Multiclass, Two-criteria Traffic Network Equilibrium Models and Vector Variational Inequalities

Systems Engineering - Theory & Practice Volume 28, Issue 1, January 2008 Online English edition of the Chinese language journal Cite this article as: SETP, 2008, 28(1): 141–145

Multiclass, Two-criteria Traffic Network Equilibrium Models and Vector Variational Inequalities LI Sheng-jie1 ,∗ , CHEN Guang-ya2 1. College of Mathematics and Science, Chongqing University, Chongqing 400044, China 2. Insitute of Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Abstract: In this article, multiclass, the two-criteria traffic network equilibrium model is dissussed. The relations between the vector and the weak vector variational inequalities and an equilibrium principle of the model are established, respectively. Key Words: traffic network equilibrium model; equilibrium principle; vector variational inequality

1 Introduction Warshop[1] first studied a single-class and singlecriteria network equilibrium problem and introduced the Warshop equilibrium principle in the sense of user optimization. Then, Quandt[2] and Schineider[3] introduced a two-criteria network equilibrium problem. Dial[4] further developed the idea of two-criteria and proposed an uncongested network equilibrium model. Dafermos[5] proposed congestion effects, investigated a multiclass, twocriteria traffic network equilibrium problem, and derived an infinite-dimensional variational inequality formulation for solving the equilibrium problem using a weighted scalarization method. In 1980, Giannessi[6] first introduced a vector variational inequality problem in a finite dimensional Euclidean space. Later on, Chen and Yang[7] studied the vector variational inequality in infinity dimensional spaces and obtained the existence of solutions and its equivalent results. In 1993, Chen and Yen[8] first proposed the vector equilibrium principle for the multicriteria network equilibrium model and studied the equivalent relations between the vector equilibrium principle and the solutions of vector variational inequality. Then, Yang and Goh[9] improved the result of reference [8] and further investigated the equivalent properties between the vector equilibrium principle and the solutions of vector variational inequality. Chen, Goh, and Yang[10] introduced a ξea -equilibrium principle by virtue of Gersterwity’s nonlinear scalarization function and discussed the equivalent relations among the weak vector equilibrium principle, the ξea equilibrium principle, and a variational inequality. In 2000, Nagurney[11] studied the multiclass, multicriteria traffic network equilibrium problems ((MMTNE), in short) in different aspects. He introduced the disutility equlibrium conditions and obtained the equivalent relation between the disutility equilibrium conditions and the solutions of variational inequality by virtue of a method of lin-

ear weigted scalarization. Motivated by the work reported above, in Section 2, we introduce the (weak) vector equilibrium principle for the multiclass, two-criteria traffic network equilibrium model ((MTTNE), in short), and discuss the relations between the equilibrium principle and the solutions of a vector variational inequality problem. In Section 3, we discuss the relations between the disutility condition of the multiclass, two-criteria traffic network equilibrium model, and the solutions of the vector and the weak vector variational inequality problem.

2

(MTTNE) model with fixed demand case

Consider a traffic network equilibrium model G = [N, L] (referred in Nagurney[11] ), where, N denotes the set of nodes in the network and L denotes the set of directed arcs. Let a denote an arc connecting a pair of nodes and p denote a path, assumed to be acyclic, consisting of a sequence of arcs connecting an origin/destination(O/D) pair of nodes. Let W be a set of O/D pairs and Pw denote the set of available paths joining the O/D pair w. Let  |Pw |. n = |L|, m = |W |, np = w∈W

Assume that there are k classes of travelers in the network with a typical class denoted by i. Let xip denote the nonnegative flow of class i on path p and xi = (xip1 , · · · , xipnp )T ∈ Rnp denote the flow vector of class i, where, p1 , · · · , pnp denote np distinct paths in the network G, respectively. Then, the flow vector xi induces an arc flow fai of class i on arc a ∈ L given by:   xip δap , (1) fai = w∈W p∈Pw

where, δap = 1, if arc a belongs to path p, and δap = 0, otherwise. For i = 1, · · · , k, let f i = (fai 1 , · · · , fai n )T , where,

Received date: December 26, 2005 ∗ Corresponding author: E-mail: [email protected] Foundation item: Supported by the National Natural Science Foundations of China (Nos.60574073 and 10471142) and Natural Science Foundtion Project

of CQ CSTC (2007BB6117) c 2008, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright 

LI Sheng-jie, et al./Systems Engineering – Theory & Practice, 2008, 28(1): 141–145

a1 , · · · , an denote n distinct arcs in L, respectively. Then, f i is an n dimensional arc flow vector and it denotes the arc flow for class i. Let f = ((f 1 )T , · · · , (f k )T )T , which denotes the kn dimensional arc flow column vector. Let x = ((x1 )T , · · · , (xk )T )T , which denotes the knp dimensional path flow vector in the network. We assume in the rest of this article that the demand of the traffic flow is fixed for each class i and O/D pair w, i.e.,  xip = diw , (2) p∈Pw

where, diw is a given number. In the l-dimensional Euclidean space Rl , we denote the orderings as follows: x≤y ⇔y−x∈

l l where, R+ is the positive quadrant of Rl and intR+ is the l interior of R+ . The orderings ≥ and > are defined similarly. The traffic flow f satisfying the demand in the network is called a feasible flow. Let K be the set of the feasible flow, i.e.,

K = {f : f ≥ 0 satisfying Eq.(1) and Eq.(2)}. Obviously, K is a closed convex set. Now, we define the index function in the network. In this article, we only consider the time and cost as the index function. For every arc, let tia denote the travel time of the flow of class i on arc a and ta denote the sum of the flow of k classes on arc a, respectively; then, k 

tia (f ), ∀a ∈ L.

(3)

i=1

cia (f )

Let denote the cost of the flow of class i on arc a and ca denote the sum of the flow of k classes on arc a, respectively; then, ca (f ) =

k 

cia (f ), ∀a ∈ L.

(4)

i=1

We introduce matrix-valued functions U i (f ) (i = 1, · · · , k) from K to R2×n and a matrix-valued function U from K to R2k×2n as follows:  i  ta1 (f ) · · · tian (f ) U i (f ) = , i = 1, · · · , k, (5) cia1 (f ) · · · cian (f ) and

2 Vpi (f ) − Vpi (f ) ∈ R+ \{0} =⇒ xip = 0.

⎛

⎞ 0 U 1 (f ) · · · ⎜ ⎟ .. .. .. U (f ) = ⎝ ⎠ . . . 0 · · · U k (f )

(6)

In the sequel, we introduce the concept of vector disutility. We define the vector disutility vpi (f ) of class i travelling on path p as   ti (f )  i a (7) δap , ∀i, p. Vp (f ) = cia (f ) a∈L

Naturally, the disutility here can be viewed as a generalized cost function. By virtue of the disutility, we can define the vector equilibrium principle, which is the generalization of the Wardrop equilibrium principle.

(8)

Definition 2.2 (Weak vector equilibrium principle) An arc flow vector f ∈ K is said to be in weak vector equilibrium if for each class i, for all O/D pairs w, and for any path p, p ∈ Pw , 2 Vpi (f ) − Vpi (f ) ∈ intR+ =⇒ xip = 0.

l R+ ;

l , x < y ⇔ y − x ∈ intR+

ta (f ) =

Definition 2.1 (Vector equilibrium principle) An arc flow vector f ∈ K is said to be in vector equilibrium if for each class i, for all O/D pairs w, and for any path p, p ∈ Pw ,

(9)

We now establish an appropriate condition for a vector equilibrium flow. Proposition 2.1 The arc flow vector f ∗ ∈ K is in vector equilibrium if f ∗ solves the following vector variational inequality problem: (VVI) find f ∗ ∈ K, s.t. 2k \{0}, ∀f ∈ K. U (f ), f − f ∗ ∈ −R+

(10)

∗

Proof. Let f ∗ ∈ K be the solution of problem Eq.(10) and for some i∗ , we have ∗

∗

2 Vqi (f ∗ ) − Vri (f ∗ ) ∈ R+ \{0}, ∗

where, q, r ∈ Pw . We shall prove (x∗q )i = 0. Construct a path flow vector x = ((x1 )T , · · · , (xk )T )T as follows: if j = i∗ , xj = (x∗ )j ; otherwise, the components of xj are: ⎧ ∗ i∗ if p = q, r ⎨ (xp ) , 0, if p = q xjp = ∗ ⎩ ∗ i∗ (xq ) + (x∗r )i , if p = r By simply verifying, we can easily prove that the arc flow vector f corresponding to x belongs to K. Then, we have U (f ∗ ), f − f ∗ =  1 ∗  ⎛  ta (f ) × (fa1 − (fa∗ )1 ) 1 ∗ a∈L ⎜ c a (f ) ⎜ ⎜ .. ⎜ ⎜  k ∗ . ⎝  ta (f ) × (fak − (fa∗ )k ) a∈L cka (f ∗ ) ⎛ ⎞ O2 ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ∗ i∗ ∗O2 i∗ ∗ ⎟ ⎜ = ⎜ (xq )r (f ) − Vq (f ) ⎟ ⎟ ⎜ ⎟ O2 ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

O2

∈

2k −R+ \{0},

(11)

LI Sheng-jie, et al./Systems Engineering – Theory & Practice, 2008, 28(1): 141–145

where,

 O2 =

0 0



∈ R2 .

Since ∗

∗

2k \{0}, Vqi (f ∗ ) − Vri (f ∗ ) ∈ R+

(12)

Proposition 3.1 If an arc flow vector f ∗ ∈ K is a solution of the (VVI) problem (10), then, there are weights w1i and w2i of class i, i = 1, · · · , k, which are nonnegative but not both equal to zero, such that the arc flow vector f ∗ ∈ K is a traffic network equilibrium flow, which satisfies equilibrium condition (14).

∗

Eq.(11) and Eq.(12) together imply that (x∗q )i = 0. Similarly, we can obtain the sufficient condition for a weak vector equilibrium flow. Proposition 2.2 The arc flow vector f ∗ ∈ K is in weak vector equilibrium if f ∗ solves the following vector variational inequality problem: (WVVI) find f ∗ ∈ K, s.t. 2k , ∀f ∈ K. U (f ∗ ), f − f ∗ ∈ −intR+

Proof. Set F = {U (f ∗ ), f ∈ R2k | f ∈ K},

F i = {U i (f ∗ ), f i ∈ R2 | f ∈ K}, i = 1, · · · , k, (16) 2k , G = U (f ∗ ), f ∗ − R+

(13)

It is noticed that we need strong restrictive conditions to obtain the necessary conditions for vector and weak vector equilibrium flow. Goh and Yang[12] proved a necessary condition that a weak vector equilibrium flow is a solution of a weak vector variational inequality problem under very strong restrictive conditions. Chen, Goh, and Yang[10] obtained a necessary and sufficient condition that a weak vector equilibrium flow is a solution of a scalar variational inequality problem by virtue of Gerstewitz’s nonlinear scalarization function. Since this problem is associated with several new concepts and theories, it has been omitted in this article.

3 Disutility conditions and vector variational inequality [11]

Nagurney presented a disutility condition as an equilibrium condition for the (MMTNE) model. In this section, we obtain the relation between the disutility equilibrium condition and a vector variational inequality. For each class of traveler i, we set two weights w1i and w2i , which are assumed to be nonnegative, but not both equal to zero. For any arc a, we construct the following function:

a∈L

Nagurney[11] introduced the following equilibrium condition for the (MMTNE) model: the arc flow pattern f ∗ is said to be in equilibrium, if for any class i, any O/D pair w, and any path p, the following condition holds:  = λiw , if (x∗p )i > 0, (14) vpi (f ∗ ) ≥ λiw , if (x∗p )i = 0, λiw

is a prearranged disutility criterion, whose value where, may not be known in advance. It is remarkable, from the following analysis, that there is no relation between the equilibrium flow pattern f ∗ and the disutility criterion. This is just the disposal of analyse technically that we obtain different traffic network equilibrium flow patterns by choosing different weights w1i and w2i . In the sequel, we consider the relations between the disutility equilibrium condition and the (VVI) problem (10) and (WVVI) problem (13), respectively.

(17)

and 2 , i = 1, · · · , k. Gi = U i (f ∗ ), (f ∗ )i − R+

(18)

It is obvious that F , F i , G, and Gi , i = 1, · · · , k are nonempty convex sets and G ∩ F = {U (f ∗ ), f ∗ }.

(19)

Then, from the definitions of U (f ∗ ) and f ∗ and Eq.(19), we get F i ∩ Gi = {U i (f ∗ ), (f ∗ )i }, i = 1, · · · , k.

(20)

It follows from the separation theorem about convex sets and Eq.(20) that there exists (w1i , w2i )T ∈ R2 \{0}, i = 1, · · · , k, such that (w1i , w2i )z1 ≥ (w1i , w2i )z2 , ∀z1 ∈ F i , z2 ∈ Gi , ∀i = 1, · · · , k.

(21)

From the definitions of F i , Gi and Eq.(21), we have

uia (f ) = w1i tia (f ) + w2i cia (f ), where, uia is called the disutility of the flow of class i on arc a. Thus, the disutility of the flow of class i on path p can be denoted as:  uia (f )δap . vpi (f ) =

(15)

(w1i , w2i )U i (f ∗ ), f i − (f ∗ )i ≥ 0, i = 1, · · · , k ∀f ∈ K,

(22)

2 \{0}, i = 1, · · · , k. (w1i , w2i ) ∈ R+

(23)

and Now, summing the inequalities of Eq.(22) for i = 1, · · · , k, we have k  

(w1i tia (f ∗ )+w2i cia (f ∗ ))×(fai −(fa∗ )i ) ≥ 0, ∀f ∈ K.

i=1 a∈L

Thus, it follows from Theorem 1 in Reference [11] and Eq.(23) that f ∗ is the traffic network equilibrium flow for the weights w1i and w2i of class i, i = 1, · · · , k, which, are nonnegative, but not both equal to zero. The proof is complete. Proposition 3.2 Suppose that the arc flow vector f ∗ ∈ K is a traffic network equilibrium flow for the weights w1i and w2i of class i, i = 1, · · · , k, which are nonnegative, but not both equal to zero. Then, this arc flow vector f ∗ ∈ K is a solution of the (WVVI) problem (13).

LI Sheng-jie, et al./Systems Engineering – Theory & Practice, 2008, 28(1): 141–145

Proof. Suppose that the arc flow vector f ∗ ∈ K is not a solution of the (WVVI) problem (13). Then, there exists an arc flow vector f ∈ K such that 2k . U (f ∗ ), f − f ∗ ∈ −intR+

Since

(w11 , w21 , · · · , w1k , w2k )T ≥ 0

and

(w11 , w21 , · · · , w1k , w2k )T = 0,

we have (w11 , w21 , · · · , w1k , w2k )U (f ∗ ), f − f ∗ < 0, i.e., k  

(w1i tia (f ∗ ) + w2i cia (f ∗ )) × (fai − (fa∗ )i ) < 0,

i=1 a∈L

which contradicts with the assumption that f ∗ ∈ K is a traffic network equilibrium flow for the weights w1i and w2i of class i, i = 1, · · · , k. The proof is complete. However, the converse result of Proposition 3.2 may not hold in general. In fact, if we add some suitable restrictive conditions, we can easily obtain the following proposition. Proposition 3.3 If an arc flow vector f ∗ ∈ K is a solution of the following vector variational inequality problems: (WVVI)i find f ∗ ∈ K, s.t. 2 , ∀f ∈ K, U (f ∗ ), f i − (f ∗ )i ∈ −intR+ i

(24)

where, i = 1, · · · , k, then, for each class i, i = 1, · · · , k, there are weights w1i and w2i such that the arc flow f ∗ ∈ K is a network equilibrium flow.

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