Research ArticleAn Intelligent Complex Event Processing with 119863Numbers underFuzzy Environment

Fuyuan Xiao

School of Computer and Information Science Southwest University Chongqing 400715 China

Correspondence should be addressed to Fuyuan Xiao xiaofuyuanswueducn

Received 24 May 2016 Revised 19 July 2016 Accepted 18 August 2016

Academic Editor Jianbing Ma

Copyright copy 2016 Fuyuan XiaoThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Efficient matching of incoming mass events to persistent queries is fundamental to complex event processing systems Eventmatching based on pattern rule is an important feature of complex event processing engine However the intrinsic uncertainty inpattern rules which are predecided by experts increases the difficulties of effective complex event processing It inevitably involvesvarious types of the intrinsic uncertainty such as imprecision fuzziness and incompleteness due to the inability of human beingssubjective judgment Nevertheless D numbers is a new mathematic tool to model uncertainty since it ignores the condition thatelements on the frame must be mutually exclusive To address the above issues an intelligent complex event processing methodwith D numbers under fuzzy environment is proposed based on the Technique for Order Preferences by Similarity to an IdealSolution (TOPSIS) method The novel method can fully support decision making in complex event processing systems Finally anumerical example is provided to evaluate the efficiency of the proposed method

1 Introduction

Nowadays there has been increasing interest in distributedapplications which require processing continuously flowingdata from geographically distributed sources to obtain timelyresponses to complex queries such as data stream processing(DSP) systems [1ndash8] and complex event processing (CEP)systems [9ndash16] In principle DSP systems differ from CEPsystems as the DSP systems focus on transforming theincoming flow of information while the CEP systems focuson detecting patterns of information that represent thehigher-level events [17 18] Because of the advantages such asexpressive rule language and efficient event detection modelCEP systems have been highly concerned in academic circlesand industry recently [19ndash25]

In CEP systems event streams are processed in or nearreal time for a variety of purposes from wireless sensornetworks to financial tickers and from traffic management toclick-stream inspection [17 18 26ndash28] In those applicationdomains accurate and effective complex event processing is

critical for dealing with real-world events whereas there aretwo possible origins for uncertainty which increases the dif-ficulties of accurate and effective complex event processingsuch as uncertainty originating at the event source and uncer-tainty resulting from event inference [29] For uncertaintyoriginating at the event source (ie raw event) there may beuncertainty associated with either the event occurrence itselfor the eventrsquos attributes due to a feature of the event sourceFor uncertainty resulting from event inference (ie derivedevents) they are based on other events and uncertaintycan propagate to the derived events Nevertheless in DSPsystems many kinds of strategies have been developed tohandle the above two types of uncertainty [29ndash33]

However most of the CEP systems are dealing withthe flow of events where the basic premise underlying thedesign of the CEP queries proposed is that the associatedpattern rules of events are predefined Moreover many ofpredefined pattern rules are based on human beings sub-jective experience so that it is not necessarily accurate inpractice While rule based reasoning system structure looks

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3713518 10 pageshttpdxdoiorg10115520163713518

2 Mathematical Problems in Engineering

simple the acquisition of knowledge is a big bottleneck in thesystem Because the rules come from experts there are noself-learning functions Such a behavior gives rise to poorquality of query results Furthermore in conventional CEPsystems it inevitably involves various types of the intrinsicuncertainty such as imprecision fuzziness and incomplete-ness due to the inability of human beings subjective judg-ment In these cases it seems more reasonable to considerfuzzy situation in the CEP systems

Due to the efficiency to handle uncertainty and fuse datasome math tools such as fuzzy sets evidence theory andprobability method are widely used in decision making [34ndash42] risk analysis [43] diagnosis [44] and optimizationproblem [45] Needless to say fuzzy sets theory introduced byZadeh is a good approach to settle with the uncertain infor-mation [35 46ndash50] The Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) [51] methodproposed by Hwang and Yoon (1981) is widely used indecisionmaking [52] Moreover the extension of the TOPSIS[51] method was developed to handle fuzzy data [53 54]The TOPSIS method is applied widely in MCDM field Andmany applications have proved its effectivenesswhereas therestill exist some shortcomings One of the open questions isthat current MCDM based on the TOPSIS method cannotadequately handle these types of uncertainties such as impre-cision fuzziness and incompleteness due to the inability ofexpertsrsquo subjective judgment D numbers proposed by Deng[55] is a new mathematic tool to model uncertainty sinceit ignores the condition that elements on the frame mustbe mutually exclusive Compared with existing methodsD numbers can efficiently represent uncertain informationand are more tallied with the actual situation There is anincreasing number of applications about D numbers such asFEMA analysis [56] failure mode analysis [57] environmentassessment [58ndash60] supplier selection [61] and productengineering [62]

To address these issues we propose an intelligent complexevent processing strategy with D numbers under fuzzy envi-ronment based on the TOPSIS method The novel methodcan deal with pattern-rule based uncertainty under fuzzyenvironment to realize accurate and effective event streamprocessing Effectiveness of the proposedmethod is evaluatedthrough a numerical experiment

The rest of this paper is organized as follows Section 2briefly introduces the preliminaries of this paper After thatSection 3 proposes an intelligent complex event processingstrategy withD numbers under fuzzy environment Section 4gives a numerical example to show the effectiveness of theproposed method Finally Section 5 gives a conclusion

2 Preliminaries

21 Dempster-Shafer Evidence Theory Dempster-Shafer evi-dent theory [63 64] also known as evidence theory is usedto handle uncertain information belonging to the categoryof artificial intelligence As a theory reasoning under theuncertain environment it needs weaker conditions thanthe Bayesian theory of probability When the probability is

confirmed Dempster-Shafer theory could convert intoBayesian theory so it is often regarded as an extension ofthe Bayesian theory Dempster-Shafer theory has the advan-tage of directly expressing the ldquouncertaintyrdquo by assigning theprobability to the subsets of the set composed of multipleobjects rather than to an individual object Besides it has theability to combine pairs of bodies of evidence or belief func-tions to derive a new evidence or belief function Based onthe Dempster-Shafer evident theory Deng proposed thegeneralized evidence theory to extend the classical evidencetheory [34] For completeness of the explanation some basicconcepts are introduced as follows

Definition 1 (frame of discernment) Let 119880 be a set of mutu-ally exclusive and collectively exhaustive indicated by

119880 = 1198641 1198642 119864

119894 119864

119873 (1)

The set 119880 is called frame of discernment The power set of 119880is indicated by 2

119880 where

2119880= 0 119864

1 119864

119873 1198641 1198642 119864

1 1198642 119864

119894

119880

(2)

and 0 is an empty set If 119860 isin 2119880 119860 is called a proposition

Definition 2 (mass function) For a frame of discernment 119880a mass function is a mapping 119898 from 2

119880 to [0 1] formallydefined by

119898 fl 2119880997888rarr [0 1] (3)

which satisfies the following condition

119898(0) = 0

sum

119860isin2119880

119898(119860) = 1(4)

In the Dempster-Shafer evident theory a mass function isalso called a basic probability assignment (BPA) If119898(119860) gt 0119860 is called a focal element the union of all focal elements iscalled the core of the mass function

Definition 3 (Dempsterrsquos rule of combination) Dempsterrsquosrule of combination also called orthogonal sum denoted by119898 = 119898

1oplus 1198982 is defined as follows

119898(119860) =

1

1 minus 119870sum

119861cap119862=119860

1198981(119861)119898

2(119862) 119860 = 0

0 119860 = 0

(5)

Mathematical Problems in Engineering 3

with

119870 = sum

119861cap119862=0

1198981(119861)119898

2(119862) (6)

where 119861 and 119862 are also elements of 2119880 and 119870 is a constant toshow the conflict between the twoBPAsNote thatDempsterrsquosrule of combination is only applicable to such twoBPAswhichsatisfy the condition119870 lt 122 D Number Theory D number theory is a generalizationof Dempster-Shafer evidence theory proposed by Deng [55]In the classical Dempster-Shafer theory there are severalstrong hypotheses on the frame of discernment and basicprobability assignment and still some shortcomings whichlimit the ability of Dempster-Shafer theory to represent sometypes of information and restrict the application in practiceD number theory as an extension and development methodis defined as follows

Definition 4 (D number) LetΩ be a finite nonempty set a Dnumber is a mapping formulated by

119863 Ω 997888rarr [0 1] (7)

with

sum

119861subeΩ

119863 (119861) le 1

119863 (0) = 0

(8)

where 119861 is a subset ofΩ

It seems that the definition of D numbers is similarto the definition of BPA However in D number theory

the elements of Ω do not require to be mutually exclusiveIn addition being contrary of the frame of discernment119880 containing overall events Ω is acceptable to incompleteinformation by sum

119861subeΩ119863(119861) le 1

Furthermore for a discrete set Ω = 1198871 1198872 119887

119894 119887

119899

where 119887119894isin 119877 and when 119894 = 119895 119887

119894= 119887119895 A special form of D

numbers can be expressed by

119863(1198871) = V

1

119863(1198872) = V

2

119863 (119887119894) = V

119894

119863 (119887119899) = V

119899

(9)

or simply denoted as 119863 = (1198871 V1) (1198872 V2) (119887

119894 V119894)

(119887119899 V119899) where V

119894gt 0 and sum

119899

119894=1V119894le 1

Definition 5 (two D numbersrsquo rule of combination) Let1198631

= (1198871

1 V11) (119887

1

119894 V1119894) (119887

1

119899 V1119899) 1198632

= (1198872

1 V21)

(1198872

119895 V2119895) (119887

2

119899 V2119899) be two D numbers the combination of

1198631and119863

2 indicated by119863 = 119863

1oplus 1198632 is defined by

119863 (119887) = V (10)

with

119887 =

1198871

119894+ 1198872

119895

2

V =

(V1119894+ V2119895) 2

119862

119862 =

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2)

119899

sum

119894=1

V1119894= 1

119898

sum

119895=1

V2119895= 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119898

sum

119895=1

(

V1119888+ V2119895

2)

119899

sum

119894=1

V1119894lt 1

119898

sum

119895=1

V2119895= 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119899

sum

119894=1

(V1119894+ V2119888

2)

119899

sum

119894=1

V1119894= 1

119898

sum

119895=1

V2119895lt 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119898

sum

119895=1

(

V1119888+ V2119895

2) +

119899

sum

119894=1

(V1119894+ V2119888

2) +

V1119888+ V2119888

2

119899

sum

119894=1

V1119894lt 1

119898

sum

119895=1

V2119895lt 1

(11)

where V1119888= 1 minus sum

119899

119894=1V1119894and V2119888= 1 minus sum

119898

119895=1V2119895

In themeanwhile an aggregation operator is proposed onthis special D number it is defined as below

4 Mathematical Problems in Engineering

0a b c

x

1

120583A(x)

Figure 1 A triangular fuzzy number

Definition 6 (D numbersrsquo integration) For 119863 = (1198871 V1) (1198872

V2) (119887

119894 V119894) (119887

119899 V119899) the integrating representation of

119863 is defined as

119868 (119863) =

119899

sum

119894=1

119887119894V119894 (12)

23 TOPSIS Method with Fuzzy Data Fuzzy set theory [65]provide an alternative and convenient framework for mod-eling of real-world fuzzy decision systems mathematically[66ndash68] A fuzzy set is any set that allows its members tohave different grades of membership in the interval [0 1] Itconsists of two components a set and amembership functionassociated with it

Definition 7 (fuzzy set [69]) Let 119883 be a collection of objectsdenoted generally by 119909 a fuzzy subset of 119883 is a set ofordered pairs

= (119909 120583(119909) | 119909 isin 119883) (13)

where 120583(119909) 119883 rarr [0 1] is called the membership function

(generalized characteristic function) which maps 119883 to themembership space 119872 Its range is the subset of nonnegativereal members whose supremum is finite

Definition 8 (triangular fuzzy number [66]) A fuzzy numberis a fuzzy subset of 119883 And a triangular fuzzy number canbe defined by a triplet (119886 119887 119888) shown in Figure 1 in which 119886119887 and 119888 are real numbers with 119886 lt 119887 lt 119888 Its membershipfunction is defined as

120583(119909) =

0 119909 lt 119886

119909 minus 119886

119887 minus 119886 119886 le 119909 le 119887

119888 minus 119909

119888 minus 119887 119887 le 119909 le 119888

0 119909 gt 119888

(14)

Definition 9 (distance between two triangular fuzzy numbers[53]) Let = (119886 119887 119888) and = (119898 119904 119899) be two triangular

fuzzy numbers then the vertexmethod is defined to calculatethe distance between them as

119889 ( ) = radic1

3[(119886 minus 119898)

2+ (119887 minus 119904)

2+ (119888 minus 119899)

2] (15)

The main idea of TOPSIS is that the best compromisesolution should have the shortest Euclidean distance fromthe positive ideal solution and the farthest Euclidean distancefrom the negative ideal solution

The procedures of TOPSIS method with fuzzy data canbe described as follows Let 119860 = 119860

119896| 119896 = 1 2 119898 be

the set of alternatives 119862 = 119904| 119904 = 1 2 119899 be the set

of criteria and 119877 = 119896119904

| 119896 = 1 2 119898 119904 = 1 2 119899 bethe performance ratings with the criteria weight vector 119882 =

119904| 119904 = 1 2 119899

Step 1 (calculate normalized ratings) Let119861 and119862 be the set ofbenefit criteria and cost criteria respectivelyThe normalizedvalue

119896119904is calculated by

119896119904

= (119886119896119904

119888+119904

119887119896119904

119888+119904

119888119896119904

119888+119904

)

where 119888+

119904= max119896

(119888119896119904) 119904 isin 119861

119896119904

= (119886minus

119904

119888119896119904

119886minus

119904

119887119896119904

119886minus

119904

119886119896119904

)

where 119886minus

119904= min119896

(119886119896119904) 119904 isin 119862

(16)

Step 2 (calculate weighted normalized ratings) In theweighted normalized decision matrix the modified ratingsare calculated by

V119896119904

= 119904times 119896119904

for 119896 = 1 2 119898 119904 = 1 2 119899 (17)

where 119904is the weight of the sth criteria

Step 3 (determine the fuzzy positive and negative idealsolutions) The elements V

119894119895 forall119894 119895 are normalized positive

triangular fuzzy numbers and their ranges belong to theclosed interval [0 1] Then the fuzzy positive ideal solution(FPIS 119860+) and the fuzzy negative ideal solution (FNIS 119860minus)are derived as follows

119860+= V+1 V+2 V+

119899

119860minus= Vminus1 Vminus2 Vminus

119899

(18)

where V+119904= (1 1 1) and Vminus

119904= (0 0 0) 119904 = 1 2 119899

Mathematical Problems in Engineering 5

Table 1 Linguistic variables for the importance weight of eachcriterion

Very poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

Table 2 The importance weight of the criteria

Expert1

Expert2

Expert3

Accuracy (1198621) 03069 02632 01478

Response time (1198622) 02255 04358 03222

Cost (1198623) 02663 00971 01269

Operability (1198624) 02013 02039 04031

Step 4 (calculate the distance of each alternative from the FPISand the FNIS) The distance of each alternative from 119860

+ and119860minus can be currently calculated as

119889+

119896=

119899

sum

119904=1

119889 (V119896119904 V+119904) 119896 = 1 2 119898

119889minus

119896=

119899

sum

119904=1

119889 (V119896119904 Vminus119904) 119896 = 1 2 119898

(19)

where 119889(sdot sdot) is the distance measurement between two fuzzynumbers

Step 5 (calculate the relative closeness coefficient to thepositive ideal solution) The relative closeness coefficient forthe alternative 119860

119896with respect to 119860

+ is

119862119896=

119889minus

119896

119889+

119896+ 119889minus

119896

119896 = 1 2 119898 (20)

Step 6 (rank the alternatives) Obviously an alternative 119860119896

is closer to the FPIS (119860+) and farther from FNIS (119860minus) as119862119896approaches 1 Therefore according to relative closeness

coefficient to the ideal alternative larger value of119862119896indicates

the better alternative 119860119896

3 The Proposed Method

Before introducing the system architecture of intelligent CEPwe first have a clear understanding of the definition of anevent [10 11] An event that represents an atomic instanceis an occurrence that is of interest at a point in timeBasically events can be classified into primitive events andcomposite events A primitive event instance is predefined asa single occurrence of interest that cannot be split into anysmall events A composite event instance that occurs overan interval is created by composing primitive or compositeevents A pattern rule is a template specifying one or morecombinations of events by the nesting of sequences (SEQ) and

conjunctions (AND) which can have negative event type(s)and their combination In the following 119864

119894denotes an event

type which can be either primitive or composite Some detailswere presented in [70]

Definition 10 A SEQ operator [13] specifies a specific orderaccording to the start time-stamps in which the event mustoccur to match the pattern and thus form a composite event

SEQ (1198641 119864

119894 119864

119899) = ⟨119890

1 119890

119894 119890

119899⟩ |

(1198901sdot 119904119905 lt sdot sdot sdot lt 119890

119894sdot 119904119905 lt sdot sdot sdot lt 119890

119899sdot 119904119905) and (119890

1sdot 119905 = 119864

1)

and sdot sdot sdot and (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot and (119890

119899sdot 119905 = 119864

119899)

(21)

For example SEQ (History taking Physical examinationLaboratory examination) consists of SEQ operator that canbe used to monitor the patients who have completed medicaldiagnosis

Definition 11 AnAND operator [13] takes a set of event typesas input and events occur within a specified time windowwithout a specified time order

AND (119864119894 119864

119896 119864

119895)

= ⟨119890119894 119890

119896 119890

119895⟩ | (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot

and (119890119896sdot 119905 = 119864

119896) and sdot sdot sdot and (119890

119895sdot 119905 = 119864

119895)

(22)

For example AND (Laboratory examination Surgery)consists of AND operator that can be used to monitor thepatientsrsquo preoperative and postoperative situations

Currently the pattern-rule base of CEP systems consist-ing of a set of pattern rules is predefined where it is based onhuman beings subjective experience so that it is not necessar-ily accurate in practice whereas the acquisition of knowledgeis a big bottleneck in the system and most researches on CEPsystems seldom discuss how to handle such kinds of prob-lems As we discussed in Section 1 because the rules comefrom experts there is no self-learning function The CEPengine based on the inaccurate predefined pattern-rule basewill generate poor quality of query results Furthermore inconventional CEP systems it inevitably involves various typesof the intrinsic uncertainty such as imprecision fuzzinessand incompleteness due to the inability of human beingssubjective judgment

In this section a novel intelligent complex event pro-cessing strategy with D numbers under fuzzy environmentis proposed based on the Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) method Theproposed method can fully support decision making in CEPsystems

The proposed system architecture of intelligent CEP isshown in Figure 2 it mainly involves two components eventcollector engine and CEP engine under D number theory

6 Mathematical Problems in Engineering

Table 3 The ratings of the three candidates by experts under all criteria

Alternative Expert1

Expert2

Expert3

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

PRA1

MG F G G VG F G G F G F GPRA2

F MG F MG VG MG VG F MG MG G FPRA3

G F G MG VG MG VG G MG G F VG

Data sources

Event collectorengine

Simple eventsCEP engine

(pattern-rule base)

Complex events

Transform

Generate new event

based on TOPSIS D number theory

Figure 2 The system architecture of intelligent CEP

Table 4The fuzzy numbers of three candidates by experts under allcriteria

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(5 7 9) (3 5 7) (7 9 10) (7 9 10)PRA2

(3 5 7) (5 7 9) (3 5 7) (5 7 9)PRA3

(7 9 10) (3 5 7) (7 9 10) (5 7 9)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(9 10 10) (3 5 7) (7 9 10) (7 9 10)PRA2

(9 10 10) (5 7 9) (9 10 10) (3 5 7)PRA3

(9 10 10) (5 7 9) (9 10 10) (7 9 10)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(3 5 7) (7 9 10) (3 5 7) (7 9 10)PRA2

(5 7 9) (5 7 9) (7 9 10) (3 5 7)PRA3

(5 7 9) (7 9 10) (3 5 7) (9 10 10)

based on TOPSIS method We will explain each componentin detail

Component 1 Event Collector Engine The event collectorengine which collects events from data sources first gener-ates a unified formal definition of the event flow

problems in detailStep 1 analyze the decision-making

Step 2 form the pattern-rule base Step 3 collect simple events by theevent collector engine

Step 4 generate complex events by the CEP engine

D number theory based on TOPSIS

Figure 3 The flowchart of the proposed method

Component 2 CEP Engine under D Number Theory Basedon TOPSIS Method Then the CEP engine processes thecollected simple events through the pattern rules in whichthe events generated at this component are called complexevents whereas the D number theory based on TOPSISmethod which is the addition to the CEP engine in view ofthe detected uncertainty event set is used for pattern-rulesanalysis to select the best one from the decision candidatesFinally the generated new events are sent out directly to par-ticular users or reused as input events or used to do furtheranalysis

The main steps of the proposed method are shown as fol-lows and the basic flowchart is given in Figure 3 for the betterunderstanding of the concept

Step 1 Make certain of the detailed information of thedecision-making problems including the goal and criteria

Mathematical Problems in Engineering 7

Table 5 The fuzzy normalized decision matrix

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(050 070 090) (030 050 070) (070 090 100) (070 090 100)PRA2

(030 050 070) (056 078 100) (030 050 070) (056 078 100)PRA3

(070 090 100) (050 070 070) (070 090 100) (050 070 090)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(090 100 100) (030 050 070) (070 090 100) (070 090 100)PRA2

(090 100 100) (056 078 100) (090 100 100) (033 056 078)PRA3

(090 100 100) (050 070 090) (090 100 100) (070 090 100)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(030 050 070) (070 090 100) (030 050 070) (070 090 100)PRA2

(050 070 090) (056 078 100) (070 090 100) (033 056 078)PRA3

(050 070 090) (070 090 100) (030 050 070) (090 100 100)

Table 6 The closeness coefficient of each alternative

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

06779 05000 08275 08275PRA2

05000 07357 05000 07357PRA3

08275 05000 08275 06779

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

09437 05000 08275 08275PRA2

09437 07357 09437 05490PRA3

09437 06779 09437 08275

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

05000 08275 05000 08275PRA2

06779 07357 08275 05490PRA3

06779 08275 05000 09437

Step 2 After determining the goal and criteria it forms thepattern-rule base by leveraging D number theory based onTOPSIS method

Step 3 For the intelligent CEP it then collects simple eventsby the event collector engine from the data sources

Step 4 Based on the pattern-rule base the intelligent CEPsystem can process the collected simple events and generatenew complex events by the CEP engine

4 Numerical Example

In this section in order to illustrate the application of theproposed method a simple numerical example is givenSuppose that there are three pattern-rule alternatives(PRA) which may be considered for further generating acomposite event in the CEP system namely PRA

1=

SEQ(1198641 119864

119894 119864

119896) PRA

2= SEQ(119864

1 119864

119895 119864

119896)

and PRA3= SEQ(119864

1 119864

119895 119864

119899) Meanwhile there are

four evaluation criteria and three experts need to select themost suitable pattern-rule candidate from them for bettersupporting decision making in CEP systems Three expertsgive different assessment results to different alternatives interms of different evaluation criteria Four benefit criteria areconsidered

(1) Accuracy (1198621)

(2) Response Time (1198622)

(3) Cost (1198623)

(4) Operability (1198624)

The proposed method is now applied to solve this decisionproblem Then computational procedure is summarized asfollows

Step 1 The experts use linguistic weighting variables to assessthe importance of the criteria which is shown in Table 1

Step 2 In this paper the criteria weights for different expertsdetermined by variation coefficient method are adopted andpresented in Table 2

Step 3 Using the linguistic rating variables of Table 1 theratings of the three decision alternatives by different expertsunder four evaluation criteria are obtained as shown in

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

[1] D J Abadi Y Ahmad M Balazinska et al ldquoThe design ofthe Borealis stream processingenginerdquo in Proceedings of the 2ndBiennial Conference on Innovative Data Systems Research pp277ndash289 2005

[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

simple the acquisition of knowledge is a big bottleneck in thesystem Because the rules come from experts there are noself-learning functions Such a behavior gives rise to poorquality of query results Furthermore in conventional CEPsystems it inevitably involves various types of the intrinsicuncertainty such as imprecision fuzziness and incomplete-ness due to the inability of human beings subjective judg-ment In these cases it seems more reasonable to considerfuzzy situation in the CEP systems

Due to the efficiency to handle uncertainty and fuse datasome math tools such as fuzzy sets evidence theory andprobability method are widely used in decision making [34ndash42] risk analysis [43] diagnosis [44] and optimizationproblem [45] Needless to say fuzzy sets theory introduced byZadeh is a good approach to settle with the uncertain infor-mation [35 46ndash50] The Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) [51] methodproposed by Hwang and Yoon (1981) is widely used indecisionmaking [52] Moreover the extension of the TOPSIS[51] method was developed to handle fuzzy data [53 54]The TOPSIS method is applied widely in MCDM field Andmany applications have proved its effectivenesswhereas therestill exist some shortcomings One of the open questions isthat current MCDM based on the TOPSIS method cannotadequately handle these types of uncertainties such as impre-cision fuzziness and incompleteness due to the inability ofexpertsrsquo subjective judgment D numbers proposed by Deng[55] is a new mathematic tool to model uncertainty sinceit ignores the condition that elements on the frame mustbe mutually exclusive Compared with existing methodsD numbers can efficiently represent uncertain informationand are more tallied with the actual situation There is anincreasing number of applications about D numbers such asFEMA analysis [56] failure mode analysis [57] environmentassessment [58ndash60] supplier selection [61] and productengineering [62]

To address these issues we propose an intelligent complexevent processing strategy with D numbers under fuzzy envi-ronment based on the TOPSIS method The novel methodcan deal with pattern-rule based uncertainty under fuzzyenvironment to realize accurate and effective event streamprocessing Effectiveness of the proposedmethod is evaluatedthrough a numerical experiment

The rest of this paper is organized as follows Section 2briefly introduces the preliminaries of this paper After thatSection 3 proposes an intelligent complex event processingstrategy withD numbers under fuzzy environment Section 4gives a numerical example to show the effectiveness of theproposed method Finally Section 5 gives a conclusion

2 Preliminaries

21 Dempster-Shafer Evidence Theory Dempster-Shafer evi-dent theory [63 64] also known as evidence theory is usedto handle uncertain information belonging to the categoryof artificial intelligence As a theory reasoning under theuncertain environment it needs weaker conditions thanthe Bayesian theory of probability When the probability is

confirmed Dempster-Shafer theory could convert intoBayesian theory so it is often regarded as an extension ofthe Bayesian theory Dempster-Shafer theory has the advan-tage of directly expressing the ldquouncertaintyrdquo by assigning theprobability to the subsets of the set composed of multipleobjects rather than to an individual object Besides it has theability to combine pairs of bodies of evidence or belief func-tions to derive a new evidence or belief function Based onthe Dempster-Shafer evident theory Deng proposed thegeneralized evidence theory to extend the classical evidencetheory [34] For completeness of the explanation some basicconcepts are introduced as follows

Definition 1 (frame of discernment) Let 119880 be a set of mutu-ally exclusive and collectively exhaustive indicated by

119880 = 1198641 1198642 119864

119894 119864

119873 (1)

The set 119880 is called frame of discernment The power set of 119880is indicated by 2

119880 where

2119880= 0 119864

1 119864

119873 1198641 1198642 119864

1 1198642 119864

119894

119880

(2)

and 0 is an empty set If 119860 isin 2119880 119860 is called a proposition

Definition 2 (mass function) For a frame of discernment 119880a mass function is a mapping 119898 from 2

119880 to [0 1] formallydefined by

119898 fl 2119880997888rarr [0 1] (3)

which satisfies the following condition

119898(0) = 0

sum

119860isin2119880

119898(119860) = 1(4)

In the Dempster-Shafer evident theory a mass function isalso called a basic probability assignment (BPA) If119898(119860) gt 0119860 is called a focal element the union of all focal elements iscalled the core of the mass function

Definition 3 (Dempsterrsquos rule of combination) Dempsterrsquosrule of combination also called orthogonal sum denoted by119898 = 119898

1oplus 1198982 is defined as follows

119898(119860) =

1

1 minus 119870sum

119861cap119862=119860

1198981(119861)119898

2(119862) 119860 = 0

0 119860 = 0

(5)

Mathematical Problems in Engineering 3

with

119870 = sum

119861cap119862=0

1198981(119861)119898

2(119862) (6)

where 119861 and 119862 are also elements of 2119880 and 119870 is a constant toshow the conflict between the twoBPAsNote thatDempsterrsquosrule of combination is only applicable to such twoBPAswhichsatisfy the condition119870 lt 122 D Number Theory D number theory is a generalizationof Dempster-Shafer evidence theory proposed by Deng [55]In the classical Dempster-Shafer theory there are severalstrong hypotheses on the frame of discernment and basicprobability assignment and still some shortcomings whichlimit the ability of Dempster-Shafer theory to represent sometypes of information and restrict the application in practiceD number theory as an extension and development methodis defined as follows

Definition 4 (D number) LetΩ be a finite nonempty set a Dnumber is a mapping formulated by

119863 Ω 997888rarr [0 1] (7)

with

sum

119861subeΩ

119863 (119861) le 1

119863 (0) = 0

(8)

where 119861 is a subset ofΩ

It seems that the definition of D numbers is similarto the definition of BPA However in D number theory

the elements of Ω do not require to be mutually exclusiveIn addition being contrary of the frame of discernment119880 containing overall events Ω is acceptable to incompleteinformation by sum

119861subeΩ119863(119861) le 1

Furthermore for a discrete set Ω = 1198871 1198872 119887

119894 119887

119899

where 119887119894isin 119877 and when 119894 = 119895 119887

119894= 119887119895 A special form of D

numbers can be expressed by

119863(1198871) = V

1

119863(1198872) = V

2

119863 (119887119894) = V

119894

119863 (119887119899) = V

119899

(9)

or simply denoted as 119863 = (1198871 V1) (1198872 V2) (119887

119894 V119894)

(119887119899 V119899) where V

119894gt 0 and sum

119899

119894=1V119894le 1

Definition 5 (two D numbersrsquo rule of combination) Let1198631

= (1198871

1 V11) (119887

1

119894 V1119894) (119887

1

119899 V1119899) 1198632

= (1198872

1 V21)

(1198872

119895 V2119895) (119887

2

119899 V2119899) be two D numbers the combination of

1198631and119863

2 indicated by119863 = 119863

1oplus 1198632 is defined by

119863 (119887) = V (10)

with

119887 =

1198871

119894+ 1198872

119895

2

V =

(V1119894+ V2119895) 2

119862

119862 =

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2)

119899

sum

119894=1

V1119894= 1

119898

sum

119895=1

V2119895= 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119898

sum

119895=1

(

V1119888+ V2119895

2)

119899

sum

119894=1

V1119894lt 1

119898

sum

119895=1

V2119895= 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119899

sum

119894=1

(V1119894+ V2119888

2)

119899

sum

119894=1

V1119894= 1

119898

sum

119895=1

V2119895lt 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119898

sum

119895=1

(

V1119888+ V2119895

2) +

119899

sum

119894=1

(V1119894+ V2119888

2) +

V1119888+ V2119888

2

119899

sum

119894=1

V1119894lt 1

119898

sum

119895=1

V2119895lt 1

(11)

where V1119888= 1 minus sum

119899

119894=1V1119894and V2119888= 1 minus sum

119898

119895=1V2119895

In themeanwhile an aggregation operator is proposed onthis special D number it is defined as below

4 Mathematical Problems in Engineering

0a b c

x

1

120583A(x)

Figure 1 A triangular fuzzy number

Definition 6 (D numbersrsquo integration) For 119863 = (1198871 V1) (1198872

V2) (119887

119894 V119894) (119887

119899 V119899) the integrating representation of

119863 is defined as

119868 (119863) =

119899

sum

119894=1

119887119894V119894 (12)

23 TOPSIS Method with Fuzzy Data Fuzzy set theory [65]provide an alternative and convenient framework for mod-eling of real-world fuzzy decision systems mathematically[66ndash68] A fuzzy set is any set that allows its members tohave different grades of membership in the interval [0 1] Itconsists of two components a set and amembership functionassociated with it

Definition 7 (fuzzy set [69]) Let 119883 be a collection of objectsdenoted generally by 119909 a fuzzy subset of 119883 is a set ofordered pairs

= (119909 120583(119909) | 119909 isin 119883) (13)

where 120583(119909) 119883 rarr [0 1] is called the membership function

(generalized characteristic function) which maps 119883 to themembership space 119872 Its range is the subset of nonnegativereal members whose supremum is finite

Definition 8 (triangular fuzzy number [66]) A fuzzy numberis a fuzzy subset of 119883 And a triangular fuzzy number canbe defined by a triplet (119886 119887 119888) shown in Figure 1 in which 119886119887 and 119888 are real numbers with 119886 lt 119887 lt 119888 Its membershipfunction is defined as

120583(119909) =

0 119909 lt 119886

119909 minus 119886

119887 minus 119886 119886 le 119909 le 119887

119888 minus 119909

119888 minus 119887 119887 le 119909 le 119888

0 119909 gt 119888

(14)

Definition 9 (distance between two triangular fuzzy numbers[53]) Let = (119886 119887 119888) and = (119898 119904 119899) be two triangular

fuzzy numbers then the vertexmethod is defined to calculatethe distance between them as

119889 ( ) = radic1

3[(119886 minus 119898)

2+ (119887 minus 119904)

2+ (119888 minus 119899)

2] (15)

The main idea of TOPSIS is that the best compromisesolution should have the shortest Euclidean distance fromthe positive ideal solution and the farthest Euclidean distancefrom the negative ideal solution

The procedures of TOPSIS method with fuzzy data canbe described as follows Let 119860 = 119860

119896| 119896 = 1 2 119898 be

the set of alternatives 119862 = 119904| 119904 = 1 2 119899 be the set

of criteria and 119877 = 119896119904

| 119896 = 1 2 119898 119904 = 1 2 119899 bethe performance ratings with the criteria weight vector 119882 =

119904| 119904 = 1 2 119899

Step 1 (calculate normalized ratings) Let119861 and119862 be the set ofbenefit criteria and cost criteria respectivelyThe normalizedvalue

119896119904is calculated by

119896119904

= (119886119896119904

119888+119904

119887119896119904

119888+119904

119888119896119904

119888+119904

)

where 119888+

119904= max119896

(119888119896119904) 119904 isin 119861

119896119904

= (119886minus

119904

119888119896119904

119886minus

119904

119887119896119904

119886minus

119904

119886119896119904

)

where 119886minus

119904= min119896

(119886119896119904) 119904 isin 119862

(16)

Step 2 (calculate weighted normalized ratings) In theweighted normalized decision matrix the modified ratingsare calculated by

V119896119904

= 119904times 119896119904

for 119896 = 1 2 119898 119904 = 1 2 119899 (17)

where 119904is the weight of the sth criteria

Step 3 (determine the fuzzy positive and negative idealsolutions) The elements V

119894119895 forall119894 119895 are normalized positive

triangular fuzzy numbers and their ranges belong to theclosed interval [0 1] Then the fuzzy positive ideal solution(FPIS 119860+) and the fuzzy negative ideal solution (FNIS 119860minus)are derived as follows

119860+= V+1 V+2 V+

119899

119860minus= Vminus1 Vminus2 Vminus

119899

(18)

where V+119904= (1 1 1) and Vminus

119904= (0 0 0) 119904 = 1 2 119899

Mathematical Problems in Engineering 5

Table 1 Linguistic variables for the importance weight of eachcriterion

Very poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

Table 2 The importance weight of the criteria

Expert1

Expert2

Expert3

Accuracy (1198621) 03069 02632 01478

Response time (1198622) 02255 04358 03222

Cost (1198623) 02663 00971 01269

Operability (1198624) 02013 02039 04031

Step 4 (calculate the distance of each alternative from the FPISand the FNIS) The distance of each alternative from 119860

+ and119860minus can be currently calculated as

119889+

119896=

119899

sum

119904=1

119889 (V119896119904 V+119904) 119896 = 1 2 119898

119889minus

119896=

119899

sum

119904=1

119889 (V119896119904 Vminus119904) 119896 = 1 2 119898

(19)

where 119889(sdot sdot) is the distance measurement between two fuzzynumbers

Step 5 (calculate the relative closeness coefficient to thepositive ideal solution) The relative closeness coefficient forthe alternative 119860

119896with respect to 119860

+ is

119862119896=

119889minus

119896

119889+

119896+ 119889minus

119896

119896 = 1 2 119898 (20)

Step 6 (rank the alternatives) Obviously an alternative 119860119896

is closer to the FPIS (119860+) and farther from FNIS (119860minus) as119862119896approaches 1 Therefore according to relative closeness

coefficient to the ideal alternative larger value of119862119896indicates

the better alternative 119860119896

3 The Proposed Method

Before introducing the system architecture of intelligent CEPwe first have a clear understanding of the definition of anevent [10 11] An event that represents an atomic instanceis an occurrence that is of interest at a point in timeBasically events can be classified into primitive events andcomposite events A primitive event instance is predefined asa single occurrence of interest that cannot be split into anysmall events A composite event instance that occurs overan interval is created by composing primitive or compositeevents A pattern rule is a template specifying one or morecombinations of events by the nesting of sequences (SEQ) and

conjunctions (AND) which can have negative event type(s)and their combination In the following 119864

119894denotes an event

type which can be either primitive or composite Some detailswere presented in [70]

Definition 10 A SEQ operator [13] specifies a specific orderaccording to the start time-stamps in which the event mustoccur to match the pattern and thus form a composite event

SEQ (1198641 119864

119894 119864

119899) = ⟨119890

1 119890

119894 119890

119899⟩ |

(1198901sdot 119904119905 lt sdot sdot sdot lt 119890

119894sdot 119904119905 lt sdot sdot sdot lt 119890

119899sdot 119904119905) and (119890

1sdot 119905 = 119864

1)

and sdot sdot sdot and (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot and (119890

119899sdot 119905 = 119864

119899)

(21)

For example SEQ (History taking Physical examinationLaboratory examination) consists of SEQ operator that canbe used to monitor the patients who have completed medicaldiagnosis

Definition 11 AnAND operator [13] takes a set of event typesas input and events occur within a specified time windowwithout a specified time order

AND (119864119894 119864

119896 119864

119895)

= ⟨119890119894 119890

119896 119890

119895⟩ | (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot

and (119890119896sdot 119905 = 119864

119896) and sdot sdot sdot and (119890

119895sdot 119905 = 119864

119895)

(22)

For example AND (Laboratory examination Surgery)consists of AND operator that can be used to monitor thepatientsrsquo preoperative and postoperative situations

Currently the pattern-rule base of CEP systems consist-ing of a set of pattern rules is predefined where it is based onhuman beings subjective experience so that it is not necessar-ily accurate in practice whereas the acquisition of knowledgeis a big bottleneck in the system and most researches on CEPsystems seldom discuss how to handle such kinds of prob-lems As we discussed in Section 1 because the rules comefrom experts there is no self-learning function The CEPengine based on the inaccurate predefined pattern-rule basewill generate poor quality of query results Furthermore inconventional CEP systems it inevitably involves various typesof the intrinsic uncertainty such as imprecision fuzzinessand incompleteness due to the inability of human beingssubjective judgment

In this section a novel intelligent complex event pro-cessing strategy with D numbers under fuzzy environmentis proposed based on the Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) method Theproposed method can fully support decision making in CEPsystems

The proposed system architecture of intelligent CEP isshown in Figure 2 it mainly involves two components eventcollector engine and CEP engine under D number theory

6 Mathematical Problems in Engineering

Table 3 The ratings of the three candidates by experts under all criteria

Alternative Expert1

Expert2

Expert3

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

PRA1

MG F G G VG F G G F G F GPRA2

F MG F MG VG MG VG F MG MG G FPRA3

G F G MG VG MG VG G MG G F VG

Data sources

Event collectorengine

Simple eventsCEP engine

(pattern-rule base)

Complex events

Transform

Generate new event

based on TOPSIS D number theory

Figure 2 The system architecture of intelligent CEP

Table 4The fuzzy numbers of three candidates by experts under allcriteria

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(5 7 9) (3 5 7) (7 9 10) (7 9 10)PRA2

(3 5 7) (5 7 9) (3 5 7) (5 7 9)PRA3

(7 9 10) (3 5 7) (7 9 10) (5 7 9)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(9 10 10) (3 5 7) (7 9 10) (7 9 10)PRA2

(9 10 10) (5 7 9) (9 10 10) (3 5 7)PRA3

(9 10 10) (5 7 9) (9 10 10) (7 9 10)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(3 5 7) (7 9 10) (3 5 7) (7 9 10)PRA2

(5 7 9) (5 7 9) (7 9 10) (3 5 7)PRA3

(5 7 9) (7 9 10) (3 5 7) (9 10 10)

based on TOPSIS method We will explain each componentin detail

Component 1 Event Collector Engine The event collectorengine which collects events from data sources first gener-ates a unified formal definition of the event flow

problems in detailStep 1 analyze the decision-making

Step 2 form the pattern-rule base Step 3 collect simple events by theevent collector engine

Step 4 generate complex events by the CEP engine

D number theory based on TOPSIS

Figure 3 The flowchart of the proposed method

Component 2 CEP Engine under D Number Theory Basedon TOPSIS Method Then the CEP engine processes thecollected simple events through the pattern rules in whichthe events generated at this component are called complexevents whereas the D number theory based on TOPSISmethod which is the addition to the CEP engine in view ofthe detected uncertainty event set is used for pattern-rulesanalysis to select the best one from the decision candidatesFinally the generated new events are sent out directly to par-ticular users or reused as input events or used to do furtheranalysis

The main steps of the proposed method are shown as fol-lows and the basic flowchart is given in Figure 3 for the betterunderstanding of the concept

Step 1 Make certain of the detailed information of thedecision-making problems including the goal and criteria

Mathematical Problems in Engineering 7

Table 5 The fuzzy normalized decision matrix

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(050 070 090) (030 050 070) (070 090 100) (070 090 100)PRA2

(030 050 070) (056 078 100) (030 050 070) (056 078 100)PRA3

(070 090 100) (050 070 070) (070 090 100) (050 070 090)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(090 100 100) (030 050 070) (070 090 100) (070 090 100)PRA2

(090 100 100) (056 078 100) (090 100 100) (033 056 078)PRA3

(090 100 100) (050 070 090) (090 100 100) (070 090 100)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(030 050 070) (070 090 100) (030 050 070) (070 090 100)PRA2

(050 070 090) (056 078 100) (070 090 100) (033 056 078)PRA3

(050 070 090) (070 090 100) (030 050 070) (090 100 100)

Table 6 The closeness coefficient of each alternative

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

06779 05000 08275 08275PRA2

05000 07357 05000 07357PRA3

08275 05000 08275 06779

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

09437 05000 08275 08275PRA2

09437 07357 09437 05490PRA3

09437 06779 09437 08275

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

05000 08275 05000 08275PRA2

06779 07357 08275 05490PRA3

06779 08275 05000 09437

Step 2 After determining the goal and criteria it forms thepattern-rule base by leveraging D number theory based onTOPSIS method

Step 3 For the intelligent CEP it then collects simple eventsby the event collector engine from the data sources

Step 4 Based on the pattern-rule base the intelligent CEPsystem can process the collected simple events and generatenew complex events by the CEP engine

4 Numerical Example

In this section in order to illustrate the application of theproposed method a simple numerical example is givenSuppose that there are three pattern-rule alternatives(PRA) which may be considered for further generating acomposite event in the CEP system namely PRA

1=

SEQ(1198641 119864

119894 119864

119896) PRA

2= SEQ(119864

1 119864

119895 119864

119896)

and PRA3= SEQ(119864

1 119864

119895 119864

119899) Meanwhile there are

four evaluation criteria and three experts need to select themost suitable pattern-rule candidate from them for bettersupporting decision making in CEP systems Three expertsgive different assessment results to different alternatives interms of different evaluation criteria Four benefit criteria areconsidered

(1) Accuracy (1198621)

(2) Response Time (1198622)

(3) Cost (1198623)

(4) Operability (1198624)

The proposed method is now applied to solve this decisionproblem Then computational procedure is summarized asfollows

Step 1 The experts use linguistic weighting variables to assessthe importance of the criteria which is shown in Table 1

Step 2 In this paper the criteria weights for different expertsdetermined by variation coefficient method are adopted andpresented in Table 2

Step 3 Using the linguistic rating variables of Table 1 theratings of the three decision alternatives by different expertsunder four evaluation criteria are obtained as shown in

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

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[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

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Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

with

119870 = sum

119861cap119862=0

1198981(119861)119898

2(119862) (6)

where 119861 and 119862 are also elements of 2119880 and 119870 is a constant toshow the conflict between the twoBPAsNote thatDempsterrsquosrule of combination is only applicable to such twoBPAswhichsatisfy the condition119870 lt 122 D Number Theory D number theory is a generalizationof Dempster-Shafer evidence theory proposed by Deng [55]In the classical Dempster-Shafer theory there are severalstrong hypotheses on the frame of discernment and basicprobability assignment and still some shortcomings whichlimit the ability of Dempster-Shafer theory to represent sometypes of information and restrict the application in practiceD number theory as an extension and development methodis defined as follows

Definition 4 (D number) LetΩ be a finite nonempty set a Dnumber is a mapping formulated by

119863 Ω 997888rarr [0 1] (7)

with

sum

119861subeΩ

119863 (119861) le 1

119863 (0) = 0

(8)

where 119861 is a subset ofΩ

It seems that the definition of D numbers is similarto the definition of BPA However in D number theory

the elements of Ω do not require to be mutually exclusiveIn addition being contrary of the frame of discernment119880 containing overall events Ω is acceptable to incompleteinformation by sum

119861subeΩ119863(119861) le 1

Furthermore for a discrete set Ω = 1198871 1198872 119887

119894 119887

119899

where 119887119894isin 119877 and when 119894 = 119895 119887

119894= 119887119895 A special form of D

numbers can be expressed by

119863(1198871) = V

1

119863(1198872) = V

2

119863 (119887119894) = V

119894

119863 (119887119899) = V

119899

(9)

or simply denoted as 119863 = (1198871 V1) (1198872 V2) (119887

119894 V119894)

(119887119899 V119899) where V

119894gt 0 and sum

119899

119894=1V119894le 1

Definition 5 (two D numbersrsquo rule of combination) Let1198631

= (1198871

1 V11) (119887

1

119894 V1119894) (119887

1

119899 V1119899) 1198632

= (1198872

1 V21)

(1198872

119895 V2119895) (119887

2

119899 V2119899) be two D numbers the combination of

1198631and119863

2 indicated by119863 = 119863

1oplus 1198632 is defined by

119863 (119887) = V (10)

with

119887 =

1198871

119894+ 1198872

119895

2

V =

(V1119894+ V2119895) 2

119862

119862 =

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2)

119899

sum

119894=1

V1119894= 1

119898

sum

119895=1

V2119895= 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119898

sum

119895=1

(

V1119888+ V2119895

2)

119899

sum

119894=1

V1119894lt 1

119898

sum

119895=1

V2119895= 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119899

sum

119894=1

(V1119894+ V2119888

2)

119899

sum

119894=1

V1119894= 1

119898

sum

119895=1

V2119895lt 1

119898

sum

119895=1

119899

sum

119894=1

(

V1119894+ V2119895

2) +

119898

sum

119895=1

(

V1119888+ V2119895

2) +

119899

sum

119894=1

(V1119894+ V2119888

2) +

V1119888+ V2119888

2

119899

sum

119894=1

V1119894lt 1

119898

sum

119895=1

V2119895lt 1

(11)

where V1119888= 1 minus sum

119899

119894=1V1119894and V2119888= 1 minus sum

119898

119895=1V2119895

In themeanwhile an aggregation operator is proposed onthis special D number it is defined as below

4 Mathematical Problems in Engineering

0a b c

x

1

120583A(x)

Figure 1 A triangular fuzzy number

Definition 6 (D numbersrsquo integration) For 119863 = (1198871 V1) (1198872

V2) (119887

119894 V119894) (119887

119899 V119899) the integrating representation of

119863 is defined as

119868 (119863) =

119899

sum

119894=1

119887119894V119894 (12)

23 TOPSIS Method with Fuzzy Data Fuzzy set theory [65]provide an alternative and convenient framework for mod-eling of real-world fuzzy decision systems mathematically[66ndash68] A fuzzy set is any set that allows its members tohave different grades of membership in the interval [0 1] Itconsists of two components a set and amembership functionassociated with it

Definition 7 (fuzzy set [69]) Let 119883 be a collection of objectsdenoted generally by 119909 a fuzzy subset of 119883 is a set ofordered pairs

= (119909 120583(119909) | 119909 isin 119883) (13)

where 120583(119909) 119883 rarr [0 1] is called the membership function

(generalized characteristic function) which maps 119883 to themembership space 119872 Its range is the subset of nonnegativereal members whose supremum is finite

Definition 8 (triangular fuzzy number [66]) A fuzzy numberis a fuzzy subset of 119883 And a triangular fuzzy number canbe defined by a triplet (119886 119887 119888) shown in Figure 1 in which 119886119887 and 119888 are real numbers with 119886 lt 119887 lt 119888 Its membershipfunction is defined as

120583(119909) =

0 119909 lt 119886

119909 minus 119886

119887 minus 119886 119886 le 119909 le 119887

119888 minus 119909

119888 minus 119887 119887 le 119909 le 119888

0 119909 gt 119888

(14)

Definition 9 (distance between two triangular fuzzy numbers[53]) Let = (119886 119887 119888) and = (119898 119904 119899) be two triangular

fuzzy numbers then the vertexmethod is defined to calculatethe distance between them as

119889 ( ) = radic1

3[(119886 minus 119898)

2+ (119887 minus 119904)

2+ (119888 minus 119899)

2] (15)

The main idea of TOPSIS is that the best compromisesolution should have the shortest Euclidean distance fromthe positive ideal solution and the farthest Euclidean distancefrom the negative ideal solution

The procedures of TOPSIS method with fuzzy data canbe described as follows Let 119860 = 119860

119896| 119896 = 1 2 119898 be

the set of alternatives 119862 = 119904| 119904 = 1 2 119899 be the set

of criteria and 119877 = 119896119904

| 119896 = 1 2 119898 119904 = 1 2 119899 bethe performance ratings with the criteria weight vector 119882 =

119904| 119904 = 1 2 119899

Step 1 (calculate normalized ratings) Let119861 and119862 be the set ofbenefit criteria and cost criteria respectivelyThe normalizedvalue

119896119904is calculated by

119896119904

= (119886119896119904

119888+119904

119887119896119904

119888+119904

119888119896119904

119888+119904

)

where 119888+

119904= max119896

(119888119896119904) 119904 isin 119861

119896119904

= (119886minus

119904

119888119896119904

119886minus

119904

119887119896119904

119886minus

119904

119886119896119904

)

where 119886minus

119904= min119896

(119886119896119904) 119904 isin 119862

(16)

Step 2 (calculate weighted normalized ratings) In theweighted normalized decision matrix the modified ratingsare calculated by

V119896119904

= 119904times 119896119904

for 119896 = 1 2 119898 119904 = 1 2 119899 (17)

where 119904is the weight of the sth criteria

Step 3 (determine the fuzzy positive and negative idealsolutions) The elements V

119894119895 forall119894 119895 are normalized positive

triangular fuzzy numbers and their ranges belong to theclosed interval [0 1] Then the fuzzy positive ideal solution(FPIS 119860+) and the fuzzy negative ideal solution (FNIS 119860minus)are derived as follows

119860+= V+1 V+2 V+

119899

119860minus= Vminus1 Vminus2 Vminus

119899

(18)

where V+119904= (1 1 1) and Vminus

119904= (0 0 0) 119904 = 1 2 119899

Mathematical Problems in Engineering 5

Table 1 Linguistic variables for the importance weight of eachcriterion

Very poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

Table 2 The importance weight of the criteria

Expert1

Expert2

Expert3

Accuracy (1198621) 03069 02632 01478

Response time (1198622) 02255 04358 03222

Cost (1198623) 02663 00971 01269

Operability (1198624) 02013 02039 04031

Step 4 (calculate the distance of each alternative from the FPISand the FNIS) The distance of each alternative from 119860

+ and119860minus can be currently calculated as

119889+

119896=

119899

sum

119904=1

119889 (V119896119904 V+119904) 119896 = 1 2 119898

119889minus

119896=

119899

sum

119904=1

119889 (V119896119904 Vminus119904) 119896 = 1 2 119898

(19)

where 119889(sdot sdot) is the distance measurement between two fuzzynumbers

Step 5 (calculate the relative closeness coefficient to thepositive ideal solution) The relative closeness coefficient forthe alternative 119860

119896with respect to 119860

+ is

119862119896=

119889minus

119896

119889+

119896+ 119889minus

119896

119896 = 1 2 119898 (20)

Step 6 (rank the alternatives) Obviously an alternative 119860119896

is closer to the FPIS (119860+) and farther from FNIS (119860minus) as119862119896approaches 1 Therefore according to relative closeness

coefficient to the ideal alternative larger value of119862119896indicates

the better alternative 119860119896

3 The Proposed Method

Before introducing the system architecture of intelligent CEPwe first have a clear understanding of the definition of anevent [10 11] An event that represents an atomic instanceis an occurrence that is of interest at a point in timeBasically events can be classified into primitive events andcomposite events A primitive event instance is predefined asa single occurrence of interest that cannot be split into anysmall events A composite event instance that occurs overan interval is created by composing primitive or compositeevents A pattern rule is a template specifying one or morecombinations of events by the nesting of sequences (SEQ) and

conjunctions (AND) which can have negative event type(s)and their combination In the following 119864

119894denotes an event

type which can be either primitive or composite Some detailswere presented in [70]

Definition 10 A SEQ operator [13] specifies a specific orderaccording to the start time-stamps in which the event mustoccur to match the pattern and thus form a composite event

SEQ (1198641 119864

119894 119864

119899) = ⟨119890

1 119890

119894 119890

119899⟩ |

(1198901sdot 119904119905 lt sdot sdot sdot lt 119890

119894sdot 119904119905 lt sdot sdot sdot lt 119890

119899sdot 119904119905) and (119890

1sdot 119905 = 119864

1)

and sdot sdot sdot and (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot and (119890

119899sdot 119905 = 119864

119899)

(21)

For example SEQ (History taking Physical examinationLaboratory examination) consists of SEQ operator that canbe used to monitor the patients who have completed medicaldiagnosis

Definition 11 AnAND operator [13] takes a set of event typesas input and events occur within a specified time windowwithout a specified time order

AND (119864119894 119864

119896 119864

119895)

= ⟨119890119894 119890

119896 119890

119895⟩ | (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot

and (119890119896sdot 119905 = 119864

119896) and sdot sdot sdot and (119890

119895sdot 119905 = 119864

119895)

(22)

For example AND (Laboratory examination Surgery)consists of AND operator that can be used to monitor thepatientsrsquo preoperative and postoperative situations

Currently the pattern-rule base of CEP systems consist-ing of a set of pattern rules is predefined where it is based onhuman beings subjective experience so that it is not necessar-ily accurate in practice whereas the acquisition of knowledgeis a big bottleneck in the system and most researches on CEPsystems seldom discuss how to handle such kinds of prob-lems As we discussed in Section 1 because the rules comefrom experts there is no self-learning function The CEPengine based on the inaccurate predefined pattern-rule basewill generate poor quality of query results Furthermore inconventional CEP systems it inevitably involves various typesof the intrinsic uncertainty such as imprecision fuzzinessand incompleteness due to the inability of human beingssubjective judgment

In this section a novel intelligent complex event pro-cessing strategy with D numbers under fuzzy environmentis proposed based on the Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) method Theproposed method can fully support decision making in CEPsystems

The proposed system architecture of intelligent CEP isshown in Figure 2 it mainly involves two components eventcollector engine and CEP engine under D number theory

6 Mathematical Problems in Engineering

Table 3 The ratings of the three candidates by experts under all criteria

Alternative Expert1

Expert2

Expert3

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

PRA1

MG F G G VG F G G F G F GPRA2

F MG F MG VG MG VG F MG MG G FPRA3

G F G MG VG MG VG G MG G F VG

Data sources

Event collectorengine

Simple eventsCEP engine

(pattern-rule base)

Complex events

Transform

Generate new event

based on TOPSIS D number theory

Figure 2 The system architecture of intelligent CEP

Table 4The fuzzy numbers of three candidates by experts under allcriteria

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(5 7 9) (3 5 7) (7 9 10) (7 9 10)PRA2

(3 5 7) (5 7 9) (3 5 7) (5 7 9)PRA3

(7 9 10) (3 5 7) (7 9 10) (5 7 9)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(9 10 10) (3 5 7) (7 9 10) (7 9 10)PRA2

(9 10 10) (5 7 9) (9 10 10) (3 5 7)PRA3

(9 10 10) (5 7 9) (9 10 10) (7 9 10)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(3 5 7) (7 9 10) (3 5 7) (7 9 10)PRA2

(5 7 9) (5 7 9) (7 9 10) (3 5 7)PRA3

(5 7 9) (7 9 10) (3 5 7) (9 10 10)

based on TOPSIS method We will explain each componentin detail

Component 1 Event Collector Engine The event collectorengine which collects events from data sources first gener-ates a unified formal definition of the event flow

problems in detailStep 1 analyze the decision-making

Step 2 form the pattern-rule base Step 3 collect simple events by theevent collector engine

Step 4 generate complex events by the CEP engine

D number theory based on TOPSIS

Figure 3 The flowchart of the proposed method

Component 2 CEP Engine under D Number Theory Basedon TOPSIS Method Then the CEP engine processes thecollected simple events through the pattern rules in whichthe events generated at this component are called complexevents whereas the D number theory based on TOPSISmethod which is the addition to the CEP engine in view ofthe detected uncertainty event set is used for pattern-rulesanalysis to select the best one from the decision candidatesFinally the generated new events are sent out directly to par-ticular users or reused as input events or used to do furtheranalysis

The main steps of the proposed method are shown as fol-lows and the basic flowchart is given in Figure 3 for the betterunderstanding of the concept

Step 1 Make certain of the detailed information of thedecision-making problems including the goal and criteria

Mathematical Problems in Engineering 7

Table 5 The fuzzy normalized decision matrix

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(050 070 090) (030 050 070) (070 090 100) (070 090 100)PRA2

(030 050 070) (056 078 100) (030 050 070) (056 078 100)PRA3

(070 090 100) (050 070 070) (070 090 100) (050 070 090)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(090 100 100) (030 050 070) (070 090 100) (070 090 100)PRA2

(090 100 100) (056 078 100) (090 100 100) (033 056 078)PRA3

(090 100 100) (050 070 090) (090 100 100) (070 090 100)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(030 050 070) (070 090 100) (030 050 070) (070 090 100)PRA2

(050 070 090) (056 078 100) (070 090 100) (033 056 078)PRA3

(050 070 090) (070 090 100) (030 050 070) (090 100 100)

Table 6 The closeness coefficient of each alternative

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

06779 05000 08275 08275PRA2

05000 07357 05000 07357PRA3

08275 05000 08275 06779

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

09437 05000 08275 08275PRA2

09437 07357 09437 05490PRA3

09437 06779 09437 08275

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

05000 08275 05000 08275PRA2

06779 07357 08275 05490PRA3

06779 08275 05000 09437

Step 2 After determining the goal and criteria it forms thepattern-rule base by leveraging D number theory based onTOPSIS method

Step 3 For the intelligent CEP it then collects simple eventsby the event collector engine from the data sources

Step 4 Based on the pattern-rule base the intelligent CEPsystem can process the collected simple events and generatenew complex events by the CEP engine

4 Numerical Example

In this section in order to illustrate the application of theproposed method a simple numerical example is givenSuppose that there are three pattern-rule alternatives(PRA) which may be considered for further generating acomposite event in the CEP system namely PRA

1=

SEQ(1198641 119864

119894 119864

119896) PRA

2= SEQ(119864

1 119864

119895 119864

119896)

and PRA3= SEQ(119864

1 119864

119895 119864

119899) Meanwhile there are

four evaluation criteria and three experts need to select themost suitable pattern-rule candidate from them for bettersupporting decision making in CEP systems Three expertsgive different assessment results to different alternatives interms of different evaluation criteria Four benefit criteria areconsidered

(1) Accuracy (1198621)

(2) Response Time (1198622)

(3) Cost (1198623)

(4) Operability (1198624)

The proposed method is now applied to solve this decisionproblem Then computational procedure is summarized asfollows

Step 1 The experts use linguistic weighting variables to assessthe importance of the criteria which is shown in Table 1

Step 2 In this paper the criteria weights for different expertsdetermined by variation coefficient method are adopted andpresented in Table 2

Step 3 Using the linguistic rating variables of Table 1 theratings of the three decision alternatives by different expertsunder four evaluation criteria are obtained as shown in

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

[1] D J Abadi Y Ahmad M Balazinska et al ldquoThe design ofthe Borealis stream processingenginerdquo in Proceedings of the 2ndBiennial Conference on Innovative Data Systems Research pp277ndash289 2005

[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

0a b c

x

1

120583A(x)

Figure 1 A triangular fuzzy number

Definition 6 (D numbersrsquo integration) For 119863 = (1198871 V1) (1198872

V2) (119887

119894 V119894) (119887

119899 V119899) the integrating representation of

119863 is defined as

119868 (119863) =

119899

sum

119894=1

119887119894V119894 (12)

23 TOPSIS Method with Fuzzy Data Fuzzy set theory [65]provide an alternative and convenient framework for mod-eling of real-world fuzzy decision systems mathematically[66ndash68] A fuzzy set is any set that allows its members tohave different grades of membership in the interval [0 1] Itconsists of two components a set and amembership functionassociated with it

Definition 7 (fuzzy set [69]) Let 119883 be a collection of objectsdenoted generally by 119909 a fuzzy subset of 119883 is a set ofordered pairs

= (119909 120583(119909) | 119909 isin 119883) (13)

where 120583(119909) 119883 rarr [0 1] is called the membership function

(generalized characteristic function) which maps 119883 to themembership space 119872 Its range is the subset of nonnegativereal members whose supremum is finite

Definition 8 (triangular fuzzy number [66]) A fuzzy numberis a fuzzy subset of 119883 And a triangular fuzzy number canbe defined by a triplet (119886 119887 119888) shown in Figure 1 in which 119886119887 and 119888 are real numbers with 119886 lt 119887 lt 119888 Its membershipfunction is defined as

120583(119909) =

0 119909 lt 119886

119909 minus 119886

119887 minus 119886 119886 le 119909 le 119887

119888 minus 119909

119888 minus 119887 119887 le 119909 le 119888

0 119909 gt 119888

(14)

Definition 9 (distance between two triangular fuzzy numbers[53]) Let = (119886 119887 119888) and = (119898 119904 119899) be two triangular

fuzzy numbers then the vertexmethod is defined to calculatethe distance between them as

119889 ( ) = radic1

3[(119886 minus 119898)

2+ (119887 minus 119904)

2+ (119888 minus 119899)

2] (15)

The main idea of TOPSIS is that the best compromisesolution should have the shortest Euclidean distance fromthe positive ideal solution and the farthest Euclidean distancefrom the negative ideal solution

The procedures of TOPSIS method with fuzzy data canbe described as follows Let 119860 = 119860

119896| 119896 = 1 2 119898 be

the set of alternatives 119862 = 119904| 119904 = 1 2 119899 be the set

of criteria and 119877 = 119896119904

| 119896 = 1 2 119898 119904 = 1 2 119899 bethe performance ratings with the criteria weight vector 119882 =

119904| 119904 = 1 2 119899

Step 1 (calculate normalized ratings) Let119861 and119862 be the set ofbenefit criteria and cost criteria respectivelyThe normalizedvalue

119896119904is calculated by

119896119904

= (119886119896119904

119888+119904

119887119896119904

119888+119904

119888119896119904

119888+119904

)

where 119888+

119904= max119896

(119888119896119904) 119904 isin 119861

119896119904

= (119886minus

119904

119888119896119904

119886minus

119904

119887119896119904

119886minus

119904

119886119896119904

)

where 119886minus

119904= min119896

(119886119896119904) 119904 isin 119862

(16)

Step 2 (calculate weighted normalized ratings) In theweighted normalized decision matrix the modified ratingsare calculated by

V119896119904

= 119904times 119896119904

for 119896 = 1 2 119898 119904 = 1 2 119899 (17)

where 119904is the weight of the sth criteria

Step 3 (determine the fuzzy positive and negative idealsolutions) The elements V

119894119895 forall119894 119895 are normalized positive

triangular fuzzy numbers and their ranges belong to theclosed interval [0 1] Then the fuzzy positive ideal solution(FPIS 119860+) and the fuzzy negative ideal solution (FNIS 119860minus)are derived as follows

119860+= V+1 V+2 V+

119899

119860minus= Vminus1 Vminus2 Vminus

119899

(18)

where V+119904= (1 1 1) and Vminus

119904= (0 0 0) 119904 = 1 2 119899

Mathematical Problems in Engineering 5

Table 1 Linguistic variables for the importance weight of eachcriterion

Very poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

Table 2 The importance weight of the criteria

Expert1

Expert2

Expert3

Accuracy (1198621) 03069 02632 01478

Response time (1198622) 02255 04358 03222

Cost (1198623) 02663 00971 01269

Operability (1198624) 02013 02039 04031

Step 4 (calculate the distance of each alternative from the FPISand the FNIS) The distance of each alternative from 119860

+ and119860minus can be currently calculated as

119889+

119896=

119899

sum

119904=1

119889 (V119896119904 V+119904) 119896 = 1 2 119898

119889minus

119896=

119899

sum

119904=1

119889 (V119896119904 Vminus119904) 119896 = 1 2 119898

(19)

where 119889(sdot sdot) is the distance measurement between two fuzzynumbers

Step 5 (calculate the relative closeness coefficient to thepositive ideal solution) The relative closeness coefficient forthe alternative 119860

119896with respect to 119860

+ is

119862119896=

119889minus

119896

119889+

119896+ 119889minus

119896

119896 = 1 2 119898 (20)

Step 6 (rank the alternatives) Obviously an alternative 119860119896

is closer to the FPIS (119860+) and farther from FNIS (119860minus) as119862119896approaches 1 Therefore according to relative closeness

coefficient to the ideal alternative larger value of119862119896indicates

the better alternative 119860119896

3 The Proposed Method

Before introducing the system architecture of intelligent CEPwe first have a clear understanding of the definition of anevent [10 11] An event that represents an atomic instanceis an occurrence that is of interest at a point in timeBasically events can be classified into primitive events andcomposite events A primitive event instance is predefined asa single occurrence of interest that cannot be split into anysmall events A composite event instance that occurs overan interval is created by composing primitive or compositeevents A pattern rule is a template specifying one or morecombinations of events by the nesting of sequences (SEQ) and

conjunctions (AND) which can have negative event type(s)and their combination In the following 119864

119894denotes an event

type which can be either primitive or composite Some detailswere presented in [70]

Definition 10 A SEQ operator [13] specifies a specific orderaccording to the start time-stamps in which the event mustoccur to match the pattern and thus form a composite event

SEQ (1198641 119864

119894 119864

119899) = ⟨119890

1 119890

119894 119890

119899⟩ |

(1198901sdot 119904119905 lt sdot sdot sdot lt 119890

119894sdot 119904119905 lt sdot sdot sdot lt 119890

119899sdot 119904119905) and (119890

1sdot 119905 = 119864

1)

and sdot sdot sdot and (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot and (119890

119899sdot 119905 = 119864

119899)

(21)

For example SEQ (History taking Physical examinationLaboratory examination) consists of SEQ operator that canbe used to monitor the patients who have completed medicaldiagnosis

Definition 11 AnAND operator [13] takes a set of event typesas input and events occur within a specified time windowwithout a specified time order

AND (119864119894 119864

119896 119864

119895)

= ⟨119890119894 119890

119896 119890

119895⟩ | (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot

and (119890119896sdot 119905 = 119864

119896) and sdot sdot sdot and (119890

119895sdot 119905 = 119864

119895)

(22)

For example AND (Laboratory examination Surgery)consists of AND operator that can be used to monitor thepatientsrsquo preoperative and postoperative situations

Currently the pattern-rule base of CEP systems consist-ing of a set of pattern rules is predefined where it is based onhuman beings subjective experience so that it is not necessar-ily accurate in practice whereas the acquisition of knowledgeis a big bottleneck in the system and most researches on CEPsystems seldom discuss how to handle such kinds of prob-lems As we discussed in Section 1 because the rules comefrom experts there is no self-learning function The CEPengine based on the inaccurate predefined pattern-rule basewill generate poor quality of query results Furthermore inconventional CEP systems it inevitably involves various typesof the intrinsic uncertainty such as imprecision fuzzinessand incompleteness due to the inability of human beingssubjective judgment

In this section a novel intelligent complex event pro-cessing strategy with D numbers under fuzzy environmentis proposed based on the Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) method Theproposed method can fully support decision making in CEPsystems

The proposed system architecture of intelligent CEP isshown in Figure 2 it mainly involves two components eventcollector engine and CEP engine under D number theory

6 Mathematical Problems in Engineering

Table 3 The ratings of the three candidates by experts under all criteria

Alternative Expert1

Expert2

Expert3

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

PRA1

MG F G G VG F G G F G F GPRA2

F MG F MG VG MG VG F MG MG G FPRA3

G F G MG VG MG VG G MG G F VG

Data sources

Event collectorengine

Simple eventsCEP engine

(pattern-rule base)

Complex events

Transform

Generate new event

based on TOPSIS D number theory

Figure 2 The system architecture of intelligent CEP

Table 4The fuzzy numbers of three candidates by experts under allcriteria

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(5 7 9) (3 5 7) (7 9 10) (7 9 10)PRA2

(3 5 7) (5 7 9) (3 5 7) (5 7 9)PRA3

(7 9 10) (3 5 7) (7 9 10) (5 7 9)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(9 10 10) (3 5 7) (7 9 10) (7 9 10)PRA2

(9 10 10) (5 7 9) (9 10 10) (3 5 7)PRA3

(9 10 10) (5 7 9) (9 10 10) (7 9 10)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(3 5 7) (7 9 10) (3 5 7) (7 9 10)PRA2

(5 7 9) (5 7 9) (7 9 10) (3 5 7)PRA3

(5 7 9) (7 9 10) (3 5 7) (9 10 10)

based on TOPSIS method We will explain each componentin detail

Component 1 Event Collector Engine The event collectorengine which collects events from data sources first gener-ates a unified formal definition of the event flow

problems in detailStep 1 analyze the decision-making

Step 2 form the pattern-rule base Step 3 collect simple events by theevent collector engine

Step 4 generate complex events by the CEP engine

D number theory based on TOPSIS

Figure 3 The flowchart of the proposed method

Component 2 CEP Engine under D Number Theory Basedon TOPSIS Method Then the CEP engine processes thecollected simple events through the pattern rules in whichthe events generated at this component are called complexevents whereas the D number theory based on TOPSISmethod which is the addition to the CEP engine in view ofthe detected uncertainty event set is used for pattern-rulesanalysis to select the best one from the decision candidatesFinally the generated new events are sent out directly to par-ticular users or reused as input events or used to do furtheranalysis

The main steps of the proposed method are shown as fol-lows and the basic flowchart is given in Figure 3 for the betterunderstanding of the concept

Step 1 Make certain of the detailed information of thedecision-making problems including the goal and criteria

Mathematical Problems in Engineering 7

Table 5 The fuzzy normalized decision matrix

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(050 070 090) (030 050 070) (070 090 100) (070 090 100)PRA2

(030 050 070) (056 078 100) (030 050 070) (056 078 100)PRA3

(070 090 100) (050 070 070) (070 090 100) (050 070 090)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(090 100 100) (030 050 070) (070 090 100) (070 090 100)PRA2

(090 100 100) (056 078 100) (090 100 100) (033 056 078)PRA3

(090 100 100) (050 070 090) (090 100 100) (070 090 100)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(030 050 070) (070 090 100) (030 050 070) (070 090 100)PRA2

(050 070 090) (056 078 100) (070 090 100) (033 056 078)PRA3

(050 070 090) (070 090 100) (030 050 070) (090 100 100)

Table 6 The closeness coefficient of each alternative

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

06779 05000 08275 08275PRA2

05000 07357 05000 07357PRA3

08275 05000 08275 06779

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

09437 05000 08275 08275PRA2

09437 07357 09437 05490PRA3

09437 06779 09437 08275

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

05000 08275 05000 08275PRA2

06779 07357 08275 05490PRA3

06779 08275 05000 09437

Step 2 After determining the goal and criteria it forms thepattern-rule base by leveraging D number theory based onTOPSIS method

Step 3 For the intelligent CEP it then collects simple eventsby the event collector engine from the data sources

Step 4 Based on the pattern-rule base the intelligent CEPsystem can process the collected simple events and generatenew complex events by the CEP engine

4 Numerical Example

In this section in order to illustrate the application of theproposed method a simple numerical example is givenSuppose that there are three pattern-rule alternatives(PRA) which may be considered for further generating acomposite event in the CEP system namely PRA

1=

SEQ(1198641 119864

119894 119864

119896) PRA

2= SEQ(119864

1 119864

119895 119864

119896)

and PRA3= SEQ(119864

1 119864

119895 119864

119899) Meanwhile there are

four evaluation criteria and three experts need to select themost suitable pattern-rule candidate from them for bettersupporting decision making in CEP systems Three expertsgive different assessment results to different alternatives interms of different evaluation criteria Four benefit criteria areconsidered

(1) Accuracy (1198621)

(2) Response Time (1198622)

(3) Cost (1198623)

(4) Operability (1198624)

The proposed method is now applied to solve this decisionproblem Then computational procedure is summarized asfollows

Step 1 The experts use linguistic weighting variables to assessthe importance of the criteria which is shown in Table 1

Step 2 In this paper the criteria weights for different expertsdetermined by variation coefficient method are adopted andpresented in Table 2

Step 3 Using the linguistic rating variables of Table 1 theratings of the three decision alternatives by different expertsunder four evaluation criteria are obtained as shown in

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

[1] D J Abadi Y Ahmad M Balazinska et al ldquoThe design ofthe Borealis stream processingenginerdquo in Proceedings of the 2ndBiennial Conference on Innovative Data Systems Research pp277ndash289 2005

[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 Linguistic variables for the importance weight of eachcriterion

Very poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

Table 2 The importance weight of the criteria

Expert1

Expert2

Expert3

Accuracy (1198621) 03069 02632 01478

Response time (1198622) 02255 04358 03222

Cost (1198623) 02663 00971 01269

Operability (1198624) 02013 02039 04031

Step 4 (calculate the distance of each alternative from the FPISand the FNIS) The distance of each alternative from 119860

+ and119860minus can be currently calculated as

119889+

119896=

119899

sum

119904=1

119889 (V119896119904 V+119904) 119896 = 1 2 119898

119889minus

119896=

119899

sum

119904=1

119889 (V119896119904 Vminus119904) 119896 = 1 2 119898

(19)

where 119889(sdot sdot) is the distance measurement between two fuzzynumbers

Step 5 (calculate the relative closeness coefficient to thepositive ideal solution) The relative closeness coefficient forthe alternative 119860

119896with respect to 119860

+ is

119862119896=

119889minus

119896

119889+

119896+ 119889minus

119896

119896 = 1 2 119898 (20)

Step 6 (rank the alternatives) Obviously an alternative 119860119896

is closer to the FPIS (119860+) and farther from FNIS (119860minus) as119862119896approaches 1 Therefore according to relative closeness

coefficient to the ideal alternative larger value of119862119896indicates

the better alternative 119860119896

3 The Proposed Method

Before introducing the system architecture of intelligent CEPwe first have a clear understanding of the definition of anevent [10 11] An event that represents an atomic instanceis an occurrence that is of interest at a point in timeBasically events can be classified into primitive events andcomposite events A primitive event instance is predefined asa single occurrence of interest that cannot be split into anysmall events A composite event instance that occurs overan interval is created by composing primitive or compositeevents A pattern rule is a template specifying one or morecombinations of events by the nesting of sequences (SEQ) and

conjunctions (AND) which can have negative event type(s)and their combination In the following 119864

119894denotes an event

type which can be either primitive or composite Some detailswere presented in [70]

Definition 10 A SEQ operator [13] specifies a specific orderaccording to the start time-stamps in which the event mustoccur to match the pattern and thus form a composite event

SEQ (1198641 119864

119894 119864

119899) = ⟨119890

1 119890

119894 119890

119899⟩ |

(1198901sdot 119904119905 lt sdot sdot sdot lt 119890

119894sdot 119904119905 lt sdot sdot sdot lt 119890

119899sdot 119904119905) and (119890

1sdot 119905 = 119864

1)

and sdot sdot sdot and (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot and (119890

119899sdot 119905 = 119864

119899)

(21)

For example SEQ (History taking Physical examinationLaboratory examination) consists of SEQ operator that canbe used to monitor the patients who have completed medicaldiagnosis

Definition 11 AnAND operator [13] takes a set of event typesas input and events occur within a specified time windowwithout a specified time order

AND (119864119894 119864

119896 119864

119895)

= ⟨119890119894 119890

119896 119890

119895⟩ | (119890119894sdot 119905 = 119864

119894) and sdot sdot sdot

and (119890119896sdot 119905 = 119864

119896) and sdot sdot sdot and (119890

119895sdot 119905 = 119864

119895)

(22)

For example AND (Laboratory examination Surgery)consists of AND operator that can be used to monitor thepatientsrsquo preoperative and postoperative situations

Currently the pattern-rule base of CEP systems consist-ing of a set of pattern rules is predefined where it is based onhuman beings subjective experience so that it is not necessar-ily accurate in practice whereas the acquisition of knowledgeis a big bottleneck in the system and most researches on CEPsystems seldom discuss how to handle such kinds of prob-lems As we discussed in Section 1 because the rules comefrom experts there is no self-learning function The CEPengine based on the inaccurate predefined pattern-rule basewill generate poor quality of query results Furthermore inconventional CEP systems it inevitably involves various typesof the intrinsic uncertainty such as imprecision fuzzinessand incompleteness due to the inability of human beingssubjective judgment

In this section a novel intelligent complex event pro-cessing strategy with D numbers under fuzzy environmentis proposed based on the Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) method Theproposed method can fully support decision making in CEPsystems

The proposed system architecture of intelligent CEP isshown in Figure 2 it mainly involves two components eventcollector engine and CEP engine under D number theory

6 Mathematical Problems in Engineering

Table 3 The ratings of the three candidates by experts under all criteria

Alternative Expert1

Expert2

Expert3

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

PRA1

MG F G G VG F G G F G F GPRA2

F MG F MG VG MG VG F MG MG G FPRA3

G F G MG VG MG VG G MG G F VG

Data sources

Event collectorengine

Simple eventsCEP engine

(pattern-rule base)

Complex events

Transform

Generate new event

based on TOPSIS D number theory

Figure 2 The system architecture of intelligent CEP

Table 4The fuzzy numbers of three candidates by experts under allcriteria

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(5 7 9) (3 5 7) (7 9 10) (7 9 10)PRA2

(3 5 7) (5 7 9) (3 5 7) (5 7 9)PRA3

(7 9 10) (3 5 7) (7 9 10) (5 7 9)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(9 10 10) (3 5 7) (7 9 10) (7 9 10)PRA2

(9 10 10) (5 7 9) (9 10 10) (3 5 7)PRA3

(9 10 10) (5 7 9) (9 10 10) (7 9 10)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(3 5 7) (7 9 10) (3 5 7) (7 9 10)PRA2

(5 7 9) (5 7 9) (7 9 10) (3 5 7)PRA3

(5 7 9) (7 9 10) (3 5 7) (9 10 10)

based on TOPSIS method We will explain each componentin detail

Component 1 Event Collector Engine The event collectorengine which collects events from data sources first gener-ates a unified formal definition of the event flow

problems in detailStep 1 analyze the decision-making

Step 2 form the pattern-rule base Step 3 collect simple events by theevent collector engine

Step 4 generate complex events by the CEP engine

D number theory based on TOPSIS

Figure 3 The flowchart of the proposed method

Component 2 CEP Engine under D Number Theory Basedon TOPSIS Method Then the CEP engine processes thecollected simple events through the pattern rules in whichthe events generated at this component are called complexevents whereas the D number theory based on TOPSISmethod which is the addition to the CEP engine in view ofthe detected uncertainty event set is used for pattern-rulesanalysis to select the best one from the decision candidatesFinally the generated new events are sent out directly to par-ticular users or reused as input events or used to do furtheranalysis

The main steps of the proposed method are shown as fol-lows and the basic flowchart is given in Figure 3 for the betterunderstanding of the concept

Step 1 Make certain of the detailed information of thedecision-making problems including the goal and criteria

Mathematical Problems in Engineering 7

Table 5 The fuzzy normalized decision matrix

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(050 070 090) (030 050 070) (070 090 100) (070 090 100)PRA2

(030 050 070) (056 078 100) (030 050 070) (056 078 100)PRA3

(070 090 100) (050 070 070) (070 090 100) (050 070 090)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(090 100 100) (030 050 070) (070 090 100) (070 090 100)PRA2

(090 100 100) (056 078 100) (090 100 100) (033 056 078)PRA3

(090 100 100) (050 070 090) (090 100 100) (070 090 100)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(030 050 070) (070 090 100) (030 050 070) (070 090 100)PRA2

(050 070 090) (056 078 100) (070 090 100) (033 056 078)PRA3

(050 070 090) (070 090 100) (030 050 070) (090 100 100)

Table 6 The closeness coefficient of each alternative

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

06779 05000 08275 08275PRA2

05000 07357 05000 07357PRA3

08275 05000 08275 06779

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

09437 05000 08275 08275PRA2

09437 07357 09437 05490PRA3

09437 06779 09437 08275

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

05000 08275 05000 08275PRA2

06779 07357 08275 05490PRA3

06779 08275 05000 09437

Step 2 After determining the goal and criteria it forms thepattern-rule base by leveraging D number theory based onTOPSIS method

Step 3 For the intelligent CEP it then collects simple eventsby the event collector engine from the data sources

Step 4 Based on the pattern-rule base the intelligent CEPsystem can process the collected simple events and generatenew complex events by the CEP engine

4 Numerical Example

In this section in order to illustrate the application of theproposed method a simple numerical example is givenSuppose that there are three pattern-rule alternatives(PRA) which may be considered for further generating acomposite event in the CEP system namely PRA

1=

SEQ(1198641 119864

119894 119864

119896) PRA

2= SEQ(119864

1 119864

119895 119864

119896)

and PRA3= SEQ(119864

1 119864

119895 119864

119899) Meanwhile there are

four evaluation criteria and three experts need to select themost suitable pattern-rule candidate from them for bettersupporting decision making in CEP systems Three expertsgive different assessment results to different alternatives interms of different evaluation criteria Four benefit criteria areconsidered

(1) Accuracy (1198621)

(2) Response Time (1198622)

(3) Cost (1198623)

(4) Operability (1198624)

The proposed method is now applied to solve this decisionproblem Then computational procedure is summarized asfollows

Step 1 The experts use linguistic weighting variables to assessthe importance of the criteria which is shown in Table 1

Step 2 In this paper the criteria weights for different expertsdetermined by variation coefficient method are adopted andpresented in Table 2

Step 3 Using the linguistic rating variables of Table 1 theratings of the three decision alternatives by different expertsunder four evaluation criteria are obtained as shown in

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

[1] D J Abadi Y Ahmad M Balazinska et al ldquoThe design ofthe Borealis stream processingenginerdquo in Proceedings of the 2ndBiennial Conference on Innovative Data Systems Research pp277ndash289 2005

[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 3 The ratings of the three candidates by experts under all criteria

Alternative Expert1

Expert2

Expert3

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

1198621

1198622

1198623

1198624

PRA1

MG F G G VG F G G F G F GPRA2

F MG F MG VG MG VG F MG MG G FPRA3

G F G MG VG MG VG G MG G F VG

Data sources

Event collectorengine

Simple eventsCEP engine

(pattern-rule base)

Complex events

Transform

Generate new event

based on TOPSIS D number theory

Figure 2 The system architecture of intelligent CEP

Table 4The fuzzy numbers of three candidates by experts under allcriteria

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(5 7 9) (3 5 7) (7 9 10) (7 9 10)PRA2

(3 5 7) (5 7 9) (3 5 7) (5 7 9)PRA3

(7 9 10) (3 5 7) (7 9 10) (5 7 9)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(9 10 10) (3 5 7) (7 9 10) (7 9 10)PRA2

(9 10 10) (5 7 9) (9 10 10) (3 5 7)PRA3

(9 10 10) (5 7 9) (9 10 10) (7 9 10)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(3 5 7) (7 9 10) (3 5 7) (7 9 10)PRA2

(5 7 9) (5 7 9) (7 9 10) (3 5 7)PRA3

(5 7 9) (7 9 10) (3 5 7) (9 10 10)

based on TOPSIS method We will explain each componentin detail

Component 1 Event Collector Engine The event collectorengine which collects events from data sources first gener-ates a unified formal definition of the event flow

problems in detailStep 1 analyze the decision-making

Step 2 form the pattern-rule base Step 3 collect simple events by theevent collector engine

Step 4 generate complex events by the CEP engine

D number theory based on TOPSIS

Figure 3 The flowchart of the proposed method

Component 2 CEP Engine under D Number Theory Basedon TOPSIS Method Then the CEP engine processes thecollected simple events through the pattern rules in whichthe events generated at this component are called complexevents whereas the D number theory based on TOPSISmethod which is the addition to the CEP engine in view ofthe detected uncertainty event set is used for pattern-rulesanalysis to select the best one from the decision candidatesFinally the generated new events are sent out directly to par-ticular users or reused as input events or used to do furtheranalysis

The main steps of the proposed method are shown as fol-lows and the basic flowchart is given in Figure 3 for the betterunderstanding of the concept

Step 1 Make certain of the detailed information of thedecision-making problems including the goal and criteria

Mathematical Problems in Engineering 7

Table 5 The fuzzy normalized decision matrix

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(050 070 090) (030 050 070) (070 090 100) (070 090 100)PRA2

(030 050 070) (056 078 100) (030 050 070) (056 078 100)PRA3

(070 090 100) (050 070 070) (070 090 100) (050 070 090)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(090 100 100) (030 050 070) (070 090 100) (070 090 100)PRA2

(090 100 100) (056 078 100) (090 100 100) (033 056 078)PRA3

(090 100 100) (050 070 090) (090 100 100) (070 090 100)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(030 050 070) (070 090 100) (030 050 070) (070 090 100)PRA2

(050 070 090) (056 078 100) (070 090 100) (033 056 078)PRA3

(050 070 090) (070 090 100) (030 050 070) (090 100 100)

Table 6 The closeness coefficient of each alternative

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

06779 05000 08275 08275PRA2

05000 07357 05000 07357PRA3

08275 05000 08275 06779

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

09437 05000 08275 08275PRA2

09437 07357 09437 05490PRA3

09437 06779 09437 08275

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

05000 08275 05000 08275PRA2

06779 07357 08275 05490PRA3

06779 08275 05000 09437

Step 2 After determining the goal and criteria it forms thepattern-rule base by leveraging D number theory based onTOPSIS method

Step 3 For the intelligent CEP it then collects simple eventsby the event collector engine from the data sources

Step 4 Based on the pattern-rule base the intelligent CEPsystem can process the collected simple events and generatenew complex events by the CEP engine

4 Numerical Example

In this section in order to illustrate the application of theproposed method a simple numerical example is givenSuppose that there are three pattern-rule alternatives(PRA) which may be considered for further generating acomposite event in the CEP system namely PRA

1=

SEQ(1198641 119864

119894 119864

119896) PRA

2= SEQ(119864

1 119864

119895 119864

119896)

and PRA3= SEQ(119864

1 119864

119895 119864

119899) Meanwhile there are

four evaluation criteria and three experts need to select themost suitable pattern-rule candidate from them for bettersupporting decision making in CEP systems Three expertsgive different assessment results to different alternatives interms of different evaluation criteria Four benefit criteria areconsidered

(1) Accuracy (1198621)

(2) Response Time (1198622)

(3) Cost (1198623)

(4) Operability (1198624)

The proposed method is now applied to solve this decisionproblem Then computational procedure is summarized asfollows

Step 1 The experts use linguistic weighting variables to assessthe importance of the criteria which is shown in Table 1

Step 2 In this paper the criteria weights for different expertsdetermined by variation coefficient method are adopted andpresented in Table 2

Step 3 Using the linguistic rating variables of Table 1 theratings of the three decision alternatives by different expertsunder four evaluation criteria are obtained as shown in

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

[1] D J Abadi Y Ahmad M Balazinska et al ldquoThe design ofthe Borealis stream processingenginerdquo in Proceedings of the 2ndBiennial Conference on Innovative Data Systems Research pp277ndash289 2005

[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 5 The fuzzy normalized decision matrix

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

(050 070 090) (030 050 070) (070 090 100) (070 090 100)PRA2

(030 050 070) (056 078 100) (030 050 070) (056 078 100)PRA3

(070 090 100) (050 070 070) (070 090 100) (050 070 090)

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

(090 100 100) (030 050 070) (070 090 100) (070 090 100)PRA2

(090 100 100) (056 078 100) (090 100 100) (033 056 078)PRA3

(090 100 100) (050 070 090) (090 100 100) (070 090 100)

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

(030 050 070) (070 090 100) (030 050 070) (070 090 100)PRA2

(050 070 090) (056 078 100) (070 090 100) (033 056 078)PRA3

(050 070 090) (070 090 100) (030 050 070) (090 100 100)

Table 6 The closeness coefficient of each alternative

(a)

Alternative Expert1

1198621

1198622

1198623

1198624

PRA1

06779 05000 08275 08275PRA2

05000 07357 05000 07357PRA3

08275 05000 08275 06779

(b)

Alternative Expert2

1198621

1198622

1198623

1198624

PRA1

09437 05000 08275 08275PRA2

09437 07357 09437 05490PRA3

09437 06779 09437 08275

(c)

Alternative Expert3

1198621

1198622

1198623

1198624

PRA1

05000 08275 05000 08275PRA2

06779 07357 08275 05490PRA3

06779 08275 05000 09437

Step 2 After determining the goal and criteria it forms thepattern-rule base by leveraging D number theory based onTOPSIS method

Step 3 For the intelligent CEP it then collects simple eventsby the event collector engine from the data sources

Step 4 Based on the pattern-rule base the intelligent CEPsystem can process the collected simple events and generatenew complex events by the CEP engine

4 Numerical Example

In this section in order to illustrate the application of theproposed method a simple numerical example is givenSuppose that there are three pattern-rule alternatives(PRA) which may be considered for further generating acomposite event in the CEP system namely PRA

1=

SEQ(1198641 119864

119894 119864

119896) PRA

2= SEQ(119864

1 119864

119895 119864

119896)

and PRA3= SEQ(119864

1 119864

119895 119864

119899) Meanwhile there are

four evaluation criteria and three experts need to select themost suitable pattern-rule candidate from them for bettersupporting decision making in CEP systems Three expertsgive different assessment results to different alternatives interms of different evaluation criteria Four benefit criteria areconsidered

(1) Accuracy (1198621)

(2) Response Time (1198622)

(3) Cost (1198623)

(4) Operability (1198624)

The proposed method is now applied to solve this decisionproblem Then computational procedure is summarized asfollows

Step 1 The experts use linguistic weighting variables to assessthe importance of the criteria which is shown in Table 1

Step 2 In this paper the criteria weights for different expertsdetermined by variation coefficient method are adopted andpresented in Table 2

Step 3 Using the linguistic rating variables of Table 1 theratings of the three decision alternatives by different expertsunder four evaluation criteria are obtained as shown in

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

[1] D J Abadi Y Ahmad M Balazinska et al ldquoThe design ofthe Borealis stream processingenginerdquo in Proceedings of the 2ndBiennial Conference on Innovative Data Systems Research pp277ndash289 2005

[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 7 The representation of119863 number for PRA1

PRA1

119863 numbersExpert

1119863

Expert1PRA1 = (06779 01478) (05000 03222) (08275 01269) (08275 04031)

Expert2

119863Expert2PRA1 = (09437 02632) (05000 04358) (08275 00971) (08275 02039)

Expert3

119863Expert3PRA1 = (05000 03069) (08275 02255) (05000 02663) (08275 02013)

Table 8 The results of119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1

119887 V 119887 V06554 00563 06637 0133807082 00477 08191 0044105000 00636 05445 0059908275 00702 06109 0060008565 00630 06783 0042706928 00366 07901 0031806997 00432 07817 0027807456 00879 07747 0047806708 00500 07153 0025007442 00230

Table 9 The ranking of alternatives

Alternative PRA1

PRA2

PRA3

119868(119863) 07000 06301 06499Ranking 1 3 2

Table 3 Furthermore the fuzzy numbers of three candidatesby experts under all criteria can be obtained as shown inTable 4

Step 4 Based on the fuzzy numbers of Table 4 the fuzzynormalized decision matrix can be constructed as shown inTable 5

Step 5 The closeness coefficient of each alternative can becalculated as shown in Table 6

Step 6 Based on Table 6 the D number for PRA1can be

represented as shown in Table 7

Step 7 The results of 119863Expert1PRA1 oplus 119863

Expert2PRA1 oplus 119863

Expert3PRA1 can be re-

presented as shown in Table 8

Step 8 The integration representation of every 119863PRA119894 iscalculated by using (12) Table 9 shows these values of 119868(119863119860

1)

119868(1198631198602) and 119868(119863119860

3) According to these values the ranking

of alternative is obtained it is PRA1≻ PRA

3≻ PRA

2 where

ldquo≻rdquo represents ldquobetter thanrdquo

5 Conclusion

In this paper we started off with identifying the uncertaintyproblems in terms of pattern rules of CEP systems Weproposed an intelligent complex event processing strategy

withD numbers under fuzzy environment which could fullysupport decision making for guaranteeing effective complexevent processing The main idea of the proposed strategy isthat we appliedD numbers based on TOPSISmethod into theCEP systems A numerical example was provided to evaluatethe effectiveness of our proposed method

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 60904099) Foundation for Funda-ment Research ofNorthwestern Polytechnical University (noJC20120235) Fundamental Research Funds for the CentralUniversities (no XDJK2015C107) and the Doctoral Programof Higher Education (no SWU115008)

References

[1] D J Abadi Y Ahmad M Balazinska et al ldquoThe design ofthe Borealis stream processingenginerdquo in Proceedings of the 2ndBiennial Conference on Innovative Data Systems Research pp277ndash289 2005

[2] D J Abadi D Carney U Cetintemel et al ldquoAurora a newmodel and architecture for data stream managementrdquo TheVLDB Journal vol 12 no 2 pp 120ndash139 2003

[3] M Balazinska H Balakrishnan S R Madden and M Stone-braker ldquoFault-tolerance in the borealis distributed stream pro-cessing systemrdquoACMTransactions onDatabase Systems vol 33no 1 article 3 2008

[4] S Chandrasekaran O Cooper A Deshpande et al ldquoTele-graphCQ continuous dataflow processing for an uncertainworldrdquo in Proceedings of the 1st Biennial Conference on Innova-tive Data Systems Research (CIDR rsquo03) Asilomar Calif USAJanuary 2003

[5] J-H Hwang Y Xing U Cetintemel and S Zdonik ldquoA coop-erative self-configuring high-availability solution for streamprocessingrdquo in Proceedings of the 23rd International Conferenceon Data Engineering (ICDE rsquo07) pp 176ndash185 Istanbul TurkeyApril 2007

[6] M A Shah J M Hellerstein and E Brewer ldquoHighly availablefault-tolerant parallel dataflowsrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo04) pp 827ndash838 Paris France June 2004

[7] F Xiao T Kitasuka and M Aritsugi ldquoEconomical and fault-tolerant load balancing in distributed stream processing sys-temsrdquo IEICE Transactions on Information and Systems vol 95no 4 pp 1062ndash1073 2012

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

[8] F Xiao K Nagano T Itokawa T Kitasuka and M Aritsugi ldquoAself-recovery technique for highly-available stream processingover local area networksrdquo in Proceedings of the IEEE Region10 Conference (TENCON rsquo10) pp 2406ndash2411 Fukuoka JapanNovember 2010

[9] Y Diao P Stahlberg and G Anderson ldquoSASE complex eventprocessing over streamsrdquo in Proceedings of the 3rd BiennialConference on Innovative Data Systems Research (CIDR rsquo07) pp407ndash411 Asilomar Calif USA 2007

[10] A J Demers J Gehrke B Panda M Riedewald V Sharmaand W White ldquoCayuga A general purpose event monitoringsystemrdquo in Proceedings of the 3rd Biennial Conference onInnovative Data Systems Research (CIDR rsquo07) pp 412ndash422Asilomar Calif USA January 2007

[11] M Liu E Rundensteiner K Greenfield et al ldquoE-Cube multi-dimensional event sequence analysis using hierarchical patternquery sharingrdquo in Proceedings of the ACM SIGMOD Interna-tional Conference on Management of Data (SIGMOD rsquo11) pp889ndash900 Athens Greece June 2011

[12] Y Mei and S Madden ldquoZstream a cost-based query processorfor adaptively detecting composite eventsrdquo in Proceedings ofthe ACM International Conference on Management of Data(SIGMOD rsquo09) pp 193ndash206 Providence RI USA July 2009

[13] EWu Y Diao and S Rizvi ldquoHigh-performance complex eventprocessing over streamsrdquo in Proceedings of the ACM SIGMODInternational Conference on Management of Data (SIGMODrsquo06) pp 407ndash418 Chicago Ill USA June 2006

[14] M Akdere U Cetintemel and N Tatbul ldquoPlan-based complexevent detection across distributed sourcesrdquo Proceedings of theVLDB Endowment vol 1 no 1 pp 66ndash77 2008

[15] J Agrawal Y Diao D Gyllstrom and N Immerman ldquoEfficientpattern matching over event streamsrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 147ndash160 ACM Vancouver CanadaJune 2008

[16] M Liu E Rundensteiner D Dougherty et al ldquoHigh-performance nested CEP query processing over event streamsrdquoinProceedings of the IEEE 27th International Conference onDataEngineering (ICDE rsquo11) pp 123ndash134 Hannover Germany April2011

[17] G Cugola and A Margara ldquoProcessing flows of informationfrom data stream to complex event processingrdquo ACM Comput-ing Surveys vol 44 no 3 article 15 2012

[18] G Cugola and A Margara ldquoLow latency complex event pro-cessing on parallel hardwarerdquo Journal of Parallel andDistributedComputing vol 72 no 2 pp 205ndash218 2012

[19] 2013 httpavidcsumassedusase[20] 2010 httpwwwcscornelledubigreddatacayuga[21] 2007 httpdbsmathematikuni-marburgdeHomeResearch

ProjectsPIPES[22] 2013 httpswwwcrunchbasecomorganizationcoral8[23] 2013 httpwwwstreambasecom[24] 2012 httpblogsoraclecomcep[25] 2013 httpstanfordhospitalorgclinicsmedServicesCOEsu-

rgicalServicesgeneralSurgerypatientEducationtestshtml[26] F Terroso-Saenz M Valdes-Vela C Sotomayor-Martınez R

Toledo-Moreo and A F Gomez-Skarmeta ldquoA cooperativeapproach to traffic congestion detection with complex eventprocessing andVANETrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 914ndash929 2012

[27] YGuG Yu andC Li ldquoDeadline-aware complex event process-ing models over distributedmonitoring streamsrdquoMathematicaland Computer Modelling vol 55 no 3-4 pp 901ndash917 2012

[28] B Ottenwalder B Koldehofe K Rothermel K Hong DLillethun and U Ramachandran ldquoMCEP a mobility-awarecomplex event processing systemrdquo ACM Transactions on Inter-net Technology vol 14 no 1 article 6 2014

[29] B Cao and J Li ldquoAn intelligent complex event processing withDS evidence theory in IT centralized monitoringrdquo in InternetandDistributed Computing Systems LectureNotes in ComputerScience pp 373ndash384 Springer Berlin Germany 2013

[30] N Khoussainova M Balazinska and D Suciu ldquoPEEX extract-ing probabilistic events from RFID datardquo in Proceedings of theInternational Conference on Data Engineering (ICDE rsquo08) pp1480ndash1482 Cancun Mexico 2008

[31] S Wasserkrug A Gal O Etzion and Y Turchin ldquoComplexevent processing over uncertain datardquo in Proceedings of the 2ndInternational Conference on Distributed Event-Based Systems(DEBS rsquo08) pp 253ndash264 Rome Italy July 2008

[32] S Wasserkrug A Gal O Etzion and Y Turchin ldquoEfficientprocessing of uncertain events in rule-based systemsrdquo IEEETransactions on Knowledge and Data Engineering vol 24 no1 pp 45ndash58 2012

[33] K Cao Y Wang and F Wang ldquoContext-aware distributedcomplex event processing method for event cloud in internet ofthingsrdquo Advances in Information Sciences and Service Sciencesvol 5 no 8 p 1212 2013

[34] YDeng ldquoGeneralized evidence theoryrdquoApplied Intelligence vol43 no 3 pp 530ndash543 2015

[35] Y Deng ldquoFuzzy analytical hierarchy process based on canonicalrepresentation on fuzzy numbersrdquo Journal of ComputationalAnalysis and Applications vol 22 no 2 pp 201ndash228 2017

[36] X Ning J Yuan X Yue and A Ramirez-Serrano ldquoInducedgeneralized Choquet aggregating operators with linguisticinformation and their application to multiple attribute decisionmaking based on the intelligent computingrdquo Journal of Intelli-gent and Fuzzy Systems vol 27 no 3 pp 1077ndash1085 2014

[37] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[38] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoThe interval-valued intuitionistic fuzzy MULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[39] C Fu J-B Yang and S-L Yang ldquoA group evidential reasoningapproach based on expert reliabilityrdquo European Journal ofOperational Research vol 246 no 3 pp 886ndash893 2015

[40] Y Deng ldquoDeng entropyrdquo Chaos Solitons amp Fractals vol 91 pp549ndash553 2016

[41] W-B Du Y Gao C Liu Z Zheng and Z Wang ldquoAdequate isbetter particle swarm optimization with limited-informationrdquoApplied Mathematics and Computation vol 268 pp 832ndash8382015

[42] S-B Tsai Y-C Lee and J-J Guo ldquoUsing modified grey fore-castingmodels to forecast the growth trends of greenmaterialsrdquoProceedings of the Institution of Mechanical Engineers Part BJournal of EngineeringManufacture vol 228 no 6 pp 931ndash9402014

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[43] W Jiang C Xie B Wei and D Zhou ldquoA modified method forrisk evaluation in failure modes and effects analysis of aircraftturbine rotor bladesrdquo Advances in Mechanical Engineering vol8 no 4 pp 1ndash16 2016

[44] W Jiang B Wei C Xie and D Zhou ldquoAn evidential sensorfusion method in fault diagnosisrdquo Advances in MechanicalEngineering vol 8 no 3 pp 1ndash7 2016

[45] X Ning J Yuan and X Yue ldquoUncertainty-based optimiza-tion algorithms in designing fractionated spacecraftrdquo ScientificReports vol 6 Article ID 22979 2016

[46] W Jiang Y Yang Y Luo andXQin ldquoDetermining basic proba-bility assignment based on the improved similarity measures ofgeneralized fuzzy numbersrdquo International Journal of ComputersCommunications amp Control vol 10 no 3 pp 333ndash347 2015

[47] W Jiang Y Luo X-Y Qin and J Zhan ldquoAn improved methodto rank generalized fuzzy numbers with different left heightsand right heightsrdquo Journal of Intelligent and Fuzzy Systems vol28 no 5 pp 2343ndash2355 2015

[48] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[49] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011

[50] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[51] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[52] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[53] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[54] B Vahdani S M Mousavi and R Tavakkoli-MoghaddamldquoGroupdecisionmaking based onnovel fuzzymodifiedTOPSISmethodrdquo Applied Mathematical Modelling vol 35 no 9 pp4257ndash4269 2011

[55] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[56] F Lolli A Ishizaka R Gamberini B Rimini and M MessorildquoFlowSort-GDSSmdasha novel group multi-criteria decision sup-port system for sorting problems with application to FMEArdquoExpert Systems with Applications vol 42 no 17-18 pp 6342ndash6349 2015

[57] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure mode andeffects analysis using D numbers and grey relational projectionmethodrdquo Expert Systems with Applications vol 41 no 10 pp4670ndash4679 2014

[58] N Rikhtegar N Mansouri A A Oroumieh A Yazdani-Chamzini E K Zavadskas and S Kildiene ldquoEnvironmentalimpact assessment based on group decision-makingmethods inmining projectsrdquo Economic Research-Ekonomska Istrazivanjavol 27 no 1 pp 378ndash392 2014

[59] G Fan D Zhong F Yan and P Yue ldquoA hybrid fuzzy evaluationmethod for curtain grouting efficiency assessment based on anAHP method extended by D numbersrdquo Expert Systems withApplications vol 44 pp 289ndash303 2016

[60] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[61] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[62] M Li Y Hu Q Zhang and Y Deng ldquoA novel distance functionof D numbers and its application in product engineeringrdquoEngineering Applications of Artificial Intelligence vol 47 pp 61ndash67 2016

[63] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo Annals of Mathematical Statistics vol38 pp 325ndash339 1967

[64] G Shafer ldquoA mathematical theory of evidencerdquo Technometricsvol 20 no 1 p 242 1978

[65] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[66] H-J Zimmermann Fuzzy Set Theory and its ApplicationsSpringer Science amp Business Media Berlin Germany 2001

[67] M Beynon ldquoDSAHPmethod amathematical analysis includ-ing an understanding of uncertaintyrdquo European Journal ofOperational Research vol 140 no 1 pp 148ndash164 2002

[68] M Bauer ldquoApproximation algorithms and decision making inthe Dempster-Shafer theory of evidencemdashan empirical studyrdquoInternational Journal of Approximate Reasoning vol 17 no 2-3pp 217ndash237 1997

[69] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Arden Shakespeare 1991

[70] F Xiao and M Aritsugi ldquoNested pattern queries process-ing optimization over multi-dimensional event streamsrdquo inProceedings of the IEEE 37th Annual Computer Software andApplications Conference (COMPSAC rsquo13) pp 74ndash83 KyotoJapan 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of