MAGDM PROBLEMS WITH CORRELATION COEFFICIENT OF TRIANGULAR FUZZY IFS
John Robinson P
Bishop Heber College, Tiruchirappalli-620017,
Tamilnadu, India.
Henry Amirtharaj E.C.
Bishop Heber College, Tiruchirappalli-620017,
Tamilnadu, India.
ABSTRACT
Correlation coefficient of Intuitionistic Fuzzy Set (IFS), Interval valued IFS, Triangular IFS and Trapezoidal IFS are already present in the literature. This paper proposes the correlation coefficient for Triangular Fuzzy Intuitionistic Fuzzy set (TrFIFS). The method on uncertain Multiple Attribute Group Decision Making (MAGDM) problems based on aggregating intuitionistic fuzzy information is investigated for TrFIFSs. The Triangular Fuzzy Intuitionistic Fuzzy Ordered Weighted Averaging (TrFIFOWA) operator is proposed for TrFIFSs and the Triangular Fuzzy Intuitionistic Fuzzy Ordered Weighted Geometric (TrFIFOWG) operator utilized for decision making models where expert weights are completely unknown. Based on these operators and the correlation coefficient defined for the TrFIFSs, new decision making model is proposed with numerical illustration. Comparisons are made with existing ranking methods for validity.
Key words: MAGDM, Triangular Fuzzy Intuitionistic Fuzzy Sets, OWA operators, Correlation of IFS.
INTRODUCTION
Intuitionistic Fuzzy Sets (IFSs) proposed by Atanassov, (1986; 1994; 1999) is a generalization of the concept of fuzzy sets. Atanassov Gargov, (1989) expanded the IFSs, using interval value to express membership and non-membership function of IFSs. Liu Yuan, (2007) introduced the concept of fuzzy number IFSs as a further generalization of IFSs. Among the works done in IFSs, Atanassov, (1986; 1994), Atanassov & Gargov, (1989), Szmidt Kacrzyk, (2000; 2002; 2003), Gerstenkorn Manko, (1991), can be mentioned. With best of our knowledge, Burillo et al., (1994) proposed the definition of intuitionistic fuzzy number (IFN) and studied the perturbations of IFN and the first properties of the correlation between these numbers. Many researchers have applied the IFS theory to the field of decision making. Recently some researches (Boren et.al., 2009; Chen et al., 2010; Liu & Yuan, 2007; Wei, 2010; Wang, 2008; Zhang & Liu, 2010) showed great interest in the fuzzy number IFSs and applied it to the field of decision making. Based on the arithmetic aggregation operators, Xu, (2006), Xu & Chen, (2007) and Wang, (2008) developed some new geometric aggregation operators and Intuitionistic Fuzzy Ordered Weighted Averaging (IFOWA) operator. Chen Tan, (1994) presented some products for dealing with multi attribute decision making (MADM) problems based on vague sets. Szmidt Kacrzyk, (2002) proposed some solution concepts in group decision making with intuitionistic (individual and social) fuzzy preference relations. Szmidt Kacrzyk, (2003) investigated the consensus-reaching process in group decision making based on individual intuitionistic fuzzy preference relations. Herrera et al., (1999) developed an aggregation process for combining numerical, interval valued and linguistic information, and then proposed different extensions of this process to deal with contexts in which can appear information such as IFSs or multi-granular linguistic information. Xu Yager, (2006) developed some geometric aggregation operators for MADM problems. Li, (2005) investigated MADM problems with intuitionistic fuzzy information and constructed several linear programming models to generate optimal weights for attributes.
Multi-attribute group decision making (MAGDM) problems are of importance in most kinds of fields such as engineering, economics and management. It is obvious that much knowledge in the real world is fuzzy rather than precise. Imprecision comes from a variety of sources such as unquantifiable information (Li & Nan, 2011). In many situations decision makers have imprecise/vague information about alternatives with respect to attributes. It is well known that the conventional decision making analysis using different techniques and tools has been found to be inadequate to handle uncertainty of fuzzy data. To overcome this problem, the concept of fuzzy approach has been used in the evaluation of decision making systems. For a long period of time, efforts have been made in designing various decision making systems suitable for the arising day-to-day problems. MAGDM problems are wide spread in real life decision making situations and the problem is to find a desirable solution from a finite number of feasible alternatives assessed on multiple attributes, both quantitative and qualitative (Power, 2013). In order to choose a desirable solution, the decision maker often provides his/her preference information which takes the form of numerical values, such as exact values, interval number values and fuzzy numbers. However, under many conditions, numerical values are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information may be stated in intuitionistic fuzzy information, especially in triangular fuzzy intuitionistic fuzzy information. Hence, MAGDM problems under intuitionistic fuzzy or triangular fuzzy intuitionistic fuzzy environment is an interesting area of study for researchers in the recent days.
In the process of MAGDM problems with triangular fuzzy intuitionistic fuzzy information, sometimes, the attribute values take the form of Triangular Fuzzy Intuitionistic Fuzzy Number (TFIFN). The information about attribute weights may sometimes be known or partially known or sometimes completely unknown. This may be because of the lack of knowledge of the data or the expert’s limited expertise about the problem domain. The MAGDM model requires that the selection be made among the decision alternatives described by their attributes. Solving a MAGDM problem having a predetermined, limited number of decision alternatives involves sorting and ranking, and can be viewed as alternative methods for combining the information in a problem’s decision matrix together with additional information from the decision maker to determine a final ranking or selection from among the alternatives. Besides the information contained in the decision matrix, all but the simplest MAGDM techniques require additional information from the decision matrix to arrive at a final ranking or selection.
It is well known that the conventional correlation analysis using probabilities and statistics has been found to be inadequate to handle uncertainty of failure data and modeling. The method to measure the correlation between two variables involving fuzziness is a challenge to classical statistical theory. Bustince Burillo, (1995), Gernstenkorn Manko, (1991), Hong Hwang, (1995), and Yu (1993) define a different correlation formula to measure the interrelation of intuitionistic fuzzy sets. In their definition, the correlation coefficient lies between 0 and 1, differing from the conventional range of [-1,1]. The work of Wang Li, (1999) suffers the same problem of the correlation lying between 0 and 1, in studying the correlation coefficient of interval valued fuzzy sets. Chiang Lin, (1999) took random samples from fuzzy sets, treating the membership grades to be crisp observations, to calculate the correlation lying in the interval [-1,1], where the sense of fuzziness is lost. Park et al., (2009) proposed the correlation coefficient of interval valued intuitionistic fuzzy sets. Robinson & Amirtharaj, (2011a; 2011b; 2012a; 2012b) proposed correlation coefficient for vague sets, interval vague sets, Triangular and Trapezoidal IFSs and a revised correlation coefficient for Triangular and Trapezoidal IFSs using graded mean integration representation.
In this paper, a novel method of correlation coefficient of Triangular Fuzzy Intuitionistic Fuzzy Sets (TrFIFS) is proposed and developed by taking into account the membership, non-membership and the hesitation degrees of TrFIFSs. The OWA operators for TrFIFSs is proposed for MAGDM problems. Aiming at the fuzzy number intuitionistic fuzzy information aggregating problems, this paper utilizes the Triangular Fuzzy Intuitionistic Fuzzy Weighted Geometric (TrFIFWG) operator and the Triangular Fuzzy Intuitionistic Fuzzy Ordered Weighted Geometric (TrFIFOWG) operator proposed by Wang, (2008). Based on these operators, an approach is suggested to solve uncertain multiple attribute group decision making problems, where the attribute values are triangular fuzzy intuitionistic fuzzy numbers and weights of decision-makers are completely unknown. A new MAGDM algorithm is developed to solve the MAGDM problems in which the correlation coefficient of TrFIFSs is used for ranking alternatives. A distance function extended from the Hamming Distance function is also proposed for TrFIFSs in ranking alternatives for MAGDM problems. A numerical illustration is provided to demonstrate the developed approach and a comparison is made with an existing method (Chen et al., 2010).
ARITHMETIC OPERATIONS FOR TrFIFSs
Definition 1: Triangular Fuzzy Number (TrFN)
is called a triangular fuzzy number, if the membership function is expressed as:
(1)
Where
Definition 2: Triangular Fuzzy Intuitionistic Fuzzy Number (TrFIFN)
Let X be a non-empty set. Then is called a Triangular Fuzzy Intuitionistic Fuzzy Number (TrFIFN) if and are triangular fuzzy numbers, which can express the membership degree and the non-membership degree of x in X, and fulfill An intuitionistic fuzzy number expressed on the basis of triangular fuzzy number is called as triangular fuzzy intuitionistic fuzzy number.
The hesitancy degree of the TrFIFN is given as follows:
If are intuitionistic fuzzy numbers, then the operation rules of the intuitionistic fuzzy numbers are as follows:
, (2)
, (3)
, (4)
. (5)
Suppose,
are two TrFIFNs, then according to the above operation rules of intuitionistic fuzzy numbers, and the operation rules of triangular fuzzy numbers, the operation rules of TrFIFNs are as follows:
(6)
(7)
(8)
(9)
For the above operation rules, the following are true:
AGGREGATION OPERATORS IN DECISION MAKING
Different types of aggregation operators are found in literature for aggregating the information (Herrera & Herrera-Viedma, 2000; Li, 2008; Li, et al., 2009; Li & Nan, 2011). A very common aggregation method is the Ordered Weighted Averaging (OWA) operator introduced by Yager, (1988), which provides a parameterized family of aggregation operators that include - as special cases - the maximum, the minimum and the average criteria. Since its appearance, the OWA operator has been used in varied applications in decision making problems. Karayiannis, (2000) and Yager, (2004) suggested a generalization of the OWA operator by using generalized means, known as the generalized OWA (GOWA) operator. With this generalization, it is possible to include the special cases found in the OWA operator like the maximum and the minimum, and the special cases found in the generalized mean such as the geometric and the harmonic mean.
When using the OWA operator, it is assumed that the available information is exact numbers or singletons. However, this may not be the real situation found in the decision making problem. Sometimes, available information is vague or imprecise and thus it is not possible to analyze it with exact numbers. Then, it is necessary to use another approach that can assess the uncertainty such as the use of Fuzzy Numbers (FNs). With the use of FNs, the best and worst possible scenario can be analyzed with the possibility that the internal values of the fuzzy interval will occur. The main characteristic of this approach is that it uses uncertain information in the aggregation represented by FNs. The FOWA operator has been studied by different authors like Merigo & Casanovas, (2008; 2009) and Yager, (2004). Recently, Merigo & Gil-Lafuente, (2009a) suggested a new model that unifies the OWA operator and the weighted average (WA) in the same formulation. The main advantage of this new model against the previous ones (Xu & Da, 2003) is that it considers the degree of importance of each concept in aggregation. Hence, it is easy to deal with situations where the WA or the OWA are very relevant. This new approach is known as the Ordered Weighted Average – Weighted Average (OWAWA) operator. Note that more recently Merigo & Casanovas, (2009) have suggested a new model that uses FNs in the decision analysis, calling it the Fuzzy OWAWA (FOWAWA) operator.
Xu, (2007b; 2007d) defined some new intuitionistic preference relations, such as the consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and studied their properties. Xu, (2007c) investigated the group decision making problems where all information provided by the decision makers is expressed as intuitionistic fuzzy decision matrices where each of the elements is characterized by intuitionistic fuzzy number, and the information about attribute weights is partially known, and may be constructed by various forms. A method based on the ideal solution was proposed by Xu, (2007a) after investigating the intuitionistic fuzzy MADM with the information about attribute weights incompletely known or completely unknown.
OWA Operator and Generalized OWA Operator
The OWA operator was introduced by Yager, (1988) and it provides a parameterized family of aggregation operators between maximum and minimum, reflecting the fact that the operator is commutative, monotonic, bounded and idempotent. Different families of OWA operators can be used by choosing a different manifestation of the weighting vector (Merigo & Gil-Lafuente, 2009a; Merigo & Casanovas, 2008; 2009; Xu, 2005; Yager, 1988; 1993; 2007). The FOWA operator is an extension of the OWA operator for uncertain situations where the available information can be assessed with FNs. The FOWA operator provides a parameterized family of aggregation operators that include the fuzzy maximum, the fuzzy minimum and the fuzzy average criteria, among others. The Generalized OWA (GOWA) operator was introduced by Karayiannis, (2000) and Yager, (2004). It generalizes a wide range of aggregation operators that includes the OWA operator with its particular cases, the Ordered Weighted Geometric (OWG) operator (Chiclana et al., 2004; Herrera et al., 2003; Xu & Da, 2003), the Ordered Weighted Harmonic Averaging (OWHA) operator and the Ordered Weighted Quadratic Averaging (OWQA) operator (Yager, 2004).
DIFFERENT CLASSES OF OPERATORS IN MAGDM PROBLEMS
Ordered Weighted Averaging (OWA) Operators
The OWA operator provides a parameterized family of aggregation operators which are used in many applications. The definition of the OWA operator was introduced by Yager, (1988).
Definition : An OWA operator of dimension n is a mapping that has an associated weighting vector of dimension n having the properties, and such that (10)