animprovediteratedlocalsearchalgorithmforthe...

Research ArticleAn Improved Iterated Local Search Algorithm for theStatic Partial Repositioning Problem in Bike-Sharing System

Qiong Tang 123 Zhuo Fu 13 Dezhi Zhang 13 Meng Qiu 4 and Minyi Li5

1School of Traffic and Transportation Engineering Central South University Changsha 410075 China2College of Economics and Management Hengyang Normal University Hengyang 421002 China3Smart Transport Key Laboratory of Hunan Province Changsha 410075 China4Institute for Data and Decision Analytics -e Chinese University of Hong Kong Shenzhen 518172 China5School of Science RMIT University Melbourne VIC 3000 Australia

Correspondence should be addressed to Zhuo Fu zhfucsueducn

Received 4 February 2020 Revised 7 August 2020 Accepted 27 October 2020 Published 17 November 2020

Academic Editor Dilum Dissanayake

Copyright copy 2020 Qiong Tang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper a single-vehicle static partial repositioning problem (SPRP) is investigated which distinguishes the user dissat-isfaction generated by different stationse overall objective of the SPRP is to minimize the weighted sum of the total operationaltime and the total absolute deviation from the target number of bikes at all stations An iterated local search is developed to solvethis problem A novel loading and unloading quantity adjustment operator is proposed to further improve the quality of thesolution Experiments are conducted on a set of instances from 30 to 300 stations to demonstrate the effectiveness of the proposedcustomized solution algorithm as well as the adjustment operator Using a small example this paper also reveals that the unitpenalty cost has an effect on the repositioning strategies

1 Introduction

As urban traffic congestion and environmental pollutionproblems become more and more serious bike-sharingsystems (BSSs) as a means of sustainable transportation caneffectively solve the ldquolast milerdquo problem of urban publictransportation and provide benefits for users in terms offinances health and a low-carbon lifestyle [1] Howeverwhen the BSS is in operation the demand and supply ofbikes in each station often become unbalanced at somestations users might not be able to find available bikes andin some others there are not enough lockers for users toreturn their bikes As a result user satisfaction decreases insuch situations As pointed out by Laporte et al [2] severalrelated problems arising at the strategic tactical and op-erational levels can be studied to improve the BSS Amongthose one of the well-known yet challenging program is thebike rebalancing problem or bike repositioning problem(BRP) Solutions to BRP aim to determine the optimalvehicle routes and the loading and unloading quantities at

stations in order to minimize the sum of travel cost and userdissatisfaction subject to operational constraints As acommon practice the BSS operators redistribute bikesamong unbalanced stations accordingly using a dedicatedfleet of trucks

Existing research on BRP are mainly divided into twocategories dynamic and static Dynamic BRP considersdaytime operations where the user demand is changing overtime while static BRP focuses on nighttime operations inwhich the temporal user demand change can be ignoredDespite that many of the recent works have focused ondynamic settings there are certainly practical advantagesand importance of investigating the static repositioningproblem For instance the static repositioning allows therepositioning truck to travel quickly in the city withoutaggravating the traffic congestion and parking problemsHence most existing works that take into account the nextday bike inventory arrangement in BSS have focused onstatic settings [3] In terms of the number of trucks usedstatic BRP studies are mainly divided into two categories

HindawiJournal of Advanced TransportationVolume 2020 Article ID 3040567 15 pageshttpsdoiorg10115520203040567

single and multiple vehicle problems However comparedwith multiple-vehicle BRP [3ndash21] it is more realistic toconsider a single-vehicle BRP for a particular street in thecity [22ndash32] In this paper we focuses on the single-vehiclestatic BRP

Often in the operation process of static BRP consideringvarious constraints of resource including operation timecapacity of the relocation trucks and the inherent imbalancebetween supply and demand of the BSS (ie the number ofsurplus bikes is not equal to the number of deficit bikes at allstations) there is no way to achieve a complete or perfectbalance among all stations [3] erefore in practice only apartial balance among some stations can be achieved isproblem is defined as the static partial repositioning problem(SPRP) [13] In the SPRP it is necessary to determine whichstation needed to be rebalanced and the loading andunloading quantities at those stations as well as the route oftrucks to minimize the sum of travel cost and userdissatisfaction

Existing models to SPRP define user dissatisfaction ei-ther as the sum of the unmet demand (only consider stationswith too few bikes) [4ndash9] or as the sum of the absolutedifferences between the current inventory and the targetinventory at all stations (including stations with too manybikes and stations with too few bikes) [10 11 32] eycommonly ignore the different level of dissatisfaction causedby different stations However existing study demonstratesthat user dissatisfaction caused by not satisfying one unit ofdemand may vary from station to station [12] For instanceat locations with high density of bike stations or convenienttransportation the level of user dissatisfaction may be lowbecause users can easily walk to other nearby stations or useother modes of transportation However it can be signifi-cantly higher in stations where bike sharing is the onlyoption available to users erefore it is important to dis-tinguish the level of user dissatisfaction generated by notsatisfying one unit of demand at different stations It is notvery scientific to quantify the total dissatisfaction of allstations by simply using the sum of the number of bikes thatare not satisfied at all stations

In this paper we build an SPRP on the model proposedby Di Gaspero et al [9] and Szeto et al [32] and developcustomized solution to the SPRP Different from existingwork our proposed model will distinguish the impact ofdifferent stations on user dissatisfaction by defining differentunit penalty costs associated with unsatisfied one bike atdifferent stations To the best of our knowledge we areamong the first to develop customized solutions to SPRP thatcould take into account the differentiation of user dissat-isfaction at different stations

e main contribution of this paper on the SPRP lit-erature can be summarized as follows

(1) A comprehensive review of the static BRP literatureis provided including the complete and partialrepositioning aspects

(2) A new SPRP is introduced where the level of userdissatisfaction could be different at different stations

(3) A heuristic algorithm based on iterated local searchis proposed to solve the problem

(4) A novel loading and unloading quantity adjustmentoperator is designed to further improve the quality ofthe solution

e remainder of this paper is organized as followsSection 2 reviews the BRP literature Section 3 describes themathematical formulation of the problem as well as thenonlinear programming model en in Section 4 wepresent the improved iterated local search algorithm to solvethe model We present experimental results in Section 5 andconclude this paper with discussion on potential futureresearch in Section 6

2 Literature Review

e bike-sharing system has become an increasingly studiedproblem in the last two decades Si et al [33] provided acomplete and insightful picture of all 208 relevant articlesabout bike-sharing research published from 2010 to 2018 Asmentioned before this paper focuses on the static bikerepositioning problem (BRP) and hence this section willmainly analyze the current literature on the static BRP

e research on the static BRP can be traced back to theseminal paper of Benchimol et al [22] in which the authorsintroduced the static single-vehicle BRP and proved it to beNP-hard For the first time Pal and Zhang [13] categorizedthe static BRP into complete and partial repositioningproblem based on the rigorous of a repositioning that needsto be performed Table 1 presents a summary of the literatureon the static BRP in terms of the type of repositioning(complete or partial) the number of trucks used (|V|) andthe objective function

21 Complete Repositioning Problem In the completerepositioning problem achieving a perfect balance amongstations is considered as a hard constraint In this case theideal inventory of each station must be met after therepositioning operation e objective of this problem ismostly to study how to arrange the order of stationsrsquo visits sothat the travel cost is minimized see eg [15ndash17 22ndash27]e exceptions are Pal and Zhang [13] and Wang and Szeto[14] where the objective is to minimize the makespan of therebalancing fleet and the total CO2 emissions respectively

Various solutions have been proposed to solve the staticsingle-vehicle complete repositioning problem for in-stances Benchimol et al [22] developed approximationalgorithms Chemla et al [24] presented a branch-and-cutmethod embedded Tabu search and Erdogan et al [26]proposed the first exact algorithm that consists of a branch-and-cut algorithm based on Benders combinatorial cutsLater on Cruz et al [25] addressed the same problem byproposing an iterated local search algorithm Comparedwith the algorithms proposed by Chemla et al [24] andErdogan et al [26] the iterated local search developed byCruz et al [25] has more advantages in term of solution time

2 Journal of Advanced Transportation

and quality Bruck et al [23] introduced static single-vehicleBRP with forbidden temporary operations and proposedthree exact algorithms based on different mathematicalformulations

ere are also studies on the static multivehicle completerepositioning problems for instance Bulhotildees et al [15]presented a branch-and-cut algorithm and an iterated localsearch DellrsquoAmico et al [16] presented branch-and-cutalgorithm DellrsquoAmico et al [17] developed a destroy andrepair metaheuristic algorithm which makes use of effectiveconstructive heuristic and of several local search procedures

It is worth mentioning that there are three BRP articlesoccurring in the BSS a bit different from most of thosecomplete repositioning problems studied in this survey Paland Zhang [13] modeled a static BRP in a free-floating bike-sharing system and presented a hybrid nested large neigh-borhood search with a variable neighborhood descent algo-rithmWang and Szeto [14] introduced a static green BRP withbroken bikes and applied CPLEX to determine the routing andloading decisions Lahoorpoor et al [27] proposed a bottom-upspatial cluster-based method to solve the static BRP withconsideration of the usersrsquo behavior in the BSS

22 Partial Repositioning Problem In the partial reposi-tioning problem achieving a perfect balance among stationsis considered as a soft constraint that is to say it is notnecessary to achieve the ideal inventory at each station afterrepositioning operation As can be seen from the Table 1with the exceptions of Ho and Szeto [28] and Alvarez-Valdes et al [18] the partial repositioning problems in theliterature are modeled as multiobjective optimizationproblems e goal of these problems is not only to pursuethe minimum transportation cost by determining routingdecisions but also to minimize user dissatisfaction by de-termining loading and unloading decisions at each stationUser dissatisfaction is commonly measured by the penaltycosts [3 12 19 28 29] the absolute deviation from the targetnumber of bikes [4ndash9] and the unmet demand [10 11 32]

Raviv et al [3] creatively defined the user dissatisfactionas penalty function which represents the expected number ofshortages at any inventory level at each station e authorsaddressed a multivehicle SPRP and proposed a two-phaseheuristic to first solve the routing subproblem then to solvethe loading and unloading subproblem en Forma et al[19] presented a 3-step math heuristic and later on Ho and

Table 1 Summary of the static BRP in the literature

References Type ofrepositioning |V| Objective function

Benchimol et al [22]

Completerepositioning

Bruck et al [23]Chemla et al [24]Cruz et al [25]Erdogan et al [26]Lahoorpoor et al [27] 1

Minimize total travel costBulhotildees et al [15]DellrsquoAmico et al[16 17] ge1

Pal and Zhang [13] ge1 Minimize the makespan of the rebalancing fleetWang and Szeto [14]

ge1 Minimize the total CO2 emissionsRaviv et al [3]Forma et al [19]Ho and Szeto [12]

Partialrepositioning

ge1 Minimize the weighted sum of the penalty cost and total travel timeHo and Szeto [28] 1 Minimize total penalty costTang et al [29]

1 e upper-level model minimizes the total penalty cost the lower-level model minimizesthe travel timeDi Gaspero et al [5 8]

Raidl et al [6]Rainer-Harbach et al[4 7] ge1 Minimize the weighted sum of the total absolute deviation from the target number of

bikes the total number of loadingunloading quantities and the total travel time

Di Gaspero et al [9] ge1 Minimize the weighted sum of the total absolute deviation from the target number ofbikes and the total travel time

Alvarez-Valdes et al[18] ge1 Minimizing the overall cost of unsatisfied demands

Erdogan et al [30] 1 Minimize total travel cost and handling costLi et al [31] 1 Minimize the sum of travel imbalance substitution and occupancy costs

Liu et al [10] ge1 Minimize the weighted sum of the total unmet demand the inconvenience of getting abike and total operational time

Szeto et al [32] 1 Minimize the weighted sum of the total unmet demand and the total operational timeSchuijbroek et al [20] ge1 Minimize maximum the sum of total travel loading and unloading costs

Szeto and Shui [20] ge1 Minimize the positive deviation from the tolerance of total demand dissatisfaction firstand then service time in form of the total service time or maximum service time

You [21] ge1 Minimize the sum of the total unmet demand and travel costs

is paper 1 Minimize the weighted sum of the total deviation from the target number of bikes andthe total operational time

Journal of Advanced Transportation 3

Szeto [12] developed a hybrid large neighborhood searchalgorithm (proven to be more effective than that proposed in[19]) to solve the same problem as in [3] Ho and Szeto [28]presented an iterated Tabu search to solve a single-vehicleSPRP Tang et al [29] proposed a bilevel programmingmodel by taking the outsourcing transportation mode intoaccount and presented an iterated local search algorithm andTabu search algorithm to solve the optimization problem

In [4] Rainer-Harbach et al measured the user dis-satisfaction based on the deviation from the target numberof bikes and addressed the multivehicle SPRP ey pro-posed a heuristic approach for the static BRP where therouting decisions are calculated by a variable neighborhoodsearch metaheuristic and the loading decisions are com-puted by a helper algorithm ere are a list of other so-lutions proposed for this problem setting for instance in[5] Di Gaspero et al introduced a combination of constraintprogramming and ant colony optimization approach in [6]Raidl et al provided an efficient method for determiningoptimal loading operations based on two maximum flowcomputations in [7] Rainer-Harbach et al proposed PI-LOT a greedy randomized adaptive search procedure alongwith a variable neighborhood search algorithm to solve thisproblem and in [8] Di Gaspero et al employed constraintprogramming to tackle this problem Last but not least in[9] Di Gaspero et al described a multivehicle SPRP with theobjective of minimizing both the weighted sum of the totalabsolute deviation from the target number of bikes as well asthe total travel time ey proposed a constraint pro-gramming approach that utilizes a smart branching strategyand large neighborhood search algorithm to solve theproblem

For the first time Szeto et al [32] measured the userdissatisfaction based on the unmet demand and addressed asingle-vehicle SPRP by proposing an enhanced chemicalreaction optimization algorithm Liu et al [10] studied amultivehicle SPRP for the free-floating bike-sharing systemey took into account the different convenience level ofgetting bikes and proposed an enhanced chemical reactionoptimization algorithm to solve the problem You [11]studied a multivehicle SPRP under a minimum service re-quirement over a planning horizon and proposed a two-phase heuristic algorithm to solve the problem based onlinear programming

In this section there are five articles differentiatethemselves from the above other studies Alvarez-Valdeset al [18] addressed a static BRP to optimize the level ofservice quality by a two-stage algorithm Erdogan et al [30]addressed a static BRP where the target inventory at eachstation must lie in a predetermined interval ey developedand implemented a Benders decomposition scheme and abranch-but-cut algorithm to solve the problem Li et al [31]investigated a static BRP with multiple bike types andproposed a combined hybrid genetic algorithm Schuijbroeket al [20] unified dual-bounded service level constraints(with added inventory flexibility) and vehicle routing forstatic BRP and proposed a cluster-first route-second heu-ristic Szeto and Shui [21] developed an enhanced artificialbee colony algorithm that incorporated a new set of optimal

loading and unloading strategies to solve the static BRP tominimize first the positive deviation from the tolerance ofthe total demand dissatisfaction and then the total servicetime of all trucks

As demonstrated above in complete repositioningproblem the loading and unloading quantities to a stationare given in advance as opposed to our partial repositioningproblem where they are decision variable e increase ofdecision variables also increased the computational com-plexity of solving partial positioning problems which is morein line with the operation of the actual BSSWhen it comes topartial positioning problems most researchers have focusedtheir objective on minimizing the user dissatisfactionmeasured by the sum of unmet demand or deviations fromthe target fill levels at all stations [4ndash11 32] However asmentioned previously the different level of user dissatis-faction caused by unmet demands or deviations at differentstations is another important aspect to take into accountalthough it has been ignored in most of the literature In thispaper we do not only pursue the overall minimization ofoperational costs and user dissatisfaction costs but alsoconsider the differences in user dissatisfaction at differentstations As summary above two works have solved thecomplete repositioning problem by iterated local search[23 25] only one work has used iterated local search to solvethe bilevel partial repositioning problem [29] It is obviousthat the iterated local search algorithm in the above threepapers cannot effectively solve the partial repositioningproblem studied in this paper Considering the character-istics and solution quality of the problem we improve thetraditional iterated local search algorithm and propose acustomized algorithm for solving the problem in this paperComputational results show that the improved algorithm iseffective

3 Mathematical Formulation

e bike-sharing system studied in this paper comprises adepot a set of stations and a truck Each station has arepositioning demand If the repositioning demand at astation is greater than 0 the station is defined as a pickupstation and can provide bikes to the delivery stationse setof pickup stations is denoted by P On the other hand if therepositioning demand at a station is smaller than 0 then thestation is defined as a delivery station and should be suppliedwith bikes from pickup stations e set of delivery stationsis denoted by D If the demand at a station is equal to 0 thenthe station is called a balanced station Balanced stations canbe ignored in the partial repositioning problem as there is novisiting or loadingunloading operations needed

We consider the scenario that only a single truck with agiven capacity is available e truck collects bikes frompickup stations and transports them to delivery stations Itstarts from and returns to the depot empty and operateswithin a given repositioning time constraint Moreover eachstation is allowed to be visited at most once and not allstations need to be visited e objective is to determine theroute of the truck and the loading and unloading quantitiesat each visited station to minimize the weighted sum of the


total operational time as well as the total absolute deviationfrom the target number of bikes at all stations

31 Notations

311 Sets

N set of stationsN0 set of nodes including the stations (indexed by ii isin N) and the depot (indexed by 0)P set of pickup stations where P sub N

D set of delivery stations where D sub N

R set of stations visited by the truck where R sub N

U set of stations not visited by the truck where U sub N

312 Parameters

di demand of station i at the beginning of the repo-sitioning operation if di gt 0 then i isin P if di lt 0 theni isin D

tij operational time that includes the travel time fromstation i to station j and the expected time required toload or unload bikes at station jQ truck capacityT repositioning time

M a very large numberwi the unit penalty cost associated with unsatisfied onebike at station i wi isin (0 1]foralli isin N

α the weight associated with the truckrsquos total opera-tional time

313 Decision Variables

xij binary variable that equal 1 if truck travels directlyfrom station i to station j and 0 otherwiseyij the number of bikes onto the truck when it travelsdirectly from station i to station jyP

i the number of bikes loaded on the truck at station

iyPi ge 0foralli isin P if i notin P then yP

i 0yD

i the number of bikes unloaded from truck at station

iyDi ge 0foralli isin D if i notin D then yD

i 0

32 Mathematical Formulation e formulation for theSPRP addressed in this study is given as follows

min α 1113944iisinN0

1113944jisinN0 inej

tijxij + 1113944iisinP

wi di minus yPi1113872 1113873 + 1113944

iisinDwi minusdi minus y

Di1113872 1113873

(1)

which is subject to

1113944jisinN

x0j 1113944jisinN

xj0 1 (2)

1113944jisinN

y0j 1113944jisinN

yj0 0 (3)

1113944jisinN0 jnei

xij le 1 foralli isin N (4)

1113944jisinN0 jnei

xij 1113944jisinN0 jnei

xji foralli isin N (5)

1113944ijisinN0 inej

tijxij leT (6)

qj ge qi + 1 minus M 1 minus xij1113872 1113873 foralli j isin N0 inej (7)

1113944jisinNo jnei

yij minus 1113944jisinNo jnei

yji yPi minus y

Di foralli isin N (8)

yij leQxij foralli j isin N0 inej (9)

1113944iisinN

yPi minus 1113944

iisinNy

Di 0 (10)

yDi lemin max minusdi 0( 1113857 middot 1113944

jisinN0 jne i

xij 1113944jisinN0 jne i

yji⎛⎝ ⎞⎠ foralli isin N (11)


yPi lemin max di 0( 1113857 middot 1113944

jisinN0 jneixij Q minus 1113944

jisinN0 jneiyji

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ foralli isin N (12)

qi ge 0 foralli isin N0 (13)

yPi ge 0 foralli isin N (14)

yDi ge 0 foralli isin N (15)

yij ge 0 foralli j isin N0 inej (16)

xij isin 0 1 foralli j isin N0 inej (17)

Objective function (1) is defined as the weighted sum ofthe total operational time and the total weighted sum ofdeviation from the target number of bikes Constraint (2)states that the truck departs from the depot only once andreturns to the depot Constraint (3) ensures that the truckstarts from and returns to the depot empty Constraints (4)ensure that the truck can visit a station at most onceConstraints (5) ensure that if the truck visits a station itmust leave that station Constraint (6) ensures that totaloperational time of the truck does not exceed the reposi-tioning time available Constraints (7) are the subtourelimination constraint [34] Constraints (8) require that thenumber of bikes loaded onto or unloaded from the truck at agiven station equals the difference between the truck loadbefore and after the station visit Constraints (9) ensure thatthe load on the vehicle cannot be greater than the truckcapacity Constraint (10) ensures that bikes picked up frompickup stations by the truck are eventually delivered todelivery stations Constraints (11) require that the unloadingquantities at a station should not be greater than the numberof bikes needed by that station and greater than the numberof bikes on the truck when the vehicle reaches stationConstraints (12) require that the loading quantities at astation should not be greater than the number of bikessupplied by that station and greater than the available spaceson the truck when it arrives at the station Constraints (13)ensure the auxiliary variable is nonnegative Constraints(14)ndash(16) ensure that loading and unloading quantities ateach station and the number of bikes on the truck arenonnegative integers Constraints (17) define xij to be abinary variable

4 Solution Method

Similar to other existing works (eg [15 25 29]) we useiterated local search (ILS) to solve the BRP problem eILS algorithm is a metaheuristic algorithm It obtains theoptimal solution through two mechanisms of local search(see Section 43) and perturbation search (see Section44) In this paper local search is executed through therandomized variable neighborhood decent (RVND)procedure Algorithm 1 illustrates the ILSSPRP algorithm

for the SPRP According to Algorithm 1 after con-structing the initial solution s0 by applying the initiali-zation procedure (see Section 42) the procedure RVND(see Section 43) and perturbation search (see Section 44)are called repeatedly until the predefined terminationcriterion (the maximum consecutive iterations withoutimprovement limit in our case) is satisfied e localsearch including 6 neighborhood structures such as Swap(N1) Insertion (N2) Delete (N3) Exchange (N4) 2-Opt(N5) and Reinsertion (N6) neighborhood structures isexecuted using an RVND procedure whereas the per-turbation procedure performs Swap (P1) Insertion (P2)Exchange (P3) and Reinsertion (P4) neighborhoodstructures

According to the characteristics of the SPRP we improvethe traditional ILS algorithm by adding a novel adjustmentoperator after RVND and perturbation search Details of theadjustment operator can be found in Section 45 and we alsoverify the impact of the proposed adjustment operator onthe ILSSPRP in Section 52 Let s represent the currentsolution for the SPRP and slowast represent the best solution thatcan be searched currently For a solution s TC (s) gives theobjective value of solution s We also denote ConsIter as thecurrent consecutive iterations without improvement andMaxConsIter as the maximum consecutive iterations with-out improvement respectively

41 Solution Representation A solution w is represented byw (x y q) as shown in Figure 1 where x is a vector thatstores the routing sequence of the truck (x0 x1 xn xn+1)where x0 xn+ 1 0 represents the depot xh(xh isin N h isin 1 n ) represents the station visited by thetruck the subscript h indicates the order of stations beingvisited by the truck y is a vector that stores the demandthat has been met at the station visited by the truckwhere positive and negative values indicate the quantities ofbikes collected and delivered respectively q is a vectorused to store the cumulative load after the truck visitsthe station According to equations (3) and (9) if a solutionis feasible then qh 0forallh isin 0 n + 1 and 0le qh leQ

forallh isin 0 1 n n + 1 where Q is the truck capacity


42 Initialization In this paper the initial feasible solutionis efficiently generated by a construction heuristic algorithm(see [12 28 29]) e basic construction heuristic algorithmwas first proposed by Ho and Szeto [28] to solve the single-vehicle BRP and Ho and Szeto [12] developed it to solve themultiple-vehicle BRP and then Tang et al [29] improved itto solve the bilevel BRP

e construction heuristic algorithm can be divided intotwo main steps

Step one Sort we first divide stations into two cate-gories pickup stations and delivery stations enstations are sorted in descending order by the rankingcriteria Similar to Tang et al [29] in this paper theranking criteria are |wid

Qi | where

dQi min(Q di) i isin P max(minusQ di) i isin D1113864 1113865

Step two Insert we first insert the pickup stations inturn into the route if no pickup station can be insertedinto the current route without exceeding the capacitylimits then delivery stations are inserted in turn in theroute We repeat this process until no station can beinserted without violating the predetermined reposi-tioning time constraint

43 Local Search After generating the initial solution weimprove this initial solution with local search technique Weconsider two kinds of neighborhood structures intersets andintraroute e former performs moves between set R (theset of stations visited by the truck) and set U (the set ofstations not visited by the truck) whereas the latter onlyconsiders moves within the route

431 Interset Neighborhood Structures

(i) Swap-N1 randomly select a station i from the route(set R) then select another station j (|di|le |dj|) ofthe same type from set U and exchange them

(ii) Insertion-N2 randomly select a station from the setU to insert into the route (set R)

(iii) Delete-N3 randomly select a station from the route(set R) and delete it

432 Intraroute Neighborhood Structures

(i) Exchange-N4 randomly select two stations from theroute and exchange them

(ii) 2-Opt-N5 randomly select two stations from theroute and exchange the access order of the stationsbetween the two stations

(iii) Reinsertion-N6 randomly select a station andreinsert it into the route

(iv) e new neighborhood solutions obtained by ap-plying above moves may no longer remain feasibleIn order to ensure that a sufficient number offeasible solutions can be obtained we selectivelyrepair some infeasible solutions Here we repairinfeasible solutions obtained by applying N2 and N3

(details to be discussed in Sections 461 and 462)In terms of the other moves only feasible solutionscan be accepted as the new neighborhood solutions

44 Perturbation Search In order to avoid falling into localoptimum and to realize global search perturbation is used toconstantly change the current local optimum solution efollowing four kinds of moves are used to implementperturbations

(i) Swap-P1 randomly select a station i from the routethen select another station j of the same type fromset U and exchange them

(ii) Insertion-P2 randomly select a station from the set Uand insert it into the route For the infeasibleneighborhood solution when using the repair oper-ator to repair Insertion-P2 randomly selects a feasiblesolution after repair as the new neighborhood solu-tion while Insertion-N2 chooses the best feasible so-lution after repair as the new neighborhood solution

(iii) Exchange-P3 randomly select two adjacent stationsfrom the route and swap them with two otheradjacent stations from the route

(iv) Reinsertion-P4 randomly select two adjacent sta-tions from the route delete them and reinsert themelsewhere on the route

(v) Here we repair infeasible solutions obtained byapplying P2 (details to be discussed in Section 463)For other moves P1 P3 and P4 only feasible so-lutions are accepted as the new neighborhoodsolutions

h

xh

yh

qh

0 8 9 6 2 1 3 5 4 0

0 5 4 ndash6 ndash2 5 ndash4 6 ndash8 0

0 5 3 1 6 2 8 0 09

0 1 2 3 4 5 6 7 8 9

Figure 1 Solution representation

s0 Initializationslowast s0 s s0 ConsIter 0while ConsIterleMaxConsIter do

sprime ≔RVND (s Nk) 1le k le 6s≔Adjustment (sprime)

if TC (s)ltTC (slowast) thenslowast s

ConsIter 0elseConsIterConsIter+ 1

endsprimeprimePerturbation search (slowast Pm) 1lem le 4sAdjustment (sprimeprime)

EndReturn slowast

ALGORITHM 1 ILSSPRP for SPRP


45 Adjustment Operator e basic idea of the adjustmentoperator is that a solution can be improved by properlyadjusting the loading and unloading quantities betweenstations in a heuristically way [10 12 28 32]

Similar to the pickup-delivery cycle proposed by Leiand Ouyang [35] we first define a pickup-delivery pair as asequence of consecutive visits that starts from a pickupstation (preceded by the depot or delivery station) and endsat the corresponding delivery station (followed by thedepot or pickup station) Figure 2 illustrates an example inwhich the truck visits two pickup-delivery pairs (withQ 10) e first pickup-delivery pair contains one pickupstation (station 1) and one delivery station (station 2) andthe second pickup-delivery pair also contains one pickupstation (station 4) and one delivery station (station 5) etruck starts from the depot with empty picks up 5 bikes atstation 1 and delivers directly to station 2 where 3 bikesare unloaded en 6 bikes are further collected at station4 delivers 8 bikes at station 5 and finally ends at the depotempty

In this paper we divide the adjustment operator into twocategories adjustment of the quantities of different types ofstations in one pickup-delivery pair and adjustment of thequantities of the same type of stations in different pickup-delivery pairs We first adjust the quantities of stationswithin each pickup-delivery pair and then adjust thequantities of the same type of stations across differentpickup-delivery pairs

451 Adjustment of the Quantities of Different Types ofStations in One Pickup-Delivery Pair In this case on thecurrent route we adjust the quantities of different type ofstations within each pickup-delivery pair until all pickup-delivery pairs are adjusted For one pickup-delivery pair weconsider one pickup station xi and one delivery station xj at atime (i and j indicate the order of stations being visited by thetruck respectively) with their loading and unloadingquantities increased r at the same time which is formulated

by r min dximinus yP

xi |dxj

| minus yDxj

Q minus qi1113882 1113883 subject to capacity

and loading constraints en the loading and unloadingquantities on station xi and station xj are updated at the sametime by yP

xi yP

xi+ r yD

xj yD

xj+ r

As shown in Figure 3(a) in the first pickup-delivery pairpickup station 1 and delivery station 2 are selected (withQ 10) to adjust the loading and unloading quantities In thiscase the increased loadingunloading quantities r min d1minus1113864

yP1 |d2| minus yD

2 Q minus q1 min 7 minus 5 6 minus 3 10 minus 5 2 enyP1 yP

1 + 2 5 + 2 7 yD2 yD

2 + 2 3 + 2 5 In thesecond pickup-delivery pair pickup station 4 and deliverystation 5 are selected as shown in Figure 3(b) r 1 thenyP4 yP

4 + 1 6 + 1 7 yD5 yD

5 + 1 8 + 1 9

452 Adjustment of the Quantities of the Same Type ofStations in Different Pickup-Delivery Pairs For two stationsof the same type adjusting the loading (for pickup sta-tions) and unloading (for delivery stations) quantitiesbetween them can improve the current solution under the

condition of ensuring the feasibility of the solution afterthe transformation Each time two stations of the sametype in different pickup-delivery pairs are selected andthe loading and unloading quantities of one station areincreased by 1 unit while that of the other station isreduced by 1 unit resulting in a decrease in the value ofthe objective function (ie the weighted sum of the op-erational time and deviation from target) e procedureis repeated until there is no loading and unloadingquantity adjustment between stations to improve thecurrent solution

Consider a current feasible solution shown in Figure 2Figures 4(a) and 4(b) show an example where pickupstation 1 (in the first pickup-delivery pair) and pickupstation 4 (in the second pickup-delivery pair) are selectedthen we shift two unit quantity between them For in-stance in Figure 4(a) we increase the loading quantities atstation 1 by two and reduce the loading quantities atstation 4 by two We do the opposite and it is shown inFigure 4(b) Under the two transformations the newsolution that could reduce the objective value will beaccepted

Similarly as shown in Figures 4(c) and 4(d) deliverystation 2 in the first pickup-delivery pair and delivery station5 in the second pickup-delivery pair are selected then weshift two unit quantities between them Figure 4(c) shows theoperation where we increase the unloading quantities atstation 2 by two and decrease the unloading quantities atstation 5 by two And Figure 4(d) shows the opposite op-eration Similarly between the two newly generated solu-tions we accept the one that could reduce the objective valueas the current solution

46 Repair Operator ere is a couple of constraints for asolution to be feasible First of all the sum of the totalsatisfied demand of stations visited by the truck on theroute is 0 (according to equation (10)) For instance afeasible solution is shown in Figure 5 suppose Q 10 andthe total satisfied demand of stations visited by the truck is0 that is 4minus3 + 2 + 2minus5 0 erefore if we want to ensurethat the neighbor solution obtained by applying the In-sertion and Delete is feasible we must make sure that thesum of the total satisfied demand of stations visited by thetruck is 0 Besides we also need to make sure that thesolution meets the capacity and loading constraints

1 5D D420 2 8 0

+5 +6 ndash8

Pickup-delivery pair

D Depot d1 = 7 d2 = ndash6

Station d4 = 8 d5 = ndash9

5

ndash3

Figure 2 Illustration of pickup-delivery pair


(according to equations (3) and (9)) In the followingsections we describe the repair procedure for infeasiblesolutions obtained by local search and perturbationsearch respectively

461 Repairing Infeasible Solutions after Applying N2Given a feasible solution if we insert a station into thesolution the obtain new neighbor solution is certainly notfeasible However the feasibility of the new neighbor so-lution can be ensured by adjusting the loadingunloadingquantity of a certain station Assuming that the currentsatisfied demand of the inserted station is ΔI then it isnecessary to reduce ΔI on the basis of the current satisfieddemand of a certain station on the route in order to balanceout the sum of the total satisfied demand on the route to be 0In this case the new neighbor solution after applying therepair procedure is feasible if the capacity and loadingconstraints are satisfied ere might be more than onefeasible adjustment scheme under which the solution withthe lowest objective value is selected as the new neighborsolution

As shown in Figure 6 if we insert station 6 (ΔI yP6 2)

into the current feasible solution (as shown in Figure 5) thetotal of all satisfied demands is 2 but not 0 en we need tochoose one station from the route and reduce its satisfieddemand by ΔI As shown in Figure 7 there are three feasibleneighbor solutions can be obtained by applying the repairprocedure and we choose the neighbor solution with thelowest objective value as the new feasible solution

462 Repairing Infeasible Solutions after Applying N3Assuming that the current satisfied demand for the deletedstation is ΔD then it is necessary to increase ΔD on the basisof the current satisfied demand of a certain station on theroute to ensure that the sum of the total satisfied demand onthe route is 0 Similarly if there are more than one feasibleneighbor solutions the solution with the lowest objectivevalue is selected as the new neighbor solution

As shown in Figure 8 if we delete station 4 (ΔD yP4 2)

from the current feasible solution (Figure 5) the total of allsatisfied demand is minus2 erefore we need to choose onestation from the route and increase its satisfied demand byΔD As shown in Figure 9 there are three feasible neighborsolutions can be obtained from the repair procedure and wechoose the neighbor solution with the lowest objective valueas the new neighbor solution

463 Repairing Infeasible Solutions after Applying P2 Inthis case the repair operation would be the same as inSection 461 the only difference is that when there is morethan one feasible neighborhood solution generated werandomly select one as the new neighbor solution

5 Computational Results

In this section we present experiments conducted to verifythe effectiveness of the proposed ILSSPRP algorithm as well asthe proposed adjustment operator We compare the per-formance of our proposed solution with the optimal solutionof Lingo 18 We also analyze through experiments the

1 5D D420 2 8 0

+7 ndash5 +6 ndash8

7

(a)

1 5D D420 2 9 0

+7 +7 ndash9ndash5

7

(b)

Figure 3 Neighbor solutions after applying the adjustment operator in one pickup-delivery pair (a) Adjustment of the loading andunloading quantities between stations in the first pickup-delivery pair (b) Adjustment of the loading and unloading quantities betweenstations in the second pickup-delivery pair

1 5D D420 4 8 0

+7 ndash3 +4 ndash8

7

(a)

1 5D D420 0 8 0

+3 ndash3 +8 ndash8

3

(b)

1 5D D420 0 6 0

+5 ndash5 +6 ndash6

5

(c)

1 5D D420 3 9 0

+5 ndash2 +6 ndash9

5

(d)

Figure 4 Neighbor solutions after applying the adjustment operator between two pickup-delivery pairs (a) Adjustment of the loadingquantities between station 1 and station 4 (b) Adjustment of the loading quantities between station 1 and station 4 (c) Adjustment of theunloading quantities between station 2 and station 5 (d) Adjustment of the unloading quantities between station 2 and station 5

31 5D D420 4 1 3 5 0

+4 ndash3 +2 +2 ndash5

d1 = 7 d2 = ndash5 d3 = 3

d4 = 3 d5 = ndash8

Figure 5 A feasible solution

31 5D D42+4 ndash3+2 +2 ndash5

6+2

0 6 3 5 7 24

Figure 6 An infeasible solution with station 6 inserted


influence of the change of weight coefficient α on theproposed ILSSPRP and the change of unit penalty cost wi onthe repositioning strategies

We follow the instances proposed by Rain-Harbach et al[7] to conduct the experiments where the size of the in-stances is ranging from 30 to 300 stations For each instancewe select the first seed of the 30 independent seeds einitial objective value the total pickup and total drop-offquantities of each instances are listed in Table 2We considerinstances with 30 and 60 stations as the small-size instancesinstances contain 120 and 180 stations as the medium-sizeinstances and instances with 240 and 300 stations as thelarge-size instances In this study unless otherwise specifiedwe follow the setting in Szeto et al [32] α 000001 ree

repositioning time constraints (in seconds) were usedT14400 s T 21600 s and T 28800 s Two truck capac-ities (in terms of the number of bikes) were consideredQ 10 and Q 20 respectively e unit penalty cost wi

associated with one unsatisfied bike at station i was ran-domly generated in the interval (0 1] foralli isin N

e experiments are conducted in Matlab on a com-puter with Intel Core i5-4590 CPU 330GHz and 4GBRAM We run each instance for 20 times the best andaverage of the solutions and the average computing time ofthose 20 runs are used to evaluate the performance of theILSSPRP

e main parameter of ILSSPRP to be calibrated is themaximum consecutive iterations without improvement(MaxConsIter) Here we set its value as a function of theinstance MaxConsItermax100 |N| similar to Bulhotildeeset al [15]

51 Comparison between Lingo and ILSSPRP In terms of thequality of the solution and the calculation efficiency Tables 3ndash5show the comparison results obtained by Lingo and ILSSPRPunder different repositioning times (T14400 s T 21600 sand T 28800 s) truck capacity (Q 10 and Q 20) andnumber of stations (|N|) GapBest and GapAvg () indicate theperformance of the ILSSPRP relative to that of Lingo based on thebest and average objective values respectively e CPU (inseconds) indicates the average computing time of the ILSSPRP orthe computing time for Lingo respectively

From Tables 3ndash5 we can see that Lingo is not able toobtain any feasible solutions for the large-size problems(where |N| 240 and |N| 300) It also generally fails toobtain any global optimal solutions within 3600 secondseven for small-size problems In contrast for the various sizeof problems from small to large the average computing timeof the ILSSPRP varies from 712 s to 3371 s

As shown in the Tables 3ndash5 almost all of the values ofGapBest and GapAvg are negative (with only one exception forthe results obtained under |N| 60 Q 10 T14400 s) eabsolute values of the average GapBest and GapAvg valuesobtained by the ILSSPRP for the scenarios with long repo-sitioning time are larger than those with short repositioningtime GapBest and GapAvg are respectively minus353 and minus331for T14400 s and minus546 and minus445 for T 21600 s butminus2359 and minus2081 for T 28800 s is implies that theILSSPRP performs better than Lingo when solving the SPRPwith a longer repositioning time

erefore we can draw the conclusion that the proposedILSSPRP can obtain better feasible solutions much faster thanLingo

52-eEffectiveness of IntroducingAdjustmentOperator intoILSSPRP In this study the adjustment operator (presented inSection 45) which considers the characteristics of the SPRP isintroduced to improve the solution quality obtained by theproposed ILSSPRP In this section we study through experimentsthe effectiveness of the proposed adjustment operator by

31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)

Figure 9 Feasible solutions after applying repair operator (a)Adjustment of the loading quantities of station 1 (b) Adjustment ofthe unloading quantities of station 2 (c) Adjustment of theunloading quantities of station 5

31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0

+4 ndash3+2 +2 ndash76

+2

4

(c)

Figure 7 Feasible solutions after applying repair operator (a)Adjustment of the loading quantities of station 1 (b) Adjustment ofthe unloading quantities of station 2 (c) Adjustment the unloadingquantities of station 5

31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5

Figure 8 An infeasible solution with station 4 deleted


Table 3 Result comparison of Lingo and ILSSPRP (T14400 s)

Instance Lingo ILSSPRP GapBest () GapAvg ()|N| Q Optimal value CPU Best value Average value CPU

30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041

120 10 2737440 3600 2712440 2719101 665 minus092 minus06720 2968180 3600 2569386 2570192 674 minus1552 minus1548


240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063

Avg 317913 712 minus353 minus331

Table 4 Result comparison of Lingo and ILSSPRP (T 21600 s)






240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081

Table 2 Characterization of the instances

|N| Initial objective value Total pickup quantities Total drop-off quantities30 992 107 10760 1703 179 179120 3307 354 354180 6064 557 557240 7604 785 785300 8522 883 883


comparing the performance of the ILSSPRP with and withoutadjustment operator using the various instances described inSection 51

Tables 6ndash8 present the results obtained by the proposedILSSPRP algorithm with and without the adjustment operatorunder different repositioning times respectively CPU denotesthe average computing time in seconds GapBest andGapAvg here

are the deviations of the bestaverage of the objective values from20 runs obtained by the ILSSPRP with the adjustment operatorfrom the corresponding optimal solution obtained by ILSSPRPwithout the adjustment operator respectively From Tables 6ndash8we can see ILSSPRP without the adjustment operator resulting ina shorter computation time than the ILSSPRP with the adjust-ment operator

Table 6 Result comparison of ILSSPRP with and without the adjustment operator (T14400 s)

Instance ILSSPRP without adjustment ILSSPRP GapBest () GapAvg ()|N| Q Best value Average value CPU Best value Average value CPU

30 10 516440 519440 206 486440 486440 427 minus617 minus67820 483386 489039 223 414386 414386 458 minus1665 minus1802





300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005

20 7940284 7940284 448 7891392 7921236 1063 minus062 minus024Avg 317 712 minus381 minus390

Table 7 Result comparison of ILSSPRP with and without the adjustment operator (T 21600 s)







300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200



However the GapBest and GapAvg values for all discussedinstances are negative which implies that the adjustmentoperator is necessary to improve the quality of the solutionAs expected when T increases the absolute average value ofGapBest and GapAvg increases is also indicates that theILSSPRP with the adjustment operator is more effective thanthe ILSSPRP without the adjustment operator to solve theSPRP with long repositioning time erefore it can beconcluded that it is beneficial to incorporate adjustmentoperator into the ILSSPRP to solve the SPRP

53 Sensitivity Analysis on α In this set of experiments weanalyze the sensitivity of the performance of the proposedapproach to the values of α Note that increasing the value ofα means that we are increasing the impact of operationaltime on the whole weighted objective function When we setα 0 it means that the objective function only considersuser dissatisfaction

We test the sensitivity of α with a small-size instanceconsisting of 30 stations (|N| 30) and consider the threerepositioning time constraints (T14400 s T 21600 s andT 28800 s) and two truck capacity constraints (Q 10 andQ 20)

Different values of α from 0 to 0001 were consideredand for each value we perform 20 runs of the experimente results obtained with Lingo 18 and the ILSSPRP werecompared in Table 9 CPU (in seconds) is the computingtime for Lingo or the average computing time of 20 runs forthe ILSSPRP GapBest and GapAvg are the deviations of thebestaverage of the objective values from 20 runs obtained bythe ILSSPRP from the corresponding optimal solution ob-tained by Lingo respectively

As shown in Table 9 almost all of the values of GapBestand GapAvg are negative (with only one exception for theresults obtained under α 0 T 21600 s Q 10) e av-erage values of GapBest and GapAvg are minus1229 and minus872respectively is then indicates that the proposed ILSSPRP

Table 9 Sensitivity analysis on α

Instance Lingo ILSSPRP GapBest () GapAvg ()α T Q Optimal value CPU Best value Average value CPU

0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503

21600 10 298000 3600 295000 300200 329 minus102 07320 223000 3600 205000 212000 505 minus878 minus519


000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450

20 363000 3600 362400 362550 2224 minus017 minus012Avg 303213 828 minus1229 minus872

Table 10 Parameter setting

Station parameter 1 2 3 4 5 6di 9 6 minus6 minus6 8 minus5wi 08 06 08 06 02 07

Table 11 e value of objective function and the route of the truck in different scenarios

Scenario e value of objective function Route and loadingunloading quantities at stations1 180648 0⟶ 2(+5)⟶ 6(minus5)⟶ 1(+6)⟶ 4(minus6)⟶ 02 76708 0⟶1(+9)⟶ 3(minus6)⟶ 2(+2)⟶ 6(minus5)⟶ 0


can always obtain better solutions than Lingo under differentvalues of α repositioning time and truck capacity eaverage computing time of the ILSSPRP is 828 s which issignificantly less than that of Lingo (303213 s)erefore wecan conclude that the ILSSPRP works well under differentvalues of α

54 Sensitivity Analysis on wi In this section a small ex-ample is constructed to investigate the sensitivity of theproposed model to the parameter wi We consider 2 sce-narios and investigate the repositioning strategiesrespectively

(i) Scenario 1 we solve the model without taking intoaccount the different values of wi

(ii) Scenario 2 we consider different values of wi andsolve the model accordingly

We select the first six stations in the instance of |N|

30 mentioned in Section 51 and set T 7200 s andQ 10 e tested values of the parameters including diand wi are shown in Table 10 Note that when di gt 0station i is a pickup station otherwise if di lt 0 station i isa delivery station e results are displayed in Table 11including the value of the objective function the route ofthe truck and the loadingunloading quantities at eachstation

As shown in Table 11 the experimental results ofscenario 1 and scenario 2 are significantly different Forpickup station 1 and 2 in scenario 2 the value of w1 isrelatively large is means that the level of user dissat-isfaction (for one bike) of station 1 is higher than that ofstation 2 erefore in order to reduce the total cost it isnecessary to increase the loading quantity at station 1 tomeet the demand of station 1 as much as possible eloading quantity of station 1 increases from 6 bikes inscenario 1 (where we disregard the different values of wi )to 9 bikes in scenario 2 (d1 9 the maximum loadingquantity of station 1 is 9 bikes) similarly the loadingquantity of station 2 decreases from 5 bikes in scenario 1to 2 bikes in scenario 2

Regarding the delivery stations 3 4 and 6 in scenario 2because w3gtw6gtw4 the priorities of meeting the demandare first given to station 3 followed by station 6 and last tostation 4 From the results we can see that the unloadingquantity of station 3 increases from 0 bike in scenario 1 to 6bikes in scenario 2 (d3 minus6 the maximum unloadingquantity of station 3 is 6 bikes) while the unloadingquantity of station 4 decreases from 6 bikes in scenario 1 to0 bike in scenario 2

From the above results it can be seen that differentvalues of wi can directly impact the loadingunloadingquantities of repositioning truck at each station as well asthe route of the truck ie the repositioning strategy of theBSS operators erefore it can be concluded that it isnecessary to take distinguishing the user dissatisfactiongenerated by different stations into account when mod-eling the bike repositioning problem which is the keyfactor to the optimization of the bike repositioning

6 Conclusions

In this paper we study the static partial repositioningproblem where the level of user dissatisfaction differs indifferent stations We propose an iterated local search al-gorithm ILSSPRP to find solutions to the problem as well as anadjustment operator to further improve the obtained so-lutions We conduct a comprehensive set of experiments tostudy the performance of the proposed approach in terms ofsolution quality and computational time compared withLingo e experimental results show that the proposedILSSPRP almost always outperforms Lingo from small- tolarge-size problems as well as under different relative im-portance set to the two objectives (ie operational time anddeviation from the target number of bikes) We also use asmall example to illustrate that the unit penalty cost wi hasan effect on the repositioning route and the loadingunloading quantities at each station

ere are potential directions for future research Forinstance unbalance demands in the bike-sharing system willnot only increase user dissatisfaction but also aggravateurban traffic congestion erefore in addition to consid-ering economic benefits we will also need to consider en-vironmental benefits in the bike repositioning problem Howto introduce environmental benefits into the objectivefunction would be an interesting topic for future researchBesides we are also interested in developing efficient andeffective algorithms to verify feasibility of a given solution

Data Availability

All data included in this study are available from the cor-responding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors acknowledge the National Natural ScienceFoundation of China (grant no 71271220) the Project ofSocial Science Achievement Review Committee of HunanProvince (grant no XSP20YBZ165) and the HengyangSocial Science Foundation Project (grant no 2020B(II)003)

References

[1] J G Shi H Si G Wu Y Su and J Lan ldquoCritical factors toachieve dockless bike-sharing sustainability in China astakeholder-oriented network perspectiverdquo Sustainabilityvol 10 no 6 2090 pages 2018

[2] G Laporte F Meunier and R Wolfler Calvo ldquoShared mo-bility systems an updated surveyrdquo Annals of OperationsResearch vol 271 no 1 pp 105ndash126 2018

[3] T Raviv M Tzur and I A Forma ldquoStatic repositioning in abike-sharing system models and solution approachesrdquo EUROJournal on Transportation and Logistics vol 2 no 3pp 187ndash229 2013


[4] M Rainer-Harbach P Papazek B Hu and G R RaidlldquoBalancing bicycle sharing systems a variable neighborhoodsearch approachrdquo in Evolutionary Computation in Combi-natorial Optimization pp 121ndash132 Springer Berlin Ger-many 2013 Lecture Notes in Computer Science 7832

[5] L Di Gaspero A Rendl and T Urli ldquoA hybrid ACO+CP forbalancing bicycle sharing systemsrdquo in hybrid Metaheuristicspp 198ndash212 Springer Berlin Germany 2013 Lecture Notesin Computer Science 7919

[6] G R Raidl B Hu M Rainer-Harbach and P PapazekldquoBalancing bicycle sharing systems improving a VNS byefficiently determining optimal loading operationsrdquo in hybridMetaheuristics pp 130ndash143 Springer Berlin Germany 2013Lecture Notes in Computer Science 7919

[7] M Rainer-Harbach P Papazek G R Raidl B Hu andC Kloimullner ldquoPILOT GRASP and VNS approaches for thestatic balancing of bicycle sharing systemsrdquo Journal of GlobalOptimization vol 63 no 3 pp 597ndash629 2015

[8] L D Gaspero A Rendl and T Urli ldquoBalancing bike sharingsystems with constraint programmingrdquo Constraints vol 21no 2 pp 318ndash348 2016

[9] L Di Gaspero A Rendl and T Urli ldquoConstraint-basedapproaches for balancing bike sharing systemsrdquo in Principlesand Practice of Constraint Programming pp 758ndash773Springer Berlin Germany 2013 Lecture Notes in ComputerScience 8124

[10] Y Liu W Y Szeto and S C Ho ldquoA static free-floating bikerepositioning problem with multiple heterogeneous vehiclesmultiple depots and multiple visitsrdquo Transportation ResearchPart C Emerging Technologies vol 92 pp 208ndash242 2018

[11] P-S You ldquoA two-phase heuristic approach to the bikerepositioning problemrdquo Applied Mathematical Modellingvol 73 pp 651ndash667 2019

[12] S C Ho and W Y Szeto ldquoA hybrid large neighborhoodsearch for the static multi-vehicle bike-repositioning prob-lemrdquo Transportation Research Part B Methodological vol 95pp 340ndash363 2017

[13] A Pal and Y Zhang ldquoFree-floating bike sharing solving real-life large-scale static rebalancing problemsrdquo TransportationResearch Part C Emerging Technologies vol 80 pp 92ndash1162017

[14] Y Wang and W Y Szeto ldquoStatic green repositioning in bikesharing systems with broken bikesrdquo Transportation ResearchPart D Transport and Environment vol 65 pp 438ndash4572018

[15] T Bulhotildees A Subramanian G Erdogan and G Laporteldquoe static bike relocation problem with multiple vehicles andvisitsrdquo European Journal of Operational Research vol 264no 2 pp 508ndash523 2018

[16] M DellrsquoAmico E HadjicostantinouM Iori and S Novellanildquoe bike sharing rebalancing problem mathematical for-mulations and benchmark instancesrdquo OMEGA-the Interna-tional Journal of Management Science vol 45 pp 7ndash19 2014

[17] M DellrsquoAmico M Iori S Novellani and T Stutzle ldquoA de-stroy and repair algorithm for the bike sharing rebalancingproblemrdquo Computer amp Operations Research vol 71pp 149ndash162 2016

[18] R Alvarez-Valdes J M Belenguer E Benavent et al ldquoOp-timizing the level of service quality of a bike-sharing systemrdquoOmega vol 62 pp 163ndash175 2016

[19] I A Forma T Raviv and M Tzur ldquoA 3-step math heuristicfor the static repositioning problem in bike-sharing systemsrdquoTransportation Research Part B Methodological vol 71pp 230ndash247 2015

[20] J Schuijbroek R C Hampshire and W-J van Hoeve ldquoIn-ventory rebalancing and vehicle routing in bike sharingsystemsrdquo European Journal of Operational Research vol 257no 3 pp 992ndash1004 2017

[21] W Y Szeto and C S Shui ldquoExact loading and unloadingstrategies for the static multi-vehicle bike repositioningproblemrdquo Transportation Research Part B Methodologicalvol 109 pp 176ndash211 2018

[22] M Benchimol P Benchimol B Chappert et al ldquoBalancingthe stations of a self service ldquobike hirerdquo systemrdquo RAIR-OmdashOperations Research vol 45 no 1 pp 37ndash61 2011

[23] B P Bruck F Cruz M Iori and A Subramanian ldquoe staticbike sharing rebalancing problem with forbidden temporaryoperationsrdquo Transportation Science vol 53 no 3pp 882ndash896 2019

[24] D Chemla F Meunier and R Wolfler Calvo ldquoBike sharingsystems solving the static rebalancing problemrdquo DiscreteOptimization vol 10 no 2 pp 120ndash146 2013

[25] F Cruz A Subramanian B P Bruck and M Iori ldquoAheuristic algorithm for a single vehicle static bike sharingrebalancing problemrdquo Computers amp Operations Researchvol 79 pp 19ndash33 2017

[26] G Erdogan M Battarra and R Wolfler Calvo ldquoAn exactalgorithm for the static rebalancing problem arising in bicyclesharing systemsrdquo European Journal of Operational Researchvol 245 no 3 pp 667ndash679 2015

[27] B Lahoorpoor H Faroqi A Sadeghi-Niaraki andS-M Choi ldquoSpatial cluster-based model for static rebalancingbike sharing problemrdquo Sustainability vol 11 no 11 p 32052019

[28] S C Ho and W Y Szeto ldquoSolving a static repositioningproblem in bike-sharing systems using iterated tabu searchrdquoTransportation Research Part E Logistics and TransportationReview vol 69 pp 180ndash198 2014

[29] Q Tang Z Fu and M Qiu ldquoA bilevel programming modeland algorithm for the static bike repositioning problemrdquoJournal of Advanced Transportation vol 2019 Article ID8641492 19 pages 2019

[30] G Erdogan G Laporte and R Wolfler Calvo ldquoe staticbicycle relocation problem with demand intervalsrdquo EuropeanJournal of Operational Research vol 238 no 2 pp 451ndash4572014

[31] Y Li W Y Szeto J Long and C S Shui ldquoA multiple typebike repositioning problemrdquo Transportation Research Part BMethodological vol 90 pp 263ndash278 2016

[32] W Y Szeto Y Liu and S C Ho ldquoChemical reaction opti-mization for solving a static bike repositioning problemrdquoTransportation Research Prat D Transport and Environmentvol 47 pp 104ndash135 2016

[33] H Si J-G Shi G Wu J Chen and X Zhao ldquoMapping thebike sharing research published from 2010 to 2018 a scien-tometric reviewrdquo Journal of Cleaner Production vol 213pp 415ndash427 2019

[34] C E Miller A W Tucker and R A Zemlin ldquoInteger pro-gramming formulation of traveling salesman problemsrdquoJournal of the ACM vol 7 no 4 pp 326ndash329 1960

[35] C Lei and Y Ouyang ldquoContinuous approximation for de-mand balancing in solving large-scale one-commodity pickupand delivery problemsrdquo Transportation Research Part BMethodological vol 109 pp 90ndash109 2018


single and multiple vehicle problems However comparedwith multiple-vehicle BRP [3ndash21] it is more realistic toconsider a single-vehicle BRP for a particular street in thecity [22ndash32] In this paper we focuses on the single-vehiclestatic BRP

Often in the operation process of static BRP consideringvarious constraints of resource including operation timecapacity of the relocation trucks and the inherent imbalancebetween supply and demand of the BSS (ie the number ofsurplus bikes is not equal to the number of deficit bikes at allstations) there is no way to achieve a complete or perfectbalance among all stations [3] erefore in practice only apartial balance among some stations can be achieved isproblem is defined as the static partial repositioning problem(SPRP) [13] In the SPRP it is necessary to determine whichstation needed to be rebalanced and the loading andunloading quantities at those stations as well as the route oftrucks to minimize the sum of travel cost and userdissatisfaction

Existing models to SPRP define user dissatisfaction ei-ther as the sum of the unmet demand (only consider stationswith too few bikes) [4ndash9] or as the sum of the absolutedifferences between the current inventory and the targetinventory at all stations (including stations with too manybikes and stations with too few bikes) [10 11 32] eycommonly ignore the different level of dissatisfaction causedby different stations However existing study demonstratesthat user dissatisfaction caused by not satisfying one unit ofdemand may vary from station to station [12] For instanceat locations with high density of bike stations or convenienttransportation the level of user dissatisfaction may be lowbecause users can easily walk to other nearby stations or useother modes of transportation However it can be signifi-cantly higher in stations where bike sharing is the onlyoption available to users erefore it is important to dis-tinguish the level of user dissatisfaction generated by notsatisfying one unit of demand at different stations It is notvery scientific to quantify the total dissatisfaction of allstations by simply using the sum of the number of bikes thatare not satisfied at all stations

In this paper we build an SPRP on the model proposedby Di Gaspero et al [9] and Szeto et al [32] and developcustomized solution to the SPRP Different from existingwork our proposed model will distinguish the impact ofdifferent stations on user dissatisfaction by defining differentunit penalty costs associated with unsatisfied one bike atdifferent stations To the best of our knowledge we areamong the first to develop customized solutions to SPRP thatcould take into account the differentiation of user dissat-isfaction at different stations

e main contribution of this paper on the SPRP lit-erature can be summarized as follows

(1) A comprehensive review of the static BRP literatureis provided including the complete and partialrepositioning aspects

(2) A new SPRP is introduced where the level of userdissatisfaction could be different at different stations

(3) A heuristic algorithm based on iterated local searchis proposed to solve the problem

(4) A novel loading and unloading quantity adjustmentoperator is designed to further improve the quality ofthe solution

e remainder of this paper is organized as followsSection 2 reviews the BRP literature Section 3 describes themathematical formulation of the problem as well as thenonlinear programming model en in Section 4 wepresent the improved iterated local search algorithm to solvethe model We present experimental results in Section 5 andconclude this paper with discussion on potential futureresearch in Section 6

2 Literature Review

e bike-sharing system has become an increasingly studiedproblem in the last two decades Si et al [33] provided acomplete and insightful picture of all 208 relevant articlesabout bike-sharing research published from 2010 to 2018 Asmentioned before this paper focuses on the static bikerepositioning problem (BRP) and hence this section willmainly analyze the current literature on the static BRP

e research on the static BRP can be traced back to theseminal paper of Benchimol et al [22] in which the authorsintroduced the static single-vehicle BRP and proved it to beNP-hard For the first time Pal and Zhang [13] categorizedthe static BRP into complete and partial repositioningproblem based on the rigorous of a repositioning that needsto be performed Table 1 presents a summary of the literatureon the static BRP in terms of the type of repositioning(complete or partial) the number of trucks used (|V|) andthe objective function

21 Complete Repositioning Problem In the completerepositioning problem achieving a perfect balance amongstations is considered as a hard constraint In this case theideal inventory of each station must be met after therepositioning operation e objective of this problem ismostly to study how to arrange the order of stationsrsquo visits sothat the travel cost is minimized see eg [15ndash17 22ndash27]e exceptions are Pal and Zhang [13] and Wang and Szeto[14] where the objective is to minimize the makespan of therebalancing fleet and the total CO2 emissions respectively

Various solutions have been proposed to solve the staticsingle-vehicle complete repositioning problem for in-stances Benchimol et al [22] developed approximationalgorithms Chemla et al [24] presented a branch-and-cutmethod embedded Tabu search and Erdogan et al [26]proposed the first exact algorithm that consists of a branch-and-cut algorithm based on Benders combinatorial cutsLater on Cruz et al [25] addressed the same problem byproposing an iterated local search algorithm Comparedwith the algorithms proposed by Chemla et al [24] andErdogan et al [26] the iterated local search developed byCruz et al [25] has more advantages in term of solution time








































31 Notations

311 Sets





312 Parameters









i 0yD



i 0



1113944jisinN0 inej


wi di minus yPi1113872 1113873 + 1113944


Di1113872 1113873

(1)

which is subject to

1113944jisinN

x0j 1113944jisinN

xj0 1 (2)

1113944jisinN

y0j 1113944jisinN

yj0 0 (3)

1113944jisinN0 jnei


1113944jisinN0 jnei




tijxij leT (6)


1113944jisinNo jnei


yji yPi minus y



1113944iisinN

yPi minus 1113944

iisinNy

Di 0 (10)


jisinN0 jne i






jisinN0 jneiyji








4 Solution Method










Qi | where




















h

xh

yh

qh

0 8 9 6 2 1 3 5 4 0


0 5 3 1 6 2 8 0 09

0 1 2 3 4 5 6 7 8 9







EndReturn slowast








xi |dxj

| minus yDxj



xi yP

xi+ r yD

xj yD

xj+ r


yP1 |d2| minus yD


1 + 2 5 + 2 7 yD2 yD


4 + 1 6 + 1 7 yD5 yD

5 + 1 8 + 1 9






1 5D D420 2 8 0

+5 +6 ndash8




5

ndash3













1 5D D420 2 8 0

+7 ndash5 +6 ndash8

7

(a)

1 5D D420 2 9 0

+7 +7 ndash9ndash5

7

(b)


1 5D D420 4 8 0

+7 ndash3 +4 ndash8

7

(a)

1 5D D420 0 8 0

+3 ndash3 +8 ndash8

3

(b)

1 5D D420 0 6 0

+5 ndash5 +6 ndash6

5

(c)

1 5D D420 3 9 0

+5 ndash2 +6 ndash9

5

(d)


31 5D D420 4 1 3 5 0


d1 = 7 d2 = ndash5 d3 = 3

d4 = 3 d5 = ndash8



6+2

0 6 3 5 7 24














31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)


31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0


+2

4

(c)


31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5





30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References












































































31 Notations

311 Sets





312 Parameters









i 0yD



i 0



1113944jisinN0 inej


wi di minus yPi1113872 1113873 + 1113944


Di1113872 1113873

(1)

which is subject to

1113944jisinN

x0j 1113944jisinN

xj0 1 (2)

1113944jisinN

y0j 1113944jisinN

yj0 0 (3)

1113944jisinN0 jnei


1113944jisinN0 jnei




tijxij leT (6)


1113944jisinNo jnei


yji yPi minus y



1113944iisinN

yPi minus 1113944

iisinNy

Di 0 (10)


jisinN0 jne i






jisinN0 jneiyji








4 Solution Method










Qi | where




















h

xh

yh

qh

0 8 9 6 2 1 3 5 4 0


0 5 3 1 6 2 8 0 09

0 1 2 3 4 5 6 7 8 9







EndReturn slowast








xi |dxj

| minus yDxj



xi yP

xi+ r yD

xj yD

xj+ r


yP1 |d2| minus yD


1 + 2 5 + 2 7 yD2 yD


4 + 1 6 + 1 7 yD5 yD

5 + 1 8 + 1 9






1 5D D420 2 8 0

+5 +6 ndash8




5

ndash3













1 5D D420 2 8 0

+7 ndash5 +6 ndash8

7

(a)

1 5D D420 2 9 0

+7 +7 ndash9ndash5

7

(b)


1 5D D420 4 8 0

+7 ndash3 +4 ndash8

7

(a)

1 5D D420 0 8 0

+3 ndash3 +8 ndash8

3

(b)

1 5D D420 0 6 0

+5 ndash5 +6 ndash6

5

(c)

1 5D D420 3 9 0

+5 ndash2 +6 ndash9

5

(d)


31 5D D420 4 1 3 5 0


d1 = 7 d2 = ndash5 d3 = 3

d4 = 3 d5 = ndash8



6+2

0 6 3 5 7 24














31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)


31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0


+2

4

(c)


31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5





30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References








































jisinN0 jneiyji








4 Solution Method










Qi | where




















h

xh

yh

qh

0 8 9 6 2 1 3 5 4 0


0 5 3 1 6 2 8 0 09

0 1 2 3 4 5 6 7 8 9







EndReturn slowast








xi |dxj

| minus yDxj



xi yP

xi+ r yD

xj yD

xj+ r


yP1 |d2| minus yD


1 + 2 5 + 2 7 yD2 yD


4 + 1 6 + 1 7 yD5 yD

5 + 1 8 + 1 9






1 5D D420 2 8 0

+5 +6 ndash8




5

ndash3













1 5D D420 2 8 0

+7 ndash5 +6 ndash8

7

(a)

1 5D D420 2 9 0

+7 +7 ndash9ndash5

7

(b)


1 5D D420 4 8 0

+7 ndash3 +4 ndash8

7

(a)

1 5D D420 0 8 0

+3 ndash3 +8 ndash8

3

(b)

1 5D D420 0 6 0

+5 ndash5 +6 ndash6

5

(c)

1 5D D420 3 9 0

+5 ndash2 +6 ndash9

5

(d)


31 5D D420 4 1 3 5 0


d1 = 7 d2 = ndash5 d3 = 3

d4 = 3 d5 = ndash8



6+2

0 6 3 5 7 24














31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)


31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0


+2

4

(c)


31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5





30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References









































Qi | where




















h

xh

yh

qh

0 8 9 6 2 1 3 5 4 0


0 5 3 1 6 2 8 0 09

0 1 2 3 4 5 6 7 8 9







EndReturn slowast








xi |dxj

| minus yDxj



xi yP

xi+ r yD

xj yD

xj+ r


yP1 |d2| minus yD


1 + 2 5 + 2 7 yD2 yD


4 + 1 6 + 1 7 yD5 yD

5 + 1 8 + 1 9






1 5D D420 2 8 0

+5 +6 ndash8




5

ndash3













1 5D D420 2 8 0

+7 ndash5 +6 ndash8

7

(a)

1 5D D420 2 9 0

+7 +7 ndash9ndash5

7

(b)


1 5D D420 4 8 0

+7 ndash3 +4 ndash8

7

(a)

1 5D D420 0 8 0

+3 ndash3 +8 ndash8

3

(b)

1 5D D420 0 6 0

+5 ndash5 +6 ndash6

5

(c)

1 5D D420 3 9 0

+5 ndash2 +6 ndash9

5

(d)


31 5D D420 4 1 3 5 0


d1 = 7 d2 = ndash5 d3 = 3

d4 = 3 d5 = ndash8



6+2

0 6 3 5 7 24














31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)


31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0


+2

4

(c)


31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5





30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References











































xi |dxj

| minus yDxj



xi yP

xi+ r yD

xj yD

xj+ r


yP1 |d2| minus yD


1 + 2 5 + 2 7 yD2 yD


4 + 1 6 + 1 7 yD5 yD

5 + 1 8 + 1 9






1 5D D420 2 8 0

+5 +6 ndash8




5

ndash3













1 5D D420 2 8 0

+7 ndash5 +6 ndash8

7

(a)

1 5D D420 2 9 0

+7 +7 ndash9ndash5

7

(b)


1 5D D420 4 8 0

+7 ndash3 +4 ndash8

7

(a)

1 5D D420 0 8 0

+3 ndash3 +8 ndash8

3

(b)

1 5D D420 0 6 0

+5 ndash5 +6 ndash6

5

(c)

1 5D D420 3 9 0

+5 ndash2 +6 ndash9

5

(d)


31 5D D420 4 1 3 5 0


d1 = 7 d2 = ndash5 d3 = 3

d4 = 3 d5 = ndash8



6+2

0 6 3 5 7 24














31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)


31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0


+2

4

(c)


31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5





30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References
















































1 5D D420 2 8 0

+7 ndash5 +6 ndash8

7

(a)

1 5D D420 2 9 0

+7 +7 ndash9ndash5

7

(b)


1 5D D420 4 8 0

+7 ndash3 +4 ndash8

7

(a)

1 5D D420 0 8 0

+3 ndash3 +8 ndash8

3

(b)

1 5D D420 0 6 0

+5 ndash5 +6 ndash6

5

(c)

1 5D D420 3 9 0

+5 ndash2 +6 ndash9

5

(d)


31 5D D420 4 1 3 5 0


d1 = 7 d2 = ndash5 d3 = 3

d4 = 3 d5 = ndash8



6+2

0 6 3 5 7 24














31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)


31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0


+2

4

(c)


31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5





30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References

















































31 5D D25 0

+6 ndash3 +2 ndash5

0 6 3

(a)

31 5D D2 5 0

+4 ndash1 +2 ndash5

0 4 3

(b)

31 5D D2 3 00 4 1

+4 ndash3 +2 ndash3

(c)


31 5D D426+2 ndash3 +2 +2 ndash5+2

0 4 1 3 5 02

(a)

31 5D D420 6 1 3 5 0


+2

4

(b)

31 5D D420 6 3 5 7 0


+2

4

(c)


31 5D D20 4 1 3 ndash2

+4 ndash3 +2 ndash5





30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References








































30 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508

60 10 1171420 3600 1176428 1177424 533 043 05120 1134440 3600 1116422 1129824 486 minus161 minus041



240 10 6900386 6900386 84720 6719338 6719338 891

300 10 7963374 7969097 89220 7891392 7921236 1063








240 10 6560130 6564622 176920 6294130 6303757 3269

300 10 7644004 7644005 172220 7431136 7441450 4651








240 10 6235814 6259040 419120 5829796 5836207 5605

300 10 7196832 7203621 539820 7112862 7122615 5728

Avg 3600 3371 minus2359 minus2081














300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References
















































300 10 7973254 7973254 409 7963374 7969097 892 minus012 minus005









300 10 7645106 7657674 795 7644004 7644005 1722 minus001 minus018









300 10 7324820 7347626 1530 7196832 7203621 5398 minus178 minus200










0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References













































0

14400 10 485000 261 485000 485000 295 000 00020 432000 3600 406000 411300 298 minus640 minus503



000001

14400 10 486440 233 486440 486440 427 000 00020 435440 3600 414386 414386 458 minus508 minus508



00001

14400 10 499400 181 499400 499400 261 000 00020 451400 3600 419860 428651 285 minus751 minus531



0001

14400 10 629000 96 629000 629000 335 000 00020 575400 3600 544600 553964 431 minus566 minus387


2880010 424000 3600 395800 405733 1191 minus712 minus450
















6 Conclusions



Data Availability




Acknowledgments


References















































6 Conclusions



Data Availability




Acknowledgments


References






































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