Transportation Research Part C 55 (2015) 191–202

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Transportation Research Part C

journal homepage: www.elsevier .com/locate / t rc

Optimality versus run time for isolated signalized intersections

http://dx.doi.org/10.1016/j.trc.2015.02.0150968-090X/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Leidos, Inc., 11251 Roger Bacon Drive, Reston, VA 20190, United States.E-mail addresses: david@ce.ufl.edu, david.k.hale@leidos.com (D.K. Hale), bpark@virginia.edu (B.B. Park), astevano@fau.edu (A. Stevanovi

su@leidos.com (P. Su), jiaqi.ma@leidos.com (J. Ma).

David K. Hale a,⇑, Byungkyu Brian Park b, Aleksandar Stevanovic c, Peng Su d, Jiaqi Ma d

a University of Florida, PO Box 116585, Gainesville, FL 32611-6585, United Statesb University of Virginia, 351 McCormick Road, Charlottesville, VA 22904, United Statesc Florida Atlantic University, 777 Glades Road, Building 36, Room 225, Boca Raton, FL 33431, United Statesd Leidos, Inc., 11251 Roger Bacon Drive, Reston, VA 20190, United States

a r t i c l e i n f o

Article history:Received 13 October 2014Received in revised form 2 January 2015Accepted 17 February 2015Available online 7 March 2015

Keywords:Traffic signal timingSimulation-based optimizationGenetic algorithmSimulated annealingHeuristic methodTabu search

a b s t r a c t

Simulation-based optimization of traffic signal timing has become pervasive and popular,in the field of traffic engineering. When the underlying simulation model is well-trustedand/or well-calibrated, it is only natural that typical engineers would want their signaltiming optimized using the judgment of that same model. As such, it becomes importantthat the heuristic search methods typically used by these optimizations are capable oflocating global optimum solutions, for a wide range of signal systems. However off-lineand real-time solutions alike offer just a subset of the available search methods. The resultis that many optimizations are likely converging prematurely on mediocre solutions. Inresponse, this paper compares several search methods from the literature, in terms of bothoptimality (i.e., solution quality) and computer run times. Simulated annealing and geneticalgorithm methods were equally effective in achieving near-global optimum solutions.Two selection methods (roulette wheel and tournament), commonly used within geneticalgorithms, exhibited similar effectiveness. Tabu searching did not provide significant ben-efits. Trajectories of optimality versus run time (OVERT) were similar for each method,except some methods aborted early along the same trajectory. Hill-climbing searchesalways aborted early, even with a large number of step-sizes. Other methods only abortedearly when applied with ineffective parameter settings (e.g. mutation rate, annealingschedule). These findings imply (1) today’s products encourage a sub-optimal ‘‘one size fitsall’’ approach, (2) heuristic search methods and parameters should be carefully selectedbased on the system being optimized, (3) weaker searches abort early along the OVERTcurve, and (4) improper choice of methods and/or parameters can reduce optimizationbenefits by 22–33%.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Simulation-based optimization of traffic signal timing has become pervasive and popular, in the field of traffic engineer-ing. This is mainly due to the robustness and reliability of prominent traffic analysis tools, in terms of their ability to evaluateexisting conditions. The traffic analysis tools used for signal optimization usually employ detailed analytical and/or macro-scopic simulation models. Despite their complexity it is possible to perform hundreds of model runs within seconds, giventhe speed of today’s computers. This is quite helpful for signal optimization, because each of these model runs can be used to

c), peng.

192 D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202

test a different timing plan alternative. Although the microscopic simulation models are also trusted, their computer runtime requirements often preclude their direct involvement in the optimization process.

Optimization tools such as PASSER (TTI), TRANSYT (TRL), TRANSYT-7F (McTrans), Synchro (Trafficware), HCS-Streets(McTrans), Vistro (PTV), and ArteryLite (AGA) employ variations of simulation-based optimization. One crucial advantageof simulation-based optimization is the ability to simulate existing conditions, and perform model calibration based on thoseexisting conditions, prior to any design or optimization. When the underlying simulation model is well-trusted and/orwell-calibrated, it is only natural that typical engineers would want their signal timing optimized using the judgment of thatsame model. Only simulation-based optimization can provide this.

1.1. Heuristic search methods used for traffic signal timing

As such, it becomes important that the heuristic search methods typically used by these optimizations are mathematical-ly capable of locating global optimum solutions, for a wide range of signal systems. However, what we see in practice is thatthe popular software solutions only offer a limited subset of the available search methods. Moreover, the software solutionsonly offer a limited subset of the available parameters, which can often improve the efficiency of those methods. The result isthat many signal optimization efforts are likely experiencing premature convergence on mediocre solutions, because theyinvolved ineffective methods and/or parameters. Some traffic signal timing studies (Oda et al., 1996) have shown geneticalgorithms to be more effective than hill-climbing methods, and one signal timing study (Agbolosu-Amison et al., 2009)has shown genetic algorithms to be more effective than Harmony search. However no known traffic signal timing studieshave examined many other available search methods and parameters.

For example, the effectiveness or viability of heuristic search methods such as simulated annealing and Tabu search in thesignal optimization process appears to be unknown. No known traffic signal timing optimization studies have compared theeffectiveness of tournament selection and roulette wheel selection, within the genetic algorithm; or the number of steps andstep-sizes, within the hill-climbing method. There is a need to perform apples-to-apples comparisons of all these searchmethods; by applying them to the same underlying evaluation model, across a wide variety of intersection geometriesand signal phasing. Finally, no known signal timing studies have focused on the ramifications of widely-varying computerrun times required by these various methods.

1.2. Importance of computer run times

The issue of computer run times appears to be particularly underrated. Although many off-line optimizations completewithin seconds, these efforts are likely producing sub-optimal solutions for small networks, analyzed across a small numberof time periods. Engineers would care more about computer run times if they knew that (a) other available heuristic searchmethods would often provide better designs for such networks and (b) future optimizations may require larger trafficnetworks, numerous time periods, and micro-simulation of all candidate timing plans. Regarding real-time optimizations,run times are important because timing plans are changing so frequently. This highlights an incentive to maximize the ratioof optimality (i.e., solution quality) over run time.

The trade-off between optimality and run time becomes clear when working with these heuristic search methods. Forexample, it is common to observe significant improvement in the first fraction of run time, followed by modest improve-ments requiring a much longer period of run time. It is common to observe the fastest-converging search methods producingreasonable and effective timing plans, whereas longer-running (without converging) search methods arrive much closer tothe global optimum solution. It would be ideal if a new search method could obtain global optimum solutions in the shortestrun times, but such a method has been elusive.

Given the trade-off between run times and optimality, it stands to reason that different optimization projects wouldbenefit from different search methods. Faster methods (e.g. hill-climbing search, greedy algorithms, equalizing degree ofsaturation) are needed for real-time adaptive control, whereas slower methods (e.g. genetic algorithms, simulated annealing,particle swarm optimization) are more acceptable for off-line applications. Faster methods are needed to process largernetworks across numerous time periods, whereas slower methods are more acceptable for small datasets. Different engi-neers have different standards regarding how much time they are willing and able to wait, during an off-line (non-adaptive)optimization run. Faster methods are needed to directly optimize computationally expensive simulation models, whereasslower methods are quite effective for simple capacity analysis (analytical) models.

1.3. Scope of this paper

To shed more light on the trade-offs between solution quality and time needed, this paper compares optimality andcomputer run times for several top optimization search methods from the literature. Given that no optimization method out-performs all others under all conditions (Ciuffo and Punzo, April 2014), this paper will not try to determine the ‘‘best’’method for all conditions. Instead the paper will present evidence that some optimization methods, when used with theright parameters, have a strong tendency to avoid premature convergence. To produce this evidence, over a thousand testoptimizations of isolated intersections were performed using one objective function, and one underlying traffic model.

D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202 193

Follow-up studies will hopefully perform similar data collection with more optimization methods, more objective functions,more traffic models, and more complex signal systems (i.e., coordinated signalized arterials and grid systems).

2. Literature review

A comprehensive review of the available signal optimization methods was provided by Yun and Park (2012). Early signaloptimization strategies involved minimizing intersection delay as predicted by the original Webster (1958) formula, and bysubsequent evolutionary versions (Akcelik, 1998; Husch, 1996; Carson and Maria, 1997) of this formula. This paper will notevaluate these early optimization strategies, because modern methods for determining delay have much better accuracy.Modern computational engines contain thousands of lines of programming code, and cannot easily be summarized into asmaller set of equations. To achieve 100% agreement with the chosen underlying model during the optimization process,it is preferable to execute the full underlying model, to evaluate candidate timing plans.

Hill-climbing optimization (Robertson, 1969) is one example of a simulation-based optimization method, and has beenemployed by multiple signal timing tools (Hale, 2009; Chang and Messer, 1991; Wallace and Courage, 1991) since thelate 20th century. Additional simulation-based optimization methods are provided in the literature review by Yun andPark (2012); who state that heuristic methods are ideal for complex engineering models, having numerous and complexsub-optimal solutions (Perentonos et al., 2002). Examples of heuristic optimization methods include hill-climb search,genetic algorithms, simulated annealing, Tabu search (Carson and Maria, 1997), and Harmony search (Geem et al.,2001). Efficiency of the genetic algorithm, which has become popular (Yun and Park, 2012) for use in signal optimization,is also known to be affected by its underlying selection methods (e.g., roulette wheel and tournament) (Chevalier, 1994).These simulation-based optimization methods from the literature are believed to be good candidates for signal timingoptimization in general.

2.1. Hill-climbing search

Hill-climbing search optimization is apparently applied in many different fields, and its use has been well-documentedin the context of signal optimization (Robertson, 1969; Hale, 2009). Hill-climbing optimization is advertised by off-lineproducts including Vistro, PASSER, and TRANSYT. Synchro provides a quasi-exhaustive method of offset optimization(Yun and Park, 2012), which is very similar to hill-climbing. On-line adaptive signal products tend to have proprietaryclassified algorithms; but during workshop presentations, some have been said to use hill-climbing methods. Other adap-tive products are said to use the greedy algorithm and/or ‘‘Equisat’’ (Akcelik et al., 1997) which, like hill-climbing, appearto be some of the fastest available methods (i.e. computationally inexpensive, in terms of computer run times) for timingsignals. Thus hill-climbing and related methods have become somewhat dominant in U.S. signal timing practices, overthe past few decades.

When the hill-climbing search method is in effect, miscellaneous jumps are made in search of the lowest objectivefunction value. When a single jump finds a better solution, another jump is made in the same direction, and using the same‘‘step-size’’. Additional jumps are made until finally a worse solution is encountered, at which time the process backtracks tothe previous best solution. The hill-climb method then switches to a different step-size, and sometimes reverses jumpdirections.

By jumping across the solution space at various step-sizes, the optimization process saves time by not evaluating eachpoint along the curve. By using a variety of step-sizes, the optimization process tries to avoid getting stuck in a local opti-mum solution. Unfortunately the number of steps and step-sizes is somewhat arbitrary, and no set of steps or step-sizes isperfect for all solution spaces.

2.2. Genetic algorithm

The genetic algorithm (GA) is an evolution-inspired (Whitley, 1994) algorithm; capable of simultaneously searchingnumerous areas of promising solution space, across numerous generations of individuals. This ability to simultaneouslysearch numerous areas of solution space gives GA an advantage over the hill-climbing method, in which one search must‘‘solve’’ the entire solution space. For traffic signal optimization, each individual within the population must be translated(Park, 1998) into a long array of bits (1’s and 0’s), which represents the DNA chromosome. These chromosomes are thenmanipulated by optimization processes such as competition, crossover, and mutation (Michalewicz, 1994).

Regarding the determination of which individuals are performing well enough to participate in crossover, GA uses tour-nament selection or roulette wheel selection. When the tournament method is in effect, pairs of individuals are randomlyselected from the population to compete with each other, and competition losers are eliminated from the gene pool.When roulette wheel selection is in effect, weaker individuals tend to have higher probabilities of being eliminated fromthe gene pool. No known signal timing studies have compared these selection methods.

Following crossover there is also a process of mutation, to randomly change a small number of characteristics within eachindividual. Thus the crossover process allows simultaneous searching across the search space, while the mutation processallows for consideration of fresh solutions. Some research (Chevalier, 1994) indicates that relatively low mutation rates

194 D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202

are more efficient when applied in conjunction with tournament selection, whereas relatively high mutation rates are moreefficient with roulette wheel selection. No known signal timing studies have examined combinations of selection methodand mutation rate, but this paper will examine various combinations.

2.3. Simulated annealing

Simulated annealing (SA) is an optimization method (Kirkpatrick et al., 1983) inspired by metallurgy; capable of evaluat-ing numerous sub-optimal solutions over time, but gradually reducing the number of sub-optimal solutions considered.Similar to the mutation process within GA, the consideration of sub-optimal solutions prevents SA from quickly getting stuckon a local optimum solution. However, the metallurgy-inspired ‘‘slow cooling’’ process ensures optimization, graduallyreducing the probability of accepting sub-optimal solutions over time.

The Boltzmann factor (Kittel and Kroemer, 1980) is used within SA to compute the probability of accepting new solutions,as shown below in Eq. (1). ‘‘Delta S’’ represents the distance between fitness of the current solution, and fitness of any newcandidate solution. ‘‘Alpha T’’ represents the annealing schedule, which will affect the speed of optimization. The existence ofDelta S causes better solutions to have higher probabilities of acceptance. Similar to the mutation rate within GA, values ofAlpha T that are too low or too high could slow down the process of optimization, or prevent optimization altogether.

PðacceptanceÞ ¼ e�Ds/Tð Þ ð1Þ

Research has shown that the success of SA ‘‘depends strongly on the choice of a suitable annealing schedule’’ (Christoph andHoffmann, 1993). Other research has shown that SA is proven to locate global optimum solutions; but only as the annealingschedule approaches infinity (TTG Incorporated, xxxx), implying a relationship between optimality and run times. No knownsignal timing studies have examined the potential effectiveness of SA, but this paper will compare effectiveness of SA againstother methods.

2.4. Tabu search

Tabu search (TS) is an optimization method (Glover, 1986) that stores its most effective solutions, or at least componentsof its most effective solutions, into short-term or long-term memory. During the optimization process, solution componentsstored within this memory are marked as ‘‘taboo’’ and cannot be used by new candidate solutions. Solution components arestored within memory in a first-in-first-out process; such that the oldest taboo components become fair game again, as newcomponents are added to the list. Similar to the mutation process within GA, the forced consideration of sub-optimal solu-tions prevents TS from quickly getting stuck on a local optimum solution. However if the Tabu memory is too long this couldslow down the process of optimization, or prevent optimization altogether. No known signal timing studies have examinedthe potential effectiveness of TS, but this paper will perform such tests.

2.5. Harmony search

Harmony search (HS) is an optimization method (Geem et al., 2001) inspired by musical orchestras; capable of evaluatingnumerous disharmonious melodies over time, but gradually converging on one harmonious melody. The original objective ofHS was to develop a new heuristic algorithm with better performance (better solutions, fewer iterations) than the prior artalgorithms (GA, SA, TS, etc.). Similar to the crossover process within GA, new harmonies are ‘‘improvised’’ by using elementsof the prior music. Similar to the competition process within GA, weaker harmonies are eliminated from consideration.Similar to the mutation process within GA, harmony memory and pitch adjustment processes force random harmonies intothe solution space, at various intervals. The HS optimization method encountered criticism (Weyland, 2010; Padberg, 2012)for not being substantially different from GA, and prior signal timing research (Oda et al., 1996) concluded that HS was lesseffective than GA. For these reasons, HS was not included within the detailed comparison of optimization methods per-formed for this paper.

3. Methodology

This research focused on signal timing optimization for maximum green times at isolated, actuated signals. Chapter 18 ofthe 2010 Highway Capacity Manual (HCM) (Transportation Research Board, 2010) contains computational proceduresdesigned to evaluate the operational efficiency of signalized intersections. No known studies have ever focused onChapter 18, HCM-based optimization. Because of this, and due to the interest level in HCM Chapter 18, the choice was madeto use these analytical procedures as the underlying evaluation model. There was no need to explicitly consider cycle lengthoptimization, because cycle lengths are an automatic output of the Chapter 18 evaluation procedures. There was also noneed to consider phasing sequence optimization, which primarily benefits coordinated intersections. Offset optimizationis not applicable to isolated intersections.

The choice of optimization objective function (Hajbabaie and Benekohal, 2013) is an important consideration, in the con-text of signal timing. However, the scope of this research was limited to comparing the effectiveness of various optimization

D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202 195

search processes from the literature. The objective function used by this research was the intersection-wide average delayper vehicle; which is currently the primary measure of effectiveness for determining level-of-service, in Chapter 18.

3.1. Implementation of optimization methods

To facilitate the comparison, several heuristic optimization methods were computer-programmed into a research versionof the HCS 2010™ signalized analysis module. This analysis module is intended to be a 100% faithful implementation of theHCM Chapter 18 procedures, and provides an apples-to-apples comparison platform for the optimization methods. Beforethe beginning of this research, the signalized analysis module already contained functionality and logic for genetic algorithm(GA) optimization, using the tournament selection method. Therefore, the implementation of new methods was performedin this sequence:

� GA optimization with roulette wheel selection.� Hill-climbing optimization.� Tabu search (implemented within a GA framework).� Simulated annealing (implemented within a hill-climbing framework).

Although the hill-climbing method seems likely to benefit from adjustments to avoid premature convergence, imple-menting Tabu search (TS) within a hill-climbing framework was too difficult, so TS was implemented within a GA frameworkinstead. Because GA already contains elements of randomness, and was thus unlikely to benefit, simulated annealing was notimplemented within a GA framework. The result of these modifications was a group of optimization methods to becompared, in terms of their optimality and run times; including:

� GA with tournament selection,� GA with roulette wheel selection,� TS with tournament selection,� TS with roulette wheel selection,� standard hill-climbing search, and� simulated annealing.

3.2. Intersection conditions

It was desirable to test a wide variety of intersection geometries and signal phasing. Table 1 illustrates eight majorcategories of test cases: protected-permitted left-turn phasing, protected-only left-turn phasing, permitted-only ‘‘green ball’’phasing, side-street split phasing, protected-permitted left-turn phasing, protected-permitted phasing from a shared lane,T-shaped intersections, I-shaped intersections, and ‘‘advanced’’ conditions. The eight major categories were further dividedinto 23 subcategories, based on various forms of left-turn treatment and intersection geometry. Finally, the 23 subcategorieswere expanded into 72 test cases by considering dual turn lanes (‘‘dual’’, ‘‘left’’, ‘‘right’’), dual thru lanes (‘‘thru’’), leading lefts(‘‘lead’’), lagging lefts (‘‘lag’’), and combinations (e.g., ‘‘lagdual’’). Table 1 contains four sets of three columns showing themajor category (left), subcategory (middle), and case name (right). Input demand volumes for these test cases were adjustedprior to the comparison experiment; until intersection-wide delays fell between 80 and 85 s per vehicle (i.e., near-capacityconditions), to allow significant potential space for optimization improvement.

The 72 test cases were not subjected to any amount of calibration or validation. Insight into the effectiveness of simula-tion-based (or ‘‘black box’’-based) optimization methods should not require accurate modeling of specific, real-world loca-tions. Instead, the most insight on optimization effectiveness should be gained through comprehensive testing of a widevariety of ‘‘typical’’ real-world locations, which is how the experiment was designed.

3.3. Choice of optimization parameters

It is difficult to design a ‘‘perfect’’ optimization experiment, due to the large number of optimization parameters, and theamount of time required for optimization. Hill-climb methods can have any number of steps and step-sizes. Simulatedannealing can have various annealing schedules. Tabu searching can be applied with short-term, medium-term, or long-termmemories. GA can be applied with different mutation rates, or different numbers of generations. Different intersectionconditions might possibly benefit from different types of optimization. Despite these challenges, experience and judgmentwere used to design a proper experiment, to shed light on the most effective optimization methods. This experimental designprocess is described below.

3.3.1. Hill-climbing parametersWith hill-climbing optimization, it was necessary to determine an effective number of steps, and distribution of

step-sizes. A trial-and-error process was employed, to determine when additional steps would not yield significant

Table 1Test cases with a wide variety of intersection geometry and signal phasing.

Subcat. Case

Protected-permittedLeading lefts 1-1

1-1dual1-1thru

Lead-lag 1-21-2dual1-2thru

Lag-lead 1-31-3dual1-3thru

Lagging lefts 1-41-4dual1-4thru

Protected onlyNo right turns 2

2dual2thru

Right turns 33left3right3thru

Shared rights 44dual4thru

Permitted onlyAll exclusive lanes 5

5left5right5thru

Shared rights 66dual6thru

Shared lefts 77dual7thru

One lane 8Split phasingAll exclusive lanes 9

9left9right9thru

Shared rights 1010dual10thru

Shared lefts 1111dual11thru

One lane 12Prot-perm-sharedExclusive thru lanes 13

13dual13thru13lag13lagdual13lagthru

One lane 1414lag

T-shapeExclusive lanes on side street 15

15left15right15thru

196 D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202

Table 1 (continued)

Subcat. Case

Single lane on side street 1616left16right16thru

I-shapeExclusive lefts 17

17dual17thru

Shared left 1818thru

AdvancedShielded right turns 19

19left19right19thru

Extensive satflow adjustments 2020dual20thru

Fig. 1. Implementations of hill-climbing search (HC) and simulated annealing (SA).

D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202 197

improvements. This evaluation process showed that making 4–6 ‘‘passes’’ through a series of 8 step-sizes would capturemost of the optimization improvement, in most cases. The hill-climbing process is illustrated in Fig. 1.

198 D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202

3.3.2. Simulated annealing parametersBecause simulated annealing (SA) was implemented within the framework of hill-climb optimization, it was necessary to

determine (1) an appropriate number of steps, (2) an effective distribution of step-sizes, and (3) an effective annealing sched-ule. Numerous combinations of steps, step-sizes, and schedules were tested, to estimate the most effective parameters. Theprocess showed that 10 annealing passes through 4 step-sizes would capture most of the optimization improvement, in mostcases. Although 10 annealing passes was more effective than 5 passes (as expected), 4 step-sizes was more effective than 8(not expected). The effective SA parameters are shown in Fig. 1. It became clear that SA would sometimes terminate on solu-tions weaker than those evaluated earlier in the process, due to forced consideration of sub-optimal solutions. Because ofthis, it was important to store the best solution observed so far during the process, so it could be retrieved after the fullannealing schedule expired.

3.3.3. Genetic algorithm parametersWith GA optimization, it was necessary to determine an appropriate mutation rate and number of generations. Mutation

rates of 1% and 4% were used, because these values have been effective for isolated intersections in the past (Hale, 2009).Regarding the number of generations, it can frequently be observed that most improvement occurs in the early generations.The evaluation process showed that 250 generations would capture most of the optimization improvement, in most cases.For example, Fig. 2 is an image generated by the HCS software. This figure illustrates percentage improvement (Y-axis) overtime (X-axis). The improvement percentage was calculated as shown in Eq. (2); and this percentage was used extensively tocompare optimization methods, as shown later. The figure illustrates that, although maximum improvement occurs after1500 generations, over 90% of this improvement occurred within the first 250 generations. This improvement rate was typi-cal of most of the intersections tested. Therefore, 250 generations was used for most GA-based optimization runs. Note thatalthough generations are somewhat synonymous with time, it took approximately 8–9 min to run 1500 generations. Giventhe population size of 10 individuals, 1500 generations represents exactly 15,000 executions of the HCM Chapter 18procedure.

PctImprovement ¼dorig � dfinal

dorig

� �� 100 ð2Þ

where dorig and dfinal are intersection-wide delays before and after optimization

3.3.4. Tabu search parametersFor Tabu search, implemented in the framework of GA, it was necessary to determine an effective Tabu memory. Low

storage could cause Tabu and GA to produce identical results. Overly-high storage could prevent solutions from improving.A trial-and-error process indicated that storing 5 chromosome bits would allow efficient optimization, in most cases. Storing10 bits made optimization less efficient in several cases. Thus whenever a new elite individual was born, as proven by com-parison to all others in the population, up to 5 bit positions (differentiating this individual from the prior elite individual)were stored into Tabu memory. These bit positions would then become unavailable for crossover and mutation by all futurecandidate individuals, at least until replaced by future additions to the Tabu memory.

Fig. 2. Plot of GA optimization improvement percentage over time.

D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202 199

4. Comparison of methods

4.1. Primary results

Fig. 3 illustrates four scatterplots, evaluating heuristic search methods in terms of optimality versus run time. Optimality(Y-axis) is expressed as an improvement percentage between the initial and final solutions. Run time (X-axis) is listed inunits of seconds. Each plotted point represents an average improvement percentage and run time based on 72 optimizationruns. In other words, the 72 basic test cases from Table 1 were optimized using a given heuristic method with given para-meter settings, and these 72 results were averaged to produce one scatter point. Fig. 4 further compares the methods bycombining four plots into one.

Regarding the given parameter settings, traditional hill-climbing search was tested with three different passes (2, 4, and6) through each step size. Simulated annealing (SA) was tested with four different annealing schedules (5, 10, 15, and 20).The genetic algorithm (GA) with tournament selection was tested with three combinations of mutation rate and number ofgenerations (1% @ 250, 4% @ 250, 4% @ 375). GA with roulette wheel selection was tested with slightly different combina-tions (1% @ 250, 4% @ 250, 1% @ 375). Thus the four plots contain a total of 13 scatter points, based on 13 ⁄ 72 = 936 opti-mization runs.

GA-based roulette wheel and tournament selection were tested with different mutation rates at 375 generations, becausethe initial data collection implied they would operate more efficiently under different mutation rates. Specifically, after 250generations, the 4% mutation rate produced a much better improvement percentage (31.4% versus 20.7%) for tournamentselection. However, also after 250 generations, the 4% mutation rate produced a worse improvement percentage (28.8% ver-sus 31.5%) for roulette selection. Therefore, when seeded with their ‘‘favorite’’ mutation rates for 375 generations, tourna-ment and roulette selection both performed quite well with 32.0% and 32.1% improvements, respectively.

4.2. Tabu search and other results

During the overall experiment there were actually 16 ⁄ 72 = 1152 optimization runs performed, requiring a total of 14.6 hof computer run time. Three points (based on 216 optimization runs) that could have been plotted were instead not includedon the graphs.

Two of these points involved Tabu searching (TS) at the 4% mutation rate, within the genetic algorithm. When TS wascombined with tournament selection, the average improvement changed from 31.4% to 31.3%. When TS was combined withroulette wheel selection, the average improvement changed from 28.8% to 29.2%. Because the impact of TS was fairly neg-ligible, these points were omitted from the graphs, in order to make the graphs more understandable.

A third discarded point involved simulated annealing (SA). In the original implementation of SA, ‘‘Delta S’’ represented thedistance between fitness of the current and candidate solutions. In an alternative implementation that produced the thirddiscarded point, ‘‘Delta S’’ was re-programmed to represent distance between fitness of the best solution encountered sofar, and candidate solutions. When using an annealing schedule of 10, this alternative implementation achieved a 26.8% aver-age improvement, compared to 28.5% for the original implementation. However, the alternative SA required 17.4 s of averagerun time, compared to 58.6 s for the original SA. Because the alternative SA (with annealing schedule 10) and original SA(with annealing schedule 5) produced nearly identical ratios of optimality to run time, there was no need to consider or plotthe alternative SA.

4.3. Interpretation of results

These results provided some evidence that changing from regular searching to TS had mixed effects, which were possiblyself-cancelling effects, on optimization efficiency. TS essentially specifies that, if a new signal setting (e.g., maximum green of29 s for phase #3) is found that works well, the optimization process must temporarily try to find optimized plans withoutusing that setting. This strategy leads to searching a wider range of signal setting values, possibly facilitating discovery of thebest values. The disadvantage is that TS spends less time examining interactions between that max green (e.g., 29 @ p3) andother max greens, possibly preventing discovery of the best overall combination. Thus TS increases the number of valuestested, but spend less time testing combinations of values known to be effective. This could explain why TS had a somewhatnegligible effect on optimization efficiency.

The recurring pattern within Figs. 2–4 is intriguing, and was not expected. The heuristic optimization methods have sig-nificantly different mathematics and algorithms. The hill-climbing methods (traditional and SA) have only one optimumsolution during each iteration, whereas the genetic algorithm has a full population of current solutions. Traditional hill-climbing (HC) contains no randomness, whereas other methods (SA, GA, TS) randomly accept weaker solutions. Methods thatdo employ randomness do so in very different ways. Even GA employs randomness in very different ways, depending onwhich selection method (roulette wheel or tournament) and mutation rate are in effect. Given these fundamental differ-ences, a consistent relationship of optimality to run time was not expected across all heuristic search methods, but wasobserved nonetheless. This relationship observed in the prior figures is generalized now within Fig. 5.

Fig. 3. Percentage improvement (Y-axis) versus seconds of run time (X-axis).

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

0.0 20.0 40.0 60.0 80.0 100.0 120.0

Impr

ovem

ent (

%)

Run Time (sec)

SA

HC

GA (T)

GA (R)

Fig. 4. Combined percentage improvement versus seconds of run time.

Fig. 5. The OVERT curve.

200 D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202

Region A is the area of significant sub-optimality. In this study, HC optimization runs were rarely able to escape Region A,whose endpoint is marked by Point X. For optimizations with very little time available, or extremely slow computers, orextremely slow underlying simulation models, or extremely high numbers of intersections and time periods, Region A mightrepresent the only feasible and practical solution space.

Region B is the area of transition, whose endpoint is marked by Point Y. For optimizations with medium time available, ormedium-speed computers, or medium-speed underlying simulation models, or medium numbers of intersections and time

D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202 201

periods, Point Y might represent the most desirable and efficient outcome. Beyond Point Y, there would presumably bediminishing returns on significant additional run times required. However, it might be a challenge to make advance predic-tions on the amount of time needed to reach Point Y.

Region C is the area of fine-tuning, whose endpoint is marked by the global optimum solution (Point Z). For optimizationswith significant time available, or extremely fast computers, or extremely fast underlying simulation models, or extremelysmall numbers of intersections and time periods, Point Z might be frequently attainable. In this study both SA and GA appearedfully capable of reaching Point Z; given a sufficient annealing schedule, or number of generations. However GA became trappedin Region A when seeded with the wrong mutation rate, and SA might do the same if seeded with ineffective step-sizes.

Although the OVERT curve illustrates results averaged over approximately a thousand optimization runs, some optimiza-tion results deviated significantly from this curve. For example in test case ‘‘1–1dual’’, both SA and GA performed poorly, withapproximately 23.0% improvement after 120 s. This represents a severe downward ‘‘bending’’ of the curve. In this same case, HCachieved 23.6% improvement within 9 s. Then in test case ‘‘19thru’’, HC was able to achieve 55.2% improvement after 10 s,which represents a severe upward bending of the curve. In case ‘‘19thru’’ SA and GA achieved 52% and 55% improvement respec-tively, but required 109 and 94 s respectively. These results imply that the OVERT curve is not accurate in 100% of all cases; andthat the ‘‘outlier’’ cases might share common characteristics, which could possibly be identified by future research.

5. Conclusions

Despite the popularity of simulation-based optimization for traffic signal timing, the software tools may unintentionallybe encouraging a ‘‘one size fits all’’ philosophy. Popular tools in the industry offer only a limited subset of the availableheuristic search methods. The result is that many signal optimization efforts are likely experiencing premature convergenceon mediocre solutions, and optimization strategies not consistent with the traffic signal system in question. No known signaltiming studies have compared several top search methods from the literature in terms of optimality (i.e., solution quality)versus computer run times, under the widest possible combination of intersection geometries and signal phasing.Experimental results from this paper may be summarized as follows:

� The OVERT curve relationship between optimality and run time appears to be consistent for several heuristic searchmethods.� Traditional hill-climbing search, which still exists within popular off-line and adaptive optimization products, consistent-

ly fell far short of reaching global optimum solutions.� When seeded with proper parameters, simulated annealing (SA) and genetic algorithm (GA) methods were equally effec-

tive at locating solutions close to the global optimum.� Despite SA and GA improving performance by 30–32% when applied with proper parameter settings, traditional hill-

climbing search produced only 22–23% improvement.� Compared to when SA or GA were applied with proper parameters, other methods and parameters reduced optimization

benefits by 22–33%.� When seeded with proper parameters, roulette wheel and tournament selection were equally effective within GA.� The best mutation rate for use in conjunction with roulette wheel selection was distinctly different from the best muta-

tion rate for use in conjunction with tournament selection.� Tabu search did not significantly improve or degrade optimization efficiency.

Although this research tested all heuristic search methods considered to be top candidates at the time, there is now moti-vation to examine additional methods. The simultaneous perturbation stochastic approximation method could be capable of‘‘bending’’ the OVERT curve, producing global optimum solutions within minimal times, and might be fast enough for imple-mentation within adaptive control products. In addition, because the greedy algorithm is used by some adaptive controlproducts, it would help to test the efficiency of this method. However on the surface, the greedy algorithm appears to bea ‘‘quick-and-dirty’’ method, similar to traditional hill-climbing.

Finally, upcoming research should investigate whether the OVERT curve relationship also holds for arterials and grids.Such studies would likely confirm that the best parameter settings for mutation rate, annealing schedule, Tabu memory,hill-climbing stepsizes, number of generations, etc. differ significantly for ‘‘problems of different complexity’’ (e.g. differentnumbers of intersections, or different numbers of traffic signal settings being optimized). The ongoing hypothesis is thatoptimization methods and parameter settings should be considered and chosen more carefully. More search methods shouldbe developed inside the optimization tools. More methods and parameter settings should be made available to the engineer.Proper guidelines should be developed on how to choose and apply these methods and settings. Expert systems could evenbe designed to automatically apply different methods and settings, based on ‘‘problem complexity.’’

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.trc.2015.02.015.

202 D.K. Hale et al. / Transportation Research Part C 55 (2015) 191–202

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