Transportation Research Part B 60 (2014) 85–106

Contents lists available at ScienceDirect

Transportation Research Part B

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

Capacity optimization of an isolated intersection underthe phase swap sorting strategy

0191-2615/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.trb.2013.12.001

⇑ Corresponding author. Tel.: +86 10 62796513.E-mail address: haijiang@tsinghua.edu.cn (H. Jiang).

Chiwei Yan a, Hai Jiang a,⇑, Siyang Xie b

a Department of Industrial Engineering, Tsinghua University, Beijing 100084, Chinab Department of Civil Engineering, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o

Article history:Received 17 May 2013Received in revised form 1 December 2013Accepted 5 December 2013

Keywords:Signal controlSignal timingSorting strategyPre-signalCapacity optimization

a b s t r a c t

It is well recognized that the left turn reduces the intersection capacity significantly,because some of the traffic lanes cannot be used to discharge vehicles during its greenphases. In this paper, we operationalize the phase swap sorting strategy (Xuan, 2011) touse most, if not all, traffic lanes to discharge vehicles at the intersection cross-section toincrease its capacity. We explicitly take into consideration all through, left- and right-turn-ing movements on all arms and formulate the capacity maximization problem as a Binary-Mixed-Integer-Linear-Programming (BMILP) model. The model is efficiently solved bystandard branch-and-bound algorithms and outputs optimal signal timings, lane alloca-tions, and other decisions. Numerical experiments show that substantially higher reservecapacity can be obtained under our approach.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

One of the major objectives in traffic signal optimization is to increase the capacity of at-grade intersections (Improta andCantarella, 1984). Literature in this field can be broadly classified into two streams: The first stream develops optimizationmodels to determine optimal lane allocations, signal phases, and signal timings for conventional intersections. Early researchin this stream includes Webster (1958) and Allsop (1972), which are later extended to group-based methods where trafficmovements are grouped into phases (Improta and Cantarella, 1984; Gallivan and Heydecker, 1988; Silcock, 1997), and morerecently to lane-based methods where lane allocations are simultaneously optimized (Wong and Wong, 2003; Wong andHeydecker, 2011). The second stream of research recognizes the fact that the left turn significantly reduces the capacityof conventional intersections, because it often requires separate green phase allocation, during which only part of the inter-section cross-section can be used to discharge vehicles (Newell, 1989). Researchers have therefore proposed a variety ofways to ban and re-route left-turning vehicles at such intersections, for example, median U-turns, jug handles, superstreetsand so on (Reid, 2004; Rodegerdts et al., 2004). These strategies increase the intersection capacity by eliminating the leftturns and the need for left turn phases.

Recently, the sorting strategy, a class of methods belonging to the second stream of research, has emerged as a promisingnew way to mitigate the negative impact of the left turn. In this strategy, pre-signals are used to re-organize traffic upstreamof the intersection, so that the entire intersection cross-section can be used to discharge vehicles during left turn phases.Xuan et al. (2011) proposes the tandem sorting strategy, where a pre-signal is installed upstream of the intersection signal(see Fig. 1), which forms a sorting area between the pre-signal and the intersection signal. Upstream of the pre-signal, lanesare marked by movement to segregate left-turning vehicles (LVs) and through vehicles (TVs) onto separate sets of lanes. In

Fig. 1. The tandem sorting strategy proposed by Xuan et al. (2011).

86 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

this example, the first set contains one lane for LVs and the second one contains two lanes for TVs. Right-turning vehicles(RVs) are not shown for simplicity of exposition. The pre-signal operates on the same cycle as the intersection signal andgives green time to the two sets of lanes alternatively. When the LVs or TVs advance into the sorting area, they use allthe lanes so that when the intersection green phases start, all lanes can be used to discharge vehicles during both throughand left turn phases. LVs and TVs are asked to form tandem batches and parade through the sorting area as well as the inter-section cross-section using all lanes. In this way, the intersection capacity can be improved. Later, Xuan et al. (2012) presentempirical evidence from an intersection in the city of Chengdu, China to prove its effectiveness. More recently, Zhou and Zhu-ang (2013) develop an optimization model to minimize the delay associated with this strategy and output optimal lane allo-cations and signal timings.

In an effort to reduce the length of the sorting area, Xuan (2011) proposes the phase swap sorting strategy where only asingle batch of LVs or TVs are allowed to queue in the sorting area. This strategy is illustrated in Fig. 2: In Fig. 2(a), the pre-signal starts its cycle by giving the green to LVs while the intersection signal is red to allow LVs to enter the sorting area andwait at the intersection stop line on all lanes. In Fig. 2(b), the pre-signal turns red for LVs and the intersection signal turnsgreen to discharge LVs queued in the sorting area until the sorting area is clear of vehicles. In Fig. 2(c), the pre-signal turnsgreen for TVs to allow them to enter the sorting area and wait at the intersection stop line on all lanes. In Fig. 2(d), the pre-signal turns red for TVs and the intersection signal turns green to discharge TVs queued in the sorting area until the sortingarea is clear of vehicles. The operation on this arm then returns to Fig. 2(a) and repeats the above steps. We want to point outthat besides the capability to work with a short sorting area, the phase swap sorting strategy has another important advan-tage for easier compliance than the original tandem sorting strategy. The phase swap sorting strategy no longer requiresvehicles to form tandem batches and parade through the sorting area, which may be difficult, if not impossible, to happendue to the heterogeneity in car following behavior among drivers. When drivers follow vehicles with different speeds andspacing, the boundaries between adjacent batches will become irregular or even disappear, causing disruptions to thesystem.

The goal of this paper is to operationalize the phase swap sorting strategy and develop an optimization model that pro-duces capacity maximization decisions. We first propose an operational paradigm for the phase swap sorting strategy thatcoordinates the pre-signals on all arms and the intersection signal. We then generalize the strategy to explicitly take into

Fig. 2. The phase swap sorting strategy proposed by Xuan (2011). (a) LVs advance into the sorting area. (b) LVs are discharged from the sorting area. (c) TVsadvance into the sorting area. (d) TVs are discharged from the sorting area.

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 87

consideration through, left- and right-turning movements on all arms of an intersection. We formulate the capacity maxi-mization problem as a Binary-Mixed-Integer-Linear-Programming (BMILP) model where the decision variables include sig-nal timings at the pre-signals and the intersection signal, the lengths of the sorting areas, and lane allocations upstream ofthe pre-signals as well as in the sorting areas. To quantify the benefit of our proposed approach, we conduct numericalexperiments to benchmark our model against the lane-based model, which is an advanced capacity optimization modelintroduced by Wong and Wong (2003) for the conventional strategy. Results show that substantially higher reserve capacitycan be obtained under our approach. We also find that when the demand mix is identical on all arms and the percentages ofthrough, left- and right-turning movements on each arm of the intersection are no more than 65%, respectively, the proposedsorting area strategy can bring significant benefit.

The remainder of this paper is organized as follows. In Section 2, we describe in detail our operational paradigm for thephase swap sorting strategy. In Section 3, we develop a BMILP model for capacity maximization that produces optimal signaltimings, lane allocations, and other decisions. In Section 4, we conduct numerical experiments to evaluate the benefit of ourapproach. Finally, we conclude the discussion in Section 5.

2. The phase swap sorting strategy at an intersection

The goal of this paper is to operationalize the phase swap sorting strategy proposed by Xuan (2011). To achieve this goal,we propose an operational paradigm where this strategy is applied on all arms of an intersection in Section 2.1. In Section 2.2,we generalize the phase swap sorting strategy to explicitly model all traffic movements on an arm. Finally, in Section 2.3, wediscuss design issues associated with our operational paradigm.

2.1. Our operational paradigm

We propose an operational paradigm where the phase swap sorting strategy is applied on all arms of an intersection. Itsoperation is illustrated by the 4-way intersection shown in Fig. 3(a). The circles with solid fill indicate that the correspondingtraffic lights are red and the circles with no fill indicate that the corresponding traffic lights are green. For example, there isone traffic light for each lane at the pre-signal on arm 3 and it controls the movement of vehicles in the corresponding trafficlane. We start from the initial state depicted in Fig. 3(a), where all pre-signals and the intersection signal are red. In this fig-ure, LVs and TVs wait at the pre-signals and all sorting areas are free of vehicles. In Fig. 3(b), the pre-signals on arms 1 and 3give the green to the TVs. These TVs advance into the sorting area using all the lanes and wait at the intersection stop lines.From now on, the pre-signals and the intersection signal cycle through the four phases shown in Fig. 3(c)–(f):

Fig. 3(c) (Phase 1): The pre-signals on arms 1 and 3 give the red to the TVs upstream. At the same time, the intersectionsignal gives the green to the TVs in the sorting areas of arms 1 and 3 until they are completely discharged. While the TVs arebeing discharged on arms 1 and 3, the pre-signals on arms 2 and 4 give the green to their TVs, so that the sorting areas onarms 2 and 4 can be filled with TVs.

Fig. 3(d) (Phase 2): The pre-signals on arms 2 and 4 give the red to the TVs upstream. At the same time, the intersectionsignal gives the green to the TVs in the sorting areas of arms 2 and 4, until they are completely discharged. While the TVs arebeing discharged on arms 2 and 4, the pre-signals on arms 1 and 3 give the green to their LVs, so that the sorting areas onarms 1 and 3 are filled with LVs.

Fig. 3(e) (Phase 3): The pre-signals on arms 1 and 3 give the red to the LVs upstream. At the same time, the intersectionsignal gives the green to the LVs in the sorting areas of arms 1 and 3, until they are completely discharged. While the LVs arebeing discharged on arms 1 and 3, the pre-signals on arms 2 and 4 give the green to their LVs, so that the sorting areas onarms 2 and 4 are filled with LVs.

Fig. 3(f) (Phase 4): The pre-signals on arms 2 and 4 give the red to the LVs upstream. At the same time, the intersectionsignal gives the green to the LVs in the sorting areas of arms 2 and 4, until they are completely discharged. While the LVs arebeing discharged on arms 2 and 4, the pre-signals on arms 1 and 3 give the green to their TVs, so that the sorting areas onarms 1 and 3 are filled with TVs.

Table 1 summarizes the status of the pre-signals and the intersection signal, and vehicles in the sorting area for the above4 phases. In this table, Column 1 shows the sequence of the four phases. Columns 2 and 3 show the status of the pre-signalson arms 1 and 3. Column 4 shows the types of vehicles (TVs or LVs) queued in the sorting areas of arms 1 and 3. Column 5shows the status of the intersection signal on arms 1 and 3. When the intersection signal is green, vehicles queued in thesorting areas are discharged. Columns 6 through 9 show the corresponding results on arms 2 and 4. For example, duringPhase 1, the pre-signals on arms 1 and 3 are red for both TVs and LVs and the intersection signal on these two arms are greento discharge TVs in the sorting area. At the same time, the pre-signals for the through movement on arms 2 and 4 are green toallow TVs to advance into the sorting areas on arms 2 and 4, and the intersection signal on these two arms are red.

We need to point out that when the operation transitions from Phase 1 to Phase 2 to discharge TVs on arms 2 and 4, thepre-signals for TVs on these two arms do not necessarily have to turn red. They can remain green to allow TVs to advanceinto the sorting area while TVs are being discharged from the sorting area. The only requirement is that the pre-signalsshould end their green to TVs sufficiently earlier than the time when the intersection signal ends the green to TVs, so thatall vehicles in the sorting area can be discharged completely.

Fig. 3. This figure illustrates how the phase swap sorting strategy operates on all arms of a 4-way intersection. A circles with solid fill means thecorresponding traffic light is red. A circle with no fill means the corresponding traffic light is green.

88 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

Table 1Signal phases for the phase swap sorting strategy at an intersection.

Phase Arms 1 and 3 Arms 2 and 4

Pre-signal Sorting Intersection Pre-signal Sorting Intersection

Left-turning Through area signal Left-turning Through area signal

1 Red Red TVs Green Red Green TVs Red2 Green Red LVs Red Red Red TVs Green3 Red Red LVs Green Green Red LVs Red4 Red Green TVs Red Red Red LVs Green

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 89

To better visualize the proposed paradigm, please visit the following url: http://www.ioptimize.org/sorting. This videocontains an animated simulation produced by TransModeler (Caliper, 2009) for a 4-way intersection. The simulation showsside by side how the intersection performs under the conventional strategy and the proposed paradigm of the phase swapsorting strategy. Under the conventional strategy, the queues build up quickly; however, under the proposed paradigm, thequeues are cleared during each green sub-phase and more vehicles are served.

2.2. Model all traffic movements on an arm

In existing literature on the sorting strategy, the right-turning movement is always excluded from the analysis for sim-plicity of exposition (Xuan et al., 2011; Xuan, 2011; Zhou and Zhuang, 2013). To operationalize the phase swap sorting strat-egy, however, we need to explicitly model all three traffic movements on an arm, that is, through, left- and right-turningmovements.

In the previous examples, both TVs and LVs use the sorting area, and we say these two movements participate in sorting.When the three traffic movements (that is, through, left- and right-turning movements) on an arm all participate in sortingas is illustrated in Fig. 4(a), this is called a full sort. In full sort, the sorting area spans all lanes in the road segment betweenthe pre-signal and the intersection signal. In partial sort, however, we allow either the right- or left-turning movement not toparticipate in sorting. Fig. 4(b) shows a situation where the right-turning movement does not participate in sorting. In thisfigure, the lane closest to the kerbside is dedicated to RVs and the sorting area only contains two lanes. RVs do not enterthe sorting area at all and they proceed to the stop line at the intersection signal directly. Fig. 4(c) shows a situation wherethe left-turning movement does not participate in sorting. Please note that the through movement has to always participatein sorting. Otherwise, the sorting area is split into two separate parts, one for the LVs and the other for the RVs, rendering itimpossible for the phase swap sorting strategy to work. This is illustrated in Fig. 4(d).

It is not an easy task to decide which movements should participate in sorting. When a movement participates in sorting,we can use more lanes to discharge vehicles following this movement at the intersection cross-section. However, we need toalso pay the cost, that is, to allocate green time for this movement at the intersection signal, which is a scarce resource when

Fig. 4. Examples for full sort and partial sort. (a) Full sort: all three movements participate in sorting. (b) Partial sort: the right-turning movement does notparticipate in sorting. (c) Partial sort: the left-turning movement does not participate in sorting. (d) The through movement splits the sorting area into twoseparate parts.

90 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

the cycle length is fixed. Note that throughout this paper, we maximize the capacity of the intersection given a fixed cyclelength. For example, in Fig. 4(a), we need to allocate separate green times for TVs, LVs, and RVs at the intersection signal.When a movement does not participate in sorting, although we cannot utilize all lanes at the intersection cross-sectionto discharge vehicles following this movement, we do not need to allocate green time dedicated to this traffic movementat the intersection signal, either. For example, in Fig. 4(b), the right-turning movement does not participate in sorting.Therefore, we can give the green to vehicles in the sorting area (TVs or LVs) as well as to the RVs at the same time.

2.3. Design considerations

The sorting area is used to store the transient queues formed by the different offsets and phases of the pre-signal and theintersection signal on an arm. The length of the sorting area on each arm is an important design parameter. On one hand, wewould like to have a sufficiently long sorting area to ensure that these transient queues do not spill back to the pre-signal(Xuan et al., 2011). On the other hand, the shorter the sorting area, the shorter the queue formed on each sorting lane (ofcourse, more sorting lanes need to be allocated so as to keep the capacity of the sorting area unchanged). Therefore, the lesstime it takes to discharge vehicles queued in the sorting area, meaning these vehicles do not need a long green time at theintersection, which is a scarce resource when the cycle length is fixed. We need to determine the optimal lengths of the sort-ing areas while making the above trade-offs.

Since vehicles following different movements need to be segregated by the pre-signals, no shared lane markings are al-lowed upstream of the stop lines at the pre-signals, that is, each lane upstream of the pre-signals is dedicated to a singlemovement. In addition, lane markings upstream of the pre-signal should be ordered such that lanes used for the left-turningmovement are to the left of lanes used for the through movement, and lanes used for the through movement are to the left oflanes used for the right-turning movement.

The traffic lanes in the sorting area are classified into sorting lanes and non-sorting lanes. The lanes used to store transientqueues in the sorting area are called sorting lanes. In full sort, the number of sorting lanes on an arm is the same as the num-ber of approaching lanes on that arm. For example, in Fig. 4(a), the number of sorting lanes is three. However, during partialsort, we need to allocate lanes to movements not participating in sorting in the road segment between the pre-signal and theintersection signal. These lanes are named non-sorting lanes. Hence, the number of sorting lanes is smaller than the numberof approaching lanes on that arm. In the example shown in Fig. 4(b), the number of sorting lanes is two and the number ofnon-sorting lanes is one, because the lane closest to the kerbside is dedicated to the right-turning movement. We need todecide the movements that participate in sorting and the number of sorting and non-sorting lanes in the sorting area.

Ideally, for a movement that participates in sorting, we would like to allow vehicles following this movement to advanceinto all sorting lanes, so that during the green phase, the maximum number of lanes can be used to discharge vehicles. How-ever, the number of sorting lanes allocated to a movement is also constrained by the number of exit lanes on the destinationarm of that movement. For example, suppose the origin arm of a movement has three sorting lanes but the destination armof this movement only has two exit lanes, then we can only use two lanes in the sorting area to sort vehicles for this move-ment. In other words, the number of sorting lanes for this movement is two.

3. The capacity optimization model

We develop the capacity optimization model for the phase swap sorting strategy under our proposed paradigm in thissection. The input to this model includes the number of arms at the intersection, the number of approaching and exit laneson each arm, the demand for each traffic movement, and the cycle length. The model then outputs capacity maximizing deci-sions: traffic movements that participate in sorting, lane markings upstream of the pre-signals, the lengths of the sortingareas, the number of sorting lanes allocated to each traffic movement in the sorting area, and the starts and durations ofgreen for the pre-signals and the intersection signal.

3.1. Notation and terminology

Consider an isolated intersection having NT traffic arms numbered in clockwise order. For each arm i 2 I ¼ f1;2; . . . ;NTg,we denote Na

i and Nei as the number of approaching and exit lanes on that arm, respectively. For the intersection shown in

Fig. 5, we have NT ¼ 4; Na1 ¼ 4; Ne

1 ¼ 4, and the four arms are numbered as 1, 2, 3, and 4. Given arm i, all other arms can alsobe numbered locally with respect to it with the immediate left arm numbered as 1, the other arms numbered consecutivelyin clockwise order, and the immediate right arm as NT � 1. For example, for arm 2, arm 3 is locally numbered as 1, arm 4 islocally numbered as 2, and arm 1 is locally numbered as 3. Unless otherwise specified, arm i refers to the arm whose globalarm number is i in this paper. Denote J ¼ f1;2; . . . ;NT � 1g. Traffic movement ði; jÞ originates from arm i and heads towardlocally numbered arm j, where i 2 I; j 2 J. To get the global arm number for the destination arm of traffic movement ði; jÞ, weintroduce the function Cði; jÞ: If iþ j 6 NT ; Cði; jÞ ¼ iþ j; otherwise, Cði; jÞ ¼ iþ j� NT . This also means that movement ði; jÞoriginates from arm i and heads toward arm Cði; jÞ. Approaching lanes on arm i are numbered consecutively from the centerline to the kerbside as: 1, 2, . . . ;Na

i .

Fig. 5. Numbering convention in a 4-way intersection.

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 91

For each arm i, let Li be the length of the sorting area, measured in passenger car units (pcu’s). We need to determine thecorresponding lane allocation plan on this arm, that is, the number of lanes allocated to each movement upstream of the pre-signal and the number of sorting and non-sorting lanes used by each movement in the sorting area. Fig. 4(a) and (b) eachrepresent a possible lane allocation plan. To avoid non-linearity in the model, we decide to enumerate all possible lane allo-cation plans for each arm. Let Ki be the set of possible lane allocation plans on arm i. For lane allocation plan k on arm i, letNu

i;j;k be the number of lanes devoted to movement ði; jÞ immediately upstream of the pre-signal and Ndi;k be the number of

sorting lanes in the sorting area. All lane allocation plans should satisfy the following restrictions: (1) For lane markings up-stream of the pre-signals, no shared lane markings are allowed, that is, each lane upstream of the pre-signal is dedicated to asingle movement; (2) Lane markings upstream of the pre-signal are ordered such that lanes used for the left-turning move-ment are to the left of lanes used for the through movement, and lanes used for the through movement are to the left of lanesused for the right-turning movement; (3) The number of sorting lanes in the sorting area is no more than the total number ofapproaching lanes on arm i, that is, Nd

i;k 6 Nai ; and (4) The number of sorting lanes is greater than the minimum among the

numbers of lanes allocated to each movement upstream of the pre-signal, that is, Ndi;k Pminj2J Nu

i;j;k

n o+1; otherwise, if Nd

i;k=minj2J Nu

i;j;k

n o, no movement can benefit from the phase swap sorting strategy.

Given the above restrictions, the number of possible lane allocation plans is surprisingly small. Table 2 shows the twopossible lane allocation plans for arm 1 in a 4-way intersection when Na

1 ¼ 3. Column 1 shows the value for Na1. Column 2

shows the index for the plans. Columns 3 through 5 show the number of lanes allocated to movements (1, 1), (1, 2), and(1, 3) upstream of the pre-signal, respectively. Because Na

1 ¼ 3, the only possible way is to allocate one lane to each trafficmovement, that is, Nu

1;1;k ¼ Nu1;2;k ¼ Nu

1;3;k ¼ 1. Columns 6 shows the values for minj¼1;2;3 Nu1;j;k

n o+1. The last column shows

the value for Ndi;k. Note that the feasibility of the above plans also depends on whether the movements on this arm participate

in sorting or not. For example, plan 2 is not possible if movement ð1;3Þ does not participate in sorting, because we need toallocate Nu

1;3;k ¼ Nu1;3;2 ¼ 1 non-sorting lane in the sorting area and the number of sorting lanes must be no more than

Na1 � Nu

1;3;k ¼ Na1 � Nu

1;3;2 ¼ 3� 1 ¼ 2. This feasibility requirement is enforced by Constraints (2) in Section 3.2. Table 3 showsthe nine possible lane allocation plans for Na

1 ¼ 4. Even when Na1 ¼ 5, the total number of possible allocation plans is 24,

which is still manageable. For arm i, binary decision variables xki are introduced to indicate which lane allocation plan among

Ki is selected: xki ¼ 1 if plan k is selected; xk

i ¼ 0, otherwise. Binary decision variable yi;j takes 1 if movement ði; jÞ participatesin sorting and zero otherwise.

We need to talk a little bit more on lane allocations in the sorting area. Let Ndi;j;k be the number of lanes used by movement

ði; jÞ in the sorting area. Ideally, we would like to set Ndi;j;k to Nd

i;k, so that we can use all sorting lanes in the sorting area to

discharge vehicles at the intersection. However, Ndi;j;k may be constrained by the number of exit lanes on arm Cði; jÞ, the des-

tination arm for movement ði; jÞ. Hence, we set Ndi;j;k ¼minfNd

i;k;NeCði;jÞg. Take the second plan in Table 2 as an example. The

Table 2Lane allocation plans for arm 1 in a 4-way intersection when Na

1 ¼ 3.

Na1 k Nu

1;1;k Nu1;2;k Nu

1;3;k minj¼1;...;NT�1 Nui;j;k

n o+1 Nd

1;k

31

1 1 1 22

2 3

Table 3Lane allocation plans for arm 1 in a 4-way intersection when Na

1 ¼ 4.

Na1 k Nu

1;1;k Nu1;2;k Nu

1;3;k minj¼1;...;NT�1 Nui;j;k

n o+1 Nd

1;k

4

12 1 1 2

22 33 44

1 2 1 22

5 36 47

1 1 2 22

8 39 4

92 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

sorting area contains Nd1;2 ¼ 3 lanes. For movement ð1;1Þ, suppose the number of exit lanes on arm Cð1;1Þ ¼ 2 is Ne

2 ¼ 2, then

the value of Nd1;1;2 is minfNd

1;2;N3Cð1;1Þg=minfNd

1;2;N32g ¼ 2. Since we enumerate all possible lane allocation plans before hand

and Nei are known for all i 2 I, it means the values for Nu

i;j;k; Ndi;k, and Nd

i;j;k are known for any i 2 I; j 2 J; k 2 Ki. These together

with the values for xki and yi;j jointly produce the final lane allocation plan on an arm.

Let Qi;j be the demand from origin arm i to destination arm j (numbered locally with respect to arm i). This demand isassumed to be evenly distributed on all lanes devoted to this movement upstream of the pre-signal and on all sorting lanesallocated to this movement, so that the degrees of saturation on any pair of adjacent lanes with the same permitted move-ment are identical (Allsop, 1972). ss is the saturation flow for a single lane at the pre-signal. The saturation flow of a turninglane at the intersection is based on the formula in Wong and Wong (2003) and is expressed as si;j ¼ �s

1þ1:5=ri;j, where �s is the lane

saturation flow for straight-ahead movement and ri;j is the radius of the turning trajectory (¼ 1 for the through movement).The intersection operates at a fixed cycle length, denoted as c. Decision variables hi;j and /i;j specify the start and duration

of green time for movement ði; jÞ at the intersection signal, respectively. These variables are expressed as a fraction of thecycle length, that is, the actual start and duration of green time are chi;j and c/i;j. Decision variables Hi;j and Ui;j, expressedas a fraction of the cycle length, specify the start and duration of green time for movement ði; jÞ at the pre-signal, respectively.The minimum duration of green for movement ði; jÞ is denoted as gi;j. Two traffic movements ði; jÞ and ðl;mÞ are said to bemutually incompatible if vehicles following these two movements are in conflict at the intersection. Let W be the set ofincompatible movements. For any pair of incompatible traffic movements ði; jÞ and ðl;mÞ in W, a clearance time, wi;j;l;m, mustbe given to separate their rights-of-way. In Appendix A, we summarize all notation used throughout this paper.

3.2. The constraints

3.2.1. Lane allocation planOf all possible lane allocation plans on arm i, only one can be selected, this is ensured by:

X

k2Ki

xki ¼ 1; 8i 2 I: ð1Þ

3.2.2. Number of sorting lanesFor lane allocation plan k on arm i, suppose movement ði; jÞ does not participate in sorting, that is, yi;j ¼ 0. Since there are

Nui;j;k lanes allocated to movement ði; jÞ upstream of the pre-signal, we have to allocate Nu

i;j;k non-sorting lanes in the sortingarea for this movement. Hence, the total number of sorting lanes in the sorting area must satisfy the following constraint,which is active if and only if xk

i ¼ 1:

Ndi;k 6 Na

i �Xj2J

ð1� yi;jÞNui;j;k

h iþMð1� xk

i Þ; 8i 2 I; k 2 Ki; ð2Þ

where M is a sufficiently large number.

3.2.3. Flow through the pre-signalsIf movement ði; jÞ participates in sorting, the number of lanes allocated to this movement upstream of the pre-signal

should be no more than the number of sorting lanes allocated to this movement in the sorting area. Otherwise, a bottleneckis created for this movement. This is ensured by:

Nui;j;k 6 Nd

i;j;k þMð1� xki Þ þMð1� yi;jÞ; 8i 2 I; j 2 J; k 2 Ki: ð3Þ

3.2.4. Start and duration of green at the intersection signalSince the signal settings at the intersection are cyclic, the starts of green variables can be arbitrary as long as they satisfy

the other constraints in the formulation. For convenience reasons, we restrict the start of green variables at the intersectionto stay between 0 and 1:

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 93

0 6 hi;j � 1; 8i 2 I; j 2 J: ð4Þ

The duration of green at the intersection signal, /i;j, is subject to a minimum value, gi;j, due to safety reasons:

gi;j=c 6 /i;j 6 1; 8i 2 I; j 2 J: ð5Þ

3.2.5. Start and duration of green at the pre-signalSince the green time for movement ði; jÞ at the pre-signal should start no later than the green time for this movement at

the intersection signal, we must have:

Hi;j 6 hi;j þMð1� yi;jÞ; 8i 2 I; j 2 J: ð6Þ

The duration of green for movement ði; jÞ at the pre-signal must satisfy the following constraints:

gi;j=c �Mð1� yi;jÞ 6 Ui;j 6 1þMð1� yi;jÞ; 8i 2 I; j 2 J: ð7Þ

Constraints (6) and (7) are active when and only when movement ði; jÞ participates in sorting.

3.2.6. Order of signal displays for incompatible movementsWhen the vehicle trajectories of two movements ði; jÞ and ðl;mÞ intersect at the intersection, these two movements are

incompatible. For example, the through movement on an arm is incompatible with the left-turning movement from theopposite arm at a 4-way intersection. The consequences are: (a) Their corresponding green times cannot overlap; and (b)A minimum clearance time, wi;j;l;m, should be given to further separate their rights-of-way for safety reasons. Followingthe same treatment in Heydecker (1992), binary decision variable Xi;j;l;m is used to indicate the order of signal displays formovements ðði; jÞ; ðl;mÞÞ 2 W. Xi;j;l;m ¼ 0 if the start of green for movement ði; jÞ precedes that of movement ðl;mÞ, that is,hi;j 6 hl;m; and Xi;j;l;m ¼ 1 if the opposite is true. We can therefore enforce the clearance time requirement for any pair ofincompatible movement in W as follows:

Xi;j;l;m þXl;m;i;j ¼ 1; 8ðði; jÞ; ðl;mÞÞ 2 W; ð8Þ

hi;j þ /i;j þwi;j;l;m=c 6 hl;m þXi;j;l;m; 8ðði; jÞ; ðl;mÞÞ 2 W: ð9Þ

3.2.7. Ensure vehicles in the sorting area are fully dischargedSuppose that movement ði; jÞ participates in sorting. In Fig. 2(a), the pre-signal on arm i turns green to allow vehicles follow-

ing movement ði; jÞ to advance into the sorting area. After some time, when the pre-signal for movement ði; jÞ turns red, the inter-section signal for movement ði; jÞ should remain green for a period of time, denoted as te

i;j, to allow vehicles in the sorting area tofully discharge (see Fig. 2(b)). Hence, we have the following constraints for all movements that participate in sorting:

Hi;j þUi;j þ tei;j=c 6 hi;j þ /i;j þMð1� yi;jÞ; 8i 2 I; j 2 J: ð10Þ

The value for teij depends on the number of vehicles that needs to be discharged from the sorting area. In the worst case,

suppose that when the pre-signal for movement ði; jÞ turns red, the sorting area on arm i is full of vehicles, that is, each lanecontains Li pcu’s. It takes Li=si;j time to discharge those vehicles. Hence, as long as te

ij is no less than Li=si;j, all vehicles can befully discharged from the sorting area:

tei;j P Li=si;j; 8i 2 I; j 2 J: ð11Þ

3.2.8. Order of signal displays for movements participating in sortingSuppose movements ðl;mÞ and ðl;nÞ both participate in sorting. They originate from arm l but head toward arms Cðl;mÞ

and Cðl;nÞ, respectively. Once the pre-signal gives the green to movement ðl;mÞ, it cannot give the green to movement ðl;nÞuntil the green for movement ðl;mÞ at the intersection signal ends. Otherwise, the sorting area will contain vehicles formovements ðl;mÞ and ðl;nÞ at the same time, violating our operational requirement. For example, the pre-signal cannot turngreen for TVs in Fig. 2(c) until the LVs have been discharged from the sorting area and the intersection signal has turned red.We develop the constraints for the following two cases:

Case 1: If hl;m 6 hl;n, we know that starting from time 0, movement ðl;mÞ receives the green prior to movement ðl;nÞ at theintersection signal. Hence, the green time for movement ðl;mÞ at the intersection signal should end prior to the start of thegreen time for movement ðl;nÞ at the pre-signal, that is, we have

hl;m þ /l;m 6 Hl;n:

Similarly, the green time for movement ðl;nÞ at the intersection signal should end prior to the start of the green time formovement ðl;mÞ at the pre-signal. This is ensured by the following constraint, where we have to add 1 to the right hand sideof the inequality because Hl;m 6 hl;m 6 hl;n 6 1:

hl;n þ /l;n 6 Hl;m þ 1:

94 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

Case 2: If hl;mihl;n, following the same logic in Case 1, we need the following constraints

hl;n þ /l;n 6 Hl;m;

hl;m þ /l;m 6 Hl;n þ 1:

We introduce Pl;m;l;n to indicate the order of signal displays for movements ðl;mÞ and ðl;nÞ at the intersection. Pl;m;l;n ¼ 0 ifhl;m 6 hl;n; and Pl;m;l;n ¼ 1 if the opposite is true. We can then summarize the constraints for the two cases in the followingequivalent form:

Pl;m;l;n þPl;n;l;m ¼ 1; 8l 2 I; m;n 2 J; m – n; ð12Þ

hl;m þ /l;m 6 Hl;n þPl;m;l;n þMð2� yl;n � yl;mÞ; 8l 2 I; m;n 2 J; m – n: ð13Þ

Note that Constraints (13) are active if and only if both traffic movements participate in sorting, that is yl;m ¼ yl;n ¼ 1.

3.2.9. Capacity of the sorting areaThe sorting area is used to store the transient queue formed between the pre-signal and the intersection signal. The

capacity of the sorting area should be large enough to ensure that these queues do not spill back to the pre-signals. Similarto Wong and Wong (2003) and Wong and Heydecker (2011), we assume that the traffic demand is multiplied by a commonflow multiplier l to represent the maximum allowable demand increase under which the intersection can still functionappropriately. Hence, the demand for movement ði; jÞ in a cycle, or the number of vehicles in the queue, is lQ i;jc. As longas the capacity of the sorting area is no less than lQi;jc, the queue will not spill back to the pre-signal:

LiNdi;j;k P lQ i;jc �Mð1� xk

i Þ �Mð1� yi;jÞ; 8i 2 I; j 2 J; k 2 Ki: ð14Þ

This constraint is active when and only when movement ði; jÞ participates in sorting and plan k is selected for arm i.

3.2.10. Maximum acceptable degree of saturationLet pi;j be the maximum permitted degree of saturation for movement ði; jÞ on a single lane of arm i. We need to ensure

that the traffic flow on each lane do not exceed pi;j. This is achieved by Constraints (15) and (17), where e is the extra effectivegreen time that devises from the difference between actual and effective greens (measured in time units, which are usuallytaken as 1 s):

lQ i;jc

Ndi;j;kð/i;jc þ eÞsi;j

6 pi;j þMð1� xki Þ þMð1� yi;jÞ; 8i 2 I; j 2 J; k 2 Ki; ð15Þ

lQ i;jcNu

i;j;kðUi;jc þ eÞss6 pi;j þMð1� xk

i Þ þMð1� yi;jÞ; 8i 2 I; j 2 J; k 2 Ki; ð16Þ

lQ i;jcNu

i;j;kð/i;jc þ eÞsi;j6 pi;j þMð1� xk

i Þ þMyi;j; 8i 2 I; j 2 J; k 2 Ki: ð17Þ

For movement ði; jÞ, either Constraints (15) and (16) or Constraints (17) are active. If movement ði; jÞ participates in sort-ing, its degree of saturation at the intersection signal is constrained by Constraints (15) and its degree of saturation at thepre-signal is constrained by Constraints (16). If movement ði; jÞ does not participate in sorting, its degree of saturation atthe intersection signal is constrained by Constraints (17). Note that when a movement does not participate in sorting, thereare Nu

i;j;k lanes allocated to movement ði; jÞ at the intersection signal.

3.3. Model formulation

Our goal is to maximize the capacity of an intersection. Based on the assumption that the traffic demands at an intersec-tion increase in proportion to the demand matrix (Allsop, 1972; Gallivan and Heydecker, 1988; Wong and Wong, 2003;Wong and Heydecker, 2011), the capacity maximization problem becomes maximizing the common flow multiplier l de-fined in Section 3.2.9, while observing all the constraints specified in Section 3.2. Therefore, we formulate the capacity max-imization model as the following BMILP model, which can be solved efficiently by standard branch-and-bound algorithms:

max l

subject to Constraints (1)–(17). Let lmax be the optimal value for the objective function. If lmax is less than 1, it implies thatthe intersection is overloaded by 100ð1� lÞ percent. If lmax is greater than 1, it implies the intersection has a reserve capac-ity of 100ð1� lÞ percent.

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 95

4. Numerical experiments

In this section, we conduct numerical experiments to quantify the benefits of the phase swap sorting strategy on a 4-wayintersection through three case studies. We also test the performance of the phase swap sorting strategy when we vary themix of traffic demands for different movements at the intersection.

4.1. Experimental setup

In the computational experiments, we compare the maximized common flow multiplier produced by our model underthe phase swap sorting strategy to that produced by a lane-based capacity optimization model under the conventional strat-egy. The lane-based model is an advanced capacity optimization model first proposed by Wong and Wong (2003). It canproduce optimal lane markings and signal timings under the conventional strategy. The exact model statement of thelane-based model used in our experiments is presented in Appendix B.

Altogether, we perform three case studies. In each Appendix case study, we let the number of approaching lanes on anarm be the same as the number of exit lanes on that arm. The number of approaching (or exit) lanes on each arm is reportedin Table 4. For example, in Case 1 there are 3 approaching lanes as well as 3 exit lanes on arm 1.

The lane saturation flow for through movement at the intersection is �s ¼ 2105 pcu/h/lane. The radius for left-turning tra-jectories is 10 meters and that for right-turning trajectories is 3 meters. The lane saturation flow at the pre-signal isss ¼ 2105 pcu/h/lane. The maximum acceptable degree of saturation for all traffic movements is 90%. The minimum greendurations are 5 s for all traffic movements. The cycle length is set to c = 120 s. The minimum clearance time matrix is given inTable 5. The traffic demands are given in Table 6.

The computer programs for the phase swap sorting strategy and the conventional strategy are both written in C++ andsolved by IBM ILOG CPLEX 12.3 (IBM, 2012). All computational tests are performed on a PC equipped with an Intel2.40 GHz CPU and 4 GB memory.

4.2. Optimization results

We report the summary statistics for the three case studies including problem sizes, CPU times, and the maximized com-mon flow multipliers (lmax) in Table 7. The problem sizes under the phase swap sorting strategy are comparable to thoseunder the conventional strategy. All models can be solved to optimality in a few seconds. It can be seen that the commonflow multiplier under the phase swap sorting strategy is much higher than that under the conventional strategy. Take Case1 as an example: the common flow multiplier is 1.0034 under the conventional strategy and the intersection barely has anyreserve capacity. Under the phase swap sorting strategy, the common flow multiplier increases to 1.3919 and the intersec-tion has a reserve capacity of 39.19%.

The optimal lane allocation plans under the conventional strategy and the phase swap sorting strategy for Case 1 areshown in Figs. 6(a) and (b), respectively. In Fig. 6(a), permitted movements on the approaching lanes of each arm are clearlyindicated, where some lanes permit only one traffic movement (that is, exclusive lanes) and some permit two or more move-ments (that is, shared lanes). In Fig. 6(b), the right-turning movements on all four arms do not participate in sorting underthe phase swap sorting strategy and therefore the number of sorting lanes is two on each arm. Upstream of the pre-signal oneach arm, lane markings are shown and the left-turning and through movements each get a single lane. The length of thesorting area on each arm measured in pcu’s is also shown. For example, the length of the sorting area on arm 1 is 11.6 pcu’s.Note that due to the difference in traffic demands and lane allocations on each arm, the lengths of the sorting area do differ.Fig. 7(a) and (b) show the optimal lane allocation plans under the two strategies for Case 2. In Fig. 7(b), the left-turningmovement on arm 2 does not participate in sorting, while the right-turning movements on all other arms do not participatein sorting. In addition, the numbers of lanes allocated to each movement upstream of the pre-signals differ. For example,there are two lanes allocated to the through movement on arm 1, while there is only one lane allocated to the through move-ment on arm 4. Figs. 8(a) and (b) show the optimal lane allocation plans under the two strategies for Case 3.

Fig. 9(a) and (b) show the optimal signal timing plans under the conventional strategy and the phase swap sorting strat-egy for Case 1, respectively. In Fig. 9(a), the solid bar corresponds to the green time of the intersection signal under the con-ventional strategy. For example, the green time at the intersection signal for movement from arm 1 to arm 2 starts at time0.0 and ends at time 25.9. In Fig. 9(b), the solid bar also corresponds to the green time of the intersection signal, while thegray bar corresponds to the green time of the pre-signal. For example, for traffic movement from arm 1 to arm 2, its green

Table 4The number of approaching (or exit) lanes on each arm of the intersection.

Arm 1 Arm 2 Arm 3 Arm 4

Case 1 3 3 3 3Case 2 4 4 4 4Case 3 4 3 4 3

Table 5Minimum clearance time matrix for conflicting movements (in seconds).

Arm, i Local arm, j 1 1 1 2 2 2 3 3 3 4 4 41 2 3 1 2 3 1 2 3 1 2 3

1 1 6 6 6 6 6 61 2 6 6 6 6 6 61 3 6 6

2 1 6 6 6 6 6 62 2 6 6 6 6 6 62 3 6 6

3 1 6 6 6 6 6 63 2 6 6 6 6 6 63 3 6 6

4 1 6 6 6 6 6 64 2 6 6 6 6 6 64 3 6 6

Table 6Traffic demand for the three case studies (in pcu/h).

Traffic demand To arm

From arm 1 2 3 4

1 – 300 500 4002 400 – 300 4003 500 200 – 4004 500 200 200 –

Table 7Optimization summary.

Case Strategy Problem size CPU time (s) lmax Reserve capacity (%)

Binary variables Continuous variables Constraints

1 Conventional 180 86 470 4.88 1.0034 0.34Phase swap sorting 164 65 448 3.34 1.3919 39.19

2 Conventional 192 98 586 13.88 1.4128 41.28Phase swap sorting 192 65 952 8.24 1.8126 81.26

3 Conventional 186 92 528 9.50 1.1892 18.92Phase swap sorting 178 65 700 7.18 1.5096 50.96

96 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

time at the pre-signal starts at time 57.8 and ends at time 83.2, while its green time at the intersection signal starts at time83.2 and ends at 106.0. Note that our optimization model may output Hi;j that takes a negative value. If this happens, we addc, the cycle length, to Hi;jc so that the starts of the green for the pre-signals fall between 0 and c. It can be seen that our modelgenerates the complicated cycle structure for both the intersection signal and the pre-signals. For any movement that par-ticipates in sorting, the pre-signal for this movement starts its green ahead of the green at the intersection so as to allowvehicles to advance into the sorting area and wait at the stop line at the intersection. The pre-signal also ends its green beforethe end of the green at the intersection so as to allow vehicles to be discharged from the sorting area completely. It is notdifficult to verify that the clearance time requirements are met and the green times for conflicting movements do not over-lap. Note that since movement from arm 1 to arm 4 does not participate in sorting, there is no gray bar for its green time atthe pre-signal. Figs. 10 and 11 show the optimal signal timing plans for Cases 2 and 3 under the two strategies.

We now report the details of assigned flows and performance indications on the traffic lanes under both strategies. Table 8reports the results under the conventional strategy for Case 1. Column 1 shows the origin arms. Column 2 shows the lanes onthe corresponding origin arms. Columns 3 through 6 list the assigned lane flows measured in pcu/h. For each arm-to-armmovement, the sum of the assigned lane flows over all lanes equals the corresponding traffic demand shown in Table 6.For example, the assigned flow for movement from arm 1 to arm 3 is 77.5 pcu/h on lane 1 and 422.5 pcu/h on lane 2. Thesum of these two numbers equals 500 pcu/h, which matches the traffic demand shown in Table 6. Column 7 converts theassigned lane flows on the same lane but head toward different destination arms (measured in pcu) into lane flows measuredin through car units (tcu’s). The conversion factor si;j is introduced in Appendix B. The conversion factors for left- and right-turning movements are 1þ 1:5

10 ¼ 1:15 and 1þ 1:53 ¼ 1:5, respectively. For example, the total lane flow on lane 1 of arm 1 is

300� 1:15þ 77:5� 1:0 ¼ 422:5 tcu/h. The duration of the green time is shown in Column 8. The last column shows the

Fig. 6. Optimal lane allocation plans for Case 1. (a) The conventional strategy. (b) The phase swap sorting strategy.

Fig. 7. Optimal lane allocation plans for Case 2. (a) The conventional strategy. (b) The phase swap sorting strategy.

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 97

degrees of saturation on all approaching lanes, which are the flow-to-capacity ratios and 0.9 (or 90%) is the allowable max-imum limit. For example, the degree of saturation on lane 1 of arm 1 is 1:0034�422:5�120

ð25:9þ1Þ�2105 ¼ 0:90.

Tables 9 and 10 report the assigned flows under the phase swap sorting strategy at the pre-signal and the intersectionsignal for Case 1, respectively. In Table 9, Column 1 shows the origin arms. Column 2 shows the lanes on the correspondingorigin arms. Columns 3 through 6 list the assigned lane flows measured in pcu/h. For each arm-to-arm movement, the sum ofthe assigned lane flows over all lanes equals the corresponding traffic demand shown in Table 6. Column 7 shows the totallane flow measured in pcu/h. The duration of the green time is shown in Column 8. The last column shows the degrees ofsaturation on all approaching lanes. For example, lane 2 on arm 1 carries traffic from arm 1 to arm 3, that is, movement

(1, 2), and its degree of saturation is lQ1;2cðU1;2cþ1Þss

¼ 1:3919�500�120ð43:1þ1Þ�2105 ¼ 0:90. Lane 3 on arm 1 does not participate in sorting, hence

the green time and the degree of saturation at the pre-signal are not available. Table 10 reports the results at the intersectionsignal. Columns 1 and 2 correspond to the origin and destination arms of a movement, respectively. Column 3 shows thedemand from the origin arm to the destination arm, which equals the corresponding traffic demand shown in Table 6. Col-umn 4 shows the number of sorting lanes for this movement. Column 5 shows the length of the sorting area. We divide thedemand in Column 3 by the number of sorting lanes in Column 4 to obtain the assigned lane flow for this movement if itparticipates in sorting, which is shown in Column 6. Column 7 shows the green time for this movement at the intersectionsignal and Column 8 shows the degree of saturation for this movement. For example, the degree of saturation on lanes fromarm 1 to arm 2 is 1:3919�150�120

ð22:8þ1Þ�2105=ð1þ1:5=10Þ ¼ 0:57. An interesting observation is that under the phase swap sorting strategy, the

Fig. 8. Optimal lane allocation plans for Case 3. (a) The conventional strategy. (b) The phase swap sorting strategy.

Fig. 9. Optimal signal plans for Case 1. (a) The conventional strategy. (b) The phase swap sorting strategy.

Fig. 10. Optimal signal plans for Case 2. (a) The conventional strategy. (b) The phase swap sorting strategy.

98 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

degrees of saturation at the pre-signals are much greater than those at the intersection signal, indicating that the bottleneckis now at the pre-signals. Tables 11–16 show the details of assigned flows and performance indications on the traffic lanes forCases 2 and 3 under both strategies.

Fig. 11. Optimal signal plans for Case 3. (a) The conventional strategy. (b) The phase swap sorting strategy.

Table 8Assigned flows under the conventional strategy for Case 1 (l ¼ 1:0034).

From arm Lane To arm (assigned lane flow, pcu/h) Total lane flow (tcu/h) Green time (s) Degree of saturation

1 2 3 4

1 1 300.0 77.5 422.5 25.9 0.901 2 422.5 422.5 25.9 0.901 3 400.0 600.0 37.1 0.90

2 1 300.0 27.5 372.5 22.7 0.902 2 372.5 372.5 22.7 0.902 3 400.0 600.0 37.1 0.90

3 1 365.2 420.0 25.7 0.903 2 380.0 34.8 420.0 25.7 0.903 3 120.0 200.0 420.0 25.7 0.90

4 1 311.6 358.4 21.7 0.904 2 188.4 141.7 358.4 21.7 0.904 3 58.3 200.0 358.3 21.7 0.90

Table 9Assigned flows under the phase swap sorting strategy for Case 1 at the pre-signal (l ¼ 1:3919).

From arm Lane To arm (assigned lane flow, pcu/h) Total lane flow (pcu/h) Green time (s) Degree of saturation

1 2 3 4

1 1 300 300.0 25.5 0.901 2 500 500.0 43.1 0.901 3 400 400.0 – –

2 1 300 300.0 25.5 0.902 2 400 400.0 34.3 0.902 3 400 400.0 – –

3 1 400 400.0 34.3 0.903 2 500 500.0 43.1 0.903 3 200 200.0 – –

4 1 500 500.0 43.1 0.904 2 200 200.0 16.6 0.904 3 200 200.0 – –

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 99

4.3. Sensitivity to the mix of traffic demands

The benefit of the phase swap sorting strategy may be affected by the mix of traffic demand at the intersection, that is, tothe percentages of through, left- and right-turning demands on each arm of the intersection. We therefore conduct the fol-lowing sensitivity analysis to measure the improvement in lmax when we vary the mix of traffic demands at the intersection.The experiments are conducted on the three intersections defined in Table 4.

Table 10Assigned flows under the phase swap sorting strategy for Case 1 at the intersection signal (l ¼ 1:3919).

Fromarm

Toarm

Demand(pcu/h)

No. of sortinglanes

Length of sorting area(pcu)

Assigned lane flow(pcu/h)

Green time(s)

Degree ofsaturation

1 2 300.0 2 11.6 150.0 22.8 0.571 3 500.0 2 11.6 250.0 21.0 0.901 4 400.0 – – 400.0 54.6 0.86

2 3 300.0 2 9.3 150.0 18.3 0.712 4 400.0 2 9.3 200.0 16.6 0.902 1 400.0 – – 400.0 56.6 0.83

3 4 400.0 2 11.6 200.0 25.2 0.703 1 500.0 2 11.6 250.0 21.0 0.903 2 200.0 – – 200.0 25.5 0.90

4 1 500.0 2 11.6 250.0 24.3 0.904 2 200.0 2 11.6 100.0 19.8 0.384 3 200.0 – – 200.0 62.7 0.37

Table 11Assigned flows under the conventional strategy for Case 2 (l ¼ 1:4128).

From arm Lane To arm (assigned lane flow, pcu/h) Total lane flow (tcu/h) Green time (s) Degree of saturation

1 2 3 4 saturation

1 1 244.9 281.6 24.2 0.901 2 55.1 218.3 281.7 24.2 0.901 3 281.7 281.7 24.2 0.901 4 400.0 600.0 53.6 0.90

2 1 215.9 248.3 21.2 0.902 2 84.1 151.7 248.4 21.2 0.902 3 248.3 248.3 21.2 0.902 4 400.0 600.0 52.7 0.90

3 1 273.9 315.0 27.2 0.903 2 170.0 126.1 315.0 27.2 0.903 3 315.0 315.0 27.2 0.903 4 15.0 200.0 315.0 27.2 0.90

4 1 224.6 258.3 22.1 0.904 2 224.6 258.3 22.1 0.904 3 50.8 200.0 258.3 22.1 0.904 4 200.0 300.0 53.3 0.43

Table 12Assigned flows under the phase swap sorting strategy for Case 2 at the pre-signal (l ¼ 1:8126).

From arm Lane To arm (assigned lane flow, pcu/h) Total lane flow (pcu/h) Green time (s) Degree of saturation

1 2 3 4

1 1 300 300.0 33.4 0.901 2 250 250.0 27.7 0.901 3 250 250.0 27.7 0.901 4 400 400.0 – –

2 1 300 300.0 – –2 2 200 200.0 22.0 0.902 3 200 200.0 22.0 0.902 4 400 400.0 44.9 0.90

3 1 400 400.0 44.9 0.903 2 250 250.0 27.7 0.903 3 250 250.0 27.7 0.903 4 200 200.0 – –

4 1 250 250.0 27.7 0.904 2 250 250.0 27.7 0.904 3 200 200.0 22.0 0.904 4 200 200.0 – –

100 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

Table 13Assigned flows under the phase swap sorting strategy for Case 2 at the intersection signal (l ¼ 1:8126).

From arm To arm Demand(pcu/h)

Lanes for sorting Length of sorting area (pcu) Assigned lane flow(pcu/h)

Green time (s) Degree of saturation

1 2 300.0 3 10.1 100.0 19.8 0.571 3 500.0 3 10.1 166.7 18.1 0.901 4 400.0 – – 400.0 67.9 0.90

2 3 300.0 – – 300.0 38.6 0.902 4 400.0 3 8.1 200.0 14.3 0.902 1 400.0 3 8.1 133.3 22.0 0.90

3 4 400.0 3 10.1 133.3 19.8 0.763 1 500.0 3 10.1 166.7 18.1 0.903 2 200.0 – – 200.0 62.7 0.49

4 1 500.0 3 10.1 166.7 21.0 0.904 2 200.0 3 10.1 66.7 17.2 0.384 3 200.0 – – 200.0 45.3 0.67

Table 14Assigned flows under the conventional strategy for Case 3 (l ¼ 1:1892).

From arm Lane To arm (assigned lane flow, pcu/h) Total lane flow (tcu/h) Green time (s) Degree of saturation

1 2 3 4

1 1 244.9 281.6 20.2 0.901 2 55.1 218.3 281.7 20.2 0.901 3 281.7 281.7 20.2 0.901 4 400.0 600.0 44.2 0.90

2 1 300.0 27.5 372.5 27.1 0.902 2 372.5 372.5 27.1 0.902 3 400.0 600.0 53.3 0.75

3 1 273.9 315.0 22.7 0.903 2 170.0 126.1 315.0 22.7 0.903 3 315.0 315.0 22.7 0.903 4 15.0 200.0 315.0 22.7 0.90

4 1 311.6 358.3 26.0 0.904 2 188.4 141.7 358.4 26.0 0.904 3 58.3 200.0 358.3 26.0 0.90

Table 15Assigned flows under the phase swap sorting strategy for Case 3 at the pre-signal (l ¼ 1:5096).

From arm Lane To arm (assigned lane flow, pcu/h) Total lane flow (pcu/h) Green time (s) Degree of saturation

1 2 3 4

1 1 150 150.0 13.3 0.901 2 150 150.0 13.3 0.901 3 500 500.0 46.8 0.901 4 400 400.0 – –

2 1 300 300.0 27.7 0.902 2 400 400.0 37.2 0.902 3 400 400.0 – –

3 1 200 200.0 18.1 0.903 2 200 200.0 18.1 0.903 3 500 500.0 46.8 0.903 4 200 200.0 – –

4 1 500 500.0 46.8 0.904 2 200 200.0 18.1 0.904 3 200 200.0 – –

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 101

We assume that the mix of traffic demands is identical across all four arms. In addition, we assume: (a) The total trafficdemands for through, left- and right-turning movements is 1000 pcu/h; and (b) Of the total traffic demands on each arm,there are 100� a% for the through movement, 100� b% for the left-turning movement, and 100� ð1� a� bÞ% for theright-turning movement, where a; b 2 0;1½ �, and aþ b 6 1. For example, when a ¼ 0:5; b ¼ 0:3, the traffic demands on eacharm for through, left- and right-turning movements are 500 pcu/h, 300 pcu/h, and 200 pcu/h, respectively.

Table 16Assigned flows under the phase swap sorting strategy for Case 3 at the intersection signal (l ¼ 1:5096).

Fromarm

Toarm

Demand(pcu/h)

Lanes for sorting area(pcu)

Length ofsorting

Assigned lane flow(pcu/h)

Green time(s)

Degree ofsaturation

1 2 300.0 3 8.4 100.0 16.5 0.571 3 500.0 3 8.4 166.7 14.9 0.901 4 400.0 – – 400.0 56.4 0.90

2 3 300.0 2 10.1 150.0 19.8 0.712 4 400.0 2 10.1 200.0 21.5 0.762 1 400.0 – – 400.0 60.6 0.84

3 4 400.0 3 8.4 133.3 16.5 0.753 1 500.0 3 8.4 166.7 14.9 0.903 2 200.0 – – 200.0 27.7 0.90

4 1 500.0 2 12.6 250.0 26.5 0.904 2 200.0 2 12.6 100.0 21.5 0.384 3 200.0 – – 200.0 27.7 0.90

Fig. 12. The value for lsmax=lc

max for Case 1.

Fig. 13. The value for lsmax=lc

max for Case 2.

102 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

Fig. 14. The value for lsmax=lc

max for Case 3.

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 103

For each combination of a; bð Þ, we solve for lcmax, the maximized common flow multiplier under the conventional strategy,

and lsmax, the maximized common flow multiplier under the phase swap sorting strategy. Fig. 12 shows the contour plot for

the ratio lsmax=lc

max when we vary a and b for the intersection shown in Case 1. The phase swap sorting strategy performs thebest when a and b are both around 1

3, that is, the demand for each movement is roughly the same. Under such demand mix,ls

max=lcmax can be as high as 1.7. Generally speaking, as long as a 6 0:65; b 6 0:65, and aþ b P 0:35, the ratio is close to or

greater than 1.0. The three solid thick lines in the figure correspond to a ¼ 0:65; b ¼ 0:65, and aþ b ¼ 0:35, respectively.This suggests that for intersections where the traffic mix is identical on all arms and the percentages of through, left- andright-turning movements are no more than 65%, respectively, it is beneficial to deploy the phase swap sorting strategy be-cause the capacity of the intersection can be increased significantly. Figs. 13 and 14 show the contour plots for Cases 2 and 3,respectively. Similar results are observed.

5. Conclusions

In conventional at-grade intersections, the left turn reduces the intersection capacity significantly, because some of thetraffic lanes cannot be used to discharge vehicles during its green phases. In this paper, we operationalize the phase swapsorting strategy on all arms of an intersection, which allows us to use most, if not all, traffic lanes to discharge vehicles atthe intersection cross-section. Unlike existing literature where the right-turning movement is excluded from the analysis,we explicitly take into consideration all three traffic movements on all arms of the intersection. The benefits and drawbacksof different sorting patterns are also discussed.

We formulate the capacity maximization problem as a BMILP model that maximizes the common flow multiplier at theintersection. The output of the model includes traffic movements that participate in sorting, lane markings upstream of thepre-signals, the lengths of the sorting areas, the number of sorting lanes allocated to each traffic movement in the sortingarea, and the starts and durations of green time for the pre-signals and the intersection signal. The model can be efficientlysolved by standard branch-and-bound algorithms.

Numerical experiments show that significantly higher reserve capacity can be obtained under the proposed approach. Wealso test the sensitivity of our model to the traffic mix at the intersection. We find that when the mix of traffic demand isidentical and the percentages of through, left- and right-turning movements are no more than 65% on all arms, respectively,the proposed sorting area strategy can bring significant benefit.

The idea of using the sorting area to increase intersection capacity is fairly new. There are many possible future researchdirections in this area. For example, it is worthwhile to test and validate the phase swap sorting strategy at intersections withleft turn bays, or analyze this strategy when applied to a network of intersections. It may also be interesting to carry outlaboratory driving experiments to see how drivers adapt to such a new strategy and identify ways that would facilitate theirlearning experience.

Appendix A. Summary of notation

Notation used throughout this paper is summarized as follows:

104 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

Data and parameters

NT Number of traffic arms at the intersection I Set f1;2; . . . ;NTg, indexed by i J Set f1;2; . . . ;NT � 1g, indexed by j Na

i

Number of approaching lanes on arm i 2 I Ne

i

Number of exit lanes on arm i 2 I ði; jÞ Denote the traffic movement from arm i to locally numbered arm j; 8i 2 I; j 2 J Cði; jÞ The global arm number for the destination arm of movement ði; jÞ; 8i 2 I; j 2 J Ki Set of possible lane allocation plans on arm i 2 I, indexed by k Nu

i;j;k

Number of lanes allocated to movement ði; jÞ upstream of the pre-signal for lane allocation plan k on armi; 8i 2 I; j 2 J; k 2 Ki

Ndi;k

Number of sorting lanes in the sorting area for lane allocation plan k on arm i, 8i 2 I; k 2 Ki

Ndi;j;k

Number of sorting lanes allocated to movement ði; jÞ in the sorting area for lane allocation plan k on arm

i; 8i 2 I; j 2 J; k 2 Ki

W

Set of incompatible movements Qi;j Demand for traffic movement ði; jÞ; 8i 2 I; j 2 J wi;j;l;m Minimum clearance time for incompatible traffic movements ðði; jÞ; ðl;mÞÞ 2 W gi;j Minimum green duration for traffic movement ði; jÞ; 8i 2 I; j 2 J ri;j Radius of turning trajectory for movement ði; jÞ; 8i 2 I; j 2 J s Saturation flow for straight-head movement at the intersection ss Sorting saturation flow at the pre-signals si;j Saturation flow for traffic movement ði; jÞ. si;j ¼ s=ð1þ 1:5=ri;jÞ; 8i 2 I; j 2 J c Cycle length pi;j Maximum acceptable degree of saturation for traffic movement ði; jÞ; 8i 2 I; j 2 J M A sufficiently large number e The extra effective green time that devises from the difference between actual and effective greens. Its value is

usually taken as 1 s

Decision variables

xk

i

Equals 1 if allocation plan k is selected on arm i; 0, otherwise, 8i 2 I; k 2 Ki

yi;j

Equals 1 if traffic movement ði; jÞ participates in sorting; 0, otherwise, 8i 2 I; j 2 J hi;j;/i;j Start and duration of the green time for traffic movement ði; jÞ at the intersection signal, expressed as a fraction

of the cycle length, 8i 2 I; j 2 J

Hi;j;Ui;j Start and duration of the green time for traffic movement ði; jÞ at the pre-signal, expressed as a fraction of the

cycle length, 8i 2 I; j 2 J

Li Length of the sorting area on arm i 2 I Xi;j;l;m Equals 0 if hi;j 6 hl;m; 1, otherwise 8ðði; jÞ; ðl;mÞÞ 2 W Pl;m;l;n Equals 0 if hl;m 6 hl;n; 1, otherwise 8l 2 I; m;n 2 J; m – n te

i;j

Time needed to fully discharge vehicles following movement ði; jÞ from the sorting area, 8i 2 I; j 2 J l Common flow multiplier

Appendix B. The capacity optimization model under the conventional strategy

In this section, we briefly present the capacity optimization model under the conventional strategy. It is largely based onthe lane-based signal timing and lane allocation model developed by Wong and Wong (2003). The input to this model is thesame as the capacity optimization model introduced in Section 3, that is, the number of arms at the intersection, the numberof approaching and exit lanes on each arm, the demand for each traffic movement, and the cycle length. The model thenoutputs capacity maximizing decisions including lane markings on each approaching lane and the start and duration of greentime for the intersection signal.

Recall that the approaching lanes on arm i are numbered consecutively from the center line to the kerbside as 1, 2, . . . ;Nai .

For the lane-based model, a movement is now defined as ði; j;hÞ for i 2 I; j 2 J, and h 2 f1;2; . . . ;Nai g, representing the traffic

movement that originates from lane h on arm i and heads toward locally numbered arm j. We introduce the followingadditional notation:

Ai

Set f1;2; . . . ;Nai g, indexed by h; 8i 2 I

pi;h

Maximum acceptable degree of saturation for lane h on arm i; 8i 2 I; h 2 Ai

si;j

¼ 1þ 1:5ri;j

, conversion factor of turning flow (measured in pcu’s) to through flow (mesured in tcu’s) at theintersection cross-section, 8i 2 I; j 2 J.

C. Yan et al. / Transportation Research Part B 60 (2014) 85–106 105

The decision variables in this model include:

di;j;h

Equals 1 if movement ði; jÞ is permitted on lane h of arm i; 8i 2 I; j 2 J; h 2 Ai

qi;j;h

Assigned lane flows (in pcu’s) for movements ði; jÞ on lane h of arm i; 8i 2 I; j 2 J; h 2 AiP qi;h ¼ j2Jsi;jqi;j;h, assigned lane flows (in tcu’s) for all movements on lane h of arm i, 8i 2 I; h 2 Ai

hi;j;/i;j

Start and duration of the green time for traffic movement ði; jÞ at the intersection signal, expressed as afraction of the cycle length, 8i 2 I; j 2 J

~Hi;h; ~Ui;h

Start and duration of the green time received on lane h in arm i; 8i 2 I; h 2 Ai expressed as a fraction of thecycle length, 8i 2 I; h 2 Ai

Xi;j;l;m

Equals 0 if hi;j 6 hl;m; 1, otherwise, 8ðði; jÞ; ðl;mÞÞ 2 W zi;h Flow factor for lane h in arm i, defined as zi;h ¼

qi;h

s ; 8i 2 I; h 2 Ai

l

Common flow multiplier

The optimization model is presented below:

max l

subject to

lQ i;j ¼Xh2Ai

qi;j;h; 8i 2 I; j 2 J ðB:1Þ

Xj2J

di;j;h P 1; 8i 2 I; h 2 Ai ðB:2Þ

Xh2Ai

di;j;h 6 NeCði;jÞ; 8i 2 I; j 2 J ðB:3Þ

di;j1 ;h þ di;j2 ;hþ1 6 1; 8i 2 I; j1; j2 2 J; j1 > j2; h 2 Ai n fNai g ðB:4Þ

0 6 hi;j 6 1; 8i 2 I; j 2 J ðB:5Þ

gi;jf 6 /i;j 6 1; 8i 2 I; j 2 J ðB:6Þ

�M 1� di;j;h� �

6 ~Hi;h � hi;j 6 M 1� di;j;h� �

; 8i 2 I; j 2 J; h 2 Ai ðB:7Þ

�M 1� di;j;h

� �6 ~Ui;h � /i;j 6 M 1� di;j;h

� �; 8i 2 I; j 2 J; h 2 Ai ðB:8Þ

Xi;j;l;m þXl;m;i;j ¼ 1; 8ðði; jÞ; ðl;mÞÞ 2 W ðB:9Þ

hi;j þ /i;j þxi;j;l;mf 6 hl;m þXi;j;l;m þMð2� di;j;h � dl;m;nÞ; 8ðði; jÞ; ðl;mÞÞ 2 W ðB:10Þ

�Mð2� di;j;h � di;j;hþ1Þ 6 zi;h � zi;hþ1 6 Mð2� di;j;h � di;j;hþ1Þ; 8i 2 I; j 2 J; h 2 Ai n fNai g ðB:11Þ

zi;h

~Ui;h þ e=c6 pi;h; 8i 2 I; h 2 Ai ðB:12Þ

qi;h ¼Xj2J

si;jqi;j;k; 8i 2 I; h 2 Ai ðB:13Þ

zi;h ¼ qi;h=�s; 8i 2 I; h 2 Ai ðB:14Þ

di;j;h 2 f0;1g; 8i 2 I; j 2 J; h 2 Ai ðB:15Þ

Xi;j;l;m 2 f0;1g; 8ðði; jÞ; ðl;mÞÞ 2 W ðB:16Þ

Constraints (B.1) are flow conservation constraints. Constraints (B.2) make sure that each lane is assigned to at least onetraffic movement. Constraints (B.3) ensure that the number of exit lanes on the destination arm of movement (i; j) is no lessthan the number of approaching lanes permitting this movement. Constraints (B.4) ensure that movements on adjacentlanes are not in conflict of each other. Constraints (B.5) and (B.6) specify the start and duration of the green time at the inter-section. Constraints (B.7) and (B.8) ensure that if an approach lane permits two or more movements through the provision of

106 C. Yan et al. / Transportation Research Part B 60 (2014) 85–106

a shared lane marking, then the involved movements are controlled by an identical signal phase, meaning, those movementsreceive their green for the same time period. Constraints (B.9) and (B.10) ensure that incompatible movements are separatedby a clearance time for safety purposes. Constraints (B.11) ensure that flow factors on adjacent lanes are the same so thattheir degrees of saturation are identical. Constraints (B.12) restrict the degree of saturation on each lane. Constraints(B.13) and (B.14) are the definitions for qi;h and zi;h. Finally, Constraints (B.15) and (B.16) restrict the values of binaryvariables.

References

Allsop, R.E., 1972. Estimating the traffic capacity of a signalized road junction. Transportation Research 6 (3), 245–255.Caliper, 2009. TransModeler User’s Guide. Caliper Corporation.Gallivan, S., Heydecker, B., 1988. Optimising the control performance of traffic signals at a single junction. Transportation Research Part B 22 (5), 357–370.Heydecker, B.G., 1992. Sequencing of traffic signals. In: Griffiths, J.D. (Ed.), Mathematics in Transport and Planning and Control. Clarendon Press, Oxford, pp.

57–67.IBM, 2012. IBM ILOG CPLEX 12.5 User’s Manual. IBM Corporation.Improta, G., Cantarella, G., 1984. Control system design for an individual signalized junction. Transportation Research Part B 18 (2), 147–167.Newell, G., 1989. Theory of highway traffic signals. Tech. rep. UCB-ITS-RR-89-7, Institute of Transportation Studies, University of California, Berkeley.Reid, J., 2004. Unconventional arterial intersection design, management and operations strategies. Tech. rep.Rodegerdts, L., Nevers, B., Robinson, B., Ringert, J., Koonce, P., Bansen, J., Nguyen, T., McGill, J., Stewart, D., Suggett, J., et al., 2004. Signalized intersections:

informational guide. Tech. rep.Silcock, J., 1997. Designing signal-controlled junctions for group-based operation. Transportation Research Part A 31 (2), 157–173.Webster, F., 1958. Traffic signal settings. Road Research Technical Paper (39).Wong, C.K., Heydecker, B.G., 2011. Optimal allocation of turns to lanes at an isolated signal-controlled junction. Transportation Research Part B 45 (4), 667–

681.Wong, C.K., Wong, S.C., 2003. Lane-based optimization of signal timings for isolated junctions. Transportation Research Part B 37 (1), 63–84.Xuan, Y., 2011. Increasing the flow capacity of signalized intersections with pre-signals: theory and case study. Ph.D. thesis, University of California,

Berkeley.Xuan, Y., Cassidy, M., Daganzo, C., 2012. Using a pre-signal to increase bus-and car-carrying capacity at intersections: theory and experiment. In:

Transportation Research Board 91st Annual Meeting. No. 12-1133.Xuan, Y., Daganzo, C.F., Cassidy, M.J., 2011. Increasing the capacity of signalized intersections with separate left turn phases. Transportation Research Part B

45 (5), 769–781.Zhou, Y., Zhuang, H., 2013. The optimization of lane assignment and signal timing at the tandem intersection with pre-signal. Journal of Advanced

Transportation.