Simulation exercise on time perception parameters in discrete-time dynamic assignment models

Pergamon 0969-6016(95)00024-7

Int. Trans. Opl Res. Vol. 2, No. 1, pp. 61 74, 1995 Elsevier Science Ltd.

Printed in Great Britain. 0969-6016/95 $9,50 + 0.00

Simulation Exercise on Time Perception Parameters in Discrete-time Dynamic Assignment Models

GIUSEPPE BELLEI*, MAURIZIO BIELLIt and MASSIMO GASTALDIt *University 'La Sapienza', Italy and tNational Research Council

In order to assess the performance of information systems to road users and of demand management measures, like staggered and/or flexible work times and road pricing, with respect to congestion prevention problems, it is necessary to simulate dynamics of traffic flows on the transport network and users' behaviour, This can be accomplished by taking into account several aspects such as users" information level, travel demand distribution over time and the expected smoothing effect of adopted measures and information systems. Dynamic assignment and arrival time choice models and computational procedures to perform this assessment for various traffic scenarios and hypotheses about users' behaviour are presented in this paper. Moreover, the indicators needed to quantify information systems and demand management effectiveness are identified together with parameters defining traffic scenarios. The main results obtained by applying these models and techniques as deterministic and probabilistic simulation tools are also presented and discussed,

INTRODUCTION

New road information systems for users, based on advanced telematics, are going to be introduced in many different countries, with the aim of providing users with an effective support in planning trips taking place in the near future. Many different systems have been developed and tested in supporting drivers with pre-trip information. In particular, within the Prometheus Project, research is being carried out on 'Pre-trip Planning' in order to design a prototype system, able to meet user requirements for both short and long distance travel (Bielli, 1990), and a decision support system for urban traffic management (Bielli, 1992).

These developments in the field of road transport have stimulated, well after the pioneering work of Merchant and Nemhauser 0978) that was followed by that of a system optimal approach by Carey (1987), several studies on time-dependent aspects of traffic phenomena. In fact, in order to forecast the effects of road transport informatics, the role of information and the motorists' reaction to demand management strategies have to be highlighted in a dynamic context. Some of these studies concentrate on modelling behavioural responses to various aspects of information production, transmission and utilization and represent these phenomena on schematic and idealised transportation system in order to address these conceptually complex issues without the burden of a fully general network approach (Arnott et al., 1991). A few reports dealing with motorists' choices modelling come from practical experience, but in these reference is generally made to some corridors, thus oversimplifying the potential spatial dimension of the problem (Mahamassani and Chang, 1988; Mahmassani and Jayakrishnan, 1991 ). Other studies addressed dynamic route choice assuming that it takes place in a full network, like Hamerslag 0989), Janson (1991), or in the framework of an integrated dynamic assignment and traffic control model (Papageorgiou, (1990).

The aim of our contribution is to develop a discrete-time equilibrium model, comprehensive enough to embody some relevant behavioural hypotheses and yet simple enough to allow the exploration of a feasible (or reasonable) range of descriptive parameters, related both to users' behaviour and to model application environment, on a full network. Following up previous work on deterministic dynamic traffic assignment (Bellei and Bielli, 1990, 1992) and being aware that non-equilibrium probabilistic models have already appeared in the scientific literature (Cascetta and Cantarella, 1991), a probabilistic equilibrium version is also developed here.

Of course, criticism of the equilibrium approach is well-founded when issues like day-to-day traffic

Correspondence: In q. Giuseppe Bellei. Department of Hydraulics. Transportation and Roads. Unieersity 'La Sapienza'. Via Eudossiana. 18-00184 Rome. Italy

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62 G. Bellei et al . -Discrete- t ime Dynamic Assiynment Models

fluctuations, real-time information and transient mobility patterns are the focus of attention (Dehoux and Toint, 1991). Nevertheless, if it can be assumed that the average users' choices are influenced only by historical information, equilibrium models can be appropriate. Also, with respect to drivers' perception to time, it is possible to define models where departure time is continuous (Vythoulkas, 1990), but, as argued by Bellei (1992), a fully continuous-time model is both unnecessary from a behavioural point of view and practically unfeasible.

Moreover, the proposed equilibrium model takes into account arrival time choice instead of departure, in contrast with usual assumptions. The idea behind this approach is that, when travel is planned, different choice mechanisms can be adopted in the real world by trip makers, with planning of arrival time prevailing in the morning peak and that of departure time in the evening. It is quite likely that arrival time choice is followed by a travel time estimate and by a consequent departure time choice, but in an equilibrium model determination of departure time it can be left as implicit, keeping a direct relationship between average traffic conditions and planned arrival time choices.

In the following, a definition of the discrete-time dynamic traffic assignment equilibrium problem is given and a heuristic solution algorithm is proposed and tested. The extension of the algorithm to a probabilistic model and its testing is also included, in order to compare results with different basic assumptions. In particular, a sensitivity analysis has been carried out with respect to perception error variance, the percentage of users planning their arrival time choices and the level of traffic congestion. Finally, the results obtained from this analysis on a real network are described and conclusions are drawn in relation to underlying modelling hypotheses and further research perspectives.

BASIC DEFINITIONS AND USER BEHAVIOUR ASSUMPTIONS

A relatively simple problem, which is an extension over time of the well-known fixed demand deterministic user traffic equilibrium assignment problem, will be defined first, where the time variables are suitably discretized. More precisely, the time of day is partitioned into periods and the arrival time at a destination constrained to coincide with a single point for each period, while travel time, as a component of travel impedance, is considered as a positive real variable.

The model for traffic delay in a link is a volume-delay function, like in static traffic assignment; this aggregate relationship between traffic flows and travel times is assumed to hold separately for each discrete time of day partition considered and supplies variable-in-time travel times if variable-in-time flows are given as inputs. Such a link performance model is clearly oversimplified and does not take into account the problem of representing queue evolution and traffic density on different link sections. It is however, in our opinion, no more arbitrary than the definition of an exit function, proposed first by Merchant and Nemhauser (1978) and utilized successively by many others, while it has the distinctive advantage that performance functions defined for static assignment can be used directly in a dynamic framework.

The problem resulting from this approach can be solved, at least heuristically, without exceeding computational efforts and the need for input data can be satisfied almost completely within the limits of current traffic engineering practice. The road network is represented, as usual, by a directed graph G(N,L) where N is a set of nodes and L a set of links. Trips originate from a subset R of N and have their destination in a subset S of N: each origin destination pair rs, made up of an element r e R and an element s ~ S, can also be thought as an element of the set R × S, which in the following will be more briefly denoted as RS. Each origin~lestination pair is connected by a set of paths Prs, while the set of all acyclic paths in the graph will be indicated by P; any path p ~ Prs is an ordered set of K(p) links {l 1 . . . . l k . . . . lr~p~ }, where the initial node of 11 is the origin r, the final node of Ir~p~ is the destination s, and the initial node of lk coincides with the final node of I k_ 1, for any k = 2 . . . . . K(p).

The time horizon on which the evolution of traffic flows is made up of Q intervals i, with i~ I, of equal length T; since trips arriving at a destination during this time horizon may need to leave their origin before it, an interval 0 of suitable length must be considered before interval 1. Arrivals at destination take place at t imesjT (ends of each interval), while any other event, like departures from origins and intermediate nodes, is located in any point of time and attributed to one of the Q + 1 intervals {0,1 . . . . . i . . . . . Q}, belonging to the extended time interval set I ÷.

International Transactions in Operational Research Vol. 2, No. 1 63

To simulate traffic conditions during the morning peak-hour, when trips are in general planned with regard to arrival times, we assume that, even if total demand of arrival time during time horizon defined by interval set I is known, a certain degree of flexibility exists for at least a fraction of users. Moreover, while developing the probabilistic version, we also assume that the transport network is utilized by two kinds of users, occasional and systematic. To give an analytical representation of arrival time and route choice, some further modelling assumptions have to be adopted.

Arrival time decision is modelled as a discrete choice because it is assumed that arrivals can take place only at times iT; arrival planning, hence, becomes the choice of the arrival interval, which can be described as the final outcome of the following three step process:

(a) determination of a first approximation activity schedule, based on optimal activities' duration and beginning times, but with a limited knowledge of travel times for trips between localization of consecutive activities;

(b) acquisition and/or consideration of more detailed information about travel times, costs and conditions at different times of the day;t and

(c) definition of the optimal travel and activity schedule, implying route and arrival time choice for all planned trips (departure times are implicitly determined by arrival and travel times).

By restricting ourselves to represent morning peak traffic, only a single activity to be performed and a single trip to reach its localization need be considered. Moreover, in the deterministic approach, we drastically simplify the formalization of the decision process by assuming that only two users' classes exist: one, the trip planners, that goes through all the steps of the decision process and the other, the non-planners, that do not take the opportunity of a schedule rearrangement as defined at step (b). In the probabilistic version of the model, this simplificative assumption is made for both occasional and systematic users, differing with respect to the knowledge of the network and to the propensity to schedule rearrangement, so increasing to four the total number of user classes.

Since arrival time defined at step (a) is at least approximately, optimal for all classes, the eventual schedule rearrangement and consequent arrival time change performed at step (b) should produce a net benefit to the individual by overcompensating for a slightly less than optimal arrival time with some travel time saving. The disutility of such a change has thus to be evaluated before being added to the travel related disutility. To this aim we note, at first, that the assumption of a discrete number of arrival times is equivalent to the definition of an 'indifference band' of length T around first approximation optimal arrival time, where utility is kept constant and rescaled to zero. The existence of such a band is justified if time localization of events is inaccurate,$ and hence there is an arrival time perception threshold. Immediately outside the indifference band it may happen that different arrival times are perceived, but do not imply a substantial schedule rearrangement; differences in utility, then, may be assumed to vary only slightly with respect to approximate optimal arrival time, defining an 'enlarged indifference band'.

Deviation from the approximately optimal schedule leads, outside indifference bands, to a larger disutility (presumably even larger in the case of late than in the case of early arrival) because of some relevant rearrangement, up to a point where it is no longer possible to realise some essential activity; larger deviations from schedule, and correspondingly earlier to later arrival times for the morning-peak travel, should then be assumed as unfeasible. The whole range of arrival times which allow realisation of essential activities will be named the 'feasibility band'.

This analysis leads to model step (b) of the arrival time decision process as the evaluation of a pointwise function ts(j,j*), supplying travel time equivalents of disutility arising from schedule rearrangement. If j 'T , end of interval j*, is an activity optimal arrival time (OAT) and iT, end of interval j, an alternative arrival time candidate as planned arrival time (PAT), the schedule rearrangement function is assumed to be:

t i t is likely that in step la) a limited knowledge should also be assumed with respect to the implications of each alternative schedule and in step {b) a more complete appraisal of consequences with respect to alternative schedules should be carried out; nevertheless it will be assumed here, for the sake of simplicity, that activity schedule is evaluated only at step (a), and the eventual rescheduling at step (b) takes into account only travel related disutilities.

Sit may also be assumed that it is less accurate than travel time comparison, as implicit in our model, where travel time is a continuous variable.

64 G. Bellei et al.-Discrete-time Dynamic Assignment Models

• zero within the interval corresponding to OAT (indifference band); • proportional to arrival the time differences absolute value outside the indifference band but within

the enlarged indifference band; • proportional to arrival the time differences absolute value, with different proportionality

coefficients for early and late arrivals outside the enlarged indifference band but within the feasibility band; and

• plus infinity, or a suitably large finite value, outside the feasibility band.

Since time of the day is represented by intervals of equal length T, proportionality to arrival interval differences (j-j*) can be assumed and ts(j,j*) defined as:

+ oe i f j < max{O,j* el}, a(j-j*) if max{O,j*-ef} < j < max{O,j*-ei}, flU-J*) if max{O,j*-ei} __< j < j*,

ts(j,j*) = 0 i f j = j* Vj*eI,jelFA(J*), (!) fl(j*-j) if min{Q,j* + li} =>j >j* , 7(J*-J) if min{Q,j* + l~} > j > min{Q,j* +/f}, + o0 i f j > min{Q,j* + If},

where :t, fl and 7 are non-negative constants, while I i e i is the width of enlarged indifference band and lf-ef of feasibility band.

The set of arrival times j T with j in the range [j* ef,j* + lf], denoted as IrA(j*), i.e. the set of feasible PATs with j* Tas OAT, is equivalent to feasibility band and schedule rearrangement function (1), with some marginal difference in notation, to the definition of schedule delay disutility given by Ben-Akiva et al. (1984).

Up to this point, however, randomness of utility is not taken into account and function (1) is defined on a discrete set; while randomness will be introduced later on, discretization of the time of day is inherent to this approach and a discrete representation of PAT shift from OAT seems at least consequent to a discrete representation of arrival time by itself. From a formal and aggregate point of view, several travel demand flows should then be distinguished, to take into account both the presence of the previously defined users' classes and the different PAT choices.

DETERMINISTIC MODEL FORMULATION

Trip planners demand flows are distinguished with respect both to their OATs j*T and, for each OAT, to the chosen PAT iT. For each origin-destination pair rs, OAT j*T and PAT j T with

g,s(J,J ) is defined, while for non-planners PAT is the same as OAT and j6IFA(j*) a demand flow 1 • . , 2 • demand flows may be more simply defined as g,s(J). Total demand flows with OAT j ' T , g*~(j*), and

PAT iT, gr~(j), are thus, respectively, given by:

' , g,~(j,j ) + gf~(j*) Yrs~RS, j*~I , (2) J~IFA(J*)

g,~(J) ~ ~ • . , = g,s(j,j ) + g2(j) Vrs~RS, j~ I . (3) j* e Ivn(j)

Travel demand with reference to PAT, g,~(j), turns out to be an endogenous variable, being defined as the sum of an exogenous component g2r~(j) and of an endogenous component, determined by the PAT

d 1 • . , choice model through minimization of trip planners individual disutility rs(J,J ) of choosing a j T PAT when OAT is j ' T , defined as:

d 1 • . $ rs(J,J ) = Cry(J) + ts(j,j*) YrseRS, j*~I , jelFA(j*) (4)

where v,~(j) is the travel from r to arrive at s at timejT, and the aggregate behavioural model reduces simply to (1)-(4), the non-negativity of the endogenous flow variables:

1 • ' * g,~(j,j ) >_ 0 Vrs~RS, j*~l , j~ lvA( j*) (5)


and the equilibrium conditions, which state that trip planners choose only minimum disutility arrival t i m e s :

d 1 • ., , ., 1 • ., d,~O )]o,,04 ) 0 [ ,s(J,J ) - - = Vrs ~ RS,j* e I , j ~ IrA(j* ), (6)

d 1 . ., , ., drs(J ) > , , ( j , j ) - _ o,

where * '* d,s( J ) is the equilibrium disutility for trip makers going from r to s with OAT j*T. Hypotheses needed to define the route choice model are as follows:

(i) vehicles travelling on a path and reaching their destination at the end of a time interval, are considered as a platoon, whose length is negligible;

(ii) the travel time this platoon takes to cross any link depends by the arrival time at the final node of the link;

(iii) link travel times are a piecewise constant function of time, i.e. they vary only from interval to interval, and keep constant during any interval;

(iv) link travel time associated with an interval depends on interval flow; and (v) interval flow is the sum of vehicles belonging to all platoons arriving at the final node of the link

during the interval, divided by interval length T.

On the basis of these assumptions, a traffic model, directly translating Wardrop's first principle in dynamical terms, can be established by defining the relationships between link and path variables. Flow and time variables are defined at link and path level as follows:

f~(i) flow arriving at final node of link 1 during interval i; hp(j) flow arriving at destination s at time j T travelling along the path p ~ P~,. ut(i) travel time from initial node to final node of link 1 during period p; and Zp(j) travel time to arrive at destination s at time j T travelling along the path p ~ P,s.

These variables are, if we ignore dependence from arrival times, the same used in static assignment problems. In the dynamic case, however, to represent the relationships between link and path variables, the vectors of link flows F z and travel times U~t with interval link flowsfz(i) and travel times ut(i) as elements and a time transformation function t f(Ut,T) supplying the time a platoon must be at initial node of link ! to arrive at final node at time t when the vector of link travel times is U~, have to be defined.

To obtain a continuous and non-decreasing tf(Ut,t), thus avoiding a violation of FIFO constraints (Carey, 1992), it can be expressed, on the basis of a discussion developed in more detail by Bellei and Bielli (1992) and Beilei (1992), as follows:

tf(Ut,t ) = max{t - ut(i'), i T - ut( i) l i~l , i < i'} VI~L. (7)

Path travel times zp(j) can be derived from link travel through (7) by first obtaining the times tlp(j ) of departure from the initial node of link 1 to arrive at destination s at t imejT, travelling along the path P ~ Pr~ via the recursive formula:

tkp(j ) = tf(U~,jT) for k = K, V p ~ P , j ~ I , (8)

tkp(j ) = tf[Uk,tk+t.p(j) ] for k = K - I,K - 2 . . . . . 2,1,

where, without any loss of generality, path p is a s s u m e d to include, in order, the links (l . . . . . k . . . . . K), and then successively determining path travel times by subtracting departure times from origin to arrival times at the destination:

zp(j) = j T - tlp(j ) V p s P , j 6 I . (9)

Also, link flows can be derived from path flows, by analytically expressing hypothesis (v) as follows:

f~(i)= ~ 3tp Y" hp(i) V l e L , i e l , (10) I~L jeltpli)

"tBecause of hypothesis (iv) above, U I is a separable function U~(Ft), each element ut(i ) being a function of the corresponding element .If(i) of F r

66 G. Bellei et al.-Discrete-time Dynamic Assiynment Models

where 6tp has the usual meaning of a binary variable equal to 1 when link 1 belongs to path p and ltp(i ) is the set of arrival times (at destination s of path p) such that arrival time (at final node of link 1 included in p), tt~(j ) + ut(i), falls within interval i.t

If we denote with U and H the vectors of link travel times for all intervals i e I ÷ and of path flows for all intervals j e l , respectively, the route choice model can be formulated as the problem of determining path flows hp(j) for any pc P, jE I, such that :

[Zp(j) -- v,,(j)]hp(j) = 0]

zp(j) - v.s(j) >= 0 VpeP.~, r s eRS , j ~ l , (11)

with :

hp(j) = g,s(J) Vrs6RS, j ~ l , (12) P~Prs

hp(j) > 0 Vp~P, j 6 I , (13)

ut(i) = ul[fl(i) ] VI6L, iEl , (14)

ul(0) = u°t VI~L,

Zp(j) depending on U by (8) and (9), for any p e P , j e l ft(i) depending on H by (10), for any IEL, i~ l .

In (14), travel times for interval 0 are assumed to be equal to free flow travel times u °, being representative of periods when no congestion is assumed to take place.

It is easy to verify that the conditions on path flows and times in (11 ) simply state that Wardrop's first principle must hold for any arrival time; they can also be integrated with the arrival time choice model by requiring that equations (1)-(14) hold together.

Hence, both the route choice model and the integrated route and arrival time choice model can be considered as a straight generalization of static user equilibrium which, unfortunately, is not so easily amenable to a nonlinear programming problem as in that case. Their close resemblance is, however, exploited by the following heuristic algorithm.

THE ALGORITHM

The most widely known method to solve static assignment problems is an adaptation of Frank-Wolfe's algorithm for nonlinear programming problems. This method exploits the network structure, as represented by a graph, to solve linear programming problems encountered at each iteration, through a shortest path tree search.

The algorithm which has been devised to deal with dynamic assignment problems tries to exploit in a similar way the network structure of the problem, performing at each iteration the search for shortest path trees with link lengths (travel times) variable with respect to time of day (Gallo and Pallottino, 1988). Existence and uniqueness of solution for this problem as well as convergence of the proposed algorithm are not addressed here:~ and no equivalent nonlinear programming problem is defined. Hence, even if a mathematical formulation of an equilibrium problem has been given, the algorithm presented should be considered as a heuristic, or a simulation tool. Nevertheless, if a self-reproducing flow pattern is attained, it can be shown to realise an equilibrium by a straightforward generalization of arguments developed by Sheffi (1985), with reference to the static case. Generation of successive link flows, on which computation of link travel times and search for shortest path trees is based, then, cannot be obtained by a linear search for the minimum value of an

tThis arrival time could be determined, for any given i, by repeatedly applying recursion (8), starting withj = 0 and increasing j until tip(j) + u#) > iT; both the relationships between link and path variables can thus be defined through the time transformation function (7).

++Existence and uniqueness proofs have been presented for substantially different specifications of dynamic assignment problems, e.g. as the one proposed by Ran et al. (1992), or for similar specifications, but with quite restrictive assumptions, the one proposed by Janson (1991).


objective function. Instead, link flows at each iteration are the average of link flows determined by the assignment of travel demand to shortest path trees generated up to that iteration, as in the method of successive averages, which are also widely used in static equilibrium problems and proven to converge to the correct solution in that case. A general description of the algorithm follows.

Step 0 (lnitialisation) Find link flowsf°(i), for each IE L and i t I, obtained by path flows satisfying equations (10), (12) and (13); set iteration index D = 0.

Step ! (Travel times calculation) Compute tentative link travel times uT(i) from current flowsft ° utilizing functions (14).

Step 2 (Shortest path trees search) Find shortest path trees from all origins to any destination s ~ S for any arrival time j ~ I, with link travel times for each interval given by uT(i), obtaining travel times v~(j) from r to s.

Step 3 (Arrival time choice) Compare for each origin-destination pair rs and for each interval j* the uT • . , total disutility of arrival at the end of intervals j~l~A(j*) ,~(j,j ) = v~(j) + ts(j,j*),

determining for each rs and j* the arrival interval j r which minimizes it. Step 4 (Shortest path trees loading) Load, for any rs ~ RS and j*~ I, the travel demand g*s{J*) to

origin-destination paths, identified by shortest path trees found in Step 2, taking as arrival times the end ofinterval j T for the travel demand qg*s(j*) and the end of interval j* for travel demand ( l - * "* f l .t q)gr~(J ), to get link flows T

Step 5 (Link flows updating) Calculate for any l E L and i ~ I the new current link flows as a linear combination of old current flows and flows f~r(i) that is: f°+l(i) = aft(i) + (1 - a)f~(i), with ct = 1/(D + I).

Step 6 (Convergence check) Verify if a measure of the difference between old and new current flows is lower than a predetermined tolerance value; if it is STOP, otherwise set D = D + 1 and go back to Step 1.

The main differences between a static type assignment algorithm and this one concern the following points:

!. the insertion of Step 3; 2. the fact that Steps 1, 4 and 5 have to be performed for each time interval; and 3. the shortest path trees search must take into account the dependence of travel times on time of

day.

To this aim a modified shortest path tree algorithm of the 'label-correcting' type is utilized, making use of the time transformation function tflUz,t); at each step it verifies if there is any out-of-tree link (here denoted by its initial and final nodes n and m) such that its insertion into the tree could improve the length of in-tree paths.

In our case the tree is rooted at a destination s for an arrival t ime jT and path length is given by travel time; at each step the shortest path tree algorithm should then verify if:

tf(U.m,tm) > t., (15)

where t m and t n are departure times from nodes m and n to arrive at destination s at t imejT. When no link n,m for which (15) can be verified is found, the current tree is a shortest path tree.

It is possible to show that this procedure reproduces, exactly, a minimum disutility path tree search on an equivalent graph representing a temporal transport network by defining time nodes n~ corresponding to node n for each time t, while links are defined in such a way that a link between nodes m', and n~' exist if and only if there is a link (m, n) in the original network and departure from m at time t' implies arrival at n at time t" (and vice versa); moreover, for each pair of successive time units t, t + 1 considered, and for each node n, a link is defined between nodes n~ and n,+ ~.

The arrival time choice performed at Step 3, together with the shortest paths tree search at Step 2, is equivalent to performing a shortest paths tree search on a further graph expansion, whose paths start from atemporal origin nodes r and arrive at OAT destination nodes s*, passing through

"tit is assumed here and in the following, for notational and computational convenience, that trip planners are a fraction q of total users for every OD pair and OAT, but the algorithm could be implemented as well with distinct trip planners and non-planners OD matrices, if known.

68 G. Bellei et al.-Discrete-time Dynamic Assignment Models

(departure) time nodes r, and the above defined equivalent graph and PAT destination nodes sj. To represent such paths, zero length links should connect r with each r, and each s, with sj such that t belongs to j. Moreover, links of length ts(j,j*) should connect each sj, with j belonging to the feasibility band for OAT j*, to s*.

THE PROBABILISTIC MODEL

The deterministic model described so far cannot represent the level of information available to trip makers, since route and arrival time choice are assumed to be always individually optimal in the equilibrium pattern. Moreover, it should be noted that, while the assessment of systems supplying real time information cannot take place in an equilibrium framework, the analysis of information systems based on historical or daily updated data does not imply representing route changing behaviour during the travel and may rely on average daily choices, which can be represented as an equilibrium state. If travel times and flows are considered as random variables non-optimal route choices may also be represented in this framework and the testing of another effect of improving information on traffic equilibria is allowed. A well-known approach to static assignment, the stochastic user equilibrium concept, can be extended into a dynamic framework to deal with non-optimal route choices. To this aim most of the previous modelling assumptions can be kept as valid and only equilibrium conditions need to be reformulated, in accordance with the hypothesis that link travel times are random, with known distribution and mean given by volume-delay function (14). Thus, OD and path travel times, together with path and link flows, become random too. The stochastic user equilibrium concept can be extended to get a probabilistic representation of arrival time choice, assuming that arrival time disutilities are also random with known distribution and given by function (1). Trip planners OAT OD flows are split among available arrival times proportionally to the probability of determining minimum individual disutility d,s(j, J ~ " "*)."

1 ' " 4 • " ~ [gr~(j ) g~(j2 .,)]pr.~d~s(j,l~ 1 . . , ) < drs(k, J 1 "*)Vk~IFA(j*),k 4:j~ ~,~(j,j )= Vrs e RS, j* e I, j e I FA(J*), (16)

which hold instead of (6), synthesising also (2) and (5); while non-planners OD flows are split, together with trip planners PAT OD flows as determined by (3), among available paths proportionally to the probability of determining minimum OD travel time:

hp(j) = gr~(j)Pr{zp(j) < zb(j) k/b • Prs,b # p} Vrs • RS, j* e I, j • IrA(j* ), (17)

which hold instead of (11), synthesizing also (12) and (13). The connection between route and arrival choice model in the probabilistic case is always given by

definition (4) of disutilities d~,~(j,j*), but with random variable v~ (j) explicitly defined as the minimum of random variables path travel times:

vrs(j ) = min {zp(j)}. (18) P ~ Prs

The interval link travel time and arrival time choice penalty distributions have been specified as normal and their variance assumed as proportional to link travel times and penalties, with coefficients varying among user classes, but constant through network links and arrival time periods. Notwithstanding this normality assumption the choice models we define are not probit since:

• because of equations and definitions (7)-(9) path travel times are not simply the sum of link travel times. Hence normality of link travel times does not imply normality of path travel times and route choice model is not probit;

• because of (18) OD travel times would not be normal even in the case where path travel times were normal. Hence, the arrival time choice model would also not be probit . t

tlt is interesting to note that, if the route choice model would be defined as logit, by directly assuming a Gumbel distribution for path travel times, the arrival time choice model would also be logit; here, the normality assumption for link travel times has been preferred as best suited to a simulation approach.


F-

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890000

880000

870000

- • Cony. erR. = 10 x 10 .5 m o Conv. erR. = 7 x I0 5

- * Cony. erR. = 5 x I0 "5 ~ / t 2

o o°v ori, = , ,05

- , Cony. crit. = 3x ! 0 -5 J ~ / , / / ~ , ~ Cony. c r i t . r ~ = 2.5 x I0 -s

I I I I I I I I I I I t 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Variance

Fig. 1. Graph of algorithm convergence.

In order to deal with occasional and systematic users previous formulas should be duplicated to take into account the splitting of demand (assumed as constant through OD pairs and arrival time periods) with the exception of (14), because they supply mean perceived link travel time, which does not vary among user classes. The changes needed to get a probabilistic algorithm reduce to the introduction, after Step 1, of a sample drawing for link travel times and penalties to be utilized, in subsequent steps; such a change corresponds to the application of a Monte Carlo method simulating the day-to-day process of adjusting planned arrival times and route choices until a stochastic equilibrium is reached.

COMPUTATIONAL TESTS

First of all, the algorithms presented have been utilized to verify convergence to equilibrium conditions. In the deterministic case convergence has been tested both at an aggregate and at a disaggregate level, verifying relative differences on total flows in successive iterations and equality of path travel times on sample paths. In the probabilistic case, because of the difficulty of testing proportionality of path flows to choice probabilities, convergence has been tested only at aggregate level, taking into account the influence of link travel time perception error. In Fig. 1 total travel times, obtained by setting the convergence criterion at several values of average relative difference between flows at successive iterations,P are reported against variance coefficient. In simulations where only systematic users are considered, the same coefficient is attributed to link times for non-planners and schedule rearrangement penalties, while perfect knowledge of link times is assumed for trip planners. Simulation results for different criteria exhibit a similar pattern, which is only slightly modified, towards lower total travel times and sensitivity to perception error, by decreasing relative difference limit.

A decrease in total time with perception error has been found for low variance coefficients. Even if it appears to be less marked for more demanding convergence criteria, it is not so rapidly vanishing that travel times monotonically increasing with perception error can be safely assumed as an asymptotic feature of the probabilistic algorithm. It looks, on the contrary, reasonable that a small perception error may slightly improve traffic conditions. In fact, non-individually optimal users' choices may reduce congestion on the most frequented routes, while it does not imply the use of completely 'wrong' routes, even when an equilibrated flow pattern is approached. It is also likely that higher perception errors reverse the trend by causing devious routes to be chosen.

i'More precisely, the average of flows at iterations D, D-I, D-2 and at iterations D-1, D-2, D-3 are computed at each iteration D and the average relative difference between them is the stopping criterion.

70 G. Bellei et a l . -Discrete- t ime Dynamic Assignment Models

The stopping criterion ofa 5 x 10-5 relative difference has, thus, been used in further simulations. In fact, fluctuations in results, which can be found when stopping the algorithm at higher difference values, disappear at this level. Moreover, variance coefficients have been set in such a way as to describe a low variance situation, that is at 5% for systematic, 15% for occasional users perception of link travel times (reduced to 2.5% for trip planners) and at 15% for schedule rearrangement penalties. A real size network represented by a 244 nodes, 38 centroids and 705 links graph has been considered, corresponding to the urban road network of a medium-sized (about 150,000 inhabitants) Italian town. The available 38 x 38 auto-driver mode OD matrix and its temporal distribution on six intervals from 6.30am to 9.30am, representative of an almost uncongested situation, have been adapted in such a way that significant congestion phenomena are simulated at a deeper level of detail. Traffic congestion was quantified through saturation indexes s(i), supplying the average network congestion experienced by trip makers during interval i:

.If(i) ]i(i)

s ( i ) - I EL CI (19)

Z Ji(i) l e L

where c I denotes physical link capacity. As a consequence, s(i) values higher than 30-35% are unrealistic for an urban traffic network. Sensitivity of trip planning effectiveness to mobility demand and, through it, to congestion level, was thus verified, with the aim of finding a situation where a non-negligible influence of trip planning on congestion could be recognized.

The result of these simulations, with regard to the probabilistic case, is illustrated in Fig. 2 where saturation index profiles are reported, comparing simulations including trip planning (identified by numbers l r-8r) with their analogues without trip planning (identified by numbers lb-8b), where the numbers from 1 to 8 refer to a demand multiplicative factor, with the same input parameters.t It can be seen that trip planning has a notable effect on congestion only if demand is higher than some critical level. This critical level, determining a congestion high enough to allow non-negligible travel time savings from trip planning, but not so high as to be unrealistic, may be fixed at slightly less than the multiplicative factor of four normally applied to observed demand in other simulations.

SIMULATION RESULTS

The results are all presented in the form of travel time savings deriving from the choice of arrival time and the reduction of link travel time perception error. More formally, defining T ~ as the total travel time in the base case (without arrival time choice), T TP and T NP as the total travel time for trip planners and non-planners, respectively, with arrival time choice, percentage travel time savings for trip planners S TP are:

SrP _ q T B - T TP q T R 100, (20)

while travel time savings for non-planners S NP and total travel time savings S T are:

s N P (1 -- q ) T a - T Ne = (l - q ) T B 100. (21)

T B _ T NP _ T TP ST = T a 100. (22)

These savings have been computed by the deterministic algorithm and in two different cases by the probabilistic one, again with only systematic user~ represented, no allowance being made by the deterministic algorithm for more user classes.

*A fictitious temporal distribution over 18 intervals was computed and a multiplicative factor, as determined in the following, was normally applied to observed demand.


0.50 F

0.45

0.40

0.35

0.30

0.25

0 . 2 0

0.15

0.10

0.05

• Ir O l b • 2r O 2b • 3r A 3b • 4r O 4 b • 5r O 5b • 6 r <> 6b • 7r A 7b • 8r o 8 b

I I t I f I I I I I I I I I I ' ~ : 1 ~ , 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18

Time intervals

Fig. 2. Sa tu ra t ion index profiles for different d e m a n d mul t ip l ica t ive factors.

30

25

20

t5

t,, [_, I0

5~

o

• Determ. (no variance) case - trip plan. o Determ. (no variance) case - non plan. Q Determ. (no variance) case - total

A

• Probab. (low variance) case- trip plan. / j , , " ~ o Probab. (low variance) case - non plan. , a / / B Probab. (low variance) case - total / l ~ • Probab. (high variance) case - trip plan. / ~ - A Probab. (high variance) case - non plan. a . , / ~

2 3 4 5 6 7 8

D e m a n d mul t ip l i ca t ive factor

Fig. 3. Travel t ime savings for different mobi l i ty demands and percept ion error variances.

No perception error, low variance and high variancet situations are, respectively simulated, letting mobility demand and trip planners percentage vary. Figure 3 reports the variation of S Te, S Ne and S T for these cases, correlated with the mobility demand. It shows that sensitivity of travel time savings to demand is, as one may expect, stronger for trip planners, being, in addition, constant over the whole range of demand factors for this class and up to reasonable demand levels for non-planners. In fact, it

o 1"Percentage of occas ional users was set at 10%, while percentage of t r ip p lanners was set at 20 '/o for occas ional and 10% for sys temat ic users, the shape of schedule r ea r rangement function was defined se t t ing • = 10, I' = 5, I f = e f = 3 for sys temat ic and • = 5, ~, = 3, I I = ef = 4 for occas ional users, I~ = e~ = I and 1 / = 2 for both, while percept ion er ror var iance coefficients were set at a low var iance level.

ITOR 2-1-F

72 G. Bellei et al.-Discrete-time Dynamic Assionment Models

18

14

"~ l0

6 i ~ .

4 i 2 I obab (low v ance) case t 2t • Probab. (high variance) case - trip plan.

A Probab. (high variance) case - non plan. A Probab. i(high varlance ) ca~e - total

0 I I I I I I 0 20 30 40 50 60 70 80 90

Trip planners (%)

Fig. 4. Travel t ime savings for different trip p lanners percentages and perception error variances.

is easy to note that S NP and S v diminish when demand becomes so high to determine unrealistic congestion, as it can be checked through comparison with Fig. 2, where congestion indexes around and over 35-40% correspond, during the peak, to demand factors from 6 to 8. Savings increase from the no perception error to the low variance situation and, more markedly, to the higher variance case, in line with expectations. Such a result, however, is due to the low (10%) trip planners percentage assumed for these simulations, while in general it does not hold.

As can be seen in Fig. 4, where the output of trip planners percentage simulations is reported, sensitivity of travel time savings to trip planners percentage is higher in the deterministic case. In fact, only for 10% of trip planners, S TP is lower in the deterministic case than in probabilistic low variance case. On the contrary, for the 90% trip planning case, deterministic S TP is higher also than high variance level S TP, as always happens for S NP. Trip planners percentage simulations show also that trip planners travel time savings are non-zero when user percentage goes to zero, then grow steadily until user percentage approaches 100%, while non-planners and total travel time savings are obviously zero when user percentage is zero, then grow at a diminishing rate. The difference between S TP and S NP, which can be taken as a measure of comparative advantage of trip planning behaviour, is thus quite large at the beginning of trip planning, then decreases markedly up to around 60% of trip planners and keeps almost constant for even higher percentages.

Figure 5 reports the same kind of analysis as Fig. 3, referring to the low variance probabilistic case alone, but taking into account also the presence of occasional users; comparison of Figs 3 and 5 do not show any substantial difference, since very large savings of occasional users cannot have a match in Fig. 3 and refer anyway to a mere 2% of total demand. On the other hand, comparing Fig. 4 to Fig. 6, it is possible to note that S Tp is at first markedly decreasing with increased trip planning for occasional users and then approaches a constant value when trip planning is widespread among them. Moreover, time savings for systematic users appear to be almost insensitive to trip planners percentage, at least up to about 45% trip planners.f

Some simulations have been carried out also to verify the influence of demand composition on travel time savings, but a substantial indifference to changes in the rate of occasional users has been evidentiated.

tVariance coefficients are 5% (low variance) and 20% (high variance) for schedule rearrangement penalties and perception of link travel times; these last being reduced to 2.5% and 5% for trip planners.

International Transactions & Operational Research Vol. 2, No. 1

40 - • Oec. users trip planners /

i n Occ. users non planners , ¢ /

35 / - -

• Syst. users trip planners i

0 Syst. users non planners 30

~' 20

lO

5

0t 2 3 4 5 6 7 8

D e m a n d mult ip l icat ive factor

Fig. 5. Travel t ime savings for different mobi l i ty demands (probabil ist ic case).

73

14

12

io

.E 8 '

6 .~.

[.... 4

2

0 I0

• O c c . users trip planners <> Syst. users non planners

- D Oec. users non planners • Total users ~ - " • - " " - " A • Syst. users trip p l a n n e r s ~

I I I I I I I I 20 30 40 50 60 70 80 90

Occas iona l users trip p lanners (%)

Fig. 6. Travel t ime savings for different trip planner percentages (probabil ist ic casel .

C O N C L U S I O N S

The simulation exercise carried out allows some indications on the impact of driver information systems and on their effectiveness in different situations. Main results obtained show that there is a level of congestion at which an intervention aimed at introducing trip planning systems is justified and that the development path of such systems looks quite smooth, without critical points. Simulations performed with the probabilistic model did, however, indicate that trip planning growth may be less favourable to users than in the case of the deterministic model, showing a tendency of time savings to settle at a value slightly lower than at system introduction.

Moreover, simulations with different variance coefficients seem to indicate that, after a suitably long learning period, users approach very similar equilibria independently from their a priori information level, at least if the initial perception error is not too large.

Of course, validation of these preliminary results needs field analysis and model calibration, which would be the most effective research to pursue further. Nevertheless, some useful elements on this

74 G. Bellei et a l . -Discrete- t ime Dynamic Assionment Models

issue could be obtained by reducing interval length and applying this method, still as a s imulat ion tool, to a network and demand truly representing a large, congested, metropol i tan area.

From the point of view of the building up of advanced mathematical models for informat ion systems and demand management assessment, these results can be considered as promising because they show that it is possible to represent all the main behavioural aspects of trip p lanning system introduct ion, at least with reference to an average si tuation, also exploiting well-consolidated

methodologies and algorithms. The fact that the existence and uniqueness of a solut ion for the above stated equi l ibr ium model have not yet been demonst ra ted does not imply that the chosen approach is

intrinsically unsuitable, since such a proof is still lacking for a viable dynamic assignment method. It is instead a further st imulus to deepen research, exploiting the relatively simple structure of the mathematical relationships.

R E F E R E N C E S

Arnott, R., de Palma, A. & Lindsey, R. (1991}. Does providing information to drivers reduce traffic congestion? Transportation Research A, Vol. 25A, pp. 309-318.

Ben-Akiva, M., Cyna, M. & de Palma, A. (1984). Dynamic model of peak period congestion. Transportation Research B, Vol. 18B, pp. 339-355.

Bielli, M. (1990). Characteristics and requirements of trip planning systems. Preprints of Prometheus Conference, London. Bielli, M. (1992). A DSS approach to urban traffic management. European Journal of Operational Research, Vol. 61, pp.

106-113. Bellei, G. & Bielli, M. (1990). Dynamic assignment for trip planning systems assessment. In Proceedint¢s of the ANIPLA

Conoress on Automation qf Transport Systems, Trieste, pp. 520-542. Bellei, G. & Bielli, M. (1992). Sensitivity analysis of a dynamic equilibrium model for route and arrival time choice. In

Proceedings of the Second Capri Seminar on Urban Traffic Networks. Berlin: Springer (in press). Bellei, G. (1992). Time discretization in dynamic traffic assignment. In Proceedinqs of the First Meetin9 ofEURO Workin9

Group on Urban Traffic and Transportation, Munich. Carey, M. (1987). Optimal time-varying flows on congested networks. Operations Research, Vol. 35, pp. 58-69. Carey, M. (1992). Nonconvexity of the dynamic traffic assignment problem. Transportation Research B, Vol. 26B, pp.

339-355. Cascetta, E. & Cantarella, G. E. (1991 I. A day-to-day and within-day dynamic stochastic assignment model. Transportation

Research A, Vol. 25A, pp. 277-291. Dehoux & Toint, L. (1991). Some comments on dynamic traffic modelling in the presence of advanced driver information

systems. In Proceedinys t~fDRIVE Conference, Brussels, pp. 964-980. Amsterdam: Elsevier. Gallo+ G. & Pallottino, S. (1988). Shortest path algorithms. Annals of Operations Research, Vol. 13, pp. 3-79 Baltzer AG. Hamerslag, R. [1989). Dynamic assignment in three-dimensional time space. Transportation Research Record 1220,

TRB-USA. Janson, B. N. ( 1991 ). Dynamic traffic assignment for urban road networks. Transportation Research B+ Vol. 25B, pp. 143-161. Mahmassani, H. S. & Chang, G. L. (1988). Travel time prediction and departure time adjustment behaviour dynamics in a

congested traffic system. Transportation Research B, Vol. 22B, pp. 217-232. Mahmassani, H.S. & Jayakrishnan, R. (1991). System performance and user response under real-time information in a

congested traffic corridor. Transportation Research A, Vol. 25A, pp. 293-308. Merchant, D.K. & Nemhauser, G.L. 11978). A model and algorithm for the dynamic traffic assignment problem.

Transportation Science, Vol. 12+ pp. 183 199. Papageorgiou, M. I 1990). Dynamic modelling, assignment and route guidance in traffic networks. Transportation Research B,

Vol. 24B, pp. 471-495. Ran, B., Boyce, D. E. & LeBlanc, L.J. (1992). Dynamic user-optimal route choice models based on stochastic route travel

times. In Proceedinys of the Second Capri Seminar on Urban Traffic Networks. Berlin: Springer (in press). Sheffi, Y. (1985). Urban Transportation Networks. Englewood Cliffs, N J: Prentice Hall. Vythoulkas, P.K. (1990). A dynamic stochastic assignment model for the analysis of general networks. Transportation

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Simulation exercise on time perception parameters in discrete-time dynamic assignment models

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