We present in this paper a normative model for simulating freight flows of ... tural and energy sectors in an international or interre- gional setting. It is not this class of models which is the main topic of this paper. The class of models which we consider are network ..... carried out after an "a priori" mode choice calculation,.
A Multimode Multiproduct Network Assignment Model for Strategic Planning of Freight Flows JACQUES GUELAT Universite de Lausanne, Lausanne, Switzerland
MICHAEL FLORIAN Centre de recherche sur les transports, Universite de Montreal, Montreal, Canada, H3C 3J7, and Departement d'Informatique et de Recherche Operationnelle, Universite de Montreal
and
TEODOR GABRIEL CRAINIC Centre de recherche sur les transports, Universite de Montreal, Montreal, Canada, H3C 3J7, and Departement des Sciences Administratives, Universite du Quebec a Montreal
We present in this paper a normative model for simulating freight flows of multiple products on a multimodal network. The multimodal aspects of the transportation system considered are accounted for in the network representation chosen. The multiproduct aspects of the model are exploited in the solution procedure, which is a Gauss-Seidel-Linear Approximation Algorithm. An important component of the solution algorithm is the computation of shortest paths with intermodal transfer costs. Computational results obtained with this algorithm on a network that corresponds to the Brazil transportation network are presented. Several applications of this model are reported as well.
INTRODUCTION
flows. This class of models determines simultaneously the flows between producing and consuming regions as well as the selling and buying prices. The transportation network is usually modeled in a simplistic way (bipartite networks) and these models rely to a large extent on the supply and demand functions of the producers and consumers respectively. The calibration of these functions is essential to the application of these models and the transportation costs are unit costs or may be functions of the flow on the network. There have been so far few multicommodity applications of this class of models, with the majority of applications having been carried out in the agricultural and energy sectors in an international or interregional setting. It is not this class of models which is the main topic of this paper. The class of models which we consider are network models which enable the prediction of multicommodity flows over a multimode network, where the physical
T
he prediction of multicommodity freight flows over a multimodal network has attracted much interest in recent years. In contrast to urban transportation, where the prediction of passenger flows over multimodal networks has been studied extensively and many of the research results have been transferred to practice (see, for instance, FLORIAN[5,6]), the study of freight flows at the national or regional level, perhaps due to the inherent difficulties and complexities of such problems, received less attention. The class of models that was well studied in the past for prediction of interregional flows is the spatial price equilibrium model and its variants. The model, stated initially by SAMUELSON[23] and extended by TAKAYAMA and JUDGE[26,27] then by FLORIAN and Los/B] FRIESZ, TOBIN and HARKER,[IO] has been used extensively for analyzing interregional commodity 25 Transportation Science
0041-1655/90/2401-0025 $01.25 © 1990 OperatIOns Research Society of America
Vol 24, No.1, February 1990
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GUELAT ET AL.
network is modeled at a level of detail appropriate for a nation or a large region, and represents the physical facilities with relatively little abstraction. The demand for the transportation services is exogenous and may originate from an input-output model, if one is available, or from other sources, such as observed demand or scaling of observed past demand. The choice of mode, or subsets of modes, used is exogenous and intermodal shipments are permitted. In this sense, these models may be integrated with econometric demand models as well. The emphasis is on the network representation and the proper representation of congestion effects in a static model aimed to serve for comparative studies or for discrete time multiperiod analyses. The Harvard Model (KRESGE and ROBERTS[20]), which is probably the first published predictive freight network model of the type that interests us, resorted to a fairly simple "direct link" representation of the physical network and congestion effects were not considered. Later, the Multi-State Transportation Corridor Model (MCGINNIS et al.,[2I] JONES and SHARP,[IS] and SHARP[24]) went a step further in representing an explicit multi modal network, but without any consideration of congestion. The Transportation Network Model (BRONZINI[2]) does not consider congestion effects either. The first model that considers congestion effects and shipper-carrier interactions is that of FRIESZ, VITON and TOBINY2] A review of shippercarrier models, both sequential and simultaneous, is given by FRIESZ and HARKER. [11] The first application of a model that considers congestion phenomena in this field is the Freight Network Equilibrium Model (FRIESZ, GOTTFRIED and MORLOK[13]). This is a sequential model which uses two network representations: an aggregate network that is perceived by the users, which serves to determine the carriers chosen by the shippers and then, more detailed separate networks for each carrier, where commodities are transported at least total cost. A generalization of the work of FRIESZ, VITON and TOBIN,[12] in which variable demand functions are considered in the shippers' submodels, is given by HARKER and FRIESZ.[16,17] They combine the variable demand modeling approach of spatial equilibrium models with a detailed description of the behavior of shippers and carriers, in mathematical formulations that are yet to be tested in a particular application. The class of models that we present in this paper do not consider shippers and carriers as distinct actors in the decisions made in shipping freight. The level of aggregation which is appropriate for strategic planning of freight flows, where origins and destinations correspond to relatively large geographical areas, leads to the specification of supplies and demands, for all
Copyright © 2001
the products considered, which represent the services provided by all the individual shippers for the same product. Often, demand for strategic freight scenario analyses are determined from data sources (national freight flow statistics, economic input-output models) which permit to identify the mode used but do not contain information about individual shippers. However, the demand data that is available and the demand forecasts made reflect the behavior of the shippers. Nevertheless, the models that we seek consider the competition among modes of transport, wherever it is present. The assumption is made that goods are shipped at minimum total generalized cost. At a strategic level of planning freight flows, where major quantities of bulk products are transported and the scenarios considered represent investments of large magnitude, this assumption is justified, even though the shipment of freight is governed by a variety of micro-circumstances which prevent the actual achievement of total cost minimization. The normative approach is particularly appropriate when certain products are captive to a mode, or a subset of modes, due to service availability or regulation. In other situations, when modes compete for the shipment of products, it is possible to include in the generalized cost function components which reflect the shippers' objectives, which may be costs, time delay or other relevant factors, keeping in mind that shippers, in this context, are aggregated by origins. As the aim of these models is to carry out scenario comparisons, at a strategic planning level, where the time horizon is medium to long term, a normative objective is appropriate, given the availability and level of aggregation of the demand and the nature of the decisions that are to be evaluated. Also, the generalized cost function offers sufficient modeling flexibility to adequately represent a wide variety of situations and circumstances. And, of course, the normative approach is the right one when our modeling framework is applied to the case of one carrier. The multimodal aspects of a national transportation system are accounted for in the network representation chosen. A link of the multimodal network is defined by its origin and destination nodes and a single mode. Parallel links are allowed between two adjacent nodes, one for each mode available to transport goods between them. The intermodal transfers at a node of the network are modeled as link to link permitted movements. Appropriate cost functions may be associated with links and intermodal transfers. The multiproduct aspects of a national transportation system are accounted for in the formulation of the predictive model and are taken advantage of in the solution procedure. The algorithm developed for
AILR~ig1!.h!.:;:ts~R~es:::.:e:::.!rv..!.e~d:!..-
_ _ _ _ _ _ _ _ _ _ _ _ _ __
A MODEL FOR STRATEGIC PLANNING OF FREIGHT FLOWS
I 27
this problem exploits the natural decomposition by product and results in a Gauss-Seidel-like procedure. Demand and mode choice are exogenous and assumed to specify, for each product, a set of OlD demand matrices, each of which may be assigned on the subnetwork corresponding to a permitted subset of modes only. The choice of the paths used through a permitted subnetwork is determined by the congestion conditions present and the particular form of the generalized cost structure. The multimode multiproduct model is formulated in the most general way, permitting, in principle, nonconvex and asymmetric cost functions. Nevertheless, certain assumptions that are made on the structure of the cost functions simplify the problem and permit the solution of large size problems in reasonable computational times. The paper, which is based in part on GUELAT'S[15] work, is organized in the following way. First, the network representation chosen to integrate modes and transfers is described in detail; then, the multimode multiproduct assignment model is formulated and the form of the average and marginal cost functions is analyzed. The solution algorithm adopted to solve the problem is presented next, followed by the statement and computational complexity analysis of the algorithm used to compute shortest paths, with intermodal transfer costs, on the network representation chosen; finally, the computational efficiency of the algorithm is demonstrated and the applicability of the model discussed.
m E M, where M is the set of modes available on the network. Parallel links are used to represent the situation where more than one mode is available for transporting goods between two adjacent nodes. In order to motivate our network modeling choice, consider the following simple example, which is shown in Figure 1. It consists of three modes: road transport, diesel rail and electrified rail. Cities A and B are served by all modes; A and Care served by diesel rail and road, whereas Band Care served only by road. In order to simplify the presentation, no intermodal transfers are permitted. The most compact representation of this network is to connect all cities by directed links and to allocate the modes as link attributes. See Figure 2a. Thus, link (A, B) has all the modes permitted, link (A, C) has the diesel and road modes, while link (B, C) has only the road mode. This network representation has some major disadvantages for the model that we seek. If a single flow is associated with a link, it must be the total flow on the link and the flows specific to each mode are not kept explicitly. If multiple flows, one for each mode, are associated with each link, then these
1. THE NETWORK REPRESENTATION
Fig. 1. Physical network.
THE PHYSICAL network infrastructure represented by the network model chosen supports the transportation of several products on several modes. A product is any commodity (collection of similar products), goods or passengers, that generates a link flow specifically associated with it. A mode is a means of transportation that has its own characteristics, such as vehicle type and capacity, as well as a specific cost function. Depending on the scope and level of detail of the contemplated strategic study, a mode may represent a carrier or a part of its network representing a particular transportation service, an aggregation of several carrier networks, or specific transportation infrastructures such as highway networks or ports. The base network is the network that consists of the nodes, links and modes that represent all the physical movements possible on the available infrastructure. The model that we have chosen defines a link as a triplet (i,j, m), where i is the origin node, i E N, where N is the set of nodes of the network,j is the destination node, j E N, and m is the mode allowed on the arc,
o :J:I:l:i:!:1:j: d i ese l
ro I l
E IIIIIII electrified rail R :;::::::::=::;: road
rf0~®IBJ
V~R
R
R
~0 (a)
rood
roil dieseL ro I l e l ec t ric
(b) Fig. 2. Network representation: (a) compact representation and (b) parallel representation.
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J. GUELAT ET AL.
may vary in number from link to link, depending on the number of modes associated with a given link. In addition, the physical difference of the infrastructure modeled is not explicit in such a representation and the specification of cost and delay functions for each mode poses the same problem as that mentioned above for the flows for each mode. In order to overcome these drawbacks, it is necessary to choose a network representation which permits easily the identification of the flow of goods by mode, as well as cost and delay functions by mode. This would be equivalent to keeping explicitly "copies" of the network, one for each mode defined in a particular application. However, this type of network representation becomes prohibitive if large networks with many modes need to be considered. An elegant way of achieving the aim of efficient representation of the multimodal network is to permit parallel links between each node, one for each mode permitted. In this way, the network model resembles the physical network, since, for example, the rail and road infrastructures are physically different. Also, if on a physical link, such as the rail facility, there are two different types of services, such as the diesel and electric train services of the example described in Figure 1, a separate link may be assigned to each service, since they would have different cost and delay functions. For this example, the network with parallel links is described in Figure 2b. In this representation, the mode is an integral part of the network and is not merely an attribute of the link. The parallel links introduced in the network have to be distinguished for the purpose of implementing network algorithms; this is accomplished by defining a link as a triplet (origin, destination, mode). Once the network representation is chosen, it is necessary, in order to model intermodal shipments, to permit and associate the appropriate costs and delays for mode transfers at certain nodes of the network. This may be accomplished by expanding (or exploding) a node where transfers occur by adding as many nodes as there are arcs entering and exiting the node and by adding transfer links between these nodes to form a bipartite graph. An example is given in Figure 3. The explicit explosion of nodes would increase the number of links and arcs of the network; also, the new nodes do not represent the physical network and the transfer arcs represent a mode to mode transfer, so they are not unimodal like the other arcs of the network. The representation chosen for transfers does not require the explicit modification of the network. Transfers are represented implicitly by a pair of arcs, one reaching the node and the other leaving the node. Transfer movements that are permitted at a node may
----)0-----)0---7 1
',,;] ,, ,,
3
2
,,'
4
,A ,
''.J
~O-----)O·····) T r ens fer (1. 4) (1. 3) road raiL dieseL ra i L e L ec t ric
(2.3) (2.4)
Fig. 3. Explosion of a node.
then be addressed, displayed or listed by referring only to the pairs of arcs defining the transfer at the node. It is important to point out several advantages of the network representation chosen. A path in this network consists of a sequence of directed links of a mode, a possible transfer to another mode, a sequence of directed links of the second mode, etc. Thus, a mode change is only possible at a transfer node. This representation also permits one to restrict the flows of certain commodities to subsets of modes (cf. iron ore may be shipped only by rail and ship) and thus capture the mode captivity and restrictions that occur in the operation of freight networks, as well as the transshipments at transfer nodes on transfer arcs. Other network representation may be used. However, we find our approach to be natural for algorithm adaptation and visualization of results. 2. THE MULTIMODE MULTIPRODUCT MODEL
IN THE context of strategic planning of freight flows on a national or regional scale, the most efficient use of the transportation infrastructure is to transport the freight at least total (generalized) cost. Even though it is reasonable to assume that, even in countries where a central authority controls and regulates the shipment of goods, a variety of micro-circumstances in fact prevent the precise achievement of the goal of shipping at least cost, the model that we formulate is based on the objective of minimizing total costs. The notion of cost is central to our model and we interpret cost in the most general way, in the sense that it may have different components, such as monetary cost, delay, energy consumption, etc. The network that we consider consists of a set of nodes N, a set of arcs A, A C N x N x M, a set of modes M and a set of transfers T, TeA x A. We denote their cardinality, nN, nA, nM and nT respectively. With each arc a, a E A, we associate a cost function sa(.) which depends on the volume of goods on the arc or, possibly, on the volume of goods on the other arcs of the network. Similarly, a cost function St (.) is associated with each transfer t E T.
Copyright © 2001 AllLl.uR>J.\ig.l.1h.l..r.tsw...L:R.s;ei,i;ls:Si:eJ..lry~ei.Ud_ _ _ _ _ _ _ _ _ _ _ _ _ _ __
A MODEL FOR STRATEGIC PLANNING OF FREIGHT FLOWS
The products transported over the multimodal network are denoted by index p, pEP, where P is the set of all products considered, which is of cardinality np. Each product is shipped from origins 0, 0 E 0 ~ N, to destinations d, d E D ~ N, of the network. The demand for each product for all origin/destination (O/D) pairs is specified by a set ofO/D matrices. The mode choice for each product is indicated by defining for each of these O/D matrices, a subset of modes which are permitted for transporting the corresponding demand. It is assumed that demand and the mode choice are determined exogenously. Let gm(p) be the demand matrix associated with product pEP, where m(p) is a subset of modes that belong to M(p), the set of all subsets of modes that are used to transport productp. The flows of product p on the multimodal network is denoted by v P and consists of the induced flows of this product on links and transfers. v
P
=
(Vn
(vf),
sP =
(s~),
(sf),
+
E E
LtET sf(v)vD
(1)
over the set of flows which satisfy the conservation of the flow and non negativity constraints. In order to write these constraints for the multiproduct multimode network defined above, the following notation is used. Let K':J(p) denote the set of paths that lead from origin 0, 0 E 0, to destination d, d E D, by using only modes of m(p) E M(p),p E P. The conservation of flow equations are then ~ h k = gad, m(p) £,.,kEK;:/p) (2)
o E 0,
dE D,
Let n be the set of flows v that satisfy (2) and (3). Since the conservation of flow equations are stated in the space of path flows, for notational convenience, the specification of n requires the relation between arc flows and path flows, which is v~ = LkEKP Oakhk,
m(p) E M(p),
pEP
a E A, pEP
(4)
where KP = Um(P)EM(p) U OEO U dED K':J(p) is the set of all paths that may be used by product p, and
o
=
ak
{10
E
if a k otherwise
is the indicator function which identifies the arcs of a particular path. Similarly, the flows on transfers are
vf
and s = (sP), pEP, is the vector of average cost functions of dimension np(nA X nT). The total cost of the flow on arc a, a E A, for the productp,p E P, is the product s~(v)v~; the total cost of the flow on transfer t, t E T, is sf (v) vf. The total cost of the flows of all products over the multi modal network is the function F that we seek to minimize F = LPEP (LaEA s~(v)v~
(3) kEK':J(p), oEO, dED, m(p)EM(p), pEP
A)
at TA)
29
where hk is the flow on path k. The nonnegativity constraints are
aE tE T .
The flow of all the products on the multimodal network is denoted by v = (v P), pEP, and is a vector of dimension np(nA + nT). The average cost functions s~(v), on links, and sf (v), on transfers, correspond to a given flow vector v. The average cost functions for product p are denoted, similar to the notation used for the flow vP, sP, v Ep, where
/
where
= LkEKP otkhk,
o
ak
=
{10
t E T, pEP
(5)
E
if t k otherwise.
The transfer t belongs to the path k if the two arcs that define the transfer belong to it. In conclusion, the system optimal multiproduct, multimode assignment model consists of minimizing (1) subject to (2)-(3) with the definitional constraints ( 4)-(5).
The model is sufficiently general to be adapted for different ways of specifying the demand. Even though typical applications of the model are likely to be carried out after an "a priori" mode choice calculation, which would allocate the demand for a product gP to a set of mode subsets, it is equally possible to permit the demand for a product to be transported over all the allowed modes, that is m (p) is the set of all modes of the network and M(p) has a single component, which in this case is m(p). Also, the model is flexible in the specification of intermodal movements. The mode to mode transfers may be restricted to occur only at specific nodes of the network, and only between specific modes. It is relevant to compare our model to other contributions made in the field of freight network models. It is clear that for strategic planning purposes, we do not seek the detail of a model that identifies shippers and carriers explicitly; rather we consider a model which is adequate for scenario comparisons when major investments are considered. Compared to the study of Friesz, Gottfried and Morlok, [13J we also assume an exogenous total demand by product but, unlike their model, our formulation does not yet include a carrier choice, or equivalently a mode choice component. We
Copyright © 2001 All Rights Reserved
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J. GUELAT ET AL.
assume therefore that the shippers' behavior is reflected in the origin/destination matrices used and the specification of the mode choice, as remarked above. Note, however, that when specific information is known concerning some shippers and their mode choice, this information may be introduced into our model by the proper definition of the permitted modal subset for the corresponding O/D matrix. The resulting flows are obtained by mode and no explicit information is available by carrier, unless a carrier and a mode coincide. In particular, we obtain the second component of the Friesz, Gottfried and Morlok model, which deals with the assignment of freight flows on carrier networks, by defining a mode for each carrier and by using the same functions and demand specification. Thus, the network model is much more refined as it allows detailed representation of the transportation infrastructure, facilities and services as well as the simultaneous assignment of multiple products on multiple modes, thus capturing the competition of products for the service capacity available, a feature which is of particular relevance when alternative scenarios of network capacity expansion are considered. While our aim is to provide a valid method for strategic planning at a national level, the model is sufficiently flexible to represent the transport infrastructure of one carrier only. In summary, the model that we propose provides a refined representation of a large multimode multiproduct transportation system for strategic planning purposes, but it may also be used to analyze the freight transportation carried out by a single carrier. We turn our attention next to discuss the cost functions and to develop a solution algorithm for this model. 3. THE COST FUNCTIONS
IN THE model formulated in the previous section, the average cost functions sa(u), a E A and St(u), t E T, may in principle depend on any or all the components of the vector u. Yet, the actual cost functions that are used to model delays and costs on the links and transfers of a freight network are usually link separable with the exception of transportation services that share the same facilities (e.g. rail services that operate in both directions on a single track). We state next the restrictive and simplifying hypotheses made on the structure of the average cost functions. In its more general form, the marginal cost for transporting product p on arc a is given by the expression c~
= s~- +
~ pEP
(asg ~ a p ug aEA Ua
+
aS f ~ a p tET Ua
uf )
and the marginal cost for transporting product p on transfer arc tis
cf-- = sf-- +
(asg ~ f i ug aEA aUt
~ pEP
+
aS f ~ fi tET aUt
uf )
(7)
We make the following assumptions regarding the separability of the average cost functions: 1. The arc cost functions do not depend on the trans-
fer flows. 2. The transfer cost functions do not depend on the arc flows. 3. The transfer cost functions at a node do not depend on flows on transfers at other nodes. These assumptions are quite natural and result in the simplification of Equation 6 to cl!
= sl! +
-
a
a
~
~
asg
-_ .
pEP aEA aU~
ug
(8)
and of Equation 7 to -
-
cj = sj +
~
asf-
fi .
pEP aUt
uf
(9)
The equivalent of assumption 3 cannot be made for cost functions on the arcs. It is often the case that two distinct modes (such as rail diesel and rail electric operating on the same track) use the same physical infrastructure between two nodes of the network. It is reasonable nevertheless to assume that the interactions are limited to links which are parallel, in both orientations, between two adjacent nodes, such as illustrated in Figure 4. We introduce some additional notation. The arc a = (i, j, m) connects node i to node j by mode m. Let Ma be the subset of modes which is considered in the computation of the marginal cost of arc a. Let Aa == la E A I a = (i, j, m') or a = (j, i, m'), m' E Ma I U Ia I be the set of arcs that must be considered in the computation of the marginal cost of arc a; thus sg(u) = sg(un, a E A a, pEP. If arc a f£ Aa then asg/au~ = o. This permits the simplification of the equation (8) to:
mode
m,
mode
m.
mode
m,
cl!
= sl! +
-
a
a
~
pEP
~ aEH.
asg
-_. au~
ug
(10)
(6) Fig. 4. Parallel links in both orientations.
______________~C~o~p~y~ri~gh~t~©~2001NJ~R~i~gllht~s~R~e~s~e~~~e~d~___________________________
A MODEL FOR STRATEGIC PLANNING OF FREIGHT FLOWS where Hii = la E A I aE Aa I. The set Hii can be obtained in general by scanning the arcs parallel to a in both orientations, which may be time consuming particularly if Ha = la 1for the majority of the arcs. Therefore we make the assumption that 4. The cost of links depends only on the flows of links that share the tail and head nodes, and in both orientations.
This assumption is equivalent to Ha = Aa for all aEA. The marginal costs of the arcs may be written then as
-
c~
-
= s~ + L L pEP aEA"
as~
f i v~
aV ii
(11)
In order to illustrate the assumption made, in reference to Figure 4, let modes mi and m3 share the same facility. Then, for link a = (i, j, ml), Ma = (ml, m3), and Aa = Ha = I (i,j, md, (j, i, ml), (i,j, m3), (j, i, m3) I· These assumptions simplify the specification of the cost functions for the model and decrease significantly the computational burden of evaluating the marginal costs. They also correspond to actual practice and data availability, since most cost functions used in the transport industry (cost, delay, energy consumption, etc.) are defined for a particular facility: highway segment, rail link, rail yard, truck breakbulk, port, etc. The total transportation "cost" from an origin to a destination is the sum ofthese link costs and, possibly, a "fixed" cost component (fixed in the sense that it varies with the volume of goods but is not a function of the distance travelled) for starting the trip on a particular mode. One can model the "fixed" cost by using link and transfer cost functions: the "fixed" cost is associated with the connector link from the origin to the first link and, when appropriate, with the transfer between different modes on the same path. In the code which implements the solution algorithm which is described in the next section, the partial derivatives required in the computation of the marginal costs are carried out by a rather precise numerical approximation procedure. The analytic form of the cost functions may be rather complex, in particular for the rail mode (see CRAINIC, FLORIAN and LEAL[4]) and, as a consequence, so is the analytic form of the partial derivatives and cost functions. Another consideration that motivates the numerical approximation procedure is that it obviates the specification of the partial derivatives by the user of the code, be they simple or complex. The method that we chose to implement in order to compute the derivative of a function at a given value,
/
31
is a polynomial approximation based on the least squares method, which appears to be superior to numerical interpolation (KELLISON,[19] pp. 150-151). We have chosen to use the orthogonal polynomial of Gram, which was adapted for our purposes starting from a code written by CASALETTO and RICE.[3] The interested reader is referred to GUELAT[14] for the details of the numerical approximation procedure for computing partial derivatives. We turn our attention next to develop a solution algorithm for the model stated in Section 2. 4. A SOLUTION ALGORITHM
THE PROBLEM that we consider may be stated in a compact form as Min
subject to
F(v),
v E Q
(P)
Since constraints (2)-(5) are linear, Q is a polytope which, if the multimodal network is strongly connected, is nonempty. Q is compact as well, if the cost functions that we consider are strictly positive, hence the paths considered do not contain cycles. It is assumed that F(v) is a convex function, once differentiable on an open space that contains Q. These assumptions are satisfied, for example, by average arc (transfer) cost functions of the type s~
= L,
lX,«(3,
lX,>O,
+
v~Y«),
(3,>0,
O~z(i)
