user equilibrium in traffic assignment problem with fuzzy n–a incidence matrix

Fuzzy Sets and Systems 107 (1999) 245–253www.elsevier.com/locate/fss

User equilibrium in tra�c assignment problem with fuzzy N–Aincidence matrix

Hsiao-Fan Wang∗, Hsueh-Ling LiaoDepartment of Industrial Engineering, National Tsing-Hua University, Hsinchu, 30043, Taiwan, ROC

Received December 1996; received in revised form August 1997

Abstract

The user equilibrium in tra�c assignment problem is to choose the minimum-cost path between every origin–destinationpair and through this process, those utilized paths will have equal costs. In other words, giving cost and demand functionfor transportation between every origin–destination pair, the solution of the problem is to provide the minimum cost ofwhich the ow is generated. In this study, we consider this problem when the N–A incidence matrix for transportationis fuzzy, in the sense that, which arcs are chosen into the desired path for traveling is uncertain. Therefore, we apply themethod and concept of the theory of variational inequality with fuzzy convex cone to establish a user equilibrium pattern.Finally, the proposed method is demonstrated with a numerical example. c© 1999 Published by Elsevier Science B.V. Allrights reserved.

Keywords: User equilibrium problem; Variational inequality; Multiple objective programming; Fuzzy set theory

1. Introduction

The tra�c assignment or network equilibrium [2] problem is to predict the steady-state ow of a trans-portation network. It is an application of variational inequality problem [1, 4]. The user equilibrium in tra�cassignment problem is to choose the minimum-cost path between every origin–destination pair and throughthis process, those utilized paths will have equal costs. Thus, in the model of user equilibrium, denoted byUE(c; t), we need to know which arcs are included in the desired paths in the transportation network; whatare the cost, c, and demand function t for transportation between every arc before solving this problem. Then,whenever the demand is satis�ed, the problem is solved in which the paths with costs higher than the mini-mum cost will have no ow. But, in the real world, which arcs are chosen into the desired path for travelingsometimes is uncertain. This leads to a fuzzy node–arc (N–A) incidence matrix, denoted by �. However, theuncertain paths will cause the imprecise prediction of cost and demand, and induce biased result. Thus, in thisstudy we propose a new approach to determining a user-optimized ow pattern which satis�es the equilibriumproperty with fuzzy N–A incidence matrix, denoted by UE(c; t; �). Mathematically, a user equilibrium problem

∗ Corresponding author.

0165-0114/99/$ – see front matter c© 1999 Published by Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(97)00298 -4

246 H.-F. Wang, H.-L. Liao / Fuzzy Sets and Systems 107 (1999) 245–253

UE(c; t) can be modeled as a variational inequality problem, and with fuzzy phenomena indicated above, thedeveloped method on the variational inequality with fuzzy convex cone and fuzzy function [6], denoted byVI(X ; f), can be used to establish a tra�c equilibrium pattern. Then an algorithm is designed for constructingthis pattern.In Section 2, we introduce some basic concepts of fuzzy set. In Section 3, after de�ning an UE(c; t; �),

we shall formulate an equivalent variational inequality model VI(X ; f). Then, through a transformation, amultiple objective programming model is derived to facilitate solution process. In Section 4, the existencetheorem and solution procedure of UE(c; t; �) will be proposed, and an algorithm is developed. In Section 5,a numerical example is presented for illustration. Finally, in Section 6, summary and conclusions are drawn.

2. Basic concepts of fuzzy set

A crisp set is de�ned in such a way as to dichotomize the individuals in some given universe of discourseinto two groups: members that certainly belong to the set and nonmembers that certainly do not. However,for a fuzzy set, a function can be generalized such that the values assigned to the elements of the universalset fall in a speci�ed range and indicate the membership grade of these elements in the set in question. Largevalues denote higher degrees of set membership. Such a function is called a membership function [4]. Let Xdenote a universal set. Then, the membership function �A of which a fuzzy set A is usually de�ned has theform

�A : X → [0; 1];

where [0; 1] denotes the interval of real numbers from 0 to 1. Hence, the element x of the fuzzy set A containstwo parts: the data of the element and the membership degree which belongs to the set, denoted by (x; �A(x)).In the model of user equilibrium, the cost and demand function for transportation may involve many

uncertainties and are di�cult to predict exactly. Therefore, in [8], we focus on solving the user equilibriumproblem in tra�c assignment with fuzzy travel cost and demand functions. Since the cost and demand functionsfor transportation are fuzzy, the induced optimal solution set is also fuzzy. An algorithm with a proposedmethod for solving the fuzzy system is designed to �nd the optimal fuzzy solution set for general case.

3. Problem formulation

First, we specify the notations de�ned by Friesz [2], which are required to formulate a tra�c assignmentproblem:G(V; A) ≡ a network graph G in which A represents the set of arcs and V the node set,W ≡ the set of origin–destination (O–D) pairs w = (i; j)∈W ,Pw ≡ the set of paths connecting O–D pair w∈W ,

p =⋃w∈W

Pw

ea ≡ the ow on arc a∈A, e=(ea: a ∈ A),hp ≡ the ow on path p∈P; h=(hp: p∈P),

�ap ≡{1 if path p∈P traverses arc a∈A;0 otherwise;

� ≡ [�ap], the arc–path incidence matrix,

H.-F. Wang, H.-L. Liao / Fuzzy Sets and Systems 107 (1999) 245–253 247

ca(e) ≡ the average transportation cost function for arc a∈A,c(e)= (ca(e): a∈A);

Cp(h) ≡ the average transportation cost function for path p∈P,C(h)= (Cp(h):p∈P);

uw ≡ the minimum transportation cost between O–D pair w∈W;u=(uw: w∈W )= min

p∈Pw{Cp(h)};

tw(u) ≡ the demand for transportation between O–D pair w∈W ,t(u)= (tw(u): w∈W );

where

e=�h; C(h)=�Tc(e):

Now, we shall de�ne a crisp user equilibrium model [2] and propose its equivalent variational inequalitymodel.

De�nition 3.1 (Friesz [2]). A ow-cost pattern (e∗; u∗) is user equilibrium UE(c; t) if it satis�es the followingcondition:

hp[Cp(h)− uw] = 0; Cp(h)− uw¿ 0; hp¿ 0 ∀w∈W; p∈Pw; (1a)∑p∈Pw

hp − tw(u)= 0; uw¿ 0 ∀w∈W: (1b)

In De�nition 3.1, (1a) means that the paths with costs higher than the minimum cost will have no ow;(1b) means that the demand is satis�ed.Let H = [Hij] be a path incidence matrix, then, if the jth path belongs to the path set Pi, then Hij =1;

otherwise, Hij = 0.By rewriting equation (1b) as

t(u)= (tw(u): w∈W )=∑p∈Pw

hp: w∈W=Hh (2)

we have the following result.

Theorem 3.1. Let t(u) be an invertible function and � be the inverse function. Then (e∗; t∗)= (�h∗; Hh∗)is user equilibrium UE(c; t) if and only if h∗ solves the following VI(X; f):

〈c(�h∗)T�− �(Hh∗)TH; h− h∗〉¿ 0 ∀h∈X; (3)

where

X = {h: h¿ 0; h 6=0; Hh¿ 0} and f(h)= c(�h)T�− �(Hh)TH:

Proof. From Dafermos [1] and Smith [5]; we know (e∗; t∗) is a user equilibrium solution of UE(c; t) if andonly if (e∗; t∗) solves the following VI(X ′; f′) problem:

c(e∗)T(e − e∗)− �(t∗)T(t − t∗)¿ 0 ∀(e; T )∈X ′; (4)


where

X ′=

(e; t): e=�h;

∑p∈Pw

hp= tw ∀w∈W; h¿ 0; t¿ 0

:

Let (e∗; t∗)= (�h∗; Hh∗); then (3) are equivalent to (4) and the theorem is true.

Next, we consider the user equilibrium problem under the condition that N–A incidence matrix is fuzzy.The N–A incidence matrix is to present the relation between nodes and arcs of the transportation network. Theelement with index ij of the N–A incidence matrix will be 1 if the jth arc travels the ith node, otherwise, 0.Since the N–A incidence matrix is fuzzy, the arc–path matrix is also fuzzy and denoted by �, in which eachelement of � represents that the possibility of that path p includes arc a and is denoted by �ap. Since whicharcs are chosen into the desired path for traveling is uncertain, the path incidence matrix is also fuzzy anddenoted by H . This causes the fuzzy convex cone fuzzy as X = {h: h¿0; h 6= 0; Hh¿0}. So does thefunction

f(h)= c(�h)T�− �(Hh)TH ; where f :Rn → F

with

F = {(f(h); �F(f(h))):f(h) :Rn→Rn; �F(f(h)) :Rn→ [0; 1]n; h∈Rn} (5)

in which (f(h); �F(f(h))) = ((f1(h); �F(f1(h))) ; : : : ; (fn(h); �F(fn(h)))) where fi(h) :Rn→R; �F(fi(h)) :R→ [0; 1]; �F(fi(h))=Min{�C(ci(�h)); ��(�i(Hh))}: Therefore, the user equilibrium with fuzzy N–A in-cidence matrix, denoted by UE(c; T; �); can be reformulated into a variational inequality with fuzzy convexcone and fuzzy function, which is denoted by VI(X ; f). Since the set X de�ned above is a convex cone,hence by Wang and Liao [7, 8], the solution set of VI(X ; f) is fuzzy. And then we obtain the followingresult.

Theorem 3.2. Let t(u) be an invertible function. Then (e∗; t ∗) = (�h∗; Hh∗) is a user equilibrium solutionof UE(c; T; �) if and only if (h∗; �X (h

∗)) solves the following VI(X ; f):

〈f(h∗); h− h∗〉¿(0; �); 06�61 ∀h∈ X ; (6)

where

X = {h: h¿0; h 6=0; Hh¿0} and f(h)= c(�h)T�− �(Hh)TH :

Now, let us transform the fuzzy convex cone X into a crisp convex cone X = {h: h¿ 0} by the followingtheorem.

Theorem 3.3. X = {h: h¿ 0; h 6=0; Hh¿ 0} is equivalent to X = {h: h¿ 0}.

Proof.

X = {h: h¿0; h 6=0; H h¿ 0}

⇔ X =

{h: h 6=0; Dh¿0; D=

[H

I

]}


⇔ X =

{h: h 6=0; Ah¿0; A=

[A′

I

]}by Wang and Liao [9]

⇔ X = {h: h¿ 0; h 6=0} (since {A′h¿0}⊇{h¿0}):

Thus, the problem of user equilibrium with fuzzy N–A incidence matrix can be reformulated into a vari-ational inequality with fuzzy function, VI(X; f). Since the set X de�ned above is a convex cone, hence byWang and Liao [7] the solution set of VI(X; f) is fuzzy. And then we obtain the following result.

Theorem 3.4. Let t(u) be an invertible function. Then (e∗; t∗)= (�h∗; Hh∗) is a user equilibrium solution ofUE(c; T; �) if and only if (h∗; �F(f(h

∗))) solves the following VI(X; f):

〈(f(h∗); �F(f(h∗))); h− h∗〉¿ (0; �); 06 �6 1 ∀h∈X; (7)

where X = {h: h¿0; h 6=0} and f(h)= c(�h)T�− �(Hh)TH .

Then, the equivalent multiple objective programming model of VI(X; f)(5) can be derived as follows [7]:(Fuzzy-MOP)

Minimize [y1h1; y2h2; : : : ; ynhn]T

subject to h¿0; h 6=0;

(f(h); �F(f(h)))¿(0; �F(f(h)));

y=(y1; y2; : : : ; yn)T = (f(h); �F(f(h)));

f(h)= c(�h)T�− �(Hh)TH:

(8)

Then, based on the equivalence between variational inequality with fuzzy function and fuzzy multipleobjective programming problem in [7], we have the relation between VI(X;f)(7) and Fuzzy-MOP(8) asbelow.

Corollary 3.4.1. Let Xe� denote the set of all e�cient solutions of a Fuzzy-MOP de�ned in (8). Thenthe point h∗=(h∗1 ; h

∗2 ; : : : ; h

∗n)T 6=0 with some �F(f(h∗)) belonging to Xe� is a solution of the MOP such

that y∗1h∗1 + y∗2h

∗2 + · · · + y∗nh

∗n =(0; �); 06 �6 1; if and only if (h∗; �F(f(h

∗))) is a solution ofVI(X; f)(7).

Since from Theorems 3.1 and 3.4, a user equilibrium problem UE(c; T; �) can be represented by anVI(X; f)(7), hence Fuzzy-MOP(8) can solve UE(c; T; �).

Corollary 3.4.2. Let t(u) be an invertible function. Then (e∗; t∗)= (�h∗; H h∗) is a user equilibrium solutionof UE(c; t; �) if and only if h∗=(h∗1 ; h

∗2 ; : : : ; h

∗n)T 6=0 and some �F(f(h∗)) belonging to X e� is a solution of

the Fuzzy-MOP(8) such that y∗1h∗1 + y

∗2h

∗2 + · · ·+ y∗nh∗n =(0; �); 06�61.


Furthermore, we can transform the multiple objective programming model into a single objective program-ming model by the weighted-sum method in Steuer [6]. So, we have

Minimize (f(h); �F(f(h)))Th

subject to h¿0; h 6=0;(f(h); �F(f(h)))¿(0; �F(f(h)));

y=(y1; y2; : : : ; yn)T = (f(h); �F(f(h)));

f(h)= c(�h)T�− �(Hh)TH:

(9)

This is a single objective model by aggregating multiple objectives in (8) with simple average. Now, let usestablish the relations between VI(X; f)(7) and model (9) in Corollary 3.4.3; and that between UE(c; t; �)and model (9) in Corollary 3.4.4 below. Then VI(X; f)(7) and UE(c; t; �) can be solved.

Corollary 3.4.3. (h∗; �F)f(h∗))) 6=(0; �F(f(0))) is a solution of model (9) de�ned above such that (f(h∗);

�F(f(h∗)))Th∗=(0; �); 06�61; if and only if (h∗; �F(f(h

∗))) solves VI(X; f)(7).

Corollary 3.4.4. Let t(u) be an invertible function. Then (e∗; t∗)= (�h∗; H h∗) is a user equilibrium so-lution of UE(c; t; �) if and only if (h∗; �F(f(h

∗))) 6=(0; �F(f(0))) is a solution of model (9) such that(f(h∗); �F(f(h

∗)))Th∗=(0; �); 06�61.

4. Solutions and algorithm

From the previous section, we know the relation between UE(c; t; �) and MOP. Next, we apply the refor-mulated model (8) to solve UE(c; t; �) as MOP to VI(X; f) in [7].

Theorem 4.1. (e∗; t∗)= (�h∗; Hh∗) is a user equilibrium solution of UE(c; t; �); that is; h∗=(h∗; �F(f(h∗)))

is an optimal solution of (8) such that (f(h∗); �F(f(h∗)))Th∗=(0; �) for some �∈ [0; 1]. Then;

(i) h∗ 6=0; (f(h∗); �F(f(h∗)))= (0; �F(f(h∗))) for some �F(f(h∗)); if and only if h∗¿0; h∗ 6=0; or;(ii) h∗ 6=0; (f(h∗); �F(f(h∗))) 6=(0; �F(f(h∗))) for all �F(f(h∗)); if and only if (1) there exists a k ∈

{1; 2; : : : ; m} such that h∗k =0; h∗i ¿0 for all i=1; 2; : : : ; n; i 6= k; and there exists a l¿0; l∈R; such that(h∗; �F(f(h

∗)))= ((0; : : : ; 0; l; 0; : : : ; 0); �F(f(h∗))) for some �F(f(h

∗)); or (2) if n¿2; there exist k1 andk2; k1 6= k2 and k1; k2 ∈{1; 2; : : : ; m} such that h∗k1 = 0; h∗k2 = 0; h∗i ¿0 for all i=1; 2; : : : ; n; i 6= k1; k2; andthere exist some l1¿0 and l2¿0; l1; l2 ∈R; such that (h∗; �F(f(h∗)))= ((0; : : : ; 0; l1; 0; : : : ; 0; l2; 0; : : : ; 0);�F(f(h

∗))) for some �F(f(h∗)).

Based on the above theorem, we proposed an algorithm for solving UE(c; t; �)

Algorithm for UE(c; t; �).Step 0. Let X ∗= ∅ be the optimal solution set of the UE(c; t; �).Step 1. Based on Theorem 4.1(i) to solve the following system:

f(h)= c(�h)T�− �(Hh)TH =(0; �F(f(h))); h¿0; h 6=0 (10)

and obtain solution (h∗; �F(f(h∗))), and then (�h∗; H h∗)∈ X ∗.

Step 2.Step 2.1. Let k =1.


Step 2.2. Let hk =0; hi¿0 for all i=1; 2; : : : ; m; i 6= k, and l¿0.Step 2.3. Based on Theorem 4.1(ii) (1) to solve the following system to obtain the solution h∗:

hk =0;

hi¿0; i 6= k; (11)

f(h)= c(�h)T�− �(Hh)TH =((0; : : : ; 0; l; 0; : : : ; 0); �F(f(h∗))):

Then, (�h∗; Hh∗)∈ X ∗.Step 2.4. If k¡m, let k = k + 1, go back to Step 3.2; otherwise, if n¿2 and m¿2, go to Step 3; else goto Step 4.

Step 3.Step 3.1. Let k1 = 1; k2 = 2.Step 3.2. Based on Theorem 4.1(ii) (2) to solve the following system (12) and �nd the solution h∗:

hk1 = 0; hk2 = 0; hi¿0; i 6= k1; k2; i∈{1; : : : ; n};

f(h)= c(�h)T�− �(Hh)TH= ((0; : : : ; 0; l1; 0; : : : ; 0; l2; 0; : : : ; 0); �F(f(h))); l1¿0; l2¿0: (12)

Then (�h∗; H h∗)∈ X ∗.Step 3.3. If k2 6=m, then k2 = k2+1 and go to Step 3.2. If k2 =m and k1¡m−1, let k1 = k1+1; k2 = k1+1and go to Step 3.2. If k2 =m and k1 =m− 1, then go to Step 4.

Step 4. Output the optimal solution set X ∗, then STOP.

5. Numerical examples

In order to illustrate the application of the proposed algorithm, we calculate the user-equilibrium tra�cpattern for a simple network shown in Fig. 1. There are two nodes x and y in the example, and are connectedby two-way streets and by one-way street with direction from x to y. However, because these connections arenot certain, let us assume that connection by arc a1 is with possibility 1, by arc a2 with possibility above 0.5,by arc a3 with possibility above 0.7, by arc b1 with possibility above 0.4, and by arc b2 with possibility 1.Then, from Fig. 1, we have

A= {a1; a2; a3; b1; b2}; V = {x; y};W = {(x; y); (y; x)}; p=Pxy ∪Pyx;Pxy = {Pa1 ; Pa2 ; Pa3}; Pyx = {Pb1 ; Pb2};e= {ea1 ; ea2 ; ea3 ; eb1 ; eb2}; h= {hPa1 ; hPa2 ; hPa3 ; hPb1 ; hPb2};

�=

1 0 0 0 0

0 [0:5; 1] 0 0 0

0 0 [0:7; 1] 0 0

0 0 0 [0:4; 1] 0

0 0 0 0 1

; H =

[1 [0:5; 1] [0:7; 1] 0 0

0 0 0 [0:4; 1] 1

];


Fig. 1. A network

ca1 (e)= 10ea1 + 5eb1 + 1000; ca2 (e)= 15ea2 + 5eb2 + 950;

ca3 (e)= 20ea3 + 3000; cb1 (e)= 20eb1 + 2ea1 + 1000;

cb2 (e)= 25eb2 + ea2 + 1300;

�(t)=[�xy(t)�yx(t)

]=

[50004800

]−[1012

]t:

Reformulate this UE(c; t; �) into VI(X; f)(3) and single objective programming model (7), we have

f(h)

=

[20hpa1 + 5hpa2 + 7hpa3 + 2hpb1 − 4000; 20hpa1 + 10hpa2 + 10hpa3 + 5hpb1− 4000][13:75hpa1 + 5hpa2 + 7hpa3 + 2:5hpb2 − 4525; 5hpa1 + 20hpa2 + 5hpa3 + 5hpb2 − 1550]

[10hpa1 + 5hpa2 + 16:8hpa3 − 2900; 7hpa1 + 7hpa2 + 27hpa3 + 2650][10:8hpa1 + 8hpb1 + 12hpb2 − 4400; 2hpa1 + 24:8hpb1 + 4:8hpb2 − 920][hpa2 + 4:8hpb1 + 27hpb2 − 3500; hpa2 + 12hpb1 + 27hpb2 − 3500]

:

Step 0: Let X ∗= ∅ be the optimal solution set of the UE(c; t; �).Step 1: In order to solve system (10), we reformulate the above fuzzy system (10) into crisp form [7] as

20hPa1 + (5 + 5�)hPa2 + (7 + 3�)hPa3 + (2 + 3�)hPb1 − 4000=0;

(13:75− 8:75�)hPa1 + (5 + 15�)hPa2 + (7− 2�)hPa3 + (2:5 + 2:5�)hPb2 − (4525− 2975�)= 0;

(10− 3�)hPa1 + (5 + 2�)hPa2 + (16:8 + 10:2�)hPa3 + (−2900 + 5550�)= 0;

(10:8− 8:8�)hPa1 + (8 + 16:8�)hPb1 + (12− 7:2�)hPb2 − (4400− 3400�)= 0;

hPa2 + (4:8 + 7:2�)hPb1 + 27hPb2 − 3500=0;

hPa1¿0; hPa2¿0; hPa3¿0; hPb1¿0; hPb2¿0; h 6=0; �∈ [0; 1]; i=1; : : : ; 5:And then we �nd that no solution is generated.


Step 2: For k =1 to 5, solve system (12). Then an optimal solution (h∗; �F(h∗))= ((0; 572; 0; 323; 61);

0:27) can be found and the solution of UE(c; t; �) is

((�h∗; Hh∗); �F(f(h∗)))=

0

[286; 572]

0

[129; 323]

61

;[[286; 572][190; 384]

]; 0:27

:

Step 3: For k1 = 1 to 5, k2 = k1 − 1 to 5, solve system (13), and no solution is generated.Step 4: Output the optimal solution set X ∗, then STOP.

Therefore, for example, when demand on each pair is {(x; y); (y; x)}= {[286; 572]; [190; 384]}, and the ow on every arc is e=(ea1 ; ea2 ; ea3 ; eb1 ; eb2 ) = (0; [286; 572]; 0; [129; 323]; 61), then the degree of satisfyinguser equilibrium condition of (6) is 0.27.

6. Summary

This study focuses on solving a user equilibrium problem in tra�c assignment when the N–A incidencematrix is fuzzy. By considering it as a variational inequality problem with fuzzy function in a convex cone, thisproblem is reformulated into a multiple objective programming model which can be solved by the proposedmethod in [7]. Since the cost and demand functions for transportation are fuzzy, the induced optimal solutionset is also fuzzy. An algorithm with a proposed method for solving the fuzzy system is designed to �nd theoptimal fuzzy solution set for general cases. When the travel cost and demand function are linear, the problemis solved in polynomial time. Theoretical evidences are illustrated by a numerical example.

Acknowledgements

This work was supported by National Science Council, Taiwan, Republic of China with the project numberNSC84-2213-E007.

References

[1] S. Dafermos, Tra�c equilibria and variational inequalities, Transp. Sci. 14 (1980) 42–54.[2] T.Z. Friesz, Network equilibrium, design and aggregation, Transp. Res. A 19 (1985) 413–427.[3] P.T. Harker, J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory,

algorithms and applications, Math. Programming 48 (1990) 161–220.[4] G.J. Klir, T.A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice-Hall, Englewood Cli�s, NJ, 1988.[5] M.J. Smith, The existence, uniqueness and stability of tra�c equilibria, Transp. Res. B 13 (1979) 295–304.[6] R.E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986.[7] H.F. Wang, H.L. Liao, Variational inequality with fuzzy function, J. Fuzzy Math. 4 (1996) 149–169.[8] H.F. Wang, H.L. Liao, Application of variational inequality to user equilibrium problem in tra�c assignment with fuzzy cost and

demand functions, J. Chinese Fuzzy Systems Assoc. 2 (1996) 39–47.[9] H.F. Wang, H.L. Liao, Variational inequality with fuzzy convex cone, J. Global Optim., submitted.

user equilibrium in traffic assignment problem with fuzzy n–a incidence matrix

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