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Models for traffic assignment
Transpn Res. Vol. 1, pp. 31-46. Pergamon Press 1967. Printed in Great Britain
MODELS FOR TRAFFIC ASSIGNMENT W. S. JEWELL Department of Industrial Engineering and Operations Research, University of California, Berkeley (Received 8 October 1966)
INTRODUCTION FROMSeptember 1965 to February 1966, the author was a Fulbright Research Scholar, on sabbatical leave in Paris, France, and spent a portion of his time as a guest of the Direction Scientiiique of the Societe d’Economie et de Mathematique AppliquQs (SEMA). Research was in progress at that time on properties of traffic assignment models, and the author was fortunate enough to have extended discussions with H. LeBoulanger and B. Roy, principal designers of the proprietary trafEc-assignment programs, ATCODE and ADECODE. This paper is a revision of a research paper (Jewell, 1966) written as a result of these discussions. At that time, several different models and algorithms were being proposed which seemed quite disparate; furthermore, it was not clear whether we could hope to separate the two concepts of model and method, nor whether stress should be placed on an exact or approximate procedure. The attempt of this paper was to provide a systematic exploration of the several different approaches, emphasizing the common features, if possible, but also contrasting their differences. We proceed slowly, in an axiomatic manner, starting with the single-route case, and gradually building up to the most general type of multi-commodity network. The ultimate aim was, of course, to fmd algorithms which were both realistic and efficient; but, in this, we were not successful. In fact, although certain concepts and model relationships are clearer than before, certain “hard-core” difficulties in traffic assignment will apparently need further research and computational testing. 1. THE SINGLE
ROUTE
CASE
1.1. Notation Consider a certain origin-destination (OD) pair, and some path which connects them; in this section we assume that all flow between this OD pair must use this single route. During some emission interval, E hours, a totalflow, Q(E) vehicles, will be generated and emitted at the origin. We suppose E is determined once and for all by certain exterior considerations (data availability, quantization of time-variations, control cycles, etc.), so that we will not henceforth emphasize the dependence of Q on E. Our tist assumption about this flow is: (Hl): The vehicles are emittedfrom the origin onto the route at a uniform and constantflow rate until the “stock” Q is depleted. Call this uniform and constant flow rate 4 vehicles/hr, and the time of depletion 8 hr. We will admit models in which 6# E, i.e. the vehicles may be sent out during a shorter or longer period than that which determined the value of Q. Relative to the route, which is D miles long (and, temporarily, one lane wide), we assume : 31
32
W.S.
JEWELL
(H2): when they are being emitted, vehicles travel at a constant velocity and headway (time spacing) along the route to the destination, thus defining: n-the maximum number of vehicles on the route at any one time, A-the maximum density of vehicles, in vehicles/miles, n-the common velocity of the vehicles, in mile@, h-the common headway between vehicles, in miles, and +-the common trajectory time for an individual vehicle to traverse the route in hours. These definitions are summariz ed in Fig. 1, in terms of a reservoir pipeline analogy. Normally, we consider the variables as Given :
CL Q 6%
and Unknown: +, 6, n, A, v, h, 7.
Maximum density-X,
Stock of 0 veh&s
veWclMmil0
Oripin
-
%e,t-
-
-
-
r,
hr
-
FIG.
-
-
d
Destination
1.
1.2. Obvious relations From hypotheses (Hl) and (H2), we can easily agree upon the five following relations among the seven unknowns: A = l/h
(1)
D = VT
(2)
n = AD = D/h
(may be noninteger)
Q= W # = n/T = hv
(3) (4)
(or AuS = Q)
(5)
This leaves two independent variables, say (A, v), or (n, v), or (6, v), etc., whose values must be known to specify a unique solution to ah others. Figures 2 and 3 then show the rate of emission, rate of reception and number of vehicles “in the pipeline” as functions of time, under our assumptions. 1.3. Velocity-density relationships The three submodels which follow have the common property that 6 is assumed to be tied by the algorithm, and the remaining ambiguity is removed by assumptions about local driver behavior in a certain velocity-density regime.
33
Models for trai& assignment
1.3.1. Local behavior model A (linear density-uelocity). We- assume : @Al): There is a maximum density of vehicles, f&. (HA2): There is a (maximum) ‘free” velocity, V,. (HA3): When not otherwise limited by (HAl) and (HA2), there is a linearly decreasing relationship which limits the values of h and v.
Rates of emission-reception
//
I
,
(
0
8
r
Emission
ReceptimT~8
)
Time,
hr
FIG. 2.
4
Number in route
vehicles
0
rt.8
8
c Time,
hr
FIG. 3.
I
X,
vehicks/mtle
Vf FIG.
Li
Y,
miles/hr
4.
Thus, under these hypotheses the shaded area in Fig. 4 represents the only allowed values of A and v. Implicit in the above is the assumption that the parameters V,, A,, AF and V_ can be measured from real data. It appears that another assumption is needed at this point to specify unique values of the variables. However, a later transformation would reintroduce some redundancy, so we shall postpone any further assumptions until that point. 3
34
W. S. JEWELL
Figure 4 may be written:
\A,,
v=
v,
It follows from (5) that a curve of Q vs. v would have one of the forms shown in Fig 5; # may have a local maximum, M, within the interval V, < v < V, [cases (a) and (b)], or the constrained maximum may lie on the boundary v = V, [case (c)l. In other words, M = min [1/2U; V,]
642)
= 4wVrhi+V~ &i-A,
G43)
where u
is the value shown in Fig. 4 as the v abscissa intercept of the straight-line approximation. Thus, an internal maximum in Fig. 5 is associated with a strong decrease in maximum density vs. velocity. vahicles/hr
v.
V*
vF
v,
milcshr
FIG. 5.
u
0
+,
vehickdhr
FIG. 6.
For reasons which will become evident later, we transform Fig. 5 to a CT,+) plot in Fig. 6, by using (2); the two major cases are shown, after renormalizing M to the same value. The values of the new parameters shown are easily derived in terms of the other given data.
35
Models for traffic assignment
We leave the discussion of a unique choice of operating-curve in Fig. 6 until after discussing model B. 1.3.2. Local behavior moa21 B (car-following). We assume (HBl) and (HB2) identical with (HAl) and (HA2), and (HB3): When not otherwise limited, vehicles maintain, as a minimum, an (excess) headway which is proportional to their common velocity. In terms of headways, this means: 06v
49 ha
H,,+ar(v-V,), t 4 = Ii, + 45
W
v,
v = v,
with tl, = l/AM, OLand VP to be determined by experiment. This gives a density-velocity region shown in Fig. 7: Ai&z?
o
A,l+A&&-v~),
032)
v,
A F = l/%,
v = v,
-r
v,
miles&r
FIG. 7.
It is easy to show that this region does not give an internal maximum to the boundary curve of 4 vs. v: therefore, the curve of T vs. 4 has the same shape as Fig. 6c. 1.3.3. Variations of model B. In an attempt to model a “more realistic” behavior of drivers, model B was extended by the following hypotheses: (HB3’) : The increase in headway is more than proportional to the common velocity [i.e. there is a quadratic, cubic, etc., component in (HB3)]. (HB4) : There is more than one lane in the road, the monber of effective lanes being a decreasing function of velocity (i.e. obstacles at the side of the road do not interfere with slowly moving vehicles, which “flow” around them, but do reduce eflective capacity at higher speeds). Except for differences in algebraic forms, none of these changes affects the shape of Fig. 5c for model B, i.e. neither gives an internal maximum to 4. 1.3.4. Resolution of ambiguities in models A and B. To have an unique representation of T vs. 4, we must invoke an “economic” or “behavioral” principle which will select an unique curve from the shaded area in Fig. 6.
W.S.
36
JEWELL
(HABl): For a givenfIow rate of vehicles, 4, the drivers will interact so as to minimize their trajectory time, 7. This assumption will result in the two solid curves shown below in Fig. 8. TP
hr i
I
I TF
i I
c
M 91 vehiclesAw FIQ. 8.
To eliminate the troublesome truncation on the right-hand side of Fig. 8, we may add the possibility of a certain “stretching” of M (alternate passage by side streets, alleys, etc.) with a very large increase in T, which would result in the vertical dashed line shown in Fig. 8. (HAB2): For values of t.#infinitesimally greater than M, T increases without limit. This possibility might occur, for example, when M was a critical point at which accidents, “jam-ups”, etc., and other drastic driver behavior occurred, forcing the seeking out of alternate secondary routes, waiting for tow-trucks, etc.--clearly an undesirable or “infeasible” (for planners) portion of the curve. 1.3.5. Local behavior model C (queueing). A different model is obtained if we interpret the flow variable, 4, and the trajectory time, T, as averages of random variables, and assume : (HCl) : Vehicles are emitted randomly (Poisson process). (HC2) : Every vehicle travels at some maximum veZocity v, over the route until it reaches the destination. (HC3): There is a “service mechanism” (trafic light, bottleneck, toll-gate, narrowing of the roadway, etc.) infront of the destination which “processes” vehicles with a mean processing time, S hr, and variance aZ. All vehicles not being “served” must wait in a queue, whose dimensions are small compared to D (implicit assumption of statistical equilibrium). Then, from the well-known Pollaczek-Khintchine formula of queueing theory: 7 =
wM+s+
+(s2+ a) 2(1_#)
K4
=Tfl+l_+S which has the shape indicated in Fig. 9. A feature of this model is the natural appearance of a limiting flow rate M= l/S beyond which there can be no equilibrium.
37
Models for traffic assignment
If there are multiple routes speaking, apply next one) is not
several bottlenecks along the route (as would be the case in considering with differing numbers of arcs), then the above theory does not, strictly to the resulting series gueae, as the output of one queue (which feeds the Poisson.
Fro.
9.
However, if all service-times are exponentially equivalent of (Cl) will be:
distributed
(OS= S*), then the series
where the index i refers to each (free stretch-bottleneck) segment. In terms of dominant behavior for 4 large, it is clear in any case that it is the minimum capacity M = [ma+ &j-l which determines the shape of the CT,4) curve, and thus (Cl) can be expected to provide an approximate form for any “real” curve for this model. “Parallel service” leads to similar remarks, after redefinition of M. 2. NETWORKS
WITH
ONE
ORIGIN-DESTINATION
PAIR
2.1. Notation Now consider a simplified trafhc network in which nodes 1 and N are the unique origin and destination for the flow Q, all other nodes being transhipment points. We now have, for each directed arc (i, j) between nodes i and j, the route parameters discussed in (1 .l), i.e. % IL,, +ii, PDF,etc., assumed constant over this arc; in what follows, we shall consider primarily the arc variables & and TV,. Traffic from node 1 to node N may follow any or all possible directed paths of arcs leading from 1 to N. 2.2. “Continuous” flow argument Because of the remarks made in Section 1.2 (particularly Fig. 3), it is clear that if many different paths are followed during the emission interval, the resulting flow-rate vs. time curve would be different in the different arcs. In fact certain critical arcs may never appear to be “stressed”, due to the fortuitous “phasing in” of arrivals over that segment. Certainly, all the flow may not reach N before the emission interval, E, is over. Because the interest of traffic assignment models is as much in the feusibiZity of certain patterns near saturation as in alternate routings, one desires to test a “steady-flow” saturation, rather than one which might or might not occur, depending upon local statistical
38
W.S.
JEWELL
fluctuations. The worst such case would occur if the source emitted at a constant rate, 4 vehicles/hr, ‘forever”. Clearly, if we can finally bring this rate up to as high as 4 = Q/E, consistent with the individual arc flow constraints, then any “one-shot” fiow pattern will always be feasible. Conversely, if this is not feasible, there is a pc,rsiMi@ that a one-shot flow pattern might not be feasible, under some unlucky combinations of dynamic flow patterns. Therefore, we define a variable emission interval, 8, in hours, and assume: (H3) : The problem to be analyzed will be a steady-statejiow problem in which the source emits (and the sink receives) at a constant rate (0 = Q/B vehiclesfhr. If a solution exists for B = E, then the flow demand will be said to be “steady-feasible”; if it exists only for some 8’ > E, then the flow demand will be said to be “steady-infeasible”. (Clearly, we can always tind a “steady-feasible” flow for d = 00.) We must remember that a “steady-infeasible” flow might be “one-shot” feasible, depending upon the dynamics of the real flow pattern. 2.3. Conservation laws Using the problem definition (H3) and the assumption: (H4): Nofrow is lost, a&d, or stored in the network, except at nodes 1 and N, we obtain the following equations relating the & and @
7 (&-+&
0,
i=l
0,
i=2,3
= I -a,
,..., N-l
(6)
i=N
which are equivalent to the Kirchoff Current Conservation Laws. From the definition of the arcs as directed, and the fact that all models in Section 1.3 show a maximum flow rate, we have also (7) OG*&%* for all (i,j). (We may consider It&, = 0 for nonexistent arcs.) Any set of & satisfying (6) and (7) simultaneously is “steady-feasible”; a “steadyinfeasible” set of (&} could satisfy (6), but would have to violate at least one inequality in (7). 2.4. Choice of routes Assume the steady-state problem is feasible; we still must postulate a model to represent driver choice-behavior when faced with alternate routes. AS might be expected, fhst considerations have revolved primarily around the values of (or the driver’s perception of the values of) the trajectory time, both locally and globally; the models of Section 1.3 then give a relation between T and 4. The simplest assumption seems to be: (HRl): The ultimate pattern offlow will make the total trajectory time the same on allpaths used morn 1 to N), and less than or equal to the tune via the unusedpaths, which is referred to in traf5c as Wardrop’s Principle, or, as we shall see, is also equivalent to Kirchoff’s Voltage Conservation Law.
Models for tratEc assignment
39
Another possible assumption is: (HR2): The ultimate pattern of jlow will make the total trajectory time approximately the same on all paths used, and substantially less than the time on the unused paths, which is clearly a “driver-perception” approximation to (HRl). A third choice which is used in the literature is that of some master overall economic criterion : (HR3): Tie ultimate pattern of flow minimize some overall economic objective for the network as a whole. This assumption appears to have little a priori behavioral motivation; however, we shall show below that (HR3) and (HRl) are, in some sense, equivaZentassumptions. 2.4.1. Dimudon of HRl . The primary objection to (HRl) is that it assumes a precision in measurement which may not be available to the average driver at a given point in the network; arguments counter to this usually assume some long-range “adaptive” behavior, which may or may not be correct for the problem. However, we can easily show that such an assumption provides a unique (or almost unique) routing of the &, @WI a set of curves of Tj, vs. &, for each (i, j), as summarizd in Fig. 10 below.
7ij
I
hr
LL
curvesfrom variousmodslr
TiJ
I
‘ii
hi,
vohkh/hr
Fro. 10.
If the used paths have equal total trajectory times, then there must exist “potentials” w, for each node, such that: w3-9
G 9&J
We- w, = T&~)
for all W) for all used (i, j)
(8.Rl)
in other words, if wl = 0, then w, is the trajectory time from node 1 to node i, via any equivalent trajectory; w, is then the total trajectory time. (The second part of (8.Rl) is essentially Kirchoff’s second law; the fist part adds the Kuhn-Tucker condition necessary for +U Z 0.) It is well known from electrical circuit theory that (6), (7) and (8.Rl) provide (if feasible) a unique solution of the &, and the w,, assuming only that Fig. 10 is a non&creuring curve; there is a possible tie only when two or more paths have the same total trajectory time over a finite range of flow-rate values. Furthermore, the “contact transformation” properties of (8.Rl) tell us that, in efict,
40
W. S.JEWELL
the resulting +$, constitute an extremizing solution of (6) and (7), which minimizes the functional :
Thus, we also have an equivalence with a model of type (HR3) in which the global objective has the dimensions of the total number of vehicles “stored” in the network at any one time; that such a local hypothesis should lead to such a global objective should not surprise us any more than learning that resistors divide current so as to minimize dissipated energy (even though we usually work with voltagecurrent curves), or that ideal static bodies minimize potential energy (even though we use forcedisplacement relations). If T&,) follows the lowest curve in Fig. 10 (Q = 7& for all 06 &
F = 7 F TSj&
subject to (6) and (7)
of linear programming, whose solution methods are well known. In particular, the Ford-Fulkerson algorithm (Ford and Fulkerson, 1962) decides, through a parametric increase of @, whether the problem is steady feasible or not. In words, the algorithm looks tist for the shortest path from node 1 to node N and allocates as much flow over it as possible, until an arc saturates. It then looks for the next shortest path, saturates it and so on. In each step, it is important to take the shortest incremental path (i.e. one which may traverse a nonempty arc in the reverse sense), thus allowing the possibility of “changing one’s mind” about earlier allocations for smaller 0. Thus, the Ford-Fulkerson algorithm does not correspond to any “behavioral” model of vehicular flow. Another algorithm, based on the simplex algorithm, establishes feasible flows, regardless of trajectory times. Then, circulatory flows (incremental flows in an arbitrary direction around a cycle of arcs) are used to reduce discrepancies in establishing values of the potentials. Other possibilitie: are described in Simonnard (1966). Nonlinear curves of Q(&) can be approximated by parallel combinations of arcs, as long as Tagis a nondecreasing function of &; some quadratic algorithms are also available. Thus, Assumption (HRl) leads to an area where much work has aheady been accomplished, and algorithms are available, at least for the single origin-destination case. The primary advantage of this model is its ease of conception; its disadvantage is the assumption (HRl) of ideal behavior. 2.4.2. Discussion of HR2. Routing assumption (HR2), on the other hand, when combined with a curve of Fig. 10, has an imprecision which must be specified in some manner -perhaps by giving the tolerances to the relations: w-w 5 4 w &tlr>
(iJ) used
w3-w, < T&~)
(i,j) not used
(gR2)
or by specifying a computational algorithm which gives the desired result. An alternate point of view is to specifv the imprecision directly on the Tsj($hij) curve, as in Fig. 11. We then have an “exact” representation of the trajectory-time tolerance to be allowed. Unfortunately, for this case, there are many infinities of possible & and a discussion of the solution to be obtained cannot be separatedfrom the algorithm which is used to calculate the flows.
41
Models for traffic assignment
Algorithms which were proposed during discussions at SEMA (as specialixed to the single origin-destination case) were composed of two major steps in each cycle of the algorithm: (1) Selection of a set of paths which are timewise competitive, on the basis of results obtained from an earlier allocation; and (2) A reallocation of flow among this set of paths. Allowed
0-
Muhi,
regim
vehick/hr
Fro. 11.
Step (1) clearly involves the selection of new “candidates” (and a possible elimination of old ones) by means of a shortest route algorithm and specitlcation of a criterion (8.R2) by which this inclusion-exclusion is to be performed. Step (2) may either be: (a) A computation de nouo; (b) A weighted combination of a new set of flows with the old; or (c) An incremental change which adds new flow, or reroutes existing flow in a cycle of arcs. Method (a) seems to pose grave oscillatory problems, while (b) may take long to converge; (c) is similar to the methods of (HRl). One must also specify a stopping rule for the algorithm, but this is usually obvious, once the mechanisms of (1) and (2) are specified. Clearly, acceptance of an algorithm for (HR2) requires either (i) an analytic investigation of its convergence properties, or (ii) extensive testing of numerical examples, if possible; it should have both. It does not seem worth while to the author to restrict one’s attention only to algorithms which “work” in the way we think drivers might adapt to new situations. Application of both models would then proceed by making a parametric decrease in B (increase in (o), as described previously, until steady feasibility was, or was not, demonstrated. 3. GENERAL
MULTI-ORIGIN-DESTINATION
NETWORKS
3.1. Notation With these preliminaries finished, we may now pass quickly to the general problem in which any node may act as both origin and destination. Then, we have a problem in “multi-commodity flow”, in which the flow with different origin and/or destinations must be kept separate from the other flows; to do this, we use double superscripts, defining: @r-flow rate (vehicles/hr) of vehicles in arc (i,j) whose origin was node k and whose ultimate destination is node 1.
42
W.S.
JwmLL
The boundary variables cOra(Q#)depend only on the emitting and receiving nodes during steady-state flow; cP”# wk, in general. We assume: (HS) : The tiine$ow characteristic T&,) (Fig. 10)f oreach arc remains the same for all types ofj?ow on the arc, Md depen& only on the totalflow rate in the arc, i.e. on:
3.2. Conservation and limit laws Our general model now follows from the discussion of Sections 2.2 and 2.3. We assume continuous, constant flow with the same value of 8 for all types offlow, in order to investigate the worst conditions under which the flow might not be steady feasible. Assumptions (H3) and (H4) give, for the general case:
(9)
From (7) and (HS), we also must have: $*j” 2 0
for all (i,j>, (k, I)
(10)
and TF+ij
% Mij
Any set of flows {$tr9 satisfying (9) and (10) is steady feasible. 3.3. Potential time-relations The relations equivalent to (8.Rl) and (8R2) can be obtained by considering the programming dual to (9) and (10). Defme potentials, wrM,for each node, for each type of flow; we then have:
(11) for all (i,j) and (k, I); wdj is a slack variable which adds a “surrogate delay” to a saturated arc to prevent it from becoming “attractive” to other types of flow. From the well-known Kuhn-Tucker conditions (complementary slackness conditions), we then have the following behavior: A. (ij) used by no flows I:
x tjijmn = 0; 423kl=
))lVa
kl wi - w,H~T~~(O);
0, for all (k, Z)
wdj =
0 for all (k, 1)
Models for tratlic assignment
B. (i,j) used by at least one&w,
43
but not saturated
for all (k, I) with
&” > 0;
wSj= 0
C. (i, j) used by at least one flow, and saturated
( % lcz- wp
+sjH> 0 = @4,~
for some (k, 1); X 2 &m” =W,) rnfa
+ w4j
for all (k, Z) with
+# > 0; w$~> 0
Thus, if model (HRl) is assumed for the multi-commodity case, theCpaths used from some node k to another node I consist of potentials in “tight” (equality) relationships, with the nonused paths consisting of arcs with potentials in “slack” relationship, or having a surrogate delay preventing use by that type of flow. If a model of type (HR2) were to be used, then (11) in its various cases could be considered as an approximate relationship, and a mechanism for arc-sharing among the different types of flow specified in much greater detail. 3.4. AZgorithmsfor multi-commodity flow The “state-of-the-art” for exact solutions of multi-commodity problems is still quite primitive, with only the linear case (Q = T*,) being that analysed in the literature. In this case, it is, of course, still a linear progr amming problem, and can be solved using any variant of the simplex method; however, it is of interest to inquire whether known algorithms for single-commodity flow can be modified to provide efficient algorithms for large networks. Jewell(l958) investigated a primal-dual type of flow algorithm which allocates all flows successively over the shortest (incremental) path, due account being taken to block arcs which become saturated. However, occasionally, when there are many such arcs blocked on parallel paths for different types of flow, it turns out that one must use a small, imbedded linear-programming subroutine to calculate optimal ways to exchange blocked capacities between the d#erent types offow. Although the dimensions of this problem are generally much smaller than that of the full problem (number of blocked paths vs. the number of blocked arcs, plus some capacity constraints), it is nevertheless a general linear programming problem, and must be solved by the simplex method, except perhaps for some obvious reductions. This “hard-core difficulty” leads to the well-known phenomenon of noninteger optimal solutions, even with integer data. At the same time, Ford and Fulkerson (1958) also presented a proposal for a multicommodity algorithm which is more closely related to the simplex method, and served as the precursor of the general decomposition method. With a current basic set of feasible path flows, they use the dual to “price out” a path which will save time for some type of flow. They then perform a (general) linear pro gramming of the old plus new paths to find a new convex combination which is still feasible, but “better”, overall. This linear program is quite similar to that obtained in the other algorithm. Proposals by other authors (Haley, 1960; Matthys, 1960) seem to lead to variants of these two approaches, and all eventually have the same irreducible difhculty in programming the optimal sharing of saturated arcs.
w. s. JEWELL
44
Perhaps, with the use of a model of type (HR2), one could avoid the above difficulties; however, there is a nonestimable loss in system capacity (giving infeasibility for too small values of the Vr) when this interchange is not performed correctly. It might be hoped that if one employed a nonlinear curve of Tu( z x C# $jmn) that all difficulties with the capacities would be eliminated, as the near-satursei arcs would be “priced out” smoothly, as the flow increased. The simple example below was designed to discourage this hypothesis, since it shows that discontinuous behavior can still occur for individual flows, even if the total flow does build up slowly. Clearly, a nonlinear algorithm must take into account such discontinuities, or else ignore them in an (HR2) approach. Some results may be found in Sakarovitch (1966). 3.5. A two-commodity, nonlinear example Figure 12 shows the network. The first type of flow originates at node 1, and can go directly to 3 via (1,3), or via a common arc (0,3) ; the second type, originating at 2, either shares the common arc or goes directly via (2,3). The time-flow relationships for these three arcs are shown in the Figure; arcs (1,O) and (2,O) are short one-way streets with zero (normal) delay. We assume the exact model (HRI).
Fm. 12.
We suppose that the boundary flows are increased smoothly in the following manner : al* = 301; @aa= 4X
Steady infeasibility occurs when (Y= 5017. Figure 13 shows the two types of flow in the common arc as a parametric function of 01. We leave the interpretations of this disconcerting behavior to the reader. Table 1 below summarixe s the total trajectory times of the two types of flow vs. 01. These curves are smooth and monotone because of the optimization. TABLE 1. TOTAL TRAJECTORY TIMESvs. 01 Total trajectory times Range of parameter 01 o-10/7 IO/74 e-2516 2516-5 s-50/7
Flow (1,3)
Flow (2,3)
IO/( 1- 701/20) 1 S/( 1- 7Q+lO) 20/(1- 301/20) lO/(l -h/20) 20/(1-3,x/20) 26/2/3/(1-2a/15) 24/(1-7a/SO)
Models for traffic assignment
flow pammater,
45
a
Fk3. 13. 4. SUMMARY
In spite of the wide diversity of original assumptions about driver behavior, it was found that the problem was ultimately one in which the arc trajectory time vs. flow-rate curve had one of the forms shown in Fig. 10. To find a unique allocation of flow, under what we have called “steady-feasible” conditions, it is necessary to assume either: (i) an exact timeequalization behavior of drivers when faced with a choice between two routes (HRl), or else (ii) a (well-defined) inexact behavior (HR2). For a linearized (HRl) model, simple algorithms are available only in the single origin-destination case, the “multi-commodity” case still requiring the use of general linear programming. The solution obtained with an (HR2) model is obviously strongly dependent upon the algorithm used, and none of the methods thus far proposed seem promising, even in the nonlinear case. This hard-core programming difficulty and the suspect behavior of a simple nonlinear example suggest strongly that a great deal of work remains to be done before a general flow-allocation model is available. The only directions which seem currently promising are : (1) The use of a smoothed combination of new and old solutions, which raises storage and convergence problems; and (2) The use of “exact” programming methods until they become diEcult, and then ignoring the exchange possibilities, or switching to a heuristic method; this raises questions of unknown losses in “steady feasibility”. In any case, any method proposed must have a thorough analytic and computational testing, given our current lack of knowledge about the behavior of large, multiple-commodity networks.
46
w. s. JEWW
AcknowZedgements-The author would like to thank Dr. B. Roy and M. H. LeBoulanger for the reception and many courtesies extended to him during his stay in the Direction Scientifique of SEMA, and for permission to publish this report.
REFERENCES FORD L. R., JR. and Furxxaso N D. R. (1958). A suggested computation for maximal multi-commodity flows. Mgmt Sci. $97-101. FORDL. R., JR. and F ULKBRSON D. R. (1962). Flows in Networks. Princeton University Press, New Jersey. HAPPY K. B. (1960). The solid transportation problem. Proc. 2nd Znt. Co& Operational Research, Aix-en-Pro&& p. 293 (Abstract).- English Universitiqs Press, London. _ JEWELL W. S. (1958). Onthnal flows throuah networks (Sc.D. Thesis). Interim Technical ReDort No. 8, Operations‘Re&arch Center, Massachusetts Institute of Technology. l,The multi-commodity flow algorithm of the report has been reissued as “A primal-dual multi-commodity flow algorithm”, Research Report 66-24, Operations Research Center, University of California, Berkeley (1966).1 JEW~LLW. S. (1966). Models for traffic assignment. Note de Travail No. 47, Direction Scientitique, So&t& d’Economie et de Mathematique Appliqutes, Paris. M~rrrivs G. (1960). Flot optimum dans un r&au a capacites de faisceaux. Proc. 2nd Znt. Co& Operational Research, A&en-Pronence, p. 809. English Universities Press, London. S~x,utovrrcn M. (1966). The multi-commodity maximum flow problem. Research Report 66-25, Operations Research Center, University of California, Berkeley. SIMONNARD M. (1966). Linear Programming. Prentice-Hall, New York.
MODELS FOR TRAFFIC ASSIGNMENT W. S. JEWELL Department of Industrial Engineering and Operations Research, University of California, Berkeley (Received 8 October 1966)
INTRODUCTION FROMSeptember 1965 to February 1966, the author was a Fulbright Research Scholar, on sabbatical leave in Paris, France, and spent a portion of his time as a guest of the Direction Scientiiique of the Societe d’Economie et de Mathematique AppliquQs (SEMA). Research was in progress at that time on properties of traffic assignment models, and the author was fortunate enough to have extended discussions with H. LeBoulanger and B. Roy, principal designers of the proprietary trafEc-assignment programs, ATCODE and ADECODE. This paper is a revision of a research paper (Jewell, 1966) written as a result of these discussions. At that time, several different models and algorithms were being proposed which seemed quite disparate; furthermore, it was not clear whether we could hope to separate the two concepts of model and method, nor whether stress should be placed on an exact or approximate procedure. The attempt of this paper was to provide a systematic exploration of the several different approaches, emphasizing the common features, if possible, but also contrasting their differences. We proceed slowly, in an axiomatic manner, starting with the single-route case, and gradually building up to the most general type of multi-commodity network. The ultimate aim was, of course, to fmd algorithms which were both realistic and efficient; but, in this, we were not successful. In fact, although certain concepts and model relationships are clearer than before, certain “hard-core” difficulties in traffic assignment will apparently need further research and computational testing. 1. THE SINGLE
ROUTE
CASE
1.1. Notation Consider a certain origin-destination (OD) pair, and some path which connects them; in this section we assume that all flow between this OD pair must use this single route. During some emission interval, E hours, a totalflow, Q(E) vehicles, will be generated and emitted at the origin. We suppose E is determined once and for all by certain exterior considerations (data availability, quantization of time-variations, control cycles, etc.), so that we will not henceforth emphasize the dependence of Q on E. Our tist assumption about this flow is: (Hl): The vehicles are emittedfrom the origin onto the route at a uniform and constantflow rate until the “stock” Q is depleted. Call this uniform and constant flow rate 4 vehicles/hr, and the time of depletion 8 hr. We will admit models in which 6# E, i.e. the vehicles may be sent out during a shorter or longer period than that which determined the value of Q. Relative to the route, which is D miles long (and, temporarily, one lane wide), we assume : 31
32
W.S.
JEWELL
(H2): when they are being emitted, vehicles travel at a constant velocity and headway (time spacing) along the route to the destination, thus defining: n-the maximum number of vehicles on the route at any one time, A-the maximum density of vehicles, in vehicles/miles, n-the common velocity of the vehicles, in mile@, h-the common headway between vehicles, in miles, and +-the common trajectory time for an individual vehicle to traverse the route in hours. These definitions are summariz ed in Fig. 1, in terms of a reservoir pipeline analogy. Normally, we consider the variables as Given :
CL Q 6%
and Unknown: +, 6, n, A, v, h, 7.
Maximum density-X,
Stock of 0 veh&s
veWclMmil0
Oripin
-
%e,t-
-
-
-
r,
hr
-
FIG.
-
-
d
Destination
1.
1.2. Obvious relations From hypotheses (Hl) and (H2), we can easily agree upon the five following relations among the seven unknowns: A = l/h
(1)
D = VT
(2)
n = AD = D/h
(may be noninteger)
Q= W # = n/T = hv
(3) (4)
(or AuS = Q)
(5)
This leaves two independent variables, say (A, v), or (n, v), or (6, v), etc., whose values must be known to specify a unique solution to ah others. Figures 2 and 3 then show the rate of emission, rate of reception and number of vehicles “in the pipeline” as functions of time, under our assumptions. 1.3. Velocity-density relationships The three submodels which follow have the common property that 6 is assumed to be tied by the algorithm, and the remaining ambiguity is removed by assumptions about local driver behavior in a certain velocity-density regime.
33
Models for trai& assignment
1.3.1. Local behavior model A (linear density-uelocity). We- assume : @Al): There is a maximum density of vehicles, f&. (HA2): There is a (maximum) ‘free” velocity, V,. (HA3): When not otherwise limited by (HAl) and (HA2), there is a linearly decreasing relationship which limits the values of h and v.
Rates of emission-reception
//
I
,
(
0
8
r
Emission
ReceptimT~8
)
Time,
hr
FIG. 2.
4
Number in route
vehicles
0
rt.8
8
c Time,
hr
FIG. 3.
I
X,
vehicks/mtle
Vf FIG.
Li
Y,
miles/hr
4.
Thus, under these hypotheses the shaded area in Fig. 4 represents the only allowed values of A and v. Implicit in the above is the assumption that the parameters V,, A,, AF and V_ can be measured from real data. It appears that another assumption is needed at this point to specify unique values of the variables. However, a later transformation would reintroduce some redundancy, so we shall postpone any further assumptions until that point. 3
34
W. S. JEWELL
Figure 4 may be written:
\A,,
v=
v,
It follows from (5) that a curve of Q vs. v would have one of the forms shown in Fig 5; # may have a local maximum, M, within the interval V, < v < V, [cases (a) and (b)], or the constrained maximum may lie on the boundary v = V, [case (c)l. In other words, M = min [1/2U; V,]
642)
= 4wVrhi+V~ &i-A,
G43)
where u
is the value shown in Fig. 4 as the v abscissa intercept of the straight-line approximation. Thus, an internal maximum in Fig. 5 is associated with a strong decrease in maximum density vs. velocity. vahicles/hr
v.
V*
vF
v,
milcshr
FIG. 5.
u
0
+,
vehickdhr
FIG. 6.
For reasons which will become evident later, we transform Fig. 5 to a CT,+) plot in Fig. 6, by using (2); the two major cases are shown, after renormalizing M to the same value. The values of the new parameters shown are easily derived in terms of the other given data.
35
Models for traffic assignment
We leave the discussion of a unique choice of operating-curve in Fig. 6 until after discussing model B. 1.3.2. Local behavior moa21 B (car-following). We assume (HBl) and (HB2) identical with (HAl) and (HA2), and (HB3): When not otherwise limited, vehicles maintain, as a minimum, an (excess) headway which is proportional to their common velocity. In terms of headways, this means: 06v
49 ha
H,,+ar(v-V,), t 4 = Ii, + 45
W
v,
v = v,
with tl, = l/AM, OLand VP to be determined by experiment. This gives a density-velocity region shown in Fig. 7: Ai&z?
o
A,l+A&&-v~),
032)
v,
A F = l/%,
v = v,
-r
v,
miles&r
FIG. 7.
It is easy to show that this region does not give an internal maximum to the boundary curve of 4 vs. v: therefore, the curve of T vs. 4 has the same shape as Fig. 6c. 1.3.3. Variations of model B. In an attempt to model a “more realistic” behavior of drivers, model B was extended by the following hypotheses: (HB3’) : The increase in headway is more than proportional to the common velocity [i.e. there is a quadratic, cubic, etc., component in (HB3)]. (HB4) : There is more than one lane in the road, the monber of effective lanes being a decreasing function of velocity (i.e. obstacles at the side of the road do not interfere with slowly moving vehicles, which “flow” around them, but do reduce eflective capacity at higher speeds). Except for differences in algebraic forms, none of these changes affects the shape of Fig. 5c for model B, i.e. neither gives an internal maximum to 4. 1.3.4. Resolution of ambiguities in models A and B. To have an unique representation of T vs. 4, we must invoke an “economic” or “behavioral” principle which will select an unique curve from the shaded area in Fig. 6.
W.S.
36
JEWELL
(HABl): For a givenfIow rate of vehicles, 4, the drivers will interact so as to minimize their trajectory time, 7. This assumption will result in the two solid curves shown below in Fig. 8. TP
hr i
I
I TF
i I
c
M 91 vehiclesAw FIQ. 8.
To eliminate the troublesome truncation on the right-hand side of Fig. 8, we may add the possibility of a certain “stretching” of M (alternate passage by side streets, alleys, etc.) with a very large increase in T, which would result in the vertical dashed line shown in Fig. 8. (HAB2): For values of t.#infinitesimally greater than M, T increases without limit. This possibility might occur, for example, when M was a critical point at which accidents, “jam-ups”, etc., and other drastic driver behavior occurred, forcing the seeking out of alternate secondary routes, waiting for tow-trucks, etc.--clearly an undesirable or “infeasible” (for planners) portion of the curve. 1.3.5. Local behavior model C (queueing). A different model is obtained if we interpret the flow variable, 4, and the trajectory time, T, as averages of random variables, and assume : (HCl) : Vehicles are emitted randomly (Poisson process). (HC2) : Every vehicle travels at some maximum veZocity v, over the route until it reaches the destination. (HC3): There is a “service mechanism” (trafic light, bottleneck, toll-gate, narrowing of the roadway, etc.) infront of the destination which “processes” vehicles with a mean processing time, S hr, and variance aZ. All vehicles not being “served” must wait in a queue, whose dimensions are small compared to D (implicit assumption of statistical equilibrium). Then, from the well-known Pollaczek-Khintchine formula of queueing theory: 7 =
wM+s+
+(s2+ a) 2(1_#)
K4
=Tfl+l_+S which has the shape indicated in Fig. 9. A feature of this model is the natural appearance of a limiting flow rate M= l/S beyond which there can be no equilibrium.
37
Models for traffic assignment
If there are multiple routes speaking, apply next one) is not
several bottlenecks along the route (as would be the case in considering with differing numbers of arcs), then the above theory does not, strictly to the resulting series gueae, as the output of one queue (which feeds the Poisson.
Fro.
9.
However, if all service-times are exponentially equivalent of (Cl) will be:
distributed
(OS= S*), then the series
where the index i refers to each (free stretch-bottleneck) segment. In terms of dominant behavior for 4 large, it is clear in any case that it is the minimum capacity M = [ma+ &j-l which determines the shape of the CT,4) curve, and thus (Cl) can be expected to provide an approximate form for any “real” curve for this model. “Parallel service” leads to similar remarks, after redefinition of M. 2. NETWORKS
WITH
ONE
ORIGIN-DESTINATION
PAIR
2.1. Notation Now consider a simplified trafhc network in which nodes 1 and N are the unique origin and destination for the flow Q, all other nodes being transhipment points. We now have, for each directed arc (i, j) between nodes i and j, the route parameters discussed in (1 .l), i.e. % IL,, +ii, PDF,etc., assumed constant over this arc; in what follows, we shall consider primarily the arc variables & and TV,. Traffic from node 1 to node N may follow any or all possible directed paths of arcs leading from 1 to N. 2.2. “Continuous” flow argument Because of the remarks made in Section 1.2 (particularly Fig. 3), it is clear that if many different paths are followed during the emission interval, the resulting flow-rate vs. time curve would be different in the different arcs. In fact certain critical arcs may never appear to be “stressed”, due to the fortuitous “phasing in” of arrivals over that segment. Certainly, all the flow may not reach N before the emission interval, E, is over. Because the interest of traffic assignment models is as much in the feusibiZity of certain patterns near saturation as in alternate routings, one desires to test a “steady-flow” saturation, rather than one which might or might not occur, depending upon local statistical
38
W.S.
JEWELL
fluctuations. The worst such case would occur if the source emitted at a constant rate, 4 vehicles/hr, ‘forever”. Clearly, if we can finally bring this rate up to as high as 4 = Q/E, consistent with the individual arc flow constraints, then any “one-shot” fiow pattern will always be feasible. Conversely, if this is not feasible, there is a pc,rsiMi@ that a one-shot flow pattern might not be feasible, under some unlucky combinations of dynamic flow patterns. Therefore, we define a variable emission interval, 8, in hours, and assume: (H3) : The problem to be analyzed will be a steady-statejiow problem in which the source emits (and the sink receives) at a constant rate (0 = Q/B vehiclesfhr. If a solution exists for B = E, then the flow demand will be said to be “steady-feasible”; if it exists only for some 8’ > E, then the flow demand will be said to be “steady-infeasible”. (Clearly, we can always tind a “steady-feasible” flow for d = 00.) We must remember that a “steady-infeasible” flow might be “one-shot” feasible, depending upon the dynamics of the real flow pattern. 2.3. Conservation laws Using the problem definition (H3) and the assumption: (H4): Nofrow is lost, a&d, or stored in the network, except at nodes 1 and N, we obtain the following equations relating the & and @
7 (&-+&
0,
i=l
0,
i=2,3
= I -a,
,..., N-l
(6)
i=N
which are equivalent to the Kirchoff Current Conservation Laws. From the definition of the arcs as directed, and the fact that all models in Section 1.3 show a maximum flow rate, we have also (7) OG*&%* for all (i,j). (We may consider It&, = 0 for nonexistent arcs.) Any set of & satisfying (6) and (7) simultaneously is “steady-feasible”; a “steadyinfeasible” set of (&} could satisfy (6), but would have to violate at least one inequality in (7). 2.4. Choice of routes Assume the steady-state problem is feasible; we still must postulate a model to represent driver choice-behavior when faced with alternate routes. AS might be expected, fhst considerations have revolved primarily around the values of (or the driver’s perception of the values of) the trajectory time, both locally and globally; the models of Section 1.3 then give a relation between T and 4. The simplest assumption seems to be: (HRl): The ultimate pattern offlow will make the total trajectory time the same on allpaths used morn 1 to N), and less than or equal to the tune via the unusedpaths, which is referred to in traf5c as Wardrop’s Principle, or, as we shall see, is also equivalent to Kirchoff’s Voltage Conservation Law.
Models for tratEc assignment
39
Another possible assumption is: (HR2): The ultimate pattern of jlow will make the total trajectory time approximately the same on all paths used, and substantially less than the time on the unused paths, which is clearly a “driver-perception” approximation to (HRl). A third choice which is used in the literature is that of some master overall economic criterion : (HR3): Tie ultimate pattern of flow minimize some overall economic objective for the network as a whole. This assumption appears to have little a priori behavioral motivation; however, we shall show below that (HR3) and (HRl) are, in some sense, equivaZentassumptions. 2.4.1. Dimudon of HRl . The primary objection to (HRl) is that it assumes a precision in measurement which may not be available to the average driver at a given point in the network; arguments counter to this usually assume some long-range “adaptive” behavior, which may or may not be correct for the problem. However, we can easily show that such an assumption provides a unique (or almost unique) routing of the &, @WI a set of curves of Tj, vs. &, for each (i, j), as summarizd in Fig. 10 below.
7ij
I
hr
LL
curvesfrom variousmodslr
TiJ
I
‘ii
hi,
vohkh/hr
Fro. 10.
If the used paths have equal total trajectory times, then there must exist “potentials” w, for each node, such that: w3-9
G 9&J
We- w, = T&~)
for all W) for all used (i, j)
(8.Rl)
in other words, if wl = 0, then w, is the trajectory time from node 1 to node i, via any equivalent trajectory; w, is then the total trajectory time. (The second part of (8.Rl) is essentially Kirchoff’s second law; the fist part adds the Kuhn-Tucker condition necessary for +U Z 0.) It is well known from electrical circuit theory that (6), (7) and (8.Rl) provide (if feasible) a unique solution of the &, and the w,, assuming only that Fig. 10 is a non&creuring curve; there is a possible tie only when two or more paths have the same total trajectory time over a finite range of flow-rate values. Furthermore, the “contact transformation” properties of (8.Rl) tell us that, in efict,
40
W. S.JEWELL
the resulting +$, constitute an extremizing solution of (6) and (7), which minimizes the functional :
Thus, we also have an equivalence with a model of type (HR3) in which the global objective has the dimensions of the total number of vehicles “stored” in the network at any one time; that such a local hypothesis should lead to such a global objective should not surprise us any more than learning that resistors divide current so as to minimize dissipated energy (even though we usually work with voltagecurrent curves), or that ideal static bodies minimize potential energy (even though we use forcedisplacement relations). If T&,) follows the lowest curve in Fig. 10 (Q = 7& for all 06 &
F = 7 F TSj&
subject to (6) and (7)
of linear programming, whose solution methods are well known. In particular, the Ford-Fulkerson algorithm (Ford and Fulkerson, 1962) decides, through a parametric increase of @, whether the problem is steady feasible or not. In words, the algorithm looks tist for the shortest path from node 1 to node N and allocates as much flow over it as possible, until an arc saturates. It then looks for the next shortest path, saturates it and so on. In each step, it is important to take the shortest incremental path (i.e. one which may traverse a nonempty arc in the reverse sense), thus allowing the possibility of “changing one’s mind” about earlier allocations for smaller 0. Thus, the Ford-Fulkerson algorithm does not correspond to any “behavioral” model of vehicular flow. Another algorithm, based on the simplex algorithm, establishes feasible flows, regardless of trajectory times. Then, circulatory flows (incremental flows in an arbitrary direction around a cycle of arcs) are used to reduce discrepancies in establishing values of the potentials. Other possibilitie: are described in Simonnard (1966). Nonlinear curves of Q(&) can be approximated by parallel combinations of arcs, as long as Tagis a nondecreasing function of &; some quadratic algorithms are also available. Thus, Assumption (HRl) leads to an area where much work has aheady been accomplished, and algorithms are available, at least for the single origin-destination case. The primary advantage of this model is its ease of conception; its disadvantage is the assumption (HRl) of ideal behavior. 2.4.2. Discussion of HR2. Routing assumption (HR2), on the other hand, when combined with a curve of Fig. 10, has an imprecision which must be specified in some manner -perhaps by giving the tolerances to the relations: w-w 5 4 w &tlr>
(iJ) used
w3-w, < T&~)
(i,j) not used
(gR2)
or by specifying a computational algorithm which gives the desired result. An alternate point of view is to specifv the imprecision directly on the Tsj($hij) curve, as in Fig. 11. We then have an “exact” representation of the trajectory-time tolerance to be allowed. Unfortunately, for this case, there are many infinities of possible & and a discussion of the solution to be obtained cannot be separatedfrom the algorithm which is used to calculate the flows.
41
Models for traffic assignment
Algorithms which were proposed during discussions at SEMA (as specialixed to the single origin-destination case) were composed of two major steps in each cycle of the algorithm: (1) Selection of a set of paths which are timewise competitive, on the basis of results obtained from an earlier allocation; and (2) A reallocation of flow among this set of paths. Allowed
0-
Muhi,
regim
vehick/hr
Fro. 11.
Step (1) clearly involves the selection of new “candidates” (and a possible elimination of old ones) by means of a shortest route algorithm and specitlcation of a criterion (8.R2) by which this inclusion-exclusion is to be performed. Step (2) may either be: (a) A computation de nouo; (b) A weighted combination of a new set of flows with the old; or (c) An incremental change which adds new flow, or reroutes existing flow in a cycle of arcs. Method (a) seems to pose grave oscillatory problems, while (b) may take long to converge; (c) is similar to the methods of (HRl). One must also specify a stopping rule for the algorithm, but this is usually obvious, once the mechanisms of (1) and (2) are specified. Clearly, acceptance of an algorithm for (HR2) requires either (i) an analytic investigation of its convergence properties, or (ii) extensive testing of numerical examples, if possible; it should have both. It does not seem worth while to the author to restrict one’s attention only to algorithms which “work” in the way we think drivers might adapt to new situations. Application of both models would then proceed by making a parametric decrease in B (increase in (o), as described previously, until steady feasibility was, or was not, demonstrated. 3. GENERAL
MULTI-ORIGIN-DESTINATION
NETWORKS
3.1. Notation With these preliminaries finished, we may now pass quickly to the general problem in which any node may act as both origin and destination. Then, we have a problem in “multi-commodity flow”, in which the flow with different origin and/or destinations must be kept separate from the other flows; to do this, we use double superscripts, defining: @r-flow rate (vehicles/hr) of vehicles in arc (i,j) whose origin was node k and whose ultimate destination is node 1.
42
W.S.
JwmLL
The boundary variables cOra(Q#)depend only on the emitting and receiving nodes during steady-state flow; cP”# wk, in general. We assume: (HS) : The tiine$ow characteristic T&,) (Fig. 10)f oreach arc remains the same for all types ofj?ow on the arc, Md depen& only on the totalflow rate in the arc, i.e. on:
3.2. Conservation and limit laws Our general model now follows from the discussion of Sections 2.2 and 2.3. We assume continuous, constant flow with the same value of 8 for all types offlow, in order to investigate the worst conditions under which the flow might not be steady feasible. Assumptions (H3) and (H4) give, for the general case:
(9)
From (7) and (HS), we also must have: $*j” 2 0
for all (i,j>, (k, I)
(10)
and TF+ij
% Mij
Any set of flows {$tr9 satisfying (9) and (10) is steady feasible. 3.3. Potential time-relations The relations equivalent to (8.Rl) and (8R2) can be obtained by considering the programming dual to (9) and (10). Defme potentials, wrM,for each node, for each type of flow; we then have:
(11) for all (i,j) and (k, I); wdj is a slack variable which adds a “surrogate delay” to a saturated arc to prevent it from becoming “attractive” to other types of flow. From the well-known Kuhn-Tucker conditions (complementary slackness conditions), we then have the following behavior: A. (ij) used by no flows I:
x tjijmn = 0; 423kl=
))lVa
kl wi - w,H~T~~(O);
0, for all (k, Z)
wdj =
0 for all (k, 1)
Models for tratlic assignment
B. (i,j) used by at least one&w,
43
but not saturated
for all (k, I) with
&” > 0;
wSj= 0
C. (i, j) used by at least one flow, and saturated
( % lcz- wp
+sjH> 0 = @4,~
for some (k, 1); X 2 &m” =W,) rnfa
+ w4j
for all (k, Z) with
+# > 0; w$~> 0
Thus, if model (HRl) is assumed for the multi-commodity case, theCpaths used from some node k to another node I consist of potentials in “tight” (equality) relationships, with the nonused paths consisting of arcs with potentials in “slack” relationship, or having a surrogate delay preventing use by that type of flow. If a model of type (HR2) were to be used, then (11) in its various cases could be considered as an approximate relationship, and a mechanism for arc-sharing among the different types of flow specified in much greater detail. 3.4. AZgorithmsfor multi-commodity flow The “state-of-the-art” for exact solutions of multi-commodity problems is still quite primitive, with only the linear case (Q = T*,) being that analysed in the literature. In this case, it is, of course, still a linear progr amming problem, and can be solved using any variant of the simplex method; however, it is of interest to inquire whether known algorithms for single-commodity flow can be modified to provide efficient algorithms for large networks. Jewell(l958) investigated a primal-dual type of flow algorithm which allocates all flows successively over the shortest (incremental) path, due account being taken to block arcs which become saturated. However, occasionally, when there are many such arcs blocked on parallel paths for different types of flow, it turns out that one must use a small, imbedded linear-programming subroutine to calculate optimal ways to exchange blocked capacities between the d#erent types offow. Although the dimensions of this problem are generally much smaller than that of the full problem (number of blocked paths vs. the number of blocked arcs, plus some capacity constraints), it is nevertheless a general linear programming problem, and must be solved by the simplex method, except perhaps for some obvious reductions. This “hard-core difficulty” leads to the well-known phenomenon of noninteger optimal solutions, even with integer data. At the same time, Ford and Fulkerson (1958) also presented a proposal for a multicommodity algorithm which is more closely related to the simplex method, and served as the precursor of the general decomposition method. With a current basic set of feasible path flows, they use the dual to “price out” a path which will save time for some type of flow. They then perform a (general) linear pro gramming of the old plus new paths to find a new convex combination which is still feasible, but “better”, overall. This linear program is quite similar to that obtained in the other algorithm. Proposals by other authors (Haley, 1960; Matthys, 1960) seem to lead to variants of these two approaches, and all eventually have the same irreducible difhculty in programming the optimal sharing of saturated arcs.
w. s. JEWELL
44
Perhaps, with the use of a model of type (HR2), one could avoid the above difficulties; however, there is a nonestimable loss in system capacity (giving infeasibility for too small values of the Vr) when this interchange is not performed correctly. It might be hoped that if one employed a nonlinear curve of Tu( z x C# $jmn) that all difficulties with the capacities would be eliminated, as the near-satursei arcs would be “priced out” smoothly, as the flow increased. The simple example below was designed to discourage this hypothesis, since it shows that discontinuous behavior can still occur for individual flows, even if the total flow does build up slowly. Clearly, a nonlinear algorithm must take into account such discontinuities, or else ignore them in an (HR2) approach. Some results may be found in Sakarovitch (1966). 3.5. A two-commodity, nonlinear example Figure 12 shows the network. The first type of flow originates at node 1, and can go directly to 3 via (1,3), or via a common arc (0,3) ; the second type, originating at 2, either shares the common arc or goes directly via (2,3). The time-flow relationships for these three arcs are shown in the Figure; arcs (1,O) and (2,O) are short one-way streets with zero (normal) delay. We assume the exact model (HRI).
Fm. 12.
We suppose that the boundary flows are increased smoothly in the following manner : al* = 301; @aa= 4X
Steady infeasibility occurs when (Y= 5017. Figure 13 shows the two types of flow in the common arc as a parametric function of 01. We leave the interpretations of this disconcerting behavior to the reader. Table 1 below summarixe s the total trajectory times of the two types of flow vs. 01. These curves are smooth and monotone because of the optimization. TABLE 1. TOTAL TRAJECTORY TIMESvs. 01 Total trajectory times Range of parameter 01 o-10/7 IO/74 e-2516 2516-5 s-50/7
Flow (1,3)
Flow (2,3)
IO/( 1- 701/20) 1 S/( 1- 7Q+lO) 20/(1- 301/20) lO/(l -h/20) 20/(1-3,x/20) 26/2/3/(1-2a/15) 24/(1-7a/SO)
Models for traffic assignment
flow pammater,
45
a
Fk3. 13. 4. SUMMARY
In spite of the wide diversity of original assumptions about driver behavior, it was found that the problem was ultimately one in which the arc trajectory time vs. flow-rate curve had one of the forms shown in Fig. 10. To find a unique allocation of flow, under what we have called “steady-feasible” conditions, it is necessary to assume either: (i) an exact timeequalization behavior of drivers when faced with a choice between two routes (HRl), or else (ii) a (well-defined) inexact behavior (HR2). For a linearized (HRl) model, simple algorithms are available only in the single origin-destination case, the “multi-commodity” case still requiring the use of general linear programming. The solution obtained with an (HR2) model is obviously strongly dependent upon the algorithm used, and none of the methods thus far proposed seem promising, even in the nonlinear case. This hard-core programming difficulty and the suspect behavior of a simple nonlinear example suggest strongly that a great deal of work remains to be done before a general flow-allocation model is available. The only directions which seem currently promising are : (1) The use of a smoothed combination of new and old solutions, which raises storage and convergence problems; and (2) The use of “exact” programming methods until they become diEcult, and then ignoring the exchange possibilities, or switching to a heuristic method; this raises questions of unknown losses in “steady feasibility”. In any case, any method proposed must have a thorough analytic and computational testing, given our current lack of knowledge about the behavior of large, multiple-commodity networks.
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AcknowZedgements-The author would like to thank Dr. B. Roy and M. H. LeBoulanger for the reception and many courtesies extended to him during his stay in the Direction Scientifique of SEMA, and for permission to publish this report.
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