networked timetable stability improvement based on a bilevel

Research ArticleNetworked Timetable Stability Improvement Based on a BilevelOptimization Programming Model

Xuelei Meng1 Bingmou Cui1 Limin Jia2 Yong Qin2 and Jie Xu2

1 School of Traffic and Transportation Lanzhou Jiaotong University PO Box 405 Anning West Road Anning District LanzhouGansu 730070 China

2 State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University No 3 Shangyuancun Haidian DistrictBeijing 100044 China

Correspondence should be addressed to Xuelei Meng mengxueleigmailcom

Received 28 November 2013 Revised 13 January 2014 Accepted 23 January 2014 Published 4 March 2014

Academic Editor Wuhong Wang

Copyright copy 2014 Xuelei Meng et al This 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

Train timetable stability is the possibility to recover the status of the trains to serve as arranged according to the original timetablewhen the trains are disturbed To improve the train timetable stability from the network perspective the bilevel programmingmodel is constructed in which the upper level programming is to optimize the timetable stability on the network level and thelower is to improve the timetable stability on the dispatching railway segments Timetable stability on the network level is definedwith the variances of the utilization coefficients of the section capacity and station capacity Weights of stations and sections aredecided by the capacity index number and the degrees The lower level programming focuses on the buffer time distribution planof the trains operating on the sections and stations taking the operating rules of the trains as constraints A novel particle swarmalgorithm is proposed and designed for the bilevel programmingmodelThe computing case proves the feasibility of themodel andthe efficiency of the algorithm The method outlined in this paper can be embedded in the networked train operation dispatchingsystem

1 Introduction

Train timetable is the fundamental file for organizing therailway traffic which determines the inbound and outboundtime of trains Railways are typically operated according toa planned (predetermined) timetable and the quality of thetimetable determines the quality of the railway service So it ismost important to map a high quality timetable for all kindsof trains

But there is a dilemma that we place as much as pos-sible trains on the timetable chart and simultaneously weshould enhance the possibility to adjust the timetable whendisruptions occur The randomly occurring disturbancesmay cause train delays and even disrupt the entire trainoperation plan In the railway network every station andsection are planned to serve the trains according to theschedule often compactly So a slightly delayed train maycause a domino effect of secondary delays over the thoroughnetwork Although the buffer times added to the minimum

running time in the sections and minimum dwell at stationsin scheduled timetables may absorb some train delays andassure some degree of timetable stability the large buffer timewill reduce the capacity of the railway

Therefore to ensure both the capacity and the order ofthe train operation a reliable stable robust timetable andthe feasible efficient rescheduling of the planned timetablemust be worked out A superior quality timetable cannotonly decide the inbound and outbound time at stations andthe more important can offer the possibility to recover theoperation according to the planned timetable when the trainsare disturbed by accidents randomly Timetable stability isthe index to measure the possibility Timetable stability isrelated to the train number assigned to the railway sectionsand the buffer time distributed to each station and sectionthe probability that the train is disrupted at the stations andin the sections In this paper the buffer time refers to the timeadded to the minimum running time in a section It equalsthe planned period of running minus the minimum running

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 290937 10 pageshttpdxdoiorg1011552014290937

2 Mathematical Problems in Engineering

time in the section And it also refers to the time added to thedwelling time at a station which equals the planned dwellingtime at the station minus the minimum dwelling time

So when assigning the trains paths and mapping out thetrain schedule the train number assigned to the sections andbuffer time distribution should be designed carefully not onlyconsidering the section capacity and station capacity but alsothe minimal running time at each section and the minimaldwell on the station

We define the networked timetable stability quantita-tively considering that the railway network with the goal isto optimize the timetable stability to offer more possibilitiesto reschedule the trains on the railway network to deal withdisturbs in the train operation process

The outline of the paper is as follows First Section 2proposes the literature review on timetable stability improve-ment Section 3 builds the bilevel optimization model forthe networked timetable stability Then Section 4 introducesthe hybrid fuzzy particle swarm algorithm improving thevelocity equation Section 5 applies hybrid fuzzy particleswarm optimization algorithm in solving the bilevel modelfor improving the networked timetable stability The com-puting case is presented in Section 6 Finally Section 7 givessome conclusions

2 Literature Review

It is a hot topic now to assure the reliability safety andstability of the traffic control system as discussed in [1ndash3]The timetable stability optimizing is a relatively new issue inthe field of railway operation researchThe research work waspublished in the 1990s

The research experienced two developing periods Inthe earliest period the focus was the timetable on thedispatching section according to the operation mode of therailway and the basis to study the train timetable stabilityis formed A discrete dynamic system model was built todescribe the timetable with the max-plus algebra based onthe discussion of the timetable periodicity and analyzed thetimetable stability as proposed by Goverde [4] Carey andCarville developed a simulation model to test the scheduleperformance and reliability for train stations in [5] Hansenpointed out that the effect of the stochastic disturbance ontrains relied on the adjustment of the running time andbuffer time in the timetable and assessed the advantages andthe disadvantages of the capacity and stability of evaluatingmodel [6] These researchers promoted the train timetablestability theory from the perspective of the running timein railway sections the dwelling time on railway stationsand the buffer time for running and dwelling De Kort etal proposed a method to evaluate the capacity determinedby the timetable and took the timetable stability as a part ofthe capacity see [7] The goal was tantamount to place trainrunning lines as much as possible while taking the timetableinto consideration at the same time Goverde presented amethod based on max-plus algebra to analyze the timetablestability He proved the feasibility of the method with dataof the Netherlands national railway timetable see [8] We

defined and qualified timetable stability and took it as a goalwhen rescheduling trains on the dispatching sections in [9]So it is easy to understand that the time is the key factorwhen studying the timetable of a dispatching railway sectionFocusing on the delay time the behind schedule ratio thebuffer time and time deviation researchers studied thetimetable adjustability equilibrium stability using statisticstheories max-plus algebra and so forth

Research on timetable stability progressively expanded tothe railway network for the study focusing on the timetablestability of dispatching cannot suit the networked timetabledesign and optimization Engelhardt-Funke and Kolonkoconsidered a network of periodically running railway linesThey built a model to analyze stability and investments inrailway networks and designed an innovative evolutionaryalgorithm to solve the problem in [10] Goverde analyzedthe dependence of the timetable on the busy degree of therailway network He again hired the max-plus algebra toanalyze the timetable stability of the railway network Onthis basis he proposed a novel method to generate thepaths for the trains on a large-scale railway network see[11] Vromans built a complex linear programming model tooptimize the timetable on the railway network level takingthe total delayed time as the optimizing goal And theydesigned the stochastic optimization algorithm for themodelsee [12] Delorme et al presented a station capacity evaluatingmodel and evaluated the stability on the key parts of therailway network stations see [13] We analyzed the complexcharacteristic analysis of passenger train flow network inthe former study work [14] and have done some researchwork to support the networked train timetable stabilityoptimization from transportation capacity calculation [15]paths generating [16] and line planning [17] which can beseen as the constraint of timetable stability optimizing

And we can see that the networked timetable stability isrelated to not only time but also the utilization coefficientcapacity of the railway network as discussed in [11ndash13] Thatis to say the networked timetable stability study requires thecombination of the railway network capacity utilization andthe buffer time distribution of the buffer time in the sectionsand at the stations However most of the publications areabout the stability of the timetable for a definite dispatchingrailway section And there are limited publications about thenetworked timetable stability Furthermore the research onthe timetable is in the stage of evaluating mostly qualitativenot the quantization of the timetable stability

3 The Bilevel OptimizationProgramming Model for NetworkedTimetable Stability Improvement

Networked timetable stability must be studied from twolevels The upper level is to study the relation between thetrains flow and the capacity of the sections and stationsand the ability to recover the timetable when an emergencyoccurs determined by the relation The lower level is to studythe distribution plan of the buffer time for each train in

Mathematical Problems in Engineering 3

the sections running process and the stations dwelling toeliminate the negative effects of the disturbs

The goal of the upper programming is to decide thenumber of trains assigned on each railway section and at thestations The fundamental restriction is that the number oftrains assigned to the sections and stations must not exceedthe capacity of the sections and the stations And the numberof trains received by the stations must be equal to the totalnumber of the trains running through the sections which areconnected to the relative station

The lower programming is to determine the buffer timedistribution plan The running time through a whole sectionplanned in the timetable is more than that it requires if it runsat its highest speed So there is a period of time called buffertime that can be distributed for the sections running andstations dwelling to absorb the delay caused by the randomdisturbances The restriction is that buffer time allocated toeach station and sectionmust be longer than or equal to zero

31 The Timetable Stability Improvement Programming onthe Network Level To define the timetable stability on thenetwork level the load on the sections and stations is the keyfactor So the load index numbers must be defined first

Definition 1 The load index number of a station on therailway network is

119885ST = Var (119890minus120588119894119908ST119894120588119894minus120588) (1)

where Var is the function to calculate the variance of a vector120588119894

is the load of the 119894th station the bigger 120588119894

is the smaller thestability value is 120588

119894

= 119865119894

119861119894

119861119894

is the receiving and sendingcapacity of the station 119865

119894

is the number of the receivingand sending trains by the 119894th station according to the trainsdistribution plan 120588 is a threshold value of a station load119908ST119894is the weight of the 119894th station and 119870 is the number of thestations on the railway network

The index number of the capacity of a station is

119868119883ST119894 =119861119894

119863119894

sum119870

119894=1

119861119894

119863119894

(2)

where119863119894

is the degree of the 119894th stationThen the station weight is

119908ST119894 =119868119883ST119894

sum119870

119894=1

119868119883ST119894 (3)

Definition 2 The load index number of a section on therailway network is

119885SE = Var (119890minus120582119894119908SE119894120582119894minus120582) (4)


is the loaf of the 119894th section the bigger 120582119894


119894

= 119866119894

119862119894

119862119894

is the capacity of the section119866119894

is the number of the trains running through 119894th sectionaccording to the trains distribution plan120582 is a threshold value

of a section load 119908119894

is the weight of the 119894th section and 119871 isthe number of the sections on the railway network

The index number of the capacity of a section is

119868119883SE119894 =119862119894

sum119871

119894=1

119862119894

(5)

Then the weight of the section is

119908SE119894 =119868119883SE119894

sum119871

119894=1

119868119883SE119894 (6)

Then with the load index numbers of the stations andsections the timetable stability on the network level is definedas

119878NET = 119890minus119885ST times 119890

minus119885SE = 119890minus(119885ST+119885SE) (7)

The goal of the upper programming is to optimize thetimetable stability on the network level so 119878NET is taken asthe optimization goal That is to say the goal is to maximizethe timetable on the network level 119878NET

Restrictions require that the number of the trains runningthrough a section cannot be greater than the number of thetrains that the section capacity allows Likewise the totaltrains number going through a station cannot exceed thestation capacity of receiving and sending off trains

And the total numbers of the trains distributed on thesections connected to the stationmust be equal to the numberof arriving trains at the station

119865119894

le 119861119894

119866119894

le 119862119894

119865119894

=

119880

sum

119897=1

119866119894119897

(8)

where 119880 is the number of sections connected to station 119894

32 The Timetable Stability Improvement Programming on theDispatching Section Level Take it for granted that there are119872trains going through section119901 which is the result of the upperprogramming The running times of all the119872 trains form avector 119879119901

119877

= 119905119901

119877119894

119872

The minimum running time of all the119872 trains forms a vector 119879119901min

119877

= 119905119901min119877119894

119872

Then the marginvector of the119872 trains isΔ119879119901

119877

= Δ119905119901

119877119894

119872

= 119905119901

119877119894

minus119905119901min119877119894

119872

Setthe 119860119901

119877

= Δ119905119901

119877119894

119905119901

119877119894

119872

to be the running adjustability vectorTo evaluate the equilibrium of the distribution of the buffertime in the sections the running adjustability dispersion isdefined as

Var (119860119901119877

) = Var(Δ119905119901

119877119894

119905119901

119877119894

) (9)

The smaller the value of theVar(119860119901119877

) is themore balancedthe buffer time distribution plan is and the timetable is morestable

Likewise take it for granted that there are119873 trains goingthrough station 119902 with stop or without stop The planned


dwelling time according to the timetable of the119873 trains formsa vector 119879119902

119863

= 119905119902

119863119894

119873

The minimal dwelling time of all the119873trains at 119902 also forms a vector 119879119901min

119863

= 119905119901min119863119894

119873

Then themargin vector of the 119873 trains is Δ119879119902

119863

= Δ119905119902

119863119894

119873

= 119905119902

119863119894

minus

119905119902min119863119894

119873

Set the 119860119902119863

= Δ119905119902

119863119894

119905119902

119863119894

119873

to be the dwelling adju-stability vector To evaluate the equilibriumof the distributionof the buffer time at stations the adjustability dispersion isdefined as

Var (119860119902119863

) = Var(Δ119905119902

119863119894

119905119902

119863119894

) (10)

The smaller the value of the Var(119860119902119863

) is the more bal-anced the buffer time distribution plan is and the timetableis more stable

On the basis of considering of the running adjustabilitydispersion and the dwelling adjustability dispersion thetimetable stability on the dispatching section level is definedas

119878DIS = 119890minusVar(119860119901

119877)timesVar(119860119902

119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

(11)

Then we take the timetable stability on the network levelas the optimizing goal of the upper programming

max 119878DIS (12)

When rescheduling the trains on the sections the mini-mum running time and the minimum dwelling time must beconsideredThe rescheduled running time and dwelling timemust be longer than the minimum time which is describedin (13) and (14) And the margins between inbound timesof different trains at the same stations must be bigger thanthe minimum interval time to ensure the safety of trainoperation This constraint is defined in (15) Likewise themargins between outbound times of the trains at a stationhave the same characteristic as shown in (16) And thenumber of the trains dwelling at a same station cannot bebigger than the number of the tracks in a station describedin (17)

119905119901

119877119894

ge 119905119901min119877119894

(13)

119905119902

119877119894

ge 119905119902min119877119894

(14)100381610038161003816100381610038161003816119886119901

119894119895

minus 119886119901

119897119895

100381610038161003816100381610038161003816gt 119868119886minus119886

119897 = 119894 (15)100381610038161003816100381610038161003816119889119901

119894119895

minus 119889119901

119897119895

100381610038161003816100381610038161003816gt 119868119889minus119889

119897 = 119894 (16)

119871119873119895

minus sum

119901isin119875

TNsum

119896=1

119899119896

119901119895

ge 0 (17)

33TheNetworkedTimetable StabilityDefinition Dependingon the analysis in Sections 31 and 32 networked timetablestability is defined with the following

119878 = 119878NET times 119878DIS (18)

We can see that the networked timetable stability isdirectly related to timetable stability on the network level andthe dispatching section level The programming in Sections31 and 32 can optimize the networked timetable stabilitythrough optimizing the stability on the two levels

4 The Hybrid Fuzzy Particle SwarmOptimization Algorithm

41 Fuzzy Particles Swarm Optimization Algorithm Con-siderable attention has been paid to fuzzy particle swarmoptimization (FPSO) recently Abdelbar et al proposed theFPSO [18] Abdelbar and Abdelshahid brought forwardthe instinct-based particle swarm optimization with localsearch applied to satisfiability in [19] Abdelshahid analyzedvariations of particle swarmoptimization and gave evaluationon maximum satisfiability see [20] Mendes et al proposeda fully informed particle swarm in [21] Bajpai and Singhstudied the problem of fuzzy adaptive particle swarm opti-mization (FAPSO) for bidding strategy in uniform price spotmarket in [22] Saber et al attempted to solve the problemof unit commitment computation by FAPSO in [23] Esminstudied the problem of generating fuzzy rules and fittingfuzzy membership functions using hybrid particle swarmoptimization (HPSO) see [24 25]

Computation in the PSO paradigm is based on a collec-tion (called a swarm) of fairly primitive processing elements(called particles) The neighborhood of each particle is theset of particles with which it is adjacent The two mostcommon neighborhood structures are 119901

119892

in which the entireswarm is considered a single neighborhood and 119901

119894

in whichthe particles are arranged in a ring and each particlersquosneighborhood consists of itself its immediate ring-neighborto the right and its immediate ring-neighbor to the left

PSO can be used to solve a discrete combinatorialoptimization problem whose candidate solutions can berepresented as vectors of bits 119894 is supposed to be a giveninstance of such a problem Let 119873 denote the number ofelements in the solution vector for 119894 Each particle 119894 wouldcontain two 119873-dimensional vectors a Boolean vector 119909

119894

which represents a candidate solution to 119894 and is calledparticle 119894rsquos state and a real vector V

119894

called the velocityof the particle In the biological insect-swarm analogy thevelocity vector represents how fast and in which directionthe particle is flying for each dimension of the problem beingsolved

Let 119870(119894) denote the neighbors of particle 119894 and let 119901119894

denote the best solution ever found by particle 119894 In each timeiteration each particle 119894 adjusts its velocity based on

V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + 1198882

1199032

(119901119892

minus 119909119894

)

119909119894+1

= 119909119894

+ V119894+1

(19)

where 120596 called inertia is a parameter within the range [0 1]and is often decreased over time as discussed in [26] 119888

1

and 1198882

are two constants often chosen so that 1198881

+ 1198882

= 4which control the degree to which the particle follows theherd thus stressing exploitation (higher values of 119888

2

) or goesits own way thus stressing exploration (higher values of 41)1199031

and 1199032

are uniformly random number generator functionthat returns values within the interval (0 1) and 119901

119892

is theparticle in 119894rsquos neighborhood with the current neighborhood-best candidate solution

Fuzzy PSO differs from standard PSO in only one respectin each neighborhood instead of only the best particle in


the neighborhood being allowed to influence its neighborsseveral particles in each neighborhood can be allowed toinfluence others to a degree that depends on their degree ofcharisma where charisma is a fuzzy variable Before buildinga model there are two essential questions that should beanswered The first question is how many particles in eachneighborhood have nonzero charisma The second is whatmembership function (MF)will be utilized to determine levelof charisma for each of the 119896 selected particles

The answer to the first question is that the 119896 best particlesin each neighborhood are selected to be charismatic where119896 is a user-set parameter 119896 can be adjusted according to therequired precision of the solution

The answer to the other question is that there are numer-ous possible functions for charisma MF Popular MF choicesinclude triangle trapezoidal Gaussian Bell and SigmoidMFs see [27] The Bell Gaussian sigmoid Trapezoidal andTriangular MFs

bell (119909 119888 120590) = 1

1 + ((119909 minus 119888) 120590)2

gaussian (119909 119888 120590) = exp(minus12(119909 minus 119888

120590)

2

)

sig (119909 119888 120590) = 1

1 + exp (minus120590 (119909 minus 119888))

triangle (119909 119886 119887 119888) = max(min(119909 minus 119886119887 minus 119886

119888 minus 119909

119888 minus 119887) 0)

119886 lt 119887 lt 119888

trapezoid (119909 119886 119887 119888 119889) = max(min(119909 minus 119886119887 minus 119886

1119889 minus 119909

119889 minus 119888) 0)

119886 lt 119887 le 119888 lt 119889

(20)

Let ℎ be one of the 119896-best particles in a given neighbor-hood and let119891(119901

119892

) refer to the fitness of the very-best particlefor the neighborhood under consideration The charisma120593(ℎ) is defined as

120593 (ℎ) =1

1 + ((119891 (119901ℎ

) minus 119891 (119901119892

)) 120573)2 (21)

if the MF is based on Bell functionThe charisma 120593(ℎ) is defined as

120593 (ℎ) = exp(minus12(

119891 (119901ℎ

) minus 119891 (119901119892

)

120573)

2

) (22)

if the MF is based on Gaussian functionThe charisma 120593(ℎ) is defined as

120593 (ℎ) =1

1 + exp (minus120573 (119891 (119901ℎ

) minus 119891 (119901119892

)))(23)

if the MF is based on Sigmoid function

The charisma 120593(ℎ) is defined as120593 (ℎ)

= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)

(24)if the MF is based on Triangle function


= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

1

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)

(25)if the MF is based on Trapezoid function

Because (119901ℎ

) le 119891(119901119892

) 120593(ℎ) is a decreasing function thatis 1 when 119891(119901

ℎ

) = 119891(119901119892

) and asymptotically approaches zeroas119891(119901

ℎ

)moves away from119891(119901119892

) To avoid dependence on thescale of the fitness function120573 is defined as120573 = 119891(119901

119892

)ℓwhereℓ is a user-specified parameter For a fixed 119891(119901

ℎ

) the largerthe value of ℓ the smaller the charisma 120593(ℎ) will be 119891(1199010

119892

)

and 119891(1199011119892

) are two functional values that decide the verge ofthe triangle and trapezoid function

In Fuzzy PSO velocity equation is

V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(119901ℎ

minus 119909119894

)

(26)where119861(119894 119896)denotes the119896-best particles in the neighborhoodof particle 119894 Each particle 119894 is influenced by its own best solu-tion 119901

119894

and the best solutions obtained by the 119896 charismaticparticles in its neighborhood with the effect of each weightedby its charisma 120593(ℎ) It can be seen that if 119896 is 1 this modelreduces to the standard PSO model

42 Hybrid Fuzzy PSO Hybrid rule requires selecting twoparticles from the alternative particles at a certain rate Thenthe intersecting operation work needs to be done to generatethe descendant particles The positions and velocities ofthe descendant particles are as follows according to theintersecting rule inheriting from the FPSO see [28]

V1119894+1

= 120596V1119894

+ 1198881

1199031

(119901119894

minus 1199091

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199011

ℎ

minus 1199091

119894

)

V2119894+1

= 120596V2119894

+ 1198881

1199031

(119901119894

minus 1199092

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199012

ℎ

minus 1199092

119894

)

V1119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+1

10038161003816100381610038161003816

V2119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+2

10038161003816100381610038161003816

1199091

119894+1

= 1199091

119894

+ V1119894+1

1199092

119894+1

= 1199092

119894

+ V2119894+1

1199091

119894+1

= 119901 times 1199091

119894+1

+ (1 minus 119901) times 1199092

119894+1

1199092

119894+1

= 119901 times 1199092

119894+1

+ (1 minus 119901) times 1199091

119894+1

(27)


Thus the HFPSO is built In the model 119909 is a positionvector with 119863 dimensions 119909119895

119894

stands for the particle 119895 ofthe 119894th generation particles 119901 is a random variable vectorwith119863 dimensions which obeys the equal distribution Eachdimension of 119901 is in [0 1]

43 Adaptability and Applicability HFPSO It can be seenthat the network timetable stability optimizing model is anonlinear one and it is an NP-hard problem Generally itis very difficult to solve the problem with mathematicalapproaches Evolutionary algorithms are often hired to solvethe problem for their characteristics First its rule of thealgorithm is easy to apply Second the particles have thememory ability which results in convergent speed and thereare various methods to avoid the local optimum Thirdlythe parameters which need to select are fewer and there isconsiderable research work on the parameters selecting

In addition the HFPSO hires the fuzzy theory and thehybrid handlingmethod when designing the algorithmThusit has the ability to improve the computing precision whensolving the optimization problem And it utilizes intersectingtactics to generate the new generation of particles to avoidthe precipitate of the solution It is adaptive to solve thetimetable stability optimization And its easy computingrule determines the applicability in the solving of timetablestability optimization problem

5 Hybrid Fuzzy Particle SwarmAlgorithm for the Model

51 The Particle Swarm Design for the Upper Level Program-ming The size of the particle swarm is set to be 30 to giveconsideration to both the calculating degree of accuracyand computational efficiency For 119865

119894

and 119866119894

are the decisionvariables 119871 is the number of the sections on the railwaynetwork 119870 is the number of the stations on the railwaynetwork So a particle can be designed as

119901119886UP = 1198651 1198652 119865119870 1198661 1198662 119866119871 (28)

52 The Particle Swarm Design for the Lower Level Program-ming The size of the particle swarm is also set to be 30Δ119905119901

119877119894

and Δ119905119902119863119894

are the decision variables So a particle can bedesigned as

119901119886LO = Δ1199051

1198771

Δ1199051

1198772

Δ1199051

1198771198721

Δ1199052

1198771

Δ1199052

1198772

Δ1199052

1198771198722

Δ119905119894

1198771

Δ119905119894

1198772

Δ119905119894

119877119872119894

Δ119905119871

1198771

Δ119905119871

1198772

Δ119905119871

119877119872119871

Δ1199051

1198631

Δ1199051

1198632

Δ1199051

1198631198721

Δ1199052

1198631

Δ1199052

1198632

Δ1199052

1198631198722

Δ119905119895

1198631

Δ119905119895

1198632

Δ119905119895

119863119872119895

Δ119905119870

1198631

Δ119905119870

1198632

Δ119905119870

119863119872119870

(29)

where119872119894

is the number of trains going through section 119894 and119872119895

is the number of the trains going through station 119895

1

2

3

4

5

6

7

a

b c

d

e

f

g

(23 30)

(23 235)(23 31)

(21 26)

(0 8)

(0 5)(0 65)

(0 65)

(21 275)

(21 21)

h

i

j

km

n

p

q

Figure 1 The railway network in the computing case and theoriginal distribution plan of the trains

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

Figure 2 Planned operation diagram on path 1-2ndash5ndash7

6 Experimental Results and Discussion

61 Computing Case Assumptions There are 22 stations and10 sections in the network as indicated in Figure 1 Weassume that the 44 trains operate on the network from station1 to station 7 The numbers in parentheses beside the edgesare the numbers of the trains distributed on the sectionsaccording to the planned timetable and the section capacity

And the planned operation diagram on path 1-2ndash5ndash7 isshown in Figure 2 with 23 trains The planned operationdiagram on path 1ndash3ndash6-7 is illustrated in Figure 3

According to the networked timetable stability definitionin Section 3 the timetable stability value of the plannedtimetable can be calculated out Detailed computing data arepresented in Table 1

According to Table 1(a) the 119885ST can be calculated with119885ST = Var(119890minus120588119894119908120588119894minus120588ST119894 ) = 1047978 while119885SC can be calculatedoutwith 119885SE = Var(119890minus120582119894119908

119894

120582119894minus120582) = 71940742Then the goal ofthe upper programming 119878NET = 119890

minus119885ST times 119890minus119885SE = 119890

minus(119885ST+119885SE) =

0000263

62 The Upper Level Computing Results and Discussion Wereallocate all the 44 trains on the modest railway networkas shown in Figure 4 according to the computing results ofthe upper level programming The numbers in parenthesesbeside the edges are the numbers of the trains allocated onthe sections according to the rescheduled timetable and thesection capacity Based on the capacity of every section andstation the computing result of the upper level programmingis presented in Table 2


Table 1 Computing results of the planned timetable stability

(a) Related computing results of 119885ST

Key nodes Trains number through station Nodes capacity Load Capacity index Nodes degree Degree index Stations weight1 44 660 06667 03099 20000 01111 022452 23 345 06667 01620 30000 01667 017603 21 315 06667 01479 30000 01667 016074 0 150 00000 00704 40000 02222 010205 23 360 06389 01690 30000 01667 018376 21 300 07000 01408 30000 01667 01531

(b) Related computing results of 119885SC

Section Trains number through section Sections capacity Load Capacity index Sections weight1-2 23 300 07667 01622 016221ndash3 21 275 07636 01486 014862ndash4 0 65 00000 00351 003512ndash5 23 235 09787 01270 012703-4 0 65 00000 00351 003513ndash6 21 210 10000 01135 011354-5 0 80 00000 00432 004324ndash6 0 50 00000 00270 002705ndash7 23 310 07419 01676 016766-7 21 260 08077 01405 01405

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

Figure 3 Planned operation diagram on path 1ndash3ndash6-7

1

2

3

4

5

6

7

a

b c

d

e

fg

(23 30)

(20 26)

(24 31)

(4 5)

(6 8)(5 65)

(18 235)

(5 65)(21 275)

(16 21)

h

ij

km

n

p

q

Figure 4 Distribution plan of the trains according to the computingresults

According to Table 2(a) the 119885ST can be calculated with119885ST = Var(119890minus120588119894119908120588119894minus120588ST119894 ) = 0000223814 while 119885SC can becalculated out with 119885SE = Var(119890minus120582119894119908

119894

120582119894minus120582) = 0002432614Then the goal of the upper programming 119878NET = 119890

minus119885ST times

119890minus119885SE = 119890

minus(119885ST+119885SE) = 0997

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

T1 T2T9984001 T3 T4 T5 T7 T8T6T998400

3

T9984002

T9984002

T9984004

T9984004

T9984005

T9984005

T9984006

Figure 5 Rescheduled timetable of path 1-2ndash5ndash7 according to thecomputing results of the lower level programming

63 The Lower Level Computing Results and DiscussionAccording to the computing results of the lower level pro-gramming we adjust the timetable moving some of therunning lines of the trains The two-dot chain line is thenewly planned running trajectory and the dotted line is thepreviously planned trajectory The timetable on path 1-2-5ndash7is shown in Figure 5 Figure 6 is the timetable of path 2ndash4-5Figures 7 and 8 are the timetables on paths 1ndash3ndash6-7 and 3-4ndash6respectively

From Figures 5 and 6 we can see that eight trains arerescheduled on path 1-2ndash5ndash7 which are numbered 119879

1

1198792

1198793

1198794

1198795

1198796

1198797

1198798

1198791

1198793

and 1198796

run on the original path asplanned but the inbound and outbound times at the stationsare changed 119879

2

1198794

1198795

1198797

and 1198798

modify the path when theyarrive at station 2They run through path 2ndash4-5 with arrivingtime 832 912 941 1025 and 1048 respectively at station 2


Table 2 Computing results of rescheduled timetable stability





2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012

Figure 6 Rescheduled timetable of path 2ndash4-5 according to thecomputing results of the lower level programming

1198792

1198794

and 1198795

leave arrive station 5 at 910 948 and 1018respectively then finish the rest travel along Sections 5ndash7

From Figures 7 and 8 we can see that eight trains arerescheduled on path 1ndash3ndash6-7 which are numbered 119879

9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916

run on the originalpath as planned but the inbound and outbound time at thestations are changed119879

9

11987910

11987911

11987912

and11987913

change the pathwhen they arrive at station 3 119879

9

11987910

11987911

and 11987913

run alongpath 3-4ndash6 with the arriving time 822 916 944 and 1043at station 3 respectively 119879

9

11987910

and 11987911

arrive at station 6 at853 947 and 1015 respectively 119879

12

runs along the path 3-4-5 It arrives at station 3 at 1030 and at station 4 at 1042 FromFigure 4 we can see that it runs along path 4-5 to finish therest travel

The timetable stability on the dispatching section level is119878DIS = 119890

minusVar(119860119901119877)timesVar(119860119902

119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15

Figure 7 Rescheduled timetable of path 1ndash3ndash6-7 according to thecomputing results of the lower level programming

Then the networked timetable is 119878 = 119878NET times 119878DIS = 0997 times0879 = 0876

7 Conclusion

The bilevel programming model is appropriate for thenetworked timetable stability optimizing It comprises thetimetable stability of the network level and the dispatchingsection level Better solution can be attained via hybrid fuzzyparticle swarm algorithm in networked timetable optimizingThe timetable is more stable which means that it is morefeasible for rescheduling in the case of disruption whenit is optimized by hybrid fuzzy particle swarm algorithmThe timetable rearranged based on the timetable stabilitywith bilevel networked programming model can make the


3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009

Figure 8 Rescheduled timetable of path 3-4ndash6 according to thecomputing results of the lower level programming

real train movements very close to if not the same withthe planned schedule which is very practical in the dailydispatching work

The results also show that hybrid fuzzy particle algorithmhas significant global searching ability and high speed and itis very effective to solve the problems of timetable stabilityoptimizing The novel method described in this paper canbe embedded in the decision support tool for timetabledesigners and train dispatchers

We can do some microcosmic research work on thetimetable optimizing based on the railway network in thefuture based on the method set out in the present paperenlarging the research field adding the inbound time andoutbound time of the trains at stations

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

This work is financially supported by National Natural Sci-ence Foundation of China (Grant 61263027) FundamentalResearch Funds of Gansu Province (Grant 620030) and NewTeacher Project of Research Fund for the Doctoral ProgramofHigher Education of China (20126204120002)The authorswish to thank anonymous referees and the editor for theircomments and suggestions

References

[1] H Guo W Wang W Guo X Jiang and H Bubb ldquoReliabilityanalysis of pedestrian safety crossing in urban traffic environ-mentrdquo Safety Science vol 50 no 4 pp 968ndash973 2012

[2] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behaviouralmodelwith various driving cha-racteristicsrdquo Transportation Research Part C vol 19 no 6 pp1202ndash1214 2011

[3] WWang Y Mao J Jin et al ldquoDriverrsquos various information pro-cess and multi-ruled decision-making mechanism a funda-mental of intelligent driving shapingmodelrdquo International Jour-nal of Computational Intelligence Systems vol 4 no 3 pp 297ndash305 2011

[4] R M P Goverde ldquoMax-plus algebra approach to railway time-table designrdquo in Proceedings of the 6th International Conference

on Computer Aided Design Manufacture and Operation in theRailway andOther AdvancedMass Transit Systems pp 339ndash350September 1998

[5] M Carey and S Carville ldquoTesting schedule performance andreliability for train stationsrdquo Journal of the Operational ResearchSociety vol 51 no 6 pp 666ndash682 2000

[6] I AHansen ldquoStation capacity and stability of train operationsrdquoin Proceeding of the 7th International Conference on Computersin Railways Computers in Railways VII pp 809ndash816 BolognaItaly September 2000

[7] A F De Kort B Heidergott and H Ayhan ldquoA probabilistic(max +) approach for determining railway infrastructure capa-cityrdquo European Journal of Operational Research vol 148 no 3pp 644ndash661 2003

[8] R M P Goverde ldquoRailway timetable stability analysis usingmax-plus system theoryrdquo Transportation Research Part B vol41 no 2 pp 179ndash201 2007

[9] X Meng L Jia and Y Qin ldquoTrain timetable optimizing andrescheduling based on improved particle swarm algorithmrdquoTransportation Research Record no 2197 pp 71ndash79 2010

[10] O Engelhardt-Funke and M Kolonko ldquoAnalysing stability andinvestments in railway networks using advanced evolutionaryalgorithmrdquo International Transactions in Operational Researchvol 11 no 4 pp 381ndash394 2004

[11] RM PGoverdePunctuality of railway operations and timetablestability analysis [PhD thesis] TRAIL Research School DelftUniversity of Technology Delft The Netherlands 2005

[12] M J Vromans Reliability of railway systems [PhD thesis]TRAIL Research School Erasmus University Rotterdam Rot-terdam The Netherlands 2005

[13] X Delorme X Gandibleux and J Rodriguez ldquoStability evalua-tion of a railway timetable at station levelrdquo European Journal ofOperational Research vol 195 no 3 pp 780ndash790 2009

[14] XMeng L Jia J Xie Y Qin and J Xu ldquoComplex characteristicanalysis of passenger train flow networkrdquo in Proceedings of theChinese Control and Decision Conference (CCDC rsquo10) pp 2533ndash2536 Xuzhou China May 2010

[15] X-L Meng L-M Jia Y Qin J Xu and L Wang ldquoCalculationof railway transport capacity in an emergency based onMarkovprocessrdquo Journal of Beijing Institute of Technology vol 21 no 1pp 77ndash80 2012

[16] X-L Meng L-M Jia C-X Chen J Xu L Wang and J-X XieldquoPaths generating in emergency on China new railway net-workrdquo Journal of Beijing Institute of Technology vol 19 no 2pp 84ndash88 2010

[17] L Wang L-M Jia Y Qin J Xu and W-T Mo ldquoA two-layeroptimizationmodel for high-speed railway line planningrdquo Jour-nal of Zhejiang University Science A vol 12 no 12 pp 902ndash9122011

[18] A M Abdelbar S Abdelshahid and D C Wunsch II ldquoFuzzyPSO a generalization of particle swarm optimizationrdquo in Pro-ceedings of the International Joint Conference onNeuralNetworks(IJCNN rsquo05) pp 1086ndash1091 Montreal Canada August 2005

[19] A M Abdelbar and S Abdelshahid ldquoInstinct-based PSO withlocal search applied to satisfiabilityrdquo in Proceedings of the IEEEInternational Joint Conference on Neural Networks pp 2291ndash2295 July 2004

[20] S Abdelshahid Variations of particle swarm optimization andtheir experimental evaluation on maximum satisfiability [MSthesis] Department of Computer Science American Universityin Cairo May 2004


[21] R Mendes J Kennedy and J Neves ldquoThe fully informed par-ticle swarm simpler maybe betterrdquo IEEE Transactions on Evo-lutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] P Bajpai and S N Singh ldquoFuzzy adaptive particle swarm opti-mization for bidding strategy in uniform price spot marketrdquoIEEE Transactions on Power Systems vol 22 no 4 pp 2152ndash2160 2007

[23] A Y Saber T Senjyu A Yona and T Funabashi ldquoUnit comm-itment computation by fuzzy adaptive particle swarm optimisa-tionrdquo IET Generation Transmission and Distribution vol 1 no3 pp 456ndash465 2007

[24] A A A Esmin ldquoGenerating Fuzzy rules from examples usingthe particle swarm optimization algorithmrdquo in Proceedings ofthe 7th International Conference on Hybrid Intelligent Systems(HIS rsquo07) pp 340ndash343 September 2007

[25] A A A Esmin andG Lambert-Torres ldquoFitting fuzzymembers-hip functions using hybrid particle swarmoptimizationrdquo inPro-ceedings of the IEEE International Conference on Fuzzy Systemspp 2112ndash2119 Vancouver Canada July 2006

[26] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 May 1998

[27] J S R Jang C T Sun and E Mizutani Neuro-Fuzzy and SoftComnputing Prentice-Hall 1996

[28] P J Angeline ldquoUsing selection to improve particle swarm opti-mizationrdquo in Proceedings of the IEEE International Conferenceon Evolutionary Computation (ICEC rsquo98) pp 84ndash89 May 1998

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

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International Journal of


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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


time in the section And it also refers to the time added to thedwelling time at a station which equals the planned dwellingtime at the station minus the minimum dwelling time

So when assigning the trains paths and mapping out thetrain schedule the train number assigned to the sections andbuffer time distribution should be designed carefully not onlyconsidering the section capacity and station capacity but alsothe minimal running time at each section and the minimaldwell on the station

We define the networked timetable stability quantita-tively considering that the railway network with the goal isto optimize the timetable stability to offer more possibilitiesto reschedule the trains on the railway network to deal withdisturbs in the train operation process

The outline of the paper is as follows First Section 2proposes the literature review on timetable stability improve-ment Section 3 builds the bilevel optimization model forthe networked timetable stability Then Section 4 introducesthe hybrid fuzzy particle swarm algorithm improving thevelocity equation Section 5 applies hybrid fuzzy particleswarm optimization algorithm in solving the bilevel modelfor improving the networked timetable stability The com-puting case is presented in Section 6 Finally Section 7 givessome conclusions

2 Literature Review

It is a hot topic now to assure the reliability safety andstability of the traffic control system as discussed in [1ndash3]The timetable stability optimizing is a relatively new issue inthe field of railway operation researchThe research work waspublished in the 1990s

The research experienced two developing periods Inthe earliest period the focus was the timetable on thedispatching section according to the operation mode of therailway and the basis to study the train timetable stabilityis formed A discrete dynamic system model was built todescribe the timetable with the max-plus algebra based onthe discussion of the timetable periodicity and analyzed thetimetable stability as proposed by Goverde [4] Carey andCarville developed a simulation model to test the scheduleperformance and reliability for train stations in [5] Hansenpointed out that the effect of the stochastic disturbance ontrains relied on the adjustment of the running time andbuffer time in the timetable and assessed the advantages andthe disadvantages of the capacity and stability of evaluatingmodel [6] These researchers promoted the train timetablestability theory from the perspective of the running timein railway sections the dwelling time on railway stationsand the buffer time for running and dwelling De Kort etal proposed a method to evaluate the capacity determinedby the timetable and took the timetable stability as a part ofthe capacity see [7] The goal was tantamount to place trainrunning lines as much as possible while taking the timetableinto consideration at the same time Goverde presented amethod based on max-plus algebra to analyze the timetablestability He proved the feasibility of the method with dataof the Netherlands national railway timetable see [8] We

defined and qualified timetable stability and took it as a goalwhen rescheduling trains on the dispatching sections in [9]So it is easy to understand that the time is the key factorwhen studying the timetable of a dispatching railway sectionFocusing on the delay time the behind schedule ratio thebuffer time and time deviation researchers studied thetimetable adjustability equilibrium stability using statisticstheories max-plus algebra and so forth

Research on timetable stability progressively expanded tothe railway network for the study focusing on the timetablestability of dispatching cannot suit the networked timetabledesign and optimization Engelhardt-Funke and Kolonkoconsidered a network of periodically running railway linesThey built a model to analyze stability and investments inrailway networks and designed an innovative evolutionaryalgorithm to solve the problem in [10] Goverde analyzedthe dependence of the timetable on the busy degree of therailway network He again hired the max-plus algebra toanalyze the timetable stability of the railway network Onthis basis he proposed a novel method to generate thepaths for the trains on a large-scale railway network see[11] Vromans built a complex linear programming model tooptimize the timetable on the railway network level takingthe total delayed time as the optimizing goal And theydesigned the stochastic optimization algorithm for themodelsee [12] Delorme et al presented a station capacity evaluatingmodel and evaluated the stability on the key parts of therailway network stations see [13] We analyzed the complexcharacteristic analysis of passenger train flow network inthe former study work [14] and have done some researchwork to support the networked train timetable stabilityoptimization from transportation capacity calculation [15]paths generating [16] and line planning [17] which can beseen as the constraint of timetable stability optimizing

And we can see that the networked timetable stability isrelated to not only time but also the utilization coefficientcapacity of the railway network as discussed in [11ndash13] Thatis to say the networked timetable stability study requires thecombination of the railway network capacity utilization andthe buffer time distribution of the buffer time in the sectionsand at the stations However most of the publications areabout the stability of the timetable for a definite dispatchingrailway section And there are limited publications about thenetworked timetable stability Furthermore the research onthe timetable is in the stage of evaluating mostly qualitativenot the quantization of the timetable stability

3 The Bilevel OptimizationProgramming Model for NetworkedTimetable Stability Improvement

Networked timetable stability must be studied from twolevels The upper level is to study the relation between thetrains flow and the capacity of the sections and stationsand the ability to recover the timetable when an emergencyoccurs determined by the relation The lower level is to studythe distribution plan of the buffer time for each train in











119894

= 119865119894

119861119894

119861119894


119894



119868119883ST119894 =119861119894

119863119894

sum119870

119894=1

119861119894

119863119894

(2)

where119863119894


119908ST119894 =119868119883ST119894

sum119870

119894=1

119868119883ST119894 (3)






119894

= 119866119894

119862119894

119862119894






119868119883SE119894 =119862119894

sum119871

119894=1

119862119894

(5)


119908SE119894 =119868119883SE119894

sum119871

119894=1

119868119883SE119894 (6)







119865119894

le 119861119894

119866119894

le 119862119894

119865119894

=

119880

sum

119897=1

119866119894119897

(8)



119877

= 119905119901

119877119894

119872


119877

= 119905119901min119877119894

119872


119877

= Δ119905119901

119877119894

119872

= 119905119901

119877119894

minus119905119901min119877119894

119872

Setthe 119860119901

119877

= Δ119905119901

119877119894

119905119901

119877119894

119872


Var (119860119901119877

) = Var(Δ119905119901

119877119894

119905119901

119877119894

) (9)






119863

= 119905119902

119863119894

119873


119863

= 119905119901min119863119894

119873


119863

= Δ119905119902

119863119894

119873

= 119905119902

119863119894

minus

119905119902min119863119894

119873

Set the 119860119902119863

= Δ119905119902

119863119894

119905119902

119863119894

119873


Var (119860119902119863

) = Var(Δ119905119902

119863119894

119905119902

119863119894

) (10)




119878DIS = 119890minusVar(119860119901

119877)timesVar(119860119902

119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

(11)


max 119878DIS (12)


119905119901

119877119894

ge 119905119901min119877119894

(13)

119905119902

119877119894

ge 119905119902min119877119894

(14)100381610038161003816100381610038161003816119886119901

119894119895

minus 119886119901

119897119895

100381610038161003816100381610038161003816gt 119868119886minus119886

119897 = 119894 (15)100381610038161003816100381610038161003816119889119901

119894119895

minus 119889119901

119897119895

100381610038161003816100381610038161003816gt 119868119889minus119889

119897 = 119894 (16)

119871119873119895

minus sum

119901isin119875

TNsum

119896=1

119899119896

119901119895

ge 0 (17)


119878 = 119878NET times 119878DIS (18)





119892


119894



119894


119894




V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + 1198882

1199032

(119901119892

minus 119909119894

)

119909119894+1

= 119909119894

+ V119894+1

(19)


1

and 1198882


+ 1198882


2


and 1199032


119892







bell (119909 119888 120590) = 1

1 + ((119909 minus 119888) 120590)2


120590)

2

)

sig (119909 119888 120590) = 1

1 + exp (minus120590 (119909 minus 119888))


119888 minus 119909

119888 minus 119887) 0)

119886 lt 119887 lt 119888


1119889 minus 119909

119889 minus 119888) 0)

119886 lt 119887 le 119888 lt 119889

(20)


119892


120593 (ℎ) =1

1 + ((119891 (119901ℎ

) minus 119891 (119901119892

)) 120573)2 (21)


120593 (ℎ) = exp(minus12(

119891 (119901ℎ

) minus 119891 (119901119892

)

120573)

2

) (22)


120593 (ℎ) =1

1 + exp (minus120573 (119891 (119901ℎ

) minus 119891 (119901119892

)))(23)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

1

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)


Because (119901ℎ

) le 119891(119901119892


ℎ

) = 119891(119901119892


ℎ

)moves away from119891(119901119892


119892


ℎ


119892

)

and 119891(1199011119892



V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(119901ℎ

minus 119909119894

)


119894



V1119894+1

= 120596V1119894

+ 1198881

1199031

(119901119894

minus 1199091

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199011

ℎ

minus 1199091

119894

)

V2119894+1

= 120596V2119894

+ 1198881

1199031

(119901119894

minus 1199092

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199012

ℎ

minus 1199092

119894

)

V1119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+1

10038161003816100381610038161003816

V2119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+2

10038161003816100381610038161003816

1199091

119894+1

= 1199091

119894

+ V1119894+1

1199092

119894+1

= 1199092

119894

+ V2119894+1

1199091

119894+1

= 119901 times 1199091

119894+1

+ (1 minus 119901) times 1199092

119894+1

1199092

119894+1

= 119901 times 1199092

119894+1

+ (1 minus 119901) times 1199091

119894+1

(27)



119894






119894

and 119866119894


119901119886UP = 1198651 1198652 119865119870 1198661 1198662 119866119871 (28)


119877119894

and Δ119905119902119863119894


119901119886LO = Δ1199051

1198771

Δ1199051

1198772

Δ1199051

1198771198721

Δ1199052

1198771

Δ1199052

1198772

Δ1199052

1198771198722

Δ119905119894

1198771

Δ119905119894

1198772

Δ119905119894

119877119872119894

Δ119905119871

1198771

Δ119905119871

1198772

Δ119905119871

119877119872119871

Δ1199051

1198631

Δ1199051

1198632

Δ1199051

1198631198721

Δ1199052

1198631

Δ1199052

1198632

Δ1199052

1198631198722

Δ119905119895

1198631

Δ119905119895

1198632

Δ119905119895

119863119872119895

Δ119905119870

1198631

Δ119905119870

1198632

Δ119905119870

119863119872119870

(29)

where119872119894



1

2

3

4

5

6

7

a

b c

d

e

f

g

(23 30)

(23 235)(23 31)

(21 26)

(0 8)

(0 5)(0 65)

(0 65)

(21 275)

(21 21)

h

i

j

km

n

p

q


1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions







119894



minus(119885ST+119885SE) =

0000263








1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions


1

2

3

4

5

6

7

a

b c

d

e

fg

(23 30)

(20 26)

(24 31)

(4 5)

(6 8)(5 65)

(18 235)

(5 65)(21 275)

(16 21)

h

ij

km

n

p

q



119894


minus119885ST times

119890minus119885SE = 119890

minus(119885ST+119885SE) = 0997

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

T1 T2T9984001 T3 T4 T5 T7 T8T6T998400

3

T9984002

T9984002

T9984004

T9984004

T9984005

T9984005

T9984006




1

1198792

1198793

1198794

1198795

1198796

1198797

1198798

1198791

1198793

and 1198796


2

1198794

1198795

1198797

and 1198798








2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012


1198792

1198794

and 1198795



9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916


9

11987910

11987911

11987912

and11987913


9

11987910

11987911

and 11987913


9

11987910

and 11987911


12




119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15



7 Conclusion



3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119894

= 119865119894

119861119894

119861119894


119894



119868119883ST119894 =119861119894

119863119894

sum119870

119894=1

119861119894

119863119894

(2)

where119863119894


119908ST119894 =119868119883ST119894

sum119870

119894=1

119868119883ST119894 (3)






119894

= 119866119894

119862119894

119862119894






119868119883SE119894 =119862119894

sum119871

119894=1

119862119894

(5)


119908SE119894 =119868119883SE119894

sum119871

119894=1

119868119883SE119894 (6)







119865119894

le 119861119894

119866119894

le 119862119894

119865119894

=

119880

sum

119897=1

119866119894119897

(8)



119877

= 119905119901

119877119894

119872


119877

= 119905119901min119877119894

119872


119877

= Δ119905119901

119877119894

119872

= 119905119901

119877119894

minus119905119901min119877119894

119872

Setthe 119860119901

119877

= Δ119905119901

119877119894

119905119901

119877119894

119872


Var (119860119901119877

) = Var(Δ119905119901

119877119894

119905119901

119877119894

) (9)






119863

= 119905119902

119863119894

119873


119863

= 119905119901min119863119894

119873


119863

= Δ119905119902

119863119894

119873

= 119905119902

119863119894

minus

119905119902min119863119894

119873

Set the 119860119902119863

= Δ119905119902

119863119894

119905119902

119863119894

119873


Var (119860119902119863

) = Var(Δ119905119902

119863119894

119905119902

119863119894

) (10)




119878DIS = 119890minusVar(119860119901

119877)timesVar(119860119902

119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

(11)


max 119878DIS (12)


119905119901

119877119894

ge 119905119901min119877119894

(13)

119905119902

119877119894

ge 119905119902min119877119894

(14)100381610038161003816100381610038161003816119886119901

119894119895

minus 119886119901

119897119895

100381610038161003816100381610038161003816gt 119868119886minus119886

119897 = 119894 (15)100381610038161003816100381610038161003816119889119901

119894119895

minus 119889119901

119897119895

100381610038161003816100381610038161003816gt 119868119889minus119889

119897 = 119894 (16)

119871119873119895

minus sum

119901isin119875

TNsum

119896=1

119899119896

119901119895

ge 0 (17)


119878 = 119878NET times 119878DIS (18)





119892


119894



119894


119894




V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + 1198882

1199032

(119901119892

minus 119909119894

)

119909119894+1

= 119909119894

+ V119894+1

(19)


1

and 1198882


+ 1198882


2


and 1199032


119892







bell (119909 119888 120590) = 1

1 + ((119909 minus 119888) 120590)2


120590)

2

)

sig (119909 119888 120590) = 1

1 + exp (minus120590 (119909 minus 119888))


119888 minus 119909

119888 minus 119887) 0)

119886 lt 119887 lt 119888


1119889 minus 119909

119889 minus 119888) 0)

119886 lt 119887 le 119888 lt 119889

(20)


119892


120593 (ℎ) =1

1 + ((119891 (119901ℎ

) minus 119891 (119901119892

)) 120573)2 (21)


120593 (ℎ) = exp(minus12(

119891 (119901ℎ

) minus 119891 (119901119892

)

120573)

2

) (22)


120593 (ℎ) =1

1 + exp (minus120573 (119891 (119901ℎ

) minus 119891 (119901119892

)))(23)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

1

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)


Because (119901ℎ

) le 119891(119901119892


ℎ

) = 119891(119901119892


ℎ

)moves away from119891(119901119892


119892


ℎ


119892

)

and 119891(1199011119892



V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(119901ℎ

minus 119909119894

)


119894



V1119894+1

= 120596V1119894

+ 1198881

1199031

(119901119894

minus 1199091

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199011

ℎ

minus 1199091

119894

)

V2119894+1

= 120596V2119894

+ 1198881

1199031

(119901119894

minus 1199092

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199012

ℎ

minus 1199092

119894

)

V1119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+1

10038161003816100381610038161003816

V2119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+2

10038161003816100381610038161003816

1199091

119894+1

= 1199091

119894

+ V1119894+1

1199092

119894+1

= 1199092

119894

+ V2119894+1

1199091

119894+1

= 119901 times 1199091

119894+1

+ (1 minus 119901) times 1199092

119894+1

1199092

119894+1

= 119901 times 1199092

119894+1

+ (1 minus 119901) times 1199091

119894+1

(27)



119894






119894

and 119866119894


119901119886UP = 1198651 1198652 119865119870 1198661 1198662 119866119871 (28)


119877119894

and Δ119905119902119863119894


119901119886LO = Δ1199051

1198771

Δ1199051

1198772

Δ1199051

1198771198721

Δ1199052

1198771

Δ1199052

1198772

Δ1199052

1198771198722

Δ119905119894

1198771

Δ119905119894

1198772

Δ119905119894

119877119872119894

Δ119905119871

1198771

Δ119905119871

1198772

Δ119905119871

119877119872119871

Δ1199051

1198631

Δ1199051

1198632

Δ1199051

1198631198721

Δ1199052

1198631

Δ1199052

1198632

Δ1199052

1198631198722

Δ119905119895

1198631

Δ119905119895

1198632

Δ119905119895

119863119872119895

Δ119905119870

1198631

Δ119905119870

1198632

Δ119905119870

119863119872119870

(29)

where119872119894



1

2

3

4

5

6

7

a

b c

d

e

f

g

(23 30)

(23 235)(23 31)

(21 26)

(0 8)

(0 5)(0 65)

(0 65)

(21 275)

(21 21)

h

i

j

km

n

p

q


1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions







119894



minus(119885ST+119885SE) =

0000263








1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions


1

2

3

4

5

6

7

a

b c

d

e

fg

(23 30)

(20 26)

(24 31)

(4 5)

(6 8)(5 65)

(18 235)

(5 65)(21 275)

(16 21)

h

ij

km

n

p

q



119894


minus119885ST times

119890minus119885SE = 119890

minus(119885ST+119885SE) = 0997

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

T1 T2T9984001 T3 T4 T5 T7 T8T6T998400

3

T9984002

T9984002

T9984004

T9984004

T9984005

T9984005

T9984006




1

1198792

1198793

1198794

1198795

1198796

1198797

1198798

1198791

1198793

and 1198796


2

1198794

1198795

1198797

and 1198798








2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012


1198792

1198794

and 1198795



9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916


9

11987910

11987911

11987912

and11987913


9

11987910

11987911

and 11987913


9

11987910

and 11987911


12




119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15



7 Conclusion



3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119863

= 119905119902

119863119894

119873


119863

= 119905119901min119863119894

119873


119863

= Δ119905119902

119863119894

119873

= 119905119902

119863119894

minus

119905119902min119863119894

119873

Set the 119860119902119863

= Δ119905119902

119863119894

119905119902

119863119894

119873


Var (119860119902119863

) = Var(Δ119905119902

119863119894

119905119902

119863119894

) (10)




119878DIS = 119890minusVar(119860119901

119877)timesVar(119860119902

119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

(11)


max 119878DIS (12)


119905119901

119877119894

ge 119905119901min119877119894

(13)

119905119902

119877119894

ge 119905119902min119877119894

(14)100381610038161003816100381610038161003816119886119901

119894119895

minus 119886119901

119897119895

100381610038161003816100381610038161003816gt 119868119886minus119886

119897 = 119894 (15)100381610038161003816100381610038161003816119889119901

119894119895

minus 119889119901

119897119895

100381610038161003816100381610038161003816gt 119868119889minus119889

119897 = 119894 (16)

119871119873119895

minus sum

119901isin119875

TNsum

119896=1

119899119896

119901119895

ge 0 (17)


119878 = 119878NET times 119878DIS (18)





119892


119894



119894


119894




V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + 1198882

1199032

(119901119892

minus 119909119894

)

119909119894+1

= 119909119894

+ V119894+1

(19)


1

and 1198882


+ 1198882


2


and 1199032


119892







bell (119909 119888 120590) = 1

1 + ((119909 minus 119888) 120590)2


120590)

2

)

sig (119909 119888 120590) = 1

1 + exp (minus120590 (119909 minus 119888))


119888 minus 119909

119888 minus 119887) 0)

119886 lt 119887 lt 119888


1119889 minus 119909

119889 minus 119888) 0)

119886 lt 119887 le 119888 lt 119889

(20)


119892


120593 (ℎ) =1

1 + ((119891 (119901ℎ

) minus 119891 (119901119892

)) 120573)2 (21)


120593 (ℎ) = exp(minus12(

119891 (119901ℎ

) minus 119891 (119901119892

)

120573)

2

) (22)


120593 (ℎ) =1

1 + exp (minus120573 (119891 (119901ℎ

) minus 119891 (119901119892

)))(23)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

1

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)


Because (119901ℎ

) le 119891(119901119892


ℎ

) = 119891(119901119892


ℎ

)moves away from119891(119901119892


119892


ℎ


119892

)

and 119891(1199011119892



V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(119901ℎ

minus 119909119894

)


119894



V1119894+1

= 120596V1119894

+ 1198881

1199031

(119901119894

minus 1199091

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199011

ℎ

minus 1199091

119894

)

V2119894+1

= 120596V2119894

+ 1198881

1199031

(119901119894

minus 1199092

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199012

ℎ

minus 1199092

119894

)

V1119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+1

10038161003816100381610038161003816

V2119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+2

10038161003816100381610038161003816

1199091

119894+1

= 1199091

119894

+ V1119894+1

1199092

119894+1

= 1199092

119894

+ V2119894+1

1199091

119894+1

= 119901 times 1199091

119894+1

+ (1 minus 119901) times 1199092

119894+1

1199092

119894+1

= 119901 times 1199092

119894+1

+ (1 minus 119901) times 1199091

119894+1

(27)



119894






119894

and 119866119894


119901119886UP = 1198651 1198652 119865119870 1198661 1198662 119866119871 (28)


119877119894

and Δ119905119902119863119894


119901119886LO = Δ1199051

1198771

Δ1199051

1198772

Δ1199051

1198771198721

Δ1199052

1198771

Δ1199052

1198772

Δ1199052

1198771198722

Δ119905119894

1198771

Δ119905119894

1198772

Δ119905119894

119877119872119894

Δ119905119871

1198771

Δ119905119871

1198772

Δ119905119871

119877119872119871

Δ1199051

1198631

Δ1199051

1198632

Δ1199051

1198631198721

Δ1199052

1198631

Δ1199052

1198632

Δ1199052

1198631198722

Δ119905119895

1198631

Δ119905119895

1198632

Δ119905119895

119863119872119895

Δ119905119870

1198631

Δ119905119870

1198632

Δ119905119870

119863119872119870

(29)

where119872119894



1

2

3

4

5

6

7

a

b c

d

e

f

g

(23 30)

(23 235)(23 31)

(21 26)

(0 8)

(0 5)(0 65)

(0 65)

(21 275)

(21 21)

h

i

j

km

n

p

q


1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions







119894



minus(119885ST+119885SE) =

0000263








1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions


1

2

3

4

5

6

7

a

b c

d

e

fg

(23 30)

(20 26)

(24 31)

(4 5)

(6 8)(5 65)

(18 235)

(5 65)(21 275)

(16 21)

h

ij

km

n

p

q



119894


minus119885ST times

119890minus119885SE = 119890

minus(119885ST+119885SE) = 0997

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

T1 T2T9984001 T3 T4 T5 T7 T8T6T998400

3

T9984002

T9984002

T9984004

T9984004

T9984005

T9984005

T9984006




1

1198792

1198793

1198794

1198795

1198796

1198797

1198798

1198791

1198793

and 1198796


2

1198794

1198795

1198797

and 1198798








2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012


1198792

1198794

and 1198795



9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916


9

11987910

11987911

11987912

and11987913


9

11987910

11987911

and 11987913


9

11987910

and 11987911


12




119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15



7 Conclusion



3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













bell (119909 119888 120590) = 1

1 + ((119909 minus 119888) 120590)2


120590)

2

)

sig (119909 119888 120590) = 1

1 + exp (minus120590 (119909 minus 119888))


119888 minus 119909

119888 minus 119887) 0)

119886 lt 119887 lt 119888


1119889 minus 119909

119889 minus 119888) 0)

119886 lt 119887 le 119888 lt 119889

(20)


119892


120593 (ℎ) =1

1 + ((119891 (119901ℎ

) minus 119891 (119901119892

)) 120573)2 (21)


120593 (ℎ) = exp(minus12(

119891 (119901ℎ

) minus 119891 (119901119892

)

120573)

2

) (22)


120593 (ℎ) =1

1 + exp (minus120573 (119891 (119901ℎ

) minus 119891 (119901119892

)))(23)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)



= max(min(119891 (119901ℎ

) minus 119891 (1199010

119892

)

119891 (119901119892

) minus 119891 (1199010119892

)

1

119891 (1199011

119892

) minus 119891 (119901ℎ

)

119891 (1199011119892

) minus 119891 (119901119892

)

) 0)


Because (119901ℎ

) le 119891(119901119892


ℎ

) = 119891(119901119892


ℎ

)moves away from119891(119901119892


119892


ℎ


119892

)

and 119891(1199011119892



V119894+1

= 120596V119894

+ 1198881

1199031

(119901119894

minus 119909119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(119901ℎ

minus 119909119894

)


119894



V1119894+1

= 120596V1119894

+ 1198881

1199031

(119901119894

minus 1199091

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199011

ℎ

minus 1199091

119894

)

V2119894+1

= 120596V2119894

+ 1198881

1199031

(119901119894

minus 1199092

119894

) + sum

ℎisin119861(119894119896)

120593 (ℎ) 1198882

1199032

(1199012

ℎ

minus 1199092

119894

)

V1119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+1

10038161003816100381610038161003816

V2119894+1

=V1119894+1

+ V2119894+1

1003816100381610038161003816V1

119894+1

+ V2119894+1

1003816100381610038161003816

10038161003816100381610038161003816V1119894+2

10038161003816100381610038161003816

1199091

119894+1

= 1199091

119894

+ V1119894+1

1199092

119894+1

= 1199092

119894

+ V2119894+1

1199091

119894+1

= 119901 times 1199091

119894+1

+ (1 minus 119901) times 1199092

119894+1

1199092

119894+1

= 119901 times 1199092

119894+1

+ (1 minus 119901) times 1199091

119894+1

(27)



119894






119894

and 119866119894


119901119886UP = 1198651 1198652 119865119870 1198661 1198662 119866119871 (28)


119877119894

and Δ119905119902119863119894


119901119886LO = Δ1199051

1198771

Δ1199051

1198772

Δ1199051

1198771198721

Δ1199052

1198771

Δ1199052

1198772

Δ1199052

1198771198722

Δ119905119894

1198771

Δ119905119894

1198772

Δ119905119894

119877119872119894

Δ119905119871

1198771

Δ119905119871

1198772

Δ119905119871

119877119872119871

Δ1199051

1198631

Δ1199051

1198632

Δ1199051

1198631198721

Δ1199052

1198631

Δ1199052

1198632

Δ1199052

1198631198722

Δ119905119895

1198631

Δ119905119895

1198632

Δ119905119895

119863119872119895

Δ119905119870

1198631

Δ119905119870

1198632

Δ119905119870

119863119872119870

(29)

where119872119894



1

2

3

4

5

6

7

a

b c

d

e

f

g

(23 30)

(23 235)(23 31)

(21 26)

(0 8)

(0 5)(0 65)

(0 65)

(21 275)

(21 21)

h

i

j

km

n

p

q


1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions







119894



minus(119885ST+119885SE) =

0000263








1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions


1

2

3

4

5

6

7

a

b c

d

e

fg

(23 30)

(20 26)

(24 31)

(4 5)

(6 8)(5 65)

(18 235)

(5 65)(21 275)

(16 21)

h

ij

km

n

p

q



119894


minus119885ST times

119890minus119885SE = 119890

minus(119885ST+119885SE) = 0997

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

T1 T2T9984001 T3 T4 T5 T7 T8T6T998400

3

T9984002

T9984002

T9984004

T9984004

T9984005

T9984005

T9984006




1

1198792

1198793

1198794

1198795

1198796

1198797

1198798

1198791

1198793

and 1198796


2

1198794

1198795

1198797

and 1198798








2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012


1198792

1198794

and 1198795



9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916


9

11987910

11987911

11987912

and11987913


9

11987910

11987911

and 11987913


9

11987910

and 11987911


12




119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15



7 Conclusion



3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894






119894

and 119866119894


119901119886UP = 1198651 1198652 119865119870 1198661 1198662 119866119871 (28)


119877119894

and Δ119905119902119863119894


119901119886LO = Δ1199051

1198771

Δ1199051

1198772

Δ1199051

1198771198721

Δ1199052

1198771

Δ1199052

1198772

Δ1199052

1198771198722

Δ119905119894

1198771

Δ119905119894

1198772

Δ119905119894

119877119872119894

Δ119905119871

1198771

Δ119905119871

1198772

Δ119905119871

119877119872119871

Δ1199051

1198631

Δ1199051

1198632

Δ1199051

1198631198721

Δ1199052

1198631

Δ1199052

1198632

Δ1199052

1198631198722

Δ119905119895

1198631

Δ119905119895

1198632

Δ119905119895

119863119872119895

Δ119905119870

1198631

Δ119905119870

1198632

Δ119905119870

119863119872119870

(29)

where119872119894



1

2

3

4

5

6

7

a

b c

d

e

f

g

(23 30)

(23 235)(23 31)

(21 26)

(0 8)

(0 5)(0 65)

(0 65)

(21 275)

(21 21)

h

i

j

km

n

p

q


1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions







119894



minus(119885ST+119885SE) =

0000263








1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions


1

2

3

4

5

6

7

a

b c

d

e

fg

(23 30)

(20 26)

(24 31)

(4 5)

(6 8)(5 65)

(18 235)

(5 65)(21 275)

(16 21)

h

ij

km

n

p

q



119894


minus119885ST times

119890minus119885SE = 119890

minus(119885ST+119885SE) = 0997

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

T1 T2T9984001 T3 T4 T5 T7 T8T6T998400

3

T9984002

T9984002

T9984004

T9984004

T9984005

T9984005

T9984006




1

1198792

1198793

1198794

1198795

1198796

1198797

1198798

1198791

1198793

and 1198796


2

1198794

1198795

1198797

and 1198798








2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012


1198792

1198794

and 1198795



9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916


9

11987910

11987911

11987912

and11987913


9

11987910

11987911

and 11987913


9

11987910

and 11987911


12




119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15



7 Conclusion



3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions


1

2

3

4

5

6

7

a

b c

d

e

fg

(23 30)

(20 26)

(24 31)

(4 5)

(6 8)(5 65)

(18 235)

(5 65)(21 275)

(16 21)

h

ij

km

n

p

q



119894


minus119885ST times

119890minus119885SE = 119890

minus(119885ST+119885SE) = 0997

1

2

5

7

800 900 1000 1100Time

a

b

c

d

Stat

ions

T1 T2T9984001 T3 T4 T5 T7 T8T6T998400

3

T9984002

T9984002

T9984004

T9984004

T9984005

T9984005

T9984006




1

1198792

1198793

1198794

1198795

1198796

1198797

1198798

1198791

1198793

and 1198796


2

1198794

1198795

1198797

and 1198798








2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012


1198792

1198794

and 1198795



9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916


9

11987910

11987911

11987912

and11987913


9

11987910

11987911

and 11987913


9

11987910

and 11987911


12




119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15



7 Conclusion



3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















2

j

q

800 900 1000 1100Time

i

4

n

5

Stat

ions

T9984004 T998400

5 T9984007 T998400

8T9984002

T99840012


1198792

1198794

and 1198795



9

11987910

11987911

11987912

11987913

11987914

11987915

11987916

11987914

11987915

and11987916


9

11987910

11987911

11987912

and11987913


9

11987910

11987911

and 11987913


9

11987910

and 11987911


12




119863)

= 119890minusVar(Δ119905119901

119877119894119905

119901

119877119894)timesVar(Δ119905119902

119863119894119905

119902

119863119894)

= 0879

1

f

6

7

800 900 1000 1100Time

e

3

g

h

Stat

ions

T9 T14 T99840014 T10 T11 T15 T12 T13 T16

T9984009

T9984009

T99840010

T99840011

T99840016T998400

15



7 Conclusion



3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3

m

800 900 1000 1100Time

p

4

k

6

Stat

ions

T99840010 T998400

11 T99840012 T998400

13T9984009







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









networked timetable stability improvement based on a bilevel

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