1994a. Nonlinear tariffs and freight network equilibrium. Transp. Sci. 28 (3), 236–245. Hurley, W.J., Petersen, E.R., 1994b. Optimal freight transport pricing and the freight network equilibrium problem. In: Proceedings of Tristn II Conference, Capri, Italy. Kornhaunser, A.L., Hornung, M., Harzony, I., Lutin, J., 1979. The Princeton railroad network model: application of computer graphics in the analysis of a changing industry. Harvard Graphics Conference, Transportation Program, Princeton University, Princeton. Kresge, D.T., Roberts, P.O., 1971. Systems analysis and simulation models. In: Meyer, J.R. (Ed.), Techniques of Transport Planning II. The Brookings Institute, Washington, DC. Lansdowne, Z.F., 1981. Rail freight traffic assignment. Transp. Res. A 15, 183–190. Petersen, E.R., Fullerton, H.V., (Eds.), 1975. The railcar network models. Canadian Institute of Guided Ground Transport. Report 75-11, QueenÕs University, Kingston, Ontario. Williams, H.C.W.L., 1977. On the formation of travel demand models and economic evaluation measures of user benefit. Environ. Planning A 9 (3), 285–344. Zangwill, W.I., 1969. Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs, NJ.