Signal optimisation using the cross entropy method

Transportation Research Part C 27 (2013) 76–88

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Signal optimisation using the cross entropy method Mike Maher ⇑, Ronghui Liu, Dong Ngoduy Institute for Transport Studies, University of Leeds, Leeds LS2 9JT, United Kingdom

a r t i c l e

i n f o

Article history: Received 26 July 2010 Received in revised form 19 May 2011 Accepted 27 May 2011

Keywords: Traffic control Combinatorial optimisation Cross-entropy method Traffic models Performance index Noisy optimisation

a b s t r a c t The problem of the optimisation of traffic signals in a network comes in a variety of forms, depending on whether the traffic model used to evaluate any proposed set timings is deterministic or Monte Carlo, whether the drivers’ routes are fixed or dependent on the signal timings, and whether the control is fixed-time or responsive. The paper deals with fixedtime control, and investigates the application of the cross-entropy method (CEM) to find the global optimum solution. It is shown that the CEM can be applied both to deterministic and Monte Carlo problems and to fixed-route or variable-route problems. Such combinatorial problems typically have a large number of local optima and therefore simple hillclimbing methods are ineffective. The paper demonstrates firstly how the cross-entropy method provides an efficient and robust approach when the traffic model that provides, for any solution x, the value of the performance index (PI) z(x) is deterministic. It then goes onto discuss the effect of noise in the evaluation process, such as arises when a Monte Carlo simulation model is used, so that the PI can be expressed as z(x) = z0(x) + e(x) where e is a random error, whose variance s2 depends inversely either on T, the length of the simulation run, or on M the number of simulation runs carried out for any solution. A second example illustrates the application of the CEM to a noisy problem, in which a Monte Carlo traffic assignment model is used to estimate drivers’ route choices in response to any proposed signal timings, and shows the principles by which the values of either T or M must be adapted through the iterative process. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Traffic signals are a vital tool in the efficient and safe use of road space and control of traffic in congested urban networks. For over 50 years, traffic engineers have been developing methods for modelling the flow of traffic, starting from the seminal work of Lighthill and Whitham (1955) and Richards (1956), and progressing through macroscopic models such as the platoon dispersion model (Grace and Potts, 1964) and the cell transmission model of Daganzo (1994, 1995, 1997) to Monte Carlo microscopic traffic simulation models such as Paramics (SIAS, 2011) and Dracula (Liu, 2005). The early work of Webster (1958) and Webster and Cobbe (1966) on the optimisation of signal timings at individual junctions was then generalised and formulated as a mathematical programming problem to minimise the total junction delay by Allsop (1971), and as a capacity-maximising problem by Allsop (1972), later extended to the time-dependent case by Han (1990, 1996). Work on the coordination and optimisation of signals in networks started with the work of Hillier and Rothery (1967), and Robertson (1968) amongst others with the aim of minimising delays in congested networks, and was extended through the groupbased optimisation approach of Heydecker (1996) and Wong (1996). Wong (1995, 1997) derived approximate derivatives

⇑ Corresponding author. Tel.: +44 (0)113 343 6610; fax: +44 (0)113 343 5334. E-mail address: [email protected] (M. Maher). 0968-090X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2011.05.018

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of the Transyt performance index which enabled the formulation of an integer programming technique which was combined with hill-climbing on the offsets, to solve the optimisation problem. Whilst the above work assumed that traffic flows were given and would remain fixed under changes to signal timings, there has been another strand of research into the optimisation of signals in networks where it is assumed that drivers will re-route in response to changes in signal timings. Early work by Allsop (1974), Allsop and Charlesworth (1977), and Maher and Akçelik (1977) has been further extended and formalised by, for example, Wong et al. (1998), Wong and Yang (1999), Chiou (2003) and Ceylan and Bell (2005). A considerable number of systems have been developed for both fixed-time and responsive traffic control both at isolated intersections and in a coordinated manner in networks of junctions. In order to find the optimal timings, a traffic model is required that will predict the traffic flow pattern, delays and stops that would result from the implementation of any proposed set of timings, or control policy. Such traffic models come in a variety of forms, macroscopic and microscopic, deterministic or stochastic; some have purely numerical outputs whilst others provide graphical displays. Whilst responsive control has become increasingly prevalent, fixed-time control is still an important and widely-used form of control in many urban networks. Fixed time plans can be set up for the demand pattern expected at different times of day. The problem of finding the optimal timings, according to the predictions from the assumed form of traffic model, is usually far from straightforward except in the case of a single intersection, since there are typically an enormous number of feasible solutions and a very large number of local minima; in addition the evaluation of the performance for any given solution is a far from trivial task. The problem of finding the global minimum is a complex combinatorial optimisation problem for which many heuristic approaches (including, for example, hill-climbing, evolutionary algorithms, ant colony optimisation, and simulated annealing) have been proposed and tried, but for which there is as yet no fully accepted method. The research described here is concerned with the application and testing of a relatively new approach to such problems: the cross entropy method, proposed by Rubinstein and Kroese (2004). The method has an appealingly simple structure and a sound theoretical underpinning. So far, applications of the method have been almost exclusively to deterministic problems, but the main purpose here is to see how it can be applied to noisy problems, where the objective function is evaluated using a Monte Carlo simulation model. To develop the ideas and motivate the application to the noisy case, the method will initially be applied to a deterministic signal optimisation problem. Once that has been achieved, the general principles of the extension to noisy problems will be presented and illustrated by means of an example in which the optimal signal settings in a network are to be optimised and where the re-routeing of drivers in response to the signal timing plan is estimated by a stochastic traffic assignment model. Whilst the results obtained are specific to this one example, general implications can be drawn for the solution of all noisy global optimisation problems.

2. Problem formulation We consider a network of signalised junctions, which are to operate under fixed-time control. The set of signal timings to be implemented consists of the green times for the various stages at each junction, and the offsets between the junctions. A set of timings will be referred to as a solution and denoted by x = (x1, x2, . . . xm), a vector made up of m elements, in some appropriate form, representing the green times or stage start times at each junction, and the relative offsets of nodes, each expressed as an integer number of seconds. Typically the set X of all possible solutions will be very large. The choice between alternative solutions will be made according to the traffic model adopted for any particular case. Running the traffic model with a solution x leads to a set of outputs, from which the value of an appropriate objective function or performance index can be calculated. This will be denoted by z(x). The aim is to determine the optimal solution x⁄ that minimises this objective function. In some versions of the problem, the traffic flows q are assumed to be known and to remain fixed, regardless of the signal timings that are implemented. In other versions, it is assumed that drivers’ route choices (and hence the flow pattern q) may be affected by the signal timings so that the aim is to determine the solution x that minimises z(x, q(x)) – see, for example, Allsop (1974), and Allsop and Charlesworth (1977). The traffic flow pattern q for any solution x will be estimated through an equilibrium traffic assignment model. The ‘‘mutually consistent’’ solution obtained by alternately (i) finding the routeing pattern q(x) given the current signal timings x, and (ii) the signal timings x(q) that minimise the delay for the current flow pattern q, is inferior to x because of the lack of recognition that the flows will change when the signal timings change. Furthermore, the traffic model may be either deterministic (such as that in the widely-used network signal optimisation software Transyt, or in the cell transmission model) or a Monte Carlo simulation model such as in Paramics or Dracula. Similarly the traffic assignment model may be deterministic (such as in Wardrop, or user equilibrium (UE) assignment), or may again be Monte Carlo in nature, such as in SUE Burrell-type assignment (Burrell, 1968). When the traffic model and/or assignment model are deterministic, we have a deterministic optimisation problem; but when either the traffic model or the assignment model are Monte Carlo in nature, we have a noisy optimisation problem, since the value of the objective function z will be subject to random error or noise. The main focus of this paper is to address the noisy case, and to investigate an extension of the recently-developed cross entropy method (see Rubinstein and Kroese, 2004) to the problem of finding the global optimum solution in such problems. However, for presentational reasons, the method will initially be applied to a deterministic problem. Once that has been done, the extension to the noisy case will be presented and illustrated.

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3. The cross entropy method The cross entropy method (CEM) is an iterative process for complex combinatorial optimization problems, suitable where there are numerous local optima, and the solution consists of a vector in discrete-valued elements (such as green times and offsets). We give first a generic description of the cross entropy (CE) method, based on that in de Boer et al. (2004), before seeing how it is to be applied to the current signal optimisation problem. The general problem is to find the solution x = x that minimises the objective function z(x), given that we can generate random solutions from a density function p(x). Then the associated stochastic estimation problem is to estimate the (small) probability l(c) that a randomly chosen solution x has a value of the objective function z(x) < c where c is close to (but greater than) c = z(x). Estimating this probability by generating solutions from p(x) will generally be very inefficient, and a more efficient approach is via importance sampling, where we generate solutions instead from a density g(x), designed to ensure that a high proportion of solutions are generated for which z(x) < c. The estimate of l(c) is then given by:

X pðxi Þ ^lðcÞ ¼ 1 Iðzðxi Þ < cÞ N i gðxi Þ where N is the sample size, and I(z(xi) < c) is an indicator variable, taking the value 1 if z(xi) < c and zero otherwise. The ideal density function from which to generate solutions would be that which only took non-zero values where z(xi) < c: that is:

g  ðxÞ ¼

IðzðxÞ < cÞpðxÞ lðcÞ

However, it is clearly not possible to construct this since, for one thing, the value of l(c) is not known. What the crossentropy method does is to inform us how to construct a density g(x), from amongst a family of distributions p(x; b), that is as close as possible to g by minimising the Kullback–Leibler (or cross entropy) measure D, which is the ‘‘distance’’ between two distributions. This family includes the base (typically a uniform or similarly broad) distribution p(x) = p(x, b(1)). It is then a matter of choosing the values of the parameters b so as to minimise D, and make the sampling as efficient as possible, by making the precision of the estimate of l(c) as good as possible, for the given sample size N. It is shown in de Boer et al. (2004) that this problem of minimising D is equivalent to the program:

Max DðbÞ ¼ Max E½IðzðxÞ < cÞ ln pðx; bÞ b

b

This leads to the following general form of the CE algorithm: 1. Initialise by setting the iteration counter k to 1, and set the parameter values b = b(1). 2. Generate a sample of solutions x1 . . . xN from the density p(x; b(k)). Sort the sample into ascending order by their values z(x), and select the best (lowest) 100q% (typically q = 0.05), so that the estimate of the 100qth quantile is c(k). 3. Using only these best solutions x (that is, those for which z(x) < c(k)), find the values of the parameters b that maximise P (k+1) . i ln pðxi ; bÞ. Denote these values by b 4. Increase k by 1, and return to step 1, unless b(k+1) = b(k) in which case stop. Note that step 3 is akin to maximum likelihood estimation of the parameters b, using only the best solutions x. As the iterations progress, the values of c(k) should initially reduce and then stabilise. The parameter values b(k) should also stabilise. It is usual in step 3 to ‘‘smooth’’ the parameter estimates by taking a weighted average of the new-obtained estimates and the previous ones (typically with a weight of 0.7 applied to the newly-obtained estimates). In our signal optimisation problems, each of the N solutions generated in an iteration is evaluated using the traffic model (or traffic assignment model) and its objective function value obtained. These N solutions are ranked and the best N identified (with N typically being 5% of N). The vector of parameters b is made up of subsets, each related to a particular element (green or offset) of the solution vector x. Using this ‘‘elite’’ sample, for each element of the solution we estimate the relevant parameters (typically in a manner akin to maximum likelihood estimation) and take a weighted average of these values and the previous ones. These are then used to generate the solutions in the next iteration. Hence the parameter estimates are updated at each iteration: b(1), b(2) . . . and so on. The process starts with non-informative estimates (typically with a uniform, or other similarly broad distribution assumed). Through the process of selection and updating of parameters, the quality of the solutions generated steadily improves, and the distributions gradually become more clustered around a small number of values until no further improvement occurs. The best solution found during the iterative process is the estimate of the global optimum. There is no guarantee, of course, that the global optimum will be found, as the process is stochastic. More detail will be given later in the paper of the precise representation of the solution x adopted for the example problems, and of the definition of the parameters b. Fuller details of the CEM may be found in Rubinstein and Kroese (2004).

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4. Previous work Previous work (Maher, 2008) has applied the CEM to the problem of optimisation of fixed-time signal timings on a six-arm signalised roundabout with the cell transmission model being used to describe the cyclic flow of traffic, and the build-up and decay of queues, under the assumption of constant OD flows. There, the traffic model was deterministic and macroscopic and there is no route choice. For that problem, it was calculated that there were of the order of 1016 feasible (nominally undersaturated) solutions. At each iteration N = 2000 solutions were generated, and it was found that the CEM worked well, with steady improvement through the iterative process and effective convergence occurring in around 15 iterations. It seemed therefore to provide an efficient and appealing approach to such combinatorial optimisation problems. Several aspects of the parameterisation and solution representation make the current work different from (and an advance on) that previous work. The authors have also carried out some preliminary work (Maher and Liu, 2010) applying the CEM to signal optimization on a signalized roundabout using the Monte Carlo microscopic traffic simulation model Dracula (Liu, 2005) and, in Liu and Maher (2010), investigating and quantifying the amounts of random error arising from various sources in such models (for example, the (Poisson) entry flows, day-to-day variation in demand levels, the random mix of vehicle types, vehicle performance and driver behaviour, and the random variation in supply side factors such as capacity and free-flow speed). In Ngoduy and Maher (2011) a comparison is made between the performances of the CEM and a genetic algorithm (GA) for finding the optimal signal timings in a network where re-routeing is described by a (deterministic) user equilibrium traffic assignment model. The CEM produces an appreciably better solution than the GA. The current work is to apply the CEM to two further types of signal optimisation problem, each being a separate and distinct extension to previous work. The first is a deterministic problem involving the well-known Transyt traffic model. Widely-used around the world, the Transyt software has become established as an industry standard for the optimisation of signals in urban networks, and its traffic model has been thoroughly tested and validated (see for example Rouphail, 1983; Castle and Bonniville, 1985; Axhausen and Korling, 1987; Manar and Baass, 1996; Yu, 2000; Farzaneh and Rakha, 2006). The second problem is a noisy one, involving signal optimization under re-routeing in a network. However, unlike Ngoduy and Maher (2011) where a deterministic assignment model was used, here a Monte Carlo stochastic user equilibrium traffic assignment model is used, so that the optimisation problem is a noisy one. 5. The cross-entropy method and Transyt The core parts of the Transyt software have remained essentially the same since it was first developed (Robertson, 1968): it uses a simple deterministic, macroscopic traffic model with a platoon dispersion model employed to describe the propagation of flow from the upstream end of a link to the downstream stop-line. Hence, for any proposed set of signal timings x, the cyclic flow profiles are calculated and the network performance index z(x) (a linear combination of delays and stops) evaluated. A hill-climbing method is used to search for the optimal timings x. Despite it popularity and success, it has been known for some years that the hill-climbing method results in a local optimum. Starting the search from different initial solutions can very often lead to different final solutions. To illustrate this deficiency, consider the 6-arm signalised motorway roundabout network shown in Fig. 1, which is taken from the Transyt user manual (Binning et al., 2009) and included as a sample input file in the software. Each node has two stages, with the entry flow having green in stage 1, and the circulating flow having green in stage 2. The user must provide, in the input file, an initial solution (typically based on ‘‘equisat’’ timings at each node, where the stage green times are such that the degrees of saturation on the two stages are equal). In the solution shown in Table 1, there are intergreens (or ‘‘all red’’ periods) of 5 s between successive stages at each node and so, at node 1, stage 1 runs from 0 to 13 (inclusive), there is an intergreen from 14 to 18, stage 2 runs from 19 to 54, and finally an intergreen runs from 55 to 59 to complete the cycle. As a test, 100 initial solutions were generated, similar to that shown above, with equisat green times at each node, and with a common cycle time of C = 60 s, but with randomly- generated offsets. The Transyt hill-climbing optimiser was run from each of these initial solutions. The 100 final solutions obtained were, remarkably, all different, and their PI values ranged from 679.7 to 865.5, with a mean of 718.3 and a standard deviation of 26.3: an average loss of 6% (percentage excess over the global minimum). A histogram showing these is shown in Fig. 2. These results demonstrate clearly (i) the extremely ‘‘bumpy’’ nature of the PI terrain, with a large number of local optima, and (ii) the weakness of the hill-climbing method and its inadequacy in finding the global optimum (which is firmly believed to be z = 679.7, as will be seen later). Use of a single run of the hill-climbing method can easily lead to a solution that is far from the optimum. Recognising this weakness, TRL have, in the latest version (version 14) of the Transyt software, provided some alternative optimisation approaches: a ‘‘shotgun start’’ method (in which the hill-climbing is started from a number of different initial solutions, and the best final outcome chosen), and a version of simulated annealing (see Binning et al., 2010). 6. Solution representation and parameterisation in the CEM There are a number of ways in which a solution can be represented. For input to Transyt, the required form is the set of stage start times at each node, as shown in Table 1. Note that the stage 1 start time at node 1 can always be set to zero, as all stage start times are relative to an arbitrary base.

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Fig. 1. Test network: a 6-arm signalized motorway roundabout (from the Transyt User Manual).

Table 1 Typical solution (C = 60 s). Node

Stage 1 start

Stage 2 start

1 2 3 4 5 6

0 29 8 26 30 44

19 53 31 40 4 8

30

Frequency

25 20 15 10 5

0 79

77 0

75 0

73 0

71 0

69 0

67

0

0

Fig. 2. Histogram of final PI values from hill-climbing method.

However, for the CEM, it is more convenient to represent the solution as the set of green times (that is, from the start of one stage to the start of the next stage, including intergreens etc.), and the set of relative offsets between the start of stage 1 at the end node and that at the start node on each link. A ‘‘link’’ here means a pair of adjacent signalised nodes with a specified direction (from one node to another). In our example motorway roundabout network, these are 1–2, 2–3, 3–4, 4–5, 5–6 and 6–1. So, the solution in Table 1 would be represented as x = (19, 41, 24, 36, 23, 37, 14, 46, 34, 26, 24, 36; 29, 39, 18, 4, 14, 16). The first 12 elements are the stage greens at the six nodes, and the last six elements are the offsets on the six links. Even though there is redundancy in this representation, it is convenient for the calculations carried out in the CEM. The parameters used are lij, the means of the stage green times at each node i, and a common node standard deviation ri. A multivariate Normal (MVN) distribution is used to generate random green values. If the vector of the n green times at node

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i is denoted by gi, then gi ¼ li þ ri e0i where e0i is a vector of correlated random errors, each with zero mean and unit variance and which sum to zero, constructed as follows from a vector of uncorrelated unit Normal errors ei:

e0i1 ¼ A1 ei1 e0ij ¼

j1 X

ak eik þ Aj eij

j ¼ 2; . . . n

k¼1

P 2 0 0 in which A1 = 1, aj = Aj/(n  j) for j = 1, . . . n and A2j ¼ 1  j1 k¼1 ak for j = 2, . . . n. The correlation between eij and eik for any j, k pair (j – k) is 1/(n  1). pffiffi pffiffi To illustrate, for n = 3, these correlated errors are constructed by: e01 ¼ e1 ; e02 ¼  12 e1 þ 23 e2 , and e03 ¼  12 e1 þ 23 e2 . It can be seen that e01 þ e02 þ e03 ¼ 0, as well as E(e0j ) = 0 and Var(e0j Þ ¼ 1 for j = 1, 2, 3. Finally, the green times thus obtained are rounded to the nearest integer value and, if any of the greens does not satisfy the minimum green requirements, a new set of greens is generated for that node. For the offsets on all links, a general discrete distribution is used: that is a set of pk(t) for each link k and each possible offset t (= 0, 1, . . . C  1) where C is the cycle time. Initially this is a uniform distribution so that pk(t) = 1/C for all k and t. Not all links’ preferred offsets can be accommodated (for example, once the offsets for the first five links 1–2, 2–3, 3–4, 4– 5 and 5–6 have been specified, that for the remaining link 6–1 is determined), so the links are dealt with in random order, and where it is possible to set that link’s offset to its desired (random) value, that is done, but if it is not then it has to have what is determined by other links. Once these N solutions have been generated, their PI values evaluated using the Transyt traffic model, and the best N solutions identified (typically N = 0.05N), the parameter updating can be carried out, using the principles of maximum likelihood referred to in step 3 of the general CE method in Section 3. For each of the stage greens, the average and standard deviation of the values in the elite sample are calculated. For the stage greens at any node, the average of these standard deviations is calculated to give the new common value. The new values of the means and common standard deviation are a weighted average of the previous values and these elite sample values, giving a weight of (1  a) to the previous estimates and a weight of a to the elite sample values. Here we have used a value of a = 0.7. For each of the link offsets, we count up the number of occurrences in the elite sample of each possible value t, to give a set of frequencies fk(t). These are used in the same way to update the discrete distributions: afk(t)/N + (1  a)pk(t) for use in the next iteration. Finally, at the end of each iteration, a number of statistics are calculated and recorded, such as the average of z for all solutions, the average z for the elite sample solutions, the 5th percentile z value and the z value of the best solution found. In addition S, the standard deviation of the z values of all solutions is calculated. 7. Results In the application of the CEM on the test network, a value of ri = 2 was used as an initial value of the common standard deviation at each node. The results are shown in Table 2. The progression follows a fairly regular pattern, in which the average PI in the (k + 1)th iteration is, approximately, a weighted average of the average and 5th percentile values in the kth iteration as shown below:

zðkþ1Þ ¼ 0:7zðkÞ ðkÞ 0:05 þ 0:3z S measures the between-solution variability in each iteration, and it can be seen in Fig. 3 that this steadily reduces in a roughly exponential decay fashion.

Table 2 Results from cross-entropy method. Iteration

zmin

z0.05

zav (sel)

zav (all)

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14

864.1 781.7 774.0 747.3 716.6 709.2 702.2 695.6 689.9 686.9 682.8 680.1 679.9 679.7

967.7 882.3 828.6 793.2 767.4 747.9 723.7 708.0 698.8 692.8 688.9 685.6 681.1 680.0

935.6 851.7 806.0 780.1 755.3 736.0 715.8 703.4 696.2 691.0 687.3 683.5 680.6 679.9

1413.6 1068.9 946.8 891.8 853.7 825.2 794.2 752.3 725.6 709.3 699.8 694.0 689.0 684.7

557.1 176.0 85.5 72.7 70.2 66.4 63.8 50.3 36.0 20.2 15.1 9.2 8.8 6.4

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1000

S

100

10

1

1

3

5

7

9

11

13

iteration Fig. 3. Plot of S (log scale) versus iteration number.

2

1.5

SD

SD3

1

SD3

SD1

SD4

SD2

SD5

0.5

0

1

3

5

7

9

11

13

15

iteration Fig. 4. Progression of standard deviations of greens through iterative process.

The green standard deviations ri also reduce steadily from their initial value of 2, as can be seen in Fig. 4. The best solution (giving z = 679.7) is shown in Table 3. The way in which the offset probability distributions for the offsets pk(t) steadily evolve over the iterative process is exemplified in Fig. 5 where the progression for link 1–2 is shown. The gradual concentration of probability around certain values is evident. In the 15th iteration, more than 500 of the 1000 solutions generated have a z value of 681.8 or less, so we obtain a large number of alternative solutions that are almost as good as the optimal solution of z = 679.7. Seven further independent runs of this implementation of the CEM also converged on the same solution z = 679.7. Therefore the CEM seems to be quite robust in finding the global optimum. The method was also applied using other values of the common cycle time. These results are not shown here, but are of a similar pattern, and show a similar performance for the method. Finally, using a beta version of Transyt 14 provided by TRL, we were able to compare the solution above obtained from the CEM with those achieved by the shotgun start and simulated annealing methods now offered as alternatives to the basic hillclimbing method. Using ten initial solutions in the shotgun start method, the best solution found gave z = 748.09, whilst the simulated annealing method gave z = 704.06. Both are clearly inferior to the solution produced by the CEM. It should be noted however that no information is available on the details of the implementation of either method. 8. Noisy signal optimisation problem: generic issues We turn now to the application of the CEM when the evaluation of any solution x is carried out using a Monte Carlo simulation model, contrary to the previous example where the traffic model was deterministic. Ideally we would like to know

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M. Maher et al. / Transportation Research Part C 27 (2013) 76–88 Table 3 Optimum solution. Node

Stage 1 start

Stage 2 start

1 2 3 4 5 6

0 49 46 4 7 17

19 13 9 18 41 42

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

0

offset 0.4

0.4 0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05 1

6

11

16

21

26

31

6

11

16

21

26

31

36

41

46

51

56

61

36

41

46

51

56

61

offset

0.35

0

1

36

41

46

51

56

61

0

1

6

11

16

21

offset

26

31

offset Fig. 5. Shape of p(t) for link 1–2 at iterations 1, 5, 10 and 14.

z0(x) for any solution x, but all we can observe is z(x) = z0(x) + e(x) where e(x) is a random error, with zero mean and with a variance s2. Hence, just because z(x1) < z(x2) does not necessarily imply that z0(x1) < z0(x2) for any two solutions x1 and x2. As pointed out by Maher (2007), the observed between-solutions variance is made up of two components: S2 ¼ S20 þ s2 where S20 is the variance in the (unknown) z0 values of the solutions. The error variance s2 is governed by the characteristics of the simulation process used to produce z(x), the noisy estimate of the objective function. To reduce s, the length T of the simulation can be increased, or M the number of repeat runs (from which the average value of z is taken) can be increased. In either case, s2 is inversely proportional to T or M. Therefore, accurate estimates can be obtained – but only at the expense of long run times or a large number of replicate runs for each solution. The CEM, as has been seen, relies on the ranking of solutions in any iteration. If we rank a set of solutions on the basis of their z values, a measure of the benefit B arising from the ranking and selection of the best N solutions is the difference between l (the mean PI of all solutions) and the mean z0 of the selected solutions. The ranking will be more efficient, and the benefit greater, the smaller the ratio r = s/S is; as r increases, the effect of the noise dominates, and the ranking eventually becomes effectively random and the benefit will approach zero. If we were to compare the benefit arising from ranking on the basis of the z values with the benefit B0 that would be produced from the ideal case of selection on the basis of the true z0 values, then B 6 B0 and the proportionate loss in benefit (B0  B)/B0 will be zero if r = 0, and approach 1 as r increases because of the inefficiency of the ranking process. In the case where the distributions of the e’s and the z0’s are both Normally distributed, so that z and z0 have a common mean of l, standard deviations of S and S0 and a correlation of pffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ 1  r2 then, from standard results for the bivariate Normal distribution (see Kotz et al., 2000), we have that:

S0 Eðz0 jz ¼ aÞ ¼ l þ q SS0 ða  lÞ

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It follows from this, that the proportionate loss (B0  B)/B0 is r2, so that when r = 0.7 (and the noise is comparable with the true between-solutions variation), the loss is approximately 0.5. Whilst it is ideally desirable to keep this loss low (by having a small value for r), this involves more computational effort, as the number of Monte Carlo iterations for each solution required to achieve this increases proportionally to 1/r2. Therefore there is a clear trade-off in the choice of appropriate value for r at any stage. We have seen in the previous problem how, in the cross entropy method, the value of S decreases steadily through the iterations. For this process to work effectively, the value of r must not be allowed to become too large (say > 0.7, which corresponds roughly to the point at which the components s2 and S20 are of equal magnitude), otherwise the method will cease to make any effective progress. The amount of noise must be reduced by increasing n or T (thereby increasing the computational effort in the simulation process) steadily through the iterative process of the CEM, keeping s approximately proportional to the observed value of S. Hence, short simulation runs provide sufficient accuracy in early iterations, but longer ones are required later. Therefore, in summary, to be able to apply the CEM to noisy problems, we need to understand the relationship between s2 and M or T, so that the required number of replications or the simulation run time to achieve any specified value of s can be estimated. Experience indicates that the value of s is reasonably constant for different solutions x. Therefore, an initial set of replications for some representative solution, using some trial number of replicates n, should be sufficient for this purpose. Secondly, the observed value of S from any iteration should be calculated and, on the basis of this, the required value of s for the next iteration is calculated in order to keep r below a value of approximately 0.7. It should be noted that precise estimates of S and s are not necessary.

9. Noisy optimisation: numerical example The example used to illustrate the application of the CEM to noisy problems is that of finding optimal signal timings in a network of signalised junctions, when drivers route themselves in accordance with stochastic user equilibrium principles (see Cascetta et al., 2006). The test network is shown in Fig. 6 and is loosely based on that used originally by Allsop and Charlesworth (1977), and used also in Bell and Ceylan (2005), and in Chiou (2003)). It consists of six signalised nodes each operating on a two-stage cycle, 19 links, and 22 OD demands. The total demand is 5000 vehs/h and the average free-flow

4 8 5

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17

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19

6

14

6

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2

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6 18

12 2

5 1

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1 1 Fig. 6. Test network for noisy optimisation problem.

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travel time is 32.4 s. We use a path-based approach for the assignment, with a specified path set (of up to five paths) for each demand. There are 58 paths in total. The combination of signal optimisation and traffic assignment makes this a problem that is likely to have many local minima, so that a global search method is required. A solution x consists of the signal timings (with the 12 elements being the stage green times at each of six nodes). Travel time is assumed constant along each link, so that the only delay occurs at the end of each link, at the signal. The formula of Doherty (1977) (as used in Maher et al., 2001 and in Cascetta et al., 2006) is used to estimate the mean delay d to vehicles on each link, and is a function of the green time g for the relevant stage, the cycle time C at the node, the link flow q and the saturation flow s:

d¼

C 1980q ð1  kÞ2 þ 2 skðsk  qÞ

d¼

C 198:55  3600 220  3600q ð1  kÞ2  þ 2 sk ðskÞ2

if

q < 0:95 sk if

q > 0:95 sk

where k is the green fraction, g/C. Unlike the previous example, we assume here that each node may operate on its own cycle time, and hence there is no attempt to coordinate adjacent nodes, and no representation of offsets. The objective is to find the solution that minimises the total network travel time z. Drivers are assumed to route according to probit-based SUE assignment. For each specified path between an OD pair, the mean travel time is calculated as the sum of the cruise times and mean signal delay on each of the links making up that path. Assignment is carried out using the Method of Successive Averages (MSA), with loading at each iteration being done using the ‘‘Burrell method’’ – that is, using Monte Carlo techniques (Burrell, 1968). At each iteration, with the current flows and hence the current mean link travel times, a random set of perceived link travel times are generated, about the current means, independently for each OD pair from a Normal distribution with standard deviation proportional to the mean. From the resulting perceived path travel times for that OD pair, the minimum cost path is identified and all the demand for that OD loaded onto that path. Once all ODs have been dealt with, the auxiliary flow is obtained as the sum of the separately loaded OD demands. The current and auxiliary flows are combined in the usual MSA manner to give the current flows for the start of the next iteration.M such MSA iterations are performed. Given the Monte Carlo nature of the loadings, the path choice proportions and the estimated total network travel time z will be subject to random error about their true, long-term values. The greater the number of iterations M, the smaller will be s2, the variance of the random error e (in fact it is observed that s2 is approximately inversely proportional to M). Hence, for each solution x, we carry out a Monte Carlo (or Burrell-type) assignment to get z(x) = z0(x) + e(x) where e(x) has zero mean and variP P ance s2 which is inversely proportional to M. The objective function is calculated as: z ¼ w qw k pwk twk where pwk is the proportion of the demand qw using path k, and twk is the mean travel time along path k for OD pair w. Because the returned value of z for any solution is subject to random error, the problem of finding the optimal solution x is a noisy optimisation problem. To implement the cross entropy method on this test problem, each of the 12 stage green times g was allowed to take any integer value between 10 and 49 s inclusive. At iteration 1, the distributions were all uniform, so each value had a probability of 0.025. Intergreens of 5 s were assumed throughout so that C = 10 + g1 + g2 at each node. In each CEM iteration N = 1000 solutions x are generated, and each is evaluated by running the assignment with M MSA iterations. After ranking, the best N = 50 solutions are identified and the frequency with which each possible value of green occurs is totalled up. As before, updating of the (discrete) probability distribution is done using a smoothing constant of a = 0.7. A sample set of results is shown in Table 4. These are for 20 iterations of the CEM, with M = 1000 iterations used in the assignment process. At each iteration, we have the same statistics as before: the minimum z value, the 5th percentile z value, the average of the z values of these best 50 solutions, the average of all solutions in that iteration, and the between-solution standard deviation S. The plot of r = s/S is shown as the solid line in Fig. 7. For M = 1000, it is found (from a number of initial replicate runs for a representative solution) that s is approximately 455 (from which the value of s for any other value of M can also be estimated given that s2 is approximately inversely proportional to M). From Table 4 and Fig. 7, it can be seen that progression is smooth until around iteration 12, when r reaches a value of approximately 0.5. After that, progression is slower and then more erratic when r approaches and then exceeds 1, and the values of S cease to reduce in any systematic manner. This shows that the value of M needs to be increased after iteration 10, in an adaptive manner. So, we now use M = 5000 for iterations 11–15, and then M = 25000 for iterations 16–18, and again M = 125,000 for iteration 18–20. For M = 5000, a set of 1000 runs (solutions) takes around 9 min and s is approximately 224 (as it decreases in inverse proportion to the square root of M). For M = 25000, a set of 1000 runs takes around 45 min and s = 91, whilst for M = 125,000 a set takes almost 4 h. The results from iteration 11 onwards are shown in Table 5 (the results for iterations 1–10 are the same as those in Table 4). We can now see in Table 5, the values of S do now continue to reduce steadily beyond iteration 12, but now start to stall when iteration 20 is reached. In Fig. 6 we can see dips in the plot of r in the adaptive case (dotted line), as the value of M is increased, firstly after iteration 10, then again after iteration 15, and again in iteration 17.

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Table 4 Results from cross entropy method (M = 1000 throughout). Iteration

zmin

z0.05

zav(sel)

zav(all)

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

392648.6 386352.1 386401.3 380885.6 380466.3 376853.7 376472.0 375852.9 374656.3 374991.6 374197.4 373822.7 373677.8 373570.4 373616.9 372803.1 373097.7 373099.2 372991.7 372981.8

438490.8 408565.1 398833.1 391065.0 385625.1 382438.9 379272.8 377569.3 376419.2 375728.9 375177.8 374821.6 374515.1 374335.7 374155.7 374031.3 373975.6 373815.6 373775.4 373689.3

419924.9 402137.9 394378.2 388296.2 384113.4 380966.7 378371.4 376959.2 375981.1 375518.1 374905.5 374556.6 374247.2 374082.4 373943.5 373785.0 373726.9 373633.7 373579.5 373509.3

12,96,731 747549.8 533303.2 458677.5 414790.4 400283.5 391060.5 384080.9 380111.9 378014.0 377080.8 376243.6 375696.0 375358.0 375134.5 374983.2 374904.0 374754.9 374671.8 374595.0

959567.7 513629.1 207948.7 134852.3 55779.5 28621.6 45206.5 11305.8 3919.5 2502.9 1589.3 1108.6 897.1 727.8 626.5 584.9 573.0 569.4 541.7 848.7

1.6 1.4 1.2

r

1 0.8 0.6 0.4 0.2 0

1

3

5

7

9

11

13

15

17

19

iteration Fig. 7. Plot of r with (solid line) constant M throughout, and (dotted line) adaptive M.

Table 5 Results from cross entropy method (with adaptive M). Iteration

zmin

z0.05

zav(sel)

zav(all)

S

11 12 13 14 15 16 17 18 19 20

374272.7 374112.9 374105.2 373910.7 373771.2 373911.7 373951.6 373853.1 373852.1 373917.9

375358.2 374865.0 374597.5 374343.7 374141.7 374194.3 374098.9 374049.8 374036.2 374036.4

375118.0 374691.7 374460.5 374198.7 374018.0 374119.2 374050.6 374006.6 373990.7 373991.0

377054.8 376225.5 375436.8 375011.1 374687.3 374523.3 374339.1 374257.7 374225.1 374202.0

1535.0 1092.9 733.0 551.0 392.3 268.5 192.3 105.0 93.9 56.0

So, we can continue to make further progress in the cross entropy method – but at the expense of longer run times as the number of MSA iterations in the Burrell assignment is increased. If we do not do this, there will come a point where the

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method ceases to make any further improvements in the solutions found and the ranking and selection process is rendered inefficient. The best solution found (after 20 iterations) is: x = (26, 27; 22, 14; 24, 38; 11, 14; 13, 20; 14, 25) and, using five independent assignment runs each with M = 250,000, a precise estimate of the total network travel time is z = 374173. This corresponds to an average travel time of 74.83 s (compared with the free-flow value of 32.4 s). To see the gradual improvement in the solutions through the iterative process, we also evaluated the best solutions found after 1, 5, 10 and 15 iterations, and find the corresponding average travel times to be respectively: 78.60, 76.16, 75.07 and 74.86 s. It can be seen that the major part of the improvement has come after ten iterations and that thereafter the improvement is only extremely marginal. We also obtained, for comparative purposes, the ‘‘mutually consistent’’ solution: obtained by alternately (i) estimating the routeing pattern q(x) for the current set of signal timings x, and (ii) the signal timings x(q) that minimise the delay (at each node separately) given the current flow pattern q, starting from a randomly generated initial set of timings. The average travel time for such mutually consistent solutions is found to be 77.07 s, demonstrating the superiority of the solution that minimises z(x, q(x)), as was discussed in Section 2. Hence the cross entropy method can be applied to noisy optimisation problems but to achieve the same precision as in the deterministic case requires considerably more computational effort. In practice, in any such noisy problem, there will be a point at which the gains from further improvements in precision (or reduction in the loss) come at a cost in computational effort that is not justifiable.

10. Conclusions and discussion The paper has set out to investigate the application of the cross-entropy method to two different types of signal optimization problems. In the first, there was no route choice and the traffic model used to evaluate any solution x was deterministic; whilst in the second, there was route choice and this was affected by the solution (the signal timings imposed). However, more importantly, the routeing pattern was estimated by a Monte Carlo traffic assignment model, so that the returned value of the objective function z was subject to random error. It was shown that both problems could be tackled by means of the cross entropy method: when applied to the deterministic problem, the method is very robust in finding the global optimum solution (and is markedly better than the existing hill-climbing method in Transyt). When applied to the noisy problem, it is necessary to steadily reduce the amount of noise through the iterative process, by increasing the length of the simulation runs or equivalently increasing the number of simulation runs used to evaluate any solution. A scheme for carrying out this adaptive process is proposed and tested. The cross entropy method has a simple and transparent structure of generation, evaluation and ranking of solutions, followed by the updating of the parameters used in the generation process. It is the ranking of solutions that is key to the extension to noisy problems, as its efficiency is clearly affected by the presence of noise in the objective function values. The effect of noise on the ranking can be quantified by a comparison of between-solution and within-solution variation. This then forces the user to recognise the need to reduce the amount of noise in the next iteration and provides the means to estimate the necessary reduction and hence the increase in the simulation run times to achieve that. Although dealing specifically with two types of signal optimisation problem, the paper is intended to address some more generic issues, relating to the choice faced by traffic engineers between the use of simplified, deterministic models and more detailed, simulation-based Monte Carlo models, for optimisation purposes. The simplified nature of the former makes it possible to evaluate a very large number of trial solutions and their deterministic nature enables, in principle, search procedures to be applied to find the optimum. On the other hand, simulation-based models are far more computationally demanding and, in addition, their Monte Carlo nature means that the returned performance index value is unreliable. So, the engineer is faced with a dilemma: should he use the macroscopic model, on the grounds that it makes optimisation practicable, even if he does not have great faith in the evaluations; or should he adopt the microscopic model to represent the system more accurately even though this makes optimisation far more difficult? The approach described here forms an interesting contrast with that taken by Osorio and Bierlaire (2010). They propose the use of a metamodel which combines information from a simulation-based model with that from a far simpler, deterministic, network model. The metamodel has a structure that lends itself to optimisation within a ‘‘trust region’’ and proposes new solutions to be evaluated by the simulation model, thereby updating and extending the information available. The structure of the metamodel is typically a quadratic polynomial which, with many decision variables, then requires a large sample number of evaluations to enable it to be fitted. Whether this approach can deal effectively with problems where offsets are included amongst the decision variables and consequently there are a large number of local optima so that the PI surface is far more bumpy and complex than can be described by low-degree polynomials, remains to be seen. Nevertheless, this approach could have the potential to offer an objective approach to the solution of the engineer’s dilemma, in integrating the use of the two types of model. Clearly, this topic is one of considerable relevance and importance, as the use of simulation-based models in transport grows rapidly. Whilst these models provide accurate measures of network performance and are suitable for the appraisal of traffic management schemes when there is a small number of alternatives, they are not as yet seen to lend themselves to scheme design through optimisation methods. The research described here is intended to make a contribution towards this objective.

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