Traffic user equilibrium and proportionality

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Trafﬁc user equilibrium and proportionality

Marlies Borchers, Paul Breeuwsma, Walter Kern, Jaap Slootbeek, Georg Still

⇑

, Wouter Tibben

Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands

a r t i c l e

i n f o

Article history:

Received 25 September 2014 Received in revised form 3 June 2015 Accepted 5 June 2015 Keywords: User-equilibrium Route ﬂows Proportionality Uniqueness

a b s t r a c t

We discuss the problem of proportionality and uniqueness for route flows in the classical traffic user equilibrium model. It is well-known that under appropriate assumptions the user equilibrium ðf ; xÞ is unique in the link flow x but typically not in the route flow f. We consider the concept of proportionality in detail and re-discuss the well-known rela-tion between the so-called bypass proporrela-tionality and entropy maximizarela-tion. We exhibit special proportionality conditions which uniquely determine the route flow f. The results are illustrated with some simple example networks.

Ó 2015 Published by Elsevier Ltd.

1. Introduction

In classical user equilibrium models under appropriate assumptions only the link ﬂow x of a user equilibrium (UE) ðf ; xÞ is uniquely determined by the equilibrium conditions but in general not the route ﬂow f. As uniqueness is a desirable property for various reasons (cf. Section2), researchers are interested in models that lead to an equilibrium ðf ; xÞ which is unique and stable also in f.

A natural way to achieve that is to compute the (unique) flow f which maximizes the so-called entropy. An alternative way is to directly look for extra conditions defining a unique solution f. In this paper we follow the latter approach. The extra conditions to be imposed should not only fix a unique route flow f but they should also have a practical interpretation. Here an important role is played by proportionality conditions.

The aim of this paper is to rediscuss the concept of proportionality and to obtain an ‘‘exact’’ set of (proportionality) con-ditions which determine the user equilibrium route ﬂow f in a unique and stable way.

We give a short overview of related work.Rossi et al. (1989)were the first to propose entropy maximization in order to obtain a unique route flow.Bar-Gera and Boyce (1999)then studied the implication of route flow entropy maximization and among others show that the corresponding flow satisfies the so-called by-pass proportionality conditions. InBar-Gera (2006) Bar-Gera studies the structure of all possible UE route flows and investigates the properties of entropy maximizing flows in particular proportionality. A primal method for computing this flow is proposed and tested on a set of real world networks. In 2010,Bar-Gera (2010)came up with a new algorithm for computing user equilibria. His traffic assignment by paired alternative segments (TAPAS) depends on the computation of the origin based user equilibrium link flows. The algorithm iterates alternatingly towards user equilibrium and entropy maximization. The paper reports on numerical experiments.

The subsequent article (Bar-Gera et al., 2012) compares TAPAS and two other trafﬁc assigment tools with respect to pro-portionality. InFlorian and Morosan (2014), Florian and Morosan have shown through numerical experiments that the Frank

http://dx.doi.org/10.1016/j.trb.2015.06.004

0191-2615/Ó 2015 Published by Elsevier Ltd. ⇑Corresponding author.

E-mail address:g.still@math.utwente.nl(G. Still).

Contents lists available atScienceDirect

Transportation Research Part B

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Wolfe algorithm as well as TAPAS generate UE route ﬂows which nearly satisfy proportionality. It also presents conditions assuring that a step of the Frank Wolfe algorithm increases the entropy.

There are many articles on stability and sensitivity of UE flows, e.g.,Tobin and Friesz (1988). InLu and Nie (2010), the authors investigate the uniqueness and stability of UE link flows as well as the uniqueness of corresponding route flows f maximizing appropriate (strictly convex) functions of f such as, e.g., the entropy function. AlsoBar-Gera et al. (2013)deals with sensitivity of UE link flows depending on different network design parameters.

Kumar and Peeta (2015)present and investigate an entropy weigthed averaging method as a generalization of entropy maximization. They prove that under standard assumtions also this extended method yields a unique route ﬂow. For further literature we refer the reader to the papers cited in the articles mentioned above.

The present paper is most closely related to the study inBar-Gera (2006)who introduces the notion of n-consistent route sets and analyses the structure of the route flows f that are feasible for the entropy maximization program. The approach in Bar-Gera (2006)is based on a primal method for computing the entropy maximizing route flow f. By contrast, in the present paper we seek to directly obtain an appropriate set of extra conditions which uniquely fix f. Our paper can be seen as a com-plement toBar-Gera (2006)in the sense that we approach the uniqueness question for f from another perspective, thereby obtaining some of the results inBar-Gera (2006)in a more direct way. We therefore will regularly compare and relate our arguments to the results inBar-Gera (2006).

Our paper is organized as follows. After a preliminary Section2, the concept of proportionality is treated in detail in Section3.1. Motivated byBar-Gera (2006), we introduce the notion of proportionality conditions of order n. In Section3.2 we review the relations between entropy maximization and so-called by-pass proportionality conditions fromBar-Gera and Boyce (1999). In Section3.3we determine proportionality conditions that uniquely determine the route ﬂows f. The results are illustrated with some simple example networks which we also analyze numerically. Section4studies the alge-braic structure of the proportionality conditions presented in Section3.3and relates them (at least for networks with only one OD-pair) to certain pairs of routes.

2. Preliminaries

We start with a short introduction into the classical Wardrop-Beckmann trafﬁc equilibrium model. Given a directed traf-ﬁc network N ¼ ðV; EÞ with node set V, directed link set E and origin–destination pairs (O–D pairs for short) ðow;twÞ 2 V V

with corresponding demands dwP0; w 2 W, we let Rwdenote the set of simple directed ow–twroutes, and R :¼Sw2WRw. A

trafﬁc ﬂow for the given demand d 2 RjWj

þ is a pair of vectors ðf ; xÞ of the form

K

f ¼ d; x ¼

D

f ; f P 0; ð2:1Þ

where the component xe;e 2 E of x 2 RjEjis the ﬂow on link e 2 E and the components fpof f 2 RjRjindicate the amount of

ﬂow on a route r 2 R. The elements of the matricesD2 RjEjjRj_;_K_{2 R}jWjjRj_{are deﬁned by}

D

er¼ 1 e is an edge of r 0 otherwise

K

wr¼ 1 r 2 Rw 0 otherwise :

We call x the link ﬂow and f the route ﬂow of ðf ; xÞ.

Let ce:Rþ! Rþbe non-negative, continuous, and non-decreasing link cost functions, deﬁning (separable) costs or travel

times ceðxeÞ; e 2 E, for a given trafﬁc ﬂow x 2 RjEjþ. We say that x 2 R

jEj _{induces the link costs c}

eðxeÞ; e 2 E and route costs

crðxÞ ¼Pe2rceðxeÞ; r 2 R.

Definition (Wardrop Equilibrium). Consider a traffic flow ðf ; xÞ as in(2.1)with corresponding induced costs ceðxÞ. Then ðf ; xÞ

is called a Wardrop equilibrium (or user equilibrium (UE)) if for all r; q 2 Rw;w 2 W the following condition is satisﬁed:

fr>0 )

crðxÞ ¼ cqðxÞ if fq>0

crðxÞ 6 cqðxÞ if fq¼ 0

(

Thus, in particular, in a Wardrop Equilibrium all used ow–twroutes have equal costs. A well-known equivalent

characteriza-tion is due to Beckmann:

Lemma 2.1. The following are equivalent for a traffic flow ðf ; xÞ: (1) The flow ðf ; xÞ is a Wardrop equilibrium.

(2) [Beckmann’s formulation] ðf ; xÞ is solution of the (convex) program:

min x;f zðxÞ :¼ X e2E Z xe 0 ceð

s

Þd

s

s:t:

K

f ¼ d

D

f x ¼ 0 f P 0 ð2:2Þ

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The next theorem states the existence of a user equilibrium ðf ; xÞ and uniqueness for the link ﬂow x (see e.g.,Yang and Huang, 2005for a proof).

Theorem 2.1 (Existence and Uniqueness). For continuous link costs ce:Rþ! Rþ, an equilibrium ﬂow ðf ; xÞ exists. Moreover, x is

unique if the link cost functions ceðxeÞ are strictly increasing.

Under the assumption ofTheorem 2.1on the functions ceðtÞ the link ﬂow part x of a UE ðf ; xÞ is uniquely determined, while

the route flow part f is not. Given the (unique) UE link flow x, any solution f of the systemKf ¼ d;Df ¼ x; f P 0 is a UE route flow f. In the following we will often use the abbreviation:

Af ¼ b instead of

K

f ¼ d

D

f ¼ x and F ¼ ff 2 R

R_{jAf ¼ b; f P 0g} _ð2:3Þ

We present a simple example:

Network A: Two O–D pairs ðA; DÞ and ðB; DÞ with demand 100 and 60, resp.. The given user equilibrium link ﬂows are x1¼ 100 on link 1, x2¼ 60 on link 2 and x3¼ 40; x4¼ 120 on links 3 and 4.

This example leads to the feasibility conditions:

f13 þ f14 ¼ 100 f23 þ f24 ¼ 60 f13 þ f23 ¼ 40 f14 þ f24 ¼ 120 a system Af ¼ b of rank A ¼ 3: ð2:4Þ

The set of feasible ﬂows (cf.(2.3)) is given by

F ¼ ff ¼ ðf13;f14;f23;f24Þ ¼ ð0; 100; 40; 20Þ þ tð1; 1; 1; 1Þj0 6 t 6 40g:

Non-uniqueness of f may cause serious problems:

Different solution methods find different equilibrium route flows f. These different solutions are difficult to compare. With one and the same solution method, a small perturbation of the network may lead to a completely different f.

From a mathematical viewpoint it is easy to force uniqueness of f. For example we could compute for given (unique) x part of a user equilibrium a feasible route ﬂow f which minimizes some appropriate objective G. For ﬁxed d; x solve:

ðPÞ min Gðf Þ st: f 2 F : ð2:5Þ

If the function G is strictly convex on F then the solution f of this program is unique (due to convexity of F ). We refer toLu and Nie (2010)for a proof of such an unicity result as well as stability statements.

Examples for strictly convex functions Gðf Þ are: Least square: Gðf Þ :¼1 2 P r2Rf 2 r

(negative) Entropy: Gðf Þ :¼P_r2Rfr logðfð rÞ 1Þ

or, alternatively: Gðf Þ ¼ Pr2Rlog fror Gðf Þ ¼

P

r2Rf1r.

Another possibility to force uniqueness is to impose additional (proportionality) conditions on f. This will be done in the next section. Before, we introduce 3 other networks which will be used as illustrative examples. In each case, we denote the corresponding constraint matrix by A ¼ K

D

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Network B: with two O–D pairs ðA; DÞ and ðB; DÞ, 6 routes f ¼ ðf13;f14;f15;f23;f24;f25Þ, and matrix

This matrix A has rank A ¼ 4.

Network C: with one O–D pair and 4 routes f ¼ ðf13;f14;f23;f24Þ, and matrix

Network D: with one O–D pair and 8 routes f ¼ ðf135;f136;f145;f146;f235;f236;f245;f246Þ, and matrix

Network E: (see alsoBar-Gera and Boyce, 1999; Bar-Gera, 2006) with 4 O–D pairs ðA; CÞ; ðB; DÞ; ðC; AÞ; ðD; BÞ and 8 routes f ¼ ðf67;f12;f78;f41;f85;f34;f56;f23Þ and matrix

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3. Proportionality and uniqueness 3.1. Proportionality conditions

Proportionality of equilibrium flow f can be seen as a natural condition in traffic models (see, e.g.,Bar-Gera et al., 2012; Aungsuyanon et al., 2013; Bar-Gera, 2006). It is often claimed that a realistic UE route flow f should satisfy proportionality. In words, proportionality is often formulated as follows.

Proportionality (Bar-Gera et al., 2012): A route ﬂow f 2 F is said to satisfy proportionality relative to a given UE link ﬂow x if

the same proportions occur for all travelers facing a choice between a pair of alternative segments, regardless of their origins and destinations.

Let us analyze the simplest form of proportionality from a mathematical point of view. We can formally express it in dif-ferent ways. For the simple Example A above we could postulate the following three conditions:

P1: The amount of travelers that start at A and pass through link 3 relative to the total amount of travelers that start at A is equal to the amount of travelers that start at B and pass through link 3 relative to the total amount of travelers that start at B.

f13 f13þ f14 ¼ f23 f23þ f24 ¼ f13þ f23 ðf13þ f23Þ þ ðf14þ f24Þ

(Equivalently, we could require the same for link 4: f14

f13þf14¼

f24

f23þf24.)

P2: Alternatively, we could formulate proportionality as:

f13

f13þ f23

¼ f14 f14þ f24

(Equivalently, this holds for the travelers that start at B: f23

f13þf23¼

f24

f14þf24.)

P3: The by-pass proportionality conditions inBar-Gera and Boyce (1999)take the form:

f13

f14

¼f23 f24

It is not difﬁcult to see that for Network A all three conditions yield the same solution of the system(2.4):

f13¼ 25; f14¼ 75; f23¼ 15; f24¼ 45:

The three formally different formulations of proportionality above can be generalized:

Two simple directed route segments s1;s2(directed paths) are called parallel if they have identical starting and end nodes,

resp. We call a route r 2 R with r s1switcheable (with respect to s1;s2) if r s1þ s22 R. We let fwðs1Þ denote the sum over

all route ﬂows f_rwhere r 2 Rwcontains s1and is switcheable, and

f ðs1Þ :¼

X

w

fwðs1Þ;

the total ‘‘switcheable’’ ﬂow through s1. Similar deﬁnitions apply to s2instead of s1. Relative to any parallel pair ðs1;s2Þ, we

consider the proportionality conditions

P1: (as formulated implicitly, e.g., on page 443 inBar-Gera and Boyce (1999)): For any w 2 W we require1

fwðs1Þ fwðs1Þ þ fwðs2Þ¼ f ðs1Þ f ðs1Þ þ f ðs2Þ or; better; fwðs1Þðf ðs1Þ þ f ðs2ÞÞ ¼ f ðs1Þðfwðs1Þ þ fwðs2ÞÞ: ð3:1Þ

P2: For any w 2 W we require

fwðs1Þ

f ðs1Þ ¼

fwðs2Þ

f ðs2Þ

or; better; fwðs1Þf ðs2Þ ¼ fwðs2Þf ðs1Þ: ð3:2Þ

P3: [By-pass conditions as inBar-Gera and Boyce (1999)]:

For any two switcheable routes r; q 2 R (possibly joining different O–D pairs) such that s1#r; q and

~ r :¼ r s1þ s2; ~q :¼ q s1þ s22 R we require that fr f~r ¼fq f~q or; better; frf~q¼ fqf~r: ð3:3Þ

Note that(3.1) and (3.2)are void if there is only one demand. The above three conditions are related as follows 1

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Theorem 3.1. For f 2 F :(3.3))(3.2) () (3.1)

Proof. Note that(3.1)can be stated equivalently as

fwðs1Þ

f ðs1Þ

¼fwðs1Þ þ fwðs2Þ

f ðs1Þ þ f ðs2Þ

;

which is, in turn, equivalent to(3.2).

To prove(3.3))(3.2), ﬁrst note that, due to(3.3), fr=f~r¼:

a

is actually independent of r. Hence, by summing over all

switcheable r 2 Rwthat contain s1resp. s2, we ﬁnd thatffwwðsðs21ÞÞ¼

a

is independent of w, and therefore

f ðs1Þ f ðs2Þ¼ P wfwðs1Þ P wfwðs2Þ¼

a

holds

as well. Thus(3.3)implies(3.2).

The by-pass condition(3.3)is closely related to the so-called 2-consistency inBar-Gera (2006). There, a set R0of routes is

called 2-consistent if for any two pairs of alternative routes ðr1;r01Þ connecting origin o1to destination t1and ðr2;r02Þ

connect-ing o2to t2such that

r1þ r2¼ r01þ r 0

2 ðviewed as vectors in R jEj_Þ;

holds, r1;r22 R0implies r01;r022 R0. In this sense our set R, comprised of all simple directed O–D routes, would not qualify

for 2-consistency. Indeed, in the example below, taken fromBar-Gera (2006), r1¼ ð1; 2; 4; 3; 5Þ and r2¼ ð1; 3; 4; 6Þ are simple

and, viewed as vectors, we have r1þ r2¼ r01þ r02with, say, r01¼ ð1; 3; 4; 3; 5Þ and r02¼ ð1; 2; 4; 6Þ.

Hence, 2-consistency would imply r0

1¼ ð1; 3; 4; 3; 5Þ 2 R, a contradiction, since r01is cyclic. Note, however, that cyclic

routes (i.e., routes containing cycles) never contribute to a UE ﬂow. This is why we focus on simple routes throughout. We also propose to modify the deﬁnition of consistency accordingly (seeFig. 1).

One of the important observations made inBar-Gera (2006)is that simple conditions of type(3.3)are not sufﬁcient to guarantee uniqueness in any case. In fact, as we will see, they can cope with Network A–D, but for Network E ‘‘higher order’’ conditions are necessary. Note that(3.3)only involves products of at most two variables (f_rf~qresp. f~rfq). Therefore these

con-ditions – corresponding to 2-consistency – might be called proportionality concon-ditions of order 2. To obtain a unique route ﬂow f in general (large) networks, higher order conditions are required, such as, e.g.,

f23f41f67f85¼ f12f34f56f78 ð3:4Þ

for Network E (cf., Section3.3). InBar-Gera (2006), such conditions are cast by the concept of n-consistency: A route set R0is

called n-consistent if, given any pairs ðr1;r01Þ; . . . ; ðrn;rn0Þ with ri;r0iconnecting oito tisuch that

r1þ þ rn¼ r01þ þ r0n

holds, then r1; . . . ;rn2 R0implies r01; . . . ;r0n2 R0. Proportionality conditions of any order n may be needed to ﬁx f in general

(large) networks. Network E with route set R0consisting of all 8 routes f23;f41;f67;f85;f12;f34;f56;f78 is 4-consistent (cf.,

Bar-Gera, 2006, Section 3.1) and the conditions(3.4)should be called proportionality conditions of order 4. 3.2. Entropy maximization and by-pass conditions

4

6

5

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Bar-Gera and Boyce (1999). Given the unique x part of a User Equilibrium ðf ; xÞ we consider for ﬁxed ðd; xÞ the program (we again make use of the abbreviation(2.3)):

ðEPÞ : min Eðf Þ :¼

X

r2R

fr logðfð rÞ 1Þ st: f 2 F

An important issue in solving this problem is to work with the ‘‘correct’’ set of routes r 2 R: We emphasize that, obviously, the above problem remains unchanged if we eliminate all variables fr(and remove the corresponding columns of A ¼ KD

) that are forced to be zero by the constraints in ðEPÞ. Let R0 R be the resulting route set. Thus, in what follows, we will assume:

r 2 R0 ) fr>0 for some f 2 F : ð3:5Þ

In particular, we remove all routes r passing through an edge e with xe¼ 0. So we assume as well that x > 0 in the following

(by removing edges e from the network). We refer to the resulting problem ðEPÞ as the reduced problem. We emphasize that

without condition(3.5)the solution of ðEPÞ cannot be computed by solving the KKT-condition (at a solution f with fr¼ 0 for

some r, the gradientrEðf Þ is not deﬁned).

Remark 3.1. When we replace R by R0 R, the notion of switcheability has to be adapted accordingly: Route r 2 R0with

s1 r is switchable if r s1þ s22 R0. In principle, this deﬁnition may be applied to any subset of R0. Indeed, there might be

(problem speciﬁc) reasons to exclude certain routes r 2 R (in addition to those excluded by(3.5)). The standard choice however, should be the set of cost minimal routes r 2 R such that(3.5)holds. Removing additional routes from R0will, of

course, in general allow less proportionality conditions and decrease the maximum entropy.

The following result has been shown inBar-Gera and Boyce (1999). We include a proof for convenience of the reader and further reference.

Theorem 3.2 Bar-Gera and Boyce, 1999, Section 3. The reduced problem has a (unique) optimal solution f > 0. Moreover, f satisﬁes the by-pass proportionality conditions(3.3).2

Proof. First note that in the reduced problem there is, for each remaining route r, a feasible ﬂow f with fr>0. Taking a

con-vex combination of all these ﬂows, we get a strictly positive feasible solution f₀>0 of the reduced problem. Now let f denote the (unique) optimal solution of the reduced problem. We claim that f > 0. Indeed, assume to the contrary, that fr¼ 0 for

some r. Then it is straightforward to show (by elementary calculus) that f þ

ðf0 f Þ would yield a lower entropy value

for sufﬁciently small

>0, a contradiction. Hence, indeed, f > 0 must hold and this solution f must satisfy the KKT condition. AsrEðf Þ ¼ logðf Þ, the KKT conditions for f > 0 read as

log f ¼

K

T

As discussed above, the order 2 by-pass proportionality conditions satisfied by the entropy maximizing flow f are not suf-ficient to fix f (e.g., in Network E). On the other hand, the KKT condition(3.7)for ðEPÞ contains too many proportionality

2

InBar-Gera and Boyce (1999)it is shown that extending the solution f of the reduced problem by adding zero components yields a feasible solution f 2 RjRj_{of the}

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conditions (also of higher order). The conditions are not all independent. So the question is which of these proportionality conditions are precisely needed to ﬁx f. In what follows, we seek to exhibit a minimal set of (independent) proportionality conditions that uniquely determine a suitable equilibrium route ﬂow f.

We formulate our problem more precisely. For fixed ðd; xÞ with x the (unique) equilibrium link flow, we consider the route flows f satisfying the network feasibility conditions f P 0 and (see(2.3))

K

f ¼ d

D

f ¼ x or Af ¼ b where again A ¼

K

D

;b ¼ d x : ð3:8Þ

Recall that any solution f of this system yields an user equilibrium ﬂow ðf ; xÞ.

Unicity problem: Which minimal set of (reasonable) proportionality conditions should be added to the feasibility con-dition(3.8)to determine f uniquely?

To answer this question we ﬁrst analyze how many conditions we need to ﬁx f. The solution set of(3.8)is of the form

S ¼ f þ ker A ¼ ff ¼ f þ hjAh ¼ 0g

for some particular solution f of(3.8). Here, ker A denotes the kernel (null space) of A; ker A ¼ fhjAh ¼ 0g. So we need exactly k :¼ dimðker AÞ many extra (independent) conditions to ﬁx f uniquely. Let h1; . . . ;hkbe a basis of ker A. Recall that the KKT condition(3.6)can be written in the form

log f ¼ AT

k

with multiplier vector k ¼ ð

r

;

l

Þ. So each basis vector hj;j ¼ 1; . . . ; k, leads to a condition for f:

½hjTlog f ¼ ½hjTATk¼ ½AhjTk¼ 0:

So we obtain the conditions

X r hrjlog fr¼ 0 or Y r fhrj r ¼ 1 ; j ¼ 1; . . . ; k: ð3:9Þ

These k (proportionality) conditions uniquely determine f:

Theorem 3.3. There is a unique solution f > 0 satisfying the reduced system(3.8)and the conditions(3.9). This solution coincides with the (unique) solution of ðEPÞ.

Proof. Let f > 0 satisfy(3.8) and (3.9). In view of the well-known property fimATg?¼ ker A, the relation(3.9)means that y :¼ log f satisﬁes

hTy ¼ 0 for all h 2 ker A: ð3:10Þ

Hence y 2 ker A?

¼ im AT, i.e.,

log f ¼ ATk for some multiplier k:

Since ðEPÞ is a convex program with linear constraints, this KKT relation is a sufﬁcient condition for f to be the (unique)

solu-tion of ðEPÞ.

For each of the example networks above we now give the basis of the kernel of A and the corresponding (proportionality) conditions from(3.9).

Network A: We obtain a basis vector of ker A

1 1 1 1 2 6 6 6 4 3 7 7 7

5and the conditions f13f 1 14f

1

23f24¼ 1 or f13f24¼ f14f23:

Note that this coincides with the proportionality condition obtained before. Network B: We obtain basis vectors of ker A

1 0 1 1 0 1 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ; 1 1 0 1 1 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 and f13f25¼ f15f23 f13f24¼ f14f23

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Network C: We obtain the basis vector of ker A

ð1; 1; 1; 1Þ and f13f24¼ f14f23

Network D: We obtain basis vectors of ker A

1 0 0 1 1 0 0 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; 1 0 1 0 1 0 1 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; 1 1 0 0 1 1 0 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; 1 1 1 1 0 0 0 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 with f135f246¼ f146f235 f135f245¼ f145f235 f135f236¼ f136f235 f135f146¼ f136f145

Network E: We obtain a basis vector of ker A

ð1; 1; 1; 1; 1; 1; 1; 1Þ with condition f23f41f67f85¼ f12f34f56f78

Remark 3.2. Instead of ðEPÞ we could also consider the program ðPÞ in(2.5)with some other strictly convex function. If we

choose e.g., Gðf Þ ¼ Prlog fr, or, Gðf Þ ¼

P

r1=fr then the KKT conditions for the (unique) solution f > 0 of ðPÞ in(2.5)is

1=fr¼ ½ATkr, or, 1=f 2 r ¼ ½A

T

kr. Then again, with a basis h 1

; . . . ;hkof ker A we would obtain k ¼ dimðker AÞ extra conditions on f, X r hrj fr ¼ 0 or X r hrj f2r ¼ 0; j ¼ 1; . . . ; k;

which together with the feasibility conditions Af ¼ b uniquely ﬁx f > 0.

We would like to add some remarks on the practical aspects of our approach. In principle it can be applied to compute the entropy maximizing ﬂow by directly computing the solution f of the system(3.8) and (3.9), e.g., by Newton’s method as fol-lows: For given d; x consider the (reduced) system Af ¼ b in(3.8)and proceed:

1. DetermineD0such that the rows of A0

¼ K

D0

form a basis of the row space of A and let b0¼ ðd; x0_{Þ be the corresponding}

right hand side of(3.8). Then(3.8), (3.9)yields a system

Hðf Þ :¼ A0f b0¼ 0 Y r fhrj r 1 ¼ 0; j ¼ 1; . . . ; k: ð3:11Þ

of rankA þ k equations in the same number of unknown components frof f.

2. Apply starting with some f0>0 the Newton iteration

fkþ1¼ fk

r

HðfkÞHðfkÞ ð3:12Þ

to compute the (unique) solution f > 0 of Hðf Þ ¼ 0. ByTheorem 3.3this yields the (unique) solution of the entropy max-imizing program ðEPÞ.

We report on some numerical experiments with the Example Networks B, D, E. For Network B the system(3.11)reads:

Hðf Þ ¼ A0_{f b}0 ¼ 0 f13f25 f15f23¼ 0 f13f24 f14f23¼ 0 with A0 ¼ 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 2 6 6 6 6 4 3 7 7 7 7 5 ð3:13Þ

For b ¼ ðd; xÞ with d ¼ ð100; 60Þ; x ¼ ð100; 60; 60; 40; 60Þ and thus b0

¼ ð100; 60; 60; 40Þ the (unique) solution f > 0 of(3.13)is f ¼ ð37:5; 25; 37; 5; 22:5; 15; 22:5Þ.

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For Network D the system(3.11)reads: Hðf Þ ¼ A0_{f b}0 ¼ 0 f135f246 f146f235¼ 0 f135f245 f145f235¼ 0 f135f236 f136f235¼ 0 f135f146 f136f145¼ 0 with A0 ¼ 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 2 6 6 6 4 3 7 7 7 5 ð3:14Þ

With a choice b ¼ ðd; xÞ; d ¼ ð100Þ; x ¼ ð60; 40; 40; 60; 60; 40Þ and thus b0

¼ ð100; 60; 40; 60Þ the (unique) solution f > 0 of (3.13)is f ¼ ð14:4; 9:6; 21:6; 14:4; 9:6; 6:4; 14:4; 9:6Þ.

Network E yields the system(3.11):

Hðf Þ ¼A 0_{f b}0 ¼ 0 f23f41f67f85 f12f34f56f78¼ 0 with A0¼ 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5

and with b ¼ ðd; xÞ; d ¼ ð80; 80; 80; 80Þ; x ¼ ð100; 100; 100; 100; 100; 60; 60; 60Þ and thus b0¼ ð80; 80; 80; 80; 100; 100; 100Þ we obtain the (unique) solution f ¼ ð30; 50; 30; 50; 30; 50; 30; 50Þ.

For each of the Networks B, D, E we have computed the solutions f of(3.11)by running Newton’s iteration with 100 ran-domly chosen starting vectors f0(each component of f0has been generated from a uniform distribution in ½1; 30).Table 1 presents the numerical results. We used the stopping criterium jfkþ1 fkj < 106_{and a maximum number of 30 Newton}

iterations.

To force positive solutions f we have modiﬁed the iteration(3.12)by the additional condition:

if fkþ1p <0 then redefine f kþ1

p ¼ 1:

For each of the networks the column a

v

r; maxnr; minnr in the table gives the average, maximum, minimum number, respectively, of Newton iterations necessary to meet the stopping criterium. The last column gives the number nrfail of cases (from 100) where the Newton method failed to reach the stopping criterium after a maximum number of 30 iterations.

Table 1These illustrative examples clearly do not answer the question whether our theoretical approach can also be suc-cessfully applied in practice for large networks. Realistic numerical tests are needed to achieve this goal. Possibly a procedure as inBar-Gera (2010)should be applied, where each iteration step of the algorithm for computing the UE link flow x is com-bined with an iteration for the solution of the entropy maximizing path flow f. In other words the computation of the UE link flow x and the solution f of(3.11)could be done in parallel.

4. Structure of the basis of the kernel of A

In the four example networks above we have found bases of ker A of a very special form: All basis vectors were f0; 1; 1g-vectors. It is rather unclear whether this holds in general. As to the simple examples presented above, a possible explanation would be total unimodularity of the matrix A (see e.g.,Papadimitriou and Steiglitz, 1998, Section 13.2for details). In general, however, the matrix A is not totally unimodular – even in case of a single O–D pair. Yet, the following result states that such bases exist whenever we have only one O–D pair. For related results pointing into this direction seeBar-Gera (2006, Lemma 2).

Proposition 4.1. In case of a single O–D pair ðo; tÞ, the kernel of A has a basis consisting of f0; þ1; 1g-vectors.

Proof. Let x > 0 be the unique optimal link ﬂow. (Recall that we may assume x > 0 by removing edges with xe¼ 0 from the

network.) First note that our network N ¼ ðV; EÞ must be cycle-free in the sense that there are no directed cycles. Indeed, assume to the contrary that C # E is a directed cycle. Then decreasing x on C by an amount of

Table 1 Numerical results.

Network avr maxnr minnr nrfail

Network B 3 3 3 0

Network D 5.9 30 4 1

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:¼ minfxeje 2 Cg > 0

would result in ~x ¼ x

C 2 RjEj

þ, which is still an o-t ﬂow, i.e., there are corresponding route ﬂows ~frP0 with ~x ¼D~f , so that

ð~f; ~xÞ is a feasible ﬂow with ~x 6 x (componentwise), and, consequently, x cannot be an optimal solution of Beckmann’s prob-lem, a contradiction. Thus, indeed, N ¼ ðV; EÞ must be cycle-free.

As a consequence, given two directed o-t routes r; q 2 R, which both pass through a certain node

v

2 V, we can construct the route rvq that starts in o, follows r up to node

v

and then switches to q. Cycle freeness of N ensures that rvq is again a simple route (not visiting any node twice). The same holds true for the route qvr that ﬁrst follows q and switches to r in node

v

.

After these preliminaries let us turn to the proof of our claim. Actually, we will show that ker A is generated by elementary vectors k 2 ker A # RjRj_{of the form}

kr¼ kq¼ 1; krvq¼ kqvr¼ 1; kj¼ 0 else:

It sufﬁces to show that every integer vector in ker A is generated in this way (as basic solutions of Ak ¼ 0 are rational). Thus let k 2 ZjRj_{\ ker A. We proceed by induction on}P

rjkrj. The case wherePrjkrj ¼ 0 is trivial, so assumePrjkrj > 0 and choose a

qvq02 ker A

with the property that ~kqvq0¼ k_q_v_q0 1 < 0 and q

v

q0shares a longer initial subroute with r. So, after ﬁnitely many

modiﬁca-tions – at latest when q

v

q0_{¼ r – we eventually end up in some}~~_k_{that differs from k by elementary vectors in fþ1; 1; 0g}jRj

with four nonzero components each, and hasPrjk~~rj <Prjkrj, so that the result follows by induction.

The argument in the proof ofProposition 4.1can be extended to the case of networks with one destination and multiple origins as well as (by arguing backwards) the case of one origin with multiple destinations. (The crucial observation is that also in this case the network can be assumed to be cycle free. All other arguments apply analoguously.) This slight general-ization might be interesting for example in connection with bush-based algorithms (see e.g.,Nie, 2010).

We do not expect this result to remain valid in the general case of multiple origins and multiple destinations, but could not ﬁnd a counterexample network either.

5. Conclusions

The aim of the paper is twofold. Firstly we re-discuss the concept of proportionality for a user equilibrium ðf ; xÞ and, sec-ondly, present a way to obtain a unique route flow f by only imposing a special set of extra (proportionality) conditions on f. These conditions depend on the basis of ker A of the problem matrix A. Each basis vector h of ker A defines a specific propor-tionality condition. The order of this condition is given by the number and size of nonzero entries of h. The set of conditions obtained provides additional insight and can be used to compute a unique route flow numerically by Newton’s method in a direct way.

Several interesting questions remain open: Can this approach be applied in practice to large networks? It is conceivable that the running time of Newton’s method is largely inﬂuenced by the choice of the basis of ker A.

How can ‘‘good’’ bases, i.e., ones that lead to low order conditions in(3.9)be computed in practice? Can we ﬁnd a basis consisting of only f 1; 0g-vectors in case such a basis exists?

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