Uncertainty Management in Smart EnvironmentView this Special Issue
Research Article | Open Access
Fuyuan Xiao, "An Intelligent Complex Event Processing with Numbers under Fuzzy Environment", Mathematical Problems in Engineering, vol. 2016, Article ID 3713518, 10 pages, 2016. https://doi.org/10.1155/2016/3713518
An Intelligent Complex Event Processing with Numbers under Fuzzy Environment
Efficient matching of incoming mass events to persistent queries is fundamental to complex event processing systems. Event matching based on pattern rule is an important feature of complex event processing engine. However, the intrinsic uncertainty in pattern rules which are predecided by experts increases the difficulties of effective complex event processing. It inevitably involves various types of the intrinsic uncertainty, such as imprecision, fuzziness, and incompleteness, due to the inability of human beings subjective judgment. Nevertheless, D numbers is a new mathematic tool to model uncertainty, since it ignores the condition that elements on the frame must be mutually exclusive. To address the above issues, an intelligent complex event processing method with D numbers under fuzzy environment is proposed based on the Technique for Order Preferences by Similarity to an Ideal Solution (TOPSIS) method. The novel method can fully support decision making in complex event processing systems. Finally, a numerical example is provided to evaluate the efficiency of the proposed method.
Nowadays, there has been increasing interest in distributed applications which require processing continuously flowing data from geographically distributed sources to obtain timely responses to complex queries, such as data stream processing (DSP) systems [1–8] and complex event processing (CEP) systems [9–16]. In principle, DSP systems differ from CEP systems as the DSP systems focus on transforming the incoming flow of information, while the CEP systems focus on detecting patterns of information that represent the higher-level events [17, 18]. Because of the advantages such as expressive rule language and efficient event detection model, CEP systems have been highly concerned in academic circles and industry recently [19–25].
In CEP systems, event streams are processed in or near real time for a variety of purposes, from wireless sensor networks to financial tickers and from traffic management to click-stream inspection [17, 18, 26–28]. In those application domains, accurate and effective complex event processing is critical for dealing with real-world events, whereas there are two possible origins for uncertainty which increases the difficulties of accurate and effective complex event processing, such as uncertainty originating at the event source and uncertainty resulting from event inference . For uncertainty originating at the event source (i.e., raw event), there may be uncertainty associated with either the event occurrence itself or the event’s attributes, due to a feature of the event source. For uncertainty resulting from event inference (i.e., derived events), they are based on other events and uncertainty can propagate to the derived events. Nevertheless, in DSP systems, many kinds of strategies have been developed to handle the above two types of uncertainty [29–33].
However, most of the CEP systems are dealing with the flow of events where the basic premise underlying the design of the CEP queries proposed is that the associated pattern rules of events are predefined. Moreover, many of predefined pattern rules are based on human beings subjective experience so that it is not necessarily accurate in practice. While rule based reasoning system structure looks simple, the acquisition of knowledge is a big bottleneck in the system. Because the rules come from experts, there are no self-learning functions. Such a behavior gives rise to poor quality of query results. Furthermore, in conventional CEP systems, it inevitably involves various types of the intrinsic uncertainty, such as imprecision, fuzziness, and incompleteness, due to the inability of human beings subjective judgment. In these cases, it seems more reasonable to consider fuzzy situation in the CEP systems.
Due to the efficiency to handle uncertainty and fuse data, some math tools such as fuzzy sets, evidence theory, and probability method are widely used in decision making [34–42], risk analysis , diagnosis , and optimization problem . Needless to say, fuzzy sets theory introduced by Zadeh is a good approach to settle with the uncertain information [35, 46–50]. The Technique for Order Preferences by Similarity to an Ideal Solution (TOPSIS)  method proposed by Hwang and Yoon (1981) is widely used in decision making . Moreover, the extension of the TOPSIS  method was developed to handle fuzzy data [53, 54]. The TOPSIS method is applied widely in MCDM field. And many applications have proved its effectiveness,whereas there still exist some shortcomings. One of the open questions is that current MCDM based on the TOPSIS method cannot adequately handle these types of uncertainties, such as imprecision, fuzziness, and incompleteness, due to the inability of experts’ subjective judgment. D numbers proposed by Deng  is a new mathematic tool to model uncertainty, since it ignores the condition that elements on the frame must be mutually exclusive. Compared with existing methods, D numbers can efficiently represent uncertain information and are more tallied with the actual situation. There is an increasing number of applications about D numbers, such as FEMA analysis , failure mode analysis , environment assessment [58–60], supplier selection , and product engineering .
To address these issues, we propose an intelligent complex event processing strategy with D numbers under fuzzy environment based on the TOPSIS method. The novel method can deal with pattern-rule based uncertainty under fuzzy environment to realize accurate and effective event stream processing. Effectiveness of the proposed method is evaluated through a numerical experiment.
The rest of this paper is organized as follows. Section 2 briefly introduces the preliminaries of this paper. After that, Section 3 proposes an intelligent complex event processing strategy with D numbers under fuzzy environment. Section 4 gives a numerical example to show the effectiveness of the proposed method. Finally, Section 5 gives a conclusion.
2.1. Dempster-Shafer Evidence Theory
Dempster-Shafer evident theory [63, 64], also known as evidence theory, is used to handle uncertain information, belonging to the category of artificial intelligence. As a theory reasoning under the uncertain environment, it needs weaker conditions than the Bayesian theory of probability. When the probability is confirmed, Dempster-Shafer theory could convert into Bayesian theory, so it is often regarded as an extension of the Bayesian theory. Dempster-Shafer theory has the advantage of directly expressing the “uncertainty” by assigning the probability to the subsets of the set composed of multiple objects, rather than to an individual object. Besides, it has the ability to combine pairs of bodies of evidence or belief functions to derive a new evidence or belief function. Based on the Dempster-Shafer evident theory, Deng proposed the generalized evidence theory to extend the classical evidence theory . For completeness of the explanation, some basic concepts are introduced as follows.
Definition 1 (frame of discernment). Let be a set of mutually exclusive and collectively exhaustive, indicated byThe set is called frame of discernment. The power set of is indicated by , whereand is an empty set. If , is called a proposition.
Definition 2 (mass function). For a frame of discernment , a mass function is a mapping from to , formally defined by which satisfies the following condition:
In the Dempster-Shafer evident theory, a mass function is also called a basic probability assignment (BPA). If , is called a focal element; the union of all focal elements is called the core of the mass function.
Definition 3 (Dempster’s rule of combination). Dempster’s rule of combination, also called orthogonal sum, denoted by , is defined as follows:withwhere and are also elements of and is a constant to show the conflict between the two BPAs. Note that Dempster’s rule of combination is only applicable to such two BPAs which satisfy the condition .
2.2. Number Theory
D number theory is a generalization of Dempster-Shafer evidence theory proposed by Deng . In the classical Dempster-Shafer theory, there are several strong hypotheses on the frame of discernment and basic probability assignment and still some shortcomings, which limit the ability of Dempster-Shafer theory to represent some types of information and restrict the application in practice. D number theory, as an extension and development method, is defined as follows.
Definition 4 ( number). Let be a finite nonempty set; a D number is a mapping formulated bywithwhere is a subset of .
It seems that the definition of D numbers is similar to the definition of BPA. However, in D number theory, the elements of do not require to be mutually exclusive. In addition, being contrary of the frame of discernment containing overall events, is acceptable to incomplete information by .
Furthermore, for a discrete set , where and when , . A special form of D numbers can be expressed byor simply denoted as , where and .
Definition 5 (two D numbers’ rule of combination). Let , be two D numbers; the combination of and , indicated by , is defined bywithwhere and .
In the meanwhile, an aggregation operator is proposed on this special D number; it is defined as below.
Definition 6 (D numbers’ integration). For , the integrating representation of is defined as
2.3. TOPSIS Method with Fuzzy Data
Fuzzy set theory  provide an alternative and convenient framework for modeling of real-world fuzzy decision systems mathematically [66–68]. A fuzzy set is any set that allows its members to have different grades of membership in the interval . It consists of two components: a set and a membership function associated with it.
Definition 7 (fuzzy set ). Let be a collection of objects denoted generally by ; a fuzzy subset of , , is a set of ordered pairs:where is called the membership function (generalized characteristic function) which maps to the membership space . Its range is the subset of nonnegative real members whose supremum is finite.
Definition 8 (triangular fuzzy number ). A fuzzy number is a fuzzy subset of . And a triangular fuzzy number can be defined by a triplet shown in Figure 1, in which , , and are real numbers with . Its membership function is defined as
Definition 9 (distance between two triangular fuzzy numbers ). Let and be two triangular fuzzy numbers; then the vertex method is defined to calculate the distance between them as
The main idea of TOPSIS is that the best compromise solution should have the shortest Euclidean distance from the positive ideal solution and the farthest Euclidean distance from the negative ideal solution.
The procedures of TOPSIS method with fuzzy data can be described as follows. Let be the set of alternatives, be the set of criteria, and be the performance ratings with the criteria weight vector .
Step 1 (calculate normalized ratings). Let and be the set of benefit criteria and cost criteria, respectively. The normalized value is calculated by
Step 2 (calculate weighted normalized ratings). In the weighted normalized decision matrix, the modified ratings are calculated bywhere is the weight of the sth criteria.
Step 3 (determine the fuzzy positive and negative ideal solutions). The elements , , are normalized positive triangular fuzzy numbers and their ranges belong to the closed interval . Then, the fuzzy positive ideal solution (FPIS, ) and the fuzzy negative ideal solution (FNIS, ) are derived as follows:where and , .
Step 4 (calculate the distance of each alternative from the FPIS and the FNIS). The distance of each alternative from and can be currently calculated aswhere is the distance measurement between two fuzzy numbers.
Step 5 (calculate the relative closeness coefficient to the positive ideal solution). The relative closeness coefficient for the alternative with respect to is
Step 6 (rank the alternatives). Obviously, an alternative is closer to the FPIS () and farther from FNIS () as approaches 1. Therefore according to relative closeness coefficient to the ideal alternative, larger value of indicates the better alternative .
3. The Proposed Method
Before introducing the system architecture of intelligent CEP, we first have a clear understanding of the definition of an event [10, 11]. An event that represents an atomic instance is an occurrence that is of interest at a point in time. Basically, events can be classified into primitive events and composite events. A primitive event instance is predefined as a single occurrence of interest that cannot be split into any small events. A composite event instance that occurs over an interval is created by composing primitive or composite events. A pattern rule is a template, specifying one or more combinations of events by the nesting of sequences (SEQ) and conjunctions (AND), which can have negative event type(s), and their combination. In the following, denotes an event type which can be either primitive or composite. Some details were presented in .
Definition 10. A SEQ operator  specifies a specific order according to the start time-stamps in which the event must occur, to match the pattern, and thus form a composite event.
For example, SEQ (History_taking, Physical_examination, Laboratory_examination) consists of SEQ operator that can be used to monitor the patients who have completed medical diagnosis.
Definition 11. An AND operator  takes a set of event types as input, and events occur within a specified time window without a specified time order.
For example, AND (Laboratory_examination, Surgery) consists of AND operator that can be used to monitor the patients’ preoperative and postoperative situations.
Currently, the pattern-rule base of CEP systems consisting of a set of pattern rules is predefined where it is based on human beings subjective experience so that it is not necessarily accurate in practice, whereas the acquisition of knowledge is a big bottleneck in the system and most researches on CEP systems seldom discuss how to handle such kinds of problems. As we discussed in Section 1, because the rules come from experts, there is no self-learning function. The CEP engine based on the inaccurate predefined pattern-rule base will generate poor quality of query results. Furthermore, in conventional CEP systems, it inevitably involves various types of the intrinsic uncertainty, such as imprecision, fuzziness, and incompleteness, due to the inability of human beings subjective judgment.
In this section, a novel intelligent complex event processing strategy with D numbers under fuzzy environment is proposed based on the Technique for Order Preferences by Similarity to an Ideal Solution (TOPSIS) method. The proposed method can fully support decision making in CEP systems.
The proposed system architecture of intelligent CEP is shown in Figure 2; it mainly involves two components: event collector engine and CEP engine under D number theory based on TOPSIS method. We will explain each component in detail.
Component 1: Event Collector Engine. The event collector engine, which collects events from data sources, first generates a unified formal definition of the event flow.
Component 2: CEP Engine under D Number Theory Based on TOPSIS Method. Then, the CEP engine processes the collected simple events through the pattern rules in which the events generated at this component are called complex events, whereas the D number theory based on TOPSIS method, which is the addition to the CEP engine, in view of the detected uncertainty event set, is used for pattern-rules analysis to select the best one from the decision candidates. Finally, the generated new events are sent out directly to particular users, or reused as input events, or used to do further analysis.
The main steps of the proposed method are shown as follows, and the basic flowchart is given in Figure 3 for the better understanding of the concept.
Step 1. Make certain of the detailed information of the decision-making problems, including the goal and criteria.
Step 2. After determining the goal and criteria, it forms the pattern-rule base by leveraging D number theory based on TOPSIS method.
Step 3. For the intelligent CEP, it then collects simple events by the event collector engine from the data sources.
Step 4. Based on the pattern-rule base, the intelligent CEP system can process the collected simple events and generate new complex events by the CEP engine.
4. Numerical Example
In this section, in order to illustrate the application of the proposed method, a simple numerical example is given. Suppose that there are three pattern-rule alternatives () which may be considered for further generating a composite event in the CEP system, namely, , , and . Meanwhile, there are four evaluation criteria and three experts need to select the most suitable pattern-rule candidate from them for better supporting decision making in CEP systems. Three experts give different assessment results to different alternatives in terms of different evaluation criteria. Four benefit criteria are considered:(1)Accuracy ().(2)Response Time ().(3)Cost ().(4)Operability ().The proposed method is now applied to solve this decision problem. Then, computational procedure is summarized as follows.
Step 1. The experts use linguistic weighting variables to assess the importance of the criteria, which is shown in Table 1.
Step 2. In this paper, the criteria weights for different experts determined by variation coefficient method are adopted and presented in Table 2.
Step 3. Using the linguistic rating variables of Table 1, the ratings of the three decision alternatives by different experts under four evaluation criteria are obtained as shown in Table 3. Furthermore, the fuzzy numbers of three candidates by experts under all criteria can be obtained as shown in Table 4.
Step 5. The closeness coefficient of each alternative can be calculated as shown in Table 6.
Step 7. The results of can be represented as shown in Table 8.
Step 8. The integration representation of every is calculated by using (12). Table 9 shows these values of , , and . According to these values, the ranking of alternative is obtained; it is , where “” represents “better than.”
In this paper, we started off with identifying the uncertainty problems in terms of pattern rules of CEP systems. We proposed an intelligent complex event processing strategy with D numbers under fuzzy environment, which could fully support decision making for guaranteeing effective complex event processing. The main idea of the proposed strategy is that we applied D numbers based on TOPSIS method into the CEP systems. A numerical example was provided to evaluate the effectiveness of our proposed method.
The author declares that he has no competing interests.
This work was supported in part by National Natural Science Foundation of China (no. 60904099), Foundation for Fundament Research of Northwestern Polytechnical University (no. JC20120235), Fundamental Research Funds for the Central Universities (no. XDJK2015C107), and the Doctoral Program of Higher Education (no. SWU115008).
- D. J. Abadi, Y. Ahmad, M. Balazinska et al., “The design of the Borealis stream processingengine,” in Proceedings of the 2nd Biennial Conference on Innovative Data Systems Research, pp. 277–289, 2005.
- D. J. Abadi, D. Carney, U. Çetintemel et al., “Aurora: a new model and architecture for data stream management,” The VLDB Journal, vol. 12, no. 2, pp. 120–139, 2003.
- M. Balazinska, H. Balakrishnan, S. R. Madden, and M. Stonebraker, “Fault-tolerance in the borealis distributed stream processing system,” ACM Transactions on Database Systems, vol. 33, no. 1, article 3, 2008.
- S. Chandrasekaran, O. Cooper, A. Deshpande et al., “TelegraphCQ: continuous dataflow processing for an uncertain world,” in Proceedings of the 1st Biennial Conference on Innovative Data Systems Research (CIDR '03), Asilomar, Calif, USA, January 2003.
- J.-H. Hwang, Y. Xing, U. Çetintemel, and S. Zdonik, “A cooperative, self-configuring high-availability solution for stream processing,” in Proceedings of the 23rd International Conference on Data Engineering (ICDE '07), pp. 176–185, Istanbul, Turkey, April 2007.
- M. A. Shah, J. M. Hellerstein, and E. Brewer, “Highly available, fault-tolerant, parallel dataflows,” in Proceedings of the ACM SIGMOD International Conference on Management of Data (SIGMOD '04), pp. 827–838, Paris, France, June 2004.
- F. Xiao, T. Kitasuka, and M. Aritsugi, “Economical and fault-tolerant load balancing in distributed stream processing systems,” IEICE Transactions on Information and Systems, vol. 95, no. 4, pp. 1062–1073, 2012.
- F. Xiao, K. Nagano, T. Itokawa, T. Kitasuka, and M. Aritsugi, “A self-recovery technique for highly-available stream processing over local area networks,” in Proceedings of the IEEE Region 10 Conference (TENCON '10), pp. 2406–2411, Fukuoka, Japan, November 2010.
- Y. Diao, P. Stahlberg, and G. Anderson, “SASE: complex event processing over streams,” in Proceedings of the 3rd Biennial Conference on Innovative Data Systems Research (CIDR '07), pp. 407–411, Asilomar, Calif, USA, 2007.
- A. J. Demers, J. Gehrke, B. Panda, M. Riedewald, V. Sharma, and W. White, “Cayuga: A general purpose event monitoring system,” in Proceedings of the 3rd Biennial Conference on Innovative Data Systems Research (CIDR '07), pp. 412–422, Asilomar, Calif, USA, January 2007.
- M. Liu, E. Rundensteiner, K. Greenfield et al., “E-Cube: multi-dimensional event sequence analysis using hierarchical pattern query sharing,” in Proceedings of the ACM SIGMOD International Conference on Management of Data (SIGMOD '11), pp. 889–900, Athens, Greece, June 2011.
- Y. Mei and S. Madden, “Zstream: a cost-based query processor for adaptively detecting composite events,” in Proceedings of the ACM International Conference on Management of Data (SIGMOD '09), pp. 193–206, Providence, RI, USA, July 2009.
- E. Wu, Y. Diao, and S. Rizvi, “High-performance complex event processing over streams,” in Proceedings of the ACM SIGMOD International Conference on Management of Data (SIGMOD '06), pp. 407–418, Chicago, Ill, USA, June 2006.
- M. Akdere, U. Çetintemel, and N. Tatbul, “Plan-based complex event detection across distributed sources,” Proceedings of the VLDB Endowment, vol. 1, no. 1, pp. 66–77, 2008.
- J. Agrawal, Y. Diao, D. Gyllstrom, and N. Immerman, “Efficient pattern matching over event streams,” in Proceedings of the ACM SIGMOD International Conference on Management of Data (SIGMOD '08), pp. 147–160, ACM, Vancouver, Canada, June 2008.
- M. Liu, E. Rundensteiner, D. Dougherty et al., “High-performance nested CEP query processing over event streams,” in Proceedings of the IEEE 27th International Conference on Data Engineering (ICDE '11), pp. 123–134, Hannover, Germany, April 2011.
- G. Cugola and A. Margara, “Processing flows of information: from data stream to complex event processing,” ACM Computing Surveys, vol. 44, no. 3, article 15, 2012.
- G. Cugola and A. Margara, “Low latency complex event processing on parallel hardware,” Journal of Parallel and Distributed Computing, vol. 72, no. 2, pp. 205–218, 2012.
- 2013, http://avid.cs.umass.edu/sase/.
- 2010, http://www.cs.cornell.edu/bigreddata/cayuga/.
- 2007, http://dbs.mathematik.uni-marburg.de/Home/Research/Projects/PIPES/.
- 2013, https://www.crunchbase.com/organization/coral8.
- 2013, http://www.streambase.com/.
- 2012, http://blogs.oracle.com/cep/.
- 2013, http://stanfordhospital.org/clinicsmedServices/COE/surgicalServices/generalSurgery/patientEducation/tests.html.
- F. Terroso-Sáenz, M. Valdés-Vela, C. Sotomayor-Martínez, R. Toledo-Moreo, and A. F. Gómez-Skarmeta, “A cooperative approach to traffic congestion detection with complex event processing and VANET,” IEEE Transactions on Intelligent Transportation Systems, vol. 13, no. 2, pp. 914–929, 2012.
- Y. Gu, G. Yu, and C. Li, “Deadline-aware complex event processing models over distributed monitoring streams,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 901–917, 2012.
- B. Ottenwälder, B. Koldehofe, K. Rothermel, K. Hong, D. Lillethun, and U. Ramachandran, “MCEP: a mobility-aware complex event processing system,” ACM Transactions on Internet Technology, vol. 14, no. 1, article 6, 2014.
- B. Cao and J. Li, “An intelligent complex event processing with DS evidence theory in IT centralized monitoring,” in Internet and Distributed Computing Systems, Lecture Notes in Computer Science, pp. 373–384, Springer, Berlin, Germany, 2013.
- N. Khoussainova, M. Balazinska, and D. Suciu, “PEEX: extracting probabilistic events from RFID data,” in Proceedings of the International Conference on Data Engineering (ICDE '08), pp. 1480–1482, Cancun, Mexico, 2008.
- S. Wasserkrug, A. Gal, O. Etzion, and Y. Turchin, “Complex event processing over uncertain data,” in Proceedings of the 2nd International Conference on Distributed Event-Based Systems (DEBS '08), pp. 253–264, Rome, Italy, July 2008.
- S. Wasserkrug, A. Gal, O. Etzion, and Y. Turchin, “Efficient processing of uncertain events in rule-based systems,” IEEE Transactions on Knowledge and Data Engineering, vol. 24, no. 1, pp. 45–58, 2012.
- K. Cao, Y. Wang, and F. Wang, “Context-aware distributed complex event processing method for event cloud in internet of things,” Advances in Information Sciences and Service Sciences, vol. 5, no. 8, p. 1212, 2013.
- Y. Deng, “Generalized evidence theory,” Applied Intelligence, vol. 43, no. 3, pp. 530–543, 2015.
- Y. Deng, “Fuzzy analytical hierarchy process based on canonical representation on fuzzy numbers,” Journal of Computational Analysis and Applications, vol. 22, no. 2, pp. 201–228, 2017.
- X. Ning, J. Yuan, X. Yue, and A. Ramirez-Serrano, “Induced generalized Choquet aggregating operators with linguistic information and their application to multiple attribute decision making based on the intelligent computing,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 3, pp. 1077–1085, 2014.
- E. K. Zavadskas, J. Antucheviciene, Z. Turskis, and H. Adeli, “Hybrid multiple-criteria decision-making methods: a review of applications in engineering,” Scientia Iranica, vol. 23, no. 1, pp. 1–20, 2016.
- E. K. Zavadskas, J. Antucheviciene, S. H. R. Hajiagha, and S. S. Hashemi, “The interval-valued intuitionistic fuzzy MULTIMOORA method for group decision making in engineering,” Mathematical Problems in Engineering, vol. 2015, Article ID 560690, 13 pages, 2015.
- C. Fu, J.-B. Yang, and S.-L. Yang, “A group evidential reasoning approach based on expert reliability,” European Journal of Operational Research, vol. 246, no. 3, pp. 886–893, 2015.
- Y. Deng, “Deng entropy,” Chaos, Solitons & Fractals, vol. 91, pp. 549–553, 2016.
- W.-B. Du, Y. Gao, C. Liu, Z. Zheng, and Z. Wang, “Adequate is better: particle swarm optimization with limited-information,” Applied Mathematics and Computation, vol. 268, pp. 832–838, 2015.
- S.-B. Tsai, Y.-C. Lee, and J.-J. Guo, “Using modified grey forecasting models to forecast the growth trends of green materials,” Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, vol. 228, no. 6, pp. 931–940, 2014.
- W. Jiang, C. Xie, B. Wei, and D. Zhou, “A modified method for risk evaluation in failure modes and effects analysis of aircraft turbine rotor blades,” Advances in Mechanical Engineering, vol. 8, no. 4, pp. 1–16, 2016.
- W. Jiang, B. Wei, C. Xie, and D. Zhou, “An evidential sensor fusion method in fault diagnosis,” Advances in Mechanical Engineering, vol. 8, no. 3, pp. 1–7, 2016.
- X. Ning, J. Yuan, and X. Yue, “Uncertainty-based optimization algorithms in designing fractionated spacecraft,” Scientific Reports, vol. 6, Article ID 22979, 2016.
- W. Jiang, Y. Yang, Y. Luo, and X. Qin, “Determining basic probability assignment based on the improved similarity measures of generalized fuzzy numbers,” International Journal of Computers Communications & Control, vol. 10, no. 3, pp. 333–347, 2015.
- W. Jiang, Y. Luo, X.-Y. Qin, and J. Zhan, “An improved method to rank generalized fuzzy numbers with different left heights and right heights,” Journal of Intelligent and Fuzzy Systems, vol. 28, no. 5, pp. 2343–2355, 2015.
- Y. Deng, “A threat assessment model under uncertain environment,” Mathematical Problems in Engineering, vol. 2015, Article ID 878024, 12 pages, 2015.
- Z. Yue, “A method for group decision-making based on determining weights of decision makers using TOPSIS,” Applied Mathematical Modelling, vol. 35, no. 4, pp. 1926–1936, 2011.
- G. R. Jahanshahloo, F. H. Lotfi, and M. Izadikhah, “An algorithmic method to extend TOPSIS for decision-making problems with interval data,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1375–1384, 2006.
- C. L. Hwang and K. Yoon, Multiple Attribute Decision Making: Methods and Applications, Springer, Berlin, Germany, 1981.
- S. Opricovic and G.-H. Tzeng, “Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS,” European Journal of Operational Research, vol. 156, no. 2, pp. 445–455, 2004.
- C.-T. Chen, “Extensions of the TOPSIS for group decision-making under fuzzy environment,” Fuzzy Sets and Systems, vol. 114, no. 1, pp. 1–9, 2000.
- B. Vahdani, S. M. Mousavi, and R. Tavakkoli-Moghaddam, “Group decision making based on novel fuzzy modified TOPSIS method,” Applied Mathematical Modelling, vol. 35, no. 9, pp. 4257–4269, 2011.
- Y. Deng, “D numbers: theory and applications,” Journal of Information and Computational Science, vol. 9, no. 9, pp. 2421–2428, 2012.
- F. Lolli, A. Ishizaka, R. Gamberini, B. Rimini, and M. Messori, “FlowSort-GDSS—a novel group multi-criteria decision support system for sorting problems with application to FMEA,” Expert Systems with Applications, vol. 42, no. 17-18, pp. 6342–6349, 2015.
- H.-C. Liu, J.-X. You, X.-J. Fan, and Q.-L. Lin, “Failure mode and effects analysis using D numbers and grey relational projection method,” Expert Systems with Applications, vol. 41, no. 10, pp. 4670–4679, 2014.
- N. Rikhtegar, N. Mansouri, A. A. Oroumieh, A. Yazdani-Chamzini, E. K. Zavadskas, and S. Kildienė, “Environmental impact assessment based on group decision-making methods in mining projects,” Economic Research-Ekonomska Istraživanja, vol. 27, no. 1, pp. 378–392, 2014.
- G. Fan, D. Zhong, F. Yan, and P. Yue, “A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers,” Expert Systems with Applications, vol. 44, pp. 289–303, 2016.
- X. Deng, Y. Hu, Y. Deng, and S. Mahadevan, “Environmental impact assessment based on D numbers,” Expert Systems with Applications, vol. 41, no. 2, pp. 635–643, 2014.
- X. Deng, Y. Hu, Y. Deng, and S. Mahadevan, “Supplier selection using AHP methodology extended by D numbers,” Expert Systems with Applications, vol. 41, no. 1, pp. 156–167, 2014.
- M. Li, Y. Hu, Q. Zhang, and Y. Deng, “A novel distance function of D numbers and its application in product engineering,” Engineering Applications of Artificial Intelligence, vol. 47, pp. 61–67, 2016.
- A. P. Dempster, “Upper and lower probabilities induced by a multivalued mapping,” Annals of Mathematical Statistics, vol. 38, pp. 325–339, 1967.
- G. Shafer, “A mathematical theory of evidence,” Technometrics, vol. 20, no. 1, p. 242, 1978.
- L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
- H.-J. Zimmermann, Fuzzy Set Theory and its Applications, Springer Science & Business Media, Berlin, Germany, 2001.
- M. Beynon, “DS/AHP method: a mathematical analysis, including an understanding of uncertainty,” European Journal of Operational Research, vol. 140, no. 1, pp. 148–164, 2002.
- M. Bauer, “Approximation algorithms and decision making in the Dempster-Shafer theory of evidence—an empirical study,” International Journal of Approximate Reasoning, vol. 17, no. 2-3, pp. 217–237, 1997.
- A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Arden Shakespeare, 1991.
- F. Xiao and M. Aritsugi, “Nested pattern queries processing optimization over multi-dimensional event streams,” in Proceedings of the IEEE 37th Annual Computer Software and Applications Conference (COMPSAC '13), pp. 74–83, Kyoto, Japan, 2013.
Copyright © 2016 Fuyuan Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.