- Email: [email protected]
Human migration networks
262
European Journal of Operational Research59 (1992)262-274 North-Holland
Theory and Methodology
Human migration networks Anna Nagurney
School of Management, University of Massachusetts, Amherst, MA 01003, USA
Jie Pan Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA Lan Zhao Department of Mathematics, SUNI~, College at Old Westbury, Old Westbury, NY 11568-308Z USA Received May 1990; revised December 1990 Abstract: In this paper we develop a new multiclass human migration network equilibrium model which explicitly incorporates the cost of movement between locations. The equilibrium conditions reflect that no individual in the system in any given class has any incentive to alter his location. The equilibrium conditions are then formulated as a variational inequality problem and qualitative properties discussed. A decomposition algorithm which takes advantage of the special network structure of the problem is then proposed and numerical results presented. This research brings and yet another application in operations research into the realm of network equilibrium problems.
Keywords: Migration; networks; economic equilibrium; variational inequalities
1. Introduction Networks have been used as the foundation for the study of large-scale competitive systems in distinct application areas of operations research. Examples include the traffic network equilibrium problem and the spatial price equilibrium problem as well as Walrasian price equilibrium, market equilibrium problems with production, and disequilibrium problems (see, e.g., [1,4-6,8,13,15,19,26]). In the latter set of applications, the underlying network is abstract in that the nodes no longer correspond to locations in space. In this paper we focus on the study of human population movements, a topic which has, heretofore, been studied principally by social scientists, including economists, demographers, geographers, and sociologists. In particular, we introduce a general multiclass network model of human migration which explicitly incorporates the cost of movement between locations. In our framework the cost of movement reflects not only the cost of transportation as a proxy for distance, but also the "psychic" costs associated with dislocation. 0377-2217/92/$05.00 © 1992 - ElsevierSciencePublishers B.V. All rights reserved
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The importance of movement costs in migration decision-making is well documented in the literature from both theoretical and empirical perspectives (see, e.g., [2,20,22,23,25]). Economic research, however, has emphasized the development of equilibrium models in which the population is assumed to be perfectly mobile and the costs of migration insignificant. In such models, individuals a n d / o r households are assumed to choose a location so that utilities are equalized across the economy ([10,11,22]). Recently, Nagurney [14] introduced a multiclass migration equilibrium model in the absence of migration costs. That model could be formulated as a network equilibrium problem with fixed demands and a special structure with each class being represented by a single origin/destination pair network connected with disjoint links representing the locations. The equilibrium conditions were then formulated as a variational inequality problem. In Section 2 we introduce a multiclass network model of human migration with movement costs, state the equilibrium conditions, and then derive the variational inequality formulation of the problem. The equilibrium conditions reflect that no individual in the system in any given class has any incentive to alter his location since no positive net gain (gain in utility minus movement cost) is possible by his unilateral decision. These equilibrium conditions are a generalization of those derived for a single class migration equilibrium model with movement costs in [16], for, which, due to the assumptions made on the utility functions, an equivalent optimization formulation exists. In that model the utility associated with a given location could depend only upon the population at that location and the migration cost between locations was assumed to be fixed. In this new model the utility of a given location as perceived by a particular class, can, in general, depend upon the population of every class at every location. The cost of movement associated with members of a class migrating from a given origin to a destination can, in turn, depend upon, in general, the migration flows of every class between every location. However, we retain the assumption that there is no repeat or chain migration. For this general model, we also discuss the qualitative properties of existence and uniqueness. In Section 3 we then describe a variational inequality algorithm which decomposes the multiclass problem into single class problems which are then solved in serial fashion and establish conditions for convergence. The resulting single class problems are characterized by linear, separable utility and movement cost functions and can be solved by a migration equilibration algorithm developed in [16] which exploits the special network structure by computing the flow out of each location for a particular class exactly and in closed form. In Section 4 we then present some numerical results for the decomposition algorithm embedded with the equilibration scheme on large-scale migration network problems. This research brings and yet another equilibrium problem in operations research within the domain of network equilibrium problems.
2. A general multiclass network model of human migration equilibrium In this section we introduce a network model of human migration equilibrium which allows for multiple classes as well as movement costs between locations. We assume a closed economy in which there are n locations, typically, denoted by i, and J classes, typically denoted by k. We further assume that the attractiveness of any location i as perceived by class k is represented by a utility u~. We let ff/k denote the initial fixed population of class k in location i, and we let pk denote the population of class k in location i. We group the utilities into a row vector u ~ R Jn and the populations into a column vector p ~ EJn. We assume no births and no deaths in the economy. We associate with each class k and each pair of locations i, j a nonnegative cost of movement c~ and we let the migration flow of class k from origin i to destination j be denoted by fi~. The movement costs are grouped into a row vector c ~ EJn~-~) and the flows into a column vector f ~ ~ J ~ - 1). We assume that the movement costs reflect not only the cost of physical movement but also the personal and psychic cost as perceived by a class in moving between locations.
264
A. N a g u r n e y et al. / H u m a n migration n e t w o r k s J
c
J J
12
u
1
o °
, °
~ J c
o
"c12,
,Jo
,
c
24
1 c 31
1
1 c
3
1 u
43
4
F i g u r e 1. T h e m u l t i c l a s s m i g r a t i o n n e t w o r k w i t h 4 l o c a t i o n s
The conservation of flow equations are given for each class k and each location i, assuming no repeat or chain migration, by P~ =P~ + E f t ~ 14=i
Y'~f~,
(1)
14=i
and E f ~
(2)
l~i
where fi~ >- 0 for all k = 1. . . . , J, 1 4~ i. Equation (1) states that the population at location i of class k is given by the initial population of class k at location i plus the migration flow into i of that class minus the migration flow out of i for that class. Equation (2) states that the flow out of i by class k cannot exceed the initial population of class k at i, since no chain migration is allowed. We now present the multiclass network model. As in the single class model outlined in Nagurney [16], we construct n nodes, i = 1. . . . . n, to represent the locations and a link (i, j ) connecting each pair of nodes. There are, hence, n nodes in the network and n ( n - 1) links. With each link (i, j ) we then associate k costs c~. and corresponding flows fi~. With each node i we associate k utilities u/k and the initial positive populations ~fi~. A graphical depiction of a four location migration network is given in Figure 1, where the k classes are layered. Of course, rather than a multiclass network, one can construct J copies of the network topology given in Figure 1 to represent the classes where the costs on the links and the utilities are defined accordingly. Note that in cases where the migration between locations is forbidden, due, for example, to government regulations, then those links can be removed from the network model, or in the case of class restrictions, the costs of the respective classes set sufficiently high. We are now ready to state the equilibrium conditions. We assume that migrants are rational and that migration will continue until no individual has any incentive to move since a unilateral decision will no longer yield a positive net gain (gain in utility minus
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265
movement cost). Mathematically, the multiclass equilibrium conditions are: For each class k, k = 1. . . . . J, and each pair of location i, j, i = 1 . . . . . n, j 4: i, we have that:
u~+c
~[ = u f - A ~ ,
if fij>O, k
>u~-A~,
iffi~=O,
(3)
and
A
14:i
= 0,
if
~-.fi~
(4)
l~i
subject to constraints (1) and (2). Equilibrium condition (3), although similar in structure to the equilibrium conditions governing the multicommodity spatial price equilibrium problem (cf. [5,15,17,24]), differs in that the indicator A~ is present. We now interpret the necessity of A~. Observe that, unlike spatial price equilibrium problems (or the related traffic network equilibrium problem with elastic demand; see, e.g., [4,13]), the level of the population/3~ may not be large enough so that the gain in utility u~ - u/k is exactly equal to the cost of movement c~.. Nevertheless, the utility gain minus the movement cost will be maximal and nonnegative. Moreover, the net gain will be equalized for all locations and classes which have a postive flow out of a location. In fact, from the equivalence proof of Theorem 1 given below, A~ is exactly the equalized net gain for all individuals of class k of location i. We first discuss the function structure and then derive the variational inequality formulation of the equilibrium conditions (3) and (4). We assume that, in general, the utility associated with a particular location and class can depend upon the population associated with every class and every location, i.e., we assume that u = u(p).
(5)
We also assume that, in general, the cost associated with migrating between two locations as perceived by a particular class can depend, in general, upon the flows of every class between every pair of locations, i.e., we assume that c = c(f).
(6)
We recall that in [16], a single class of migrant was assumed, and the utility functions (5) were assumed to be separable and linear, whereas the migration costs were assumed to be fixed. That model had an equivalent optimization formulation of the equilibrium conditions, which was also the approach utilized in the classical models of Samuelson [21] and Takayama and Judge [24] in spatial price equilibrium and Beckmann, McGuire, and Winsten [1] in traffic network equilibrium. The variational inequality formulation of the migration equilibrium conditions is given by: T h e o r e m 1. A population and migration flow pattern (p, f ) satisfies equilibrium conditions (3) and (4) subject to constraints (1) and (2), if and only if it satisfies the variational inequality problem
-u(p).(p'-p)+c(f).(f'-f)>O
forall ( p ' , f ' ) ~ K
(7)
where K - {(p', f ' ) l f ' > O, (p', f ' ) satisfy (1), (2)}. Proof. We first show that if a pattern (p, f ) satisfies equilibrium conditions (3) and (4), subject to constraints (1) and (2), then it also satisfies the variational inequality in (7). Suppose that (p, f ) satisfies the equilibrium conditions. Then fi~ > O, and E~ + i fi~ < Pf, for i, j and k.
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For fixed class k we define F~ = {llfi~ > 0} and Fzk = {llfi~ = 0}. Then
E [u/k(P) ff'ck(f) -- u k ( p ) ] " [fi f f t - f i if] l~i
= E
Jil --.tit l
I~F~
+ ~ [u/k(P)+c~(f)-u/k(P)]'[fi~'-f~] t~r~ ~-~ --*ki E
=-Ai
(lift--fir) "}- E (--*k)(fif')
Eli,"l*i
fi k
(8)
~ O, if E f , Y=P/*, l~i
holds for all such locations i. Therefore, for this class k and all locations i, we have that fi*t > O, Et.if~f < , ~ , and
Z E
i l~i
+ c (f)
Jit
(9)
Jit I >- O.
But (9) holds for each k; hence,
E E E [uk(p) "}-ck(f)--uk(p)] k i l~i
" [fikt--fi k] ~ 0 .
(10)
Observe now that (10) can be rewritten as
E Ellk(P)'((j~lflkt-k 1
Efjftl--(j~lf, j4=t /
k- EL'f)) jstl
+ E E Eek(f)'(fif'--fif) k i 14=i
~0. (11)
Using now constraint (1), and substituting it into (11), we conclude that
- E Y'~U/k(P)'(P~'--Pk) + E Y'~ E c ~ ( f ) ' ( f i k ' - - f i ~ ) k I k i I~i
>0,
(12)
or, equivalently, in vector notation:
-u(p).(p'-p)+c(f).(f'-f)>O
for a l l ( p ' , f ' ) ~ K .
(13)
We now show that if a pattern (p, f ) ~ K satisfies variational inequality (7), then it also satisfies equilibrium conditions (3) glad (4). Suppose that (p, f ) ~l[|~fies variational inequality (7). We shall prove that (p, f ) satisfies equilibrium conditions (3) and (4) by eonsidering two different cases; the first, locations i where the population is not exhausted, and the second, for locations where the respective populations are exhausted. We fix the class k.
Case 1. ~-ls~'ifif < pik. Construct a feasible (p', f ' ) such that fih.'=fh J JtJ
for a l l i , j , j ~ i
and h4=k.
For h = k, let
fikt -fu, __ k
ifl4=j,
ftikt =fly,k
ifl4=i.
(14)
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267
Then variational inequality (7) reduces to
iuf(p) +c~(f) --u~(p))'(fiks'--fi~) >0.
(15)
Since Y"t,;fi~
uki(p) + c ~ . ( f ) - u ~ ( p ) = O ,
iff/~>0,
uk(p)+ck(f)--u~(p)>O,
iffi~=0.
(16)
In other words, h;k _- 0, and, hence, (3) holds.
Case 2. Et.ifi~ =pi g (a) fi~ > O. Construct the same feasible (p', f ' ) as constructed in Case 1. Since E l ~ i fi~ = ~/k, we must have that fi~' < fi~. Therefore,
u~(p)
+cD(f) -u~(p) <0.
(17)
Next we establish that for locations l, j with fi k > 0 and fikj > 0, the net gains for the (i, j) pair and the (i, I) pair must be equal for this class, i.e.,
uki(p) +c~.(f) - u~(p) = uki(p) + c ~ ( f ) - u~(p).
(18)
Let a = uki(p) + c~(f) -- uf(p), b = uki(p) + C~.(f) -- Utf(p). If a, b were not equal, say a < b < O, then construct a feasible pattern (p', f ' ) such that
fsh,'=f~
forallhek,
sei
forallt-es,
f~'=fi~
forallt-~l,j.
(19)
E (uf( p) + cg( f ) -- uk( p) ) " ( fif' -- fik ) = a( fik' -- fik ) -- b( fi~' -- fi~ ) ~ 0
(20)
We then have that
t
where f/k'+ f,~'< fi~ + fik; iik'> O, fi~'> O. Let e be such that 0 < e < f/~ and let fi k'= fi k + e, fi k'= fi~ - e. Then
a(fff--fik)+b(fik'--fik)=(a--b)e
<0.
(21)
But, (21) contradicts (20). Therefore, a must be equal to b. We denote this common value by -h~. (b) = 0. We now need to show that, for any j, such that fis* - O,
uki ( p) + c~.( f ) - u ~ ( p ) > -X~.
(22)
For this purpose, let a feasible flow f ' be defined such that
fh'=fh
forallh4:k,
s-~i, t ~ s ;
(23a)
fi~' = 0 for all 1 4:j,
(23b)
fi k" =~p and f k ' = 0 for all s :~ i, l 4=s.
(23c)
Then
E (uf(p) + 4(f)
l~i
'~ E I~F l
- .f(p))
(--•ki)(fik'--fik)"}-
E I~F 2
(u~(p) + c ~ ( f ) - u ~ ( p ) ) ' f ; ~ ' .
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We first observe that the first term on the right hand side of the above equality is equal to A ki p- ki . ;tkt:kr k-k Adding now ,-t~r2"iJil -AiPi, which by the above construction, is equal to zero, to the right hand side of the expression, we obtain:
~.,(Uki(p) + c k ( f ) --u~(p))" (fitk' --fit)k l~i
= X~fi/* + Y'~ (X~ + uki(p) + c~(f) - u*t(p))"fit'- AkiPik IEY2
= (A~ + uki(p) + c~(f) -- U~( p))~i ~ >__O.
(24)
Since ~ > 0, therefore,
u (p)
(25)
>_
Summarizing the above, we have
uki(p) + c ~ ( f )
=u](p)-A~, > u~(p)--a ki,
iffi~>O, i f f i ky = 0 ,
(26)
where >0,
if ~ f i g = p ~ ,
(27)
l~i
X~
0,
if E fi k < ffik, l~i
which holds for each class k. Thus, we have established that variational inequality (7) implies that equilibrium conditions (3) and (4) hold. The proof of the Theorem is complete. Existence of at least one solution to variational inequality (7) follows from the standard theory of variational inequalities (see Kinderlehrer and Stampacchia [12] Theorem 3.1) under the sole assumption of continuity of the utility and migration cost functions u and c, since the feasible convex set K is compact. Uniqueness of the population and migration flow pattern (p, f ) follows under the assumption that the utility and movement cost functions are strictly monotone, i.e.,
_[u(pl) _U(p2)]. [pX__p2] + [c(fX)__c(f2)]. [ f l _ f 2 ] > 0 for
all (pl,
f l ) , (p2, f 2 ) ~ K such that ( p ' ,
fl) :~ (p2, f2).
(28)
We now interpret monotonicity condition (28) in terms of the applications. Under reasonable economic situations, the monotonicity condition (28) can be verified. Essentially, we assume that the system is subject to congestion; hence, the utilities are decreasing with larger populations_, and the movement costs are increasing with larger migration flows. Furthermore, we assume that each utility function u~(p) depends mainly on the population p~, and each movement cost c~.(f) depends mainly on the flow fi~- Mathematically, the strict monotonicity condition will hold, for example, when - Vu and Vc are diagonally dominant.
3. The decomposition algorithm We now present the variational inequality decomposition algorithm for the multiclass human migration equilibrium problem. The decomposition algorithm is based crucially on the special structure of the underlying network (cf. Figure 1).
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In particular, we note that the feasible set K for variational inequality (7) can be expressed as the Cartesian product J
r = 1-I Kk
(29)
k=l
where K k - - { ( p e , f k ) [ p k = { p i k, i= l . . . . . n}, f k = { f i ~ , i= l , . . . , n , j = l . . . . . n, j #i}}, and satisfying (1) and (2). We can, hence, decompose the variational inequality governing the multielass migration network equilibrium problem into J simpler variational inequalities in lower dimensions. Each variational inequality in the decomposition corresponds to a particular class which, after linearizing, is equivalent to a quadratic programming problem and can be solved by the migration equilibration algorithm developed in [16]. That algorithm is a relaxation scheme and proceeds from location (node) to location (node), at each step computing the migratory flow out of the location exactly and in closed form. This can be accomplished because the special network structure of the problem lies in that each of the paths from an origin location to the n - 1 potential destination locations are disjoint. Equilibration schemes which exploit the same special network feature in other equilibrium problems have been applied to compute solutions to spatial price equilibrium problems ([7,9,17]), market disequilibrium problems ([19]), and constrained matrix problems ([18]). The statement of the decomposition algorithm by classes is as follows: Step O. Given an initial feasible solution (p0, f0), set t = 0 and k = 1. Step 1. Solve for (pk)t+l, (fk),+l in the following separable variational inequality: E (~/ik __ (p/k)'+ 1) . ( __ u/k (( p 1xt+l . . . . . i
( p k - 1 ) t+l , ( p ) k t . . . ..
.....
+
g~'-{fi,)
+
~C~-
1)t+l,.. "
(p')')
....
)'\ci&,f
'+
) 1,
, k t
(y),.
, (fk-1)'
(f,),,..., (f,),) \
..
>_0 (30)
for all qk >__0, gk>_o, such that ~,y~igk<~ k and q k = ~ k _ y~y#i(g~_gjki). If k < J , then let k := k + 1, and go to Step 1, else, go to Step 2. Step 2. Verify convergence. If equilibrium conditions (3) and (4) hold for a given tolerance parameter e, then stop; otherwise, let t := t + 1, and go to Step 1. We now state the global convergence proof for the above linearized decomposition algorithm by specializing a more general convergence theorem (see Proposition 5.8 in Section 3.5 in [3]). We also give sufficient conditions that guarantee the convergence. Let IAI( p, f) A(p,f)
(31)
=
.4,(p, f)
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A. Nagurney et aL / Human migration networks
where
~uf Opf ~u.~ ap.~
Ak(P, f ) =
(32)
ac,~ n2Xn 2
and (p, f ) is feasible. Then the iteration mapping T determined by A(p, f ) is defined as follows: Given (p, f ) feasible, we solve T(p, f ) = ((Tik(p), Ti~(p, f)): i = 1. . . . . n, j ~ i , k --- 1 , . . . , J ) in
E(q~-~k(p)) i
(
• -u/~(p)
-
)
o--2-,~(p)(r,~(p)-p,.*) "
(
• c~(f)+~i~(f).(~,~(p,f)
+ E E ( g ~ - Tq(p,
f))
i j~i
o
>__
for all (q/k), (g~.) feasible. Note that if we take for each k
(P,f) = ((pl)t+l,...,(pk-l)t+l(pk)t,...,(pj)t,
(fl) ,+1 ..... (fk-1) t+l, (fk)t,...,(fJ)t),
then T defines precisely the linearized Gauss-Seidel decomposition algorithm by classes. Now we are ready to state the following convergence theorem, which holds since T is a constrction (see Section 3.5 in [31). Theorem 2. Suppose that there exist symmetric positive definite matrices G k such that Ak(P, f ) - - G k is semi-positive definite for all feasible (p, f ) and that there exists an a ~ (0, 1) such that
G[ 1 -uf( p) + uf( q) + ~p~ ( q) " ( pf - q~ ),..., --ukn( p) + Uk,( q) + ~pk ( q) " ( pkn -- qkn)
.....
c~
ij( f )
- c~( ~ ) - ~-~, ( g ) . ( f,~ - g~ ) . . .
_<,~-max,(pf-qt ..... Pk-q,~,, ...,fi~-~/~,..-).~",
3~;j
n2 k
(33)
where II ° II k = ('TGk")1/2. Then the linearized Gauss-Seidel decomposition algorithm by classes converges to the unique solution of the variational inequality geometrically. In the case when - u , c are separable, i.e.,
= Ui (Pi),
ck'(f) = Cij( fij ),
(34)
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271
the semi-positive definiteness of Ak(P, f ) -- G k is equivalent to the strong monotonicity of ( - u g, c k) for each block k. In fact, if Ak(p, f ) - G k is semi-positive definite, then
E ( - - u k i ( p ) +Uki(q))'(pik--qik)+ i
~_, ( c ~ ( f ) - - C ~ ( g ) ) ' ( f i k ) - gkij) i,j4-i
= ~i
-~Pi k(~7) "(Pik-q('
+
>a ~_,(pik-qi~)z+ ~-,(fi~
gii
, i
~-" ~ikJ (~)'(fikj-gikj)2 i,j 4=i
(35)
i,j
i.e., ( - u ~, c k) is strongly monotone. The converse is clear from the above inequality. The norm inequality condition is actually a measure of linearity of - u and c. In particular, when - u, c are linear and separable, the inequality is automatically satisfied, since the left hand size is zero. Of course, the variational inequality can be solved for each class by the migration equilibration algorithm in this extremal case. A not too large perturbation from this case means not too strong interactions among classes and locations. Finally, under the same condition as stated in the above theorem, the corresponding linearized Jacobi decomposition algorithm also converges to the unique solution of varaiational inequality (7).
4. Numerical results
In this Section we consider multiclass migration network problems and we present numerical results for the decomposition algorithm described in Section 3. The algorithm was implemented in F O R T R A N and compiled using the FORTVS compiler, optimization level 3. The special-purpose migration equilibration outlined in [16] was used for the embedded quadratic programming problems. The system used was the IBM 3090J at the Cornell National Supercomputer Facility. All of the CPU times reported are exclusive of i n p u t / o u t p u t times, but include initialization times. The initial pattern for all the runs was set to (p0, f 0 ) = (~, 0). The convergence tolerance used was e = 0.01. We first considered examples with asymmetric and linear utility and movement cost functions, that is, the utility functions were of the form:
uki ( p) = -- ~.,a~/pJ + bik,
(36)
l,j
and the movement cost functions were of the form: c k ( f ) = E gik'lrsf/s -t'- h k. . I~TS
(37)
The data was generated randomly and uniformly in the ranges as as follows: -aR t ~ [1, 10], b/~ ~ [10, 100], kk ~ [0.1, 0.5], and h/k1~ [1, 5] for all i, j, k, with the diagonal terms generated so that strict diagonal giji; dominance of the respective Jacobians of the utility and movement cost functions held, thus, guaranteeing uniqueness of the equilibrium pattern (p, f ) . The number of cross-terms for the functions (36) and (37) was set at five. The initial population ~ was generated randomly and uniformly in the range [10, 30], for all i, k. In Table 1 we varied the number of locations from 10 through 50, in increments of 10, and fixed the number of classes at 5 and 10.
A. Nagurney et al. / Human migration networks
272
Table 1 Numerical results for the migration decomposition algorithm on linear multiclass migration networks Number of locations
Number of classes 5
I0
10 20 30 40 50
0.23 (3) 1.57 (4) 3.74 (4) 8.83 (5) 15.36 (4)
0.32 (3) 2.08 (3) 8.58 (3) 16.29 (4) 32.01 (4)
CPU time in seconds (# of iterations)
Table 2 Numerical results for the migration decomposition algorithm on nonlinear multiclass migration networks Number of locations
Number of classes 5
10
10 20 30 40 50
0.24 (4) 1.18 (4) 3.87 (4) 8.73 (4) 16.22 (5)
0.41 (3) 2.38 (4) 9.73 (4) 17.01 (5) 33.07 (4)
CPU time in seconds (# of iterations) In Table 2, we then considered multiclass migration problems with quadratic utility and movement cost functions where the utility functions were of the form:
uki ( p ) = - - a ~tlk [\ p.k'~ 2 -- ~S"a~'!p( + bik, I ] tJ J
(38)
l,j
and the movement cost functions were of the form:
c ~ . ( f ) - -- ")/ijij( k f i jk ) 2 "}- Egiklrsf/s + hi~.
(39)
l,rs
The linear cross-terms in (38) and (39) were generated in the same manner as for the preceding linear examples, as were the initial populations. The quadratic coefficient Y~ij was generated in the range [0.1, 0.5] × 10 -6 for all i, j, k. The numerical results for the quadratic examples are presented in Table 2. As can be seen from the two tables, the decomposition algorithm by classes required only several iterations for convergence. As expected, the problems with 10 classes required, typically, at least twice the CPU time for computation as did the problems with 5 classes. The nonlinear examples required, typically, more CPU time than the linear ones, although not substantially more so. Finally, we note that, although the decomposition algorithm by classes implemented here was a Gauss-Seidel serial algorithm, the Jacobi version converges under the same conditions as given in Theorem 2. Hence, the Jacobi analogue allows for implementation on parallel computers.
5. Summary and conclusions In this paper we introduced a new multiclass human migration network equilibrium model which explicitly incorporates the cost of movement associated with migrating between two locations. Heretofore, migration equilibrium problems in a general setting have emphasized the case where movement
A. Nagurney et al. / Human migration networks
273
costs have been assumed to be insignificant, despite the empirical evidence to the contrary. The equilibrium conditions given in [16] for the single class model were then generalized to the multiclass case and the variational inequality formulation of the problem derived. The equilibrium conditions were also contrasted with those governing traffic network equilibrium problems and spatial price equilibrium problems. Qualitative results of existence and uniqueness were then given using variational inequality theory. A linearization decomposition algorithm by classes was then proposed for the computation of the equilibrium pattern and conditions for convergence given. Finally, numerical results on multiclass migration network problems were presented for both linear and nonlinear examples. This research brings and yet another application in operations research within the domain of network equilibrium problems.
Acknowledgements This paper is dedicated to the memory of Stella Dafermos, who passed away on April 5, 1990. The first and third authors were her graduate students. The authors would like to thank the referees for their careful reading of the manuscript. The first author's research was supported by NSF Grant: RII-8800361 under the sponsorship of the NSF VPW program while the author was visiting the Transportation Systems Division and the Operations Research Center at MIT and by a University Faculty Fellowship from the University of Massachusetts, Amherst, while she was visiting the Sloan School at MIT. The hospitality and cordiality of the host institution are warmly appreciated. This research was conducted at the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering at Cornell University, which is funded in part by the National Science Foundation, New York State, and IBM Corporation.
References [1] Beckmann, M.J., McGuire, C.B., and Winsten, C.B., Studies in the Economics of Transportation, Yale University Press, New Haven, CT, 1956. [2] Bennett, D.G., and Gade, O., Geographic Perspective in Migration Research - A Bibliographic Survey, University of North Carolina at Chapel Hill, Department of Geography, Studies in Geography No. 12, 1979. [3] Bertsekas, D.P., and Tsitsiklis, J.N., Parallel and Distributed Computation, Prentice-Hall, N J, 1989. [4] Dafermos, S., "The general multimodal network equilibrium problem with elastic demand", Networks 12 (1982) 57-72. [5] Dafermos, S., "Isomorphic multielass spatial price and multimodal traffic network equilibrium models", Regional Science and Urban Economics 16 (1986) 197-209. [6] Dafermos, S., and Nagurney, A., "A network formulation of market equilibrium problems and variational inequalities", Operations Research Letters 3 (1984) 247-250. [7] Dafermos, S., and Nagurney, A., "Supply and demand equilibration algorithms for a class of market equilibrium problems", Transportation Science 23 (1989)118-124. [8] Dafermos, S., and Sparrow, F.T., "The traffic assignment problem for a general network", Journal of Research of the National Bureau of Standards, Section B 73 (1969) 91-118. [9] Eydeland, A., and Nagurney, A., "Progressive equilibration algorithms: the case of linear transaction costs", Computer Science in Economics and Management 2 (1989) 197-219. [10] Greenwood, M.J., "Research on internal migration in the United States: a survey", Journal of Economic Literature 13 (1975) 397-433. [11] Greenwood, M.J., "Human migration: theory, models, and empirical studies", Journal of Regional Science 25 (1985) 521-544. [12] Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities, Academic Press, New York, 1980. [13] Nagurney, A., "Competitive equilibrium problems, variational inequalities, and regional science", Journal of Regional Science 27 (1987) 503-513. [14] Nagurney, A., "Migration equilibrium and variational inequalities", Economics Letters 31 (1989) 109-112. [15] Nagurney, A., "The formulation and solution of large-scale multicommodity equilibrium problems over space and time", European Journal of Operational Research 42 (1989) 166-177.
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[16] Nagurney, A., "A network model of migration equilibrium with movement costs", Mathematical and Computer Modelling 13 (1990) 79-88. [17] Nagurney, A., and Kim, D.S., "Parallel and serial variational inequality decomposition algorithms for multicommodity market equilibrium problems", The International Journal of Supercomputer Applications 3 (1989) 34-59. [18] Nagurney, A., Kim, D.S., and Robinson, A.G., "Serial and parallel equilibration of large-scale constrained matrix problems with application to the social and economic sciences", The International Journal of Supercomputer Applications 4 (1990) 49-71. [19] Nagurney, A., and Zhao, L., "A network equilibrium formulation of market disequilibrium and variational inequalities", Networks(1991) 109-132. [20] Rohrlich, G.F., "Economic cost-benefit approach to migration", in: Migration and Social Welfare, National Association of Social Workers, Inc., Silver Spring, MD, 1970, 55-70. [21] Samuelson, P.A., "A spatial price equilibrium and linear programming", American Economic Review 42 (1952) 283-303. [22] Shaw, R.P., Migration Theory and Fact, a Review and Bibliography of Current Literature, Regional Science Research Institute, Philadelphia, PA, 1975. [23] Sjaastad, L.A., "The costs and returns of human migration", Journal of Political Economy, Vol nr. (1962) part 2, 80-93. [24] Takayama, T., and Judge, G.G., Spatial and Temporal Price and Allocation Models, North-Holland, Amsterdam, 1971. [25] Tobler, W.R., "A model of geographical movement", GeographicalAnalysis 13 (1981) 1-20. [26] Zhao, L., "Variational inequalities in general equilibrium: analysis and computation", Ph.D. Thesis, Division of Applied Mathematics, Brown Univeristy, Providence, RI, 1989.
European Journal of Operational Research59 (1992)262-274 North-Holland
Theory and Methodology
Human migration networks Anna Nagurney
School of Management, University of Massachusetts, Amherst, MA 01003, USA
Jie Pan Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA Lan Zhao Department of Mathematics, SUNI~, College at Old Westbury, Old Westbury, NY 11568-308Z USA Received May 1990; revised December 1990 Abstract: In this paper we develop a new multiclass human migration network equilibrium model which explicitly incorporates the cost of movement between locations. The equilibrium conditions reflect that no individual in the system in any given class has any incentive to alter his location. The equilibrium conditions are then formulated as a variational inequality problem and qualitative properties discussed. A decomposition algorithm which takes advantage of the special network structure of the problem is then proposed and numerical results presented. This research brings and yet another application in operations research into the realm of network equilibrium problems.
Keywords: Migration; networks; economic equilibrium; variational inequalities
1. Introduction Networks have been used as the foundation for the study of large-scale competitive systems in distinct application areas of operations research. Examples include the traffic network equilibrium problem and the spatial price equilibrium problem as well as Walrasian price equilibrium, market equilibrium problems with production, and disequilibrium problems (see, e.g., [1,4-6,8,13,15,19,26]). In the latter set of applications, the underlying network is abstract in that the nodes no longer correspond to locations in space. In this paper we focus on the study of human population movements, a topic which has, heretofore, been studied principally by social scientists, including economists, demographers, geographers, and sociologists. In particular, we introduce a general multiclass network model of human migration which explicitly incorporates the cost of movement between locations. In our framework the cost of movement reflects not only the cost of transportation as a proxy for distance, but also the "psychic" costs associated with dislocation. 0377-2217/92/$05.00 © 1992 - ElsevierSciencePublishers B.V. All rights reserved
A. Nagurney et al. / Human migration networks
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The importance of movement costs in migration decision-making is well documented in the literature from both theoretical and empirical perspectives (see, e.g., [2,20,22,23,25]). Economic research, however, has emphasized the development of equilibrium models in which the population is assumed to be perfectly mobile and the costs of migration insignificant. In such models, individuals a n d / o r households are assumed to choose a location so that utilities are equalized across the economy ([10,11,22]). Recently, Nagurney [14] introduced a multiclass migration equilibrium model in the absence of migration costs. That model could be formulated as a network equilibrium problem with fixed demands and a special structure with each class being represented by a single origin/destination pair network connected with disjoint links representing the locations. The equilibrium conditions were then formulated as a variational inequality problem. In Section 2 we introduce a multiclass network model of human migration with movement costs, state the equilibrium conditions, and then derive the variational inequality formulation of the problem. The equilibrium conditions reflect that no individual in the system in any given class has any incentive to alter his location since no positive net gain (gain in utility minus movement cost) is possible by his unilateral decision. These equilibrium conditions are a generalization of those derived for a single class migration equilibrium model with movement costs in [16], for, which, due to the assumptions made on the utility functions, an equivalent optimization formulation exists. In that model the utility associated with a given location could depend only upon the population at that location and the migration cost between locations was assumed to be fixed. In this new model the utility of a given location as perceived by a particular class, can, in general, depend upon the population of every class at every location. The cost of movement associated with members of a class migrating from a given origin to a destination can, in turn, depend upon, in general, the migration flows of every class between every location. However, we retain the assumption that there is no repeat or chain migration. For this general model, we also discuss the qualitative properties of existence and uniqueness. In Section 3 we then describe a variational inequality algorithm which decomposes the multiclass problem into single class problems which are then solved in serial fashion and establish conditions for convergence. The resulting single class problems are characterized by linear, separable utility and movement cost functions and can be solved by a migration equilibration algorithm developed in [16] which exploits the special network structure by computing the flow out of each location for a particular class exactly and in closed form. In Section 4 we then present some numerical results for the decomposition algorithm embedded with the equilibration scheme on large-scale migration network problems. This research brings and yet another equilibrium problem in operations research within the domain of network equilibrium problems.
2. A general multiclass network model of human migration equilibrium In this section we introduce a network model of human migration equilibrium which allows for multiple classes as well as movement costs between locations. We assume a closed economy in which there are n locations, typically, denoted by i, and J classes, typically denoted by k. We further assume that the attractiveness of any location i as perceived by class k is represented by a utility u~. We let ff/k denote the initial fixed population of class k in location i, and we let pk denote the population of class k in location i. We group the utilities into a row vector u ~ R Jn and the populations into a column vector p ~ EJn. We assume no births and no deaths in the economy. We associate with each class k and each pair of locations i, j a nonnegative cost of movement c~ and we let the migration flow of class k from origin i to destination j be denoted by fi~. The movement costs are grouped into a row vector c ~ EJn~-~) and the flows into a column vector f ~ ~ J ~ - 1). We assume that the movement costs reflect not only the cost of physical movement but also the personal and psychic cost as perceived by a class in moving between locations.
264
A. N a g u r n e y et al. / H u m a n migration n e t w o r k s J
c
J J
12
u
1
o °
, °
~ J c
o
"c12,
,Jo
,
c
24
1 c 31
1
1 c
3
1 u
43
4
F i g u r e 1. T h e m u l t i c l a s s m i g r a t i o n n e t w o r k w i t h 4 l o c a t i o n s
The conservation of flow equations are given for each class k and each location i, assuming no repeat or chain migration, by P~ =P~ + E f t ~ 14=i
Y'~f~,
(1)
14=i
and E f ~
(2)
l~i
where fi~ >- 0 for all k = 1. . . . , J, 1 4~ i. Equation (1) states that the population at location i of class k is given by the initial population of class k at location i plus the migration flow into i of that class minus the migration flow out of i for that class. Equation (2) states that the flow out of i by class k cannot exceed the initial population of class k at i, since no chain migration is allowed. We now present the multiclass network model. As in the single class model outlined in Nagurney [16], we construct n nodes, i = 1. . . . . n, to represent the locations and a link (i, j ) connecting each pair of nodes. There are, hence, n nodes in the network and n ( n - 1) links. With each link (i, j ) we then associate k costs c~. and corresponding flows fi~. With each node i we associate k utilities u/k and the initial positive populations ~fi~. A graphical depiction of a four location migration network is given in Figure 1, where the k classes are layered. Of course, rather than a multiclass network, one can construct J copies of the network topology given in Figure 1 to represent the classes where the costs on the links and the utilities are defined accordingly. Note that in cases where the migration between locations is forbidden, due, for example, to government regulations, then those links can be removed from the network model, or in the case of class restrictions, the costs of the respective classes set sufficiently high. We are now ready to state the equilibrium conditions. We assume that migrants are rational and that migration will continue until no individual has any incentive to move since a unilateral decision will no longer yield a positive net gain (gain in utility minus
A. Nagurney et al. / Human migration networks
265
movement cost). Mathematically, the multiclass equilibrium conditions are: For each class k, k = 1. . . . . J, and each pair of location i, j, i = 1 . . . . . n, j 4: i, we have that:
u~+c
~[ = u f - A ~ ,
if fij>O, k
>u~-A~,
iffi~=O,
(3)
and
A
14:i
= 0,
if
~-.fi~
(4)
l~i
subject to constraints (1) and (2). Equilibrium condition (3), although similar in structure to the equilibrium conditions governing the multicommodity spatial price equilibrium problem (cf. [5,15,17,24]), differs in that the indicator A~ is present. We now interpret the necessity of A~. Observe that, unlike spatial price equilibrium problems (or the related traffic network equilibrium problem with elastic demand; see, e.g., [4,13]), the level of the population/3~ may not be large enough so that the gain in utility u~ - u/k is exactly equal to the cost of movement c~.. Nevertheless, the utility gain minus the movement cost will be maximal and nonnegative. Moreover, the net gain will be equalized for all locations and classes which have a postive flow out of a location. In fact, from the equivalence proof of Theorem 1 given below, A~ is exactly the equalized net gain for all individuals of class k of location i. We first discuss the function structure and then derive the variational inequality formulation of the equilibrium conditions (3) and (4). We assume that, in general, the utility associated with a particular location and class can depend upon the population associated with every class and every location, i.e., we assume that u = u(p).
(5)
We also assume that, in general, the cost associated with migrating between two locations as perceived by a particular class can depend, in general, upon the flows of every class between every pair of locations, i.e., we assume that c = c(f).
(6)
We recall that in [16], a single class of migrant was assumed, and the utility functions (5) were assumed to be separable and linear, whereas the migration costs were assumed to be fixed. That model had an equivalent optimization formulation of the equilibrium conditions, which was also the approach utilized in the classical models of Samuelson [21] and Takayama and Judge [24] in spatial price equilibrium and Beckmann, McGuire, and Winsten [1] in traffic network equilibrium. The variational inequality formulation of the migration equilibrium conditions is given by: T h e o r e m 1. A population and migration flow pattern (p, f ) satisfies equilibrium conditions (3) and (4) subject to constraints (1) and (2), if and only if it satisfies the variational inequality problem
-u(p).(p'-p)+c(f).(f'-f)>O
forall ( p ' , f ' ) ~ K
(7)
where K - {(p', f ' ) l f ' > O, (p', f ' ) satisfy (1), (2)}. Proof. We first show that if a pattern (p, f ) satisfies equilibrium conditions (3) and (4), subject to constraints (1) and (2), then it also satisfies the variational inequality in (7). Suppose that (p, f ) satisfies the equilibrium conditions. Then fi~ > O, and E~ + i fi~ < Pf, for i, j and k.
266
A. Nagurney et aL / Human migration networks
For fixed class k we define F~ = {llfi~ > 0} and Fzk = {llfi~ = 0}. Then
E [u/k(P) ff'ck(f) -- u k ( p ) ] " [fi f f t - f i if] l~i
= E
Jil --.tit l
I~F~
+ ~ [u/k(P)+c~(f)-u/k(P)]'[fi~'-f~] t~r~ ~-~ --*ki E
=-Ai
(lift--fir) "}- E (--*k)(fif')
Eli,"l*i
fi k
(8)
~ O, if E f , Y=P/*, l~i
holds for all such locations i. Therefore, for this class k and all locations i, we have that fi*t > O, Et.if~f < , ~ , and
Z E
i l~i
+ c (f)
Jit
(9)
Jit I >- O.
But (9) holds for each k; hence,
E E E [uk(p) "}-ck(f)--uk(p)] k i l~i
" [fikt--fi k] ~ 0 .
(10)
Observe now that (10) can be rewritten as
E Ellk(P)'((j~lflkt-k 1
Efjftl--(j~lf, j4=t /
k- EL'f)) jstl
+ E E Eek(f)'(fif'--fif) k i 14=i
~0. (11)
Using now constraint (1), and substituting it into (11), we conclude that
- E Y'~U/k(P)'(P~'--Pk) + E Y'~ E c ~ ( f ) ' ( f i k ' - - f i ~ ) k I k i I~i
>0,
(12)
or, equivalently, in vector notation:
-u(p).(p'-p)+c(f).(f'-f)>O
for a l l ( p ' , f ' ) ~ K .
(13)
We now show that if a pattern (p, f ) ~ K satisfies variational inequality (7), then it also satisfies equilibrium conditions (3) glad (4). Suppose that (p, f ) ~l[|~fies variational inequality (7). We shall prove that (p, f ) satisfies equilibrium conditions (3) and (4) by eonsidering two different cases; the first, locations i where the population is not exhausted, and the second, for locations where the respective populations are exhausted. We fix the class k.
Case 1. ~-ls~'ifif < pik. Construct a feasible (p', f ' ) such that fih.'=fh J JtJ
for a l l i , j , j ~ i
and h4=k.
For h = k, let
fikt -fu, __ k
ifl4=j,
ftikt =fly,k
ifl4=i.
(14)
A. Nagurney et aL / Human migration networks
267
Then variational inequality (7) reduces to
iuf(p) +c~(f) --u~(p))'(fiks'--fi~) >0.
(15)
Since Y"t,;fi~
uki(p) + c ~ . ( f ) - u ~ ( p ) = O ,
iff/~>0,
uk(p)+ck(f)--u~(p)>O,
iffi~=0.
(16)
In other words, h;k _- 0, and, hence, (3) holds.
Case 2. Et.ifi~ =pi g (a) fi~ > O. Construct the same feasible (p', f ' ) as constructed in Case 1. Since E l ~ i fi~ = ~/k, we must have that fi~' < fi~. Therefore,
u~(p)
+cD(f) -u~(p) <0.
(17)
Next we establish that for locations l, j with fi k > 0 and fikj > 0, the net gains for the (i, j) pair and the (i, I) pair must be equal for this class, i.e.,
uki(p) +c~.(f) - u~(p) = uki(p) + c ~ ( f ) - u~(p).
(18)
Let a = uki(p) + c~(f) -- uf(p), b = uki(p) + C~.(f) -- Utf(p). If a, b were not equal, say a < b < O, then construct a feasible pattern (p', f ' ) such that
fsh,'=f~
forallhek,
sei
forallt-es,
f~'=fi~
forallt-~l,j.
(19)
E (uf( p) + cg( f ) -- uk( p) ) " ( fif' -- fik ) = a( fik' -- fik ) -- b( fi~' -- fi~ ) ~ 0
(20)
We then have that
t
where f/k'+ f,~'< fi~ + fik; iik'> O, fi~'> O. Let e be such that 0 < e < f/~ and let fi k'= fi k + e, fi k'= fi~ - e. Then
a(fff--fik)+b(fik'--fik)=(a--b)e
<0.
(21)
But, (21) contradicts (20). Therefore, a must be equal to b. We denote this common value by -h~. (b) = 0. We now need to show that, for any j, such that fis* - O,
uki ( p) + c~.( f ) - u ~ ( p ) > -X~.
(22)
For this purpose, let a feasible flow f ' be defined such that
fh'=fh
forallh4:k,
s-~i, t ~ s ;
(23a)
fi~' = 0 for all 1 4:j,
(23b)
fi k" =~p and f k ' = 0 for all s :~ i, l 4=s.
(23c)
Then
E (uf(p) + 4(f)
l~i
'~ E I~F l
- .f(p))
(--•ki)(fik'--fik)"}-
E I~F 2
(u~(p) + c ~ ( f ) - u ~ ( p ) ) ' f ; ~ ' .
268
A. Nagurney et al. / Human migration networks
We first observe that the first term on the right hand side of the above equality is equal to A ki p- ki . ;tkt:kr k-k Adding now ,-t~r2"iJil -AiPi, which by the above construction, is equal to zero, to the right hand side of the expression, we obtain:
~.,(Uki(p) + c k ( f ) --u~(p))" (fitk' --fit)k l~i
= X~fi/* + Y'~ (X~ + uki(p) + c~(f) - u*t(p))"fit'- AkiPik IEY2
= (A~ + uki(p) + c~(f) -- U~( p))~i ~ >__O.
(24)
Since ~ > 0, therefore,
u (p)
(25)
>_
Summarizing the above, we have
uki(p) + c ~ ( f )
=u](p)-A~, > u~(p)--a ki,
iffi~>O, i f f i ky = 0 ,
(26)
where >0,
if ~ f i g = p ~ ,
(27)
l~i
X~
0,
if E fi k < ffik, l~i
which holds for each class k. Thus, we have established that variational inequality (7) implies that equilibrium conditions (3) and (4) hold. The proof of the Theorem is complete. Existence of at least one solution to variational inequality (7) follows from the standard theory of variational inequalities (see Kinderlehrer and Stampacchia [12] Theorem 3.1) under the sole assumption of continuity of the utility and migration cost functions u and c, since the feasible convex set K is compact. Uniqueness of the population and migration flow pattern (p, f ) follows under the assumption that the utility and movement cost functions are strictly monotone, i.e.,
_[u(pl) _U(p2)]. [pX__p2] + [c(fX)__c(f2)]. [ f l _ f 2 ] > 0 for
all (pl,
f l ) , (p2, f 2 ) ~ K such that ( p ' ,
fl) :~ (p2, f2).
(28)
We now interpret monotonicity condition (28) in terms of the applications. Under reasonable economic situations, the monotonicity condition (28) can be verified. Essentially, we assume that the system is subject to congestion; hence, the utilities are decreasing with larger populations_, and the movement costs are increasing with larger migration flows. Furthermore, we assume that each utility function u~(p) depends mainly on the population p~, and each movement cost c~.(f) depends mainly on the flow fi~- Mathematically, the strict monotonicity condition will hold, for example, when - Vu and Vc are diagonally dominant.
3. The decomposition algorithm We now present the variational inequality decomposition algorithm for the multiclass human migration equilibrium problem. The decomposition algorithm is based crucially on the special structure of the underlying network (cf. Figure 1).
269
A. Nagurney et al. / Human migration networks
In particular, we note that the feasible set K for variational inequality (7) can be expressed as the Cartesian product J
r = 1-I Kk
(29)
k=l
where K k - - { ( p e , f k ) [ p k = { p i k, i= l . . . . . n}, f k = { f i ~ , i= l , . . . , n , j = l . . . . . n, j #i}}, and satisfying (1) and (2). We can, hence, decompose the variational inequality governing the multielass migration network equilibrium problem into J simpler variational inequalities in lower dimensions. Each variational inequality in the decomposition corresponds to a particular class which, after linearizing, is equivalent to a quadratic programming problem and can be solved by the migration equilibration algorithm developed in [16]. That algorithm is a relaxation scheme and proceeds from location (node) to location (node), at each step computing the migratory flow out of the location exactly and in closed form. This can be accomplished because the special network structure of the problem lies in that each of the paths from an origin location to the n - 1 potential destination locations are disjoint. Equilibration schemes which exploit the same special network feature in other equilibrium problems have been applied to compute solutions to spatial price equilibrium problems ([7,9,17]), market disequilibrium problems ([19]), and constrained matrix problems ([18]). The statement of the decomposition algorithm by classes is as follows: Step O. Given an initial feasible solution (p0, f0), set t = 0 and k = 1. Step 1. Solve for (pk)t+l, (fk),+l in the following separable variational inequality: E (~/ik __ (p/k)'+ 1) . ( __ u/k (( p 1xt+l . . . . . i
( p k - 1 ) t+l , ( p ) k t . . . ..
.....
+
g~'-{fi,)
+
~C~-
1)t+l,.. "
(p')')
....
)'\ci&,f
'+
) 1,
, k t
(y),.
, (fk-1)'
(f,),,..., (f,),) \
..
>_0 (30)
for all qk >__0, gk>_o, such that ~,y~igk<~ k and q k = ~ k _ y~y#i(g~_gjki). If k < J , then let k := k + 1, and go to Step 1, else, go to Step 2. Step 2. Verify convergence. If equilibrium conditions (3) and (4) hold for a given tolerance parameter e, then stop; otherwise, let t := t + 1, and go to Step 1. We now state the global convergence proof for the above linearized decomposition algorithm by specializing a more general convergence theorem (see Proposition 5.8 in Section 3.5 in [3]). We also give sufficient conditions that guarantee the convergence. Let IAI( p, f) A(p,f)
(31)
=
.4,(p, f)
270
A. Nagurney et aL / Human migration networks
where
~uf Opf ~u.~ ap.~
Ak(P, f ) =
(32)
ac,~ n2Xn 2
and (p, f ) is feasible. Then the iteration mapping T determined by A(p, f ) is defined as follows: Given (p, f ) feasible, we solve T(p, f ) = ((Tik(p), Ti~(p, f)): i = 1. . . . . n, j ~ i , k --- 1 , . . . , J ) in
E(q~-~k(p)) i
(
• -u/~(p)
-
)
o--2-,~(p)(r,~(p)-p,.*) "
(
• c~(f)+~i~(f).(~,~(p,f)
+ E E ( g ~ - Tq(p,
f))
i j~i
o
>__
for all (q/k), (g~.) feasible. Note that if we take for each k
(P,f) = ((pl)t+l,...,(pk-l)t+l(pk)t,...,(pj)t,
(fl) ,+1 ..... (fk-1) t+l, (fk)t,...,(fJ)t),
then T defines precisely the linearized Gauss-Seidel decomposition algorithm by classes. Now we are ready to state the following convergence theorem, which holds since T is a constrction (see Section 3.5 in [31). Theorem 2. Suppose that there exist symmetric positive definite matrices G k such that Ak(P, f ) - - G k is semi-positive definite for all feasible (p, f ) and that there exists an a ~ (0, 1) such that
G[ 1 -uf( p) + uf( q) + ~p~ ( q) " ( pf - q~ ),..., --ukn( p) + Uk,( q) + ~pk ( q) " ( pkn -- qkn)
.....
c~
ij( f )
- c~( ~ ) - ~-~, ( g ) . ( f,~ - g~ ) . . .
_<,~-max,(pf-qt ..... Pk-q,~,, ...,fi~-~/~,..-).~",
3~;j
n2 k
(33)
where II ° II k = ('TGk")1/2. Then the linearized Gauss-Seidel decomposition algorithm by classes converges to the unique solution of the variational inequality geometrically. In the case when - u , c are separable, i.e.,
= Ui (Pi),
ck'(f) = Cij( fij ),
(34)
A. Nagurney et al. / Human migration networks
271
the semi-positive definiteness of Ak(P, f ) -- G k is equivalent to the strong monotonicity of ( - u g, c k) for each block k. In fact, if Ak(p, f ) - G k is semi-positive definite, then
E ( - - u k i ( p ) +Uki(q))'(pik--qik)+ i
~_, ( c ~ ( f ) - - C ~ ( g ) ) ' ( f i k ) - gkij) i,j4-i
= ~i
-~Pi k(~7) "(Pik-q('
+
>a ~_,(pik-qi~)z+ ~-,(fi~
gii
, i
~-" ~ikJ (~)'(fikj-gikj)2 i,j 4=i
(35)
i,j
i.e., ( - u ~, c k) is strongly monotone. The converse is clear from the above inequality. The norm inequality condition is actually a measure of linearity of - u and c. In particular, when - u, c are linear and separable, the inequality is automatically satisfied, since the left hand size is zero. Of course, the variational inequality can be solved for each class by the migration equilibration algorithm in this extremal case. A not too large perturbation from this case means not too strong interactions among classes and locations. Finally, under the same condition as stated in the above theorem, the corresponding linearized Jacobi decomposition algorithm also converges to the unique solution of varaiational inequality (7).
4. Numerical results
In this Section we consider multiclass migration network problems and we present numerical results for the decomposition algorithm described in Section 3. The algorithm was implemented in F O R T R A N and compiled using the FORTVS compiler, optimization level 3. The special-purpose migration equilibration outlined in [16] was used for the embedded quadratic programming problems. The system used was the IBM 3090J at the Cornell National Supercomputer Facility. All of the CPU times reported are exclusive of i n p u t / o u t p u t times, but include initialization times. The initial pattern for all the runs was set to (p0, f 0 ) = (~, 0). The convergence tolerance used was e = 0.01. We first considered examples with asymmetric and linear utility and movement cost functions, that is, the utility functions were of the form:
uki ( p) = -- ~.,a~/pJ + bik,
(36)
l,j
and the movement cost functions were of the form: c k ( f ) = E gik'lrsf/s -t'- h k. . I~TS
(37)
The data was generated randomly and uniformly in the ranges as as follows: -aR t ~ [1, 10], b/~ ~ [10, 100], kk ~ [0.1, 0.5], and h/k1~ [1, 5] for all i, j, k, with the diagonal terms generated so that strict diagonal giji; dominance of the respective Jacobians of the utility and movement cost functions held, thus, guaranteeing uniqueness of the equilibrium pattern (p, f ) . The number of cross-terms for the functions (36) and (37) was set at five. The initial population ~ was generated randomly and uniformly in the range [10, 30], for all i, k. In Table 1 we varied the number of locations from 10 through 50, in increments of 10, and fixed the number of classes at 5 and 10.
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Table 1 Numerical results for the migration decomposition algorithm on linear multiclass migration networks Number of locations
Number of classes 5
I0
10 20 30 40 50
0.23 (3) 1.57 (4) 3.74 (4) 8.83 (5) 15.36 (4)
0.32 (3) 2.08 (3) 8.58 (3) 16.29 (4) 32.01 (4)
CPU time in seconds (# of iterations)
Table 2 Numerical results for the migration decomposition algorithm on nonlinear multiclass migration networks Number of locations
Number of classes 5
10
10 20 30 40 50
0.24 (4) 1.18 (4) 3.87 (4) 8.73 (4) 16.22 (5)
0.41 (3) 2.38 (4) 9.73 (4) 17.01 (5) 33.07 (4)
CPU time in seconds (# of iterations) In Table 2, we then considered multiclass migration problems with quadratic utility and movement cost functions where the utility functions were of the form:
uki ( p ) = - - a ~tlk [\ p.k'~ 2 -- ~S"a~'!p( + bik, I ] tJ J
(38)
l,j
and the movement cost functions were of the form:
c ~ . ( f ) - -- ")/ijij( k f i jk ) 2 "}- Egiklrsf/s + hi~.
(39)
l,rs
The linear cross-terms in (38) and (39) were generated in the same manner as for the preceding linear examples, as were the initial populations. The quadratic coefficient Y~ij was generated in the range [0.1, 0.5] × 10 -6 for all i, j, k. The numerical results for the quadratic examples are presented in Table 2. As can be seen from the two tables, the decomposition algorithm by classes required only several iterations for convergence. As expected, the problems with 10 classes required, typically, at least twice the CPU time for computation as did the problems with 5 classes. The nonlinear examples required, typically, more CPU time than the linear ones, although not substantially more so. Finally, we note that, although the decomposition algorithm by classes implemented here was a Gauss-Seidel serial algorithm, the Jacobi version converges under the same conditions as given in Theorem 2. Hence, the Jacobi analogue allows for implementation on parallel computers.
5. Summary and conclusions In this paper we introduced a new multiclass human migration network equilibrium model which explicitly incorporates the cost of movement associated with migrating between two locations. Heretofore, migration equilibrium problems in a general setting have emphasized the case where movement
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costs have been assumed to be insignificant, despite the empirical evidence to the contrary. The equilibrium conditions given in [16] for the single class model were then generalized to the multiclass case and the variational inequality formulation of the problem derived. The equilibrium conditions were also contrasted with those governing traffic network equilibrium problems and spatial price equilibrium problems. Qualitative results of existence and uniqueness were then given using variational inequality theory. A linearization decomposition algorithm by classes was then proposed for the computation of the equilibrium pattern and conditions for convergence given. Finally, numerical results on multiclass migration network problems were presented for both linear and nonlinear examples. This research brings and yet another application in operations research within the domain of network equilibrium problems.
Acknowledgements This paper is dedicated to the memory of Stella Dafermos, who passed away on April 5, 1990. The first and third authors were her graduate students. The authors would like to thank the referees for their careful reading of the manuscript. The first author's research was supported by NSF Grant: RII-8800361 under the sponsorship of the NSF VPW program while the author was visiting the Transportation Systems Division and the Operations Research Center at MIT and by a University Faculty Fellowship from the University of Massachusetts, Amherst, while she was visiting the Sloan School at MIT. The hospitality and cordiality of the host institution are warmly appreciated. This research was conducted at the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering at Cornell University, which is funded in part by the National Science Foundation, New York State, and IBM Corporation.
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