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A probit-based stochastic user equilibrium assignment model
Trunspn Res.-8, Vol. 31. No. 4, pp. 341 355. 1997 c 1997 Elsevier Science Ltd
All rights reserved. Printed in Great Britain 0191-2615197 $17.00+0.00
Pergamon PII: SO191-2615(96)00028-8
A PROBIT-BASED STOCHASTIC USER EQUILIBRIUM ASSIGNMENT MODEL M. J. MAHER* and P. C. HUGHES Department
of Civil and Transportation (Received
22 Februarv
Engineering,
Napier
1995; in revised/inn
Ilniversity. 23 April
Edinburgh,
Scotland
1996)
Abstract--Stochastic methods of traffic assignment have received much less attention in the literature than those based on deterministic user equilibrium (UE). The two best known methods for stochastic assignment are those of Burrell and Dial, both of which have certain weaknesses which have limited their usefulness. Burrell’s is a Monte Carlo method, whilst Dial’s logit method takes no account of the correlation, or overlap. between alternative routes. This paper describes, firstly, a probit stochastic method (SAM) which does not suffer from these weaknesses and which does not require path enumeration. While SAM has a different routefinding methodology to Burrell. it is shown that assigned flows are similar. The paper then goes on to show how, by incorporating capacity restraint (in the form of link-based cost-flow functions) into this stochastic loading method, a new stochastic user equilibrium (SUE) model can be developed. The SUE problem can be expressed as a mathematical programming problem. and its solution found by an iterative search procedure similar to that of the Frank-Wolfe algorithm commonly used to solve the UE problem. The method is made practicable because quantities calculated during the stochastic loading process make the SUE objective function easy to compute. As a consequence, at each iteration. the optimal step length along the search direction can be estimated using a simple interpolation method. The algorithm is demonstrated by applying it successfully to a number of test problems, in which the algorithm shows good behaviour. It is shown that, as the values of parameters describing the variability and degree of capacity restraint are varied, the SUE solution moves smoothly between the UE and pure stochastic solutions. ~(5 I997 Elsevier Science Ltd
I. INTRODUCTION
It is commonly observed in empirical studies of driver route choice that a variety of routes are chosen between any origin+lestination (O-D) pair. Traditionally, traffic assignment models fall into two classes, depending on the form of mechanism used to produce this ‘multi-routeing’ or ‘route spreading’ effect. The first class consists of models which incorporate the effect of congestion through the use of capacity restraint and which aim to find the User Equilibrium (UE) solution, in accordance with Wardrop’s First Principle (Wardrop, 1952). They are deterministic in nature, assuming that drivers are perfectly rational and identical, and have complete and perfect knowledge of network conditions. The problem of finding the Wardrop equilibrium for separable link cost-flow functions can be posed as a minimization problem (Beckmann et al., 1956). Most implementations of UE minimize the Beckmann objective function, using the method due to Frank and Wolfe (1956). The second class consists of probabilistic or stochastic methods, which aim to model the variations in driver perceptions or preferences, and reflect the imperfect knowledge drivers have of conditions throughout the network; drivers choose different routes through the network based on these different perceptions. Pure stochastic methods do not include any capacity restraint. Recently, a third class has received some attention. This class contains ‘time-cost’ models which derive their multi-routing behaviour from the idea that the value of time has a probability distribution, rather than taking a single value. The models, put forward by Leurent (1994) and Dial (1995), extend and generalise the notion of multiple user classes, each with its own form of generalised cost. UE models are widely used in application to congested urban networks, but perform less well in less congested, inter-urban networks. Stochastic methods are more suitable for lightly congested networks, typically inter-urban networks. However, it would be far better to have one method *Author
for correspondence.
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which incorporated both congestion and stochastic effects and which could be applied equally well to all types of network. As a consequence, several authors have, in recent years, developed techniques for the solution of the stochastic user equilibrium (SUE) problem. Daganzo and Sheffi (1977) first formulated SUE, showing it as a generalisation of user equilibrium, and Sheffi and Powell (1982) showed that the SUE problem could be posed as a mathematical programming problem. Fisk (1980) proposed an alternative formulation for the logit-based problem. Most implementations of SUE have used the Method of Successive Averages (MSA), though Chen and Alfa (1991) Bell et al. (1993), Leurent (1995) and Akamatsu (1995) have each made significant progress towards the development of algorithms which involve the evaluation and minimisation of the Fisk function. Most authors agree that probit-based models are preferable to logit-based ones, which take no account of overlapping, or correlated, routes. Their development in SUE algorithms has been restricted, however, by the apparent need for complete path enumeration or Monte Carlo techniques. This paper will describe a new SUE algorithm which suffers from neither of these disadvantages, formed by incorporating capacity restraint (in the form of link-based cost-flow functions) in a recently-developed probit-based stochastic method. The paper has the following structure. Section 2 will review previous pure stochastic methods, and the new probit stochastic model will be outlined in Section 3. Section 4 describes the phenomenon of ‘deadlock’ and how it is overcome, and Section 5 will show the development of the new SUE algorithm. The application of the algorithm to test networks will be described in Section 6, comparisons between the SUE, UE and pure stochastic solutions in Section 7, and tests on a larger network are reported in Section 8. Finally, Section 9 gives a summary and points the way to possible extensions of the model. 2. STOCHASTIC
METHODS
Two main mechanisms have been used for stochastic assignment, based on methods due to Burrell (1968) and Dial (1971). Burrell-type methods are Monte Carlo, depending on repeated sampling of link cost distributions, either uniform or normal, and loading portions of the demand, all-or-nothing, based on the sampled costs [Burrell originally used uniform distributions, while Sheffi and Powell (1982) suggested to use the Normal distribution]. As Van Vuren (1994) pointed out, this suffers from the disadvantage that successive runs of a program, with different random seeds, will give different answers, unless considerable computation time is spent in getting the answers to converge (see Section 3). The property of repeatability is important when comparing two cases (for instance a network with and without a new road scheme), when the estimation of benefit is partially obscured by the noise. Dial’s STOCH method is based on the use of a logistic function for the split between any pair of routes. This gives rise to a conceptually simple and efficient method which gives repeatable results. It suffers however, from one considerable weakness. and that is that there is no way to represent the correlations between route cost distributions when routes overlap. So. for instance, slight variations on a route through a town will all be allocated equal traffic, and a bypass route will be under-assigned in comparison. Both of the main existing stochastic assignment models, Burrell and Dial, have certain disadvantages, then, which have limited their usefulness. A method which is based on the assumption of normal link cost distributions (a ‘probit’ model) is, in principle, preferable to both, since it can take account of correlations between routes and, as it does not rely on Monte Carlo techniques, is repeatable. Until recently, probit-based methods relied on the enumeration of all possible routes and did not, therefore, offer an efficient, computationally feasible approach for networks of practical size. 3. THE SAM STOCHASTIC
METHOD
The SAM (Stochastic Assignment Method) model was first described in Maher (1992) and owes its origins to the SCATA method of Robertson and Kennedy (1979), although this method assumed a uniform distribution for link costs and was never developed to cope with anything other than very simple network configurations. In SAM link costs are assumed to be Normally
A probit-based
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distributed and independent with specified means and variances. For simplicity, we assume a constant variability parameter /? (the ratio of variance to mean) for all links in the network, although this parameter could be allowed to take different values for different links, with no alterations to the basic method. The full details of the method have been given in Maher (1992) and will not be repeated here, but the basic steps for loading the demand, origin by origin, are as follows: 1. Scan outwards from the origin to the ends of all links which leave the origin. This gives the travel time distributions from the origin to the ends of those links (these random variables will be referred to as Yi, for any link i in the network). 2. Find a node n, such that the distributions of the Yi variables have been determined for all links i which enter node n (the case in which no such node can be found will be discussed later). ‘Merge’ at this node, i.e. determine the distribution of the travel time IV,, from the origin to this node. The random variable W, is defined as: W,, = min C( Y;)
(1)
ieB.
3. 4.
5.
6.
where B,, is the set of all links which enter node n (the ‘before’ links). The merge calculations also give the probabilities pin that traffic entering node n from the origin would have travelled on each of the before links i. These probabilities will later be used to split the traffic between the entering links. The node n is now said to be ‘complete’. The calculations are based on Clark’s approximation (Clark, 1961): the minimum of two Normal random variables is itself approximately Normal. As well as the mean and variance of W,,, all relevant covariances are found. Scan out from node n, which was completed in step 2, estimating the distributions of the Yj for all jEA,, the set of ‘after’ links for node n. If there are any nodes which are not complete, return to 2. Steps l-4 comprise the forward pass for the current origin, in which all the required probabilities Pi, are calculated. We then carry out the backward pass to load the traffic. Using the splitting probabilities Pi, from step 2, load the traffic, starting at the last node at which merging took place, and work back through each node n towards the origin, in the reverse order to which merging took place. Repeat steps l-5 for each origin in turn,
For each origin node r, the expected value of the minimum perceived travel cost to destination node s, E( W,), will be denoted by S,,, sometimes referred to as the ‘satisfaction function’. Several authors have carried out work on alternatives to the Clark approximation (Horowitz et al., 1982; Kamakura, 1989), but Clark was used here because of its simplicity. Future work may involve comparing the performance of the alternatives, implemented in the SAM model. The network of Fig. 1 can be used to show how SAM represents the correlation between routes, succeeding where Dial is deficient. Figure 2 shows the loading on the upper link for different values of X. As expected, the loading changes from 50 to 33% as x increases. The SAM stochastic loading method, then, can be seen as an alternative to the Burrell and Dial methods. It is systematic and easy to apply, and it has some important advantages over
Fig.
I. Network with correlated
routes.
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Flow on Upper Link (%) 50
SAM 45
30’
0
I 1
2
3
4
5
6
7
8
9
10
X
Fig. 2. Comparison between Dial and SAM. Network with correlated routes.
those methods: it is numerical rather than Monte Carlo, and it is probit-based rather than logitbased. Although it is tempting to think of the SAM method as a numerical version of a Burrell-type method with Normal link costs [as proposed by Sheffi and Powell (Sheffi and Powell, 1982)], there is a subtle difference in the underlying routeing principle. This may be seen from the following simple example. The 5-link network in Fig. 3 has three routes; each route cost distribution has a mean of (a + b) units. The link cost random variables are denoted by XABetc. and the route costs by:
TI =
XAB
+
XED
T2
=
XAB
+
XBC
T3
=
XAC
+
XCD
+
XCD
(2)
In Burrell, the choice is made between whole routes, so that by symmetry the two outer routes, 1 and 3, will receive the same amount of traffic, and links AB and CD will have the same flow, as will links AC and BD. In SAM, however, the method progresses through the network using a ‘Markov’ assumption; that is, the pattern of route choice from an intermediate node N to the destination is independent of the pattern of route choice from the origin to N. When we merge at node C, the two incoming routes are seen to be of equal cost. The split will be 50:50; flows on links AC and BC will be equal, both being half the flow on link CD. For traffic destined for node D which chooses to pass through node C the split between the two routes to C is the same as it would be if C were the final destination. It is this feature of SAM which permits it to work efficiently, by moving progressively through the network, in a step by step, Markovian, manner and without the need for path enumeration. As a consequence, however, in a case such as that in Fig. 3, the loading is asymmetric. It is certainly not clear which, if either, of the routeing assumptions implicit in SAM and Burrell is the ‘correct’ one. Burrell ensures that a route decision is based on the cost of the whole route, while SAM ensures that the choice between routes to intermediate points is consistent with
Fig. 3. 5-link network.
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Proportion on Route 3 0.4
. . . .. . .. .. .
,,
Burrell
pm.e-T{ ..” .,...
0.3
) 0.2
0.1
I
Fig. 4. Differences
between SAM and Burrell.
S-link network
that to those same points treated as destinations. In the literature on routeing strategies, from a geographical or psychological point of view (see for example Gould, 1989; Kuipers, 1978; Mark and McGranaghan, 1986 and Stern and Leiser, 1988), the distinction is accepted between different kinds of spatial knowledge. The most basic level is ‘landmark’ knowledge, where the individual knows several landmarks, without much spatial relationship between them. Secondly, there is ‘route’ knowledge, where the individual is capable of linking locations in his or her travels, but lacks an overall understanding of the spatial organization. This level may be purely enactive, that is, the travelling individual can recognize what to do at every intersection, but is incapable of describing the entire route from memory (Stern and Leiser, 1988). The third level of knowledge, which Stern and Leiser’s surveys in Beer Sheva found was only truly exhibited by professional drivers, is ‘survey’ knowledge. At this level, the individual has a proper spatial understanding, and comes to think of routes as links between locations, rather than a sequence of direction choices. In this analysis, the SAM routeing strategy corresponds quite closely to route knowledge, the incomplete knowledge exhibited by a large section of the population, which is quite sufficient for day-to-day use, but less efficient than the taxi-driver’s store of short-cuts. Route decisions made on whole routes, as in Burrell, correspond more closely to survey knowledge. This is more logical and is how we would like to think we make decisions, but there is evidence that if anything, a routeing strategy based on short term decisions is closer to the behaviour of an average driver. Figure 4 shows the comparison of assigned flows on route 2 (and hence on link BC) from the Burrell and SAM methods for all possible values of the ratio a/b. As the value of a increases from zero, until it reaches the value of b, the network changes from being equivalent to a straight threelink equal cost choice, to being equivalent to a figure-of-eight network, where a cross-over is possible from the upper to the lower route (but not the reverse). SAM and Burrell loadings are identical when a/b is zero, the loadings being a third on each of the three routes, but diverge as u/b tends to 1, as is shown in Fig. 4. The differences, however, are not great. Also shown in Fig. 4 are the assigned flows from Dial which are l/3 for all values of a/b. To compare the computing times required to carry out Burrell and SAM loadings, tests were carried out on a realistic sized network (the Sioux Falls network which will be referred to again in Section 8). An overall statistic (to be defined in Section 6) was used to measure the mean difference between (i) two Burrell loadings, using different random number seeds and (ii) a Burrell loading and a SAM loading. It was found that in order to obtain similar magnitudes for these two values, it was necessary to use around 10,000 samplings for each origin in the Burrell loadings. As it took 10.33 s for such a Burrell loading compared with 0.66 s for a SAM loading (the timings being on a Pentium 90 MHz PC), it was concluded that the SAM loading process was appreciably faster than a comparable Burrell loading. 4. DEADLOCK
AND LOOPING
When the SAM method for stochastic loading was described in Section 3 it was assumed that step 2 (finding a node at which to merge) could always be applied. However, when it is applied to a
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network which contains loops, it reaches a point where no further progress can be made in the forward pass through the network, because there is no node at which a ‘merge’ operation can be carried out. Consider, for example, the network shown in Fig. 5. After scanning from the origin at A, it is not possible to merge at either node B or node C. We refer to this condition as ‘deadlock’. There are three main approaches which may be adopted to tackle deadlock. The first is to delete links as necessary (according to some criterion, so that the deleted links are those judged to be likely to carry little or no traffic from that origin). This may be thought of as somewhat similar to the pruning of the network in Dial’s STOCH algorithm by the application of an ‘efficiency’ criterion. Certainly, in most practical networks, a large proportion of links could be eliminated from the network for any particular origin with only minimal effect, as they would not be likely to carry significant traffic. An alternative approach is to make an informed estimate of the distribution of certain Yj in order that the deadlock may be broken. For example, in the network of Fig. 5, one could estimate the distribution of Y,. This would then enable a merge to be carried out at B, and the forward pass to continue, by scanning from B and merging at C, scanning from C and so on. After scanning from C, it is possible to reassess the estimate of Y,, and so this suggests that a third alternative would be to carry out this process of reassessment in an inner iterative loop. The problem of deadlock appears to be unavoidable in the application to any network which contains loops of a stochastic assignment process which does not identify all feasible routes. It is a direct consequence of the nature of the merging process in the forward pass: the proportions of traffic from a given origin arriving at a node on each of its before links are independent of the link used to leave the node. So, for example, in Fig. 5, the proportions of traffic entering node B on links 1 and 3 apply both to traffic leaving on link 5 and on link 4, even though we would wish to exclude the possibility of traffic using both links 3 and 4. Instead of storing splits in this way, however, the splits that apply to each exiting link can be stored separately; that is, by storing turning proportions. This extension enables all two-link loops to be dealt with, so that no traffic is assigned to them. Higher order loops (ones containing three or more links) could be dealt with in a similar way, by storing 3, 4, or 5-dimensional turning proportions, in effect starting to store path histories, but this approach would quickly become infeasible in terms of computing time. Such an approach has been proposed for logit-based assignment by Lam et al. (1996). The method currently implemented in SAM is intended to give a reasonable compromise between the various possibilities; two-link loops are eliminated by using turning proportions, as in the above paragraph, and higher order loops are dealt with by estimating the required distributions, as described in the paragraph before that. The estimation process is not iterated, to preserve the efficiency of the one-pass algorithm. The merging process in SAM is the feature of the algorithm which makes it computationally feasible by not requiring complete path enumeration. The advantages of the algorithm, in allowing the development of a computationally feasible probit-based SUE method, outweigh the disadvantages arising from the deadlock problem, it is felt. A further paper will give a fuller explanation and comparison of the various alternatives, and their merits and disadvantages. This problem of loops and the Markov nature of the algorithm is discussed, albeit in the context of logit-based SUE, in Bell (1994) and Akamatsu (1995). /
@ 1’?
1 ,,’
‘,\
,c’ 8A
31
‘1
1..
;
\:
1,
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Fig. 5. Network
_5 \\
1’ $3
/
showing
deadlock.
A probit-based SUE
model
347
S. STOCHASTIC USER EQUILIBRIUM
The existing SAM method is a pure stochastic assignment model. This section describes the development of SAM into a stochastic user equilibrium (SUE) method, by incorporating capacity restraint in the form of link-based cost-flow functions. The method will be seen to be analogous to the Frank-Wolfe method for UE, in that it consists of the minimization of an objective function. The link cost-flow functions relate the mean link cost c, to the link flow x,; we use cost functions that depend solely on the flow on the link itself. Various forms could be used but for present purposes we adopt the BPR cost-flow functions (U.S. Bureau of Public Roads, 1964):
cdx,) =
cm[ +k(-$q] 1
where c,(O) is the free-flow cost on link a, and x,“~J’IS the ‘capacity’ of link a (the flow at which the link cost is (1 + k) times the free-flow cost). The parameter k is used later as a measure of the degree of capacity restraint. Clearly, a vector x of link flows will define a vector c of link costs; if a stochastic assignment is carried out based on c (used as mean link costs), the new flow vector will generally be different from the original s. SUE has been reached when the link costs and flows are consistent, so that the vector c leads back to the original x. Now with ,@) being the flow vector at iteration n, let J @)be the auxiliary flow vector, found by carrying out a stochastic loading based on mean link c&s c (‘1. Following the approach used for the UE problem, the new flow pattern will be given by:
The next question is clearly how to choose the step length h,. Previous SUE algorithms have relied on the Method of Successive Averages (MSA) with, typically, h, = l/(n + 1); it is known to converge to the SUE solution (see Sheffi, 1985) but is usually very slow. A better method is to choose the ‘best’ h, at each iteration to speed the convergence. Using an objective function which has its minimum at the SUE solution, the point with the lowest value of this function along the search direction from g(n) to y(“) is then chosen as the new current solution -x(“+‘). One such function is that of Sheffi and Poweli (1982):
--Cd= -
c j ca(4~~+ cu x,cdxu) - cTS T&s
(5)
a 0
where T,, is the demand and S,, is the expected perceived minimum cost (the satisfaction function) between O-D pair (r,s). The first term in the above is the function which is minimized at the UE solution; the second term is the total cost of travel in the network, and so would be minimized at the system optimal solution (Wardrop’s Second Principle). The third term is the expected value of the total perceived network cost, and is generally the most difficult part of the function to calculate. However, as was noted earlier, the SAM method calculates the S,, for all O-D pairs as part of the loading process, so that the third term can be found. The loading process also gives the information for the gradient of the function (along the search direction) to be calculated at the same time without extra effort. The cost in computation terms of evaluating the function and gradient once is thus equivalent to running the SAM assignment routine once. We now have the basis for a SUE algorithm based on the SAM stochastic loading method: 1. Choose initial costs c(O),usually free-flow costs. Find an initial feasible flow pattern x(‘) by carrying out, for example, a pure stochastic loading using mean costs c co). Set the iteration count n to 1. 2. Given x(“), the current flow pattern, calculate the current costs ~(~1using the link cost-flow relations. Find the auxiliary flow pattern y- (“1, by carrying out a pure stochastic loading based on the mean costs &“).
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3. Carry out a line search to find the choice of h, which minimizes z(x) along the search y(“) - z(“) . Calculate the new current solution:&+‘) = ,tn) + k n (y_‘“’> (Increase n by 1. 4. If $‘) and y(“) - are sufficiently close, stop. Otherwise repeat steps 2 and 3. direction
,(n)).
The method can be seen to be analogous to the Frank-Wolfe algorithm for finding the UE solution, referred to earlier. The important differences are (i) that the objective function is different and (ii) that the auxiliary flow pattern is from an all-or-nothing loading in the case of UE but a stochastic loading in the case of SUE. In the UE case, the auxiliary flow pattern is always an extreme solution, relatively far away from the current pattern, even at the final solution. In the SUE case, the current and auxiliary flow patterns are the same in the limit, as costs and flows become consistent. Therefore, there is no expectation for the optimal step length to diminish as equilibrium is approached. Furthermore, the difference between the current and auxiliary flow patterns can be used as a guide to convergence in the SUE case, rather than just the change from the last flow pattern. For the line search, an interpolation method is used. The derivative of the objective function with respect to a link flow variable x, is (see Sheffi, 1985): ip
= ”
(x,
_
ya)y
(6) a
in which y is the auxiliary flow vector which would arise from a stochastic loading using costs corresponding to the flow vector x. from the current solution x(“), let us In carrying out the search in the direction y cn)- $) > ( denote z (# + h (z(“) - .x(“))) by z(h). Then, -the gradient g at any point x along this search direction is given by: (7) where _x@) and y(“) are the points at either end of the interval, x is any intermediate point, and y- is the auxiliary found using the costs at x_. Therefore, at the current solution x @)(A.= 0), where the function value is zo, the gradient go is:
By carrying out a further stochastic loading using costs corresponding to the auxiliary flow vector y(“), we can also calculate the gradient gl (and the value of the objective function zi) at the step length h = 1. Then, using the values ZO,zl, go and gl, a cubic can be fitted and the point at which this cubic is minimized taken as the estimate of the optimal step length h,. Alternatively, a quadratic could be fitted, using only go and gl, to estimate the step length at which the gradient is zero. The methodology described above, however, depends on the use of the Sheffi and Powell objective function, the validity of which relies on the assumption of the drivers’ choices being between whole routes whereas, as has been seen in Section 4, the SAM loading routine is based on a Markovian process which avoids the need to construct routes. Furthermore, it should be recalled that the calculations in SAM are not exact but use the Clark approximation. As a consequence, the algorithm which has been described is strictly a heuristic. Close examination of the objective function along a search direction reveals that the function value z and the gradient g are slightly inconsistent, although this generally only becomes apparent close to the equilibrium point. The point at which the current and auxiliary solutions are identical is therefore not necessarily the point at which the function z is minimized. The procedure we have adopted is to use cubic interpolation for the line searches at the start of the iterative process but to switch to quadratic interpolation in the later stages, so as to ensure that the equilibrium point x = y is found. An alternative is to use quadratic interpolation throughout.
A probit-based
SUE model
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There are two more points which should be mentioned. The first is that, as Sheffi (1985, p. 3 13) points out, the link variances should be fixed at the start of the iterative process and not allowed to change as the mean link cost changes. Following Sheffi, we set the link variance to be /? times the mean free-flow link cost. The second point is that a slight change in the link flows may result in a difference in the order in which the nodes are merged during a SAM loading. This may lead to discontinuities in the objective function. A similar phenomenon arises in a SUE algorithm which uses Dial loadings: a small change in link flows may lead to a different set of links being declared to be ‘inefficient’, a point which was noted by Thomas (1991). The procedure we have adopted is the following: once the algorithm is close to convergence, the order in which the nodes and links are merged is fixed so as to ensure the continuity thereafter of the objective function. Notwithstanding its heuristic nature the proposed probit-based SUE algorithm appears to be a significant advance on previous approaches. Sheffi (1985, pp. 323-324) concluded that such an algorithm ‘cannot be easily carried out for the SUE program’ of eqn (5), since (i) a ‘probit loading can only be carried out by using Monte Carlo simulation’, and (ii) ‘the move size cannot be optimized since the objective function itself is difficult to calculate’. The proposed algorithm obviates the need for path enumeration, is numerical rather than Monte Carlo, and enables the objective function z to be calculated and the step length to be optimized. The next sections demonstrate its application to a number of test networks. 6. APPLICATIONS
OF THE SUE ALGORITHM
A simple network with three parallel links can be used to give a graphical demonstration of how the iterative method works in practice. The cost functions of the three links are as follows:3 +x1 2 + 2x2 (9) = 2.5 + 1.5.X, c3 If we assume a single unit of demand travels from origin to destination, all feasible flow patterns will be in a plane triangle in 3D space, between the three extreme (all-or-nothing) solutions (O,O,1), (0,l ,O) and (O,O,l). The triangle can be plotted in two dimensions, and the progress of the iterative method shown. In Fig. 6, a few iterations of the ‘hard speed change’ (I,,= 1) method are shown, with characteristic zig-zagging behaviour. Figure 7 shows MSA, which tends smoothly towards the solution, and Fig. 8 shows the optimization method, which finds the solution more quickly, in just two iterations, up to the resolution of the diagram. In all three cases, the iterations start from the all-or-nothing solution for free-flow costs. The value for /? is 0.25. Cl
=
Q
=
Fig. 6. Five iterations
of hard speed change.
3-link network
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Fig. 8. Two iterations of optimization method.
The second application of the SUE algorithm is to a 4x4 grid network based on that in Dial (1971) (shown in Fig. 9). In that paper, there was no capacity restraint imposed. Here, however, we set the free flow cost of a standard link at 1 unit, and the free flow cost of the motorway links (those central four links shown in Fig. 9 by thicker lines) as 0.5 units; also, the capacity of the motorway links was taken to be larger (168) than that of the other links (100). A power of 4 was adopted for the BPR function, and a value of 1 for k. There is a single 0-D pair, from top left corner to bottom right corner, with a demand of 300 units. The value of the variability parameter b was set to 0.25. Figure 10 shows, for one search direction, a plot of the objective function z(x) against the step length, together with the approximating cubic function, the latter being shown as a dashed line. It may be seen that the fitted cubic agrees well with the true function, and the step length which minimizes the cubic gives a good estimate of the true optimal step length. We now show how the interpolation method converges. Figure 11 shows the objective function which, it can be seen, decreases rapidly. The RMS (root mean square) difference between the current and auxiliary solutions is a convenient measure of convergence since, at equilibrium, the two should be identical. This measure is defined as:
u”r--yyy
RMS(C,A)=&)qy
+y>
where N is the number of links in the network. This is also plotted in Fig. 11. The method can be seen to converge quickly and efficiently to a value of zero.
A probit-basedSUE model
Fig. 9. Grid network
-1,600’ 0
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
I
I
I
0.6
0.7
0.0
I
0.9
1
Step Length Fig. 10. z(x)over a search direction. Grid network-2nd iteration, demand 300.
RMS(C,A)
Z
4
-1,550
3
Iterations Fig. 1I. Covergence of cubic interpolation method. Grid network.
M. J. Maher
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7. COMPARISONS
and P. C. Hughes
WITH UE AND PURE STOCHASTIC
METHODS
The SUE method includes the stochastic effect (through the variability parameter fi) and the capacity restraint effect (through the cost-flow functions and summarized by the k value in the BPR functions). It is clear that as k tends to zero, the SUE solution should tend towards the pure stochastic solution, and that as p tends to zero, it should tend towards the UE solution. This is confirmed by the following set of tests carried out again on the grid network. Whilst keeping the values of the demand and the capacities x car constant, the SUE solution was found for different values of @and k. The difference between the SUE solution &SUEand pure stochastic solution 3 is measured by the root mean square function RMS(SUE, S) and is shown is Fig. 12, from which it can be seen that, as k-+0, zsUE --f ,xs and that, for fixed k, the difference decreases as B increases. Similarly, the difference between the SUE solution and the UE solution xun is measured by RMS(SUE, UE) and this is shown in Fig. 13. In this it can be seen that as j?-0, SUE -+ gun and that, for fixed 8, the difference decreases as k increases. RMS (SUE,S)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
k Fig. 12. SUE/SAM
differences.
Grid network.
RMS (SUE,UE)
6 c----------
---
k=O
k=2
2 1 OF 0
I
0.05
0.1
0.15
0.2
Fig. 13. SUE/UE
0.25 B
0.3
differences.
0.35
0.4
Grid network.
0.45
0.5
SUE model
A probit-based
353
LOG(RMS(C,A)) 21
I
-lOi ’ 0 2
)
4
’
6
’
1
’ ! ’
’
’ ’ ’ ’ ’
8 1012141618202224262830
’
Iteration Fig. 14. MSA and optimization
method.
Table Required
RMS(C.A)
MSA (s) Optimization
(s)
Sioux Falls network
I.
10-l
10-2
10-J
10-4
7.20 20.49
62.47 34.29
617.91 49.89
6277.42 70.49
These tests demonstrate that the SUE algorithm behaves as expected and, in particular, that it allows a smooth transition between the extremes of UE and pure stochastic assignment. 8. TESTS ON A LARGER
NETWORK
In this section, tests on a larger network are described, concentrating on the comparison between SAM and MSA assignments, especially in terms of the speed of convergence. The Sioux Falls network which is used here has been used by several previous authors: for example, LeBlanc (1975) and Vythoulkas (1990). It has 76 links, 24 nodes and 24 origins, and a value of B=O.S was used. The rates of convergence for MSA as well as the new optimization-based SUE algorithm are displayed in Fig. 14, which shows the value of the logarithm of the RMS(C,A) statistic at successive iterations. It can be seen that for the first few iterations the two methods follow a very similar course but that they then diverge markedly. It should also be recalled that the new method requires at least two stochastic loadings per search direction, whereas MSA requires only one. Table 1 compares the computation time required (on a Pentium 90 MHz PC) to achieve various levels of convergence. This shows that MSA is a viable approach provided strict convergence is not required but its slow rate of convergence makes it unattractive for more stringent levels of accuracy. This in turn suggests a hybrid approach, in which the MSA step lengths are used for the first few iterations (requiring only one stochastic loading per iteration), after which optimal step lengths are found at each iteration using the new algorithm. 9. SUMMARY
After an overview of previous developments in stochastic assignment, this paper has described SAM, a new probit method for pure stochastic assignment, and demonstrated its advantages in relation to Dial and Burrell, the two main methods available at present. The SAM loading process avoids the need for path enumeration by its assumption of a ‘Markovian’ routeing strategy, in support of which there is empirical evidence on drivers’ network knowledge and route choice. By combining SAM with capacity restraint, in the form of link cost-flow functions, a new SUE algorithm has been developed, using the mathematical programming formulation first proposed by
354
M. J. Maher and P. C. Hughes
Sheffi and Powell. Carrying out a SAM loading at a current solution g enables not only the auxiliary solution y to be found but also the value of the objective function Z(X) and the gradient of z along the search direction. By carrying out a second SAM loading at y, the optimal step length may be determined using either quadratic or cubic interpolation. It was shown that this new SUE algorithm is strictly a heuristic and that it can be used to determine either the solution which minimizes z or that at which there is equilibrium: _u= -1, but not the two simultaneously. The current implementation adopts quadratic interpolation in the later stages in order to find the equilibrium solution. The algorithm was applied to a number of networks, ranging from a simple 3-link network to the 76 link Sioux Falls network. In each case, convergence was smooth and efficient, and it was shown that, as the the values of parameters describing the variability and degree of capacity restraint are varied the SUE solution moves smoothly between the extremes of UE and pure stochastic solutions. Comparison with the Method of Successive Averages (MSA) showed that the new algorithm is considerably superior to MSA if anything other than a coarse level of convergence is required. The new algorithm appears to offer an attractive new method for SUE assignment: it is probit-based and therefore allows for overlapping routes, it does not require Monte Carlo simulations, it calculates and minimizes the Sheffi and Powell objective function, and it converges quickly because of the determination of the optimal step length at each iteration. Work has already been carried out extending the new SUE algorithm to the case of multiple user classes (Maher and Hughes, 1995), applying it to the estimation of the benefits from ATT systems which may be characterized as reducing the value of the uncertainty parameter /? for drivers of vehicles equipped with the ATT system. Current work is investigating in more depth alternative procedures for dealing with deadlock and loops, and it is planned to extend the model further by allowing for elastic demand. AcknoM,ledgemen,s-Thanks are due to the U.K. Engineering and Physical Sciences Research Council for their financial support of this research through research grant GR/K54502 as part of their Built Environment programme, and to two anonymous referees who made several very valuable and constructive comments on an earlier version of this paper.
REFERENCES Akamatsu. T. (1995) Cyclic flows, Markov process and stochastic traffic assignment. Transporration Research B 30(5), 369386.
Beckmann, M., McGuire, C. B. and Winsten, C. B. (1956) Studies in the Economics of Transportation. Cowles Commission Monograph, Yale University Press, New Haven. Bell, M. G. H. (1994) A simple alternative to Dial’s logit assignment algorithm. Transporfarion Research B 29(4), 287-295. Bell, M. G. H., Lam, W. H. K., Ploss, G. and Inaudi, D. (1993) Stochastic user equilibrium assignment and iterative balancing. Proceedings of the 12th Inrernurionul Symposium on Transportation and Trufic Theory, University of California, Berkeley. Burrell, J. E. (1968) Multiple route assignment and its application to capacity restraint. Proceedings of rhe 4th Infernufionul Symposium on Transportation and Traffic Theory, University of Karlsruhe. Chen, M. and Alfa, A. S. (1991) Algorithms for solving Fisk’s stochastic traffic assignment model. Transports/ion Research B 25(6), 40.5412.
Clark, C. E. (1961) The greatest of a finite set of random variables. Operalions Research 9, 145-162. Daganzo, C. F. and Sheffi, Y. (1977) On stochastic models of traffic assignment. Trunsporrution Science l](3), 253-274. Dial, R. B. (1971) A probabilistic multipath traffic assignment model which obviates path enumeration. Transportation Research 5, 83-l I I. Dial, R. B. (1995) Multicriteria equilibrium traffic assignment basic theory and elementary algorithms. Part I: T2, the bicriteria model. Presented at the 74th annual meeting of the Transportation Research Board, Washington DC. Fisk, C. (1980) Some developments in equilibrium traffic assignment. Transportation Research B 14(3), 243-255. Frank, M. and Wolfe, P. (I 956) An algorithm for quadratic programming. Naval Research Logisfics Quarterly 3( I -2), 95-I IO. Gould. M. D. (1989) Considering individual cognitive ability in the provision of usable navigation assistance. Proceedings 1st VNIS Conference, IEEE, Toronto, pp.443-447. Horowitz, J. L.. Sparmann, J. M. and Daganzo, C. F. (1982) An investigation of the accuracy of the Clark approximation for the multinomial probit model. Transportation Science 16(3), 382401. Kamakura, W. A. (1989) The estimation of multinomial probit models: a new calibration algorithm. Transportation Science 23(4). 253-265.
Kuipers, B. (1978) Modeling spatial knowledge. Cognirive Science 2, 129-153. Lam, W. H. K., Chan, E. K. S. and Bell, M. G. H. (1996) The treatment of cycles in stochastic user equilibrium assignment. To be presented at the 13th International Symposium on Transportation and Traffic Theory, Lyon. LeBlanc, L. J. (1975) An algorithm for the discrete network design problem. Transportation Science 9, 183-199. Leurent, F. M. (1994) Cost versus time equilibrium over a network. Presented at the 73rd annual meeting of the Transportation Research Board, Washington DC.
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Leurent. F. M. (1995) Contributions to the logit assignment model. Presented at the 74th annual meeting of the Transportation Research Board, Washington DC. Maher, M. J. (1992) SAM - A stochastic assignment model. In Mafhemarics in Trnnsporr Planning and Control, ed. J. D. Griffiths. Oxford University Press, Oxford. Maher. M. J. and Hughes, P. C. (1995) Estimation of the potential benefits from an ATT system using a multiple user class stochastic user equilibrium assignment model. In Applicakms of Advanced Technologies in Transportation Engineering. eds Y. J. Stephanedes and F. Filippi. American Society of Civil Engineers, New York. Mark. D. M. and McGranaghan. M. (1986) Effective provision of navigation assistance to drivers: a cognitive science approach. In Proceedings of Conference Auto-Carto. London, pp.399-408. Robertson, D. 1. and Kennedy, J. V. (1979) The Choice of Route. Mode, Origin and Desrination by Calcularion und Simulufion. TRRL Report LR877. Transport Research Laboratory, Crowthorne, Berkshire, U.K. Sheffi, Y. (1985) Urban Transportation Neluaorks: Equilibrium Analysis with Mathematical Programming Methods. PrenticeHall, Englewood Cliffs, NJ. ShetIi, Y. and Powell. W. B. (1982) An algorithm for the equilibrium assignment problem with random link times. Nrtbcorks 12(2), 191-207. Stern. E. and Leiser, D. (1988) Levels of spatial knowledge and urban travel modeling. Geographical Analysis 20(2). 140-l 55. Thomas, R. (1991) TraJic Assignment Techniques. Avebury Technical, Aldershot, U.K. U.S. Bureau of Public Roads (1964) Traffic Assignment Manual. U.S. Department of Commerce, Washington DC. Van Vuren, T. (1994) The trouble with SUE - stochastic assignment options in practice. Proceedings of Seminar H, 22nd PTRC European Transport Forum, Warwick University, pp.41-52. Vythoulkas, P. C. (1990) A dynamic stochastic assignment model for the analysis of general networks. Transportarion Research B 24(6), 453469.
Wardrop, J. G. (1952) Some theoretical aspects of road traffic research. Proceedings. InsUution qf Civil Engineers II 1. 325.-378.
All rights reserved. Printed in Great Britain 0191-2615197 $17.00+0.00
Pergamon PII: SO191-2615(96)00028-8
A PROBIT-BASED STOCHASTIC USER EQUILIBRIUM ASSIGNMENT MODEL M. J. MAHER* and P. C. HUGHES Department
of Civil and Transportation (Received
22 Februarv
Engineering,
Napier
1995; in revised/inn
Ilniversity. 23 April
Edinburgh,
Scotland
1996)
Abstract--Stochastic methods of traffic assignment have received much less attention in the literature than those based on deterministic user equilibrium (UE). The two best known methods for stochastic assignment are those of Burrell and Dial, both of which have certain weaknesses which have limited their usefulness. Burrell’s is a Monte Carlo method, whilst Dial’s logit method takes no account of the correlation, or overlap. between alternative routes. This paper describes, firstly, a probit stochastic method (SAM) which does not suffer from these weaknesses and which does not require path enumeration. While SAM has a different routefinding methodology to Burrell. it is shown that assigned flows are similar. The paper then goes on to show how, by incorporating capacity restraint (in the form of link-based cost-flow functions) into this stochastic loading method, a new stochastic user equilibrium (SUE) model can be developed. The SUE problem can be expressed as a mathematical programming problem. and its solution found by an iterative search procedure similar to that of the Frank-Wolfe algorithm commonly used to solve the UE problem. The method is made practicable because quantities calculated during the stochastic loading process make the SUE objective function easy to compute. As a consequence, at each iteration. the optimal step length along the search direction can be estimated using a simple interpolation method. The algorithm is demonstrated by applying it successfully to a number of test problems, in which the algorithm shows good behaviour. It is shown that, as the values of parameters describing the variability and degree of capacity restraint are varied, the SUE solution moves smoothly between the UE and pure stochastic solutions. ~(5 I997 Elsevier Science Ltd
I. INTRODUCTION
It is commonly observed in empirical studies of driver route choice that a variety of routes are chosen between any origin+lestination (O-D) pair. Traditionally, traffic assignment models fall into two classes, depending on the form of mechanism used to produce this ‘multi-routeing’ or ‘route spreading’ effect. The first class consists of models which incorporate the effect of congestion through the use of capacity restraint and which aim to find the User Equilibrium (UE) solution, in accordance with Wardrop’s First Principle (Wardrop, 1952). They are deterministic in nature, assuming that drivers are perfectly rational and identical, and have complete and perfect knowledge of network conditions. The problem of finding the Wardrop equilibrium for separable link cost-flow functions can be posed as a minimization problem (Beckmann et al., 1956). Most implementations of UE minimize the Beckmann objective function, using the method due to Frank and Wolfe (1956). The second class consists of probabilistic or stochastic methods, which aim to model the variations in driver perceptions or preferences, and reflect the imperfect knowledge drivers have of conditions throughout the network; drivers choose different routes through the network based on these different perceptions. Pure stochastic methods do not include any capacity restraint. Recently, a third class has received some attention. This class contains ‘time-cost’ models which derive their multi-routing behaviour from the idea that the value of time has a probability distribution, rather than taking a single value. The models, put forward by Leurent (1994) and Dial (1995), extend and generalise the notion of multiple user classes, each with its own form of generalised cost. UE models are widely used in application to congested urban networks, but perform less well in less congested, inter-urban networks. Stochastic methods are more suitable for lightly congested networks, typically inter-urban networks. However, it would be far better to have one method *Author
for correspondence.
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M. J. Maher and P. C. Hughes
which incorporated both congestion and stochastic effects and which could be applied equally well to all types of network. As a consequence, several authors have, in recent years, developed techniques for the solution of the stochastic user equilibrium (SUE) problem. Daganzo and Sheffi (1977) first formulated SUE, showing it as a generalisation of user equilibrium, and Sheffi and Powell (1982) showed that the SUE problem could be posed as a mathematical programming problem. Fisk (1980) proposed an alternative formulation for the logit-based problem. Most implementations of SUE have used the Method of Successive Averages (MSA), though Chen and Alfa (1991) Bell et al. (1993), Leurent (1995) and Akamatsu (1995) have each made significant progress towards the development of algorithms which involve the evaluation and minimisation of the Fisk function. Most authors agree that probit-based models are preferable to logit-based ones, which take no account of overlapping, or correlated, routes. Their development in SUE algorithms has been restricted, however, by the apparent need for complete path enumeration or Monte Carlo techniques. This paper will describe a new SUE algorithm which suffers from neither of these disadvantages, formed by incorporating capacity restraint (in the form of link-based cost-flow functions) in a recently-developed probit-based stochastic method. The paper has the following structure. Section 2 will review previous pure stochastic methods, and the new probit stochastic model will be outlined in Section 3. Section 4 describes the phenomenon of ‘deadlock’ and how it is overcome, and Section 5 will show the development of the new SUE algorithm. The application of the algorithm to test networks will be described in Section 6, comparisons between the SUE, UE and pure stochastic solutions in Section 7, and tests on a larger network are reported in Section 8. Finally, Section 9 gives a summary and points the way to possible extensions of the model. 2. STOCHASTIC
METHODS
Two main mechanisms have been used for stochastic assignment, based on methods due to Burrell (1968) and Dial (1971). Burrell-type methods are Monte Carlo, depending on repeated sampling of link cost distributions, either uniform or normal, and loading portions of the demand, all-or-nothing, based on the sampled costs [Burrell originally used uniform distributions, while Sheffi and Powell (1982) suggested to use the Normal distribution]. As Van Vuren (1994) pointed out, this suffers from the disadvantage that successive runs of a program, with different random seeds, will give different answers, unless considerable computation time is spent in getting the answers to converge (see Section 3). The property of repeatability is important when comparing two cases (for instance a network with and without a new road scheme), when the estimation of benefit is partially obscured by the noise. Dial’s STOCH method is based on the use of a logistic function for the split between any pair of routes. This gives rise to a conceptually simple and efficient method which gives repeatable results. It suffers however, from one considerable weakness. and that is that there is no way to represent the correlations between route cost distributions when routes overlap. So. for instance, slight variations on a route through a town will all be allocated equal traffic, and a bypass route will be under-assigned in comparison. Both of the main existing stochastic assignment models, Burrell and Dial, have certain disadvantages, then, which have limited their usefulness. A method which is based on the assumption of normal link cost distributions (a ‘probit’ model) is, in principle, preferable to both, since it can take account of correlations between routes and, as it does not rely on Monte Carlo techniques, is repeatable. Until recently, probit-based methods relied on the enumeration of all possible routes and did not, therefore, offer an efficient, computationally feasible approach for networks of practical size. 3. THE SAM STOCHASTIC
METHOD
The SAM (Stochastic Assignment Method) model was first described in Maher (1992) and owes its origins to the SCATA method of Robertson and Kennedy (1979), although this method assumed a uniform distribution for link costs and was never developed to cope with anything other than very simple network configurations. In SAM link costs are assumed to be Normally
A probit-based
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SUE model
distributed and independent with specified means and variances. For simplicity, we assume a constant variability parameter /? (the ratio of variance to mean) for all links in the network, although this parameter could be allowed to take different values for different links, with no alterations to the basic method. The full details of the method have been given in Maher (1992) and will not be repeated here, but the basic steps for loading the demand, origin by origin, are as follows: 1. Scan outwards from the origin to the ends of all links which leave the origin. This gives the travel time distributions from the origin to the ends of those links (these random variables will be referred to as Yi, for any link i in the network). 2. Find a node n, such that the distributions of the Yi variables have been determined for all links i which enter node n (the case in which no such node can be found will be discussed later). ‘Merge’ at this node, i.e. determine the distribution of the travel time IV,, from the origin to this node. The random variable W, is defined as: W,, = min C( Y;)
(1)
ieB.
3. 4.
5.
6.
where B,, is the set of all links which enter node n (the ‘before’ links). The merge calculations also give the probabilities pin that traffic entering node n from the origin would have travelled on each of the before links i. These probabilities will later be used to split the traffic between the entering links. The node n is now said to be ‘complete’. The calculations are based on Clark’s approximation (Clark, 1961): the minimum of two Normal random variables is itself approximately Normal. As well as the mean and variance of W,,, all relevant covariances are found. Scan out from node n, which was completed in step 2, estimating the distributions of the Yj for all jEA,, the set of ‘after’ links for node n. If there are any nodes which are not complete, return to 2. Steps l-4 comprise the forward pass for the current origin, in which all the required probabilities Pi, are calculated. We then carry out the backward pass to load the traffic. Using the splitting probabilities Pi, from step 2, load the traffic, starting at the last node at which merging took place, and work back through each node n towards the origin, in the reverse order to which merging took place. Repeat steps l-5 for each origin in turn,
For each origin node r, the expected value of the minimum perceived travel cost to destination node s, E( W,), will be denoted by S,,, sometimes referred to as the ‘satisfaction function’. Several authors have carried out work on alternatives to the Clark approximation (Horowitz et al., 1982; Kamakura, 1989), but Clark was used here because of its simplicity. Future work may involve comparing the performance of the alternatives, implemented in the SAM model. The network of Fig. 1 can be used to show how SAM represents the correlation between routes, succeeding where Dial is deficient. Figure 2 shows the loading on the upper link for different values of X. As expected, the loading changes from 50 to 33% as x increases. The SAM stochastic loading method, then, can be seen as an alternative to the Burrell and Dial methods. It is systematic and easy to apply, and it has some important advantages over
Fig.
I. Network with correlated
routes.
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Flow on Upper Link (%) 50
SAM 45
30’
0
I 1
2
3
4
5
6
7
8
9
10
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Fig. 2. Comparison between Dial and SAM. Network with correlated routes.
those methods: it is numerical rather than Monte Carlo, and it is probit-based rather than logitbased. Although it is tempting to think of the SAM method as a numerical version of a Burrell-type method with Normal link costs [as proposed by Sheffi and Powell (Sheffi and Powell, 1982)], there is a subtle difference in the underlying routeing principle. This may be seen from the following simple example. The 5-link network in Fig. 3 has three routes; each route cost distribution has a mean of (a + b) units. The link cost random variables are denoted by XABetc. and the route costs by:
TI =
XAB
+
XED
T2
=
XAB
+
XBC
T3
=
XAC
+
XCD
+
XCD
(2)
In Burrell, the choice is made between whole routes, so that by symmetry the two outer routes, 1 and 3, will receive the same amount of traffic, and links AB and CD will have the same flow, as will links AC and BD. In SAM, however, the method progresses through the network using a ‘Markov’ assumption; that is, the pattern of route choice from an intermediate node N to the destination is independent of the pattern of route choice from the origin to N. When we merge at node C, the two incoming routes are seen to be of equal cost. The split will be 50:50; flows on links AC and BC will be equal, both being half the flow on link CD. For traffic destined for node D which chooses to pass through node C the split between the two routes to C is the same as it would be if C were the final destination. It is this feature of SAM which permits it to work efficiently, by moving progressively through the network, in a step by step, Markovian, manner and without the need for path enumeration. As a consequence, however, in a case such as that in Fig. 3, the loading is asymmetric. It is certainly not clear which, if either, of the routeing assumptions implicit in SAM and Burrell is the ‘correct’ one. Burrell ensures that a route decision is based on the cost of the whole route, while SAM ensures that the choice between routes to intermediate points is consistent with
Fig. 3. 5-link network.
A probit-based
345
SUE model
Proportion on Route 3 0.4
. . . .. . .. .. .
,,
Burrell
pm.e-T{ ..” .,...
0.3
) 0.2
0.1
I
Fig. 4. Differences
between SAM and Burrell.
S-link network
that to those same points treated as destinations. In the literature on routeing strategies, from a geographical or psychological point of view (see for example Gould, 1989; Kuipers, 1978; Mark and McGranaghan, 1986 and Stern and Leiser, 1988), the distinction is accepted between different kinds of spatial knowledge. The most basic level is ‘landmark’ knowledge, where the individual knows several landmarks, without much spatial relationship between them. Secondly, there is ‘route’ knowledge, where the individual is capable of linking locations in his or her travels, but lacks an overall understanding of the spatial organization. This level may be purely enactive, that is, the travelling individual can recognize what to do at every intersection, but is incapable of describing the entire route from memory (Stern and Leiser, 1988). The third level of knowledge, which Stern and Leiser’s surveys in Beer Sheva found was only truly exhibited by professional drivers, is ‘survey’ knowledge. At this level, the individual has a proper spatial understanding, and comes to think of routes as links between locations, rather than a sequence of direction choices. In this analysis, the SAM routeing strategy corresponds quite closely to route knowledge, the incomplete knowledge exhibited by a large section of the population, which is quite sufficient for day-to-day use, but less efficient than the taxi-driver’s store of short-cuts. Route decisions made on whole routes, as in Burrell, correspond more closely to survey knowledge. This is more logical and is how we would like to think we make decisions, but there is evidence that if anything, a routeing strategy based on short term decisions is closer to the behaviour of an average driver. Figure 4 shows the comparison of assigned flows on route 2 (and hence on link BC) from the Burrell and SAM methods for all possible values of the ratio a/b. As the value of a increases from zero, until it reaches the value of b, the network changes from being equivalent to a straight threelink equal cost choice, to being equivalent to a figure-of-eight network, where a cross-over is possible from the upper to the lower route (but not the reverse). SAM and Burrell loadings are identical when a/b is zero, the loadings being a third on each of the three routes, but diverge as u/b tends to 1, as is shown in Fig. 4. The differences, however, are not great. Also shown in Fig. 4 are the assigned flows from Dial which are l/3 for all values of a/b. To compare the computing times required to carry out Burrell and SAM loadings, tests were carried out on a realistic sized network (the Sioux Falls network which will be referred to again in Section 8). An overall statistic (to be defined in Section 6) was used to measure the mean difference between (i) two Burrell loadings, using different random number seeds and (ii) a Burrell loading and a SAM loading. It was found that in order to obtain similar magnitudes for these two values, it was necessary to use around 10,000 samplings for each origin in the Burrell loadings. As it took 10.33 s for such a Burrell loading compared with 0.66 s for a SAM loading (the timings being on a Pentium 90 MHz PC), it was concluded that the SAM loading process was appreciably faster than a comparable Burrell loading. 4. DEADLOCK
AND LOOPING
When the SAM method for stochastic loading was described in Section 3 it was assumed that step 2 (finding a node at which to merge) could always be applied. However, when it is applied to a
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M. J. Maher and P. C. Hughes
network which contains loops, it reaches a point where no further progress can be made in the forward pass through the network, because there is no node at which a ‘merge’ operation can be carried out. Consider, for example, the network shown in Fig. 5. After scanning from the origin at A, it is not possible to merge at either node B or node C. We refer to this condition as ‘deadlock’. There are three main approaches which may be adopted to tackle deadlock. The first is to delete links as necessary (according to some criterion, so that the deleted links are those judged to be likely to carry little or no traffic from that origin). This may be thought of as somewhat similar to the pruning of the network in Dial’s STOCH algorithm by the application of an ‘efficiency’ criterion. Certainly, in most practical networks, a large proportion of links could be eliminated from the network for any particular origin with only minimal effect, as they would not be likely to carry significant traffic. An alternative approach is to make an informed estimate of the distribution of certain Yj in order that the deadlock may be broken. For example, in the network of Fig. 5, one could estimate the distribution of Y,. This would then enable a merge to be carried out at B, and the forward pass to continue, by scanning from B and merging at C, scanning from C and so on. After scanning from C, it is possible to reassess the estimate of Y,, and so this suggests that a third alternative would be to carry out this process of reassessment in an inner iterative loop. The problem of deadlock appears to be unavoidable in the application to any network which contains loops of a stochastic assignment process which does not identify all feasible routes. It is a direct consequence of the nature of the merging process in the forward pass: the proportions of traffic from a given origin arriving at a node on each of its before links are independent of the link used to leave the node. So, for example, in Fig. 5, the proportions of traffic entering node B on links 1 and 3 apply both to traffic leaving on link 5 and on link 4, even though we would wish to exclude the possibility of traffic using both links 3 and 4. Instead of storing splits in this way, however, the splits that apply to each exiting link can be stored separately; that is, by storing turning proportions. This extension enables all two-link loops to be dealt with, so that no traffic is assigned to them. Higher order loops (ones containing three or more links) could be dealt with in a similar way, by storing 3, 4, or 5-dimensional turning proportions, in effect starting to store path histories, but this approach would quickly become infeasible in terms of computing time. Such an approach has been proposed for logit-based assignment by Lam et al. (1996). The method currently implemented in SAM is intended to give a reasonable compromise between the various possibilities; two-link loops are eliminated by using turning proportions, as in the above paragraph, and higher order loops are dealt with by estimating the required distributions, as described in the paragraph before that. The estimation process is not iterated, to preserve the efficiency of the one-pass algorithm. The merging process in SAM is the feature of the algorithm which makes it computationally feasible by not requiring complete path enumeration. The advantages of the algorithm, in allowing the development of a computationally feasible probit-based SUE method, outweigh the disadvantages arising from the deadlock problem, it is felt. A further paper will give a fuller explanation and comparison of the various alternatives, and their merits and disadvantages. This problem of loops and the Markov nature of the algorithm is discussed, albeit in the context of logit-based SUE, in Bell (1994) and Akamatsu (1995). /
@ 1’?
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_5 \\
1’ $3
/
showing
deadlock.
A probit-based SUE
model
347
S. STOCHASTIC USER EQUILIBRIUM
The existing SAM method is a pure stochastic assignment model. This section describes the development of SAM into a stochastic user equilibrium (SUE) method, by incorporating capacity restraint in the form of link-based cost-flow functions. The method will be seen to be analogous to the Frank-Wolfe method for UE, in that it consists of the minimization of an objective function. The link cost-flow functions relate the mean link cost c, to the link flow x,; we use cost functions that depend solely on the flow on the link itself. Various forms could be used but for present purposes we adopt the BPR cost-flow functions (U.S. Bureau of Public Roads, 1964):
cdx,) =
cm[ +k(-$q] 1
where c,(O) is the free-flow cost on link a, and x,“~J’IS the ‘capacity’ of link a (the flow at which the link cost is (1 + k) times the free-flow cost). The parameter k is used later as a measure of the degree of capacity restraint. Clearly, a vector x of link flows will define a vector c of link costs; if a stochastic assignment is carried out based on c (used as mean link costs), the new flow vector will generally be different from the original s. SUE has been reached when the link costs and flows are consistent, so that the vector c leads back to the original x. Now with ,@) being the flow vector at iteration n, let J @)be the auxiliary flow vector, found by carrying out a stochastic loading based on mean link c&s c (‘1. Following the approach used for the UE problem, the new flow pattern will be given by:
The next question is clearly how to choose the step length h,. Previous SUE algorithms have relied on the Method of Successive Averages (MSA) with, typically, h, = l/(n + 1); it is known to converge to the SUE solution (see Sheffi, 1985) but is usually very slow. A better method is to choose the ‘best’ h, at each iteration to speed the convergence. Using an objective function which has its minimum at the SUE solution, the point with the lowest value of this function along the search direction from g(n) to y(“) is then chosen as the new current solution -x(“+‘). One such function is that of Sheffi and Poweli (1982):
--Cd= -
c j ca(4~~+ cu x,cdxu) - cTS T&s
(5)
a 0
where T,, is the demand and S,, is the expected perceived minimum cost (the satisfaction function) between O-D pair (r,s). The first term in the above is the function which is minimized at the UE solution; the second term is the total cost of travel in the network, and so would be minimized at the system optimal solution (Wardrop’s Second Principle). The third term is the expected value of the total perceived network cost, and is generally the most difficult part of the function to calculate. However, as was noted earlier, the SAM method calculates the S,, for all O-D pairs as part of the loading process, so that the third term can be found. The loading process also gives the information for the gradient of the function (along the search direction) to be calculated at the same time without extra effort. The cost in computation terms of evaluating the function and gradient once is thus equivalent to running the SAM assignment routine once. We now have the basis for a SUE algorithm based on the SAM stochastic loading method: 1. Choose initial costs c(O),usually free-flow costs. Find an initial feasible flow pattern x(‘) by carrying out, for example, a pure stochastic loading using mean costs c co). Set the iteration count n to 1. 2. Given x(“), the current flow pattern, calculate the current costs ~(~1using the link cost-flow relations. Find the auxiliary flow pattern y- (“1, by carrying out a pure stochastic loading based on the mean costs &“).
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M. J. Maher and P. C. Hughes
3. Carry out a line search to find the choice of h, which minimizes z(x) along the search y(“) - z(“) . Calculate the new current solution:&+‘) = ,tn) + k n (y_‘“’> (Increase n by 1. 4. If $‘) and y(“) - are sufficiently close, stop. Otherwise repeat steps 2 and 3. direction
,(n)).
The method can be seen to be analogous to the Frank-Wolfe algorithm for finding the UE solution, referred to earlier. The important differences are (i) that the objective function is different and (ii) that the auxiliary flow pattern is from an all-or-nothing loading in the case of UE but a stochastic loading in the case of SUE. In the UE case, the auxiliary flow pattern is always an extreme solution, relatively far away from the current pattern, even at the final solution. In the SUE case, the current and auxiliary flow patterns are the same in the limit, as costs and flows become consistent. Therefore, there is no expectation for the optimal step length to diminish as equilibrium is approached. Furthermore, the difference between the current and auxiliary flow patterns can be used as a guide to convergence in the SUE case, rather than just the change from the last flow pattern. For the line search, an interpolation method is used. The derivative of the objective function with respect to a link flow variable x, is (see Sheffi, 1985): ip
= ”
(x,
_
ya)y
(6) a
in which y is the auxiliary flow vector which would arise from a stochastic loading using costs corresponding to the flow vector x. from the current solution x(“), let us In carrying out the search in the direction y cn)- $) > ( denote z (# + h (z(“) - .x(“))) by z(h). Then, -the gradient g at any point x along this search direction is given by: (7) where _x@) and y(“) are the points at either end of the interval, x is any intermediate point, and y- is the auxiliary found using the costs at x_. Therefore, at the current solution x @)(A.= 0), where the function value is zo, the gradient go is:
By carrying out a further stochastic loading using costs corresponding to the auxiliary flow vector y(“), we can also calculate the gradient gl (and the value of the objective function zi) at the step length h = 1. Then, using the values ZO,zl, go and gl, a cubic can be fitted and the point at which this cubic is minimized taken as the estimate of the optimal step length h,. Alternatively, a quadratic could be fitted, using only go and gl, to estimate the step length at which the gradient is zero. The methodology described above, however, depends on the use of the Sheffi and Powell objective function, the validity of which relies on the assumption of the drivers’ choices being between whole routes whereas, as has been seen in Section 4, the SAM loading routine is based on a Markovian process which avoids the need to construct routes. Furthermore, it should be recalled that the calculations in SAM are not exact but use the Clark approximation. As a consequence, the algorithm which has been described is strictly a heuristic. Close examination of the objective function along a search direction reveals that the function value z and the gradient g are slightly inconsistent, although this generally only becomes apparent close to the equilibrium point. The point at which the current and auxiliary solutions are identical is therefore not necessarily the point at which the function z is minimized. The procedure we have adopted is to use cubic interpolation for the line searches at the start of the iterative process but to switch to quadratic interpolation in the later stages, so as to ensure that the equilibrium point x = y is found. An alternative is to use quadratic interpolation throughout.
A probit-based
SUE model
349
There are two more points which should be mentioned. The first is that, as Sheffi (1985, p. 3 13) points out, the link variances should be fixed at the start of the iterative process and not allowed to change as the mean link cost changes. Following Sheffi, we set the link variance to be /? times the mean free-flow link cost. The second point is that a slight change in the link flows may result in a difference in the order in which the nodes are merged during a SAM loading. This may lead to discontinuities in the objective function. A similar phenomenon arises in a SUE algorithm which uses Dial loadings: a small change in link flows may lead to a different set of links being declared to be ‘inefficient’, a point which was noted by Thomas (1991). The procedure we have adopted is the following: once the algorithm is close to convergence, the order in which the nodes and links are merged is fixed so as to ensure the continuity thereafter of the objective function. Notwithstanding its heuristic nature the proposed probit-based SUE algorithm appears to be a significant advance on previous approaches. Sheffi (1985, pp. 323-324) concluded that such an algorithm ‘cannot be easily carried out for the SUE program’ of eqn (5), since (i) a ‘probit loading can only be carried out by using Monte Carlo simulation’, and (ii) ‘the move size cannot be optimized since the objective function itself is difficult to calculate’. The proposed algorithm obviates the need for path enumeration, is numerical rather than Monte Carlo, and enables the objective function z to be calculated and the step length to be optimized. The next sections demonstrate its application to a number of test networks. 6. APPLICATIONS
OF THE SUE ALGORITHM
A simple network with three parallel links can be used to give a graphical demonstration of how the iterative method works in practice. The cost functions of the three links are as follows:3 +x1 2 + 2x2 (9) = 2.5 + 1.5.X, c3 If we assume a single unit of demand travels from origin to destination, all feasible flow patterns will be in a plane triangle in 3D space, between the three extreme (all-or-nothing) solutions (O,O,1), (0,l ,O) and (O,O,l). The triangle can be plotted in two dimensions, and the progress of the iterative method shown. In Fig. 6, a few iterations of the ‘hard speed change’ (I,,= 1) method are shown, with characteristic zig-zagging behaviour. Figure 7 shows MSA, which tends smoothly towards the solution, and Fig. 8 shows the optimization method, which finds the solution more quickly, in just two iterations, up to the resolution of the diagram. In all three cases, the iterations start from the all-or-nothing solution for free-flow costs. The value for /? is 0.25. Cl
=
Q
=
Fig. 6. Five iterations
of hard speed change.
3-link network
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M. J. Maher and P. C. Hughes
Fig. 8. Two iterations of optimization method.
The second application of the SUE algorithm is to a 4x4 grid network based on that in Dial (1971) (shown in Fig. 9). In that paper, there was no capacity restraint imposed. Here, however, we set the free flow cost of a standard link at 1 unit, and the free flow cost of the motorway links (those central four links shown in Fig. 9 by thicker lines) as 0.5 units; also, the capacity of the motorway links was taken to be larger (168) than that of the other links (100). A power of 4 was adopted for the BPR function, and a value of 1 for k. There is a single 0-D pair, from top left corner to bottom right corner, with a demand of 300 units. The value of the variability parameter b was set to 0.25. Figure 10 shows, for one search direction, a plot of the objective function z(x) against the step length, together with the approximating cubic function, the latter being shown as a dashed line. It may be seen that the fitted cubic agrees well with the true function, and the step length which minimizes the cubic gives a good estimate of the true optimal step length. We now show how the interpolation method converges. Figure 11 shows the objective function which, it can be seen, decreases rapidly. The RMS (root mean square) difference between the current and auxiliary solutions is a convenient measure of convergence since, at equilibrium, the two should be identical. This measure is defined as:
u”r--yyy
RMS(C,A)=&)qy
+y>
where N is the number of links in the network. This is also plotted in Fig. 11. The method can be seen to converge quickly and efficiently to a value of zero.
A probit-basedSUE model
Fig. 9. Grid network
-1,600’ 0
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
I
I
I
0.6
0.7
0.0
I
0.9
1
Step Length Fig. 10. z(x)over a search direction. Grid network-2nd iteration, demand 300.
RMS(C,A)
Z
4
-1,550
3
Iterations Fig. 1I. Covergence of cubic interpolation method. Grid network.
M. J. Maher
352
7. COMPARISONS
and P. C. Hughes
WITH UE AND PURE STOCHASTIC
METHODS
The SUE method includes the stochastic effect (through the variability parameter fi) and the capacity restraint effect (through the cost-flow functions and summarized by the k value in the BPR functions). It is clear that as k tends to zero, the SUE solution should tend towards the pure stochastic solution, and that as p tends to zero, it should tend towards the UE solution. This is confirmed by the following set of tests carried out again on the grid network. Whilst keeping the values of the demand and the capacities x car constant, the SUE solution was found for different values of @and k. The difference between the SUE solution &SUEand pure stochastic solution 3 is measured by the root mean square function RMS(SUE, S) and is shown is Fig. 12, from which it can be seen that, as k-+0, zsUE --f ,xs and that, for fixed k, the difference decreases as B increases. Similarly, the difference between the SUE solution and the UE solution xun is measured by RMS(SUE, UE) and this is shown in Fig. 13. In this it can be seen that as j?-0, SUE -+ gun and that, for fixed 8, the difference decreases as k increases. RMS (SUE,S)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
k Fig. 12. SUE/SAM
differences.
Grid network.
RMS (SUE,UE)
6 c----------
---
k=O
k=2
2 1 OF 0
I
0.05
0.1
0.15
0.2
Fig. 13. SUE/UE
0.25 B
0.3
differences.
0.35
0.4
Grid network.
0.45
0.5
SUE model
A probit-based
353
LOG(RMS(C,A)) 21
I
-lOi ’ 0 2
)
4
’
6
’
1
’ ! ’
’
’ ’ ’ ’ ’
8 1012141618202224262830
’
Iteration Fig. 14. MSA and optimization
method.
Table Required
RMS(C.A)
MSA (s) Optimization
(s)
Sioux Falls network
I.
10-l
10-2
10-J
10-4
7.20 20.49
62.47 34.29
617.91 49.89
6277.42 70.49
These tests demonstrate that the SUE algorithm behaves as expected and, in particular, that it allows a smooth transition between the extremes of UE and pure stochastic assignment. 8. TESTS ON A LARGER
NETWORK
In this section, tests on a larger network are described, concentrating on the comparison between SAM and MSA assignments, especially in terms of the speed of convergence. The Sioux Falls network which is used here has been used by several previous authors: for example, LeBlanc (1975) and Vythoulkas (1990). It has 76 links, 24 nodes and 24 origins, and a value of B=O.S was used. The rates of convergence for MSA as well as the new optimization-based SUE algorithm are displayed in Fig. 14, which shows the value of the logarithm of the RMS(C,A) statistic at successive iterations. It can be seen that for the first few iterations the two methods follow a very similar course but that they then diverge markedly. It should also be recalled that the new method requires at least two stochastic loadings per search direction, whereas MSA requires only one. Table 1 compares the computation time required (on a Pentium 90 MHz PC) to achieve various levels of convergence. This shows that MSA is a viable approach provided strict convergence is not required but its slow rate of convergence makes it unattractive for more stringent levels of accuracy. This in turn suggests a hybrid approach, in which the MSA step lengths are used for the first few iterations (requiring only one stochastic loading per iteration), after which optimal step lengths are found at each iteration using the new algorithm. 9. SUMMARY
After an overview of previous developments in stochastic assignment, this paper has described SAM, a new probit method for pure stochastic assignment, and demonstrated its advantages in relation to Dial and Burrell, the two main methods available at present. The SAM loading process avoids the need for path enumeration by its assumption of a ‘Markovian’ routeing strategy, in support of which there is empirical evidence on drivers’ network knowledge and route choice. By combining SAM with capacity restraint, in the form of link cost-flow functions, a new SUE algorithm has been developed, using the mathematical programming formulation first proposed by
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M. J. Maher and P. C. Hughes
Sheffi and Powell. Carrying out a SAM loading at a current solution g enables not only the auxiliary solution y to be found but also the value of the objective function Z(X) and the gradient of z along the search direction. By carrying out a second SAM loading at y, the optimal step length may be determined using either quadratic or cubic interpolation. It was shown that this new SUE algorithm is strictly a heuristic and that it can be used to determine either the solution which minimizes z or that at which there is equilibrium: _u= -1, but not the two simultaneously. The current implementation adopts quadratic interpolation in the later stages in order to find the equilibrium solution. The algorithm was applied to a number of networks, ranging from a simple 3-link network to the 76 link Sioux Falls network. In each case, convergence was smooth and efficient, and it was shown that, as the the values of parameters describing the variability and degree of capacity restraint are varied the SUE solution moves smoothly between the extremes of UE and pure stochastic solutions. Comparison with the Method of Successive Averages (MSA) showed that the new algorithm is considerably superior to MSA if anything other than a coarse level of convergence is required. The new algorithm appears to offer an attractive new method for SUE assignment: it is probit-based and therefore allows for overlapping routes, it does not require Monte Carlo simulations, it calculates and minimizes the Sheffi and Powell objective function, and it converges quickly because of the determination of the optimal step length at each iteration. Work has already been carried out extending the new SUE algorithm to the case of multiple user classes (Maher and Hughes, 1995), applying it to the estimation of the benefits from ATT systems which may be characterized as reducing the value of the uncertainty parameter /? for drivers of vehicles equipped with the ATT system. Current work is investigating in more depth alternative procedures for dealing with deadlock and loops, and it is planned to extend the model further by allowing for elastic demand. AcknoM,ledgemen,s-Thanks are due to the U.K. Engineering and Physical Sciences Research Council for their financial support of this research through research grant GR/K54502 as part of their Built Environment programme, and to two anonymous referees who made several very valuable and constructive comments on an earlier version of this paper.
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