INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET)

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1 INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET) ISSN (Print) ISSN (Online) Volume 3, Issue 2, July- September (2012), pp IAEME: Journal Impact Factor (2012): (Calculated by GISI) IJCET I A E M E FUZZY METAGRAPH BASED KNOWLEDGE REPRESENTATION OF DECISION SUPPORT SYSTEM ABSTRACT A.THIRUNAVUKARASU 1 Dr. S. UMA MAHESWARI 2 1 Visiting Faculty, Department of Computer Science and Engineering, Anna University of Technology Madurai- Ramanathapuram campus, Tamilnadu, India. 2 Associate Professor, Department of Electronics and Communication Engineering, Coimbatore Institute of Technology, Coimbatore, Tamilnadu, India. 1 thsa07@gmail.com, 2 sumacit@rediffmail.com This paper proposes a Fuzzy Metagraph based Knowledge representation of Decision Support System (DSS). This system will help users to make correct decision with very low risk. Fuzzy Metagraph is an emerging technique widely used for real world applications. Fuzzy Metagraph is used to form the rule base to support inference system to make correct decision. This method can be used in many real world applications like E- commerce, share market, disease analysis. Key words: Metagraph, Fuzzy Metagraph, DSS, Fuzzy production rules, Fuzzy Expert System. INTRODUCTION Graphs play an important role in the design of most information processing system. The graph may be simple graph or ordered graph, A graph is defined by a pair G={X, E}, where X ={x 1, x 2, x 3 x n } is a finite set of vertices and E a collection of edges that happen to connect vertices. There are several diagrammatic system design tools whose construction is based on the graph concept, such as data flow diagram, functional dependency diagram, Petri nets, and semantic nets. Hyper graph shows connectivity between set of elements. Metagraph is graphical hierarchical structure in which every node is a set having one or more elements. It has all the properties of graphs. In a metagraph, there is set to set mapping in place of node to node as in a conventional graph structure [1, 2]. Metagraph is a graphical model that not only visualized the process of any system but also their formal analysis where the analysis will be accomplished by means of an algebraic representation of the graphical structure. The graphical structure can be represented by the adjacency and incidence matrix of a 157

2 metagraph. Fuzzy metagraph is an emerging technique used in the design of many information processing systems like transaction processing systems, decision support systems, and workflow Systems Conventional rule-based expert systems use human expert knowledge to solve real-world problems that normally would require human intelligence. Expert knowledge is often represented in the form of rules or as data within the computer. Depending upon the problem requirement, these rules and data can be recalled to solve problems. Rule-based expert systems have played an important role in modern intelligent systems and their applications in strategic goal setting, planning, design, scheduling, fault monitoring, diagnosis and so on[14]. A graph-based approach provides a two dimensional language suitable for visualization and a mathematical analysis scheme for the analysis of system structures [13]. Because of their role in reducing computations and improving visualization and problem understanding, graph and net theory have become important computational paradigms for representing and analyzing intelligent systems [13]. In crisp graphs, a number of new constructs that are geared toward set-to-set mappings have been proposed, e.g., directed hypergraphs, higraphs, and metagraphs. In particular, the metagraph has shown its capability of graphic presentation as well as its efficiency of algebraic analysis. Directed hypergraphs possess some similar properties to metagraphs. Both of them can be regarded as a combination of directed graphs and hypergraphs. Metagraphs, however, demonstrate a great potential for mathematical analysis. The rest of the paper is organized as follows. Section 2 gives the related work. Section 3 points out Fuzzy Metagraph technique. Section 4 deals with Fuzzy Metagraph based knowledge representation. Fuzzy Expert System architecture is presented in section 5. In section 6 concludes the paper. 2. RELATED WORKS Deepti Gaur, Aditya Shastri and Ranjit Biswas have proposed a model for metagraph data structure. They have used to store data inside the computer memory either in the form of Adjacency matrix or in Adjacency list so it has been used efficiently [3], they have proposed metagraph based substructure pattern mining technique. they have develop an algorithm which adapts the depth-first search strategy to mine frequent connected sub metagraph efficiently [2]. They have proposed fuzzy metagraph method of clustering to find the similar fuzzy nodes in a fuzzy metagraph. They have used T-norms (Triangular Norms) functions and join two or more T norms to cluster the fuzzy nodes [4]. Pankaj Dashore and Suresh Jain have used rule based system and fuzzy metagraph for real world application (Online Transaction such as banking, E commerce and share market,) to make correct decision. It has been constructed by projection operation of a fuzzy metagraph to provide high level view that reduces the unnecessary details. The projection of fuzzy metagraph is more dominant because there is less number of edges [7, 8, 9, 10 11]. 158

3 S. S. Hashemin has used fuzzy metagraph tool for scheduling and control of fuzzy projects. It uses constrained renewable resource allocation in fuzzy metagraphs. The metagraph edges are based on minimum fuzzy slack time when the available resource is renewable and constrained for scheduling [12]. Zheng-Hua Tan has proposed a Fuzzy Metagraph (FM) based knowledge representation. The FM has been applied to fuzzy rule-based systems for knowledge representation and reasoning. In the format of algebraic representation and FM closure matrix [13]. 3. FUZZY METAGRAPH Basu and Blanning introduced the concept of metagraph [1]. A metagraph S = {X, E} is a graphical representation consisting of two tuples X and E. Here X is its generating set and E is the set of edges defined on generating sets. The generating set X of the metagraph S i.e. the set of elements X = {x1, x2, x3,,xn} represents variables and occurs in the edges of the metagraph. Fig 1. Example of Metagraph Figure 1, shows an example of metagraph. X = {x1, x 2 x 3, x 4,x 5,x 6,x 7 } is the generating set, and E = {e1, e 2, e 3, e 4 } is the set of edges. The edge set can be specified as E = {<{x1, x 2}, {x 4}>, <{x 2, x 3}, {x 5}>, < {x 4, x 5}, {x 6, x 7}>, < {x 5}, {x 7}>}. In-vertex is a function having one argument which can find out the internal vertices from a given set. In-vertex (< {x 4, x 5}, {x 6, x 7}>) = {x 4, x 5}. Out-vertex is another function having one argument which can find out what are the out vertices from the given set. Out-vertex (< {x 4, x 5}, {x 6, x 7}>) = {x 6, x 7}. Two more functions of metagraph are the co-input and Co output functions each have two arguments. Co input function gives he co-input from a given set. Co-input {x 4, < {x 4, x 5 }, {x 6, x 7 }>} = {x 5 },Co-output{x6, < {x 4, x 5},{x 6, x 7}>} = {x 7}.Generally the edges of the metagraph are labeled as e1 = <{x1, x 2}, {x 4}>,e 2 = <{x 2, x 3}, {x 5}>, e 3 = <{x 4, x 5 }, {x 6, x 7 }>,e 4 = < {x 5 }, {x 7 }>. Metagraph is graphical hierarchical structure in which every node is a set having one or more elements. It has all the properties of graphs. In a metagraph, there is set to set mapping in place of node to node as in a conventional graph structure [6]. The concept of a fuzzy graph is the fuzzification of the crisp graphs using fuzzy sets. A fuzzy graph G ~ can be defined as a triple {X, X ~, E ~ }, where X ~ is a fuzzy set on X and E ~ is a fuzzy relation on X X. 159

4 A fuzzy set X on X is completely characterized by its membership function µ:x [0, 1] for each x є X, µ(x) illustrates the truth value of the statement of x belongs to X ~. The fuzzy metagraph is the concept of Fuzzification of the crisp Metagraph using fuzzy generating set. Fuzzy generating set is the node set of all the elements of fuzzy metagraph [13]. Consider a finite set X={x1, x2, x3,, x n }. A fuzzy metagraph is a triple S ~ ={X, X ~, E ~ } in which X ~ is a fuzzy set on X and E ~ is a fuzzy edge set { e ~ m,m=1, 2, 3,.. m}. Each component e ~ in E ~ is characterized by an ordered pair <V ~ m,w ~ m>. In the pairv ~ m subset of X ~ is the in-vertex of e ~ m and W ~ m subset ofw ~ is the out-vertex. Fig 2.Fuzzy Metagraph Often, the membership value of an edge is also called certainty factor (CF) of the edge. For simplicity, assign X ~ i denoting (X i, µ ( X ~ i) ) and e~ k denoting (e k, CF k ). Figure 2 shows fuzzy metagraph whose element set is X = { X ~ 1, X ~ 2,..., X ~ 6 } is known as fuzzy Meta Node and whose edge set consists of: e ~ 1 = <{ X ~ 1, X ~ 2 }, { X ~ 3 }> and e ~ 2 = <{ X ~ 3, X ~ 4 }, { X ~ 5, X ~ 6 }>. The in-vertex and out-vertex of e ~ 1 are { X~ 1, X ~ 2} and { X ~ 3 }. 4. FUZZY METAGRAPH BASED KNOWLEDGE REPRESENTATION Knowledge representation is one of the most important and actively investigated areas in artificial intelligence [13]. Fuzzy production rules (FPRs) among others are widely used in expert systems to represent fuzzy, imprecise, and vague concepts. In this section, we address issues of applying Fuzzy Metagraph to FPRs. In Fuzzy Metagraph-based knowledge representation, each edge represents a rule in which the in-vertex represents the antecedent of the rule and the outvertex represents the consequent. Furthermore, each path a sequence of edges represents a reasoning chain. 4.1 Fuzzy Metagraph-Based Representation of Rules The uncertainty of an elementary rule can be modeled by an FM as shown in Fig.3.a. The figure illustrates the following rule: IF X ~ 1 THEN X ~ 2 (CF1). According to the rule, the truth value of the consequent is the product of the truth value of the antecedent and the CF of the rule. A single node is transformed to a new node after processing or transformation. 160

5 Fig 3.a Fuzzy Metagraph based representation of processing (basic) rule An FPR is called a compound FPR if its antecedent part and/or consequent part AND or OR connectors. Fig. 3 b describes the following rule: IF X ~ 3 AND X ~ 4 THEN X ~ 5 (CF 2 ). There is conjunction operation in this rule. Zadeh proposed to use the operators min = ^ for conjunction, and max = v for union. Two or more kinds of nodes are assembled to a new node. Fig 3.b Fuzzy Metagraph based representation of assembling rule. Figure 3.c describes the following two rules: 1) IF X ~ 6 THEN X ~ 8 (CF 3 ) and 2) IF X ~ 7THEN X ~ 8 (CF 4 ). In this case, the truth value of consequent equals the maximal one among those obtained from different rules. Fig 3.c Fuzzy Metagraph based representation of multi supply source processing rule. When the two rules have the same CF, i.e., CF 3 =CF 4 =CF, the combined rule illustrated by the FM is IF X ~ 6 OR X ~ 7 THEN X ~ 8 (CF). If there are several union antecedents, the overall truth value of the antecedents equals the maximal one. 161

6 Fig 3.d Fuzzy Metagraph based representation of single supply source and processing rule. Fig 3.d illustrates the following rule: IF X ~ 9 THEN X ~ 10 AND X ~ 11 (CF 5 ). In Fig. 4d, two separate basic rules are described. We do not model a rule like: IF X ~ 12 THEN X ~ 13 OR X ~ 14 (CF 6 ) since it is not allowed in a rule base, some kinds of nodes come from different supplier(source), so there are many options available. The graphic representation is of help in understanding the system yet difficult for computers to process [13]. Fig 3.e Fuzzy Metagraph based representation of single supply source and two destination rule. 5. FUZZY EXPERT SYSTEM The world of information is surrounded by uncertainty and imprecision. The human reasoning process can handle inexact, uncertain, and vague concepts in an appropriate manner. Usually, the human thinking, reasoning, and perception process cannot be expressed precisely. These types of experiences can rarely be expressed or measured using statistical or probability theory. Fuzzy logic provides a framework to model uncertainty, the human way of thinking, reasoning, and the perception process. Fuzzy systems were first introduced by Zadeh (1965). A fuzzy expert system is simply an expert system that uses a collection of fuzzy membership functions and rules, instead of Boolean logic, to reason about data (Schneider et al., 1996). Fuzzy expert system consists of Fuzzification, inference system, rule base, Defuzzification units. It has the capability to solve decision making problems for which no exact algorithm exists. Fuzzy expert systems are well to problems that exhibit uncertainty resulting from inexactness, vagueness or subjectivity. Fuzzy expert system architecture used in this system is shown in the figure

7 Fig 4. Basic architecture of Fuzzy Expert System Fuzzification is the process of converting crisp input to fuzzy value. Membership Functions (MFs) are used to convert crisp inputs into fuzzy value. The MF maps each element of input to a membership grade (or membership value) between zero and one [14]. A rule-based system consists of if-then rules, a set of facts, and an interpreter controlling the application of the rules, given the facts. These if-then rule statements are used to formulate the conditional statements that comprise the complete knowledge base. A single if-then rule assumes the form if x is A then y is B and the if-part of the rule x is A is called the antecedent or premise, while the then-part of the rule y is B is called the consequent or conclusion. Fuzzy expert system modeling can be pursued using the following steps. 1. Select relevant input and output variables. Determine the number of linguistic terms associated with each input/output variable. Also, choose the appropriate family of membership functions, fuzzy operators, reasoning mechanism, and so on. 2. Choose a specific type of fuzzy inference system (for example, Mamdani, Takagi Sugeno etc.). In most cases, the inference of the fuzzy rules is carried out using the min and max operators for fuzzy intersection and union. 3. Design a collection of fuzzy if-then rules (knowledge base). To formulate the initial rule base, the input space is divided into multidimensional partitions and then actions are assigned to each of the partitions. In most applications, the partitioning is achieved using one dimensional membership functions using fuzzy if-then rules as illustrated in Figure 5. The consequent parts of the rule represent the actions associated with each partition. It is evident that the MFs and the number of rules are tightly related to the partitioning. 163

8 Fig 5. Example of how the two-dimensional spaces are partitioned. Two triangular membership functions (MFs) for each input variable and four triangular MFs for the output variable (power). Figure 5, shows that how the two-dimensional spaces are partitioned. A simple if-then rule will appear as If input-1 is medium and input 2 is large, then rule R8 is fired. Using the grid partitioning method (Figure 5), four if-then rules were developed. The rule base consisted of nine if-then rules. min and max were used as T-norm. 6. CONCLUSION This paper proposed a Fuzzy Metagraph based knowledge representation of decision support system. Fuzzy metagraph and Fuzzy Expert System are discussed in detail. This method can be used in many real world applications like E- commerce, share market, disease analysis. Future works may concentrate on optimization techniques applied for tuning the input parameters for enhancement of the performance of the system. REFERENCES [1]AmitBasu, RobertW. Blanning(2007), metagraph and their applications. [2] Deepti Gaur, Aditya Shastri(2008), Metagraph-Based Substructure Pattern mining International Conference on Advanced Computer Theory and Engineering, pp

9 [3] Deepti Gaur, AdityaShastri(2008), Metagraph: A new model of Data Structure, IEEE International Conference on Computer Science and Information Technology, pp [4] Deepti Gaur,AdityaShastri(2008), Fuzzy Meta Node Fuzzy Metagraph and its Cluster Analysis,Journal of Computer Science, 4 (11), pp , India. [5] Deepti Gaur,AdityaShastri(2009), Vague Metagraph, International Journal of Computer Theory and Engineering, Vol. 1, No.2, pp ,india. [6] DashoreP(2007), Uncertainty Knowledge Representation Through Fuzzy Metagraph International Journal of computer Application (IJCA), vol 2, pp [7] PankajDashore, Suresh Jain (2009), Fuzzy Rule Based System and metagraph for Risk Management in Electronic banking activities, International Journal of Engineering and Technology, Volume 1 No.1, pp. ( ), India. [8] PankajDashore, Suresh Jain (2010), Fuzzy Metagraph and Rule Based System for Decision Making in Share Market, International Journal of Computer Applications, Volume 6 No.2, pp. ( ), India. [9] PankajDashore, Suresh Jain (2010), Fuzzy Rule Based Expert System to Represent Uncertain Knowledge of E-commerce, International Journal of Computer Theory and Engineering, Vol.2, No.6, pp [10]PankajDashore, Suresh Jain (2011), Fuzzy Metagraph and Hierarchical modeling, International Journal on Computer Science and Engineering, Volume 3 No.1, pp [11]PankajDashore, Suresh Jain, Fuzzy Rule Based metagraph model of Air Quality Index to Suggest outdoor activities, International Journal on Computer Science and Engineering, Volume 2 No.1, ISSN: [12] S. S. Hashemin (2011), Constrained Renewable Resource Allocation in Fuzzy Metagraphs via Min Slack, International Journal of Applied Operational Research, Ardabil, Iran. [13] Zheng-Hua Tan, Senior Member (2006), Fuzzy Metagraph and Its Combination with the Indexing Approach in Rule-Based Systems, IEEE transactions on knowledge and data Engineering, vol. 18, no. 6, China. [14]AjithAbraham(2005), Rule-based Expert Systems,Handbook of Measuring System Design, Oklahoma State University, Stillwater,USA, pp [15] R.W. Blanning,AmitBasu (1994), Metagraphs: A Tool for Modeling DSS, Management Science, vol. 40, pp [16] Vaidehi.V,Monica.S(2008), A Prediction System Based on Fuzzy Logic Proceedings of The World Congress on Engineering and Computer Science, October 22-24, San Francisco, USA. 165

10 ABOUT THE AUTHORS Mr.A.Thirunavukarasu completed his B.E Degree in Computer Science and Engineering from Coimbatore Institute of Technology, Coimbatore in the year 2006 and M.E. Degree in Computer Science and Engineering from Anna University of Technology, Coimbatore in the year Currently he is pursuing PhD degree from Anna University of Technology, Coimbatore. He is working as a Visiting Faculty, Department of Computer Science and Engineering in Anna University of Technology, Madurai-Ramanathapuram Campus. He is having more than 3 years of teaching experience. He has published technical papers in national /international conferences. His areas of specialization include Data Structures and Algorithms, Compilers, Theory of computation, Data mining and Database Security, and Metagraph. Dr. S. Uma Maheswari received her B.E Degree in Electronics and Communication Engineering from Government College of Technology, Coimbatore in the year 1985 and M.E (Applied Electronics) from Bharathiar University in She received her Ph.D degree in the area of Biometrics from Bharathiar University, Coimbatore in the year She is Associate Professor of Electronics and Communication Engineering department in Coimbatore Institute of Technology. She is having more than 20 years of teaching experience. She has published technical papers in national /international conferences/ journals. Her special fields of interest are Digital Image Processing and Digital Signal Processing. She is a Member of IE (India), Life Member in Indian Society for Technical Education (India), Life Member in Systems Society of India, and Life Member in Council of Engineers (India). 166