research article a new accident analysis method based on...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 437428 9 pageshttpdxdoiorg1011552013437428

Research ArticleA New Accident Analysis Method Based on Complex Networkand Cascading Failure

Ziyan Luo Keping Li Xin Ma and Jin Zhou

State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Ziyan Luo zyluobjtueducn

Received 7 June 2013 Accepted 8 October 2013

Academic Editor Xiaohua Ding

Copyright copy 2013 Ziyan Luo 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

Anew accident causationmodel is proposed for accident analysis based on the complex network theory By employing the cascadingfailure scheme a new accident investigation method is performed on the associated new model by which we can reveal keycausation factors and key causation factor chains that lead to the final accident The efficiency of a network is introduced forevaluating the severity of the damage of the whole network and hence the severity of the accident if it happens All these canprovide the government or associations with recommendations for accident prediction and prevention

1 Introduction

Accident causation models are tools to describe scenarios foraccident occurrences explain possible causationmechanismsof accidents provide conceptual or theoretical basis foraccident investigation methods and hence give evidence toformulate specific recommendations for accident preventionAs a fundamental but essential task of accident analysisthe modelling of accident causation mechanisms has con-centrated great interests of researchers and engineers inmany fields especially in those high-risk industries such asaviation nuclear plants and railway system As Svenson [1]has stated that ldquoan accident can be explained in differentways depending on the accident analysis model that is usedrdquodifferent models focus of different aspects on the accidentoccurrences and provide different recommendations forimproving measurements

To get a clear understanding of the accidents a numberof different accident causation models have been proposedwhich can be roughly divided into three major groupsaccording toHollnagelrsquos classification [2]Thefirst group alsothe earliest one is termed as the ldquosequential accident modelrdquo[3] with the well-known Domino theory [4] as a typicalexample In this group of models accidents are regardedas a one-dimensional sequence of events that happened in

a specific order The second group is called the ldquoepidemio-logical accident modelrdquo [5] in which accidents are regardedas analog to the spreading of epidemiological diseases withthe Swiss Cheese model [6 7] as a major contribution to thisgroup The third group also known as the most modernone is the ldquosystemic accident modelrdquo (eg see [8ndash11]) inwhich accident processes are described as a complex andinterconnected network of events rather than a simple cause-effect chain of events as in the first two groups Rasmussenrsquos[11] risk management model and Levesonrsquos [10] STAMP(Systems-TheoreticAccidentModel andProcesses)model aretwo notable examples in the third group which endeavoredto model the dynamics of complex sociotechnical systems ofaccidents

Accident causation models can provide theoretical basisor conceptualization for accident investigation methodsWith the development of accident causation models a num-ber of methods for accident investigation have emerged andevolved Examples include the widely used fault tree analysis[3] management oversight and risk tree [12] multilinearevents sequencing [13] systematic cause analysis technique[14] causal tree method [15] occupational accident researchunit [16] TRIPOD [17] accident evolution and barrier func-tion [18] integrated safety investigation methodology [19]work accidents investigation technique [20] control change

2 Discrete Dynamics in Nature and Society

cause analysis [21] Accimap [22] HFACS [23] CREAM [8]FRAM [9] and STAMP [10] to just name a few The detailedalignments for the previous methods and causation modelsare well discussed [24 25]

With the rapid development in technology and automa-tion the underlying sociotechnical systems are becomingmore and more complex and of high risk This motivatesus to tailor more powerful accident causation models tocapture the complexity of the highly technological systemsfrom a broad systemic view for understanding characteristicsof accidents

Mentioning that causation factors and their relationshipsfor accidents are always complex in terms of uncertaintyrandomness abstractness fuzziness and other propertiesit would be a nice try to employ the complex networktheory [26ndash28] to reveal the involved complexity in accidentscausation analysis The primary purpose of this paper isto construct an accident causation network for causalityanalysis based on the complex theory With the influenceor relation between accident causation models and accidentinvestigation methods as mentioned above we apply thecascading failure scheme to characterize the process of theaccident occurrence performed on the proposed model as aninvestigation approach Together with the help of the networkefficiency of the underlying accident causation networkwe can evaluate the injury severity of the whole systemwith unexpected disturbances from technical human socialorganizational and environmental aspects of the wholesystem

This paper is organized as follows Some selected funda-mental concepts in complex network are recalled in Section 2The new accident causation network model is constructedin Section 3 and the cascading failure scheme is appliedto characterize the evolution of the proposed causationnetwork in Section 4 Simulation based on our proposedaccident causation analysis method is described in Section 5Conclusions are drawn in Section 6

2 Basic Concepts in Complex Network

The complex network is a graph with complex topologicalfeatures thatmay not occur in simple networks such as latticesor random graphs but often occur in real graphs The studyof complex networks has attracted great interest inspiredlargely by the empirical study of real-world networks suchas computer networks and social networks In mathematicalterms a network is represented by a graph A graph is apair of sets (119881 119864) where 119881 is a set of 119873 nodes (or vertices)V1 V2 V

119873and 119864 is a set of edges (or links) that connect

two elements of 119881 Graphs are usually represented as aset of dots each corresponding to a node two of thesedots being joined by a line if the corresponding nodes areconnected Usually we use 119873(V

119894) to denote the set of all

nodes in 119881 that are connected to node V119894 A path in a

graph is a sequence of edges which connect a sequence ofvertices The shortest path length of two nodes is definedas the smallest number of edges that connect these twonodes

1

2

3

5

4

Figure 1 Illustration of a graph with 119873 = 5 nodes and6 edges where the edge set is 119864 = (V

1 V2) (V1 V3) (V1 V5)

(V2 V3) (V2 V5) (V3 V5) and119873(V

1) = V

2 V3 V5119873(V

2) = V

1 V3 V5

119873(V3) = V

1 V2 V5 119873(V

4) is an empty set and 119873(V

5) = V

1 V2 V3

The shortest path length of nodes V1and V

2is 1

2

31

4 10

5

9

6 78

Figure 2 Illustration for the random network with probability 119901 =015 for every pair of nodes being connected 119873(V

1) = V

8 and

(V1) = V

6 V7

Figure 1 shows an illustration of a graph with 119873 = 5

nodes and 6 edges With the complexity of real networksthe edge sets are sometimes not determined which meansthere are some pairs of nodes with a random or uncertainlink such as the random network shown in Figure 2 withprobability 119901 = 015 for every pair of nodes being connected[29] To distinguish the adjacent nodes with deterministicconnections and those with probability less than 1 we use(119894) to denote the set of all nodes in 119881 that are connectedto node V

119894with probability less than 1

3 Accident Causation Network

Causation factors and their relationships for sociotechni-cal system accidents are always complex with uncertaintyrandomness abstractness fuzziness and other propertiesFor example the relation between two causation factors

Discrete Dynamics in Nature and Society 3

Table 1 Causation factors of the 723 China Yongwen railway accident [30]

Ministry of RailwaysA1 seek quick success and benefits A2 week management and incomplete rule standardsA3 unclear job responsibilities and functions A4 inadequate inspection and supervision for ShanghaiRailway Bureau

Department ofTechnologies FoundationDepartment Science andTechnology Division CRSC

B1 lack of careful supervision on the bidding of the equipment in Hening-Hewu Yongwen line traincontrol center B2 poor management of the operation on new products B3 not enough examination ofthe LKD2-T1 B4 without clear regulations on the technical review B5 no valid or regular technicalprereview on the equipment LKD2-T1 for train control center B6 illegal approval from the Science andTechnology Division approved to use the LKD2-T1 B7 inadequate inspection and supervision of thequality management by CRSC B8 little supervision or inspection from CRSC who fully transmit theproject to the local design institute B9 cursory decision on the bidding for the Hening-Hewu line controlequipment B10 unware of the illegal change of version of the train control center equipment in Hefeistation

Shanghai Railway Bureauand the signaling designinstitute

C1 not enough safety education and training C2 not sufficient inspection and supervisionC3 not sensitive safety awareness not efficient measures to avoid or alleviate the accident C4 notappropriate accident handling C5 unwise decision on update of the LKD2-T1 C6 lack of the technicalreview on the development of the equipment for train control center C7 lack of responsibility onscientific research management and inefficient control and supervision of the local companies on theproduct quality

Vehicle depot electricitydepot engineering systemand train control institute

D1 poor travel management and emergency handling D2 not efficient supervision on the safetyproduction management and train service work lack of supervision to Wenzhou south station D3 poorsupervision on the dispatching institute and the vehicle depot system D4 insufficient education andtraining for the staff D5 lack of job responsibilities of the electricity emergency management D6 cursorydesign of the equipment LKD2-T1 D7 poor equipment research and development management in thetrain control center D8 the redesign of the equipment LKD2-T1 by the train control institute

The attendantsrsquo behaviorsand process

E1 failure of following further situation of red band by the dispatcher in Shanghai Railway Bureau E2careless monitoring on the situation of D3115 E3 no reminder of the emergency to D301 E4 no in timecontact with the D301 driver E5 no record of the circuit failure of the 5829AG E6 no record of thereplacement of some equipments of the track circuit besides 5829AG E7 illegal behaviors E8 the mistaketo inform D3115 to switch to the visual driving mode if the signal was red E9 D3115 stopped by the ATPE10 D3115 failed to drive in visual mode 3 times E11 D3115 failed to report to the dispatcher E12 D3115switched to the visual driving mode but still in the 5829AG E13 D301 left Yongjia station E14 D301rear-ended D3115 E15 illegal to open the protection net for work

Equipment andenvironment

F1 the damage of 4 sender boxes F2 the damage of 2 receiver boxes F3 the damage of 1 attenuator F4the fuse of F2 in LKD2-T1 F5 the design flaw in PIO of LKD2-T1 F6 the activation of ATP on the D3115F7 thunder strike F8 failure of the ATP on D301 which did not take any action F9 the reduction of CANtotal resistance F10 unavailable communication between 5829AG and the train control center F11 wrongdisplays on the terminal F12 abnormal track circuit signal F13 a red band F14 wrong signal whichmaintained green for the faulted track section F15 the sending of the unoccupied signal to D301

might be related under some special circumstances which canbe regarded as a dash-line edge between them in the networkwith some associated conditional probabilityThis might be aclue for us to employ the complex network to characterize thiscomplex system We call this model the accident causationnetwork which can be viewed as an undetermined graphconsisting of nodes connected by edges with the nodes andedges representing those causation factors and their possiblecausal or relevant relationships respectively

In order to get a relatively comprehensive and completeextraction of causation factors and their relationships toconstruct this railway accident causation network we canemploy some classification approach for specific accidentsWe take the 723 China Yongwen railway accident as an exam-ple to illustrate the proposed accident causation network Byutilizing Rasmussenrsquos hierarchical sociotechnical framework[11] causation factors of the 723 China Yongwen railwayaccident are distributed into the following six hierarchies the

ministry of railways the Railway Bureau train control centertrain dispatcher train drivers and driving environmentincluding line environment and the natural environment aspresented in Table 1

Evidently the above classification approach covers causa-tion factors with respect to human equipment environmentand organizational management which form a complexsystem Figure 3 shows the causation network of Yongwenrailway accident which happened in July 23 2011 in China

4 The Accident Cascading Failure Process

From the systemic theory perspective any accident can beregarded as a result of a series of unsatisfied constraints orfactors which are out of control These failures or incidentscan be spread andmight eventually lead to an accident In this


A A

AA

BBBBB

BB

B

B B

CCCCCC

C

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

D D DD

DD

DDEE

EE

EEEE

EE EEEE E

F FF

F

F

F

F

FF

FF

F FF

F

9

9

9

10

10

10

11

11

12

12

8

8

8

8

7

7

7

7

13

13

14

1415

15

7

Figure 3 The causation network of the 723 China Yongwen railway accident

regard the failure cascading scheme might be applicable tocharacterize the evolution process on our proposed accidentcausation network With some unexpected disturbances inthe system this network will be evolved to a determinednetwork or graph which might lead to an accident

The cascading scheme for accident analysis based on theproposed accident causation network is elaborated as followsLet 119866(119905) be the graph of the underlying accident causationnetwork at time 119905 119905 = 0 1 and119873 is the number of nodes(ie the number of possible causation factors) of the networkFor any 119894 = 1 119873 119871

119894(119905) is the load of node 119894 at time 119905 and

119888119894= 120572119894119871119894(0) is the tolerance of node 119894 For any distinct 119894 and

119895 119890119894119895(119905) denotes the efficiency between nodes 119894 and 119895 at time 119905

with 119890119894119895(0) = 1 When the load of node 119894 exceeds its capacity

that is 119871119894(119905) gt 119888

119894 that is the constraint at this node fails to

hold then the associated efficiency between node 119894 and anyother node 119895 will be reduced Assume that it evolves in thefollowing simple manner

119890119894119895(119905 + 1) =

119890119894119895(0)

119888119894

119871119894(119905) if 119888

119894lt 119871119894(119905)

119890119894119895(0) otherwise

(1)

Define the efficiency of the whole network at time 119905 as

119890 (119866 (119905)) =1

119873 (119873 minus 1)sum119894 = 119895

119890119894119895(119905) (2)

It is trivial that if any failure or incident happens insome nodes this efficiency will be reduced Therefore thisquantity can to some extent indicate how badly the under-lying accident causation network is damaged at any time 119905Moreover it could provide an index for defining the severitylevel of the accident if it finally happensThere are some basicassumptions involved

Assumption 1 Let node 119897 be defined as the accident indicatorwith the meaning that the accident happens at time 119905 if 119888

119897lt

119871119897(119905) The evolution stops once the accident happens

Assumption 2 Each node has its shortest path length to node119897 as its capacity and the load evolves in the following mannerwith equal spreading loads

119871119894(119905 + 1) = 119871

119894(119905) + sum119895isin119861119894(119905)

119871119895(119905)

10038161003816100381610038161003816119905(119895)10038161003816100381610038161003816

(3)

with 119861119894(119905) = 119895 isin

119905(119894) 119888

119895lt 119871119895(119905)

119905(119894) being the set of

all adjacent nodes which are connected to node 119894with dashedline at time 119905

Assumption 3 Each node 119894with 119888119894lt 119871119894(119905)will have solid lines

to its adjacent nodes after time 119905

See Figure 4 as a simple illustration for the evolution ofan accident causation network

From the previous assumptions we can find that if node119894 is out of control at time 119905 that is 119888

119894lt 119871119894(119905) then it will affect

all its adjacent nodes at time 119905 + 1 since the correspondingconditional probabilities increase These growing loads willadd the burden of its adjacent nodes and might lead tocascading failures or even accident in the future For examplein Figure 4 if node 9 is the accident indicator then theaccident happens at time 119905 = 5 with an original attack onnode 2

Assumption 1 provides a direct way to predict an accidentby calculating the load 119871

119873(119905) In this case we can analyze

key causation factors for accidents by testing each failurenode in terms of the occurrence of accidents Specifically ifonly node 119894 fails at the beginning and it leads to the failureof node 119873 according to the above revolution rules then itis reasonable to say it is a key causation factor for the final


1

11

11 22

22

2

33

333

44

444

55

555

66

666

77

777

88

888

99

999

t = 1 t = 2 t = 3

t = 4t = 5

Figure 4 Illustration for the evolution of an accident causationnetwork

accidentThose paths formed by failure nodes to the accidentnode during the whole cascading process are called the keycausation chains In Figure 4 node 2 is a key causation factorand the corresponding key causation chains are

2mdash1mdash3mdash8mdash92mdash5mdash9

Denote 119864 be the 119899 times 119899matrix with all entries 1 and 119864119894119895 be the119899 times 119899 matrix with its 119894119895th and 119895119894th entries 1 and 0 elsewhereFor any 119894 isin 1 119873 and any time 119905 define

120575119894(119905) =

1 minus119888119894

119871119894(119905) if 119888

119894lt 119871119894(119905) for some 119905 le 119905

0 otherwise(4)

By direct calculation we can obtain the evolution formu-las for the efficiency of the network at each time period asfollows

119890 (119866 (119905 + 1)) = tr[119864119873

sum119894=1

(119868 + 120575119894(119905) 119864119894119894) 119864 (119868 + 120575

119894(119905) 119864119894119894)]

119905 = 0 1

(5)

where sgn(sdot) is the sign function tr(sdot) is the trace operator ofmatrix 120575

119894(119905) is defined as in (4) and 119868 is the identity matrix

This could provide a way to quantify the accident severitylevel in terms of the corresponding efficiency matrix whichcan be calculated as in (5)

5 A Case Study

The ldquo723rdquo Yongwen railway accident is chosen as a case studyhere to test the efficiency of our proposed accident causationmodel-method Based on its accident causation network asconstructed in Figure 3 we perform our cascading evolutionprocess as follows

Step 1 The capacity or tolerance of node 119894 is chosenas the shortest path length 119897

119894of node 119894 to node E14

(the accident indicator) for simplicity which is shown inTable 2 by direct calculation This assumption is reasonablesince the further the factor away from the accident indicatorthe less impact (or more robust) of leading to the accident

Step 2 The initial load for each node is chosen as 119871119894(0) =

(12)119888119894 that is 120572

119894= 2 for each 119894 = A1 F15 It is realistic

to choose a normal and safe state as a start

Step 3 Disturbances

Case I (4 times of the capacity) (1) Take the hub node F14 asthe first attacking point with the attacking load 8 (4 times ofits capacity) at time 119905 = 1The evolution process is performedas follows

119905 = 1 Attacking the hub node F14 with a load 8

119871F14 (1) = 8 gt 2 = 119888F14

1(F14) = E6E12E13E15 F4 F5 F6 F8 F11 F15

(6)

119905 = 2 Changing all dashed lines connected to F14 to solid

119871E6 (2) = 15 + 08 lt 3 = 119888E6

119871E12 (2) = 05 + 08 gt 1 = 119888E12

2(E6) = E9E10E11E14 F15

119871E13 (2) = 05 + 08 gt 1 = 119888E13

2(E13) = E3E4E14 F15

119871E15 (2) = 15 + 08 lt 3 = 119888E15

119871F4 (2) = 15 + 08 lt 3 = 119888F4

119871F5 (2) = 15 + 08 lt 3 = 119888F5

119871F6 (2) = 15 + 08 lt 3 = 119888F6

119871F8 (2) = 05 + 08 gt 1 = 119888F8

2(F8) = E14

119871F11 (2) = 15 + 08 lt 3 = 119888F11

119871F15 (2) = 1 + 08 lt 2 = 119888F15

(7)

119905 = 3 Changing all dashed lines connected to E12 E13 andF8 to solid

119871E14 (3) = 185 +13

5+13

4+ 13 gt 37

= 119888E14 (accident happens)

119871E9 (3) = 1 +25

5+ 35 gt 2 = 119888E9

(8)

The evolution stops at time 119905 = 3 by Assumption 1 sincethe accident happens By (3) the efficiency of the whole


Table 2 The shortest path length 119888119894of every node to the E14

119897119894

119894

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Level

A 4 5 5 5B 5 5 4 4 3 4 5 5 5 5C 4 5 5 5 4 4 4D 3 5 4 4 4 3 3 3E 4 3 2 2 5 3 4 3 2 2 2 1 1 37 3F 5 5 5 3 3 3 4 1 4 3 3 3 4 2 2

network turns out to be 119890(119866(3)) = 093The load distributionsof each vertex in the whole process are illustrated below Forsimplicity nodes A1 to F15 are renumbered as 1 to 59 inFigure 5 and similarly in Figures 6 7 and 8

(2) Take the natural environment node F7 as anotherattacking point with the load 40 which is also 4 times of itscapacity119905 = 1 Attacking node F7 with a load 16

119871F7 (1) = 16 gt 4 = 119888F14 1198731(F7) = F1 F2 F3 F4

(9)


119871F1 (2) = 25 + 4 gt 5 = 119888F1 1198732(F1) = F9

119871F2 (2) = 25 + 4 gt 5 = 119888F2 1198732(F2) = F9

119871F3 (2) = 25 + 4 gt 5 = 119888F3 1198732(F3) = F9

119871F4 (2) = 15 + 4 gt 3 = 119888F4 1198732(F4) = F14

(10)

119905 = 3 Changing all dashed lines connected to F7 F1 F2 F3F4 to solid

119871F9 (3) = 2 + 65 + 65 + 65 gt 4 = 119888F9

3(F9) = F10

119871F14 (3) = 1 + 55 gt 2 = 119888E9

3(F14) = F5 F6 F8 F11 F15E6E12E13E15

(11)

119905 = 4 Changing all dashed lines connected to F9 F14 to solid

119871F10 (4) = 15 + 215 gt 3 = 119888F10

1198734(F10) = F12 F13E11

119871F5 (4) = 15 +65

9lt 3 = 119888F5

119871F6 (4) = 15 +65

9lt 3 = 119888F6

119871F8 (4) = 05 +65

9gt 1 = 119888F8

1198734(F8) = E14

119871F11 (4) = 15 +65

9lt 3 = 119888F11

119871F15 (4) = 1 +65

9lt 2 = 119888F10

119871E6 (4) = 15 +65

9lt 3 = 119888E6

119871E12 (4) = 05 +65

9gt 1 = 119888E12

4(E12) = E9E10E11E14 F15

119871E13 (4) = 05 +65

9gt 1 = 119888E13

4(E13) = E3E4E14E15

119871E15 (4) = 15 +65

9lt 3 = 119888E15

(12)

119905 = 5 Changing all dashed lines connected to F8 F10 E12and E13 to solid

119871E14 (5) = 185 + (05 +65

9) (1 +

1

4+1

5)

= 362 lt 37 = 119888E14 (accident will not happen)

119871F12 (5) = 15 +23

3gt 3 = 119888F12 119873

5(F12) = E10

119871F13 (5) = 2 +23

3gt 4 = 119888F13 119873

5(F13)= empty set

119871E11 (5) = 1 +23

3gt 2 = 119888E11 119873

5(E11) = E2

(13)

The evolution stops at time 119905 = 5 by Assumption 1 and theefficiency of the whole network turns out to 119890(119866(5)) = 077The load distributions of all vertices in the whole process areillustrated in Figure 6

Case II (10 times of the capacity) (3) Take the hub node F14as the attacking point as a load 20 (10 times of its capacity)at time 119905 = 1 which largely exceeds its capacity Similar tocase I the evolution process can be described in Figure 7


0

0

1

2

3

4

5

6

7

8

10 20 30 40 50 60

Vertices

Load

Capacity distributiont = 0

t = 1

t = 2

t = 3

Figure 5 Load distributions of the evolution process in Case I(1)

00

5

10

10

20 30 40 50 60

Vertices

Load

Capacity distribution

t = 0

t = 1

t = 2

t = 3

t = 4

t = 5

15

20

25


The load distributions of each vertex in the whole process areillustrated in Figure 7

We can also get 119871E14(3) = 185 + 255 + 254 + 2 gt

37 = 119888E14 (accident happens) and the efficiency of the wholenetwork 119890(119866(3)) = 084

(4) Take the natural environment node F7 as anotherattacking point with the load 40 which is also 10 timesof its capacity Similarly we can obtain the following loaddistributions of the whole evolution process

By direct calculation we have 119871E14(5) = 459 gt 37 = 119888E14(accident happens) with the efficiency 119890(119866(5)) = 066

From the analysis for case I it indicates that F14 is akey causation factor to the accident with respect to a 4times capacity attack with the efficiency loss 007 while forthe same severity of attack on F7 the accident indicatorwill not get a heavy load larger than its capacity whichmeans that the accident will not happen This tells us thatwith a 4 times attacking load node F7 could not be a keycausation factor for the occurrence of the accident This isreasonable since the thunder strike (F7) might be a triggerand may play a role in the 723 Yongwen accident but is

Vertices


t = 1

t = 2

t = 3

0

20

18

16

14

12

10

8

6

4

2

0

Load

10 20 30 40 50 60

Figure 7 Load distributions of the evolution process in Case II(3)

Vertices


t = 0

t = 1

t = 2

t = 3

0 10 20 30 40 50 60

40

35

30

25

20

15

10

5

0

Load

t = 4

t = 5


not essential while the equipmentsrsquo failure (F14) is the keycausation factor However with the analysis of case II it isalso worth mentioning that with bad natural disasters suchas hurricanes or earthquakes which result in a super heavyattacking load the environment factor would turn to a keycausation factor as well Meanwhile from the comparison ofcases I and II it is easy to see that the heavier the attack isthe larger the efficiency loss is and hence the higher severitythe accident is On the other hand as we can find in cases(1) (3) and (4) the involved key causation chains containE8 E12 and E13 which are all related to the control flaws ofthe train operation system This tells us that more attentionshould be paid to the control flaws to prevent or encumberthe spreading of cascading failure which is essential to theaccident occurrence

It is known that nodes with large degrees play an impor-tant role in the cascading failure for a network Thus thosenodes with largest degrees in each level are chosen and theircritical loads to lead an accident by means of the proposedcascading failure scheme are calculated To characterize theirsensitivities and also for the sake of comparison the ratio of


140

120

100

80

60

40

20

A1 B5 C1 C5 D1 D4 D7 D8 E6 E13 F140

Critical loadcapacity


Figure 9 Ratios of critical loads and capacities

the critical load to the capacity of each of them is illustratedin Figure 9

As we can see in Figure 9 E6 and F14 possess smallratios which means that they are quite sensitive for theaccident occurrence with a small attacking intensity Thusit is important to prevent failures made by the related staffand also the control equipment In contrast A1 and C5 haverelatively large ratios This tells us that the culture of seekingquick success and benefits in the ministry of railways andthe unwise decision on update of the LKD2-T1 are not thatsensitive but they do have effect on the accident With anyattacking intensity larger than their corresponding ratios itwill lead to an accident in the cascading failure processThusit is urgent to build a healthy safe and sustainable culturefor the railway development in China and the design of theequipment must enforce the safety constraints in face of anenvironment disturbance or other factorsrsquo failure

6 Conclusions

In this paper we have introduced an accident causationnetwork model based on the complex network theory Byutilizing the cascading failure scheme the evolution processof the proposed causation network has been described andkey causations of accidents have been explored and analyzedBased on some reasonable assumptions the sensitivities ofsome important key causation factors for accident occurrenceare characterized and compared The severity of the accidenthas been characterized via the network efficiency of theevolved network quantitatively as well It is worth pointingout that the accuracy of the method can be greatly improvedby a relatively comprehensive and complete extraction ofcausation factors and their relationships for the causationnetwork construction and by the expert knowledge andreliable statistical results for the cascading failure processApproaches to improve the accuracy desire further investi-gation which is our future research topic

Research Highlights

(i) We propose a new network model for accident causa-tion analysis

(ii) We regard the accident occurrence as a cascadingfailure

(iii) We reveal key causation factors and key causationchains

(iv) We design a way for accident severity evaluation

Acknowledgments

This work was supported by Research Foundations of StateKey Laboratory of Rail Traffic Control and Safety (noRCS2012ZQ001 and no RCS2012ZZ001) Beijing JiaotongUniversity and the National High Technology Research andDevelopment Program of China (no 2011AA110502)

References

[1] O Svenson ldquoOn models of incidents and accidentsrdquo in Pro-ceedings of the 7th European Conference on Cognitive ScienceApproaches to Process Control pp 169ndash174 Villeneuve drsquoAscqFrance September 1999

[2] E Hollnagel ldquoUnderstanding accidents-from root causes toperformance variabilityrdquo in Proceedings of the 7th Conferenceon Human Factors Meeting pp 1ndash6 Scottsdale Ariz USASeptember 2002

[3] T S Ferry Ed Modern Accident Investigation and AnalysisJohn Wiley amp Sons New York NY USA 1988

[4] W H Heinrich Ed Industrial Accident Prevention McGraw-Hill New York NY USA 1941

[5] E Hollnagel Ed Barriers and Accident Prevention AshgateHampshire UK 2001

[6] J Reason EdHuman Error University Press Cambridge UK1990

[7] J Reason Ed Managing the Risks of Organisational AccidentsAshgate Aldershot UK 1997

[8] E Hollnagel Ed Cognitive Reliability and Error AnalysisMethod CREAM Elsevier San Diego Calif USA 1998

[9] E Hollnagel Barriers and Accident Prevention Ashgate Hamp-shire UK 2004

[10] N G Leveson ldquoA new accident model for engineering safersystemsrdquo Safety Science vol 42 no 4 pp 237ndash270 2004

[11] J Rasmussen ldquoRisk management in a dynamic society amodelling problemrdquo Safety Science vol 27 no 2-3 pp 183ndash2131997

[12] W G Johnson Ed MORT Safety Assurance Systems MarcelDekker New York NY USA 1980

[13] L Benner Jr ldquoAccident investigations multilinear eventssequencing methodsrdquo Journal of Safety Research vol 7 no 2pp 67ndash73 1975

[14] U Kjellen and J Hovden ldquoReducing risks by deviationcontrolmdasha retrospection into a research strategyrdquo Safety Sciencevol 16 no 3-4 pp 417ndash438 1993

[15] J Leplat ldquoAccident analyses and work analysesrdquo Journal ofOccupational Accidents vol 1 no 4 pp 331ndash340 1978


[16] U Kjellen and T J Larsson ldquoInvestigating accidents andreducing risksmdasha dynamic approachrdquo Journal of OccupationalAccidents vol 3 no 2 pp 129ndash140 1981

[17] W A Wagenaar J Groeneweg P T W Hudson and J TReason ldquoPromoting safety in the oil industryrdquo Ergonomics vol37 no 12 pp 1999ndash2013 1994

[18] O Svenson ldquoAccident Analysis and Barrier Function (AEB)Method Manualfor Incident Analysisrdquo SKI Project 97176Stockholm University 2000 httpwwwiriskserefhtm

[19] M Ayeko ldquoIntegrated safety investigation method (ISIM)mdashinvestigating for riskmitigationrdquo in Proceedings of theWorkshopon Investigation and Reporting of Incidents and Accidents pp115ndash126 Glasgow UK July 2002

[20] C Jacinto and E Aspinwall ldquoWork accidents investigationtechnique (WAIT)mdashpart Irdquo Safety Science Monitor vol 7 no1 article IV-2 2003

[21] J Kingston ldquo3CA-Investigatorsquos Manual NRI-3rdquo 2007 httpwwwnrieucomNRI3pdf

[22] J Clarkson A Hopkins and K Taylor ldquoReport of the boardof inquiry into F-111 (Fuel tank) desealreseal and spray sealprogramsrdquo vol I Canberra ACT Royal Australian Air Force

[23] D A Wiegmann and S A Shappell A Human Error Approachto Aviation Accident Analysis The Human Factors Analysis andClassification System Ashgate Hampshire UK 2003

[24] Z H Qureshi A Review of Accident Modeling Approachesfor Complex Social-Technical Systems Australian ComputerScience Sydney Australia 2007

[25] P Katsakiori G Sakellaropoulos and E Manatakis ldquoTowardsan evaluation of accident investigation methods in terms oftheir alignment with accident causationmodelsrdquo Safety Sciencevol 47 no 7 pp 1007ndash1015 2009

[26] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[27] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999

[28] S H Strogatz ldquoExploring complex networksrdquo Nature vol 410no 6825 pp 268ndash276 2001

[29] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[30] ldquoThe state investigation team of the China-Yongwen railwayaccidentrdquo The investigation report on the ldquo7 23rdquo Yongwen linemajor railway accident 2011 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


cause analysis [21] Accimap [22] HFACS [23] CREAM [8]FRAM [9] and STAMP [10] to just name a few The detailedalignments for the previous methods and causation modelsare well discussed [24 25]

With the rapid development in technology and automa-tion the underlying sociotechnical systems are becomingmore and more complex and of high risk This motivatesus to tailor more powerful accident causation models tocapture the complexity of the highly technological systemsfrom a broad systemic view for understanding characteristicsof accidents

Mentioning that causation factors and their relationshipsfor accidents are always complex in terms of uncertaintyrandomness abstractness fuzziness and other propertiesit would be a nice try to employ the complex networktheory [26ndash28] to reveal the involved complexity in accidentscausation analysis The primary purpose of this paper isto construct an accident causation network for causalityanalysis based on the complex theory With the influenceor relation between accident causation models and accidentinvestigation methods as mentioned above we apply thecascading failure scheme to characterize the process of theaccident occurrence performed on the proposed model as aninvestigation approach Together with the help of the networkefficiency of the underlying accident causation networkwe can evaluate the injury severity of the whole systemwith unexpected disturbances from technical human socialorganizational and environmental aspects of the wholesystem

This paper is organized as follows Some selected funda-mental concepts in complex network are recalled in Section 2The new accident causation network model is constructedin Section 3 and the cascading failure scheme is appliedto characterize the evolution of the proposed causationnetwork in Section 4 Simulation based on our proposedaccident causation analysis method is described in Section 5Conclusions are drawn in Section 6

2 Basic Concepts in Complex Network

The complex network is a graph with complex topologicalfeatures thatmay not occur in simple networks such as latticesor random graphs but often occur in real graphs The studyof complex networks has attracted great interest inspiredlargely by the empirical study of real-world networks suchas computer networks and social networks In mathematicalterms a network is represented by a graph A graph is apair of sets (119881 119864) where 119881 is a set of 119873 nodes (or vertices)V1 V2 V

119873and 119864 is a set of edges (or links) that connect

two elements of 119881 Graphs are usually represented as aset of dots each corresponding to a node two of thesedots being joined by a line if the corresponding nodes areconnected Usually we use 119873(V

119894) to denote the set of all

nodes in 119881 that are connected to node V119894 A path in a

graph is a sequence of edges which connect a sequence ofvertices The shortest path length of two nodes is definedas the smallest number of edges that connect these twonodes

1

2

3

5

4

Figure 1 Illustration of a graph with 119873 = 5 nodes and6 edges where the edge set is 119864 = (V

1 V2) (V1 V3) (V1 V5)

(V2 V3) (V2 V5) (V3 V5) and119873(V

1) = V

2 V3 V5119873(V

2) = V

1 V3 V5

119873(V3) = V

1 V2 V5 119873(V

4) is an empty set and 119873(V

5) = V

1 V2 V3

The shortest path length of nodes V1and V

2is 1

2

31

4 10

5

9

6 78

Figure 2 Illustration for the random network with probability 119901 =015 for every pair of nodes being connected 119873(V

1) = V

8 and

(V1) = V

6 V7

Figure 1 shows an illustration of a graph with 119873 = 5

nodes and 6 edges With the complexity of real networksthe edge sets are sometimes not determined which meansthere are some pairs of nodes with a random or uncertainlink such as the random network shown in Figure 2 withprobability 119901 = 015 for every pair of nodes being connected[29] To distinguish the adjacent nodes with deterministicconnections and those with probability less than 1 we use(119894) to denote the set of all nodes in 119881 that are connectedto node V

119894with probability less than 1

3 Accident Causation Network

Causation factors and their relationships for sociotechni-cal system accidents are always complex with uncertaintyrandomness abstractness fuzziness and other propertiesFor example the relation between two causation factors





















A A

AA

BBBBB

BB

B

B B

CCCCCC

C

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

D D DD

DD

DDEE

EE

EEEE

EE EEEE E

F FF

F

F

F

F

FF

FF

F FF

F

9

9

9

10

10

10

11

11

12

12

8

8

8

8

7

7

7

7

13

13

14

1415

15

7








that is 119871119894(119905) gt 119888



119890119894119895(119905 + 1) =

119890119894119895(0)

119888119894

119871119894(119905) if 119888

119894lt 119871119894(119905)

119890119894119895(0) otherwise

(1)


119890 (119866 (119905)) =1

119873 (119873 minus 1)sum119894 = 119895

119890119894119895(119905) (2)



119897lt



119871119894(119905 + 1) = 119871

119894(119905) + sum119895isin119861119894(119905)

119871119895(119905)

10038161003816100381610038161003816119905(119895)10038161003816100381610038161003816

(3)

with 119861119894(119905) = 119895 isin

119905(119894) 119888

119895lt 119871119895(119905)













1

11

11 22

22

2

33

333

44

444

55

555

66

666

77

777

88

888

99

999

t = 1 t = 2 t = 3

t = 4t = 5





120575119894(119905) =

1 minus119888119894

119871119894(119905) if 119888

119894lt 119871119894(119905) for some 119905 le 119905

0 otherwise(4)


119890 (119866 (119905 + 1)) = tr[119864119873

sum119894=1

(119868 + 120575119894(119905) 119864119894119894) 119864 (119868 + 120575

119894(119905) 119864119894119894)]

119905 = 0 1

(5)




5 A Case Study






(12)119888119894 that is 120572



Step 3 Disturbances



119871F14 (1) = 8 gt 2 = 119888F14

1(F14) = E6E12E13E15 F4 F5 F6 F8 F11 F15

(6)


119871E6 (2) = 15 + 08 lt 3 = 119888E6

119871E12 (2) = 05 + 08 gt 1 = 119888E12

2(E6) = E9E10E11E14 F15

119871E13 (2) = 05 + 08 gt 1 = 119888E13

2(E13) = E3E4E14 F15

119871E15 (2) = 15 + 08 lt 3 = 119888E15

119871F4 (2) = 15 + 08 lt 3 = 119888F4

119871F5 (2) = 15 + 08 lt 3 = 119888F5

119871F6 (2) = 15 + 08 lt 3 = 119888F6

119871F8 (2) = 05 + 08 gt 1 = 119888F8

2(F8) = E14

119871F11 (2) = 15 + 08 lt 3 = 119888F11

119871F15 (2) = 1 + 08 lt 2 = 119888F15

(7)


119871E14 (3) = 185 +13

5+13

4+ 13 gt 37


119871E9 (3) = 1 +25

5+ 35 gt 2 = 119888E9

(8)




119897119894

119894

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Level

A 4 5 5 5B 5 5 4 4 3 4 5 5 5 5C 4 5 5 5 4 4 4D 3 5 4 4 4 3 3 3E 4 3 2 2 5 3 4 3 2 2 2 1 1 37 3F 5 5 5 3 3 3 4 1 4 3 3 3 4 2 2



119871F7 (1) = 16 gt 4 = 119888F14 1198731(F7) = F1 F2 F3 F4

(9)


119871F1 (2) = 25 + 4 gt 5 = 119888F1 1198732(F1) = F9

119871F2 (2) = 25 + 4 gt 5 = 119888F2 1198732(F2) = F9

119871F3 (2) = 25 + 4 gt 5 = 119888F3 1198732(F3) = F9

119871F4 (2) = 15 + 4 gt 3 = 119888F4 1198732(F4) = F14

(10)


119871F9 (3) = 2 + 65 + 65 + 65 gt 4 = 119888F9

3(F9) = F10

119871F14 (3) = 1 + 55 gt 2 = 119888E9

3(F14) = F5 F6 F8 F11 F15E6E12E13E15

(11)


119871F10 (4) = 15 + 215 gt 3 = 119888F10

1198734(F10) = F12 F13E11

119871F5 (4) = 15 +65

9lt 3 = 119888F5

119871F6 (4) = 15 +65

9lt 3 = 119888F6

119871F8 (4) = 05 +65

9gt 1 = 119888F8

1198734(F8) = E14

119871F11 (4) = 15 +65

9lt 3 = 119888F11

119871F15 (4) = 1 +65

9lt 2 = 119888F10

119871E6 (4) = 15 +65

9lt 3 = 119888E6

119871E12 (4) = 05 +65

9gt 1 = 119888E12

4(E12) = E9E10E11E14 F15

119871E13 (4) = 05 +65

9gt 1 = 119888E13

4(E13) = E3E4E14E15

119871E15 (4) = 15 +65

9lt 3 = 119888E15

(12)


119871E14 (5) = 185 + (05 +65

9) (1 +

1

4+1

5)


119871F12 (5) = 15 +23

3gt 3 = 119888F12 119873

5(F12) = E10

119871F13 (5) = 2 +23

3gt 4 = 119888F13 119873

5(F13)= empty set

119871E11 (5) = 1 +23

3gt 2 = 119888E11 119873

5(E11) = E2

(13)




0

0

1

2

3

4

5

6

7

8

10 20 30 40 50 60

Vertices

Load


t = 1

t = 2

t = 3


00

5

10

10

20 30 40 50 60

Vertices

Load


t = 0

t = 1

t = 2

t = 3

t = 4

t = 5

15

20

25








Vertices


t = 1

t = 2

t = 3

0

20

18

16

14

12

10

8

6

4

2

0

Load

10 20 30 40 50 60


Vertices


t = 0

t = 1

t = 2

t = 3

0 10 20 30 40 50 60

40

35

30

25

20

15

10

5

0

Load

t = 4

t = 5





140

120

100

80

60

40

20

A1 B5 C1 C5 D1 D4 D7 D8 E6 E13 F140






6 Conclusions


Research Highlights





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























A A

AA

BBBBB

BB

B

B B

CCCCCC

C

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

D D DD

DD

DDEE

EE

EEEE

EE EEEE E

F FF

F

F

F

F

FF

FF

F FF

F

9

9

9

10

10

10

11

11

12

12

8

8

8

8

7

7

7

7

13

13

14

1415

15

7








that is 119871119894(119905) gt 119888



119890119894119895(119905 + 1) =

119890119894119895(0)

119888119894

119871119894(119905) if 119888

119894lt 119871119894(119905)

119890119894119895(0) otherwise

(1)


119890 (119866 (119905)) =1

119873 (119873 minus 1)sum119894 = 119895

119890119894119895(119905) (2)



119897lt



119871119894(119905 + 1) = 119871

119894(119905) + sum119895isin119861119894(119905)

119871119895(119905)

10038161003816100381610038161003816119905(119895)10038161003816100381610038161003816

(3)

with 119861119894(119905) = 119895 isin

119905(119894) 119888

119895lt 119871119895(119905)













1

11

11 22

22

2

33

333

44

444

55

555

66

666

77

777

88

888

99

999

t = 1 t = 2 t = 3

t = 4t = 5





120575119894(119905) =

1 minus119888119894

119871119894(119905) if 119888

119894lt 119871119894(119905) for some 119905 le 119905

0 otherwise(4)


119890 (119866 (119905 + 1)) = tr[119864119873

sum119894=1

(119868 + 120575119894(119905) 119864119894119894) 119864 (119868 + 120575

119894(119905) 119864119894119894)]

119905 = 0 1

(5)




5 A Case Study






(12)119888119894 that is 120572



Step 3 Disturbances



119871F14 (1) = 8 gt 2 = 119888F14

1(F14) = E6E12E13E15 F4 F5 F6 F8 F11 F15

(6)


119871E6 (2) = 15 + 08 lt 3 = 119888E6

119871E12 (2) = 05 + 08 gt 1 = 119888E12

2(E6) = E9E10E11E14 F15

119871E13 (2) = 05 + 08 gt 1 = 119888E13

2(E13) = E3E4E14 F15

119871E15 (2) = 15 + 08 lt 3 = 119888E15

119871F4 (2) = 15 + 08 lt 3 = 119888F4

119871F5 (2) = 15 + 08 lt 3 = 119888F5

119871F6 (2) = 15 + 08 lt 3 = 119888F6

119871F8 (2) = 05 + 08 gt 1 = 119888F8

2(F8) = E14

119871F11 (2) = 15 + 08 lt 3 = 119888F11

119871F15 (2) = 1 + 08 lt 2 = 119888F15

(7)


119871E14 (3) = 185 +13

5+13

4+ 13 gt 37


119871E9 (3) = 1 +25

5+ 35 gt 2 = 119888E9

(8)




119897119894

119894

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Level

A 4 5 5 5B 5 5 4 4 3 4 5 5 5 5C 4 5 5 5 4 4 4D 3 5 4 4 4 3 3 3E 4 3 2 2 5 3 4 3 2 2 2 1 1 37 3F 5 5 5 3 3 3 4 1 4 3 3 3 4 2 2



119871F7 (1) = 16 gt 4 = 119888F14 1198731(F7) = F1 F2 F3 F4

(9)


119871F1 (2) = 25 + 4 gt 5 = 119888F1 1198732(F1) = F9

119871F2 (2) = 25 + 4 gt 5 = 119888F2 1198732(F2) = F9

119871F3 (2) = 25 + 4 gt 5 = 119888F3 1198732(F3) = F9

119871F4 (2) = 15 + 4 gt 3 = 119888F4 1198732(F4) = F14

(10)


119871F9 (3) = 2 + 65 + 65 + 65 gt 4 = 119888F9

3(F9) = F10

119871F14 (3) = 1 + 55 gt 2 = 119888E9

3(F14) = F5 F6 F8 F11 F15E6E12E13E15

(11)


119871F10 (4) = 15 + 215 gt 3 = 119888F10

1198734(F10) = F12 F13E11

119871F5 (4) = 15 +65

9lt 3 = 119888F5

119871F6 (4) = 15 +65

9lt 3 = 119888F6

119871F8 (4) = 05 +65

9gt 1 = 119888F8

1198734(F8) = E14

119871F11 (4) = 15 +65

9lt 3 = 119888F11

119871F15 (4) = 1 +65

9lt 2 = 119888F10

119871E6 (4) = 15 +65

9lt 3 = 119888E6

119871E12 (4) = 05 +65

9gt 1 = 119888E12

4(E12) = E9E10E11E14 F15

119871E13 (4) = 05 +65

9gt 1 = 119888E13

4(E13) = E3E4E14E15

119871E15 (4) = 15 +65

9lt 3 = 119888E15

(12)


119871E14 (5) = 185 + (05 +65

9) (1 +

1

4+1

5)


119871F12 (5) = 15 +23

3gt 3 = 119888F12 119873

5(F12) = E10

119871F13 (5) = 2 +23

3gt 4 = 119888F13 119873

5(F13)= empty set

119871E11 (5) = 1 +23

3gt 2 = 119888E11 119873

5(E11) = E2

(13)




0

0

1

2

3

4

5

6

7

8

10 20 30 40 50 60

Vertices

Load


t = 1

t = 2

t = 3


00

5

10

10

20 30 40 50 60

Vertices

Load


t = 0

t = 1

t = 2

t = 3

t = 4

t = 5

15

20

25








Vertices


t = 1

t = 2

t = 3

0

20

18

16

14

12

10

8

6

4

2

0

Load

10 20 30 40 50 60


Vertices


t = 0

t = 1

t = 2

t = 3

0 10 20 30 40 50 60

40

35

30

25

20

15

10

5

0

Load

t = 4

t = 5





140

120

100

80

60

40

20

A1 B5 C1 C5 D1 D4 D7 D8 E6 E13 F140






6 Conclusions


Research Highlights





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

11

11 22

22

2

33

333

44

444

55

555

66

666

77

777

88

888

99

999

t = 1 t = 2 t = 3

t = 4t = 5





120575119894(119905) =

1 minus119888119894

119871119894(119905) if 119888

119894lt 119871119894(119905) for some 119905 le 119905

0 otherwise(4)


119890 (119866 (119905 + 1)) = tr[119864119873

sum119894=1

(119868 + 120575119894(119905) 119864119894119894) 119864 (119868 + 120575

119894(119905) 119864119894119894)]

119905 = 0 1

(5)




5 A Case Study






(12)119888119894 that is 120572



Step 3 Disturbances



119871F14 (1) = 8 gt 2 = 119888F14

1(F14) = E6E12E13E15 F4 F5 F6 F8 F11 F15

(6)


119871E6 (2) = 15 + 08 lt 3 = 119888E6

119871E12 (2) = 05 + 08 gt 1 = 119888E12

2(E6) = E9E10E11E14 F15

119871E13 (2) = 05 + 08 gt 1 = 119888E13

2(E13) = E3E4E14 F15

119871E15 (2) = 15 + 08 lt 3 = 119888E15

119871F4 (2) = 15 + 08 lt 3 = 119888F4

119871F5 (2) = 15 + 08 lt 3 = 119888F5

119871F6 (2) = 15 + 08 lt 3 = 119888F6

119871F8 (2) = 05 + 08 gt 1 = 119888F8

2(F8) = E14

119871F11 (2) = 15 + 08 lt 3 = 119888F11

119871F15 (2) = 1 + 08 lt 2 = 119888F15

(7)


119871E14 (3) = 185 +13

5+13

4+ 13 gt 37


119871E9 (3) = 1 +25

5+ 35 gt 2 = 119888E9

(8)




119897119894

119894

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Level

A 4 5 5 5B 5 5 4 4 3 4 5 5 5 5C 4 5 5 5 4 4 4D 3 5 4 4 4 3 3 3E 4 3 2 2 5 3 4 3 2 2 2 1 1 37 3F 5 5 5 3 3 3 4 1 4 3 3 3 4 2 2



119871F7 (1) = 16 gt 4 = 119888F14 1198731(F7) = F1 F2 F3 F4

(9)


119871F1 (2) = 25 + 4 gt 5 = 119888F1 1198732(F1) = F9

119871F2 (2) = 25 + 4 gt 5 = 119888F2 1198732(F2) = F9

119871F3 (2) = 25 + 4 gt 5 = 119888F3 1198732(F3) = F9

119871F4 (2) = 15 + 4 gt 3 = 119888F4 1198732(F4) = F14

(10)


119871F9 (3) = 2 + 65 + 65 + 65 gt 4 = 119888F9

3(F9) = F10

119871F14 (3) = 1 + 55 gt 2 = 119888E9

3(F14) = F5 F6 F8 F11 F15E6E12E13E15

(11)


119871F10 (4) = 15 + 215 gt 3 = 119888F10

1198734(F10) = F12 F13E11

119871F5 (4) = 15 +65

9lt 3 = 119888F5

119871F6 (4) = 15 +65

9lt 3 = 119888F6

119871F8 (4) = 05 +65

9gt 1 = 119888F8

1198734(F8) = E14

119871F11 (4) = 15 +65

9lt 3 = 119888F11

119871F15 (4) = 1 +65

9lt 2 = 119888F10

119871E6 (4) = 15 +65

9lt 3 = 119888E6

119871E12 (4) = 05 +65

9gt 1 = 119888E12

4(E12) = E9E10E11E14 F15

119871E13 (4) = 05 +65

9gt 1 = 119888E13

4(E13) = E3E4E14E15

119871E15 (4) = 15 +65

9lt 3 = 119888E15

(12)


119871E14 (5) = 185 + (05 +65

9) (1 +

1

4+1

5)


119871F12 (5) = 15 +23

3gt 3 = 119888F12 119873

5(F12) = E10

119871F13 (5) = 2 +23

3gt 4 = 119888F13 119873

5(F13)= empty set

119871E11 (5) = 1 +23

3gt 2 = 119888E11 119873

5(E11) = E2

(13)




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140

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A1 B5 C1 C5 D1 D4 D7 D8 E6 E13 F140






6 Conclusions


Research Highlights





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119897119894

119894

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Level

A 4 5 5 5B 5 5 4 4 3 4 5 5 5 5C 4 5 5 5 4 4 4D 3 5 4 4 4 3 3 3E 4 3 2 2 5 3 4 3 2 2 2 1 1 37 3F 5 5 5 3 3 3 4 1 4 3 3 3 4 2 2



119871F7 (1) = 16 gt 4 = 119888F14 1198731(F7) = F1 F2 F3 F4

(9)


119871F1 (2) = 25 + 4 gt 5 = 119888F1 1198732(F1) = F9

119871F2 (2) = 25 + 4 gt 5 = 119888F2 1198732(F2) = F9

119871F3 (2) = 25 + 4 gt 5 = 119888F3 1198732(F3) = F9

119871F4 (2) = 15 + 4 gt 3 = 119888F4 1198732(F4) = F14

(10)


119871F9 (3) = 2 + 65 + 65 + 65 gt 4 = 119888F9

3(F9) = F10

119871F14 (3) = 1 + 55 gt 2 = 119888E9

3(F14) = F5 F6 F8 F11 F15E6E12E13E15

(11)


119871F10 (4) = 15 + 215 gt 3 = 119888F10

1198734(F10) = F12 F13E11

119871F5 (4) = 15 +65

9lt 3 = 119888F5

119871F6 (4) = 15 +65

9lt 3 = 119888F6

119871F8 (4) = 05 +65

9gt 1 = 119888F8

1198734(F8) = E14

119871F11 (4) = 15 +65

9lt 3 = 119888F11

119871F15 (4) = 1 +65

9lt 2 = 119888F10

119871E6 (4) = 15 +65

9lt 3 = 119888E6

119871E12 (4) = 05 +65

9gt 1 = 119888E12

4(E12) = E9E10E11E14 F15

119871E13 (4) = 05 +65

9gt 1 = 119888E13

4(E13) = E3E4E14E15

119871E15 (4) = 15 +65

9lt 3 = 119888E15

(12)


119871E14 (5) = 185 + (05 +65

9) (1 +

1

4+1

5)


119871F12 (5) = 15 +23

3gt 3 = 119888F12 119873

5(F12) = E10

119871F13 (5) = 2 +23

3gt 4 = 119888F13 119873

5(F13)= empty set

119871E11 (5) = 1 +23

3gt 2 = 119888E11 119873

5(E11) = E2

(13)




0

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t = 2

t = 3


00

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Load


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t = 1

t = 2

t = 3

t = 4

t = 5

15

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Vertices


t = 1

t = 2

t = 3

0

20

18

16

14

12

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6

4

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Load

10 20 30 40 50 60


Vertices


t = 0

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t = 2

t = 3

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40

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30

25

20

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10

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0

Load

t = 4

t = 5





140

120

100

80

60

40

20

A1 B5 C1 C5 D1 D4 D7 D8 E6 E13 F140






6 Conclusions


Research Highlights





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0

1

2

3

4

5

6

7

8

10 20 30 40 50 60

Vertices

Load


t = 1

t = 2

t = 3


00

5

10

10

20 30 40 50 60

Vertices

Load


t = 0

t = 1

t = 2

t = 3

t = 4

t = 5

15

20

25








Vertices


t = 1

t = 2

t = 3

0

20

18

16

14

12

10

8

6

4

2

0

Load

10 20 30 40 50 60


Vertices


t = 0

t = 1

t = 2

t = 3

0 10 20 30 40 50 60

40

35

30

25

20

15

10

5

0

Load

t = 4

t = 5





140

120

100

80

60

40

20

A1 B5 C1 C5 D1 D4 D7 D8 E6 E13 F140






6 Conclusions


Research Highlights





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










140

120

100

80

60

40

20

A1 B5 C1 C5 D1 D4 D7 D8 E6 E13 F140






6 Conclusions


Research Highlights





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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