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1 This article was downloaded by: [Whan Uniersity] On: 5 Febrary 202, At: 9:06 Pblisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Nmber: Registered office: Mortimer Hose, 37-4 Mortimer Street, London WT 3JH, UK International Jornal of Geographical Information Science Pblication details, inclding instrctions for athors and sbscription information: Reliable shortest path finding in stochastic networks with spatial correlated link trael times Bi Y Chen a b, William H.K. Lam a, Agachai Smalee a & Zhi-lin Li c a Department of Ciil and Strctral Engineering, The Hong Kong Polytechnic Uniersity, Hong Kong, PR China b State Key Laboratory of Information Engineering in Sreying, Mapping and Remote Sensing, Whan Uniersity, Whan, PR China c Department of Land Sreying & Geo-Informatics, The Hong Kong Polytechnic Uniersity, Hong Kong, PR China Aailable online: 25 Oct 20 To cite this article: Bi Y Chen, William H.K. Lam, Agachai Smalee & Zhi-lin Li (202): Reliable shortest path finding in stochastic networks with spatial correlated link trael times, International Jornal of Geographical Information Science, 26:2, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Fll terms and conditions of se: This article may be sed for research, teaching, and priate stdy prposes. Any sbstantial or systematic reprodction, redistribtion, reselling, loan, sb-licensing, systematic spply, or distribtion in any form to anyone is expressly forbidden. The pblisher does not gie any warranty express or implied or make any representation that the contents will be complete or accrate or p to date. The accracy of any instrctions, formlae, and drg doses shold be independently erified with primary sorces. The pblisher shall not be liable for any loss, actions, claims, proceedings,

2 International Jornal of Geographical Information Science Vol. 26, No. 2, Febrary 202, Reliable shortest path finding in stochastic networks with spatial correlated link trael times Bi Y Chen a,b *, William H.K. Lam a, Agachai Smalee a and Zhi-lin Li c a Department of Ciil and Strctral Engineering, The Hong Kong Polytechnic Uniersity, Hong Kong, PR China; b State Key Laboratory of Information Engineering in Sreying, Mapping and Remote Sensing, Whan Uniersity, Whan, PR China; c Department of Land Sreying & Geo-Informatics, The Hong Kong Polytechnic Uniersity, Hong Kong, PR China Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 (Receied 9 Febrary 20; final ersion receied 5 Jne 20) This article proposes an efficient soltion algorithm to aid traelers rote choice decisions in road network with trael time ncertainty, in the context of adanced traeler information systems (ATIS). In this article, the trael time of a link is assmed to be spatially correlated only to the neighboring links within a local impact area. Based on this assmption, the spatially dependent reliable shortest path problem (SD-RSPP) is formlated as a mlticriteria shortest path-finding problem. The dominant conditions for the SD-RSPP are established in this article. A new mlticriteria A algorithm is proposed to sole the SD-RSPP in an eqialent two-leel hierarchical network. A case stdy sing real-world data shows that link trael times are, indeed, only strongly correlated within the local impact areas; and the proposed limited spatial dependence assmption can well approximate path trael time ariance when the size of the impact area is sfficiently large. Comptational reslts demonstrate that the size of the impact area wold hae a significant impact on both accracy and comptational performance of the proposed soltion algorithm. Keywords: reliable shortest path problem; trael time ncertainty; spatial correlation; trael time reliability. Introdction Shortest path problems hae been intensiely stdied owing to their broad applications in arios science and engineering disciplines. In the field of geographical information science (GIS), sbstantial attention has been gien to the deelopment of efficient shortest path algorithms for rote gidance systems (RGS) (Hang et al. 2007, Zeng and Chrch 2009). The ale of RGS is most eident when real-time traffic information, generated by adanced traeler information systems (ATIS), is incorporated. It has been recognized that ATIS-based RGS can not only help traelers to make better rote choice decisions in congested road networks, bt also improe oerall network traffic conditions (Li et al. in press). Most existing RGS assme that link trael times are deterministic. Howeer, link trael times in rban road networks are highly stochastic, de to random traffic demand flctations and capacity degradations (Li et al. in press). In addition, link trael times are *Corresponding athor. chen.biy@gmail.com ISSN print/issn online 202 Taylor & Francis

3 366 B.Y. Chen et al. Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 spatially correlated (Chan et al. 2009). For example, a traffic accident on a major rban road may also case significant trael delays on that road s pstream links. The trael time ariations and spatial correlations hae been measred (in terms of ariance coariance matrices) in ATIS as an important data sorce for trael time estimation and prediction (Tam and Lam 2008, Chan et al. 2009, El Esawey and Sayed 20). Therefore, traelers path trael time is not deterministic bt a random ariable, the distribtion of which is a joint distribtion of all link trael times along that path. Many empirical stdies hae fond that trael time ncertainties hae a significant impact on traelers rote choice behaior (Lam and Small 200, Tam et al. 2008). These stdies reealed that traelers indeed consider trael time ncertainties as a risk, when planning for important eents. Clearly large trael time ariations may case late arrials and the sbseqent imposition of high penalties for traelers (e.g., missed flights). As a reslt, traelers tend to depart from the jorney origin early, so that they can arrie at the destination with a gien on-time arrial probability, termed trael time reliability in the literatre. Hence trael time reliability concerns are necessary inclsions in sophisticated ATIS-based RGS applications. In order to sole the reliable shortest path problem (RSPP), Frank (969) introdced the concept of the most reliable path, in other words, a path that maximizes on-time arrial probability for a gien trael time bdget. As an alternatie, Chen and Ji (2005) proposed the concept of the alpha-shortest path, the aim of which is to minimize the trael time bdget reqired to ensre a gien on-time arrial probability threshold. Based on the aboe two definitions, many soltion algorithms hae been deeloped for soling the RSPP in road networks with stochastic and independent link trael times. Shao et al. (2004) proposed a heristic method for finding the most reliable path, based on the relationship between link trael time mean and standard deiation. Chen and Ji (2005) presented a simlation-based genetic algorithm for finding the most reliable path and the alpha-shortest path. Lim (2008) and Nikoloa (2009) deeloped a parametric approach to determine the most reliable path for risk-aerse traelers. Howeer, this method cannot proide a soltion for risk-seeking traelers, whose trael time bdget is less than the least expected trael time between origin destination (O D) nodes. Nie and W (2009a) proposed a label-correcting algorithm (dynamic programming approach) to find the most reliable path by generating all nondominated paths nder the first-order stochastic dominant (FSD) condition. Howeer, spatial correlations of link trael times were not considered in these stdies. To writers knowledge, the spatially dependent RSPP (SD-RSPP) has not receied mch attention in the literatre. Nie and W (2009b) stdied the SD-RSPP by assming that the probability density fnction (PDF) of the trael time of a link is conditional on the state of traelers arriing at the tail node of the link. The FSD condition and the dynamic programming approach can be employed, based on this assmption, to sole the SD-RSPP. Howeer, it is a formidable task for ATIS to generate sch probabilities for a large nmber of links in real road networks. As an alternatie approach, Ji et al. (20) formlated the spatial correlations as ariance coariance matrices which can be directly obtained from ATIS. Based on this formlation, a simlation-based method was proposed to sole the SD-RSPP. Neertheless, the simlation-based method is comptationally expensie and the precision of reslts is dependent on the maximm nmber of simlations. Using the same formlation of ariance coariance matrix, Nikoloa (2009) proposed a network transformation techniqe to sole the SD-RSPP sing the parametric approach. Howeer, this parametric method cannot proide a soltion for risk-seeking traelers; and sch method

4 International Jornal of Geographical Information Science 367 Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 may enconter infinite negatie cycles in the transformed network de to the negatie trael time coariance. The aim of this article is to inestigate the SD-RSPP in the context of ATIS-based RGS applications, so as to aid arios traelers (inclding risk-aerse, risk-netral, and risk-seeking traelers) make their rote choice decisions nder trael time ncertainties. Ths, the link trael time correlations are represented by ariance coariance matrices. In this article, the trael time of a link is assmed to be spatially correlated only with its neighboring links within a local impact area, in which the topological distance (measred by nmber of links) between any two links is less than or eqal to k. The consideration of this k-limited spatial dependence in the SD-RSPP extends the work of Nie and W (2009b) and Nikoloa (2009), which considers only trael time correlations on the adjacent links (k = ). To some extent, sch k-limited spatial dependence can be interpreted as Tobler s First Law of Geography that all things are related, bt nearby things are more related than distant things (Tobler 970). Empirical stdies based on field obserations also fond trael times, among adjacent links, to be strongly correlated (Gajewski and Rilett 2003). The correlation is sally ery low for links that are spatially distant, een on the same street (El Esawey and Sayed 20). Based on this assmption, an efficient soltion algorithm is proposed in this article to exactly sole the SD-RSPP. The main contribtions of this article are smmarized as follows. () Seeral dominant conditions for the SD-RSPP are established, in which trael time correlations among neighboring links are considered and represented by ariance coariance matrices. These established dominant conditions enable the se of efficient dynamic programming approaches to sole the SD-RSPP. Ths, the established dominant conditions hae important implications regarding the algorithm design. (2) A new mlticriteria A algorithm is proposed to sole the SD-RSPP in an eqialent two-leel hierarchical network. The proposed hierarchical network can well represent trael time correlations among neighboring links and facilitate reliable shortest path findings based on established dominant conditions. Based on this hierarchical network, an efficient mlticriteria A algorithm is proposed to sole exactly the SD-RSPP. The optimality of the proposed algorithm can be formally proed. Therefore, the proposed algorithm has significant adantages oer preios simlation-based method. (3) A comprehensie case stdy sing data from a real-world ATIS in Hong Kong is carried ot. The nmerical reslts show that link trael times are strongly correlated only within local impact areas. The proposed limited spatial dependence assmption can well approximate path trael time standard deiation when the size of impact area is sfficiently large. Comptational reslts demonstrate that the size of the impact area has significant impacts on both the accracy and comptational performance of the proposed soltion algorithm. These key findings from the case stdy hae sefl implications in the deelopment of RGS applications. The rest of this article is organized as follows. The definition of the SD-RSPP sed in this research is presented in Section 2. The dominant conditions and properties of the SD-RSPP are introdced in Section 3. The proposed two-leel hierarchical network is presented in Section 4. The soltion algorithm for soling the SD-RSPP is gien in Section 5. The nmerical examples sing data from a real-world ATIS in Hong Kong are reported

5 368 B.Y. Chen et al. in Section 6. Finally, conclsions and recommendations for frther stdy are gien in Section 7. Downloaded by [Whan Uniersity] at 9:06 5 Febrary Problem statement Let G = (N, A, ) be a directed network consisting of a set of nodes N, a set of links A, and a set of moements. Each link a ij A has a tail node i N, a head node j N, and a random trael time T ij. The mean and standard deiation (SD) of link trael time are denoted by t ij and σ ij, respectiely. Each node i has a set of sccessor nodes SCS(i) = {j : a ij A} and a set of predecessor nodes PDS(i) ={k : a ki A}. The moement ψ ijk = (a ij, a jk ) represents an allowed moement (e.g., throgh-moement or right-trn) at node j. A moement ψ ijk / means that the moement is restricted at node j (e.g., no left-trn or no U-trn). Sppose that the nodes r N and s N represent the O D nodes. Let p rs = {a,..., a m,..., a λ } be a path from the origin r to the destination s, consisting of λ consectie links. The path trael time, denoted by T rs, is the sm of the related link trael times along the path as λ = Tij m () T rs m= where Tij m is the trael time distribtion of a m (the mth link along path p rs ). As preiosly indicated, the trael time of link a ij in this article is assmed ) to be spatially correlated only within a local impact area, denoted by Gij (N k = ij k, Ak ij, k ij.letd qw ij be the topological distance (measred by nmber of links) between links a ij and a qw. A link a qw is said to be a k-neighboring link of link a ij if and only if d qw ij = k. With this concept, the impact area Gij k can be formally defined as a sb-network of G, satisfying d qw ij k, a qw A k ij. In this article, link trael times are also assmed to follow normal distribtions. The normality assmption of link trael times is common in the stdies of stochastic shortest path problems (Chang et al. 2005). Recent empirical stdies based on field obserations fond that the se of normal distribtions appears to reflect obsered trael time distribtions (Rakha et al. 2006), and the normality assmption cold be sfficient from a practical standpoint, gien its comptational simplicity (Lim 2008). Under these two assmptions, the path trael time T rs follows a mltiariate normal distribtion. Its mean and SD, respectiely, denoted by t rs rs and σ, can be calclated as σ rs = λ = tij m (2) t rs λ (σ m ) 2 + m= m= k λ n 2co(a m, a m+n ) (3) n= m= where co(a m, a m+n ) is the trael time coariance between links a m and a m+n. Let rs, (α) be the inerse of the cmlatie distribtion fnction (CDF) of path trael time T rs at α confidence leel. It can be expressed as

6 International Jornal of Geographical Information Science 369 rs, (α) = trs + z ασ rs (4) where z α is the inerse CDF of standard normal distribtion at α confidence leel. The confidence leel α (0, ) is the probability of traelers arriing at the destination within the trael time bdget rs, (α). The on-time arrial probability α represents traelers attitdes toward risks of being late (α > 0.5, α = 0.5, and α < 0.5 for risk-aerse, risk-netral, and risk-seeking attitdes, respectiely). The ale of α can be predetermined based on traelers trip prposes. According to Chen and Ji (2005), the problem of finding the reliable shortest path which minimizes the trael time bdget rs, (α), reqired to ensre a gien on-time arrial probability α, can be formally expressed as the following optimization problem Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 sbject to j SCS(i) x rs ij Min x rs ij T rs k PDS(i) x rs = a ij A rs, (α) (5) { ki = 0, T ij x rs ij (6) i = r i = r ; i = s i = s (7) x rs ij {0, }, a ij A (8) ψ ijk, ψ ijk p rs (9) where x rs ij is the decision ariable regarding the link path incidence relationship; x rs ij = means that the link a ij is on the path p rs, and otherwise xrs ij = 0. Eqation (5) represents the trael time bdget which traelers want to minimize. Eqation (6) defines the path trael time as mentioned in Eqations () (3). Eqation (7) ensres that the reliable shortest path is feasible. Eqation (8) is concerned with the link path incidence ariables which shold be binary in natre. Eqation (9) ensres that all moements along the reliable shortest path are feasible. The aboe formlation of SD-RSPP can be regarded as a generalization of the traditional shortest path problem. The optimal soltion of the SD-RSPP depends on predetermined on-time arrial probability α. For risk-netral scenarios (α = 0.5), Z α = 0 and ths the SD-RSPP becomes the traditional shortest path problem, which is, namely,. For risk-aerse scenarios (α >0.5), rs and σ.forriskseeking scenarios (α <0.5), traelers tend to choose the optimal path with smaller t rs bt to find the path with least mean trael time t rs traelers make rote choice decisions by simltaneosly minimizing t rs larger σ rs. Figre illstrates the aboe concept, by means of a small network. In this figre, all link trael times follow normal distribtions. The mean link trael times are shown on the links while the link trael time ariance and coariance are gien in the matrix. In the ariance coariance matrix, elements along the diagonal are the ariance of link trael times and off-diagonal elements are the trael time coariance between two links. As the matrix is symmetric, only a lower trianglar matrix is shown in the figre.

7 370 B.Y. Chen et al. Figre. An illstratie example. Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 As shown in Figre, three paths, p 5 = a 4 a 45, p 5 2 = a 3 a 35, and p 5 3 = a 2 a 23 a 35, are aailable from Node to Node 5, where is a path concatenation operator (p 5 = a 4 a 45 means that p 5 goes throgh a 4 and a 45 ). When α = 0. (Z α =.28), traelers prefer to take a risk by sing path p 5 3 which has a large trael time ariation so as to assign a small trael time bdget 5,3 (0.) = When α = 0.5 (Z α = 0), risk-netral traelers prefer to choose path p 5 2 with the least mean trael time t5 2 = 7 and set this mean trael time as the time bdget for their trael. When α = 0.9 (Z α =.28), traelers become risk-aerse. They tend to se the more reliable path p 5 with a small trael time standard deiation and assign a larger trael time bdget 5, (0.9) = Therefore, the optimal soltion of the SD-RSPP depends on predetermined traelers attitdes toward risk of being late (α). It can also be obsered from Figre that the SD-RSPP is nonadditie, as the trael time bdget cannot be calclated by the sm of the related link costs. For example, when α = 0.9, the cost of path p 5 = a 4 a 45 is not eqal to the aggregate cost of links a 4 and a 45, ( 5, (0.9) = 0.22 < a 4 (0.9) + a 45 (0.9) = 0.56 ). In this case, the SD-RSPP cannot be soled easily by traditional shortest path algorithms (e.g., Dijkstra s algorithm) which bild on the additie assmption. In following sections, the SD-RSPP is soled sing mlticriteria shortest path approach. 3. Mlticriteria shortest path approach for soling SD-RSPP The SD-RSPP can be formlated and soled as a mlticriteria shortest path problem where the additie mean and ariance of link trael times are considered as two independent ariables. The soltion algorithms, for soling the mlticriteria shortest path problem, typically rely on the dominant conditions to determine a set of Pareto-optimal paths (or nondominated paths) instead of single optimal path. For the SD-RSPP, the nondominated paths can be defined as follows. Definition 3. Gien an on-time arrial probability α and two paths = p P, dominates p (denoted by p ), if and only if pjl(α) < p jl(α), pjl P jl, l N. Definition 3.2 A path P is a nondominated path, if and only if is not dominated by P. Based on the aboe two definitions, it is obios that the reliable shortest path is one of the nondominated paths between O D nodes. The nondominated paths can be fond by

8 International Jornal of Geographical Information Science 37 Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 recrsie path extensions from the origin to the destination sing generalized dynamic programming approach (Carraway et al. 990). Dring this path search process, all dominated paths can be eliminated withot frther consideration, since they cannot be parts of the reliable shortest path between the O D nodes. In the following, dominant conditions to identify dominated paths are introdced, when limited trael time correlations are considered. Let p ij,k ={a,..., a k } be a path from node i to node j consisting of k consectie links, and p,k+λ = p ri,λ p ij,k be a path from origin r to node j going throgh the sb-path p ij,k. The FSD condition for the SD-RSPP can be formally defined as below. Proposition 3.,η+k if p,k+λ Proof See Appendix. (FSD condition). Gien two paths,λ+k and,k+η satisfy (y) < (y) y (0, ). =,η+k P,,λ+k In addition to the FSD condition, following mean ariance (M V) dominant condition exists: Proposition 3.2 two paths,λ+k Proof () t t and Z ασ (2) t < t and Z ασ (M V dominant condition) Gien an on-time arrial probability α and =,η+k P, p,λ+k,η+k if,k+λ and,k+η satisfy either See Appendix. < Z ασ Z ασ or The FSD condition and the M V dominant condition are interrelated. The FSD can identify dominated paths satisfying (y) < (y) y (0, ). These FSD-dominated paths can be discarded withot frther consideration in the path search process regardless of traelers desired on-time arrial probability (α). In contrast to the FSD condition, the M V dominant condition can be adopted only when the on-time arrial probability is pre-gien. For risk-aerse traelers (α > 0.5), M V-dominated paths, satisfying (y) < (y) y [0.5, ), can be identified. For risk-seeking traelers (α <0.5), M V-dominated paths can be determined when (y) < (y) y (0, 0.5] is satisfied. Figre 2 illstrates the aboe two established dominant conditions in a simple network when k = 3 is considered. As shown in this figre, for paths from Node to Node 8 go throgh the same sb-path p 58,3 = a 56 a 67 a 78 with three links. The mean and trael time standard deiation of these for paths are gien in Figre 2a, and the CDF of for path trael time distribtions are illstrated in Figre 2b. It can be obsered from Figre 2b that the path p 8,5 4 is FSD dominated by the path p 8,5 since (y) < (y) y (0, ). p 8 p 8 4 Ths, for all traelers with different on-time arrial probabilities, p 8,5 4 can be discarded in the path search process. The other three paths (p 8,5, p 8,5 2, and p 8,6 3 ) shold be maintained as FSD nondominated paths. If a traeler is risk-aerse, p 8,6 3 can be frther eliminated as an M V-dominated path since (y) < (y) y [0.3, ). Similarly, according to the p 8 p 8 3 M V dominant condition, the path p 8,5 2 can be also determined as the M V-dominated path for a risk-seeking traeler, since (y) < (y) y (0, 0.68]. Therefore, with p 8 p 8 2 the gien on-time arrial probability, the M V dominant condition can help determine potential dominated paths which may not be identified nder the FSD condition.

9 372 B.Y. Chen et al. Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 Figre 2. An illstration of dominant conditions (a) a simple network (b) path trael time distribtions. Another important SD-RSPP property is that the ales of objectie fnction (, (α)) are monotonically increasing dring the path search process as follows: Proposition 3.3, (α) α always holds. Gien two paths and prl = p a jl, the relationship rl, (α) > ( Proof rl, (α), (α) = t jl + Z α (σ ) 2 + 2co(, a jl ) + σ 2 ( ) α 0.5, we hae rl, (α), (α) t jl + Z α (σ σ jl ) 2 σ jl σ ). When = t jl Z α σ jl.de to the nonnegatie property of trael time of link a jl,wehaet jl Z α σ jl > 0 and ths (α) (α) > 0. Similarly, we can proe (α) (α) > 0 when α<0.5. rl,, It is well known that the optimal path in traditional shortest path problems is acyclic withot passing the same node more than once. In the SD-RSPP, the following acyclic property holds. For conenience, a path withot passing the same link twice is hereafter referred to as the acyclic path in the SD-RSPP. Proposition 3.4 Proof link a ij twice. There exists a path p rs Since p rs rl, The reliable shortest path mst not pass the same link more than once. Sppose p rs = pri a ij a ij p js, is the reliable shortest path passing the passing the link a ij only once. (α) > rs, (α) α is not the reliable shortest path according to is the reliable shortest path. = pri a ij p js passes more links in sb-path a ij a ij,wehae rs, according to Proposition 3.3. Therefore, p rs Definition 3., contradicting the assmption that p rs 4. A two-leel hierarchical network In this section, a two-leel hierarchical network is proposed to represent road networks with spatial correlated link trael times. For clarity, the network G presented in Section 2 is hereafter referred to as the primal network. As preiosly indicated, the SD-RSPP can be formlated sing a generalized dynamic programming approach, and a set of nondominated paths is maintained and ealated at the same sb-path with k consectie links in the primal network. This dynamic programming approach, howeer, may not be easily implemented in the primal network, becase the sb-paths with k links are not explicitly represented in this network.

10 International Jornal of Geographical Information Science 373 Figre 3. An illstration of hierarchical network: (a) primal network G, (b) grond hierarchy H g and (c) top hierarchy H t. Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 To facilitate sch a path search approach, a two-leel hierarchical network, denoted by HG = ( H g, H t), is proposed sing k consectie primal links as basis network elements. The proposed hierarchical network has two hierarchies. The grond hierarchy H g consists of N directed-in-trees, where N is the nmber of nodes in the primal network G. For each primal node i G, a directed-in-tree Ḡ g i = ( N i, Ā i )is constrcted rooting at this primal node. In each directed-in-tree Ḡ g i, a grond hierarchical node nij,λ Ḡ g i represents a primal path p ij,λ G with λ consectie primal links (λ k always holds). In this way, all primal paths p ij,λ G with λ links (λ k ) can be represented in the grond hierarchy. Figre 3b illstrates the constrction of the grond hierarchy, when k = 3, from the primal network, shown in Figre 3a. As illstrated in Figre 3b, there are nine directed-intrees constrcted for all nodes in the primal network. In Figre 3b, a node in these trees corresponds to a path in the primal network. For example, node n AB in Figre 3b represents the path p 5,2 = a A a B in the primal network. The nodes in the grond hierarchy can be classified into two categories: border nodes and local nodes. The border nodes for a tree Ḡ g i, denoted by BORDER(Ḡg i ) ={ nij,λ, λ = k }, are defined as those nodes responding to k consectie primal links. The other nodes in the tree Ḡ g i can be defined as local nodes denoted by LOCAL(Ḡ g i ) = N i /BORDER(Ḡ g i ). In the example shown in Figre 3b, nine nodes (denoted in gray) are classified as border nodes, since they correspond to primal paths in Figre 3a with two links (λ = k = 2). These nine border nodes can be pshed into the top hierarchy as nodes in Figre 3c. The other nodes (λ = 0, ) in Figre 3b are classified as local nodes withot consideration in the top hierarchy. ( The top hierarchy H t HG has only one network, denoted by Ĝ t = ˆN, Â, ˆ ). As mentioned, all top hierarchical nodes are the border nodes from the grond hierarchy, denoted as ˆN ={BORDER(Ḡ g i ), Ḡg i }. Each top hierarchical link â iw, Â represents a primal path p iw,k ={a ij,..., ak lw } with k consectie links. For the top hierarchical link â iw,, its tail node n ij,k and head node n iw,k, respectiely, represent the first and last k consectie primal links of p iw,k (i.e., p ij,k and p jw,k ). Each top hierarchical moement ˆψ iq, ={â iw,, â jq, } ˆ corresponds to a primal path p iq,k+ ={a ij,..., ak lw, ak+ wq } passing k + consectie links. Similarly, this top hierarchical moement s tail link â iw, and head link â jq,, respectiely, represent the first and last k consectie primal links of p iq,k+ p iw,k and p jq,k ). Figre 3c depicts the constrction of the top hierarchy, where k = 3, from the same primal network in Figre 3a. As shown in Figre 3c, all nodes of the top hierarchical network are from the border nodes in the grond hierarchy. It can also be fond from Figre 3 that the top hierarchical network consists of links and moements which, respectiely, (i.e.,

11 374 B.Y. Chen et al. represent the paths in Figre 3a with three and for primal links. For instance, a top hierarchical moement ˆψ ABGH ={â ABG, â BGH } in Figre 3c, corresponds to the primal path p 7,4 = a A a B a G a H in Figre 3a. This moement s tail link â ABG represents the primal path p 6,3 = a A a B a G in Figre 3a. It can be obsered from Figre 3 that all acyclic primal paths (k 4) in Figre 3a are represented in the hierarchical network (Figre 3b and c). Accordingly, this hierarchical network can be said to be a k = 3 complete dal network of the primal network in Figre 3a. The concept of the k complete dal network can be formally defined as follows. Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 Definition 4. A two-leel hierarchical network HG = ( H g, H t) is a k complete dal network of the primal network G if and only if (i) any acyclic path p ij,λ G with λ k links has a corresponding node n ij,λ H g ; (ii) any acyclic path p iw,k G with k links has a corresponding link â iw, H t ; and (iii) any acyclic path p iq,k+ G with k + links has a corresponding moement ˆψ iq, H t. In the proposed hierarchical network, trael times and their correlations are stored in both grond and top hierarchies. In the grond hierarchy H g HG, each node n ij,λ maintains a trael time distribtion T ij,λ for its corresponding primal path p ij,λ G. In the top hierarchy H t HG, each link â iw, H t has a link trael time distribtion T âiw,.lettâiw, and ( ) σiw, â 2 be mean and ariance of T âiw,, respectiely. They can be calclated from the primal network as tâiw, = tk (0) ( )2 σiw, â = ( σ k) 2 k + 2co ( a n, a k) () where a n and a k are nth and kth links in the corresponding primal path p iw,k G, respectiely. The trael time coariance between two adjacent top hierarchical links â iw, and â jq, is stored as an attribte of the corresponding top hierarchical moement ˆψ iq, = {â iw,, â jq, }. This trael time coariance co(â iw,, â jq, ) can be expressed as ) co (â iw,, â jq, = co ( a, a k+) (2) n= where a and a k+ are the first and last links of the moement s corresponding primal path p iq,k+ G, respectiely. Let p rs,λ+k ={a ri,..., aλ+k qs } G be a primal path passing λ + k consectie links. Obiosly, the primal path p rs,λ+k contains λ primal sb-paths {p ri,k,..., pjw,k, p lq,k,..., p θs,k }with consectie k links. These λ primal sb-paths correspond to λ top hierarchical links {â ri,,..., âm jw,, âm lq,,..., âλ θs, } in the top hierarchy. Let be the top hierarchical path consisting of these λ top hierarchical links. Its mean and ˆp rs,λ trael time ariance, denoted by t ˆp rs, and ( σ ˆp rs,) 2, respectiely are defined in this article as t ˆp rs, = tri,k + λ m= tâ,m lq, (3)

12 International Jornal of Geographical Information Science 375 ( ) σrs, ˆp 2 = ( σ ri,k ) 2 + λ m= ( σ â,m lq,) 2 + λ m=2 ( ) 2co â m jw,, âm lq, (4) Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 where t ri,k first node n ri,λ along the path ˆp rs,λ and ( ) 2, σ ri,k respectiely, are mean and trael time ariance stored in the of the top hierarchical path ˆp rs,λ ; and â m jw, are two adjacent links and the primal path p rs,λ+k and âm lq,. Using this setting, it can be proed that the top hierarchical path ˆp rs,λ hae an identical trael time distribtion as below. Proposition 4. Gien a top hierarchical path ˆp rs,λ H t, its trael time distribtion is eqialent to that of the corresponding primal path p rs,λ+k G. Proof See Appendix. A simple path in Figre 3 can be sed to illstrate the Proposition 4.. A top hierarchical path ˆp 8,3 =â ABG â BGH â GHI H t in Figre 3c corresponds to a primal path p 8,6 = a A a B a G a H a I G in Figre 3a. The mean trael time of ˆp 8,3 is eqialent to that of p 8,6 as tˆp 8, = t5,2 + (tâ, 6, + tâ,2 27, + tâ,3 58, ) = t A + t B + (t G + t H + t I ) = t 8,5. The paths ˆp 8,3 and p 8,6 also hae the same trael time ariance as follows: ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 σ ˆp 8, = σ 5,2 + σ â, 6, + σ â,2 27, + σ â,3 ) 58, + 2co (â6,, â 27, + 2co ( ) â 7,, â 58, = ( σa 2 + σ B 2 + 2co(a A, a B ) ) + ( σg 2 + 2co(a A, a G ) + 2co(a B, a G ) ) + ( σh 2 + 2co(a B, a H ) + 2co(a G, a H ) + ( σi 2 + 2co(a G, a I ) + 2co(a H, a I ) ) + (2co(a A, a H )) ( + (2co(a B, a I )) = σ 8,5 With Proposition 4., following two important lemmas hold: Lemma 4. the path extension p rq,m+k â lq, in top hierarchy is eqialent to a wq in the primal network, where top hierarchi- G; top hierarchical link G; and primal link a wq G is the last link cal path ˆp rw,m A path extension ˆp rq,m = p rw,m+k 2 ) 2 =ˆp rw,m H t corresponds to the primal path p rw,m+k 2 â lq, H t corresponds to the primal path p lq,k of the primal path p lq,k. Proof It can be easily followed by Proposition 4.. Lemma 4.2 Gien a path p rs,λ G, it can be determined either a node n rs,λ H g or a path ˆp rs,λ k+ H t with the same trael time distribtion as p rs,λ G, if the hierarchical network HG = ( H g, H t) is a k complete dal network of the primal network G. Proof When λ k, according to Definition 4., there exists a node n rs,λ H g representing the path p rs,λ. According to the definition of the hierarchical network, the attribte stored at the node n rs,λ is eqialent to trael time of p rs,λ. When λ>k,p rs,λ contains λ k + sb-paths with consectie k links. According to Definition 4., all these sb-paths, with consectie k links, in the primal network are represented as links in the top hierarchy. Sch top hierarchical links form a path ˆp rs,λ k+. According to Proposition 4., ˆp rs,λ k+ and p rs,λ hae an identical path trael time distribtion.

13 376 B.Y. Chen et al. With the aboe two lemmas, the SD-RSPP in the primal network can be eqally soled in the proposed two-leel hierarchical network sing the generalized dynamic programming approach. As each top hierarchical link represents a primal path with k consectie links, the nondominated paths passing the same k consectie links (according to Propositions 3.2 and 3.3) can be directly maintained and ealated at each top hierarchical link. In addition, as the path extension in the top hierarchy is eqialent to the path extension in the primal network, the monotonic increasing property (Proposition 3.3) and the acyclic property (Proposition 3.4) are also satisfied in the top hierarchical network. These properties contribte to the deelopment of efficient soltion algorithms gien in the next section. Downloaded by [Whan Uniersity] at 9:06 5 Febrary Soltion algorithm This section presents a mlticriteria hierarchical A algorithm, named SDRSP-HA,to sole the SD-RSPP in the proposed hierarchical network. Unless otherwise stated, the hierarchical network HG = ( H g, H t) sed hereafter is a k complete dal network of the primal network. Similar to traditional A algorithm (Zeng ) and Chrch 2009), the SDRSP-HA algorithm ses a heristic alation fnction F (ˆp =, (α) + h(j) as a label for the top hierarchical path ˆ Ht, where h(j) is a trael time bdget estimate from node j G to destination s G, and h(s) = 0 at the destination. By sing this heristic alation fnction, a higher priority can be assigned to the nodes closer to the destination, so as to redce the nmber of examined nodes and speed p the search process. As indicted aboe, the SD-RSPP in the primal network can eqally be soled in the proposed two-leel hierarchical network. Each top hierarchical link â ij H t represents a primal path p ij,k G with k consectie links. Ths, M V nondominated paths = pri p ij,k G passing the same p ij,k can be represented as corresponding top hierarchical paths ˆ =ˆpri â ij H t.letp ={ˆ,..., ˆp } be a set of nondominated paths maintained at the top hierarchical link â ij. The nondominated paths in ˆP are sorted in ascending order by mean trael time t ˆp,. Nondominated paths from all top hierarchical links are maintained in a scan eligible set, denoted by SE ={ˆ,..., ˆprw }. The) nondominated paths in SE are ordered by increasing ale of the heristic fnction, F (ˆp. ) At each iteration, nondominated path ˆ at the top of SE (with minimm F(ˆp )is selected from SE for path extensions. A temporary acyclic path is constrcted by extending the selected path ˆ to its sccessor link â qw H t, denoted by ˆp rw := ˆ â qw. The dominant relationship between the newly generated path ˆp rw and the set of nondominated paths ˆP rw at link â qw is determined sing the M V dominant condition (Proposition 3.2). If ˆp rw is a M V nondominated path at link â qw, it is then inserted into ˆP rw and SE. The newly generated path ˆp rw may also dominate a set of paths in ˆP rw, denoted by ˆP D rw. These dominated paths in ˆP D rw can be eliminated from ˆP rw and SE. The algorithm contines this path search process ntil the destination is reached or SE becomes empty. The steps of SDRSP-HA algorithm are gien below. Algorithm SDRSP-HA Inpts: O D nodes (r, s) and on-time arrial probability α Retrns: the reliable shortest path

14 International Jornal of Geographical Information Science 377 Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 Step. Initialization: For each border node n ri,k Ḡ r (Ḡ r H g denotes the tree rooted at origin r) For each top hierarchical link â ij emanating from node n ri,k ) Generate a new path ˆ := nri,k â ij and calclate h(j) and F (ˆp. Set ˆP :={ˆ } and SE := SE {ˆp }. End for End for If destination s Ḡ r, then insert all paths p rs Ḡ r into SE. Step 2. Path selection: If SE = φ, then Stop; otherwise, contine. Select ˆ at the top of SE and set SE := SE\{ˆp }. If j = s, then Stop; otherwise contine. Step 3. Path extension: For eery moement ˆψ ijw ={â ij,, â qw } (â ij, denotes the last link of ˆ Generate a new path ˆp rw If p rw Call procedre ˆP rw If ˆp rw End for Go to Step 2. := ˆ â qw and calclate h(w) and F(ˆp rw is acyclic, then contine; otherwise, scan next moement. D := CheckDominance ( ˆp rw, ˆP rw ). is a nondominated path, then set SE := SE {ˆprw } and SE := SE\ ˆP D rw. Procedre: CheckDomin ance Inpts: A path ˆ and a set of nondominated path ˆP Retrns: ˆP D storing the set of paths dominated by ˆp, and pdated ˆP Step : Initialization If α>0.5, then β: = If α = 0.5, then β: = 0.5. If α<0.5, then β: = Set ˆP D := φ and : =. Step 2: Dominant relationship determination While ˆP D and t > t ( ˆP D is the nmber of paths in ˆP D ) If, (β) >, (β), then retrn ˆP D. Set : = +. End while If t = t Insert ˆ and, (β) >, (β), then retrn ˆP D. into ˆP at th position and set : = + (by defalt ˆP While ˆP D and, (β) Set ˆP := ˆP \{ˆ } and ˆP D := ˆP Set : = +. End while Retrn ˆP D., (β) D {ˆp }. D ) ). := ˆP D +). The heristic fnction F(ˆ ) is admissible if the following ineqality is satisfied F(ˆ â qw ) = rw, (α) + h(w) F(ˆp ) = (α) + h(j) (5),

15 378 B.Y. Chen et al. Eqation (5) indicates that the heristic fnction ale of F(ˆ ) shold monotonically increase with path extensions. A common admissible h(j) is the Eclidean distance fnction h(j) = e js /V max, where e js is the Eclidean distance from node j to destination s and V max is maximm trael speed (or design speed) of the network. If the heristic fnction is admissible, it can be proed that the SDRSP-HA algorithm can obtain the optimal soltion for the SD-RSPP as follows. Proposition 5. If the heristic fnction sed is admissible, the SDRSP-HA algorithm can determine the reliable shortest path when the destination node is reached. Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 Proof Let P rs G be the set of paths containing all nondominated paths between O D nodes. When destination node was reached, the path ˆp rs is selected from SE. The selected path ˆp rs can be either a node in the grond hierarchy or a path ˆprs Ht in the hierarchical network. As at each iteration the path with minimm F(ˆ ) is selected from SE, the heristic fnction ale of ˆp rs (F(ˆprs )) is the minimm heristic fnction ale among all paths in SE. Since all paths in P rs = P rs \{ˆp rs } are extended from SE and the heristic fnction ale monotonically increase with path extensions, the heristic fnction ale of ˆp rs is less than that of any path in P rs.ash(s) = 0, F(ˆp rs ) = rs, (α)istheminimmtraeltime bdget in P rs and ths ˆp rs is the reliable shortest path between O D nodes. The performance of SDRSP-HA algorithm depends on the qality of h(j) sed. The better the trael time bdget h(j) estimates, the better the comptational performance of the SDRSP-HA algorithm. When h(j) = 0, the SDRSP-HA algorithm redces to the labelsetting algorithm which ses trael time bdget, path ˆ (α) as the heristic fnction ale for instead of F(ˆp ). With the implementation of SE sing F-heap data strctre (Fredman and Taan 987), in worse case the label-setting algorithm reqires O ( ˆ ˆP 2 +  ˆP Log(  ˆP ) where  and ˆ are the nmber of links and moements in the top hierarchical network, and ˆP is maximm nmber of nondominated paths associated with a top hierarchical link. It is clear that this comptational complexity depends on the size of ˆP. Theoretically, ˆP grows exponentially with the network size. In practice, Nie and W (2009b) fond that the nmber of nondominated paths is mch smaller than the maximm possible size, especially for sparse transportation networks. 6. A case stdy in Hong Kong A real-world case stdy, to demonstrate the applicability of the proposed soltion algorithm, is described in this section. In Hong Kong, real-time traffic information on major rban roads is proided by a Real-time Trael Information System (RTIS) ( (Tam and Lam 2008). In the RTIS, radio-freqency identification technology is adopted to collect real-time traffic data. Offline link trael times and ariance coariance matrices, generated by traffic flow simlators (Lam et al. 2002), are also adopted for the RTIS. With the se of real-time and offline traffic data, the RTIS can proide trael time estimates eery 5 mintes for both links, either with or withot real-time data. As shown in Figre 4, the RTIS network consists of 367 nodes, 3655 links, and,849 moements at road intersections. In this article, the RTIS data (inclding mean and ariance coariance matrix of link trael times) were collected at a morning peak hor (08:00 09:00) on 23 September 200 (Thrsday). ),

16 International Jornal of Geographical Information Science 379 Nodes: 367 Links: 3655 Moement:,849 Airport Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 Figre 4. Hong Kong real-time trael information system (RTIS) network. Cross-harbor tnnel Cross bsiness district To normalize the link trael time coariance in the whole network, a correlation coefficient for eery two links, a ij and a qw, was calclated as The ale of ρ qw ij is between and +; ρ qw ij =+ is the case of perfect positie cor- = is the case of perfect negatie correlation. Let NL k ij be the set of relation and ρ qw ij ρ qw ij = co(a ij, a qw ) σ ij σ qw (6) k-neighboring links for link a ij. The mean absolte ale (MAV) of correlation coefficients for all k-neighboring links, denoted by E(ρ k ), can be calclated as abs(p qw ij ) E(ρ k a ij A ) = NL k ij, a qw NL k ij (7) a ij A where abs(p qw ij ) is the absolte ale of ρ qw ij for links a ij and a qw ; and NL k ij is the nmber of k-neighboring links for link a ij.thealeofe(ρ k ) is between 0 and +. The larger the E(ρ k ) ale, the stronger the trael time correlations among k-neighboring links. The E(ρ k ) ale can be adopted as an indicator to measre trael time correlations among k-neighboring links. Figre 5 shows sch E(ρ k ) ales for the RTIS network dring the peak hor. In this figre, the x-axis refers to k ales and the y-axis at the left hand side represents E(ρ k ) ales. It can be seen from Figre 5 that with the increase of the k ale the trael time

17 380 B.Y. Chen et al. Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 Figre 5. Trael time correlations among k-neighboring links. correlations significantly decrease. For instance, when the k ale increases from to 4, E(ρ k ) decreases from 0.29 to 0.04, a redction of abot 86.2%. This obseration is consistent with the empirical findings of preios stdies (Gajewski and Rilett 2003): the trael time correlation is sally ery low for links that are spatially distant. From the aboe obseration, the k-limited spatial dependence assmption, that the trael time of a link correlates only with its neighboring links within a local impact area, seems alid. To jstify this assmption, the approximation accracy of path trael time standard deiation (SD) was examined sing a typical cross-harbor jorney from the central bsiness district (CBD) to Hong Kong International Airport (HKIA). As shown in Figre 4, the rote passing the cross-harbor tnnel (CHT) was selected for this case stdy, becase the CHT was proed to be the most freqently sed tnnel in Hong Kong and also, not srprisingly, had a large trael time ariation. Figre 5 illstrates this SD approximation accracy nder different k ales dring the peak hor (refers to the y-axis at right-hand side). The actal path trael time SD was.64 mintes. It can be fond from Figre 5 that the path trael time SD can be nderestimated by 26.5% when link trael time correlations are ignored (k = 0). This SD approximation accracy can be improed by increasing the k ale. For example, the approximation accracy of path trael time SD can be significantly improed to 87.%, when trael time correlations among adjacent links (k = ) are considered. This SD approximation accracy can be frther improed to 99.% when the k ale increases to 4. Therefore, for this case stdy, the path trael time SD can be well approximated by sing the k-limited spatial dependence assmption with a sfficiently large k ale (e.g., k = 4). Table gies the sizes of the impact area nder different k ales. The impact area size is measred by the aerage nmber of links, denoted by E( A k ij ). It can be calclated as E( A k ij ) = A k ij a ij A A (8) where A k ij is the nmber of links of the impact area Gk ij for a primal link a ij G and A is the nmber of links in the primal network G. It can be obsered from Table that the size of the impact area exponentially increases with k ales. For instance, when k increases

18 International Jornal of Geographical Information Science 38 Table. The sizes of impact area and hierarchical network nder different k ales. k Vale Impact area (links) Top hierarchy Grond hierarchy (nodes) Nodes Links Moements 3.24,367 3,655, ,022 3,655,849 34, ,87,849 34,555 06, ,426 34,555 06,942 35, ,00 06,942 35,72 958,54 Downloaded by [Whan Uniersity] at 9:06 5 Febrary 202 from to 4, the A k ij ale grows by 27 times, from 3.24 to This reslt can be sed to interpret the alidation of the k-limited spatial dependence assmption. When k is large enogh, the impact area can maintain a considerable nmber of links with correlated trael times; and ths the majority of link trael time correlations, along the path, can be captred within that impact area. The proposed two-leel hierarchical network was constrcted sing different k ales (see Table ). It can be seen from Table that the size of constrcted hierarchical network also exponentially increases with k ales, similar to the size of impact area. For instance, when k increases from to 4, the nmber of links in the top hierarchical network grows by 29 times from 3655 to 06,942. This growth rate is close to that of the impact area size (abot 27 times). Therefore, the increase in the k ale can improe the approximation accracy of path trael time distribtions, bt at the cost of increasing the SD-RSPP problem size. It shold be noted that when k = (only correlations among adjacent links are considered), the primal network G can be sed directly for soling the SD-RSPP. The trael time correlations among adjacent links can be maintained as an attribte of each moement in the primal network. The comptational performance of the proposed SDRSP-HA algorithm sing different k ales was tested on the RTIS network. The SDRSP-HA algorithm was coded in Visal C# programming langage. The scan eligible (SE) was implemented sing the F-heap data strctre (Fredman and Taan 987). The h(j) sed was the Eclidean distance fnction. All experiments were condcted in the compter with a for-core Intel Xeon 3.2 GHz CPU (only one core was sed) and a Windows Serer 2003 operation system. Table 2 reports the comptational performance of the proposed SDRSP-HA algorithm sing different k ales. Three risk-taking scenarios, inclding risk-aerse (α = 0.9), risk-seeking (α = 0.), and risk-netral (α = 0.5), were tested for each k ale. All reported reslts were the aerage of 00 compter rns, sing different O D nodes in each rn. The 00 O D nodes were randomly selected and the same O D node set was sed for eery test performed on all hierarchical networks. Table 2 gies the comptational performance of the SDRSP-HA algorithm nder different k ales. As expected, the comptational time of the SDRSP-HA algorithm exponentially increases with respect to the k ales. It can be also obsered from Table 2 that the SDRSP-HA algorithm rns mch faster at the risk-netral scenario than at the other two risk-taking scenarios. For example, when k = 4, the comptational time reqired by a risk-aerse scenario is abot 58 times (79.085/.365) larger than that reqired by the risk-netral scenario. This is becase when traelers are risk-netral, the SDRSP-HA algorithm becomes, essentially, a traditional A algorithm. In this case, the least nmber of nondominated paths is generated in the search process, since only a single path is kept at