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The spatial price equilibrium problem with path variables
Socio-hon. Phn. Printed
in Great
Sci. Vol. 20, No. 5, pp. 299-310,
0038-0121/86
1986
THE SPATIAL PRICE EQUILIBRIUM WITH PATH VARIABLES PATRICK Department
$3.00 + 0.00
Pergamon Journals Ltd
Britain
of Decision
Sciences,
PROBLEM
T. HARKERt
The Wharton School, University PA 19104, U.S.A.
of Pennsylvania,
Philadelphia,
(Received 21 November 1985; in revised form 24 February 1986)
Abstract-Most
of the recent literature dealing with the solution of the spatial price equilibrium model has dealt with the arc variable formulation; that is, the situation where the flows on the paths connecting an origindestination pair in a network with trans-shipment nodes are not explicitly considered. This paper presents a clarification of the main assumption underlying this arc formulation and presents an argument for the need for using a mode1 with path variables in certain applications. After this discussion, a new solution algorithm for the model with path variables is stated and compared against the linear approximation algorithm first proposed by Florian and Los [I] for this problem. Numerical examples are presented which illustrate the features of this new algorithm.
1. INTRODUCTION The equilibrium of the regional prices of commodities and the resulting regional supplies and demands and inter-regional flows has been at the heart of many freight movement studies. Ever since Samuelson’s [2] mathematical programming formulation of the spatial price equilibrium concept [which some attribute to Cournot [3] in the 19th century, but may in fact have been stated even earlier], a vast body of literature has been devoted to extensions of the original theoretical framework, alternative mathematical formulations, solution algorithms, and empirical applications. Takayama and Judge [4,5] extended Samuelson’s model to multiple commodities, MacKinnon [6,7] put the problem into the form of a fixed point problem, Florian and Los [l] considered nonlinear transportation costs, and Friesz, Tobin, Smith and Harker [8] formulated the spatial price equilibrium concept as a nonlinear complementarity problem. Smith [9] has recently used this complementarity formulation to present a general proof of existence of a spatial price equilibrium. Other theoretical extensions include the work by Thore [lo, 111, the partial equilibrium model using fuzzy set theory by Ponsard [12], Puu’s [13] model of equilibrium over a continuous plane, Sheppard and Curry’s [14] critique of the assumptions underlying the spatial price equilibrium concept, especially the omission of local spatial monopoly considerations, and Harker’s [ 151 extension to include profit-maximizing transportation firms. Algorithmic work includes the papers by Friesz et al. [8] and MacKinnon [6, 71, as well as the work bv Rowse 1161. Asmuth. Eaves and Peterson [17], Pang [18, 19j, &ng and Lee [20], Pang and Chan 1211. Jones et al. 122. 231. Takavama and Uri [24], aid Friesz et al. [25j. Also:‘Irwin Bnd Yang [26], Dafermos and Nagurney [27] and Chao and Friesz [28] have been studying the application of tFormer1y at the Department of Geography, California at Santa Barbara.
University
of 299
sensitivity analysis to this class of models. Empirical applications of this concept abound; the book by Takayama and Judge [29] and the recent paper by Harker [30] chronicle many of these studies. The purpose of this paper is 2-fold. First, in formulations of the spatial price equilibrium concept which incorporate trans-shipment nodes in the network (nodes at which no traffic is originated or terminated; that is, nodes for which there are no supply or demand functions defined), there are two general categories of models and associated algorithms. In the first category, the models are stated with path variables; that is, paths between each origin and destination must be explicitly considered. In the second category, the models are stated only in terms of arc flows. This paper will explicitly state the assumption which underlies the path variable formulations and critique this assumption. The next section will state the various mathematical formulations of the spatial price equilibrium concept and addresses this first issue. The second purpose of this paper is to present an Evans [31] type algorithm for the spatial price equilibrium model with path variables which overcomes some of the difficulties associated with applying a linear approximation or Frank-Wolfe type algorithm to this problem, as several authors have suggested or done (Florian and Los [l], Harker [15]). Section 3 presents the algorithm along with a statement of the linear approximation algorithm. Numerical tests on both small- and large-scale examples are used to illustrate the properties of each algorithm presented in Section 4, and the conclusions from this study are stated in Section 5. 2. THE SPATIAL
PRICE EQUILIBRIUM PROBLEM
Consider a network G[N, A] composed of a set of N of nodes and a set A of arcs, where N and A denote the cardinality of the respective sets. Let L E N be the set of L centroids of the regions comprising the study area. Multiple commodities are treated in this frame-
300
PATRICKT. HARKER
work through the use of Aashtiani’s [32] multi-copy network concept in which each element of the network is associated with both a physical location and a specific commodity. Thus, a separate network is defined for each commodity. For each centroid, 1EL, let us define a price 7t, and the vector of prices L = (K,, ) 7c,, , q)‘, where t denotes the transpose operation. Let us define a supply function S,(n) and demand function D,(rr) for each IsL, and S=(S ,,..., S ,,..., S)‘and D=(D ,,..., D ,,..., D)‘. Also, let us define W = {w = (ij); isL, jsL} to be the set of origin-destination (O-D) pairs, T,%.to wcw, be the flow between O-D pair , T,-)‘, and u,. to be the minimum T = (T,, . . . ,T,., cost to transport a unit of a commodity between O-D pair w EW. Finally, let us define fi, to be the set of GD pairs which enter region I EL including the O-D pair w = (II) and define E, to be the set of O-D pairs which leave region 1eL including w = (II); that is ii,={w
=(U)eW:
i=l}
E,=(w
=(ij)cW:
i=l}.
c=(c,,. .,c,,. .,CA)‘, h, = the flow on path p E P, h=(h ,,..., h,,. ..,h,#)‘, c, = the average transportation path p E P, and s
= OY,
Using the above notation, can be stated:
D,(Z) = 1 T,. WEI?, (ii) economic
VlcL
(1)
V/EL
(2)
rationality
if T, > 0, if JT,+ uy > x,,
then n,+ui,=n, then T,, = 0
V(ij)sW V (ij) EW.
Conditions (i) and (ii) were originally a mathematical program by defining
formulated
Y,(S) = the inverse of the supply function ICL,
as
at region
and O,(D) = the inverse of the demand region 1EL.
function
at
Later, conditions (i) and (ii) were treated directly by formulating them as a complementarity problem. When trans-shipment nodes are added, L becomes a proper subset of N and the set of arcs A no longer equals W. The trans-shipment nodes represent major point of interest from a transportation point of view, such as yards where railroads interline, ports, motor carrier terminals, etc. That is, no traffic is generated or terminated at these points, but there is some activity worthy of study occurring at these locations. For this case, let us define P = the set of P paths in the network, p = an index of paths, p E P, P,. = the set of paths between O-D pair w E W, f, = the flow on arc a E A, f=(f I)..., f,,. .,fA)‘, c,(f) = the average cost per unit of transporting a commodity on arc a E A.
arc
a E
the following
f,=
1 6,h, PEP
cp=
1 &G”(f) atA
T, = c h, PEP,
A
relationships
VaEA,
(3)
VP 6 P,
(4)
V w E W,
(5)
and
x
{cp}.
(6)
In this case, equilibrium conditions (i) and (ii) are expanded to include a Wardropian User Equilibrium (Wardrop [33], Fernandez and Friesz [34]) on all utilized paths between every O-D pair (see Friesz et al. [8] and Florian and Los [l]): (iii)
c T, MB,
1, if path p E P traverses 0 otherwise
u,,, = $n
The original Samuelson/Takayama-Judge model assumed that every node was a regional centroid (L = N) and that there were no trans-shipment nodes, so that A = W. In this case, the equilibrium conditions for this model can be written as: (i) flow conservation S,(Z)=
i
cost per unit on
then c, = u,,. V w E W, p E P,,
if h, > 0,
if c, > u,,., then h, = 0
V M’E W, p E P,(
Conditions (it(iii) can be written as either a complementarity problem (Friesz et al. [8], Harker [l5]) or a variational inequality problem (Florian and Los [l], Harker [15]) which leads to an equivalent optimization formulation under certain assumptions (namely, integrability of the inverse supply and demand functions and the arc cost functions). The introduction of the path variables in condition (iii) causes difficulties in the solution of this problem, as anyone familiar with the standard traffic assignment problem (Fernandez and Friesz [34]) understands; Florian and Los [I] highlight a few of these difficulties. Recently Tobin and Friesz [35] and Friesz et al. [8] have been able to overcome the use of path variables by noting that each trans-shipment node, while not having a supply or demand function defined on it, still has an implicit price which reaches an equilibrium value. Thus, conditions (ii) and (iii) have been rewritten as (iii)’
if f, > 0, Va
=
then r-r,+ c,,(f) = rr,
(i,j)eA;
if 7~,+ c,(f) > n,, Va
=
(i,j)~A;
i,i E N then f, = 0 i,j E N.
This arc variable formulation is at the heart of the development of efficient solution procedures, such as those by Pang and Lee [20] and Jones et al. [22,23]. The efficient solution procedures developed for the arc variable formulation brings to question the necessity of the path variable formulation, for if the two formulations are equivalent, there is no need to deal with paths. Unfortunately, the arc variable formu-
301
The spatial price equilibrium problem lation makes a rather strong assumption which may not hold in general. In the path variable formulation, we are assured that a movement from i toj will indeed go from i to j by the use of paths. In the arc variable formulation, however, each trans-shipment node is considered a “market” where incoming shipments to the node are “auctioned” off to the neighboring nodes. Thus, there is no guarantee that a shipment starting at region i will in fact arrive at region j; the only guarantee is that supplies and demand will be met in each region. For example, consider the network depicted in Fig. 3 and an O-D flow patterns of T (z ,) = To.,, = To,,, = 10. If only arc flow conservation were enforced, then a flow pattern of 20 units on arcs 3, 13 and 5 would satisfy a net flow of 20 units out of node 2 and 20 units into node I. Obviously, this arc flow pattern does not satisfy the @D flow pattern stated above. In order to guarantee that the C&D flow pattern is satisfied, the paths between each O-D pair must be explicitly incorporated into the model. For some commodities and/or networks with few trans-shipment nodes, the use of the arc formulation may not be a bad approximation given the high efficiency of the algorithms which have been developed for this problem. However, for commodities such as coal and for networks with many transshipment nodes, the use of paths is imperative. Thus, the efficiencies gained from the arc variable formulation are achieved at the cost of making a rather strong assumption. Possibly due to ignorance of the above fact, there has been relatively little work done to develop ejficient algorithms for the path variable formulation which can be used for large-scale problems. Florian and Los [l] suggest the use of a linear approximation (or Frank-Wolfe) algorithm, and Harker [15] has implemented this algorithm for his generalization of the spatial price equilibrium concept on a large-scale problem (960 nodes and 6993 arcs for the pure spatial price equilibrium portion of the problem). Section 3.1 will detail the application of the linear approximation method and its associated difficulties. Due to these difficulties, a new algorithm in the spirit of Evans’ [3 I] algorithm for urban traffic distribution and assignment will be developed and tested. However, before beginning our discussion of solution algorithms, we must first state the equilibrium conditions (it(iii) in a usable mathematical form. Let x = (S’, D’, T’, h’, f’)’ and define the feasible set for x as
and R” denotes n-dimensional Euclidean space, such that for the vector function F: R” + R” the following condition holds for all z E K: F(y)‘(-
- y) > 0.
THEOREM
I
x* = [(S*)‘, (D*)‘, (T*)‘, (h*)‘, (f*)]’ E fi satisfies the equilibrium conditions of the spatial price equilibrium model with path variables if and only if x* is the solution to the following variational inequality problem:
1 @,@*I 0%- D:)
,;; Y,(S*) (S,- W -
It
+
I.
C c,(f*) (4 -C’) 20
IJeA
V 1%’ E W;
A variational inequality problem is the problem of finding a vector y E K c R”, where K is a feasible set
(8)
for all x E 6. If each function in (8) were separable and monotone, then it is well-known (e.g. Kinderlehrer and Stampacchai [36]) that the variational inequality is equivalent to a convex optimization problem. Even if the functions are not separable, the use of the diagonalization/relaxation algorithm described by Dafermos [37] and Pang and Chan [21] would create a sequence of separable subproblems whose solutions will converge to the solution of the original variational inequality problem. Thus, we shall focus on the solution of the equivalent optimization problem which arises from (8) without loss of generality. Before doing this, however, let us first consider some additional constraints which need to be added to R due to practical considerations. The solution algorithms which will be presented for the solution of the equivalent optimization problem arising from (8) assume that the functions in (8) have continuous derivatives. However O,(D,) may have a discontinuous derivative at D, = 0 and at the point where O,(D,) = 0. Consider the linear function depicted in Fig. 1. At these points, there are kinks in O,(D,) due to the fact that negative prices and negative demands are infeasible from an economic point of view. Thus, we should bound D, to lie
A T,< = 1 h, Pt r,,
(7)
The following theorem shows that the spatial price equilibrium model with path variables can be cast into the form of a variational inequality problem; the proof of this theorem can be found in the Appendix [see also Dafermos and Nagurney [27] or Friesz et al. [25]:
@,(D,)
Fig. 1
PATRICK T, HARKER
302
between 0 and IJF to insure that O,(D,) has continuous derivatives over the feasible set. Also, some regional centroids may act as export points where a given amount of a commodity is to be demanded for export out of the study area. In this case the demands must be bounded from below by this fixed quantity, say Mf). Similarly, the regional supplies may need to be bounded due to continuity of the derivatives of Y,(S) and due to the possibility of limited production capacity in each region. Thus, the added constraints are My
V/EL,
M:
(9)
V/EL,
(10)
where Mf), Up, Ms and Us are given constants. In an analogous manner we may wish to bound interregional flows due to capacity limits and fixed commitments to move a certain amount of a commodity on an SD pair. Thus, the O-D flows can be constrained by Mj3
VIEW.
(11)
Finally, it will prove to be somewhat easier to deal with constraints (1) and (2) if we remove the C%D pair which goes from every region to itself. In this case, (1) and (2) can be written as (see Lemma 3.5 on p. 83 of Harker [15]): D,-S,+
1 T,*cE,
c T,=O FE&
VIcL
(12)
where E, = 8, - ((II)) H, = H, - {(ll)). Given these new constraints, rewritten as: R=
x:D,-SS,+
c T,WEE,
i T,=
the feasible set can be
c h, PEP,
1 T,=O WEH,
VIeL;
Kuhn-Tucker conditions of (13) to convince himself or herself that the solution of (13) does indeed yield the desired equilibrium conditions. The remainder of this paper deals with the solution of (13) for large-scale problems. As was discussed earlier, the solution algorithms developed for (13) are more general than for just problems with separable functions through the use of a diagonalizatiom relaxation algorithm. However, before closing this section, we must address the issue of choosing the variational inequality/equivalent optimization formulation over a nonlinear compl~enta~ty formulation and solution algorithm for the spatial price equilibrium problem with path variables. Nonlinear complementarity formulations and solution algorithms for the arc variable formulation of the spatial price equilibrium problem have received a great deal of attention in recent years. It is possible to formulate the path variable model as a nonlinear complementarity problem f15], but there are 2 main difficulties in doing so. First, the dependence on paths makes all the current complementarity algorithms infeasible for large-scale problems. Aashtiani [32] and Harker [IS] both illustrate that the algorithms for complementarity problems are extremely efficient but, unfortunately, their computer storage requirements are prohibitive for all but small test examples. Second, the type of constraints arising out of practical considerations, such as (9)-(1 I), seem much more “natural” in the variational inequality/ equivalent optimization framework. It is possible to add these constraints in the complementarity formulations, but most likely the introduction of these constraints will destroy the special structure dealt with in most of the efficient complementarity algorithms. Thus, the need to solve large-scale problems and the addition of constraints (9)-(1 I) necessitate the use of the variational inequality/equivalent optimization formulation.
VweW;
3. SOLUTION
ALGORITHMS
In this section, 2 solution algorithms for the equivalent optimization problem (13) are to be presented. Since both are feasible direction algorithms, let us first present a general statement of a feasible direction algorithm. Consider the following problem: minimize f(x) .reX
.
h>O,f>O
1 Using the feasible set R, the equivalent optimization formulation of the spatial price equilibrium problem with path variables which arises out of the variational inequality (8) is O,(s) ds
+c, ueA
f”
J
c,(s) ds
(13)
0
subject to x = (S’, D’, T’, h’, P)‘ER. The interested
reader
is invited
to analyze
the
(14)
where f(x) is a convex continuously differentiable function over the closed connected feasible set X c R”. A feasible direction algorithm for the solution of (14) consists of the following steps (e.g. Avriel [38], Luenberger [39]). Step 0
Choose an initial xeX,
Step 1
Select a feasible descent direction d”+’ such that
set k = 0.
(dk+‘) Vf(xk) < 0
(15)
and such that there exists a scalar value &+, > 0 satisfying xk + Q+, dk+‘eC for every 0 C zkcl < 6&+,.
The spatial price equilibrium problem Step 2
Determine tion of
xt+, such that a$+, is the solu-
minimize
f(x” + ak+‘dk+‘)
(16)
0 < ak+l < $+, (usually done by a bisection linesearch in the standard Frank-Wolfe traffic assignment algorithm).
If ]x,-x:] 5.6, a preset tolerance, for i = 1,. ,n; stop, x ‘+’ is the optimal solution. Else set, k = k + 1 and return to Step 1. Convergence of this algorithm has been proven under very mild assumptions given that the direction of descent is computed properly and the stepsize cr:+r is chosen appropriately (see Avriel [38] or Luenberger [39] for a further discussion of the convergence properties of this algorithm). The 2 solution algorithms to be presented differ in their selection of dk+‘, the direction of descent. Let us begin with a description of the linear approximation algorithm. 3.1. Linear approximation algorithm The use of the linear approximation or Frank-Wolfe [40] algorithm for the path variable formulation of the price equilibrium problem first suggested by Florian and Los [I] and used by Harker [15] in his study of coal movements in the United States. The linear approximation algorithm defines the search direction at each iteration to be equal to
where x* program
is the
solution
minimize XEX
_ Xk
(17)
to the
following
x’ V f(xk).
linear
(18)
Applying this algorithm to (13), we have the following linear program at each iteration k: minimize
c Y,(Sf)S, - c O,(Df)D, ,el_ /EL +
1
%V%
(19)
lIEA
subject
to x~fi.
Letting u,. denote the minimum cost between pair WOW, problem (19) can be reduced to: minimize
1
Y,(SF).S-
c
IEL
ISL
1
u,,T,.
to c T,,* EE,
M:
c T,=O *Err,
VlaL
M,D
VIcL
MT,
VwoW.
c S,--ISL 1 D,=O,
(21)
and applying an efficient algorithm such as the network simplex or out-of-kilter algorithms; Harker [I 51 discusses more fully the solution of this subproblem. The solution of the search direction step is then completed by loading T,* onto the shortest path between O-D pair w for each w E W. While being able to compute the search direction at each iteration quickly, the linear approximation algorithm suffers from 2 serious problems. First, as Florian and Los [l] point out, each @D flow T_ must have an upper bound assigned to it even if the bound must be set arbitrarily. The reason for this is that if Us = Up = cc for all ZEL, the solution to (20) lies at either T, = ML or T, = U; for all w EW. When Us and LJF are less than infinity for some subset of regions but not for all I EL, then it is still possible for T, = II& and thus we must bound all O-D flows to insure a bounded descent direction for the original problem. The “bang-bang” nature of the solution to (20) severely affects the convergence speed of this algorithm, as Harker [15] illustrates. This is the same problem addressed by LeBlanc and Farhangian [41] in their study of solution algorithms for the combined urban trip distribution and assignment problem. In this study they found that the speed of convergence of this type of algorithm is very sensitive to the value of the bounds. In practice, it is very difficult to know a priori what are good bounds to use. Coupled with the above problem is the sublinear convergence rate of the Frank-Wolfe algorithm (Dunn [42], Cannon and Cullum [43]). In fact, the convergence speed is so slow that in most practical applications the algorithm is never run to convergence, but is typically only run until a certain number of iterations have been performed. These 2 difficulties with the linear approximation algorithm have motivated the search for a more efficient algorithm which can be used on the same large-scale problems as can be addressed with the linear approximation algorithm. 3.2. Evans-type algorithm
W,E w
D,-S,+
SD
O,(Df)D,
+
subject
If there were no upper bounds for S and D and no lower bounds different from zero for T, S and D, then Florian and Los [l] show that (20) can be solved by inspection. However, when these bounds are present (20) can be solved by turning (20) into a linear network flow problem through the introduction of the following constraint.
/EL
Set xk+’ = xk + g:+,dk+’
dk+’ =x*
303
VIGL
(20)
Evans [31] noted the same type of difficulties as stated above in the solution of the urban distribution-assignment problem. Her algorithm to compute a search direction linearizes only half of the objective function and she finds that an analytic expression can be found for the solution of the resulting subproblem. In the spirit of Evans’ work, let us again consider problem (14) but let x and f(x) be decomposed into x = ( y’, z’)’ and f(x) = g(y) + h(z). In this case, problem (14) becomes mini$ze
f(x) = g(y) + h(z).
(22)
304
PATRICK T. HARKER
Let us define the search direction dk+’ zx* --k where x* is the solution mini$ze The following of descent.
(23)
of
g(y) + Z’ V h(z).
theorem
THEOREM
to be
(24)
shows that dkC’ is a direction
2
Let g(y) be a strictly convex function and let d’+’ be given by (23~(24) then dk+’ is a direction of descent. PROOF The vector dk-&’ is a direction of descent if (dk+‘)l V f(.xk) < 0,
(25)
where
(26) Since x* is the solution
of (24), it must be the case
that
NE. node 4 IS an artlficlat a network flow
node added problem
to create
Fig. 2
g(y*)+(~*)‘Vh(z”)
(27)
for all x = (y’, z ‘)‘EX. If g(y) is strictly convex,
then
g(.r*) > g(vk) + (Y * - Yk)’ v gW) or g(Y*) - g(.Y”) > (Y* -YkI’V for y* # y”. Substituting ranging terms leads to
(28) into
&Yk). (27) and
(28) rear-
(zk - z*)‘V h(z“) 2 g(y*) - g(yk) >(4’* -.yk)lVg(yk) Or
(y*-yk)‘Vg(yk)+(z*-zk)‘Vh(z)k
(29)
which is equal to (d”+ ‘) V f(xk) < 0. Thus, dkf’ is a descent direction. q Applying this method for computing a descent direction to (13) where the integral of the arc cost functions are the only functions to be linearized leads to the following subproblem at each iteration:
s
C let
0
s DI
Sl
mit$ttize
Y,(s)ds
O,(s) ds - C IGL 0 + c
c,(C)f,.
(30)
ilEA
Again letting u,, be the minimum transportation between O-D pair w E W, (30) is reduced to si n1 e,(s) ds minimize C yv,(s) d.s - C iGL s 0 ,sL s 0 + c
u,T,v
CEW
subject
to
D,-S,+ I@
c T,.c T,,.=O i\’EE, II.EH Vl6L
Mp
V~EL
M;f
VIY~W,
VIEL
cost
(31)
Problem (31) is a nonlinear network Row problem on the type of network depicited in Fig. 2. While not as easy to solve as the linear network flow subproblem in the linear approximation algorithm, many efficient algorithms have been recently developed for largescale nonlinear network flow problems. The reducedgradient algorithms by Dembo and Khncewicz [44] and Beck ef al. [45] and more recently the quadratically convergent Newton algorithm by Klincewicz [46] and truncated Newton algorithm by Mulvey [47] are examples of the efficient algorithms which exist for the solution of (31). The last 2 algorithms are particularly noteworthy not only for their quadratic convergence rate, but also because in many practical applications the functions tu,(S,) and O,(D,) will be linear in which case (31) becomes a quadratic program which can be solved in essentially one step by these algorithms. Thus, (31) can be solved efficiently for large-scale problems. There are 3 major advantages which this algorithm has over the linear approximation algorithm. First, based upon LeBlanc and Farhangian’s [41] experience with Evans’ algorithm, this algorithm should converge faster than the linear approximation algorithm. Even though each iteration will take longer with this algorithm, the total number of iterations should be less. Second, the bounding of all O-D flows is no longer necessary due to the fact that (31) has retained the full info~ation contained in the inverse supply and demand function. This too should help the Evans-type algorithm to converge much faster than the linear approximation algorithm. Also, the difficulty of choosing appropriate bounds is avoided. Finally, each iteration of the Evans-type algorithm computes an exact solution for equilibrium conditions {if and (ii). In the linear approximation algorithm, none of the equilibrium conditions is met until the final convergence of the algorithm occurs. This feature of the Evans-type algorithm is extremely important in large-scale applications since it is unlikely that either algorithm will be run to exact convergence due to the high computational costs
305
The spatial price equilibrium problem
to be solved-on the order of 200 regions, 5000 nodes and lO-20,000 arcs, which is equivalent to a 24 million variable mathematical program. Thus, the Evans-type algorithm is a good compromise between speed and storage requirements in that it requires approximately the same storage as the linear approximation algorithm. 4. NUMERICAL
EXAMPLES
In this section 4 numerical examples are presented in order to illustrate the differences in the algorithms presented in Section 3. In all examples, the following functional forms are used: Y,(S) = m/+ P,S,
(32)
@,(D,) = P, - n,D,
(33)
c,(f,)
= K, + v,c
(34)
and v, are constants. The where aI, B,, P/, n,, problems were run on a DEC-10 computer and the nonlinear network code NLPNET by Dembo [48] was used to solve the subproblems in the Evans-type algorithm. Figure 3 depicts the network used in Example 1 and Table 1 contains the data used for this example; the results of this example are summarized in Table 2 and Fig. 4. As Table 2 shows, the Evans-type algorithm outperforms the linear approximation algorithm in both speed and in the value of the objective function which obtains [Z in Fig. 41. Also, Table 2 illustrates that the equilibrium conditions (iHiii) are approximately met by the solution obtained from this algorithm. Although the rate of convergence of both algorithms is at best linear (resulting in the long flat curve in Fig. 4), the Evans-type algorithm reaches the objective function asymptote much faster than the linear approximation K,
Fig. 3
involved. In this situation, one more iteration of the Evans-type algorithm’s subproblem (31) can be run to assure that the equilibrium conditions (i) and (ii) are exactly met; whereas in using the linear approximation algorithm, we are not sure if any of the equilibrium conditions are close to convergence. Hence, the Evans-type algorithm, will always yield a solution which is a spatial price equilibrium and an approximation to the Wardropian User Equilibrium on the paths when the sequence generated by this algorithm is terminated prior to convergence. Finally, it will be the case that more sophisticated second-order methods are faster than the Evans-type algorithm proposed in this section. However, these methods cannot deal with the size problem which is
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lteratlans
Fig. 4
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306
PATRICKT. HARKER
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44000
43000
Fig. 5 algorithm. This same type of behavior is also exhibited in the larger example (Example 2) summarized in Table 3 and Fig. 5. Again, the Evans-type algorithm converges faster and with a lower objective function value. Tables 4 and 5 and Figs 6 and 7 also confirm these results for even larger examples, and Table 6 presents a comparison of the CPU time necessary to achieve the linear approximation objective function value. This table and Figs 4-7 clearly show that the Evans-type algorithm is far superior to the linear approximation algorithm in terms of its computational speed. Finally, on the final iteration the spatial price equilibrium conditions (i) and (ii) are exactly met with the Evans-type algorithm, whereas with the linear approximation algorithm the solution is not guaranteed to meet these conditions when the sequence generated by the algorithm is truncated. In summary, the examples presented in this section illustrate that the Evans-type algorithm is superior to the linear approximation algorithm in terms of speed, is more efficient in terms of the minimum value of the objective function achieved, and is superior in terms of achieving a solution to equilibrium conditions (i) and (ii), while approximating a solution to (iii). 5. CONCLUSIONS This paper has shown that, although related, the arc variable and path variable formulations of the spatial price equilibrium model with trans-shipment nodes have a fundamental conceptual difference which necessitates the use of the more difficult path variable formulation in certain circumstances. Also, this paper has presented an alternative algorithm to the linear approximation algorithm and has shown this solution procedure to be superior in terms of computational effort and in terms of the minimum objective function value which is achieved and the
convergence of the spatial price equilibrium conditions (i) and (ii). Recent research on efficient solution procedures for the traffic assignment problem, such as the simplicial Lawmethod proposed by decomposition phongpanich and Heat-n [49] and Pang and Yu [50], may yield significant advances over the algorithm discussed in this paper. However, these new procedures are not yet well tested and developed. The Evans-type algorithm presented in this paper is very easy to implement; only minor modifications to a currently operating linear approximation or spatial price equilibrium software package need to be made. Also, this algorithm requires very little additional computer storage over the linear approximation
Table No. No. No. No.
1. Data for example
of producing/consuming of nodes of arcs of SD pairs Region I EL a/ 1 2 3 Arc aeA
I 2 3 4 5 6 7 8 9 10 II 12 13 14
I
regions
3 6 = 14 =6
8,
PI
“I
1.0 2.0 I.5 K,
1.0 0.8 0.6 “,
19.0 27.0 30.0
0.20 0.01 0.30
1.0 2.0 3.0 1.0 2.0 1.0 1.0 3.0 2.0 1.0 2.0 2.0 1.0 3.0
0.50 0.20 0.30 0.40 0.30 0.10 0.10 0.50 0.20 1.00 0.25 0.20 0.90 0.80
The spatial
price equilibrium
Table 2. Results of example
CPU set per iteration Min objective
function
Results
2 3
12.15 34.23 31.18
15.49 30.56 31.51 u,
Y’,+ “(, - 0,
Path p
(l,2)
3.34
IO.11
-0.06
I
DEC-l0/90,
of of of of
producing/consuming nodes arcs O-D pairs
Fortran
CPU see per iteration Min objective function value a DEC-l0/90,
2
Evans-type Algorithm 55.78 89 0.627 -46318.038
-46300.284 Fortran
Composed arcs
of h,
2 3 4
1.7, II 2, IO, I I l,6, IO, II 2,9,7, I I
2.17 I.17 0.00 0.00
cP IO.11 10.20 12.15 II.15
I
4, II
0.33
6.45
method and, hence, it is capable of solving very large problems. It is not yet clear whether or not the simplicial decomposition technique can compete with this method in terms of storage requirements on the very large problems which are encountered in the real-life application of this model. In summary, the Evans-type algorithm presented in this paper is an efficient and easily implementable procedure which has the desirable property of always converging to the solution of the spatial price equilibrium conditions (i) and (ii) while approximating a solution to condition (iii).
4 9 = 32 = I2
Linear Approx Algorithm 60.70 II0 0.552
*i 16.49 26.45 20.40
16.57 26.66 20.65
77
regions
CPU sect Iterations
ton
+2.54 + 15.88 + 10.80 + 7.88 +0.19
6.70 6.00 5.00 4.05 6.45
0,
-3.34 + 3.67 -0.33
T,
0.00 0.00 0.00 0.00 0.33
Algorithm
D,rS,
GD pair w =(i,j)sW
Table 3. Data and results of exam&e No. No. No. No.
- 986.406
from the Evans-type
S,
(133) (2.1) (2,3) (3,l) (332)
0.496
0.483
D,
I
Evans-type Algorithm 10.92 22
-984.361
value
Node leL
a
I
Linear Approx Algorithm 12.55 26
CPU sect Iterations
ton
307
problem
77.
-Z 2700 ___--___
_cc__________________--_________
I
/
2600
/’
.-
:
-_)
: :
2500
:
I’
1’ / 2400
2300
2200
/ : : : : : : : : 2
4
6
8
10
12
14
16
18
Iterations
Fig. 6
20
22
24
26
28
30
32
34
36
b
PATRICK T. HARKER
308
-Z
i
6200
/HC
/--
__-_--- _-_--- __-__--
___-------
/’
,/--
/’
6000
I’ : :
I :
5800
I : : : : : : I I I I I
5600
5400
5200
2
I1
4
I,
6
8
I1
10
12
14
I
I1
16
18
11
20
22
1,
24
26
11
28
30
32
.
lteratlans
Fig. 7
Table 4. Data and results of example 3 No. No. No. No.
of of of of
producing/consuming nodes arcs O-D pairs
a DEC-10/90,
= = = =
Linear Approx Algorithm 35.18 32 1.099
CPU sect Iterations CPU set per iteration Min objective function value ton
regions
-2670.816
Fortran
10 22 56 58
Evans-type Algorithm 51.97 35 1.485 - 2675.221
77.
Table 5. Data and results of eXa”IDk 4 No. of producing/consuming No. of nodes No. of arcs
= I5 = 38 = I08 = 86
regions
No. of C&D pairs
CPU sect Iterations CPU set per iteration
Linear Approx Algorithm 193.06 124 I.557
Min objective function value
- 6259.564
Evans-type Algorithm 34.12 I6 2.133
-6262.013
ton a DEC-10/90, Fortran 77.
Table 6. Comparison
of CPU tnnes to achieve objective
linear amxoximation
% Savings
Linear Example
I 2 3 4
Approximation 12.55 60.70 35.18 193.06
value of
aleorithm Evans-type
in CPU Time
I .98 34.49 7.43 8.53
84.2 43.2 79.0 95.6
Acknowledgements-This research has been supported by a University Research Grant from the University of California at Santa Barbara, by funds from The Wharton School, and by the National Science Foundation under grant CEE840392; their support is gratefully acknowledged.
REFERENCES 1. M. Florian and M. Los. A new look at static spatial price equilibrium models. Reg. Sci. Urban Econ. 12, 579-597 (1982). 2. P. Samuelson. Spatial price equilibrium and linear programming. Am. Econ. Reu. 42, 283-304 (1952). 3. A. A. Cournot. Researches into the Mathematical Principles of the Theory of Wealth. (Translated by N. T. Bacon). Kelley, New York (1960). 4. T. Takayama and G. G. Judge. Equilibrium among spatially separated markets: a reformulation. Econometrica 32, 510-524 (1964). 5. T. Takayama and G. G. Judge. Spatial and Temporal Price and ANocation Models. North-Holland, Amsterdam (1971). An algorithm for the generalized 6. J. G. MacKinnon. transportation problem. Reg. Sci. Urban Econ. 5, 445464 (1975). 7. J. G. MacKinnon. A technique for the solution of spatial equilibrium models. J. Reg. Sci. 16, 293-307 (1976). 8. T. L. Friesz, R. L. Tobin, T. E. Smith and P. T. Harker. A nonlinear complementarity formulation and solution procedure for the general derived demand network equilibrium problem. J. Reg. Sci. 23, 337-359 (1983). 9. T. E. Smith. A solution condition for complementarity problems: with an application to spatial price equilibrium. Appl. Math. Comput. 15, 62-69 (1984). 10. S. Thore. The Takayama-Judge model of spatial equilibrium extended to convex production sets. J. Reg. Sci. 22, 203-212 (1982a). Il. S. Thore. The Takayama-Judge model with endo-
The spatial
12. 13. 14. 15.
16. 17.
18.
19.
20.
21.
22.
23.
24
2.5
26.
27.
28. 29.
30.
31.
32.
33.
34.
35.
price equilibrium
genous income. Reg. Sci. Urban Econ. 12, 351-364 (1982b). C. Ponsard. Partial spatial equilibria with fuzzy constraints, J. Reg. Sci. 22, 1599175 (1982). T. Puu. The general equilibrium of a spatially extended market economy. Geopr. Analysis 14, 1455154 (1982). E. Sheppard and L. -Curry. Spatial price equilibria. Geogr. Analysis 14, 2799304 (1982). P. T. Harker. Prediction of intercity freight flows: theory and application of a generalized spatial price eauilibrium model. Ph.D. dissertation, Department of Cjvil Engineering, University of Pennsylvania (1983). J. Rowse. Solving the generalized transportation problem. Reg. Sci. Urban Econ. 11, 57-68 (1981). R. Asmuth, B. Curtis Eaves and E. L. Peterson. Computing economic equilibria on affine networks with Lemke’s algorithm, Maths Ups Res. 4, 2099214 (I 979). J.-S. Pang. A hybrid method for the solution of some multicommodity spatial equilibrium problems. Mgmt Sci. 27, 114221157 (1981). J.-S. Pang. Solution of the general multi-commodity spatial equilibrium problem by variational and complementarity methods. J. Reg. Sci. 24(3), 403414 (1984). J.-S. Pang and P. S. Lee. A parametric linear complementarity technique for the computation of equilibrium prices in single commodity spatial models. Math. Prog. 20, 81-102 (1981). J.-S. Pang and D. Chan. Iterative methods for variational and complementarity problems. Math1 Progm. 24, 284313 (1981). P. C. Jones, R. Saigal and M. H. Schneider. A variable dimension homotopy for computing spatial equilibria. Ops Res. Left. Forthcoming (1985a). P. C. Jones, R. Saigal and M. H. Schneider. Computing nonlinear network equilibria. Math/ Progm. 31, 57766 (1985b). T. Takayama and N. Uri. A note on spatial and temporal price and allocation modeling: quadratic programming or linear complementarity programming? Reg. Sci. Urban Econ. 13, 455470 (1983). T. L. Friesz, P. T. Harker and R. L. Tobin. Alternative algorithms for the general network spatial price equilibrium problem. J. Reg. Sci., 24(4), 475-507 (1984). C. L. Irwin and C. W. Yang. Iteration and sensitivity for a spatial equilibrium problem with linear supply and demand functions. Ops Res. 30, 319-335 (1981). S. Dafermos and A. Naguney. Sensitivity analysis for the general spatial economic equilibrium problem. Ops Res. 32, 1069-1086 (1984). G. S. Chao and T. L. Friesz. Spatial price equilibrium sensitivity analysis. Transp. Res. B, 4233440 (1984). T. Takayama and G. G. Judge. Studies In Economic Planning Oper Space and Time. North-Holland, Amsterdam (1973). P. T. Harker. The state of the art in the predictive analysis of freight transportation systems. Tramp. Rev. S(2), 1433164 (1985). S. Evans. Derivation and analysis of some models for combining trip distribution and assignment, Tramp. Res. 10, 37757 (1976). H. Z. Aashtiani. The multi-model traffic assignment problem. Ph.D. dissertation, Sloan School of Managment, M.I.T. (1979). J. G. Wardrop. Some theoretical aspects of road traffic research. Proc. Inst. Civil Engineers Part 2, 1, 3255378 (1952). J. E. Fernandez and T. L. Friesz. Equilibrium predictions in transportation markets: the state of the art. Tramp. Res. 178, 1555172 (1983). R. L. Tobin and T. L. Friesz. Formulating and solving the network spatial price equilibrium problem with
36.
37. 38. 39. 40. 41.
42.
43.
44.
45.
46.
47. 48.
49.
50.
problem
309
trans-shipment in terms of arc variables. J. Reg. Sci. 23, 187-198 (1983). D. Kinderlehrer and G. Stampacchia. An Inrroduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). S. Dafermos. An iterative scheme for variational ineoualities. Math1 Progm. 26, 40-47 (1983). M. Avriel. NonlinearProgramming. Prentice Hall, Englewood Cliffs, NJ (1976). D. G. Luenberger. Introduction to Linear and Nonlinear Programming. Addison-Wesley, Reading, MA (1973). M. Frank and P. Wolfe. An algorithm for quadratic programming. Nav. Res. Logist. Q. 3, 95-l 10 (1956). L. J. LeBlanc and K. Farhaneian. Efficient algorithms for solving elastic demand modal split-assignment programming problems. Tramp. Sci. 15, 306317 (1981). J. C. Dunn. Rates of convergence of conditional gradient algorithms near singular and nonsingular extremals. SIAM J. Control and Optimization 17, 187-211 (1979). M. D. Cannon and C. D. Cullum. A tight upper bound on the rate of convergence of the FrankkWolfe algorithm. SIAM J. Control and Optimization 6, 187-211 (1968). R. Dembo and J. Klincewicz. A scaled reduced gradient algorithm for network flow problems with convex separable costs. Math1 Progm. 15, 125-147 (1981). P. Beck, L. Lasdon and M. Engquist. A reduced gradient algorithm for nonlinear network problems. ACM Trans. Math. Software 9, 57-70 (1983). J. G. Klincewicz. A Newton method for convex separable network flow nroblems. Networks 13. 427452 (1983). J. Mulvey. Private communication, Princeton University (1983). Dembo R. NLPNET User’s Manual. Yale School of Organization and Management, New Haven, CT (1983). S. Lawphongpanich and D. W. Hearn. Simplicial decomposition of the asymmetric traffic assignment problem. Transp. Res. 18B, 123-134 (1984). J.-S. Pang and C. S. Yu. Linearized simplicial decomposition methods for computing traffic equilibria on networks. School of Management and Administration, The University of Texas at Dallas, Richardson, TX. (1983). APPENDIX Proof of Theorem I
Let us first prove necessity. Inverting the supply demand functions, we can rewrite condition (ii) as if T, > 0,
(ii)’
then Y,(S) + a!, = O,(D)
and
V (i, j)eW V(i,j)sW
if’?,(S)+u,,>@,(D),thenT,,=O
where Y,(S) is the supply price at region ieL and Q,(D) is the demand price at region jeL. Also, we can combine conditions (ii’) and (iii) to form the following equivalent equilibrium condition: (iv)
if h, > 0,
then ‘f’<(S) + c,, = O,(D) v(i,j)EW,
if Y,(S) + c,, > O,(D),
V(i.j)EW, Condition
PEP,,
then h, = 0 PEP,,.
(iv) implies that
c,*>O,(D*)-Y,(S*)
v(i,j)~W.
pop,,,
(Al)
where the asterisk denotes an equilibrium solution. Consider the term (hr - hf) where h: is the equilibrium flow on path p EP and h,, is any other feasible flow. If (h,, - hi) < 0, this implies that h: > 0. which in turn implies by condition (iv)
310
PATRICKT. HARKER
that c; = O,(D*) - Y,(S*). Thus, let us multiply (Al) by (h, - h,*) and sum the resulting function over all PEP. The direction of the inequality in (Al) is unchanged in this operation due to the above fact that if (h, - h,*) < 0. then c,* is equal to 0,(D*) - Y,(S*). This manipulation leads to the following condition:
Since (A3) must hold for all feasible path flows h, let h = h’ except in one component h, = h,++ c, where c is a small positive number. This flow vector will remain feasible by the appropriate adjustments to T, and D, and S, for this path PEP,,. Thus condition (A3) implies that
[YJS*) -i-c; - O,(D*)]c > 0 or
-
c c
‘J’,(S*)0+,-h,*)
642)
[Y,(S*) + cf - O,(D*)] > 0.
(A4)
WW,j?EP,
Using the relationships in Q plus (4), (A2) can be rewritten as
aTA %(f*) (f, - f,‘) 2
c O,(D*)
,Cl.
(D, - D:)
-
,gLY,(s*) (S,- w,
which is condition (8). Thus, necessity is proven. To prove sufficiency, assume condition (8) is obtained. By appropriate substitutions from a and equation (4), (8) can be written as (A2). which can also be rewritten as
,.jjw,;
[‘y,(s*) + c,”- @,@*)I h,’
Therefore, the term in the bracket in (A4) is nonnegative for each path by applying the above procedure to each PEP. Also, the flow vector h =0 is a feasible flow vector (x = 0 E a). This implies by (A3) that ,,;wps
II
[yi(S*)+ c*P - O,(D*)] h; $0.
(A5)
Since the expression in the bracket and h,* are non-negative. (A5) must hold with equality. This implies that if hf > 0,
then Y,(S*) + c; = O,(D*)
if Y,(S*)-tc,t>O,(D*),
then h,*=O
vow,
PGP,,
V(g)oW,
PEP,,,
”
< C C [V,(s*) +c; - @,(D*)lh,. (MewPEP”
(A3)
which is a statement
sufftciency is proven.
of the equilibrium
condition
(iv). Thus
in Great
Sci. Vol. 20, No. 5, pp. 299-310,
0038-0121/86
1986
THE SPATIAL PRICE EQUILIBRIUM WITH PATH VARIABLES PATRICK Department
$3.00 + 0.00
Pergamon Journals Ltd
Britain
of Decision
Sciences,
PROBLEM
T. HARKERt
The Wharton School, University PA 19104, U.S.A.
of Pennsylvania,
Philadelphia,
(Received 21 November 1985; in revised form 24 February 1986)
Abstract-Most
of the recent literature dealing with the solution of the spatial price equilibrium model has dealt with the arc variable formulation; that is, the situation where the flows on the paths connecting an origindestination pair in a network with trans-shipment nodes are not explicitly considered. This paper presents a clarification of the main assumption underlying this arc formulation and presents an argument for the need for using a mode1 with path variables in certain applications. After this discussion, a new solution algorithm for the model with path variables is stated and compared against the linear approximation algorithm first proposed by Florian and Los [I] for this problem. Numerical examples are presented which illustrate the features of this new algorithm.
1. INTRODUCTION The equilibrium of the regional prices of commodities and the resulting regional supplies and demands and inter-regional flows has been at the heart of many freight movement studies. Ever since Samuelson’s [2] mathematical programming formulation of the spatial price equilibrium concept [which some attribute to Cournot [3] in the 19th century, but may in fact have been stated even earlier], a vast body of literature has been devoted to extensions of the original theoretical framework, alternative mathematical formulations, solution algorithms, and empirical applications. Takayama and Judge [4,5] extended Samuelson’s model to multiple commodities, MacKinnon [6,7] put the problem into the form of a fixed point problem, Florian and Los [l] considered nonlinear transportation costs, and Friesz, Tobin, Smith and Harker [8] formulated the spatial price equilibrium concept as a nonlinear complementarity problem. Smith [9] has recently used this complementarity formulation to present a general proof of existence of a spatial price equilibrium. Other theoretical extensions include the work by Thore [lo, 111, the partial equilibrium model using fuzzy set theory by Ponsard [12], Puu’s [13] model of equilibrium over a continuous plane, Sheppard and Curry’s [14] critique of the assumptions underlying the spatial price equilibrium concept, especially the omission of local spatial monopoly considerations, and Harker’s [ 151 extension to include profit-maximizing transportation firms. Algorithmic work includes the papers by Friesz et al. [8] and MacKinnon [6, 71, as well as the work bv Rowse 1161. Asmuth. Eaves and Peterson [17], Pang [18, 19j, &ng and Lee [20], Pang and Chan 1211. Jones et al. 122. 231. Takavama and Uri [24], aid Friesz et al. [25j. Also:‘Irwin Bnd Yang [26], Dafermos and Nagurney [27] and Chao and Friesz [28] have been studying the application of tFormer1y at the Department of Geography, California at Santa Barbara.
University
of 299
sensitivity analysis to this class of models. Empirical applications of this concept abound; the book by Takayama and Judge [29] and the recent paper by Harker [30] chronicle many of these studies. The purpose of this paper is 2-fold. First, in formulations of the spatial price equilibrium concept which incorporate trans-shipment nodes in the network (nodes at which no traffic is originated or terminated; that is, nodes for which there are no supply or demand functions defined), there are two general categories of models and associated algorithms. In the first category, the models are stated with path variables; that is, paths between each origin and destination must be explicitly considered. In the second category, the models are stated only in terms of arc flows. This paper will explicitly state the assumption which underlies the path variable formulations and critique this assumption. The next section will state the various mathematical formulations of the spatial price equilibrium concept and addresses this first issue. The second purpose of this paper is to present an Evans [31] type algorithm for the spatial price equilibrium model with path variables which overcomes some of the difficulties associated with applying a linear approximation or Frank-Wolfe type algorithm to this problem, as several authors have suggested or done (Florian and Los [l], Harker [15]). Section 3 presents the algorithm along with a statement of the linear approximation algorithm. Numerical tests on both small- and large-scale examples are used to illustrate the properties of each algorithm presented in Section 4, and the conclusions from this study are stated in Section 5. 2. THE SPATIAL
PRICE EQUILIBRIUM PROBLEM
Consider a network G[N, A] composed of a set of N of nodes and a set A of arcs, where N and A denote the cardinality of the respective sets. Let L E N be the set of L centroids of the regions comprising the study area. Multiple commodities are treated in this frame-
300
PATRICKT. HARKER
work through the use of Aashtiani’s [32] multi-copy network concept in which each element of the network is associated with both a physical location and a specific commodity. Thus, a separate network is defined for each commodity. For each centroid, 1EL, let us define a price 7t, and the vector of prices L = (K,, ) 7c,, , q)‘, where t denotes the transpose operation. Let us define a supply function S,(n) and demand function D,(rr) for each IsL, and S=(S ,,..., S ,,..., S)‘and D=(D ,,..., D ,,..., D)‘. Also, let us define W = {w = (ij); isL, jsL} to be the set of origin-destination (O-D) pairs, T,%.to wcw, be the flow between O-D pair , T,-)‘, and u,. to be the minimum T = (T,, . . . ,T,., cost to transport a unit of a commodity between O-D pair w EW. Finally, let us define fi, to be the set of GD pairs which enter region I EL including the O-D pair w = (II) and define E, to be the set of O-D pairs which leave region 1eL including w = (II); that is ii,={w
=(U)eW:
i=l}
E,=(w
=(ij)cW:
i=l}.
c=(c,,. .,c,,. .,CA)‘, h, = the flow on path p E P, h=(h ,,..., h,,. ..,h,#)‘, c, = the average transportation path p E P, and s
= OY,
Using the above notation, can be stated:
D,(Z) = 1 T,. WEI?, (ii) economic
VlcL
(1)
V/EL
(2)
rationality
if T, > 0, if JT,+ uy > x,,
then n,+ui,=n, then T,, = 0
V(ij)sW V (ij) EW.
Conditions (i) and (ii) were originally a mathematical program by defining
formulated
Y,(S) = the inverse of the supply function ICL,
as
at region
and O,(D) = the inverse of the demand region 1EL.
function
at
Later, conditions (i) and (ii) were treated directly by formulating them as a complementarity problem. When trans-shipment nodes are added, L becomes a proper subset of N and the set of arcs A no longer equals W. The trans-shipment nodes represent major point of interest from a transportation point of view, such as yards where railroads interline, ports, motor carrier terminals, etc. That is, no traffic is generated or terminated at these points, but there is some activity worthy of study occurring at these locations. For this case, let us define P = the set of P paths in the network, p = an index of paths, p E P, P,. = the set of paths between O-D pair w E W, f, = the flow on arc a E A, f=(f I)..., f,,. .,fA)‘, c,(f) = the average cost per unit of transporting a commodity on arc a E A.
arc
a E
the following
f,=
1 6,h, PEP
cp=
1 &G”(f) atA
T, = c h, PEP,
A
relationships
VaEA,
(3)
VP 6 P,
(4)
V w E W,
(5)
and
x
{cp}.
(6)
In this case, equilibrium conditions (i) and (ii) are expanded to include a Wardropian User Equilibrium (Wardrop [33], Fernandez and Friesz [34]) on all utilized paths between every O-D pair (see Friesz et al. [8] and Florian and Los [l]): (iii)
c T, MB,
1, if path p E P traverses 0 otherwise
u,,, = $n
The original Samuelson/Takayama-Judge model assumed that every node was a regional centroid (L = N) and that there were no trans-shipment nodes, so that A = W. In this case, the equilibrium conditions for this model can be written as: (i) flow conservation S,(Z)=
i
cost per unit on
then c, = u,,. V w E W, p E P,,
if h, > 0,
if c, > u,,., then h, = 0
V M’E W, p E P,(
Conditions (it(iii) can be written as either a complementarity problem (Friesz et al. [8], Harker [l5]) or a variational inequality problem (Florian and Los [l], Harker [15]) which leads to an equivalent optimization formulation under certain assumptions (namely, integrability of the inverse supply and demand functions and the arc cost functions). The introduction of the path variables in condition (iii) causes difficulties in the solution of this problem, as anyone familiar with the standard traffic assignment problem (Fernandez and Friesz [34]) understands; Florian and Los [I] highlight a few of these difficulties. Recently Tobin and Friesz [35] and Friesz et al. [8] have been able to overcome the use of path variables by noting that each trans-shipment node, while not having a supply or demand function defined on it, still has an implicit price which reaches an equilibrium value. Thus, conditions (ii) and (iii) have been rewritten as (iii)’
if f, > 0, Va
=
then r-r,+ c,,(f) = rr,
(i,j)eA;
if 7~,+ c,(f) > n,, Va
=
(i,j)~A;
i,i E N then f, = 0 i,j E N.
This arc variable formulation is at the heart of the development of efficient solution procedures, such as those by Pang and Lee [20] and Jones et al. [22,23]. The efficient solution procedures developed for the arc variable formulation brings to question the necessity of the path variable formulation, for if the two formulations are equivalent, there is no need to deal with paths. Unfortunately, the arc variable formu-
301
The spatial price equilibrium problem lation makes a rather strong assumption which may not hold in general. In the path variable formulation, we are assured that a movement from i toj will indeed go from i to j by the use of paths. In the arc variable formulation, however, each trans-shipment node is considered a “market” where incoming shipments to the node are “auctioned” off to the neighboring nodes. Thus, there is no guarantee that a shipment starting at region i will in fact arrive at region j; the only guarantee is that supplies and demand will be met in each region. For example, consider the network depicted in Fig. 3 and an O-D flow patterns of T (z ,) = To.,, = To,,, = 10. If only arc flow conservation were enforced, then a flow pattern of 20 units on arcs 3, 13 and 5 would satisfy a net flow of 20 units out of node 2 and 20 units into node I. Obviously, this arc flow pattern does not satisfy the @D flow pattern stated above. In order to guarantee that the C&D flow pattern is satisfied, the paths between each O-D pair must be explicitly incorporated into the model. For some commodities and/or networks with few trans-shipment nodes, the use of the arc formulation may not be a bad approximation given the high efficiency of the algorithms which have been developed for this problem. However, for commodities such as coal and for networks with many transshipment nodes, the use of paths is imperative. Thus, the efficiencies gained from the arc variable formulation are achieved at the cost of making a rather strong assumption. Possibly due to ignorance of the above fact, there has been relatively little work done to develop ejficient algorithms for the path variable formulation which can be used for large-scale problems. Florian and Los [l] suggest the use of a linear approximation (or Frank-Wolfe) algorithm, and Harker [15] has implemented this algorithm for his generalization of the spatial price equilibrium concept on a large-scale problem (960 nodes and 6993 arcs for the pure spatial price equilibrium portion of the problem). Section 3.1 will detail the application of the linear approximation method and its associated difficulties. Due to these difficulties, a new algorithm in the spirit of Evans’ [3 I] algorithm for urban traffic distribution and assignment will be developed and tested. However, before beginning our discussion of solution algorithms, we must first state the equilibrium conditions (it(iii) in a usable mathematical form. Let x = (S’, D’, T’, h’, f’)’ and define the feasible set for x as
and R” denotes n-dimensional Euclidean space, such that for the vector function F: R” + R” the following condition holds for all z E K: F(y)‘(-
- y) > 0.
THEOREM
I
x* = [(S*)‘, (D*)‘, (T*)‘, (h*)‘, (f*)]’ E fi satisfies the equilibrium conditions of the spatial price equilibrium model with path variables if and only if x* is the solution to the following variational inequality problem:
1 @,@*I 0%- D:)
,;; Y,(S*) (S,- W -
It
+
I.
C c,(f*) (4 -C’) 20
IJeA
V 1%’ E W;
A variational inequality problem is the problem of finding a vector y E K c R”, where K is a feasible set
(8)
for all x E 6. If each function in (8) were separable and monotone, then it is well-known (e.g. Kinderlehrer and Stampacchai [36]) that the variational inequality is equivalent to a convex optimization problem. Even if the functions are not separable, the use of the diagonalization/relaxation algorithm described by Dafermos [37] and Pang and Chan [21] would create a sequence of separable subproblems whose solutions will converge to the solution of the original variational inequality problem. Thus, we shall focus on the solution of the equivalent optimization problem which arises from (8) without loss of generality. Before doing this, however, let us first consider some additional constraints which need to be added to R due to practical considerations. The solution algorithms which will be presented for the solution of the equivalent optimization problem arising from (8) assume that the functions in (8) have continuous derivatives. However O,(D,) may have a discontinuous derivative at D, = 0 and at the point where O,(D,) = 0. Consider the linear function depicted in Fig. 1. At these points, there are kinks in O,(D,) due to the fact that negative prices and negative demands are infeasible from an economic point of view. Thus, we should bound D, to lie
A T,< = 1 h, Pt r,,
(7)
The following theorem shows that the spatial price equilibrium model with path variables can be cast into the form of a variational inequality problem; the proof of this theorem can be found in the Appendix [see also Dafermos and Nagurney [27] or Friesz et al. [25]:
@,(D,)
Fig. 1
PATRICK T, HARKER
302
between 0 and IJF to insure that O,(D,) has continuous derivatives over the feasible set. Also, some regional centroids may act as export points where a given amount of a commodity is to be demanded for export out of the study area. In this case the demands must be bounded from below by this fixed quantity, say Mf). Similarly, the regional supplies may need to be bounded due to continuity of the derivatives of Y,(S) and due to the possibility of limited production capacity in each region. Thus, the added constraints are My
V/EL,
M:
(9)
V/EL,
(10)
where Mf), Up, Ms and Us are given constants. In an analogous manner we may wish to bound interregional flows due to capacity limits and fixed commitments to move a certain amount of a commodity on an SD pair. Thus, the O-D flows can be constrained by Mj3
VIEW.
(11)
Finally, it will prove to be somewhat easier to deal with constraints (1) and (2) if we remove the C%D pair which goes from every region to itself. In this case, (1) and (2) can be written as (see Lemma 3.5 on p. 83 of Harker [15]): D,-S,+
1 T,*cE,
c T,=O FE&
VIcL
(12)
where E, = 8, - ((II)) H, = H, - {(ll)). Given these new constraints, rewritten as: R=
x:D,-SS,+
c T,WEE,
i T,=
the feasible set can be
c h, PEP,
1 T,=O WEH,
VIeL;
Kuhn-Tucker conditions of (13) to convince himself or herself that the solution of (13) does indeed yield the desired equilibrium conditions. The remainder of this paper deals with the solution of (13) for large-scale problems. As was discussed earlier, the solution algorithms developed for (13) are more general than for just problems with separable functions through the use of a diagonalizatiom relaxation algorithm. However, before closing this section, we must address the issue of choosing the variational inequality/equivalent optimization formulation over a nonlinear compl~enta~ty formulation and solution algorithm for the spatial price equilibrium problem with path variables. Nonlinear complementarity formulations and solution algorithms for the arc variable formulation of the spatial price equilibrium problem have received a great deal of attention in recent years. It is possible to formulate the path variable model as a nonlinear complementarity problem f15], but there are 2 main difficulties in doing so. First, the dependence on paths makes all the current complementarity algorithms infeasible for large-scale problems. Aashtiani [32] and Harker [IS] both illustrate that the algorithms for complementarity problems are extremely efficient but, unfortunately, their computer storage requirements are prohibitive for all but small test examples. Second, the type of constraints arising out of practical considerations, such as (9)-(1 I), seem much more “natural” in the variational inequality/ equivalent optimization framework. It is possible to add these constraints in the complementarity formulations, but most likely the introduction of these constraints will destroy the special structure dealt with in most of the efficient complementarity algorithms. Thus, the need to solve large-scale problems and the addition of constraints (9)-(1 I) necessitate the use of the variational inequality/equivalent optimization formulation.
VweW;
3. SOLUTION
ALGORITHMS
In this section, 2 solution algorithms for the equivalent optimization problem (13) are to be presented. Since both are feasible direction algorithms, let us first present a general statement of a feasible direction algorithm. Consider the following problem: minimize f(x) .reX
.
h>O,f>O
1 Using the feasible set R, the equivalent optimization formulation of the spatial price equilibrium problem with path variables which arises out of the variational inequality (8) is O,(s) ds
+c, ueA
f”
J
c,(s) ds
(13)
0
subject to x = (S’, D’, T’, h’, P)‘ER. The interested
reader
is invited
to analyze
the
(14)
where f(x) is a convex continuously differentiable function over the closed connected feasible set X c R”. A feasible direction algorithm for the solution of (14) consists of the following steps (e.g. Avriel [38], Luenberger [39]). Step 0
Choose an initial xeX,
Step 1
Select a feasible descent direction d”+’ such that
set k = 0.
(dk+‘) Vf(xk) < 0
(15)
and such that there exists a scalar value &+, > 0 satisfying xk + Q+, dk+‘eC for every 0 C zkcl < 6&+,.
The spatial price equilibrium problem Step 2
Determine tion of
xt+, such that a$+, is the solu-
minimize
f(x” + ak+‘dk+‘)
(16)
0 < ak+l < $+, (usually done by a bisection linesearch in the standard Frank-Wolfe traffic assignment algorithm).
If ]x,-x:] 5.6, a preset tolerance, for i = 1,. ,n; stop, x ‘+’ is the optimal solution. Else set, k = k + 1 and return to Step 1. Convergence of this algorithm has been proven under very mild assumptions given that the direction of descent is computed properly and the stepsize cr:+r is chosen appropriately (see Avriel [38] or Luenberger [39] for a further discussion of the convergence properties of this algorithm). The 2 solution algorithms to be presented differ in their selection of dk+‘, the direction of descent. Let us begin with a description of the linear approximation algorithm. 3.1. Linear approximation algorithm The use of the linear approximation or Frank-Wolfe [40] algorithm for the path variable formulation of the price equilibrium problem first suggested by Florian and Los [I] and used by Harker [15] in his study of coal movements in the United States. The linear approximation algorithm defines the search direction at each iteration to be equal to
where x* program
is the
solution
minimize XEX
_ Xk
(17)
to the
following
x’ V f(xk).
linear
(18)
Applying this algorithm to (13), we have the following linear program at each iteration k: minimize
c Y,(Sf)S, - c O,(Df)D, ,el_ /EL +
1
%V%
(19)
lIEA
subject
to x~fi.
Letting u,. denote the minimum cost between pair WOW, problem (19) can be reduced to: minimize
1
Y,(SF).S-
c
IEL
ISL
1
u,,T,.
to c T,,* EE,
M:
c T,=O *Err,
VlaL
M,D
VIcL
MT,
VwoW.
c S,--ISL 1 D,=O,
(21)
and applying an efficient algorithm such as the network simplex or out-of-kilter algorithms; Harker [I 51 discusses more fully the solution of this subproblem. The solution of the search direction step is then completed by loading T,* onto the shortest path between O-D pair w for each w E W. While being able to compute the search direction at each iteration quickly, the linear approximation algorithm suffers from 2 serious problems. First, as Florian and Los [l] point out, each @D flow T_ must have an upper bound assigned to it even if the bound must be set arbitrarily. The reason for this is that if Us = Up = cc for all ZEL, the solution to (20) lies at either T, = ML or T, = U; for all w EW. When Us and LJF are less than infinity for some subset of regions but not for all I EL, then it is still possible for T, = II& and thus we must bound all O-D flows to insure a bounded descent direction for the original problem. The “bang-bang” nature of the solution to (20) severely affects the convergence speed of this algorithm, as Harker [15] illustrates. This is the same problem addressed by LeBlanc and Farhangian [41] in their study of solution algorithms for the combined urban trip distribution and assignment problem. In this study they found that the speed of convergence of this type of algorithm is very sensitive to the value of the bounds. In practice, it is very difficult to know a priori what are good bounds to use. Coupled with the above problem is the sublinear convergence rate of the Frank-Wolfe algorithm (Dunn [42], Cannon and Cullum [43]). In fact, the convergence speed is so slow that in most practical applications the algorithm is never run to convergence, but is typically only run until a certain number of iterations have been performed. These 2 difficulties with the linear approximation algorithm have motivated the search for a more efficient algorithm which can be used on the same large-scale problems as can be addressed with the linear approximation algorithm. 3.2. Evans-type algorithm
W,E w
D,-S,+
SD
O,(Df)D,
+
subject
If there were no upper bounds for S and D and no lower bounds different from zero for T, S and D, then Florian and Los [l] show that (20) can be solved by inspection. However, when these bounds are present (20) can be solved by turning (20) into a linear network flow problem through the introduction of the following constraint.
/EL
Set xk+’ = xk + g:+,dk+’
dk+’ =x*
303
VIGL
(20)
Evans [31] noted the same type of difficulties as stated above in the solution of the urban distribution-assignment problem. Her algorithm to compute a search direction linearizes only half of the objective function and she finds that an analytic expression can be found for the solution of the resulting subproblem. In the spirit of Evans’ work, let us again consider problem (14) but let x and f(x) be decomposed into x = ( y’, z’)’ and f(x) = g(y) + h(z). In this case, problem (14) becomes mini$ze
f(x) = g(y) + h(z).
(22)
304
PATRICK T. HARKER
Let us define the search direction dk+’ zx* --k where x* is the solution mini$ze The following of descent.
(23)
of
g(y) + Z’ V h(z).
theorem
THEOREM
to be
(24)
shows that dkC’ is a direction
2
Let g(y) be a strictly convex function and let d’+’ be given by (23~(24) then dk+’ is a direction of descent. PROOF The vector dk-&’ is a direction of descent if (dk+‘)l V f(.xk) < 0,
(25)
where
(26) Since x* is the solution
of (24), it must be the case
that
NE. node 4 IS an artlficlat a network flow
node added problem
to create
Fig. 2
g(y*)+(~*)‘Vh(z”)
(27)
for all x = (y’, z ‘)‘EX. If g(y) is strictly convex,
then
g(.r*) > g(vk) + (Y * - Yk)’ v gW) or g(Y*) - g(.Y”) > (Y* -YkI’V for y* # y”. Substituting ranging terms leads to
(28) into
&Yk). (27) and
(28) rear-
(zk - z*)‘V h(z“) 2 g(y*) - g(yk) >(4’* -.yk)lVg(yk) Or
(y*-yk)‘Vg(yk)+(z*-zk)‘Vh(z)k
(29)
which is equal to (d”+ ‘) V f(xk) < 0. Thus, dkf’ is a descent direction. q Applying this method for computing a descent direction to (13) where the integral of the arc cost functions are the only functions to be linearized leads to the following subproblem at each iteration:
s
C let
0
s DI
Sl
mit$ttize
Y,(s)ds
O,(s) ds - C IGL 0 + c
c,(C)f,.
(30)
ilEA
Again letting u,, be the minimum transportation between O-D pair w E W, (30) is reduced to si n1 e,(s) ds minimize C yv,(s) d.s - C iGL s 0 ,sL s 0 + c
u,T,v
CEW
subject
to
D,-S,+ I@
c T,.c T,,.=O i\’EE, II.EH Vl6L
Mp
V~EL
M;f
VIY~W,
VIEL
cost
(31)
Problem (31) is a nonlinear network Row problem on the type of network depicited in Fig. 2. While not as easy to solve as the linear network flow subproblem in the linear approximation algorithm, many efficient algorithms have been recently developed for largescale nonlinear network flow problems. The reducedgradient algorithms by Dembo and Khncewicz [44] and Beck ef al. [45] and more recently the quadratically convergent Newton algorithm by Klincewicz [46] and truncated Newton algorithm by Mulvey [47] are examples of the efficient algorithms which exist for the solution of (31). The last 2 algorithms are particularly noteworthy not only for their quadratic convergence rate, but also because in many practical applications the functions tu,(S,) and O,(D,) will be linear in which case (31) becomes a quadratic program which can be solved in essentially one step by these algorithms. Thus, (31) can be solved efficiently for large-scale problems. There are 3 major advantages which this algorithm has over the linear approximation algorithm. First, based upon LeBlanc and Farhangian’s [41] experience with Evans’ algorithm, this algorithm should converge faster than the linear approximation algorithm. Even though each iteration will take longer with this algorithm, the total number of iterations should be less. Second, the bounding of all O-D flows is no longer necessary due to the fact that (31) has retained the full info~ation contained in the inverse supply and demand function. This too should help the Evans-type algorithm to converge much faster than the linear approximation algorithm. Also, the difficulty of choosing appropriate bounds is avoided. Finally, each iteration of the Evans-type algorithm computes an exact solution for equilibrium conditions {if and (ii). In the linear approximation algorithm, none of the equilibrium conditions is met until the final convergence of the algorithm occurs. This feature of the Evans-type algorithm is extremely important in large-scale applications since it is unlikely that either algorithm will be run to exact convergence due to the high computational costs
305
The spatial price equilibrium problem
to be solved-on the order of 200 regions, 5000 nodes and lO-20,000 arcs, which is equivalent to a 24 million variable mathematical program. Thus, the Evans-type algorithm is a good compromise between speed and storage requirements in that it requires approximately the same storage as the linear approximation algorithm. 4. NUMERICAL
EXAMPLES
In this section 4 numerical examples are presented in order to illustrate the differences in the algorithms presented in Section 3. In all examples, the following functional forms are used: Y,(S) = m/+ P,S,
(32)
@,(D,) = P, - n,D,
(33)
c,(f,)
= K, + v,c
(34)
and v, are constants. The where aI, B,, P/, n,, problems were run on a DEC-10 computer and the nonlinear network code NLPNET by Dembo [48] was used to solve the subproblems in the Evans-type algorithm. Figure 3 depicts the network used in Example 1 and Table 1 contains the data used for this example; the results of this example are summarized in Table 2 and Fig. 4. As Table 2 shows, the Evans-type algorithm outperforms the linear approximation algorithm in both speed and in the value of the objective function which obtains [Z in Fig. 41. Also, Table 2 illustrates that the equilibrium conditions (iHiii) are approximately met by the solution obtained from this algorithm. Although the rate of convergence of both algorithms is at best linear (resulting in the long flat curve in Fig. 4), the Evans-type algorithm reaches the objective function asymptote much faster than the linear approximation K,
Fig. 3
involved. In this situation, one more iteration of the Evans-type algorithm’s subproblem (31) can be run to assure that the equilibrium conditions (i) and (ii) are exactly met; whereas in using the linear approximation algorithm, we are not sure if any of the equilibrium conditions are close to convergence. Hence, the Evans-type algorithm, will always yield a solution which is a spatial price equilibrium and an approximation to the Wardropian User Equilibrium on the paths when the sequence generated by this algorithm is terminated prior to convergence. Finally, it will be the case that more sophisticated second-order methods are faster than the Evans-type algorithm proposed in this section. However, these methods cannot deal with the size problem which is
A -2
990
980
970
960
950
940
2
4
6
8
IO
12
14
16
18
lteratlans
Fig. 4
20
22
. 24
26
306
PATRICKT. HARKER
-Z
46000
45000
44000
43000
Fig. 5 algorithm. This same type of behavior is also exhibited in the larger example (Example 2) summarized in Table 3 and Fig. 5. Again, the Evans-type algorithm converges faster and with a lower objective function value. Tables 4 and 5 and Figs 6 and 7 also confirm these results for even larger examples, and Table 6 presents a comparison of the CPU time necessary to achieve the linear approximation objective function value. This table and Figs 4-7 clearly show that the Evans-type algorithm is far superior to the linear approximation algorithm in terms of its computational speed. Finally, on the final iteration the spatial price equilibrium conditions (i) and (ii) are exactly met with the Evans-type algorithm, whereas with the linear approximation algorithm the solution is not guaranteed to meet these conditions when the sequence generated by the algorithm is truncated. In summary, the examples presented in this section illustrate that the Evans-type algorithm is superior to the linear approximation algorithm in terms of speed, is more efficient in terms of the minimum value of the objective function achieved, and is superior in terms of achieving a solution to equilibrium conditions (i) and (ii), while approximating a solution to (iii). 5. CONCLUSIONS This paper has shown that, although related, the arc variable and path variable formulations of the spatial price equilibrium model with trans-shipment nodes have a fundamental conceptual difference which necessitates the use of the more difficult path variable formulation in certain circumstances. Also, this paper has presented an alternative algorithm to the linear approximation algorithm and has shown this solution procedure to be superior in terms of computational effort and in terms of the minimum objective function value which is achieved and the
convergence of the spatial price equilibrium conditions (i) and (ii). Recent research on efficient solution procedures for the traffic assignment problem, such as the simplicial Lawmethod proposed by decomposition phongpanich and Heat-n [49] and Pang and Yu [50], may yield significant advances over the algorithm discussed in this paper. However, these new procedures are not yet well tested and developed. The Evans-type algorithm presented in this paper is very easy to implement; only minor modifications to a currently operating linear approximation or spatial price equilibrium software package need to be made. Also, this algorithm requires very little additional computer storage over the linear approximation
Table No. No. No. No.
1. Data for example
of producing/consuming of nodes of arcs of SD pairs Region I EL a/ 1 2 3 Arc aeA
I 2 3 4 5 6 7 8 9 10 II 12 13 14
I
regions
3 6 = 14 =6
8,
PI
“I
1.0 2.0 I.5 K,
1.0 0.8 0.6 “,
19.0 27.0 30.0
0.20 0.01 0.30
1.0 2.0 3.0 1.0 2.0 1.0 1.0 3.0 2.0 1.0 2.0 2.0 1.0 3.0
0.50 0.20 0.30 0.40 0.30 0.10 0.10 0.50 0.20 1.00 0.25 0.20 0.90 0.80
The spatial
price equilibrium
Table 2. Results of example
CPU set per iteration Min objective
function
Results
2 3
12.15 34.23 31.18
15.49 30.56 31.51 u,
Y’,+ “(, - 0,
Path p
(l,2)
3.34
IO.11
-0.06
I
DEC-l0/90,
of of of of
producing/consuming nodes arcs O-D pairs
Fortran
CPU see per iteration Min objective function value a DEC-l0/90,
2
Evans-type Algorithm 55.78 89 0.627 -46318.038
-46300.284 Fortran
Composed arcs
of h,
2 3 4
1.7, II 2, IO, I I l,6, IO, II 2,9,7, I I
2.17 I.17 0.00 0.00
cP IO.11 10.20 12.15 II.15
I
4, II
0.33
6.45
method and, hence, it is capable of solving very large problems. It is not yet clear whether or not the simplicial decomposition technique can compete with this method in terms of storage requirements on the very large problems which are encountered in the real-life application of this model. In summary, the Evans-type algorithm presented in this paper is an efficient and easily implementable procedure which has the desirable property of always converging to the solution of the spatial price equilibrium conditions (i) and (ii) while approximating a solution to condition (iii).
4 9 = 32 = I2
Linear Approx Algorithm 60.70 II0 0.552
*i 16.49 26.45 20.40
16.57 26.66 20.65
77
regions
CPU sect Iterations
ton
+2.54 + 15.88 + 10.80 + 7.88 +0.19
6.70 6.00 5.00 4.05 6.45
0,
-3.34 + 3.67 -0.33
T,
0.00 0.00 0.00 0.00 0.33
Algorithm
D,rS,
GD pair w =(i,j)sW
Table 3. Data and results of exam&e No. No. No. No.
- 986.406
from the Evans-type
S,
(133) (2.1) (2,3) (3,l) (332)
0.496
0.483
D,
I
Evans-type Algorithm 10.92 22
-984.361
value
Node leL
a
I
Linear Approx Algorithm 12.55 26
CPU sect Iterations
ton
307
problem
77.
-Z 2700 ___--___
_cc__________________--_________
I
/
2600
/’
.-
:
-_)
: :
2500
:
I’
1’ / 2400
2300
2200
/ : : : : : : : : 2
4
6
8
10
12
14
16
18
Iterations
Fig. 6
20
22
24
26
28
30
32
34
36
b
PATRICK T. HARKER
308
-Z
i
6200
/HC
/--
__-_--- _-_--- __-__--
___-------
/’
,/--
/’
6000
I’ : :
I :
5800
I : : : : : : I I I I I
5600
5400
5200
2
I1
4
I,
6
8
I1
10
12
14
I
I1
16
18
11
20
22
1,
24
26
11
28
30
32
.
lteratlans
Fig. 7
Table 4. Data and results of example 3 No. No. No. No.
of of of of
producing/consuming nodes arcs O-D pairs
a DEC-10/90,
= = = =
Linear Approx Algorithm 35.18 32 1.099
CPU sect Iterations CPU set per iteration Min objective function value ton
regions
-2670.816
Fortran
10 22 56 58
Evans-type Algorithm 51.97 35 1.485 - 2675.221
77.
Table 5. Data and results of eXa”IDk 4 No. of producing/consuming No. of nodes No. of arcs
= I5 = 38 = I08 = 86
regions
No. of C&D pairs
CPU sect Iterations CPU set per iteration
Linear Approx Algorithm 193.06 124 I.557
Min objective function value
- 6259.564
Evans-type Algorithm 34.12 I6 2.133
-6262.013
ton a DEC-10/90, Fortran 77.
Table 6. Comparison
of CPU tnnes to achieve objective
linear amxoximation
% Savings
Linear Example
I 2 3 4
Approximation 12.55 60.70 35.18 193.06
value of
aleorithm Evans-type
in CPU Time
I .98 34.49 7.43 8.53
84.2 43.2 79.0 95.6
Acknowledgements-This research has been supported by a University Research Grant from the University of California at Santa Barbara, by funds from The Wharton School, and by the National Science Foundation under grant CEE840392; their support is gratefully acknowledged.
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The spatial
12. 13. 14. 15.
16. 17.
18.
19.
20.
21.
22.
23.
24
2.5
26.
27.
28. 29.
30.
31.
32.
33.
34.
35.
price equilibrium
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Let us first prove necessity. Inverting the supply demand functions, we can rewrite condition (ii) as if T, > 0,
(ii)’
then Y,(S) + a!, = O,(D)
and
V (i, j)eW V(i,j)sW
if’?,(S)+u,,>@,(D),thenT,,=O
where Y,(S) is the supply price at region ieL and Q,(D) is the demand price at region jeL. Also, we can combine conditions (ii’) and (iii) to form the following equivalent equilibrium condition: (iv)
if h, > 0,
then ‘f’<(S) + c,, = O,(D) v(i,j)EW,
if Y,(S) + c,, > O,(D),
V(i.j)EW, Condition
PEP,,
then h, = 0 PEP,,.
(iv) implies that
c,*>O,(D*)-Y,(S*)
v(i,j)~W.
pop,,,
(Al)
where the asterisk denotes an equilibrium solution. Consider the term (hr - hf) where h: is the equilibrium flow on path p EP and h,, is any other feasible flow. If (h,, - hi) < 0, this implies that h: > 0. which in turn implies by condition (iv)
310
PATRICKT. HARKER
that c; = O,(D*) - Y,(S*). Thus, let us multiply (Al) by (h, - h,*) and sum the resulting function over all PEP. The direction of the inequality in (Al) is unchanged in this operation due to the above fact that if (h, - h,*) < 0. then c,* is equal to 0,(D*) - Y,(S*). This manipulation leads to the following condition:
Since (A3) must hold for all feasible path flows h, let h = h’ except in one component h, = h,++ c, where c is a small positive number. This flow vector will remain feasible by the appropriate adjustments to T, and D, and S, for this path PEP,,. Thus condition (A3) implies that
[YJS*) -i-c; - O,(D*)]c > 0 or
-
c c
‘J’,(S*)0+,-h,*)
642)
[Y,(S*) + cf - O,(D*)] > 0.
(A4)
WW,j?EP,
Using the relationships in Q plus (4), (A2) can be rewritten as
aTA %(f*) (f, - f,‘) 2
c O,(D*)
,Cl.
(D, - D:)
-
,gLY,(s*) (S,- w,
which is condition (8). Thus, necessity is proven. To prove sufficiency, assume condition (8) is obtained. By appropriate substitutions from a and equation (4), (8) can be written as (A2). which can also be rewritten as
,.jjw,;
[‘y,(s*) + c,”- @,@*)I h,’
Therefore, the term in the bracket in (A4) is nonnegative for each path by applying the above procedure to each PEP. Also, the flow vector h =0 is a feasible flow vector (x = 0 E a). This implies by (A3) that ,,;wps
II
[yi(S*)+ c*P - O,(D*)] h; $0.
(A5)
Since the expression in the bracket and h,* are non-negative. (A5) must hold with equality. This implies that if hf > 0,
then Y,(S*) + c; = O,(D*)
if Y,(S*)-tc,t>O,(D*),
then h,*=O
vow,
PGP,,
V(g)oW,
PEP,,,
”
< C C [V,(s*) +c; - @,(D*)lh,. (MewPEP”
(A3)
which is a statement
sufftciency is proven.
of the equilibrium
condition
(iv). Thus