research article floyd-a algorithm solving the least...

Research ArticleFloyd-Alowast Algorithm Solving the Least-Time Itinerary PlanningProblem in Urban Scheduled Public Transport Network

Yu Zhang1 Jiafu Tang12 Shimeng Lv1 and Xinggang Luo1

1 Department of Systems Engineering State Key Lab of Synthetic Automation of Process Industries Northeastern UniversityShenyang 110004 China

2 College of Management Science and Engineering Dongbei University of Finance and Economics (DUFE)Shahekou Dalian 116025 China

Correspondence should be addressed to Jiafu Tang jftangmailneueducn

Received 26 October 2013 Revised 5 March 2014 Accepted 10 March 2014 Published 27 April 2014

Academic Editor Cristian Toma

Copyright copy 2014 Yu Zhang et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider an ad hoc Floyd-Alowast algorithm to determine the a priori least-time itinerary from an origin to a destination givenan initial time in an urban scheduled public transport (USPT) network The network is bimodal (ie USPT lines and walking)and time dependent The modified USPT network model results in more reasonable itinerary results An itinerary is connectedthrough a sequence of time-label arcs The proposed Floyd-Alowast algorithm is composed of two procedures designated as ItineraryFinder and Cost Estimator The Alowast-based Itinerary Finder determines the time-dependent least-time itinerary in real time aidedby the heuristic information precomputed by the Floyd-based Cost Estimator where a strategy is formed to preestimate the time-dependent arc travel time as an associated static lower bound The Floyd-Alowast algorithm is proven to guarantee optimality in theoryand demonstrated through a real-world example in Shenyang City USPT network to be more efficient than previous proceduresThe computational experiments also reveal the time-dependent nature of the least-time itinerary In the premise that lines runpunctually ldquojust boardingrdquo and ldquojust missingrdquo cases are identified

1 Introduction

When a traveler plans to travel from one place (origin) toanother (destination) beginning at a given initial time (orimposed by a deadline) in a real-world urban scheduled pub-lic transport (USPT) network it can be difficult to determinethe a priori least-time (LT) itinerary Typically speaking theitinerary should specify the USPT services (eg metro linesbus lines) that combine the vehicle trips to take the roads towalk and the stops at which to transfer in order to arrive ata destination consuming the least time The aforementionedleast-time itinerary planning problem in an urban scheduledpublic transport network (LTIP-USPT) is a common decisionproblem for travelers in a city travelling from an origin to adestination tomake a date attend a conference participate ina party and most often to go to work

Unlike the well-studied shortest path problems in staticnetworks LTIP-USPT is more difficult to address concerning

the following reasons (i) the topological structures andtimetables of the public transport services in a city areprescheduled (ii) One or more transfers are unavoidable inmost cases resulting in walking time and waiting time aspenalties (iii) In addition to spatial connectivity and dis-tance travelers must also consider temporal and operationalfactors In a word public-transport travelers as opposed toprivate-vehicle travelers cannot wander in a USPT networkfreely without any constraints (formore detailed explanationplease see [1]) Moreover in real-world applications the solu-tion procedure should be as fast as possible It is a challengingtask because in the time-dependent USPT network variousnodes and lines are interconnected thus leading to numerouscombinations of lines vehicle trips walks and transfersmaking a difficult combination optimization problem

The headway-based public transport services were con-sidered in early researches (see [2ndash4]) Recent years wit-ness a boom in the development of schedule-based public

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 185383 15 pageshttpdxdoiorg1011552014185383

2 Mathematical Problems in Engineering

transport Correspondingly in the transportation researchcommunity the focuses have shifted from headway-basedservices to schedule-based ones The main difference comesfrom the evaluating of each transferwaiting timeThe transferwaiting time is typically assumed to be half of the headwayconcerning headway-based services However for schedule-based services it could be precisely evaluated dependingon the combined timetables Thus the network travel times(especially transfer waiting times) shift from deterministicto time dependent resulting in the needs of methodologicalchanges for many transportation problems for exampleitinerary planning (see [5ndash8]) and traffic assignment (see [9ndash11]) And these methodologies appear more promising andcompeting against those for the traditional headway-basedservices Note that in reality the ldquoschedule-based linesrdquo are ageneralized concept that might include bus lines and metrolines In this sense the bimodal (USPT lines and walking)network considered in this paper could also be interpreted asa multimodal one which is a typical urban public transportnetwork in real world

There are numerous related works in the existing litera-tures Tong and Richardson [12] were among the first to studythe scheduled public transport they developed a branch-and-bound type algorithm for the minimum itinerary Alongthis vein research community extended the itinerary plan-ning problems by introducing different real-world consider-ations Horn introduced multimodal transport services anda Dijkstra-based algorithm was considered for minimizinggeneralized travel costs [13] Tan et al [7] developed a recur-sive algorithm for finding reasonable paths that satisfy thedefined acceptable time criterion and transfer-walk criterionwhere travelersrsquo preferences were required to give Not onlythe LT itinerary but also the k-LT itinerary need to be deter-mined for travelers Accordingly Xu et al [6] and Canca etal [5] studied the k-shortest path problems in schedule-basedtransit networks Androutsopoulos and Zografos considereda dynamic programming based algorithm [14 15] for theitinerary planning problem in the context where travelerimposes time windows on nodes multiobjective was alsoconsidered These works have significantly contributed toreal-world derived itinerary planning problems for differenttypes of travelersrsquo requirements However their focuses weremainly on the considerations of real-world factors Withregard to the algorithmic efficiency they only cared aboutwhether a query could be completed within a short timeIn query-intensive scenarios the query should be as fastas possible which motivated researchers in recent years toexamine heuristic methods to speed up computing

Fu et al reviewed the heuristic shortest path algorithmsfor transportation applications [16] noting that Alowast-basedalgorithms were widely usedThe heuristic Alowast algorithm wasfirst proposed by Hart et al [17] typically for the shortestpath problems in static networks The performance of Alowastalgorithm depends primarily on the strategy of estimatingthe travel time of a partial path A well-designed strategyleads to a considerable savings of computational time whileassuring an optimal solution Chabini and Lan adapted Alowast

for the fastest path in deterministic discrete-time dynamicnetworks [18] they proposed three estimating strategies Sub-sequently numerous works appeared in the literatures thatprimarily applied Alowast-based algorithms to LT path problemsin unscheduled time-dependent road networks (see [19 20])Chen et al proposed an Alowast-based integrated approach com-bining offline precomputation and online path retrieval for aroad navigation system [21] This work introduced precom-putation (see [22 23]) technology and significantly improvedthe efficiency However determining the LT itinerary in ascheduled network is more challenging than that of unsched-uled network But relatively little related works were found

In fact two notable works had studied the itineraryplanning problem in scheduled public transport networkespecially focusing on the speed-up technologies [24 25]They showed that the time-dependent model is superior tothe time-expanded model in the sense of itinerary-findingefficiency Alowast-based and other strategies are demonstrated tobe capable of speeding up computing Following the previousworks this paper goes beyond in the following aspects(i) modelling a modified bimodal (ie USPT services andwalking) time-dependent USPT network This model isobserved to be more applicable in that the results over themodified network model intrinsically have smaller numberof transfer times (ii) An ad hoc Floyd-Alowast algorithm isdeveloped to solve the LTIP-USPT where transit vehicles areassumed to run punctually A novel approach to estimatingtravel time of the partial itinerary is embodied in Floyd-AlowastTo implement the approach we generate a slacked networkand let the arc travel time be a static tight lower bound of theassociated real time-dependent arc travel time Precomputingtechnology is also usedThe algorithm is proven to be optimalin theory and was demonstrated with a real-world example tobe very applicable The Floyd-Alowast procedure outperforms theprevious procedures It reduces the averaged computationaltime by 639 compared with a conventional Dijkstra-like procedure (iii) From the management perspective anillustrated example reveals the time-dependent nature of theleast-time itinerary In the premise that lines run punctuallythe solution aids travelers in avoiding ldquojust missingrdquo cases

The remainder of this paper is organized as followsSection 2 formulates the modified USPT network modeland itinerary structure develops the formula for each time-dependent time-label arc and subsequently formulates theLTIP-USPT followed by hypotheses In Section 3 we proposean ad hoc Floyd-Alowast algorithm composed of two proceduresthat is Floyd-based Cost Estimator and Alowast-based ItineraryFinder The Cost Estimator precomputes the estimated traveltime of destination-ended partial itineraries as heuristicinformation The Itinerary Finder heuristically determinesthe LT itinerary in real-time Floyd-Alowast is mathematicallyproven to be admissible and efficient Furthermore an illus-trated example is presented in Section 4 that reveals the time-dependent nature of an LT itinerary and provides guidancefor travelers in determining the initial time to begin travelMeanwhile through both a numerical example and a real-world case the Floyd-Alowast procedure is proven to be moreefficient than two other conventional procedures that is

Mathematical Problems in Engineering 3

Table 1 A timetable example

119896

119899

1198971

1198972

1198973

1198974

1198975

1198991

1198992

1198993

1198994

1198991

1198992

1198994

1198995

1198996

1198997

1198991

1198997

1 602 614 625 605 618 630 605 615 602 616 600 6122 608 621 631 615 629 640 610 620 610 623 603 6153 613 625 636 625 639 651 615 624 618 632 606 6174 618 629 639 633 646 658 620 630 626 640 609 6205 622 634 644 641 654 707 623 633 630 645 612 624

Dijkstra-like and Plain-Alowast procedures Finally concludingremarks and future works are discussed in Section 5

2 Formulation of the Least-Time ItineraryPlanning Problem in Urban ScheduledPublic Transport Network (LTIP-USPT)

In modeling the urban scheduled public transport (USPT)network both USPT lines and walking modes should beconsidered An itinerary should encompass spatial temporaland operational features In an USPT network a passengerwho plans to travel from an origin to a destination at a giveninitial time will select the USPT lines to take the roads towalk and the stops at which to transfer in order to arriveat their destination as quickly as possible We identify thisproblem hereafter as LTIP-USPT To solve a LTIP-USPT achallenge is to model a more applicable USPT network anda reasonable itinerary structure these are developed in Sec-tions 21 and 22 respectively In Section 23 we formulate theLTIP-USPT as the least-time itinerary planning problem in adeterministic bimodel time-dependent scheduled network

21 A Modified Scheduled Network Model IntuitivelyFigure 1(a) shows an example of a physical USPT networkand the associated modified network model 119866 = (119873119860 119871)

is shown in Figure 1(b) The advantage of this model beyondprevious scheduled networkmodel can be found in Remark 1Table 1 gives an associated timetable example

Let 119897 = (119873119897 119860119897 119879119897) isin 119871 denote the scheduled directed

USPT line that operates on the USPT network The USPTline typically refers to (but not constrained to) the bus lineor metro line that runs on fixed road and runs through apredetermined serious of nodes 119873

119897based on a timetable 119879

119897

There are numerous vehicle trips within a single day 120579n119897119896

element of 119879119897represents the scheduled time when the 119896th

vehicle trip of 119897 arrives or departs at node 119899 isin 119873119897 Table 1

shows a timetable example where for example 120579119899211989712= 6 21

Any move along a specific line is not necessarily betweentwo adjacent nodes but may pass through one or more inter-mediate nodes Correspondingly 1198622

|119873119897|arcs can be generated

by line 119897 The set of these arcs is formulated as119860119897= (119899119894 119899119895)119897|

0 lt 120588

119899119894

119897lt 120588

119899119895

119897le |119873119897| where 120588119899

119897denotes the sequence number

that line 119897 passes through node 119899 Let 120588119899119897= 0 if 119897 does not pass

through 119899 Obviously for any node 119899 120588119899119897isin 0 1 2 |119873

119897|

For example 1198971of Figure 1 associates with the set of arcs

1198601198971

= (1198991 1198992)1198971

(1198991 1198993)1198971

(1198992 1198993)1198971

In this model it shouldbe noted that only one arc in 119860

119897 rather than two or more

connected arcs is traversed from the travelerrsquos boarding toalighting a vehicle of line 119897

Remark 1 One of the challenges of this problem is theexistence of multiple solutions and it is fairly easily thatan algorithm is trapped into some poor local minimumsThis Remark elaborates this phenomenon and the solutionmethod In previous related works the arcs of a USPT linetypically only exist between the adjacent nodes that is 119860

119897=

(119899119894 119899119895)119897| 0 lt 120588

119899119894

119897= 120588

119899119895

119897minus 1 lt |119873

119897| In practice the

disadvantage is shown as follows by an example Considerthe two USPT network models in Figure 2 where Figure 2(a)shows a previous network model and Figure 2(b) a modifiedone A traveler goes from 119899

1to 1198993starting at 900 There are

the following two alternative itineraries

Itinerary 1 Wait 5 minutes and start at origin 1198991 traveling by

1198972directly to the destination 119899

3

Itinerary 2 First travel by line 1198971to 1198992 and wait 5 minutes and

then travel by 1198972to 1198994

They both arrive at destination at 950 so they are boththe least-time itineraries in theory But real-world travelertypically prefers Itinerary 1 because Itinerary 2 containsa transfer activity In the model of Figure 2(a) a label-setting algorithm (see Dijkstra 1959) will obviously chooseItinerary 2 because 925 is earlier than 930 regarding node1198992 that is to say arc(119899

1 1198992)1198971

dominates arc(1198991 1198992)1198972

Incomparison executing a label-setting algorithm in the modelof Figure 2(b) will lead to the choice of Itinerary 1 In factonce the arc(119899

1 1198993)1198972

is searched it will never be dominatedbecause there is no way to reach 119899

3earlier than 950 It

is observed that the aforementioned phenomenon existscomprehensively in our experiments The results over themodified network model intrinsically have smaller numberof transfer times thus it is more applicable

In most real-world cases the destination node 119899119889cannot

be reached by using only one line from the origin node 119899119900

so transfer is necessary Transferring does not always occurat just one node a traveler may have to walk some distance


l2

l3

l1

l5

l4n7

n4

n1 n2

n6

n5n3

USPT serviceStopRoad

(a)

l2

l2

l2

l3

l1

l1

l1

l5

l4n7

n4

n1 n2

n6

n5n3

NodeArc

ww

w

w

(b)

Figure 1 A USPT network example

900

905

925

930 950

l1

l2 l2n1

n2 n3

(a)

900

905

925

930 950

l1

l2l2

l2n1n2 n3

(b)

Figure 2 USPT network example

to another node in order to transfer The tolerable walkingdistance is constrained by a constant upper bound 119863 Thusthe set of walking arcs is formulated as 119860

119908= (119899

119894 119899119895)119908

|

119899119894= 119899119895 dist(119899

119894 119899119895) lt 119863 Using the USPT network of Figure 1

as an example (1198993 1198995)119908isin 119860119908while (119899

1 1198992)119908notin 119860119908because

dist(1198991 1198992) gt 119863 For denotation convenience let USPT

service s denote either a line 119897 or the walk 119908 that is 119904 isin

119871⋃119908In summary with regard to USPT network 119866 = (119873119860

119871) 119873 = ⋃119897isin119871

119873119897 and 119860 = ⋃

119897isin119871119860119897⋃119860119908 In general a

node 119899 isin 119873 in this network represents a bus stop or metrostation An arc(119899

119894 119899119895)119904isin 119860 shows an available move In any

given specific arc the arc travel time does not always remainconstant which is actually dependent on the initial start timeThis makes the USPT network a single-layer bimodal andtime-dependent network

22 Itinerary Structure and Timing The Itinerary is repre-sented as a sequence of orderly arcs or nodes in a staticnetwork though there must be some adaptation in a USPTnetwork context An arc could not describe the temporalfactor so we define the time-label arc in Definition 2 Anitinerary in a USPT network could be represented as asequence of time-label arcs

Definition 2 A time-label arc (t-arc for short) is defined as a 4-tuple (119899

119894 119899119895 119904119894 119905119894) link representing a passengerrsquos move from

a tail node 119899119894to a head node 119899

119895by means of a transport service

119904119894at a given initial time 119905

119894 This representation is legitimate

if and only if there exists an available transport service 119904119894isin

119871⋃119908 for a passenger who is located at node 119899119894isin 119873 at time

119905119894(maybewith somewaiting time) tomove towards node 119899

119895isin

119873 Using Figure 1 as an example 119905-arc(1198991 1198992 1198971 10) denotes

that a passenger arrives at node 1198991at the initial time 71000

and travels to node 1198992by line 119897

1 In addition subscripts of the

service 119904119894and the initial time 119905

119894are kept consistent with the

tail node 119899119894 If necessary superscripts are used to distinguish

the different services and initial times

When a passenger travels from 119899119900to 119899119889at a given initial

time 119905119900 there can be numerous eligible itineraries The set of

these itineraries is denoted by119875119905119900(119899119900 119899119889) whose elements can

be represented by a sequence of connective 119905-arcs shown in

119901119905119900(119899119900 119899119889) = (119899

119900 1198992 119904119900 119905119900) (1198992 1198993 1199042 1199052)

(119899|119901| 119899119889 119904|119901| 119905|119901|)

(1)

where |119901| is the number of 119905-arcs that compose 119901119905119900(119899119900 119899119889)

A passenger may be concerned about the total travel timeof itinerary 119901119905119900(119899

119900 119899119889) which is the accumulated travel time

of each component 119905-arc The elapsed travel time thereforeacts as the cost (weight) of each 119905-arc There are threecomponents of travel time as follows

(1) in-vehicle timemdashelapsed during vehicular travel onthe line

(2) walking timemdashelapsed during walking between twonodes for transfer purposes

(3) waiting timemdashelapsed at node waiting for the arrivingtransfer vehicle

Let120587 be an operator that times each 119905-arc or itineraryThecomputing method to time 119905-arc (119899

119894 119899119895 119904119894 119905119894) depends on the


associated transport service 119904119894 If 119904119894= 119908 the 119905-arc is traversed

by walking It calculates the fixed walking time cost as shownin

120587 (119899119894 119899119895 119908 119905119894) =

dist (119899119894 119899119895)

Vwalk (2)

The arrival time at 119899119895is then easily calculated in

119905119895= 119905119894+ 120587 (119899

119894 119899119895 119908 119905119894) (3)

If 119904119894= 119897 the 119905-arc is traversed by line 119897 Both the in-vehicle

time and the waiting time must be considered Thereforethe associated travel time is not fixed but time dependent ascalculated in

120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

119897119896minus 119905119894 (4)

Because the passenger will board the first arriving vehicleof the transferred line 119897 119896 of Formula (4) is determinedby Formula (5) the waiting time and the in-vehicle timein this process are 120579119899119894

119897119896minus 119905119894and 120579

119899119895

119897119896minus 120579

119899119894

119897119896 respectively The

corresponding arrival time at 119899119895is calculated with Formula

(6)

119896 = arg min119896

(120579

119899119894

119897119896minus 119905119894| 120579

119899119894

119897119896minus 119905119894gt 0) (5)

119905119895= 120579

119899119895

119897119896 (6)

In any specific USPT network once the initial time 119905119894of

each 119905-arc (119899119894 119899119895 119904119894 119905119894) is known the travel time of this 119905-arc

120587(119899119894 119899119895 119904119894 119905119894) and the associated arrival time 119905

119895can be easily

calculated With respect to any itinerary 119901119905119900(119899119900 119899119889) the first

initial time 119905119900is predetermined by the passenger and the

subsequent times can be calculated recursively by Formula(3) or (6) In other words the initial time of a specific 119905-arcis equal to the arrival time of the upstream 119905-arc In this casethe travel time of the itinerary formulated in Expression (1)can be written as

120587 (119901119905119900(119899119900 119899119889)) = 120587 (119899

119900 1198992 119904119900 119905119900) + 120587 (119899

2 1198993 1199042 1199052)

+ sdot sdot sdot + 120587 (119899|119901| 119899119889 119904|119901| 119905|119901|)

(7)

Alongwith spatial and temporal features practical opera-bility should also be considered from the passengerrsquos perspec-tive Some properties of the itinerary that describe operabilityare given below

Property 1 Two 119905-arcs that are traversed by walking cannotbe adjacent due to the hypothesis that a walking distancebetween two nodes cannot be larger than 119863 In other wordswhen 119904

119894= 119908 we have 119904

119894+1= 119908 where 119894 = 119900 2 |119901| minus 1

Property 2 During the travel process if a line has beenalready used as a transport service a passenger will not likelyreuse this line or its inverted line (see Definition 3) in hishersubsequent travel process In other words when 119904

119894= 119908 we

have 119904119895

= 119904119894and 119904119895

= 119904119894 where 119894 119895 = 119900 2 3 |119901| and 119894 = 119895

Property 3 In reality a passenger is not likely to travel anitinerary that goes through a specific node twice Thereforewe have 119899

119894= 119899119895 where 119894 119895 = 119900 2 3 |119901| 119889 and 119894 = 119895

Definition 3 With regard to a specific line 119897 there usuallyexists an inverted line 119897 that runs on almost the same roadsegments of 119897 but in inverted directions 119897 is also the invertedline of 119897 that is 119897 = 119897 Intuitively 119897

3is the inverted line of 119897

4in

the USPT network shown in Figure 1 (1198973= 1198974)

Take the USPT network of Figure 1 as an example whoseassociated timetables are provided in Table 1 A passengerarrives at 119899

2at 610 waits for 4minutes takes the first available

vehicle trip of 1198971towards 119899

3 arrives at 625 walks to 119899

6

using 180m15ms = 2(min) waits for 3 minutes boardsthe vehicle on the 5th trip of 119897

4at 630 and finally arrives

at 1198997at 645 This itinerary is represented by 11990110(119899

2 1198997) =

(1198992 1198993 1198971 6 10) (119899

3 1198996 119908 6 25) (119899

6 1198997 1198974 6 27)

and consumes 35 minutes in total thus 120587(119901610(1198992 1198997)) =

35(min)

23 Problem Formulation In any specific USPT network apassenger decides to travel from an origin 119899

119900to a destination

119899119889 at an initial time 119905

119900 The problem is determining a

connected itinerary among the large volume of available choi-ces that requires a minimum of travel time This can bemathematically formulated as follows

min 120587 (119901119905119900(119899119900 119899119889))

st 119901119905119900(119899119900 119899119889) isin 119875119905119900(119899119900 119899119889)

(8)

The travel time of the 119905-arc traversed by walking is fixedwhile that traversed by a line is time dependent thusleading to a time-dependent rather than static USPT net-work The LTIP-USPT pertains to the least-time itineraryplanning problem in a bimodal time-dependent schedulednetwork The traditional shortest path algorithms do notapply Through the adaptation of the Alowast algorithm Section 3develops an ad hoc Floyd-Alowast algorithm to address the LTIP-USPT The following hypotheses are assumed and summa-rized as follows

(1) Line vehicles run punctually(2) The vehicle capacities are infinite(3) The road network is noncongested(4) The vehicle departs immediately after arriving at a

specific node(5) The origins and destinations are all located just at

nodes(6) One walking distance cannot be greater than the

tolerable upper bound119863

The findings of this research can be widely used they canassist passengers in arranging their travel and be integratedinto traffic assignment models They can also verify theaccessibility of a USPT network and help in the design oftimetables contributing both theoretically and practically


3 Floyd-Alowast Algorithm for LTIP-USPT

To solve the LTIP-USPT efficiently an ad hoc Floyd-Alowast algo-rithm is developed that is composed of two procedures thatis an Alowast-based Itinerary Finder and a Floyd-based CostEstimator The basic scheme of the Floyd-Alowast algorithm isshown in Figure 3

The Cost Estimator precalculates the estimated traveltimes of itineraries between any two nodes in a slacked USPTnetwork where static arc travel time is given as the lowerbound of the associated time-dependent actual travel timeThese values are stored in Table H This is accomplishedby a Floyd-based algorithm [26] which is a well-knownall-to-all shortest paths algorithm Once complete the CostEstimator is no longer required unless there is an updateto the USPT network The Alowast-based Itinerary Finder makesuse of the Table H obtained by the Cost Estimator asheuristic information determining the least-time itineraryIn the case that traveler inputs a triad of (119899

119900 119899119889 119905119900) only the

Itinerary Finder conducts a real-time computationThese twoprocedures are expounded in detail in Sections 31 and 32respectively Section 33mathematically proves its admissibil-ity and analyzes the corresponding computing efficiency bycomparing it with Plain-Alowast and Dijkstra-like procedures

Remark 4 Speed-up technologies such as ldquoAvoiding BinarySearchrdquo and ldquoFurther Speedup When Modeling with TrainRoutesrdquo discussed in the work of Pyrga et al [25] may furthercontribute to a higher efficiency However this paper onlyconcerns a more efficient Alowast-based search (also known asgoal-directed search) which could coexist with other speed-up technologies to further speed up computing

31 Least-Time Itinerary-Finder Procedure Assuming thattypical readers may not be familiar with the Alowast algorithmthis searching processwill be explained in detail Given a triadof origin destination and initial time (119899

119900 119899119889 119905119900) to determine

an LT itinerary 119901119905119900(119899119900 119899119889) isin 119875119905119900(119899119900 119899119889) the Itinerary Finder

expands promising origin-rooted partial itineraries (partialitinerary for short) in a node-to-node manner Beginningwith 119899

119900 each successor 119899

119900+is expanded by searching for

each 119905-arc (119899119900 119899119900+ 119904119900 119905119900) in the first round Each of these 119905-

arcs (partial itineraries) may contribute to the LT itineraryDuring the second round wemust determinewhich terminalnode of partial itinerary among several candidates is themostpromising one

Let each node 119899119894be associated with a state denoted by

state(119899119894) There are three states of node 119899

119894

(1) NEW node 119899119894has not been expanded up to now

(2) OPEN node 119899119894has been expanded and acts as a

candidate to expand to another node in the nextsearching process That is to say for each partialitinerary 119901119905119900(119899

119900 119899119894) thus far state(119899

119894) = OPEN

(3) CLOSED node 119899119894has been expanded and has already

expanding to another node In other words for anynode 119899

119895= 119899119894that has gone through by any current

partial itinerary 119901119905119900(119899119900 119899119894) state(119899

119895) = CLOSED

Procedure of Itinerary

Least-time itinerary

Output

Input

Inquiry

Output

Procedure Cost Estimator(Floyd-based)

Table H

(Real-time computation)(Precalculation and storage)

The slacked network data

Input

Return hrsquo(n)

User input no nd 120591d

(Alowast-based)Finder

Figure 3 Scheme of Floyd-Alowast procedure

As defined above the nodes associated with the stateOPEN are candidates for expanding partial itineraries Forconvenience we use relative time rather than absolute timehereafter Using the USPT network of Figure 1 as an examplelet 119899119900= 1198991 119899119889= 1198995and 119905119900= 2 (minutes after 600) Figure 4

combined with Table 2 shows part of the searching processWefirst initialize the state of origin 119899

1asOPENandothers

as NEW by default (see Figure 4(a)) In the first expansionround (see Figure 4(b)) 119899

2 1198993 and 119899

7are expanded by

searching for 119905-arcs (1198991 1198992 1198971 2) (119899

1 1198992 1198972 2) (119899

1 1198993 1198971 2)

and (1198991 1198997 1198975 2) At the same time 119899

1becomes CLOSED 119899

2

1198993 and 119899

7turn from NEW to OPEN The next paragraph

shows that (1198991 1198997 1198975 2) is the most promising partial

itinerary and 1199057= 15 thus we should continue the second

expansion round for themost promising node 1198997 and only 119899

4

is expanded by searching for a 119905-arc (1198997 1198994 119908 15) This time

1198997becomes CLOSED and 119899

4turns toOPEN (see Figure 4(c))

The searching process continues by similar means Note thatthe state of a node may turn from NEW to OPEN fromOPEN to CLOSED or remain the same However a CLOSEDnode can never re-OPEN (see Theorem 10) for example(1198994 1198991 1198972 16) is searched in the 3rd searching round (see

Figure 4(d)) but the state of 1198991unconditionally remains

CLOSEDThe exposition above focuses on the changing states

of nodes during the expansion of partial itineraries Todetermine the most promising OPEN node among severalcandidates 1198911015840(119899

119894) is defined as the estimated travel time

of an LT itinerary 119901119905119900(119899119900 119899119894 119899119889) For each partial itinerary

119901119905119900(119899119900 119899119894) the terminal node(s) 119899

119894whose 119891

1015840

(119899119894) isare the

minimum one(s) among those of all OPEN nodes isareidentified as the most promising one(s) If there is more thanone you may choose the first expanded one

The actual travel time of the LT itinerary119901119905119900(119899119900 119899119894 119899119889) can

be the summation of two parts calculated as

120587 (119901119905119900(119899119900 119899119894 119899119889)) = 120587 (119901

119905119900(119899119900 119899119894)) + 120587 (119901

119905119894(119899119894 119899119889))

(9)

However it is difficult to calculate 120587(119901119905119900(119899119900 119899119894)) and

120587(119901119905119894(119899119894 119899119889)) in real-time within an acceptable computing

time Because of the time-dependence factor they are notable to be precalculated and stored as fixed values This is a


Table 2 The changing labels of nodes associated with Figure 4

Node Labels119878119905119886119905119890(119899) 119905

119894= 1198921015840

(119899) + 119905119900

1198921015840

(119899) ℎ1015840

(119899) 1198911015840

(119899) 119875119903119890(119899) Updated

Initialization

1198991

OPEN 2 0 21 21 nil lowastradic

1198992

NEW infin infin infin infin nil1198993





NEW infin infin infin infin nil

1st searching round

1198991

CLOSED 2 0 21 21 nil lowast

1198992

OPEN min14 30 = 14 12 115 235 (1198991 1198992 1198971 2) lowast

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2) lowast

1198994




OPEN 15 13 10 23 (1198991 1198997 1198975 2) lowastradic

2nd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN 14 12 115 235 (1198991 1198992 1198971 2)

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

OPEN 15 + 1 = 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowastradic

1198995



CLOSED 15 13 10 23 (1198991 1198997 1198975 2) lowast

3rd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN min14 51 = 14 12 115 235 (1198991 1198992 1198971 2) lowastradic

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

CLOSED 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowast

1198995

OPEN 30 28 0 28 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198995 1198973 16) lowast

1198996


CLOSED 15 13 10 23 (1198991 1198997 1198975 2)

radicThe node to be COLSED in the next searching round lowastThe node whose labels are updated in the searching round

different situation from a static network context Therefore1198921015840

(119899119894) and ℎ

1015840

(119899119894) are defined to estimate them respectively

1198911015840

(119899119894) is their summation calculated as

1198911015840

(119899119894) = 1198921015840

(119899119894) + ℎ1015840

(119899119894) (10)

The Alowast-based Itinerary Finder utilizes the minimumtravel time of the partial itinerary 119901119905119900(119899

119900 119899119894) determined to

this point as 1198921015840(119899119894) the strategy for estimating ℎ1015840(119899

119894) will be

addressed in Section 32 To illustrate the process for selectingthe most promising node we again use the USPT network ofFigure 1 as an example A traveler first predetermines 119905

1= 2

In the first searching round (see Figure 4(b))120587(1198991 1198992 1198971 2) =

12 (min) 120587(1198991 1198992 1198972 2) = 28 120587(119899

1 1198993 1198971 2) = 23 and

120587(1198991 1198997 1198975 2) = 13 can be easily determined with Formula

(4) One can easily determine that 1198921015840(1198992) = min12 28 =

12 1198921015840(1198993) = 23 and 119892

1015840

(1198997) = 13 As for the heuristic

information yielded by the Cost Estimator ℎ1015840(1198992) = 115

ℎ1015840

(1198993) = 15 and ℎ

1015840

(1198997) = 10 Thus 1198911015840(119899

2) = 119892

1015840

(1198992) +

ℎ1015840

(1198992) = 12 + 115 = 235 Similarly we have 1198911015840(119899

3) = 245

and 1198911015840

(1198997) = 23 Dijkstra-based approaches only consider

the performances of origin-rooted partial itineraries and

thus identify (1198991 1198992 1198972 2) as the most promising partial

itinerary due to 1198921015840

(1198992) lt 119892

1015840

(1198997) lt 119892

1015840

(1198993) The Alowast-based

approaches however are goal-directed by the heuristics andthe Itinerary Finder selects 119899

7as a result of 1198911015840(119899

7) lt 1198911015840

(1198992) lt

1198911015840

(1198993) Similarly the second searching round selects the

terminal node of (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) that is 119899

4 as

the most promising node and so forth Note that in the 3rdsearching round the 119905-arc (119899

4 1198992 1198972 16) is searched We have

1199052= 51 calculated by Formula (4) meaning that 1198921015840(119899

2) of

OPEN node 1198992will be potentially turned to 51 minus 119905

119900= 49

However because the previous value of 1198921015840(1198992) is 12 and 49 gt

12 the value of 1198921015840(1198992) is not updated but remains 12 In

another words the partial itinerary (1198991 1198992 1198971 2) dominates

(1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198992 1198972 16) referring to the

partial itinerary from 1198991to 1198992 The destination 119899

5is also

expanded in this searching round the associated state turnsto OPENThe searching process will continue however untilstate(119899

5) = CLOSED

Through the scheme that is recursively expanding com-paring and selecting promising partial itineraries the algo-rithm is terminated once the state of destination turns to


l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www

State of nodeNEWOPENCLOSED

State of arcNot searchedSearched

(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)

Figure 4 Partial searching process by Floyd-Alowast

CLOSED If an algorithm is guaranteed to determine anoptimal itinerary from origin to destination we designate itas admissible The Itinerary Finder is proven to be admissiblein Section 33The Itinerary Finder places OPEN nodes in anOPEN list and CLOSED nodes in a CLOSED list If the stateof the node cannot be placed either in the OPEN or CLOSEDlist it is regarded as NEW as default In a summary of theabove analysis the outline of the Itinerary Finder is presentedin Algorithm 1

If more detailed information (eg waiting time in-vehi-cle time) is required with respect to a specific arc this canbe obtained by simply adding to pre(119899

119894) in the associated

iteration

32 Cost-Estimator Procedure for a Tighter Lower BoundThe Itinerary-Finder procedure must be well informed whenmaking a choice to expand partial itineraries Expanding anunlikely part of an LT itinerary is a waste of computationaltime whilemissing a promising partial itinerarymay lead to afailure in determining the LT itineraryTherefore the strategyof estimating the travel time of a destination-ended partialitinerary is viewed as the key to improving the efficiency of theItinerary Finder Meanwhile the estimated travel time must

be a lower bound of the real travel time Note that a tighterlower bound results in higher efficiency

The travel time of an itinerary is composed of the traveltime during walking between two nodes waiting at nodes fora transfer and traveling in vehiclesThewalking time betweentwo specific nodes is fixed The waiting time varies in dif-ferent cases If fortunate a traveler can transfer without wait-ing time The in-vehicle time depends on the timetable ofdifferent lines combined with their different vehicle tripsThis paper proposes a strategy to estimate the travel timebetween two nodes as a tight static lower bound of thisreal time-dependent value The basic concept is shown bygenerating an associated slacked network (see Definition 5)of the USPT network the minimum travel time of itineraryin this SUSPTnetwork is the associated estimated value in theUSPT network

Definition 5 A slacked USPT network (SUSPT network forshort) is defined to share the same topological structure as theUSPT network However each arc of the SUSPT network isassigned a static travel time as a lower bound of the associatedreal travel time of the arc in the USPT network The arcin the SUSPT network is timed by explicitly slacking theassociated real travel time by using the following 3 rules


Step 0 (Initialization)Set OPEN list = and CLOSED list =Set 1198921015840(119899

119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873

Add 119899119900to OPEN list set 1198921015840(119899

119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899

119900) is pre-calculated by Cost-Estimator

Step 1 (Expanding partial itinerary)while OPEN list = do

Select any node 119899119894isin 119899119894| min(1198911015840(119899

119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else

Move 119899119894from OPEN list to CLOSED list

end iffor all t-arc (119899

119894 119899119894+ 119904119894 119905119894) do

if 119899119894+in CLOSED list then

continueend ifif 119904119894= 119904119894minus= 119908 then Set 119904

119900minus= 119899119894119897 previously

continueend ifif 119904119894= 119904119898or 119904119894= 119904119894where 119904

119898= 119904119900 119904119900+ 119904

119894minusthen

continueend ifCalculate 120587(119899

119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then

continueelse if 119899

119894+not in OPEN list then

Add 119899119894+to OPEN list

end ifCalculate corresponding 119905

119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while

Step 2 (Reconstructing LT itinerary)Reconstruct LT itinerary 119901119905119900 (119899

119900 119899119889) by recursively recalling 119901119903119890(119899

119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)

Algorithm 1 Procedure of Itinerary Finder (119899119900 119899119889 119905119900)

Figure 5 shows the associated SUSPT network of the USPTnetwork of Figure 1

Rule 1 Walking times remain the same

Rule 2 Ignore all waiting times

Rule 3 Let the minimum travel time among those traversedby different lines combined with different vehicle tripsbetween two specific nodes be the estimated travel time

Obviously there exist no temporal concepts in the staticSUSPT network therefore let each initial time of 119905-arc in theSUSPT network be nil Let 1205871015840 denote the operator to time the119905-arc in SUSPT network Rule 1 can be reflected in Formula(11) Rules 2 and 3 are interpreted in Formula (12)

1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)

For this problem typical all-to-all shortest paths algo-rithms are qualified This paper chooses a well-known FloydalgorithmWe assume typical readers have already known thealgorithm so there is no detailed exposition here

The outline of the procedure is shown in Algorithm 2

Remark 6 In this work the Itinerary Finder obtains ℎ1015840(119899119894)

from table 119867 outputted by the Cost Estimator while theprevious related works substituted ℎ

1015840

(119899119894) with Formula (13)

which was calculated in an online wayThe strategy proposedin this work is proved to generate a tighter lower bound andthus leads the Floyd-Alowast algorithm to be more efficient bothin theory and in computation experiments (see Sections 3341 and 42)

ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)


Step 0 (Initialize the SUSPT network)for all 119899

119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for

Step 1 (Calculate costs of all-to-all shortest paths)for all 119899

119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for

Algorithm 2 Procedure of Cost Estimator

NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10

Figure 5 Associated SUSPT network of the USPT network ofFigure 1

33 Admissibility and Efficiency Analysis The admissibilityand efficiency of the Floyd-Alowast algorithm are discussed inthis section Hart et al [17] established how to determine theadmissibility of an Alowast algorithm which is primarily affectedby the travel time estimating strategy of the destination-ended partial itineraries shown in Lemma 7 On this basisTheorem 8 establishes the admissibility of the ItineraryFinder

Lemma7 If ℎ1015840(119899119894) le 120587(119901

119905119894(119899119894 119899119889)) thenAlowast is admissible [17]

Theorem 8 The Cost Estimator guarantees that the ItineraryFinder is admissible

Proof The Itinerary Finder is Alowast-based where ℎ1015840(119899119894) is com-

puted by the Cost Estimator To prove Theorem 8 we learnfrom Lemma 7 that it is equivalent to prove that the CostEstimator guarantees each ℎ1015840(119899

119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899

119889minus 119899119889 119904119889minus 119905119889minus) be a

destination-ended partial itinerary of the actual LT itinerary119901119905119900(119899119900 119899119889) Therefore

120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899

119889minus 119899119889 119904119889minus 119905119889minus)

(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

denotes the LT itinerary in the SUSPT network Note thatthe topological structures of 119901119905119894(119899

119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are

not necessarily the sameIf 119904119894= 119908 then for any 119905-arc (119899

119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)

In contrast 120587(119899119894 119899119895 119897 119905119894) can be calculated by Formula (4)

combined with Formula (5) that is

120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)

In summation for any 119905-arc (119899119894 119899119895 119904119894 119905119894) 1205871015840(119899

119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)

Thus Theorem 8 is proven

If Inequality (20) is satisfied we deem this a consistencyassumption for the Itinerary Finder The definition of this


assumption helps to explain why the Itinerary Finder neverre-OPENs a CLOSED nodeThe explanation can be found inTheorem 10120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)

Lemma 9 Assuming that the consistency assumption is satis-fied Alowast needs never to re-OPEN a CLOSED node [17]

Theorem 10 The Cost Estimator assures that the ItineraryFinder needs never to re-OPEN a CLOSED node

Proof To prove Theorem 10 we learn from Lemma 9 that itis equivalent to prove that the estimating strategy proposedin the Cost Estimator satisfies the consistency assumption

Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)

One can prove that1205871015840(119899119894 119899119895 119904119894 119899119894119897) le 120587(119899

119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)

In other words the consistency assumption is satisfiedTheorem 10 is thus proven

Previous related works had developed two variants ofItinerary-Finder procedure that is the Plain-Alowast procedurein which the value ℎ1015840(119899

119894) is revised by calculating in Formula

(13) and theDijkstra-likeprocedure inwhichℎ1015840(119899119894) is replaced

by constant 0 Similarly it is not difficult to prove thatthe Dijkstra-like and Plain-Alowast both satisfy the consistencyassumption and are thus admissibleWe show the comparisonamong the Floyd-Alowast and the two procedures as follows

Lemma 11 Consider the set of lower bounds verifying the con-sistency assumption If a node is selected by the Alowast algorithmfor a given lower bound then this node will be selected by theAlowast algorithm using any smaller lower bound [18]

Let 119873DA 119873SA and 119873DL denote the sets of expandednodes by Floyd-Alowast Plain-Alowast and Dijkstra-like respectivelyAccording to Lemma 11 Theorem 12 refers to their relation-ships

Theorem 12 119873DA sube 119873SA sube 119873DL

Proof To prove Theorem 12 one can equivalently prove thatthe estimated travel time values of Floyd-Alowast Plain-Alowast andDijkstra-like are each a smaller lower bound of the real costthan the next that is 120587(119901119905119894(119899

119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0

Theorem 8 has proved 120587(119901119905119894(119899119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

and it is obvious that dist(119899119894 119899119889)Vmax ge 0 because dist(119899

119894 119899119889)

and Vmax are both positiveWe therefore need only prove that

1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840

(119899119894 119899119895 119904119894 119899119894119897) can be calculated as len(119899

119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)

With regard to dist(119899119894 119899119889)Vmax the numerator dist(119899

119894

119899119889) is obviously not greater than the real distance of any itin-

erary from 119899119894to 119899119889 and the denominator Vmax is not less than

any velocity observed by walking bus and metro Then

1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge

len (119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + len (119899

119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)

Thus the theorem is proven

Corollary 13 |119873DA| le |119873SA| le |119873DL|

Under the premises of Theorem 12 Corollary 13 canbe easily determined meaning that the total number ofexpanded nodes from the Dijsktra-like Plain-Alowast and Floyd-Alowast algorithms are each no less than the next Correspond-ingly their efficiencies increase orderly

In summation the searching scopes of the three proce-dures intuitively seem to be as shown in Figure 6 and theirefficiencies are shown as tested in Sections 41 and 42

4 Example Illustration and Analysis

A numerical example and a real-world USPT networkinstance are presented to demonstrate the suitability andefficiency of the proposed Floyd-Alowast algorithm as well as theinstructive significance for travelers For this purpose theexperiments are composed of four parts Section 41 showsthe efficiency of the Floyd-Alowast algorithm through a numericalexample in comparisonwith the two other conventional pro-cedures that is the Dijkstra-like and Plain-Alowast procedures Areal-world instance is tested to demonstrate applicability andefficiency of the Floyd-Alowast algorithmwhen solving large-scale


USPT networkDijkstra-like

Plain-Alowast

noFloyd-Alowast nd

Figure 6 Searching scope of three procedures

network instance which is given in Section 42 The experi-ments on time-dependent nature of the least-time itineraryand the phenomenon ldquojustmissingrdquo and ldquojust boardingrdquo casesare presented in Sections 43 and 44 respectivelyThe exper-iments ran in a MATLAB environment on an HP Compaq8280 Elite CMT PC with Intel Core i5-2400 CPU 31GHzand 4GB memory (RAM)

The USPT network of the numerical example shown inFigure 7 is formed by 30 nodes and 103 arcs There are 10lines including 2 metro lines and 8 bus lines where 119897

4=

1198978 10 corresponding timetables are also provided Node 119899

5

is traversed by 1198971 1198972 and 119897

3 each of another ten nodes is

simultaneously traversed by 2 lines Specific data are omitteddue to the limited space

41 High Efficiency of the Floyd-Alowast Algorithm An itineraryplanning assistant is capable of determining the LT itinerarythrough real-time querying Efficiency is the ultimate goalFor testing 1000 triads of 119899

119900 119899119889 and initial time 119905

119900are

randomly generated with the distance between each pair of119899119900 119899119889no less than 5000meters all pairs are connectable Note

that in the generating process cases exist where no itineraryfrom 119899

119900to 119899119889was foundTherefore the Floyd-Alowast algorithm is

capable of verifying the connexity of a USPT network Giveneach triad (119899

119900 119899119889 and 119905

119900) each of the three procedures (ie

Floyd-Alowast Plain-Alowast and Dijkstra-like aforementioned inSection 33) are used to solve the LTIP-USPT The Dijkstra-like and Plain-Alowast procedures are traditional methods forsolving these types of problemsWeutilize two indicators oneaveraged the running time during the calculating of the LTitinerary by a specific procedure and the other averaged theexpansion times of nodes during the searching process Usingthe performance of Dijkstra-like procedure as a referencethe relative reductions of the two indicators are shown inTable 3 as well In addition the results outputted by differentprocedures in a specific instance are exactly the sameConsidering that the three procedures are all admissiblewe learn from Table 3 that Floyd-Alowast procedure reduced therunning time by 333 and the expansion times of nodes by6158 compared with the Dijkstra-like procedure while thetwo corresponding values were 1284 and 2534 savings

Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15

Figure 7 An USPT network example for illustration

from the Plain-Alowast procedure Floyd-Alowast procedure proposedin this paper is superior to both the Plain-Alowast and theDijkstra-like conventional procedures in terms of efficiency

42 Applicability of the Floyd-Alowast Algorithm for Real-WorldInstance To verify the applicability and efficiency of theFloyd-Alowast procedure in a real-world network we implementand test the three procedures in a Visual Studio 2010environment on the aforementioned PC using the real-world public transport data of Shenyang City the central cityof northeastern China The main urban zone of ShenyangCity has a size of more than 700 square kilometers and apopulation of more than 5 million until the year 2010 Thereare totally 446 directed USPT lines which are composed of2 metro lines and 444 bus lines The modeled Shenyang CityUSPT network (within the main urban zone) is formed by2812 nodes (after aggregating) and 184178 arcs Similar to theexperiments performed in Section 41 1000 triads of 119899

119900 119899119889

and initial time 119905119900are randomly generated the performances

are shown in Table 4 The real-world LTIP-USPT can besolved by the Floyd-Alowast procedure in a more efficient way itreduces the averaged running time by 639 compared withthat solved by the Dijkstra-like procedureTherefore we con-cluded that the Floyd-Alowast procedure is significantly superiorto the previous related work that is both the Plain-Alowast andthe Dijkstra-like procedures with reference to efficiency

In reality faced with such a large network local citizensand tourists are difficult to determine an optimal itinerarywithout an itinerary planning system To benefit the travelersthe Floyd-Alowast algorithm module is implemented and embed-ded in a Shenyang City Public Transport Query Systemshown in Figure 8 The system is implemented in a VisualStudio 2010 environment combined with the geographyinformation system TransCAD In a case that a traveler wantsthe least-time travel from the Bainaohui Stop to theWanquanPark Stop given the initial starting time 910 the systemreturns the solution that the traveler should cost 24 minutes


Table 3 Efficiency comparison of three procedures for LTIP-USPT in Figure 7

ProceduresItem

Running time (ms) Relative reduction ofrunning time ()

Expanding times of nodes(sec)

Relative reduction ofexpansion times ()

Dijkstra-like 20422 0 5364 0Plain-Alowast 178 1284 40049 2534Floyd-Alowast 13621 333 20611 6158

Table 4 Efficiency comparison of three procedures for LTIP inShenyang City USPT network

ProceduresItem


Dijkstra-like 435 0Plain-Alowast 342 214Floyd-Alowast 157 639

(including in-vehicle time and waiting time) traveling fromthe Bainaohui Stop to the EPA Stop by Line 222 walking1 minute to another EPA Stop and finally arriving at theWanquan Park Stop by Line 118 in 15 minutes Note that thetwo EPA Stops are geographically different but close Theinterface and the LT itinerary of the example are given asshown in Figure 8 It appears to be applicable and efficientafter numerous experiments It finally turns out that Floyd-Alowast can potentially be used into many large-scale real-worldUSPTnetworks for LT itinerary planning useNote that it alsohas the potential to be applied in interurban context giventhat all services are schedule-based

43 Time-Dependent Nature In a static public transportnetwork that does not consider a timetable it is obvious thatgiven an origin and destination pair the optimal itinerary(also referred as path) will consider objectives such as theleast transfer time and the lowest financial expense In otherwords the solution does not depend on the departure timewhile the situation is different when considering a timetable

In the case of a specified origin and destination whengiven different initial time 119905

119900 the proposed computation

method returns a different LT itinerary 119901119905119900(119899119900 119899119889) and corre-

sponding travel time 120587(119901119905119900(119899119900 119899119889)) Using the USPT network

of Figure 7 as an example 119899119900= 1198991and 119899

119889= 11989924

are pre-determined when given a different initial time for example119905119900= 55 and 119905

119900= 60 the itinerary 11990155(119899

1 11989924) and itinerary

11990160

(1198991 11989924) foundwith the Itinerary-Finder procedure are LT

itineraries in these two cases respectively These results areshown in Figure 9 where the horizontal axis represents thetime of day and the vertical axis represents the accumulatedtravel distance of the itinerary The circles represent nodesand the links are explained in the legend It is not difficultto see that the slope of the link represents the correspondingvelocity and the curve must be monotonically increasing

Figure 8 An example of system interface of LT itinerary planningin USPT network

55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000

Time of day (min) (after 70000)

Accu

mul

ated

trav

el d

istan

ce (m

) Least-time itineraries

WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6

Figure 9 Two LT itineraries with different initial times

11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976

713) (11989918 11989924 119908 84) costs 331 minutes traveling 12336

meters the itinerary 11990160(1198991 11989924) = (119899

1 1198992 1198971 60) (119899

2 11989910 1198975


metersThe Spatial itinerary is defined as an itinerary with the

temporal factors deleted The spatialitineraries of itineraries11990155

(1198991 11989924) and 11990160(119899

1 11989924) are represented as sp

1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as

shown in Figure 10 If we neglect the waiting time at transferthe static itinerary sp

1(1198991 11989924) intuitively appears more likely

to cost less time than sp2(1198991 11989924) because about half the


Totally 12336 meters


n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2

Figure 10 Two static itineraries

distance of sp1(1198991 11989924) is traversed by the metro which is

much faster than a bus and the total distances of the twoitineraries are very close How can sp

2(1198991 11989924) sometimes

cost less time than sp1(1198991 11989924) for example when 119905

119900= 60

To answer this question the corresponding itineraries ofsp1(1198991 11989924) and sp

2(1198991 11989924) both given an initial time of 60

are compared in Figure 11 The waiting time of the formeritinerary is 37 minutes longer than the latter one while thevalue of total travel time is only 24 minutes longer Similarresults can be found in other cases Therefore we concludethat the complex timetables that lead to waiting times duringtransfers are variable and almost uncontrollable primarilyresulting in the time-dependent nature of an LT itineraryin a USPT network Obviously these results could not bedetermined without considering timetables

44 Just Missing and Just Boarding Recall that the USPTlines are assumed to run punctually In this premise thephenomena of ldquojust missingrdquo and ldquojust boardingrdquo can beevaluated with the proposed algorithm Again we let 119899

119900= 1198991

and 119899119889= 11989924 When 119905

119900= 98 (a ldquojust boardingrdquo case) and

119905119900= 98 + 120585 (a ldquojust missingrdquo case) the approach determines

LT itineraries 11990198(1198991 11989924) and 119901

98+120585

(1198991 11989924) respectively as

shown in Figure 12 There is no waiting time at 1198991associated

with 11990198

(1198991 11989924) while with 119901

98+120585

(1198991 11989924) the traveler must

wait for 6 minutes to board the vehicle of 1198971 Furthermore we

learn that when 119905119900varies in a continuous interval (98 104]

the solutions are nearly the same only differing in waitingtime at 119899

119900 For example 11990198+120585(119899

1 11989924) costs 40 minutes to

get to 11989924 while it only costs 34 minutes in 119901

104

(1198991 11989924)

They both arrive at 11989924at 138 but experience different waiting

times at 1198991 If these results are preknown by the traveler

he might adjust earlier to 119905119900= 98 and take 32 minutes to

arrive at 119899119889or postpone to 119905

119900= 104 In fact the waiting

time for transfer between two lines is almost uncontrollabledepending entirely on complex timetables while the timespent at 119899

119900is controllable depending on both 119905

119900and the

timetable These findings can significantly help travelersdetermine an ideal initial time to begin travel by meetinga ldquojust boardingrdquo case and avoiding a ldquojust missingrdquo casethereby saving time These two categories of special casescould not be determined without considering a timetable

60 65 70 75 80 85 90 95 100 105

Two paths with the same initial time

0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60

Figure 11 Two different itineraries with the same initial time

95 100 105 110 115 120 125 130 135 140

Least-time itineraries

0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)

Figure 12 ldquoJust missingrdquo and ldquojust boardingrdquo cases

5 Conclusions and Future Work

This paper has presented an ad hoc Floyd-Alowast algorithm todetermine the least-time itinerary from origin to destinationin an urban scheduled public transportation network whengiven initial time to start the travel Amodified representationof the USPT network and the travel itinerary was proposedThe itinerary in a bimodal time-dependent USPT networkwas composed of time-label arcs whose timing methodwas explicitly specified Traveler operability was given toconstrain the itinerary structure A methodology to estimatetravel time between two nodes as a sufficiently tight staticlower bound of the corresponding real travel time wasproposed as a key contribution for high efficiency The adhoc Floyd-Alowast procedure was mathematically proven to becorrect and more efficient than the Plain-Alowast and Dijkstra-like procedures which appeared in previous related worksMeanwhile through an illustrated example and a real-worldexample we showed that the Floyd-Alowast algorithm appears tobe very suitable and efficient for LTIP-USPT These resultsreflect the time-dependent nature of the least-time itineraryin a scheduled network and can serve as guidance for travelersin predetermining an ideal initial time by meeting ldquojustboardingrdquo cases while avoiding ldquojust missingrdquo cases in the


premise that lines run punctuallyThe efficiency performancewas numerically tested to be superior to both Plain-Alowast andDijkstra-like procedures

It should be noted that the proposed approach for LTIP-USPT is primarily based on the hypothesis that vehiclesof lines run in absolute compliance with their timetablesHowever it is evident that early arrivals or delays can occurat each node during vehicle trips and the uncertainties aredynamically revealed One way to mitigate these uncertaineffects is to consider the stochastic optimization or robustoptimization methodologies Floyd-Alowast may not be suitablefor direct application at that time but its adaptation (maybewith a parallel program) should be considered These topicscertainly constitute a motivation for future works

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is financially supported by the National NaturalScience Foundation of China (71021061) the FundamentalResearch Funds for the Central Universities (N090204001N110404021 N110204005) and the National College StudentInnovative Experimental Project of China The authors alsogratefully acknowledge the insightful comments and sugges-tions made by the anonymous referees

References

[1] H Bast ldquoCar or public transportmdashtwo worldsrdquo Efficient Algo-rithms vol 5760 pp 355ndash367 2009

[2] R B Dial ldquoTransit pathfinder algorithmrdquo Highway ResearchRecord vol 205 pp 67ndash85 1967

[3] H Spiess andM Florian ldquoOptimal strategies a new assignmentmodel for transit networksrdquoTransportation Research Part B vol23 no 2 pp 83ndash102 1989

[4] S C Wong and C O Tong ldquoEstimation of time-dependentorigin-destination matrices for transit networksrdquo Transporta-tion Research B vol 32 no 1 pp 35ndash48 1998

[5] D Canca A Zarzo P L Gonzlez-R E Barrena and E AlgabaldquoA methodology for schedule-based paths recommendationin multimodal public transportation networksrdquo Journal ofAdvanced Transportation vol 47 no 3 pp 319ndash335 2013

[6] W Xu S He R Song and S S Chaudhry ldquoFinding the K short-est paths in a schedule-based transit networkrdquo Computers ampOperations Research vol 39 no 8 pp 1812ndash1826 2012

[7] M-C Tan C O Tong S CWong and J-M Xu ldquoAn algorithmfor finding reasonable paths in transit networksrdquo Journal ofAdvanced Transportation vol 41 no 3 pp 285ndash305 2007

[8] R Huang ldquoA schedule-based pathfinding algorithm for transitnetworks using pattern first searchrdquo GeoInformatica vol 11 no2 pp 269ndash285 2007

[9] A Nuzzolo U Crisalli and L Rosati ldquoA schedule-based assign-ment model with explicit capacity constraints for congestedtransit networksrdquo Transportation Research C Emerging Tech-nologies vol 20 no 1 pp 16ndash33 2012

[10] Y Hamdouch H W Ho A Sumalee and G Wang ldquoSchedule-based transit assignment model with vehicle capacity and seatavailabilityrdquo Transportation Research B Methodological vol 45no 10 pp 1805ndash1830 2011

[11] M H Poon S C Wong and C O Tong ldquoA dynamic schedule-based model for congested transit networksrdquo TransportationResearch B Methodological vol 38 no 4 pp 343ndash368 2004

[12] C O Tong and A J Richardson ldquoA computer model for findingthe time-dependent minimum path in a transit system withfixed schedulesrdquo Journal of Advanced Transportation vol 18 no2 pp 145ndash161 1984

[13] M E T Horn ldquoAn extended model and procedural frameworkfor planning multi-modal passenger journeysrdquo TransportationResearch B vol 37 no 7 pp 641ndash660 2003

[14] K N Androutsopoulos and K G Zografos ldquoSolving the multi-criteria time-dependent routing and scheduling problem ina multimodal fixed scheduled networkrdquo European Journal ofOperational Research vol 192 no 1 pp 18ndash28 2009

[15] K G Zografos and K N Androutsopoulos ldquoAlgorithms foritinerary planning in multimodal transportation networksrdquoIEEE Transactions on Intelligent Transportation Systems vol 9no 1 pp 175ndash184 2008

[16] L Fu D Sun and L R Rilett ldquoHeuristic shortest path algo-rithms for transportation applications state of the artrdquoComput-ers amp Operations Research vol 33 no 11 pp 3324ndash3343 2006

[17] P E Hart N J Nilsson and B Raphael ldquoA formal basis forthe heuristic determination of minimum cost pathsrdquo IEEETransactions on Systems Science and Cybernetics vol 4 no 2pp 100ndash107 1968

[18] I Chabini and S Lan ldquoAdaptations of the Alowast algorithm forthe computation of fastest paths in deterministic discrete-timedynamic networksrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 3 no 1 pp 60ndash74 2002

[19] G Nannicini D Delling D Schultes and L Liberti ldquoBidirec-tional Alowast search on time-dependent road networksrdquo Networksvol 59 no 2 pp 240ndash251 2012

[20] M Yu Y Ni Z Wang and Y Zhang ldquoDynamic route guidanceusing improved genetic algorithmsrdquoMathematical Problems inEngineering vol 2013 Article ID 765135 6 pages 2013

[21] Y Chen M G H Bell and K Bogenberger ldquoReliable pretripmultipath planning and dynamic adaptation for a centralizedroad navigation systemrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 8 no 1 pp 14ndash20 2007

[22] A V Goldberg ldquoPoint-to-point shortest path algorithms withpreprocessingrdquo LectureNotes in Computer Science vol 4362 pp88ndash102 2007

[23] J Maue P Sanders and D Matijevic ldquoGoal-directed shortest-path queries using precomputed cluster distancesrdquo Journal ofExperimental Algorithmics vol 14 article 2 2009

[24] MMuller-Hannemann F Schulz DWagner andC ZaroliagisldquoTimetable information models and algorithmsrdquo AlgorithmicMethods for Railway Optimization vol 4359 pp 67ndash90 2007

[25] E Pyrga F Schulz D Wagner and C Zaroliagis ldquoEfficientmodels for timetable information in public transportationsystemsrdquo ACM Journal of Experimental Algorithmics vol 12article 24 2008

[26] R W Floyd ldquoAlgorithm 97 shortest pathrdquo Communications ofACM vol 5 no 6 p 345 1962

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


transport Correspondingly in the transportation researchcommunity the focuses have shifted from headway-basedservices to schedule-based ones The main difference comesfrom the evaluating of each transferwaiting timeThe transferwaiting time is typically assumed to be half of the headwayconcerning headway-based services However for schedule-based services it could be precisely evaluated dependingon the combined timetables Thus the network travel times(especially transfer waiting times) shift from deterministicto time dependent resulting in the needs of methodologicalchanges for many transportation problems for exampleitinerary planning (see [5ndash8]) and traffic assignment (see [9ndash11]) And these methodologies appear more promising andcompeting against those for the traditional headway-basedservices Note that in reality the ldquoschedule-based linesrdquo are ageneralized concept that might include bus lines and metrolines In this sense the bimodal (USPT lines and walking)network considered in this paper could also be interpreted asa multimodal one which is a typical urban public transportnetwork in real world

There are numerous related works in the existing litera-tures Tong and Richardson [12] were among the first to studythe scheduled public transport they developed a branch-and-bound type algorithm for the minimum itinerary Alongthis vein research community extended the itinerary plan-ning problems by introducing different real-world consider-ations Horn introduced multimodal transport services anda Dijkstra-based algorithm was considered for minimizinggeneralized travel costs [13] Tan et al [7] developed a recur-sive algorithm for finding reasonable paths that satisfy thedefined acceptable time criterion and transfer-walk criterionwhere travelersrsquo preferences were required to give Not onlythe LT itinerary but also the k-LT itinerary need to be deter-mined for travelers Accordingly Xu et al [6] and Canca etal [5] studied the k-shortest path problems in schedule-basedtransit networks Androutsopoulos and Zografos considereda dynamic programming based algorithm [14 15] for theitinerary planning problem in the context where travelerimposes time windows on nodes multiobjective was alsoconsidered These works have significantly contributed toreal-world derived itinerary planning problems for differenttypes of travelersrsquo requirements However their focuses weremainly on the considerations of real-world factors Withregard to the algorithmic efficiency they only cared aboutwhether a query could be completed within a short timeIn query-intensive scenarios the query should be as fastas possible which motivated researchers in recent years toexamine heuristic methods to speed up computing

Fu et al reviewed the heuristic shortest path algorithmsfor transportation applications [16] noting that Alowast-basedalgorithms were widely usedThe heuristic Alowast algorithm wasfirst proposed by Hart et al [17] typically for the shortestpath problems in static networks The performance of Alowastalgorithm depends primarily on the strategy of estimatingthe travel time of a partial path A well-designed strategyleads to a considerable savings of computational time whileassuring an optimal solution Chabini and Lan adapted Alowast

for the fastest path in deterministic discrete-time dynamicnetworks [18] they proposed three estimating strategies Sub-sequently numerous works appeared in the literatures thatprimarily applied Alowast-based algorithms to LT path problemsin unscheduled time-dependent road networks (see [19 20])Chen et al proposed an Alowast-based integrated approach com-bining offline precomputation and online path retrieval for aroad navigation system [21] This work introduced precom-putation (see [22 23]) technology and significantly improvedthe efficiency However determining the LT itinerary in ascheduled network is more challenging than that of unsched-uled network But relatively little related works were found

In fact two notable works had studied the itineraryplanning problem in scheduled public transport networkespecially focusing on the speed-up technologies [24 25]They showed that the time-dependent model is superior tothe time-expanded model in the sense of itinerary-findingefficiency Alowast-based and other strategies are demonstrated tobe capable of speeding up computing Following the previousworks this paper goes beyond in the following aspects(i) modelling a modified bimodal (ie USPT services andwalking) time-dependent USPT network This model isobserved to be more applicable in that the results over themodified network model intrinsically have smaller numberof transfer times (ii) An ad hoc Floyd-Alowast algorithm isdeveloped to solve the LTIP-USPT where transit vehicles areassumed to run punctually A novel approach to estimatingtravel time of the partial itinerary is embodied in Floyd-AlowastTo implement the approach we generate a slacked networkand let the arc travel time be a static tight lower bound of theassociated real time-dependent arc travel time Precomputingtechnology is also usedThe algorithm is proven to be optimalin theory and was demonstrated with a real-world example tobe very applicable The Floyd-Alowast procedure outperforms theprevious procedures It reduces the averaged computationaltime by 639 compared with a conventional Dijkstra-like procedure (iii) From the management perspective anillustrated example reveals the time-dependent nature of theleast-time itinerary In the premise that lines run punctuallythe solution aids travelers in avoiding ldquojust missingrdquo cases

The remainder of this paper is organized as followsSection 2 formulates the modified USPT network modeland itinerary structure develops the formula for each time-dependent time-label arc and subsequently formulates theLTIP-USPT followed by hypotheses In Section 3 we proposean ad hoc Floyd-Alowast algorithm composed of two proceduresthat is Floyd-based Cost Estimator and Alowast-based ItineraryFinder The Cost Estimator precomputes the estimated traveltime of destination-ended partial itineraries as heuristicinformation The Itinerary Finder heuristically determinesthe LT itinerary in real-time Floyd-Alowast is mathematicallyproven to be admissible and efficient Furthermore an illus-trated example is presented in Section 4 that reveals the time-dependent nature of an LT itinerary and provides guidancefor travelers in determining the initial time to begin travelMeanwhile through both a numerical example and a real-world case the Floyd-Alowast procedure is proven to be moreefficient than two other conventional procedures that is



119896

119899

1198971

1198972

1198973

1198974

1198975

1198991

1198992

1198993

1198994

1198991

1198992

1198994

1198995

1198996

1198997

1198991

1198997

1 602 614 625 605 618 630 605 615 602 616 600 6122 608 621 631 615 629 640 610 620 610 623 603 6153 613 625 636 625 639 651 615 624 618 632 606 6174 618 629 639 633 646 658 620 630 626 640 609 6205 622 634 644 641 654 707 623 633 630 645 612 624









119897








0 lt 120588

119899119894

119897lt 120588

119899119895

119897le |119873119897| where 120588119899




119897|


1198601198971

= (1198991 1198992)1198971

(1198991 1198993)1198971

(1198992 1198993)1198971





119897=

(119899119894 119899119895)119897| 0 lt 120588

119899119894

119897= 120588

119899119895

119897minus 1 lt |119873







3




1 1198992)1198971

dominates arc(1198991 1198992)1198972


1 1198993)1198972








l2

l3

l1

l5

l4n7

n4

n1 n2

n6

n5n3


(a)

l2

l2

l2

l3

l1

l1

l1

l5

l4n7

n4

n1 n2

n6

n5n3

NodeArc

ww

w

w

(b)


900

905

925

930 950

l1

l2 l2n1

n2 n3

(a)

900

905

925

930 950

l1

l2l2

l2n1n2 n3

(b)



119908= (119899

119894 119899119895)119908

|

119899119894= 119899119895 dist(119899



1 1198992)119908notin 119860119908because




119871) 119873 = ⋃119897isin119871

119873119897 and 119860 = ⋃

119897isin119871119860119897⋃119860119908 In general a














119895isin













119901119905119900(119899119900 119899119889) = (119899

119900 1198992 119904119900 119905119900) (1198992 1198993 1199042 1199052)

(119899|119901| 119899119889 119904|119901| 119905|119901|)

(1)









119894 119899119895 119904119894 119905119894) depends on the




120587 (119899119894 119899119895 119908 119905119894) =

dist (119899119894 119899119895)

Vwalk (2)


119905119895= 119905119894+ 120587 (119899

119894 119899119895 119908 119905119894) (3)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

119897119896minus 119905119894 (4)


119897119896minus 119905119894and 120579

119899119895

119897119896minus 120579

119899119894



(6)

119896 = arg min119896

(120579

119899119894

119897119896minus 119905119894| 120579

119899119894

119897119896minus 119905119894gt 0) (5)

119905119895= 120579

119899119895

119897119896 (6)




119895can be easily




120587 (119901119905119900(119899119900 119899119889)) = 120587 (119899

119900 1198992 119904119900 119905119900) + 120587 (119899

2 1198993 1199042 1199052)

+ sdot sdot sdot + 120587 (119899|119901| 119899119889 119904|119901| 119905|119901|)

(7)



119894= 119908 we have 119904

119894+1= 119908 where 119894 = 119900 2 |119901| minus 1


119894= 119908 we

have 119904119895

= 119904119894and 119904119895

= 119904119894 where 119894 119895 = 119900 2 3 |119901| and 119894 = 119895


119894= 119899119895 where 119894 119895 = 119900 2 3 |119901| 119889 and 119894 = 119895



4in






6




2 1198997) =

(1198992 1198993 1198971 6 10) (119899

3 1198996 119908 6 25) (119899

6 1198997 1198974 6 27)


35(min)






min 120587 (119901119905119900(119899119900 119899119889))

st 119901119905119900(119899119900 119899119889) isin 119875119905119900(119899119900 119899119889)

(8)











119900 119899119889 119905119900) only the




119900 119899119889 119905119900) to determine









119894




119900 119899119894) thus far state(119899

119894) = OPEN





119895) = CLOSED



Output

Input

Inquiry

Output


Table H



Input

Return hrsquo(n)






1asOPENandothers


2 1198993 and 119899

7are expanded by


1 1198992 1198972 2) (119899

1 1198993 1198971 2)



2

1198993 and 119899





4











119894whose 119891

1015840

(119899119894) isare the




120587 (119901119905119900(119899119900 119899119894 119899119889)) = 120587 (119901

119905119900(119899119900 119899119894)) + 120587 (119901

119905119894(119899119894 119899119889))

(9)






Node Labels119878119905119886119905119890(119899) 119905

119894= 1198921015840

(119899) + 119905119900

1198921015840

(119899) ℎ1015840

(119899) 1198911015840

(119899) 119875119903119890(119899) Updated

Initialization

1198991


1198992







1st searching round

1198991


1198992

OPEN min14 30 = 14 12 115 235 (1198991 1198992 1198971 2) lowast

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2) lowast

1198994





2nd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN 14 12 115 235 (1198991 1198992 1198971 2)

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

OPEN 15 + 1 = 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowastradic

1198995



CLOSED 15 13 10 23 (1198991 1198997 1198975 2) lowast

3rd searching round

1198991

CLOSED 2 0 21 21 nil1198992


1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

CLOSED 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowast

1198995

OPEN 30 28 0 28 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198995 1198973 16) lowast

1198996


CLOSED 15 13 10 23 (1198991 1198997 1198975 2)



(119899119894) and ℎ

1015840


1198911015840


1198911015840

(119899119894) = 1198921015840

(119899119894) + ℎ1015840

(119899119894) (10)


119900 119899119894) determined to


119894) will be


1= 2


12 (min) 120587(1198991 1198992 1198972 2) = 28 120587(119899

1 1198993 1198971 2) = 23 and



12 1198921015840(1198993) = 23 and 119892

1015840



ℎ1015840

(1198993) = 15 and ℎ

1015840

(1198997) = 10 Thus 1198911015840(119899

2) = 119892

1015840

(1198992) +

ℎ1015840


3) = 245

and 1198911015840





(1198992) lt 119892

1015840

(1198997) lt 119892

1015840



7as a result of 1198911015840(119899

7) lt 1198911015840

(1198992) lt

1198911015840


terminal node of (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) that is 119899

4 as




2) of


119900= 49




(1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198992 1198972 16) referring to the


5is also


5) = CLOSED



l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)





iteration







119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896

119899

1198971

1198972

1198973

1198974

1198975

1198991

1198992

1198993

1198994

1198991

1198992

1198994

1198995

1198996

1198997

1198991

1198997

1 602 614 625 605 618 630 605 615 602 616 600 6122 608 621 631 615 629 640 610 620 610 623 603 6153 613 625 636 625 639 651 615 624 618 632 606 6174 618 629 639 633 646 658 620 630 626 640 609 6205 622 634 644 641 654 707 623 633 630 645 612 624









119897








0 lt 120588

119899119894

119897lt 120588

119899119895

119897le |119873119897| where 120588119899




119897|


1198601198971

= (1198991 1198992)1198971

(1198991 1198993)1198971

(1198992 1198993)1198971





119897=

(119899119894 119899119895)119897| 0 lt 120588

119899119894

119897= 120588

119899119895

119897minus 1 lt |119873







3




1 1198992)1198971

dominates arc(1198991 1198992)1198972


1 1198993)1198972








l2

l3

l1

l5

l4n7

n4

n1 n2

n6

n5n3


(a)

l2

l2

l2

l3

l1

l1

l1

l5

l4n7

n4

n1 n2

n6

n5n3

NodeArc

ww

w

w

(b)


900

905

925

930 950

l1

l2 l2n1

n2 n3

(a)

900

905

925

930 950

l1

l2l2

l2n1n2 n3

(b)



119908= (119899

119894 119899119895)119908

|

119899119894= 119899119895 dist(119899



1 1198992)119908notin 119860119908because




119871) 119873 = ⋃119897isin119871

119873119897 and 119860 = ⋃

119897isin119871119860119897⋃119860119908 In general a














119895isin













119901119905119900(119899119900 119899119889) = (119899

119900 1198992 119904119900 119905119900) (1198992 1198993 1199042 1199052)

(119899|119901| 119899119889 119904|119901| 119905|119901|)

(1)









119894 119899119895 119904119894 119905119894) depends on the




120587 (119899119894 119899119895 119908 119905119894) =

dist (119899119894 119899119895)

Vwalk (2)


119905119895= 119905119894+ 120587 (119899

119894 119899119895 119908 119905119894) (3)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

119897119896minus 119905119894 (4)


119897119896minus 119905119894and 120579

119899119895

119897119896minus 120579

119899119894



(6)

119896 = arg min119896

(120579

119899119894

119897119896minus 119905119894| 120579

119899119894

119897119896minus 119905119894gt 0) (5)

119905119895= 120579

119899119895

119897119896 (6)




119895can be easily




120587 (119901119905119900(119899119900 119899119889)) = 120587 (119899

119900 1198992 119904119900 119905119900) + 120587 (119899

2 1198993 1199042 1199052)

+ sdot sdot sdot + 120587 (119899|119901| 119899119889 119904|119901| 119905|119901|)

(7)



119894= 119908 we have 119904

119894+1= 119908 where 119894 = 119900 2 |119901| minus 1


119894= 119908 we

have 119904119895

= 119904119894and 119904119895

= 119904119894 where 119894 119895 = 119900 2 3 |119901| and 119894 = 119895


119894= 119899119895 where 119894 119895 = 119900 2 3 |119901| 119889 and 119894 = 119895



4in






6




2 1198997) =

(1198992 1198993 1198971 6 10) (119899

3 1198996 119908 6 25) (119899

6 1198997 1198974 6 27)


35(min)






min 120587 (119901119905119900(119899119900 119899119889))

st 119901119905119900(119899119900 119899119889) isin 119875119905119900(119899119900 119899119889)

(8)











119900 119899119889 119905119900) only the




119900 119899119889 119905119900) to determine









119894




119900 119899119894) thus far state(119899

119894) = OPEN





119895) = CLOSED



Output

Input

Inquiry

Output


Table H



Input

Return hrsquo(n)






1asOPENandothers


2 1198993 and 119899

7are expanded by


1 1198992 1198972 2) (119899

1 1198993 1198971 2)



2

1198993 and 119899





4











119894whose 119891

1015840

(119899119894) isare the




120587 (119901119905119900(119899119900 119899119894 119899119889)) = 120587 (119901

119905119900(119899119900 119899119894)) + 120587 (119901

119905119894(119899119894 119899119889))

(9)






Node Labels119878119905119886119905119890(119899) 119905

119894= 1198921015840

(119899) + 119905119900

1198921015840

(119899) ℎ1015840

(119899) 1198911015840

(119899) 119875119903119890(119899) Updated

Initialization

1198991


1198992







1st searching round

1198991


1198992

OPEN min14 30 = 14 12 115 235 (1198991 1198992 1198971 2) lowast

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2) lowast

1198994





2nd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN 14 12 115 235 (1198991 1198992 1198971 2)

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

OPEN 15 + 1 = 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowastradic

1198995



CLOSED 15 13 10 23 (1198991 1198997 1198975 2) lowast

3rd searching round

1198991

CLOSED 2 0 21 21 nil1198992


1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

CLOSED 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowast

1198995

OPEN 30 28 0 28 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198995 1198973 16) lowast

1198996


CLOSED 15 13 10 23 (1198991 1198997 1198975 2)



(119899119894) and ℎ

1015840


1198911015840


1198911015840

(119899119894) = 1198921015840

(119899119894) + ℎ1015840

(119899119894) (10)


119900 119899119894) determined to


119894) will be


1= 2


12 (min) 120587(1198991 1198992 1198972 2) = 28 120587(119899

1 1198993 1198971 2) = 23 and



12 1198921015840(1198993) = 23 and 119892

1015840



ℎ1015840

(1198993) = 15 and ℎ

1015840

(1198997) = 10 Thus 1198911015840(119899

2) = 119892

1015840

(1198992) +

ℎ1015840


3) = 245

and 1198911015840





(1198992) lt 119892

1015840

(1198997) lt 119892

1015840



7as a result of 1198911015840(119899

7) lt 1198911015840

(1198992) lt

1198911015840


terminal node of (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) that is 119899

4 as




2) of


119900= 49




(1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198992 1198972 16) referring to the


5is also


5) = CLOSED



l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)





iteration







119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










l2

l3

l1

l5

l4n7

n4

n1 n2

n6

n5n3


(a)

l2

l2

l2

l3

l1

l1

l1

l5

l4n7

n4

n1 n2

n6

n5n3

NodeArc

ww

w

w

(b)


900

905

925

930 950

l1

l2 l2n1

n2 n3

(a)

900

905

925

930 950

l1

l2l2

l2n1n2 n3

(b)



119908= (119899

119894 119899119895)119908

|

119899119894= 119899119895 dist(119899



1 1198992)119908notin 119860119908because




119871) 119873 = ⋃119897isin119871

119873119897 and 119860 = ⋃

119897isin119871119860119897⋃119860119908 In general a














119895isin













119901119905119900(119899119900 119899119889) = (119899

119900 1198992 119904119900 119905119900) (1198992 1198993 1199042 1199052)

(119899|119901| 119899119889 119904|119901| 119905|119901|)

(1)









119894 119899119895 119904119894 119905119894) depends on the




120587 (119899119894 119899119895 119908 119905119894) =

dist (119899119894 119899119895)

Vwalk (2)


119905119895= 119905119894+ 120587 (119899

119894 119899119895 119908 119905119894) (3)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

119897119896minus 119905119894 (4)


119897119896minus 119905119894and 120579

119899119895

119897119896minus 120579

119899119894



(6)

119896 = arg min119896

(120579

119899119894

119897119896minus 119905119894| 120579

119899119894

119897119896minus 119905119894gt 0) (5)

119905119895= 120579

119899119895

119897119896 (6)




119895can be easily




120587 (119901119905119900(119899119900 119899119889)) = 120587 (119899

119900 1198992 119904119900 119905119900) + 120587 (119899

2 1198993 1199042 1199052)

+ sdot sdot sdot + 120587 (119899|119901| 119899119889 119904|119901| 119905|119901|)

(7)



119894= 119908 we have 119904

119894+1= 119908 where 119894 = 119900 2 |119901| minus 1


119894= 119908 we

have 119904119895

= 119904119894and 119904119895

= 119904119894 where 119894 119895 = 119900 2 3 |119901| and 119894 = 119895


119894= 119899119895 where 119894 119895 = 119900 2 3 |119901| 119889 and 119894 = 119895



4in






6




2 1198997) =

(1198992 1198993 1198971 6 10) (119899

3 1198996 119908 6 25) (119899

6 1198997 1198974 6 27)


35(min)






min 120587 (119901119905119900(119899119900 119899119889))

st 119901119905119900(119899119900 119899119889) isin 119875119905119900(119899119900 119899119889)

(8)











119900 119899119889 119905119900) only the




119900 119899119889 119905119900) to determine









119894




119900 119899119894) thus far state(119899

119894) = OPEN





119895) = CLOSED



Output

Input

Inquiry

Output


Table H



Input

Return hrsquo(n)






1asOPENandothers


2 1198993 and 119899

7are expanded by


1 1198992 1198972 2) (119899

1 1198993 1198971 2)



2

1198993 and 119899





4











119894whose 119891

1015840

(119899119894) isare the




120587 (119901119905119900(119899119900 119899119894 119899119889)) = 120587 (119901

119905119900(119899119900 119899119894)) + 120587 (119901

119905119894(119899119894 119899119889))

(9)






Node Labels119878119905119886119905119890(119899) 119905

119894= 1198921015840

(119899) + 119905119900

1198921015840

(119899) ℎ1015840

(119899) 1198911015840

(119899) 119875119903119890(119899) Updated

Initialization

1198991


1198992







1st searching round

1198991


1198992

OPEN min14 30 = 14 12 115 235 (1198991 1198992 1198971 2) lowast

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2) lowast

1198994





2nd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN 14 12 115 235 (1198991 1198992 1198971 2)

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

OPEN 15 + 1 = 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowastradic

1198995



CLOSED 15 13 10 23 (1198991 1198997 1198975 2) lowast

3rd searching round

1198991

CLOSED 2 0 21 21 nil1198992


1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

CLOSED 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowast

1198995

OPEN 30 28 0 28 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198995 1198973 16) lowast

1198996


CLOSED 15 13 10 23 (1198991 1198997 1198975 2)



(119899119894) and ℎ

1015840


1198911015840


1198911015840

(119899119894) = 1198921015840

(119899119894) + ℎ1015840

(119899119894) (10)


119900 119899119894) determined to


119894) will be


1= 2


12 (min) 120587(1198991 1198992 1198972 2) = 28 120587(119899

1 1198993 1198971 2) = 23 and



12 1198921015840(1198993) = 23 and 119892

1015840



ℎ1015840

(1198993) = 15 and ℎ

1015840

(1198997) = 10 Thus 1198911015840(119899

2) = 119892

1015840

(1198992) +

ℎ1015840


3) = 245

and 1198911015840





(1198992) lt 119892

1015840

(1198997) lt 119892

1015840



7as a result of 1198911015840(119899

7) lt 1198911015840

(1198992) lt

1198911015840


terminal node of (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) that is 119899

4 as




2) of


119900= 49




(1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198992 1198972 16) referring to the


5is also


5) = CLOSED



l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)





iteration







119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120587 (119899119894 119899119895 119908 119905119894) =

dist (119899119894 119899119895)

Vwalk (2)


119905119895= 119905119894+ 120587 (119899

119894 119899119895 119908 119905119894) (3)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

119897119896minus 119905119894 (4)


119897119896minus 119905119894and 120579

119899119895

119897119896minus 120579

119899119894



(6)

119896 = arg min119896

(120579

119899119894

119897119896minus 119905119894| 120579

119899119894

119897119896minus 119905119894gt 0) (5)

119905119895= 120579

119899119895

119897119896 (6)




119895can be easily




120587 (119901119905119900(119899119900 119899119889)) = 120587 (119899

119900 1198992 119904119900 119905119900) + 120587 (119899

2 1198993 1199042 1199052)

+ sdot sdot sdot + 120587 (119899|119901| 119899119889 119904|119901| 119905|119901|)

(7)



119894= 119908 we have 119904

119894+1= 119908 where 119894 = 119900 2 |119901| minus 1


119894= 119908 we

have 119904119895

= 119904119894and 119904119895

= 119904119894 where 119894 119895 = 119900 2 3 |119901| and 119894 = 119895


119894= 119899119895 where 119894 119895 = 119900 2 3 |119901| 119889 and 119894 = 119895



4in






6




2 1198997) =

(1198992 1198993 1198971 6 10) (119899

3 1198996 119908 6 25) (119899

6 1198997 1198974 6 27)


35(min)






min 120587 (119901119905119900(119899119900 119899119889))

st 119901119905119900(119899119900 119899119889) isin 119875119905119900(119899119900 119899119889)

(8)











119900 119899119889 119905119900) only the




119900 119899119889 119905119900) to determine









119894




119900 119899119894) thus far state(119899

119894) = OPEN





119895) = CLOSED



Output

Input

Inquiry

Output


Table H



Input

Return hrsquo(n)






1asOPENandothers


2 1198993 and 119899

7are expanded by


1 1198992 1198972 2) (119899

1 1198993 1198971 2)



2

1198993 and 119899





4











119894whose 119891

1015840

(119899119894) isare the




120587 (119901119905119900(119899119900 119899119894 119899119889)) = 120587 (119901

119905119900(119899119900 119899119894)) + 120587 (119901

119905119894(119899119894 119899119889))

(9)






Node Labels119878119905119886119905119890(119899) 119905

119894= 1198921015840

(119899) + 119905119900

1198921015840

(119899) ℎ1015840

(119899) 1198911015840

(119899) 119875119903119890(119899) Updated

Initialization

1198991


1198992







1st searching round

1198991


1198992

OPEN min14 30 = 14 12 115 235 (1198991 1198992 1198971 2) lowast

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2) lowast

1198994





2nd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN 14 12 115 235 (1198991 1198992 1198971 2)

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

OPEN 15 + 1 = 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowastradic

1198995



CLOSED 15 13 10 23 (1198991 1198997 1198975 2) lowast

3rd searching round

1198991

CLOSED 2 0 21 21 nil1198992


1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

CLOSED 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowast

1198995

OPEN 30 28 0 28 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198995 1198973 16) lowast

1198996


CLOSED 15 13 10 23 (1198991 1198997 1198975 2)



(119899119894) and ℎ

1015840


1198911015840


1198911015840

(119899119894) = 1198921015840

(119899119894) + ℎ1015840

(119899119894) (10)


119900 119899119894) determined to


119894) will be


1= 2


12 (min) 120587(1198991 1198992 1198972 2) = 28 120587(119899

1 1198993 1198971 2) = 23 and



12 1198921015840(1198993) = 23 and 119892

1015840



ℎ1015840

(1198993) = 15 and ℎ

1015840

(1198997) = 10 Thus 1198911015840(119899

2) = 119892

1015840

(1198992) +

ℎ1015840


3) = 245

and 1198911015840





(1198992) lt 119892

1015840

(1198997) lt 119892

1015840



7as a result of 1198911015840(119899

7) lt 1198911015840

(1198992) lt

1198911015840


terminal node of (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) that is 119899

4 as




2) of


119900= 49




(1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198992 1198972 16) referring to the


5is also


5) = CLOSED



l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)





iteration







119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119900 119899119889 119905119900) only the




119900 119899119889 119905119900) to determine









119894




119900 119899119894) thus far state(119899

119894) = OPEN





119895) = CLOSED



Output

Input

Inquiry

Output


Table H



Input

Return hrsquo(n)






1asOPENandothers


2 1198993 and 119899

7are expanded by


1 1198992 1198972 2) (119899

1 1198993 1198971 2)



2

1198993 and 119899





4











119894whose 119891

1015840

(119899119894) isare the




120587 (119901119905119900(119899119900 119899119894 119899119889)) = 120587 (119901

119905119900(119899119900 119899119894)) + 120587 (119901

119905119894(119899119894 119899119889))

(9)






Node Labels119878119905119886119905119890(119899) 119905

119894= 1198921015840

(119899) + 119905119900

1198921015840

(119899) ℎ1015840

(119899) 1198911015840

(119899) 119875119903119890(119899) Updated

Initialization

1198991


1198992







1st searching round

1198991


1198992

OPEN min14 30 = 14 12 115 235 (1198991 1198992 1198971 2) lowast

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2) lowast

1198994





2nd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN 14 12 115 235 (1198991 1198992 1198971 2)

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

OPEN 15 + 1 = 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowastradic

1198995



CLOSED 15 13 10 23 (1198991 1198997 1198975 2) lowast

3rd searching round

1198991

CLOSED 2 0 21 21 nil1198992


1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

CLOSED 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowast

1198995

OPEN 30 28 0 28 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198995 1198973 16) lowast

1198996


CLOSED 15 13 10 23 (1198991 1198997 1198975 2)



(119899119894) and ℎ

1015840


1198911015840


1198911015840

(119899119894) = 1198921015840

(119899119894) + ℎ1015840

(119899119894) (10)


119900 119899119894) determined to


119894) will be


1= 2


12 (min) 120587(1198991 1198992 1198972 2) = 28 120587(119899

1 1198993 1198971 2) = 23 and



12 1198921015840(1198993) = 23 and 119892

1015840



ℎ1015840

(1198993) = 15 and ℎ

1015840

(1198997) = 10 Thus 1198911015840(119899

2) = 119892

1015840

(1198992) +

ℎ1015840


3) = 245

and 1198911015840





(1198992) lt 119892

1015840

(1198997) lt 119892

1015840



7as a result of 1198911015840(119899

7) lt 1198911015840

(1198992) lt

1198911015840


terminal node of (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) that is 119899

4 as




2) of


119900= 49




(1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198992 1198972 16) referring to the


5is also


5) = CLOSED



l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)





iteration







119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Node Labels119878119905119886119905119890(119899) 119905

119894= 1198921015840

(119899) + 119905119900

1198921015840

(119899) ℎ1015840

(119899) 1198911015840

(119899) 119875119903119890(119899) Updated

Initialization

1198991


1198992







1st searching round

1198991


1198992

OPEN min14 30 = 14 12 115 235 (1198991 1198992 1198971 2) lowast

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2) lowast

1198994





2nd searching round

1198991

CLOSED 2 0 21 21 nil1198992

OPEN 14 12 115 235 (1198991 1198992 1198971 2)

1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

OPEN 15 + 1 = 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowastradic

1198995



CLOSED 15 13 10 23 (1198991 1198997 1198975 2) lowast

3rd searching round

1198991

CLOSED 2 0 21 21 nil1198992


1198993

OPEN 25 23 15 245 (1198991 1198993 1198971 2)

1198994

CLOSED 16 14 9 23 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) lowast

1198995

OPEN 30 28 0 28 (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198995 1198973 16) lowast

1198996


CLOSED 15 13 10 23 (1198991 1198997 1198975 2)



(119899119894) and ℎ

1015840


1198911015840


1198911015840

(119899119894) = 1198921015840

(119899119894) + ℎ1015840

(119899119894) (10)


119900 119899119894) determined to


119894) will be


1= 2


12 (min) 120587(1198991 1198992 1198972 2) = 28 120587(119899

1 1198993 1198971 2) = 23 and



12 1198921015840(1198993) = 23 and 119892

1015840



ℎ1015840

(1198993) = 15 and ℎ

1015840

(1198997) = 10 Thus 1198911015840(119899

2) = 119892

1015840

(1198992) +

ℎ1015840


3) = 245

and 1198911015840





(1198992) lt 119892

1015840

(1198997) lt 119892

1015840



7as a result of 1198911015840(119899

7) lt 1198911015840

(1198992) lt

1198911015840


terminal node of (1198991 1198997 1198975 2) (119899

7 1198994 119908 15) that is 119899

4 as




2) of


119900= 49




(1198991 1198997 1198975 2) (119899

7 1198994 119908 15) (119899

4 1198992 1198972 16) referring to the


5is also


5) = CLOSED



l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)





iteration







119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(a)

l3

l4

n4

n7w

l1

l1

l1l2

l2

l2l5

n1 n2

n3

n5

n6 www



(b)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3n4 n5

n6n7w www



(c)

l1

l1

l1l2

l2

l2

l3

l4

l5

n1 n2

n3

n4 n5

n6n7w www



(d)





iteration







119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894) = infin ℎ

1015840

(119899119894) = infin119891

1015840

(119899119894) = infin and 119901119903119890(119899

119894) = 119899119894119897 where 119899

119894isin 119873


119900) = 0

Calculate 1198911015840(119899119900) = 1198921015840

(119899119900) + ℎ1015840

(119899119900) where ℎ1015840(119899




119894)) 119904119905119886119905119890(119899

119894) = 119874119875119864119873

if 119899119894= 119899119889then

Go to Step 2else



119894 119899119894+ 119904119894 119905119894) do





119898= 119904119900 119904119900+ 119904

119894minusthen


119894 119899119894+ 119904119894 119905119894) by formula (2) or (4)

if 1198921015840(119899119894) + 120587(119899

119894 119899119894+ 119904119894 119905119894) ge 1198921015840

(119899119894+) then





119894+by formula (3) or (6)

Update 1198921015840(119899119894+) with 1198921015840(119899

119894+) + 120587(119899

119894 119899119894+ 119904119894 119905119894)

Calculate 1198911015840(119899119894+) = 1198921015840

(119899119894+) + ℎ1015840

(119899119894+)

Record 119901119903119890(119899119894+) = (119899

119894 119899119894+ 119904119894 119905119894)

end forend while



119894) where 119899

119894= 119899119889 119899119889minus 119899

119900+

Return 119901119905119900 (119899119900 119899119889)







1205871015840

(119899119894 119899119895 119908 119899119894119897) =

dist (119899119894 119899119895)

Vwalk (11)

1205871015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (12)





1015840

(119899119894) with Formula (13)


ℎ1015840

(119899119894) =

dist (119899119894 119899119889)

Vmax (13)



119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894isin 119873 do

for all 119899119895isin 119873 do

119867(119894 119895) = min1199041205871015840

(119899119894 119899119895 119904 119899119894119897)

end forend for


119898isin 119873 do

for all 119899119894isin 119873 do

for all 119899119895isin 119873 do

if 119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895) then119867(119894 119895) gt 119867(119894 119896) + 119867(119896 119895)

end ifend for

end forend for


NodeArc

9

1 1

n1 n2

n3

n4 n5

n6n7

13

13

25 21

11

11

11

1525

10



Lemma7 If ℎ1015840(119899119894) le 120587(119901





119894) le 120587(119901

119905119894(119899119894 119899119889))

Let 119901119905119894(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119905119894) (119899



120587 (119901119905119894(119899119894 119899119889)) = 120587 (119899

119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


(14)

119901nil(119899119894 119899119889) = (119899

119894 1198991015840

119894+ 1199041015840

119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)


119894 119899119889) and 119901

119899119894119897

(119899119894 119899119889) are


119894 119899119895 119908 119905119894) 1205871015840(119899

119894 119899119895 119908

119899119894119897) = 120587(119899119894 119899119895 119908 119905119894)

otherwise 119904119894= 119897 120587

1015840

(119899119894 119899119895 119897 119899119894119897) = min

119896(120579

119899119895

119897119896minus 120579

119899119894

119897119896) (15)



120587 (119899119894 119899119895 119897 119905119894) = 120579

119899119895

1198971198960

minus 119905119894 (16)

where

1198960= arg min

119896

(120579

119899119894

119897119896minus 119905 | 120579

119899119894

119897119896minus 119905119894gt 0) (17)

Obviously

1205871015840

(119899119894 119899119895 119897 nil) = min

119896

(120579

119899119895

119897119896minus 120579

119899119894

119897119896) le 120579

119899119895

1198971198960

minus 120579

119899119894

1198971198960

le 120579

119899119895

1198971198960

minus 119905119894= 120587 (119899

119894 119899119895 119897 119905119894)

(18)


119894 119899119895 119904119894

119899119894119897) le 120587(119899119894 119899119895 119904119894 119905119894)

Furthermore

ℎ1015840

(119899119894) = 1205871015840

(119901119899119894119897

(119899119894 119899119889)) le 120587

1015840

(119901119905119894(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

le 1205871015840

(119899119894 119899119894+ 119904119894 119905119894) + sdot sdot sdot + 120587 (119899


= 120587 (119901119905119894(119899119894 119899119889))

(19)





(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119901119899119894119897

(119899119895 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889))

(20)




Let119901119899119894119897 (119899119895 119899119889) = (119899

119895 119899119895+ 119904119895 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

and119901119899119894119897 (119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

1015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

(21)


119894 119899119895 119904119894 119905119894)Thus

120587 (119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119901119899119894119897

(119899119895 119899119889))

= 120587 (119899119894 119899119895 119904119894 119905119894) + 1205871015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119895 119904119894 119899119894119897) + 120587

1015840

(119899119895 119899119895+ 119904119895 119899119894119897)

+ sdot sdot sdot + 1205871015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

ge 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(1198991015840

119889minus 119899119889 1199041015840

119889minus 119899119894119897)

= 1205871015840

(119901119899119894119897

(119899119894 119899119889))

(22)










119894 119899119889)) ge 120587

1015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist(119899119894 119899119889)Vmax ge 0


1015840

(119901119899119894119897

(119899119894 119899119889))


119894 119899119889)


1205871015840

(119901119899119894119897

(119899119894 119899119889)) ge

dist (119899119894 119899119889)

Vmax (23)

where 119901119899119894119897(119899119894 119899119889) = (119899

119894 119899119894+ 119904119894 119899119894119897) (119899

119889minus 119899119889 119904119889minus 119899119894119897)

1205871015840


119894 119899119895 119904119894 119899119894119897)

V(119899119894 119899119895 119904119894 119899119894119897)


119894




1205871015840

(119901119899119894119897

(119899119894 119899119889))

= 1205871015840

(119899119894 119899119894+ 119904119894 119899119894119897) + sdot sdot sdot + 120587

1015840

(119899119889minus 119899119889 119904119889minus 119899119894119897)

=

len (119899119894 119899119894+ 119904119894 119899119894119897)

V (119899119894 119899119894+ 119904119894 119899119894119897)

+ sdot sdot sdot +

len (119899119889minus 119899119889 119904119889minus 119899119894119897)

V (119899119889minus 119899119889 119904119889minus 119899119894119897)

ge


119889minus 119899119889 119904119889minus 119899119894119897)

Vmax

ge

dist (119899119894 119899119889)

Vmax

(24)









Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Plain-Alowast

noFloyd-Alowast nd




4=


5





119900 119899119889 and initial time 119905

119900are





119900 119899119889 and 119905



Bus lineMetro line

NodeRoad

l2

l3l5 l10

l8

l9

l6

l4

l1

l7

n10

n1 n11

n14

n13

n16

n30n24

n28

n26

n22

n25

n20

n29

n27

n21

n12

n23

n18

n19

n17

n5

n4 n7

n6

n9

n3

n2

n8

n15




119900 119899119889






ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ProceduresItem






ProceduresItem










119889= 11989924


119900= 60 the itinerary 11990155(119899


11990160




55 60 65 70 75 80 85 90 95 1000

2000

4000

6000

8000

10000

12000

14000


Accu

mul

ated

trav

el d

istan

ce (m


WalkingWaiting

In busIn metro

p55(n1 n24)

p60(n1 n24)n3

n1 n1 n1

n2n2

n1

n18

n15

n15

n10n24 n24

l5

l1

l1

l6


11990155

(1198991 11989924) = (119899

1 1198993 1198971 55) (119899

3 11989915 119908 70) (119899

15 11989918 1198976



1 1198992 1198971 60) (119899

2 11989910 1198975




(1198991 11989924) and 11990160(119899


1(1198991 11989924) =

(1198991 1198993 1198971) (1198993 11989915 119908) (119899

15 11989918 1198976) (11989918 11989924 119908) and sp

2(1198991

11989924) = (119899

1 1198992 1198971) (1198992 11989910 1198975) (11989910 11989924 119908) respectively as







n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












n24

n24

n10

n15

n1 n1 n2

n3

n18

l1l1

l6

l5l5

Bus lineMetro line

NodeWalk

sp1

sp2




2(1198991 11989924) sometimes


119900= 60





119900= 1198991

and 119899119889= 11989924 When 119905




98+120585



with 11990198

(1198991 11989924) while with 119901

98+120585





119900 For example 11990198+120585(119899



104

(1198991 11989924)








119900and the


60 65 70 75 80 85 90 95 100 105


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n1 n1

n2n2

n3n15 n15

n10n18

n24

n24

l1

l1

l5 l6

sp2 to = 55sp2 to = 60


95 100 105 110 115 120 125 130 135 140


0

2000

4000

6000

8000

10000

12000

14000

Accu

mul

ated

trav

el d

istan

ce (m

)


WalkingWaiting

In busIn metro

n24n24 n18

n15

n15n15

n15 n3

n1n1

n3

n18

l6l6

l1l1

p98(n1 n24)

p981(n1 n24)









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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