Decomposition of Path Choice Entropy inGeneral Transport Networks

TAKASHIAKAMATSU

Department ofKnowledge-Based Information Engineering, Toyohashi University of Technology,Toyohashi, Aichi 441, Japan

This paper shows that the LOGIT type stochastic assignment / stochastic user equilibriumassignment can be represented as an optimization problem with only link variables. Theconventional entropy function defined by path flows in the objective can be decomposed into afunction consisting only of link flows. The idea of the decomposed formulation is derived froma consideration ofthe most likely link flow patterns over a network. Then the equivalence ofthedecomposed formulation to LOGIT assignment is proved by using the Markov properties thatunderlie Dial's algorithm. Through the analyses] some useful properties ofthe entropy functionand its conjugate dual function (expected minimum cost function) have been derived. Finally]it is discussed that the derived results have a potential impact on the development of efficientalgorithms for the stochastic user equilibrium assignment.

requires the prohibitively time-consuming task ofpathenumeration over a general network.

The purpose of this paper is to show that theLOGIT-based stochastic assignmentlstochastic userequilibrium assignment can be formulated as thelink-based equivalent program, Le., the entropyfunction in the objective can be decomposed into linkvariables and does not require path enumeration.The derived program takes the form of the standardlink-node formulation, which is convenient for ap­plying the state-of-the-art algorithms for a conven­tional network programming problem such as New­ton projection method or its variants. Therefore, itseems reasonable to suppose that the derived resultin this paper has a potential impact on the develop­ment of efficient algorithms to solve the stochasticuser equilibrium assignment.

The analyses in this paper also disclose that thedecomposed formulation of the LOGIT assignmentis closely related to the "most likely flow pattern"approach that was originally developed by WILSON(1969) and SASAKI (1969) in the context of trip dis­tribution models. It is also shown that the decompo­sition of the entropy function stems from the"Markov properties" of the LOGIT-based assign­ment model that underlie Dial's algorithm (Thereader who is not famili~r with Dial's algorithm isadvised to read Appendix 2 or the paper by Dial

Transportation ScienceVol. 31, No. 4, November 1997

0041-1655/97/3104-0349 $01.25© 1997 Institute for Operations Research and the Management Sciences

rER

Now, add the following "total (or average) cost" con­straint:

max In z(f) = q(ln q - 1) - L fr(ln fr - 1)rER

(1)

(2)

(3)

(4)

rER

q = L fr

where R denotes the set of paths between the O-Dpair, and fr denotes the flow on rth path in R.

The number of states that result in the assign­ment pattern {fr} is given by

q!z(f)=ll (".

rER r·

The set of path flows that is most likely to occur isthe set with the maximum ofz(f) subject to the flowconservation constraint. Taking the logarithm of (2)and approximating by Stirling's formula, we have

volves the link flow pattern, which seems to besomewhat new. The considerations in this sectioncan be also utilized in a certain application problemthat will be discussed in Section 5.

Throughout this paper, we consider a directedgraph with a finite set N of nodes and a finite set Lof links. Sequential numbers from 1 to N are allo­cated to N nodes. The number of links is L and a linkfrom node i to j is denoted as link ij. To avoid thenotational complexity, we· focus on the case of asingle O-D pair through Section 1 and Section 2.

1.1 Path Flow Pattern

Given q O-D trips, we want to find the most likelypath flow pattern {fr} satisfying the flow constraint:

where Cr denotes the cost of path r, and C denotesthe observed total travel cost. Then, the solution ofthe optimization problem, {f~}, is given by the well­known LOGIT formula:

350 / TAKASHI AKAMATSU

(1971)). Through the derivation of the link-basedprogram, some useful insights into the properties ofthe entropy function and the related function havebeen obtained. These properties are also discussed.

This paper is organized as follows: Section 1 in­troduces the concept of path choice entropy, whichstems from the problem of finding the most likelypath flow pattern. Next, we present the problem offinding the most likely state with respect to linkflow. The question about the relation between thesetwo problems is posed here. To consider this prob­lem, we first analyze the Markov property of theLOGIT model in Section 2. As a result, it is shownthat the path choice entropy in the LOGIT-basedassignment .can be decomposed into a more conve­nient form where the variables are not concernedwith paths but only links or nodes. To investigatethe properties of the path choice entropy function indepth, we consider, in Section 3, the conjugate (dual)function of the entropy, which is also known as the"expected minimum cost." Since this function, aswell as the entropy function, includes the path vari­ables, it is difficult to evaluate in a general network.Nevertheless, we can find a method for evaluating it,using the decomposition properties of the entropyfunction. In Section 4, we construct an optimizationproblem with only link variables, which is a slightmodification of the program presented in Section 1.We then establish the uniqueness of the solutionand the equivalence to the LOGIT-based stochasticassignment. The proof uses the properties of thepath choice entropy and its dual function derived inthe previous sections. Section 5 shows two examplesof the useful extension of this decomposition ap­proach. First, the approach is extended to the flowdependent case, i.e., the link-based equivalent pro­gram for the stochastic user equilibrium assignmentis derived. The discussions on the development ofthe efficient algorithms are also presented. Second,we consider "the most likely O-D link uses problem"that is presented in ROSSI et al. (1989) and JANSON(1993). Though the algorithm for this problem hasbeen an open problem, we solve it by the decompo­sition approach. Finally, various results obtained inthis paper are summarized.

(5)1. MOST LIKELY NETWORK FLOW PATTERN ANDENTROPY FUNCTION

IN THIS SECTION, we show the decomposition of thepath choice entropy by comparing the two differentresults of the classical "most likely pattern ap­proach." One is concerned with the path flow pat­tern, and the results are well known since the worksof Wilson (1969) and Sasaki (1969). The other in-

*exp[ - BCr]

f = q Vrr LrER exp[ -OCr]

where parameter 0 is the Lagrange multiplier forthe constraint inequality (4).

1.2 Link Flow Pattern

The flow pattern over a network is representednot only by the path flow {fr}, but also the link flow

(8)

(6)

{Xij}. Accordingly, it is natural to consider the mostlikely flow pattern with respect to the link flow.

Suppose q generation trips from an origin areassigned to all the links oj emanating from the ori­gin node by Xoj. The number of states that result inthis assignment pattern is given by

q!/~ Xoj!.

Similarly, the number of states that the trips enter­ing into node k are assigned to the links kj emanat­ing from node k by Xkj are given by

(~Xik)!/~ Xk) for 'Vk "* d, (7a)

(~Xik)!/q! = 1 for k = d. (7b)

Thus, the number of combinations for the assign­ment pattern {Xij} which satisfies the flow conserva­tion law in each node is represented by

DECOMPOSITION OF PATH CHOICE ENTROPY I 351

where the constraint equations (lOb) are the flowconservation law in each node, and the constraintinequality (10c) corresponds to the total cost con­straint (4) in the program in Section 1.1. Note thatthe objective function (lOa) can be interpreted as"link-choice entropy" minus "node-choice entropy."

Recall now that both {f~} and {x!j} are the mostlikely flow pattern under the same constraints. It isnatural to speculate that the link flow pattern re­sulting from {f~} and {xij} is precisely the same. Inother words, we can conjecture that the solution ofthe link-based program above yields the LOGIT typeassignment pattern.

For the full examination of the conjecture, theanalytical solution of the program above is required.Prior to the direct analyses, we will explore the prop­erties ofLOGIT-based assignment in Sections 2 and 3.

2. MARKOV PROPERTIES OF LOGIT MODEL ANDDECOMPOSITION OF ENTROPY

FOR THE CONVENIENCE of exposition, we describe theflow pattern by the "probability" variables

Pr=fr1q, Pij=xijlq.

The LOGIT-based stochastic network loading con­sists of the following path choice model:

That is, we can obtain the most likely link flowpattern by solving the following program:

max Z(x) == - L Xkj In Xkj

kj

The set of link flows that is most likely to occur,{xij }, is the set with the maximum of z(x). Takingthe logarithm of (8) and approximating by Stirling'sformula, we have

In z = q In q - L Xkj In Xkj

kj(12)

(13)er = L t ij8ij,r,ij

and the relationships between path variables andlink variables over a network:

where (J is the dispersion parameter in the LOGITmodel, t ij is the travel cost on link ij, and. 8ij ,r is thetypical element of the path-link incidence matrix,Le., it is defined to be 1 if link ij belongs to path rand 0 otherwise.

The flow pattern (p, P) generated by the LOGIT­based stochastic assignment satisfies the followingequation representing the Markov property:

(lOa)

subject to

LXik - LXkj + 80kq - 8dk q = 0 VkENj

L Xijt ij ~ CijEL

(lOb)

(lOc)

(lOd)

Pr = n Pr(ijlj)8ij,rijEL

where Pr(ij/J) == Pij/~Pmj, (14)

The reason equation (14) holds may be easily under­stood by the fact that the link flow pattern generated

352 / TAKASHI AKAMATSU

by Dial's algorithm is equivalent to the LOGIT typeassignment: Le., (11)-(13). In considering theMarkov property, however, we must draw attentionto the definition ofthe path set for loading flows. Theproperty does not necessarily hold for a set ofsimplepaths, Le., paths which do not pass through samelink more than once. The reason is that the defini­tion of the path set and the Markov property (14)contradict each other. Consider a path set consistingof simple paths, and suppose that the flows on allthe links in a certain cycle are not zero. This canoccur by appropriate overlaps of various simplepaths even if the path set does not contain explicitcycles. The choice-probability of the path, includingthe cycle, is not zero according to (14). On the otherhand, such a path (cycle) cannot be included in theset of simple paths according to the definition.

To avoid such a contradiction, we assume in thelater sections that the path set for the assignmentconsists of (a) Dial's efficient paths or (b) all the possi­ble paths with no restriction (which may possibly in­clude infinite cycles). In case (a), the elements of thepath set are restricted to the paths where flows areassigned in Dial's algorithm; see Appendix 1 or Dial(1971). Since Dial's efficient paths form no apparentcycles, they always satisfy (14). In case (b), the path setis not restricted, that is, the path set can include notonly simple paths but also all the possible cycles (pos­sibly infinite cycles). It is clear that the latter path setalso satisfies (14). For further discussions on the pathset and Markov property, see AKAMATSU (1996), BELL(1995), and SASAKI (1965). Although the Markov prop­erty ofthe LOGIT model may hold for other definitionsof the path set, we employ the definitions in (a) or (b)throughout this paper.

Using the Markov property of the LOGIT-model,we can obtain the following proposition, whichstates the decomposition of path choice entropy.

PROPOSITION 1. Given a stochastic assignmentmodel satisfying the Markov property (14), let P bethe path choice probability determined by the assign­ment model. Then, an entropy function with respectto P, HP (P), can be decomposed into the followingrepresentation:

Proof. Substitute (14) into the path choice entro­py:

Considering the relationship between P and p (12)and the definition of Pr(ijlj), we obtain

- LPrlnPr = - LPijlnPr(ijli)ij

Pii=-LPijln~

ij LJmPmj

r

3. CALCULATION OF EXPECTED MINIMUM COSTWITHOUT PATH ENUMERATION

WE HAVE ANALYZED the properties of an entropyfunction so far.. To investigate the related propertiesin depth, let us consider the conjugate (dual) func­tion. This function is also known as the "expectedminimum cost (maximum utility)" or the "satisfac­tion," which plays an important role in the random

= - ~Pij lnPij + ~(~Pij)ln(~Pij)."} }" "

(18)

DIn (15), HL(p) means an entropy function with

respect to "link choice probability Pij," and the sec­ond term, HN(p), is one with respect to "node choiceprob~bility L! Pij." The link choice probability, Pij,

consIstent WIth a LOGIT-type stochastic assign­ment, can be obtained by Dial's algorithm withoutpath enumeration (we exclude cyclic flows from thepath set for assigning flows). Hence, we see that thevalue of the path choice entropy over a general net­work can be computed without path enumeration.

The decomposed entropy function has the follow­ing property, which is used for the analysis of anequivalent optimization problem of the stochasticloading in Section 4.

PROPOSITION 2. H(p) == HL(p) - HN(p) is astrictly concave function with respect to link choiceprobability. .

Proof. See Appendix 3.

-~ PrInPr = -~[PrIn(gpr(ij1j)&jr}]

=- L[Pr~ 8ij,r In Pr(ij! j)]r "}

=- ~[L(Pr8ij,r)lnpr(ij1j)] (17)"} r

HL == -L lnpijlnpij,ij

r

HP(P) = HL(p) - HN(p) (15)

HP == - L Pr In Pr, (16)where

(19)

utility .,ry; see, for example, WILLlAMS (1976,1977) _GANZO (1980, 1982), MIYAGI (1986). The"expected minimum cost" in the context of the sto­chastic assignment is the expectation of the mini­mum path cost defined as follows:

Sod(Cod) == E[ m!n{C~ + E~}]'

where C~d is the cost of rth path between O-D pairod and €~d is the random error term. This functionalso appears in the objective of the following dualproblem for the stochastic equilibrium assignment:

min Z(t) = 2: Jtii

ti/(v) dv - 2: qodSod (20)ij tij (0) od

where tijl(.) is the inverse of a travel time func­tion for link ij, qod is the flow from origin 0

to destination d.

Despite the importance ofevaluating the expectedminimum cost, it is difficult to compute the value ina general network. The reason is that the definitionof the function contains the path cost variables,which are almost impossible to enumerate over ageneral network. The decomposition formula (15)derived in the previous section, however, enables usto devise a method for evaluating the function in theLOGIT model without path enumeration. In addi­tion, we also demonstrate below that "link-weight"in Dial's algorithm can be utilized for the computa­tion. This relationship is also useful for the analysisof an equivalent optimization problem discussed inSection 4.

3.1 Entropy Function and ExpectedMinimum Cost

We first show a conjugate relation between theexpected minimum cost function and the entropyfunction in a straightforward manner. A more so­phisticated derivation of the relation based onROCKAFELLER's (1970) conjugate-dual theory can beseen in Miyagi (1986). The expected minimum costfunction in the LOGIT-based assignment can be rep­resented as

1Sod(Cod) = -0 In 2: exp[-BC~]

r

1 exp[-OC~]- --In- 0 pod

r

1= C~ + "0 In P~ \fr, \fod (21)

DECOMPOSITION OF PATH CHOICE -ENTROPY I 353

where p~d denotes the probability that path r inO-D pair od is chosen, and C~d denotes the path costfor the path r in O-D pair od. Multiplying a pathchoice probability to both sides of (21), and summingup with respect to all the paths, we have

r

(22)

The first term of r.h.s. in (22) can be convertedinto the representation by link variables, becausethis means an average (or total) travel cost. Thesecond term is the path choice entropy, to which thedecomposition formula can be applied. Hence, wesee that the expected minimum cost function is eas­ily evaluated by calculating the link choice probabil­ity using Dial's algorithm, and using the followingequation:

Sod(Cod) = 2: piJtij

ij

+ ~(~pijd Inprj - f( ~prj)In(~prj))(23)

3.2 Dial's Algorithm and Expected MinimumCost

Since (21) holds for an arbitrary path, the equa­tion for a minimum cost path

also holds, where C~n denotes a minimum path costfor O-D pair od, and P~n is the probability that theminimum cost path for O-D pair od .is chosen.Hence, if the minimum path cost and the probabilityare obtained, we can easily evaluate the expectedminimum cost.

The probability that a minimum cost path is cho­sen can be computed from the ''link-weight" in Dial'salgorithm. In Dial's algorithm, denoting the nodesincluded in the minimum cost path as (0 ~ A ~

B ~ · · ·~ y ~ Z ~ d), this probability is,given by

od W[Z ~ d] Wry ~ Z]

P roin = Lm W[m ~ d] Lm W[m ~ Z]

W[A ~ B] W[o ~ A]

· · · 2:m W[m ~ B] 2:m W[m ~ A]

354 / TAKASHI ·AKAMATSU

. 1pod - (26)

min - ~m W[m ~ dJ'

1= Lm W[m ~ d] L[Z ~ d]L[Y ~ Z]

· · · L[A ~ B]L[o ~ A] (25)

This equation holds not only for a destination nodebut also for arbitrary traversal nodes. Accordingly,we can easily evaluate the expected minimum costsfor all the nodes from an origin by implementingDial's algorithm and using (27).

where L [i ~ j] denotes a "link-likelihood" for link ij,and W[i ~ j] denotes a "link-weight" for link ij inDial's algorithm (see Appendix 2). From the defini­tion, the value of "link-likelihood" for the links in­cluded in a minimum cost path is equal to 1. Hence,(25) can now be written as

C in the constraint. The solutions of both programstake the same form with respect to link flows.

For the fundamental property of [SA-arc], we canstate .the following proposition, which is easily un­derstood from Proposition 2.

PROPOSITION 3. The globally optimum solution forproblem [SA-arc] can be uniquely determined.

Proof. See Appendix 4.

A consideration of the propositions stated so farand the properties of "expected minimum cost" pre­sented in the previous section lead us to establishthe following proposition.

PROPOSITION 4. The problem [SA-arc] is equivalentto the LOGIT-based stochastic assignment with asingle O-D pair, where the path set is defined as (aJor (b) described in Section 2.

Proof Define the Lagrangian function for the pro­gram [SA-arc] as follows:

L(p, J.L) = Z(p) + ~ J.Lkgk(P) (31)k

Because the program [SA-arc] is a convex program,the necessary and sufficient conditions for optimal­ity are given by the following Kuhn-Tucker condi­tions:

aL aLPij a,n ..= 0, -a- ~ 0, and Pij ~ ° Vij E L

r£j 'Pij(32)

~ Pij - ~ Pkj + Oak - Ddk = 0 Vk E N. (33)j

(27)od 1 ~Sod = Cmin - Oln L.J W[m ~ d].m

and the expected minimum cost is given by

4. EQUIVALENT OPTIMIZATION PROBLEM BY LINKFLOWS

WE ARE NOW in a position to analyze the equivalentprogram for stochastic loading by using only linkflows. We begin with the program for the case of asingle O-D pair with unit O-D flow, as discussed inprevious sections. Next, we extend the program to ageneral case of many-to-many O-D pairs.

4.1 Single O-D Pair Case

Consider the "following minimization problem:[SA-arc]

1min Z(p) = ~ Pi/ii - (j {HL(p) - HN(p)}

ij(28)

Suppose Pij =1= 0, then (32) becomes

(34)

Multiplying both sides of this equation for an arbi­trary path r, we obtain

subject to

gk(P) == ~Pik - ~Pkj + Sok - Sdk = 0, Vk E Nj

This program is almost the same as the one pre­sented in Section 1.2. In [SA-arc], the parameter (J isexternally given, instead of giving total travel time

Pij ~ 0 Vij E L

(29)

(30) On the other hand, considering the flow conserva­tion law in (34),~ the following equation should hold:

~ exp[ -(J(tij - J.Li + J.Lj)J = 1 Vj E N, (36)

DECOMPOSITION OF PATH CHOICE ENTROPY / 355

Le., On the other hand, as we derived in the previoussection, the expected minimum cost can be repre­sented by the link-weight

Substituting (45) into (35) results in the LOGIT­based stochastic loading represented by link flows

1/Ld - /La = -Sod = 7i In L exp[ -OCr]. (45)

r

Consequently, we can see that the difference be­tween a Lagrange multiplier for origin, Mo, and onefor destination, Md' determines the expected mini­mum cost:

Vj E N, j =1= o. (37)

This equation means that the value of the Lagrangemultiplier in nodej, J.Lj' is determined by the valuesof J.L in node i, J.Li' which has the links entering intonode} (Le., node i is a predecessor node for node}).Therefore, evaluating (37) from the destinationnode to the origin node in sequence results inexp[ - O(J.Ld - J.Lo)]. For the evaluation, we classifythe problem into two cases based on the definition ofthe path set assumed in the model:

a. the path set consists of Dial's efficient paths,b. the path set includes all possible paths, with no

restrictions.

. 1 "Soj = C%in - (j In LJ W[m ~ j].m

(44)

We first consider the case (a). We use Dial's algo­rithm for the evaluation of (32). From the definitionin Dial's algorithm, "link-likelihood" and "link­weight" are given by

L[i ~ j] = exp[O(C%in - C~in - tu)], (38)

W[i ~ j] = L[i ~ j]L W[m ~ i], (i =t= 0)

i.e.,

(P .. ) 8ij.r exp[-eC]IT tJ • r,

ij Lm Pmj Lr1exP[ -eCr,](46)

(47)

m(39)

where C%:in denotes the minimum path cost from anorigin 0 to a nodej. Define a "node-weight," Wj' as asum of the "link-weight" for the links entering intonodej:

Next, we consider the case (b) where the feasibleregion defined by (29) and (30) includes all possiblepaths (and possibly includes all cycles). It is enoughfor the proof to show that (37) is equivalent to (45).

Let {Vi} and {Wij} be defined as follows:

Wj == L W[m ~ i] and Wo == 1. (40) (48a)m

Summing over i and substituting the definition of''link-likelihood" and "link-weight" (Le., (38) and(39») into (40), and rearranging it, yields

exp[ -OC%in]Wj = L exp[ -Otij]exp[ -OC:in]Wi.

{exp[ -Otij] Vii E L

W ij - 0 if link ij does not exist.(48b)

Then (37) can be represented as a system of linearequations. In a matrix-vector form it is written as

(41)

Comparing the equation above and (37), we see thatJ.L and Whave the following relationship:

exp[ -OC%in]Wj = exp[OJ.Lj]. (42)

Therefore, a solution to (37) can be obtained by set­ting Mo = 0 and for j =1= 0 choosing J.Lj to satisfy

. 1 .J.Lj = -C~in + (j In Wj = -C~in

V=[I+W+W2 +···]Wo• (51)

(49)

(50)

By the well-known transformation of the inversematrix, (50) can be represented as

where V == [VI, V2' ••• , v n _ 1]T, W is a (n - 1) X

(n - 1) matrix whose (j, i) element is Wij (i =1= 0 andi =1= 0), Wo == [W o l' W o2, ••• , won]T and n denotesthe number of nodes. If the inverse of [I - W] exists,we have

(43)1

+ 7i In L W[m ~ j].m

where HL(xod) == - ~ xC?c! In xC?c!LJ"J "J'ij

1min Z(x) = ~ xi}ti} - fi ~{HL(xOO) - HN(xod

)}

ij od

(54)

356 / TAKASHI AKAMATSU

Note that the (j, i) element ofWn, w!J], yields

w~j] = ~ exp[ -eC~~n] (52)rER?i

(56)

(58)

(59)x .. = ~ xC?· Vi)· E L"J LJ 1,J

o

~ (~od xii)= LJ L.J Xij In ~ od.od ij LJm xTnJ

~{HL(xod) - HN(xod)}

od

Thus, we can establish the following proposition.

PROPOSITION 5. The LOGIT-based stochastic assign­ment with many-to-many O-D pairs is equivalent tothe following prograln:

[SA-arc(MM)]

subject to

~xik - ~xkj + ~qod8ok - qod8dk = 0 Vo, Vk E Nj d

1min Z(x) = ~ xiAi - fi ~{HL(xO) - HN(X')}

ij 0

(57)

o

(53)Vd = ~ exp[ -eC~],rERod

which implies (45).

4.2 Many-to-Many O-D Pair Case

The discussion above holds for each O-D pair inthe case ofmany-to-many O-D pairs. Therefore, rep­resenting the link flows by each O-D pair and sum­ming up the objectives for each O-D pair, we canconstruct the objective function of the equivalentoptimization problem for the many-to-many O-Dpair case. That is, the objective is given by

where Rij is the set of paths which co~ect node iand} by passing through n links, and C;n is the costof rth path belonging to Rij. '

Define Rod as the path set whose elements are allthe possible paths connecting node 0 and d. Then itis clear that Rod consists of {R~d: n = 1, 2, 3, ...}.Hence, we see that (51) means that v~ is given by

where x~. == Ld x~t;l is the flow on link ij with origino. Using this relationship, the entropy function in(54) takes the form

Xijd denotes the flow on link ij with O-Dpair od.

Furthermore, as we will show, we can replace theobjective function with a more "compact" one wherethe unknown variables are not the link flows by O-Dpair, xod , but the link flows by origin, xo.

Consider the proportion of trips choosing a node iconditional on choosing a successor node}. As the"backward pass" in Dial's algorithm indicates, thisproportion does not depend on the destination of theflow with some origin in the LOGIT-based stochasticassignment. That is,

(60)xij ~ 0 Vo, Vi} E L,

where the functions HL and HN are defined as

HL(XJ) == - ~ xij In xij, (61)ij

This program can be solved by Dial's algorithmwithout path enumeration when we regard the fea­sible link flow patterns defined by (58), (59), and (60)as excluding all possible cyclic flows. On the otherhand, when we consider that the feasible regionincludes all possible cycles, the solution is clearlydifferent from the flow .pattern generated by Dial'salgorithm. Recently, the solution's properties wereconsidered by Bell (1995) and Akamatsu (1996). Us­ing the Markov properties of the model, they alsodeveloped a method to obtain the solution withoutpath enumeration (we call the method MCA:Markov Chain Assignment).

(55)Vo,d, i =1= d

5. SOME EXTENSIONS

THE RESULTS PRESENTED in the previous sectionscan be easily extended to some useful applications.We consider two important problems below. Thefirst problem is the flow-dependent stochastic as­signment (Le., the stochastic user equilibrium as­signment). The derived result has a potential impacton the development of efficient algorithms for solv­ing the assignment. The second example is the prob­lem of determining a unique path flow pattern (orO-D-specific link uses) from given link flows; thisproblem was recently considered by Rossi et al(1989) and Janson (1993). A consideration of thisproblem indicates another aspect of the theoreticallinkage between the LOGIT-type stochastic assign­ment and the combinatorial considerations shown inSection 1.

5.1 Stochastic User Equilibrium Assignment

Suppose that the cost for each link ij E L, t ij, isrepresented by the increase function of the link flowXij'

(63)

For the stochastic assignment with flow-dependentlink costs (Le., the stochastic user equilibrium as­signment), we can construct an equivalent optimi­zation proWem, slightly modifying the result pre­sented in the previous section.

PROPOSITION 6. The LOGIT-based stochastic userequilibrium assignment is equivalent to the follow­ing program:

[SUE-arc(MM)]

min Z(x) = ~ f%ii tij(w) dw

"J 0

1- e2:{HL(xO) - HN(xO)} (64)o

subject to (58), (59), and (60).

The difference between [SA-arc(MM)] and [SUE­arc(MM)] is only the link cost term in the objectivefunctions. Therefore, proving the equivalency of the[SUE-arc(MM)] to the stochastic user equilibriumassignment parallels the proof in [SA-arc]. It mustbe noted, however, that the equivalency holds onlyfor the path set assumed so far: (a) the path set thatexcludes all possible cyclic flows or (b) the path setthat consists of all possible paths (and possibly in­cludes infinite cycles). Unlike the flow independentcase (Le., [SA-arc(MM)]), determining the path set

DECOMPOSITION OF PATH CHOICE ENTROPY / 357

(a) is not'self-evident. In [SA-arc(MM)], the corre­sponding path set can be easily determined as Dial'sefficient paths. In [SUE-arc(MM)], however, thepath set cannot be determined before solving theprogram, because efficient paths are dependent onthe link cost pattern, which is given after the equi­librium solution is obtained.

Although the program [SUE-arc(MM)] seems tohave the issue related to the path set mentionedabove, the program has a potential impact on thedevelopment of efficient algorithms for the LOGIT­based stochastic user equilibrium assignment. Theprogram does not include the path variables, whichmeans that the objective and the constraints of theprogram can be explicitly dealt with. Consequently,various solution methods for standard convex net­work programming problems, such as the partiallinearization method (EVANS, 1976; PATRICKSSON,1993), Frank-Wolfe algorithm/modified Frank­Wolfe algorithm (LEBLANC et al., 1975, 1985; FuKU­SHIMA, 1984), subgradient algorithm (AKAMATSUand MATSillvIOTO, 1989) are applicable.

Indeed" AKAMATSU et al. (1990a,b) implementedthe partiallinearization (PL) method for solving theprogram [SUE-arc(MM)], and found that the methodconverged very rapidly and outperformed by far theFrank-Wolfe (FW) algorithm for solving the Ward­ropian equilibrium problem. The algorithmic outlineof the PL is similar to the FW. The difference is thesubprogram that determines the descent direction.The objective of the subprogram for PL is not alinear function as in FW, but a partially linear one.In solving [SUE-arc(MM)], the first term of (64) isapproximated by the linearized function and thesecond term remains unchanged. Then the subpro­gram reduces to the problem with the same form as[SA-arc(MM)] (Le., a flow-independent LOGIT-typestochastic assignment). In the implementation byAkamatsu et al., Dial's algorithm was used for solv­ing the subprogram. Unlike the method of succes­sive averages (Sheffi and Powell, 1982), the exactline search step based on the objective function (64)was implemented, which accelerated the conver­gence.

Strictly speaking, however, the method abovesometimes does not yield an exact solution, but anapproximate one. The reason is that the path setgenerated by Dial's algorithm in each iteration doesnot necessarily coincide with that for the equilib­rium cost pattern. To obtain an exact solution, thesub-program in PL should be solved by fixing thepath set so that it remains unchanged through iter­ations. Alternatively, the MCA method by Bell(1995) and Akamatsu (1996) also can be employed,where the path set for the equilibrium point includes

358 / TAKASHI AKAMATSU

5.2 Most Likely O-D Link Uses Problem

Let us turn to the second example of the applica­tions of the entropy decomposition approach. Re­cently, Rossi et al. (1989) considered the problemthat determines the unique path flow solution of aWardrop equilibrium assignment. The Wardropequilibrium assignment has a unique solution interms of aggregated (total) link flows, but is gener­ally not unique in terms of path flows or O-D-spe­cific link flows. Rossi et al. determined the mostlikely path flow pattern given link flows by a War­drop equilibrium assignment. Although the link flowsare assumed to be given by a Wardrop equilibriumassignment in their original problem, it is also possibleto assume that the link flows are given by an arbitrarymethod (e.g., observed traffic counts).

By the combinatorial consideration shown in Sec­tion 1.1 ofthis paper, Rossi et al. (1989) proposed thefollowing formulation of the problem:[ML-path(MM)]

min Z(f) ='2: 2: trod In t:d (65)od r

all possible paths (Le., "path set (b)" is assumed).Because MCA solves the stochastic assignmentproblem for the same path set regardless of the linkcost pattern, the path set used in the algorithm re­mains unchanged and it will converge to an equilib­rium point. Further discussions on using MCA insteadof Dial's algorithm and the related numerical experi­ments will be reported in a subsequent paper.

Finally, we should mention the other approachesfor improving the method for solving [SUE­arc(MM)]. It seems reasonable to suppose that themethod can be further improved with regard to boththe convergence speed and the storage required. It ispossible to apply recent, more efficient network pro­gramming techniques such as the Newton projectionmethod or its variants, because the [SUE-arc(MM)]takes the form of the standard link-node formula­tion that is convenient for the method; see, for ex­ample, KLINCEWICZ (1983), DEMBO (1987), DEMBOand TULOWITZKI (1988), KATSURA et al. (1989),IBARAKI et al. (1991), ZENIOS and PINAR (1992). Al­though the discussion on this topic is interesting, wedo not go into detail here because its implementa­tion is outside the scope of this paper.

(70)

Vod, Vk E N

x?J ~ 0 Vod, Vij E L (72)

Janson argued that the program above is equivalentto the Rossi program [ML-path(MM)]. He did notshow, however, the "correct" combinatorial consider­ations of this problem. The simple formula in (71)should not be used to count the number of possiblelink flow patterns, because the link flows in a net­work cannot be mutually independent, owing to the.flow conservation at each node. Thus, it is clear thatthe program above is not equivalent to [ML­path(MM)]. Janson argues that the optimality con­dition for the two programs yields the same result.In his proof, each optimality condition for Rossi'sprogram and Janson's is shown as Eq. (A!) and (A6),_;respectively. Although he ~howed that the righthand sides of (AI) and (A6) take· identical form, hedid not show that the left ·hand sides of (AI) and (A6)

2: x?t - 2: Xkt + q od0od - q od0dk =0j

subject to

xij = 2: 2: xrJ Vij E L,o d

where tij denotes the Lagrange multiplier for con-straint (66).

Although the solution takes an apparently simpleform, it is difficult to apply this model to a largescale network because the program above is repre­sented by path flows (In fact, the Rossi applicationwas limited to a very small network).

For this problem, Janson (1993) proposed the fol­lowing link flow-based formulation where the pathflows are replaced by the O-D-specific link uses xijd:

min Z({xod: Vod}) = 2: 2: 2: x?t ln x?J (71)

ij 0 d

where Xii denotes the given (total) link flow forlink ij,r:d denotes the flow on rth path for O-Dpair od,oitr is 1 if link ij belongs to rth path forO-D pair od and

ootherwise.As shown in their paper, the problem's solutionyields the following LOGIT-type model:

exp[ -C:d]

fr;J = qod = 2: - d 'r:Ir, od, (69)r' ERod exp[- c:., ]

(66)

(67)

(68)

subject to Xii = 2: 2: f~oiJ,rod r

coincide with each other. The proof is therefore in­complete.

In order to obtain the correct equivalent program,recall the discussions in Section 1. The program[ML-path(MM)] is similar to the program presentedin Section 1.1, and both programs are solved accord­ing to the LOGIT-type route choice model. The dif­ference is only the definition of the "link costs."Furthermore, to obtain the "most likely" O-D linkuses, the formulation should be derived from a com­binatorial consideration of link flow patterns. Con­sidering these facts, we see that the link-based prob­lem that is equivalent to [ML-path(MM)] shouldcorrespond to the program derived in Section 1.2.Specifically, it can be stated as follows:

PROPOSITION 7. The link-based program for the mostlikely O-D link uses (i.e., the equivalent program for[ML-path(MM)]) is given by

[ML-arc(MM)]

min Z({xo: Vo}) = - L{HL(xO

) - HN(xO)} (73)

subjectto Xij= L xij Vij E L, (74)o

(58) and (60),

where the definition ofpath set parallels the discus­sion in Section 5.1.

The transformation from the path-based programto the link-based program provides various methodsto solve the problem without path enumeration. Anatural approach to solve [ML-arc(MM)] is to usethe following dual program, which can be obtainedby the slight modification of the standard dual pro­gram of the stochastic user equilibrium assignment(Le., (20) in Section 3):

min Z(x) = L tiftij - L qodSod(Cod(t)) (75)ij od

where Sod == -In L exp[ -C~d].r

The function Sod in the objective can be evaluatedwithout path enumeration by using the method dis­cussed in Section 3.1. Therefore, this program maybe solved by appropriate optimization techniques.After obtaining the solution ("link cost" t), we cancalculate"the O-D-specific link flows by implement­ing the LOGIT-type stochastic assignment based onthe ''link cost" t. The stochastic assignment is easilyimplemented by Dial's algorithm when the loopflows are excluded from the path set. Alternatively,

DECOMPOSITION OF PATH CHOICE ENTROPY / 359

the Markov Chain Assignment can be used when wedo not restrict the assignment path set.

6. CONCLUDING REMARKS

THIS PAPER SHOWED that the LOGIT type stochasticassignment can be represented as an optimizationproblem with only link variables. Namely, the con­ventional entropy function defined by path flows inthe objective was decomposed into the function con­sisting ofonly link flows. The idea ofthe decomposedformulation was first derived from a consideration ofthe most likely link flow patterns over a network,compared with the most likely path flow patterns,which parallels the classical works ofWilson (1969)and Sasaki (1969). Then the equivalency of the de­composed formulation to the LOGIT assignmentwas proved by using the Markov properties implic­itly indicated by Dial's algorithm. Through the anal­yses, some useful properties of entropy function andits conjugate dual function have also been derived.Finally, the entropy decomposition approach wasextended to two important problems: the stochasticuser equilibrium assignment and the problem ofdetermining the most likely O-D link uses. In theformer problem, a link-based program similar tothat in the flow-independent case was obtained. Thelatter problem was also represented by the decom­posed program with only link variables, which canbe solved without path enumeration.

This paper concentrated on showing the theoreti­cal foundation concerning a family of LOGIT-basedstochastic assignment problem by link variablesrather than by implementing an algorithm. It isworth mentioning, however, that the derived resultscan be used for developing efficient algorithms tosolve the problems as suggested in Section 5. Asimple and efficient method to solve the decomposedequivalent program is to apply a partial lineariza­tion method. In the implementation, the MarkovChain Assignment method developed· by Bell (1995)and Akamatsu (1996) can be effectively used to solvethe subproblem. Another promising approach is todevise efficient implementation techniques of thevariants of the projected Newton method, whoseconvergence rate is quadratic. More precise discus­sions on these matters are interesting topics andwill be reported in subsequent papers.

APPENDIX 1: ALGORITHMS FOR THE STOCHASTICUSER EQUILmRIUM ASSIGNMENT

THE METHOD OF successive averages, where the pre­determined step size is employed and Dial's algo­rithm is used to find a direction vector, was proposedby Sheffi and Powell (1982), Powell and Sheffi

L[i ~ j]

APPENDIX 3: PROOF OF PROPOSITION 2 (THESTRICT CONCAVITY OF H(p»

THE FUNCTION H(p) == HL(p) - HN(p) can bedecomposed by each node:

\:/j E NFiJ

where NFi is the set of final nodes of all linksleaving node i, and NTi is the set ofstarting nodesof all links arriving at node i.

Step 2: ("Backward Pass") Starting with the mostdistant destination node, examine each node j EN in descending order of C~in. Assign a link flowXi} to each link i ~ j:

{p[ Ll (coj COi t)] if Com!i

1' n > Com

i1'n_ ex (} min - min - ij

- 0 otherwise

Step 1: ("Forward Pass") By exa~iningall node i EN in ascending order of C~iIl' calculate "linkweight":

W[i ~ j]

{

L[i ~ j] if i = 0

= L[i ~ j] ~ W[k ~ i] otherwisekENT;

(AI)

ifj=dED

H(p) = ~ H jj

W[i ~ j]

qod ~kENTjW[k ~ j]

~. W[i ~ j]L.J Xjk "" W[k .] otherwise

kENFj L.JkENTj ~ J

Vi E NTj

where D is the set of destination nodes, and qod isthe O-D flow from origin node 0 to destinationnode d. When the origin node is reached, stop.

Restricting the path set for loading flows is an-other important feature of this algorithm. More spe­cifically, each link in the path to which the algo­rithm assigns flows ("efficient path") must increasethe minimum cost from the origin; see the definitionof"link likelihood" in Step o. The "efficient path" hasan interesting property: the set of paths forms nocycles (loops) in general. In other words, Dial's algo­rithm solves the equivalent program, [SA-arc], ex­cluding the cyclic flows from the feasible regionformed by (58), (59) and (60). For further discussionson these properties, see Akamatsu (1996).

360 / TAKASHI AKAMATSU

(1982), and Daganzo (1982). Although the algorithmcan avoid path enumeration over a network, its con­vergence rate is very slow because it does not eval­uate the objective function. The algorithm has an­other drawback in that the convergence is notguaranteed if the paths for loading flows are notfixed through the iterations. Recently, Chen andAlfa (1991) proposed the algorithm that utilizes thegeneralized inverse for the link-path incidence ma­trix. It is not, however, suitable for large scale prob­lems. The difficulty of applying this algorithm isdescribed in Bell et aI. (1993). More recently, Bell etaI. (1993) propo'sed the modified Frank-Wolfe algo­rithm utilizing Bregman's iterative balancingmethod. This algorithm avoids enumerating all pos­sible paths, and it therefore may be applicable tolarge scale networks. With regard to the efficiency ofthe algorithm, we cannot make any definite conclu­sions because no implementations of this algorithmfor large scale networks have been reported. Never­theless, it seems that the method leaves room for avariety of improvements from the theoretical viewpoint. The reason is that the convergence rate of thisalgorithm is not comparable to the state-of-the-artalgorithms for a conventional nonlinear networkprogramming problem; the convergence rate ofFrank-Wolfe algorithm is sublinear, which meansvery slow convergence. On the other hand, that ofthe recent algorithms such as the projected Newtonmethod or its variants (see, for example, Klincewicz,1983; Dembo, 1987; Dembo and Tulowitzki, 1988;Katsura et al., 1989; Ibaraki et al., 1991; Zenios andPinar, 1992) is super-linear or quadratic, whichmeans very fast convergence.

APPENDIX 2: DIAL'S ALGORITHM

DIAL'S ALGORITHM IS THE procedure that assignsflows over a network according to a LOGIT-typeroute choice model. The algorithm does not requirepath enumeration, because it operates only withlink/node variables. The reason this algorithm gen­erates the link flow pattern consistent with theLOGIT-type route choice model is described in Dial(1971). In essence, the feature stems from theMarkov property of LOGIT-type assignment models(Le. (14». The algorithm for a one-to-many O-Dpattern is summarized as follows:

Step o.a. For each node i E N calculate the minimum

travel time from an origin node 0, C:in, basedon link cost {tij}.

b. For each link i ~ j E L calculate the ''linklikelihood," L [i ~ j], according to the followingformula:

where

(A2)

HLj - - 2:Pij In Pij, H~ == - (f Pij)In(f Pij)­

(A3)

Hence, to prove the strict concavity of Hj for eachnode} with respect to {P:i} means the strict concavityof the H(p) with respect to p.

First, we show the concavity ofH by showing thatHessian ofH j is a negative semidefinite matrix. Thetypical element of Hessian matrix ofHj with respectto p is given by

aH· a ( )a / = -a- -In P ij + In 2: P mjPij Pkj Pkj m

1 1= -- 8[i, k] + " (A4)

Pkj LimPmj

where 8[i, k] means Kronecker's delta. Hence, thequadratic form of Hessian matrix with respect to areal vector r is

(A5)

Thus, H is a concave function with respect to p.Next, we show the strict concavity of H. Suppose

the vector p is represented by convex combination oftwo vectors pa and pb(* kpa, \;fk E RI):

Define H(A) as the function of A:

The second derivative of H(A) is

Since pa - pb * P for \;f(pa, pb, p), (AB) is alwaysnegative. From (A5), we can see that (AB) becomeszero only for pa - pb = p. That is, H(A) is strictlyconcave with respect to A. Consequently, the func­tion H(p) is strictly concave with respect to p. 0

DECOMPOSITION OF PATH CHOICE ENTROPY / 361

APPENDIX 4: PROOF OF PROPOSITION 3 (THEUNIQUENESS OF SA-arc)

THE FIRST TERM in the objective function for problem[SA-arc] is a linear function; it is a convex function.The remaining term is strictly convex with respectto the link choice probability p from Proposition 2.Hence, the objective function, which is a sum of aconvex function and a strictly convex function, is astrictly convex function.

Inasmuch as the linear equations in the con­straints have nonnegative solutions, because of themodularity of the coefficients, the solution of theconstraints with the linear equations and nonnega­tivities exists. Hence, the feasible region of problem[SA-arc] is a closed, nonempty convex set.

Thus, the problem [SA-arc] is a convex program­ming which has a strictly convex objective function,and the globally optimum solution can be uniquelydetermined if the solution exists.

REMARKS. The feasible region of problem [SA-arc] isnot necessarily bounded above due to the existenceof cyclic flows. Therefore, there may be a possibilitythat the solution does not exist, where the objectivefunction decreases as cyclic flows become infinite.The existence condition is described in Akamatsu(1996).

ACKNOWLEDGMENT

I WOULD LIKE to thank the associate editor and threeanonymous referees for their helpful suggestions,which greatly improved this paper's form and con­tent. I also thank Masazo Kuwahara of the Univer­sity of Tokyo for his helpful comments.

REFERENCES

T. AKAMATSU, "Cyclic Flows, Markov Process and Stochas­tic Traffic Assignment," Transp. Res. ·gOB, 369-386(1996).

T. AKAMATSU AND Y. MATSUMOTO, "A Stochastic NetworkEquilibrium Model with Elastic Demand and Its Solu­tion Method" (in Japanese), JSCE J. Infrastruct. Plan­ning & Management 401, 109-118 (1989).

T. AKAMATSU AND Y. TSUCHIYA, "Parallel Distributed Pro­cessing on Neural Network for Some TransportationEquilibrium Assignment Problems," in Proceedings ofthe 11th International Symposium on the Theory ofTraffic Flow and Transportation, M. Koshi Ced.),Elsevier, Yokohama, pp. 307-323, 1990.

T. AKAMATSU, Y. TSUCHIYAAND N. KAwAKAMI,"A Compar­ison of Algorithms for Solving the Stochastic Equilib­rium Assignment," (in Japanese) Infrastruct. PlanningRev. 8, 89-96 (1990a).

T. AKAMATSU, Y. TSUCHIYA AND N. KAWAKAMI, "An Effi-

362 / TAKASHI AKAMATSU

cient Algorithm for the Stochastic Equilibrium Assign­ment," (in Japanese) Traffic Eng. 26, 51-58 (1990b).

M. G. H. BELL, "Alternatives to Dial's LOGIT AssignmentAlgorithm," Transp. Res. 29B, 287-296 (1995).

M. G. H. BELL, W. H. K. LAM, G. PLOSS AND D. INAUDI,"Stochastic User Equilibrium Assignment and Itera­tive Balancing," in Proceedings of the 12th Interna­tional Symposium on the Theory of Traffic Flow andTransportation, C. F. Daganzo Ced.), Berkeley, pp. 427­439,1993.

M. CHEN AND A. S. ALFA, "Algorithms for Solving Fisk'sStochastic Traffic Assignment Model," Transp. Res.25B, 405-412 (1991).

C. F. DAGANZO, Multinomial Probit: Theory and Its Appli­cation to Demand Forecasting, Academic Press, NewYork, 1980.

C. F. DAGANZO, "Unconstrained Extremal Fonnulation ofSome Transportation Equilibrium Problems," Transp.Sci. 16, 332-360 (1982).

C. F. DAGANZO AND Y. SHEFFI, "On Stochastic Models ofTraffic Assignment," Transp. Sci. 11, 253-274 (1977).

R. S. DEMBO, "A Primal Truncated Newton Algorithmwith Application to Large-Scale Nonlinear NetworkOptimization," Math. Programming Stud. 31, 43-71(1987).

R. S. DEMBO AND U. TULOWITZKI, "Computing Equilibriaon Large Multicommodity Networks: Application ofTruncated Quadratic Programming Algorithms," Net­works 18, 273-284 (1988).

R. B. DIAL, "A Probabilistic Multipath Traffic AssignmentAlgorithm which Obviates Path Enumeration,"Transp. Res. 5, 83-111 (1971).

S. P. EVANS, "Derivation and Analysis of Some Models forCombining Trip Distribution and Assignment,"Transp. Res. 10, 37-57 (1976).

C. S. FISK, "Some Developments in Equilibrium TrafficAssignment," Transp. Res. 14B, 243-255 (1980).

M. FUKUSHlMA, "A Modified Frank-Wolfe Algorithm forSolving the Traffic Assignment Problem," Transp. Res.18B, 169-177 (1984).

S. IBARAKI, M. FUKUSHlMA AND T. IBARAKI, "Dual BasedNewton Methods for Nonlinear Minimum Cost Net­work Flow Problems," J. Opera Res. Soc. Japan 34,263-286 (1991).

B. N. JANSON, "Most Likely Origin-Destination Link Usesfrom Equilibrium Assignment," Transp. Res. 27B,333-350 (1993).

R. KATSURA, M. FUKUSHlMA AND T. IBARAKI, "InteriorMethods for Nonlinear Minimum Cost Network FlowProblems," J. Opera Res. Soc. Japan 32, 174-199(1989).

J. G. KLINCEWICZ, "A Newton Method for Convex Separa..

hIe Network Flow Problems," Networks 13, 427-442(1983).

L. J. LEBLANC, E. MORLOK AND W. PIERSKELLA, "An Effi­cient Approach to Solving the Road Network Equilib­rium Assignment Problem," Transp. Res. 9, 309-318(1975).

L. J. LEBLANC, R. V. HELGASON AND D. E. BOYCE, "Im­proved Efficiency of the Frank-Wolfe Algorithm forConvex Network Programs," Transp. Sci. 19, 445-462(1985).

T. MIYAGI, "On the Formulation ofa Stochastic User Equi­librium l\tlodel Consistent with the Random UtilityTheory," Proceedings of the 4th World Conference onTransportation Research, pp. 1619-1635, 1986.

M. PATRICKSSON, "Partial Linearization Methods in Non­linear Programming," J. Optim. Theory Appl. 78, 227­246 (1993).

W. POWELL AND Y. SHEFFI, "The Convergence of Equilib­rium Algorithm with Predetermined Step Sizes,"Transp. Sci. 16, 45-55 (1982).

R. T. ROCKAFELLER, Convex Analysis, Princeton Univer­sity Press, Prlnceton, NJ, 1970.

T. F. ROSSI, S. McNEIL AND C. HENDRICKSON, "EntropyModel for Consistent Impact Fee Assessment," J. Ur­ban Plan. Devel./ASCE 115, 51-63 (1989).

T. SASAKI, "Theory of Traffic Assignment through Absorb­ing Markov Process," (in Japanese), Proc. JSCE 121,28-32 (1965).

T. SASAKI, "Probabilistic Models for Trip Distribution,"Proceedings of the 4th International Symposium on theTheory of Traffic Flow, 205-210, 1969.

Y. SHEFFI AND W. B. POWELL, "An Algorithm for the Equi­librium Assignment Problem with Random LinkTimes," Networks 12, 191-207 (1982).

H. C. W. L. WILLIAMS, "Travel Demand Models, DualityRelations and User Benefit Analysis," J. Regional Sci.16, 147-166 (1976).

H. C. W. L. WILLIAMS, "On the Formation of Travel De­mand Models and Economic Evaluation Measure ofUser Benefit," Environment and Planning A9, 285­344 (1977).

A. G. WILSON, "The Use of Entropy Maximizing Models inthe Theory of Trip Distribution, Mode Split and RouteSplit," J. Transport Econ. Policy 3, 108-126 (1969).

S. A. ZENIOS AND M. Q. PINAR, "Parallel Block-Partitioningof Truncated Newton for Nonlinear Network Optimi­zation," SIAM J. Sci. Statist. Comput. 13, 1173-1193(1992).

(Received: February 1995; revisions received: July 1995, October1995, December 1996; accepted: January 1997)