
Research Article


An Intuitionistic Fuzzy AHP Based on Synthesis of Eigenvectors and its Application 

Hai Wang,
Gang Qian
and
Xiangqian Feng



ABSTRACT

With regard to multi Criteria Decision Making (MCDM) problems, performances of the Analytical Hierarchy Process (AHP) are prominent. Fuzzy AHP, an extension of AHP, serves as a grateful approach due to its outstanding advantage when dealing with uncertainties. Based on advantages of Intuitionistic Fuzzy Sets (IFSs) in expressing information of preferences, this study presents an intuitionistic fuzzy AHP (IFAHP) approach. The proposed IFAHP synthesizes eigenvectors of Intuitionistic Fuzzy Comparison Matrix (IFCM) in which all the information for decision are represented by Intuitionistic Fuzzy Values (IFVs). The IFAHP approach enables to handle MCDM problems without loss of information or defuzzification and represent arbitrary hesitation in interval [0, 1]. Firstly, Intuitionistic Fuzzy (IF) matrix and IFCM associated with its consistency and satisfactory consistency are defined after some relative basal knowledge are introduced. Secondly, the eigenvector and eigenvalue of IFCM is defined and a linear program model is presented to obtain it as the priority of relative criteria. Furthermore, methods for comparisons of IFVs are proposed in order to rank alternatives utilizing eigenvectors. And then, a integrate procedure of IFAHP involving comparison and rating is presented and illustrated by two applied examples cited from literatures. An involved decision support system can be setup according to the procedure. Comparing with some existing methods, the proposed approach gives both rational global priorities and robust final decision.





Received: June 12, 2011;
Accepted: July 22, 2011;
Published: September 19, 2011


INTRODUCTION
The Analytical Hierarchy Process (AHP) approach which was developed by Saaty
(1980), Saaty (1977) possesses distinct advantage
of dealing with subjective information of Decision Makers (DMs). As one of the
most outstanding Multi Criteria Decision Making (MCDM) approach (Saaty
et al., 2007; Saaty, 2008) or a weight estimation
technique (Vaidya and Kumar, 2006; Mahdavi
et al., 2008a), AHP can be applied in many areas such as selection
(Parsakhoo and Lotfalian, 2009; Mohammaditabar
and Teimoury, 2008), evaluation (Huang et al.,
2011; Wu et al., 2010; Abdullah
et al., 2009), planning and development (MaskaniJifroudi
et al., 2009), decision making (Mahdavi et
al., 2008b), forecasting and so on. Pairwise comparison makes it simple
to express DM’s preferences and meanwhile the consistent ratio insures
the validity of judgments. Interestingly, the pairwise comparison methods are
based on crisp real number (Saaty, 2006) while DMs’
assessments in pairwise comparison always include uncertainty in reality so
that DMs’ may sometimes feel more confident to provide fuzzy judgment than
crisp comparisons (Wang et al., 2008).
In order to deal with its inability in handling the uncertain and imprecise
decisionmaking problems, recent years, scholars extended the real comparison
matrix to fuzzy comparison matrix and then fuzzyAHP was proposed on the basis
of the concepts of the fuzzy set theory (Zadeh, 1965).
An example of application of fuzzy logic can be seen by Binwahlan
et al. (2009). Although, Saaty and Tran (2007)
argued the invalidity of using fuzzy number to improve the outcome from judgments,
the convenience of express uncertainty of both vagueness and ignorance is apparent.
Existing versions of fuzzy AHP usually focus on the comparison matrices and
resultant priorities.
Atanassov (1986) and Atanassov (1999)
extended the concept of Zadeh’s fuzzy sets (Zadeh, 1965)
and introduced Intuitionistic Fuzzy Sets (IFSs), whose prominent characteristic
is that it assigns to each element a membership degree and a nonmembership
degree. So it gives a powerful tool to deal with uncertainty in real applications
especially when to express a pairwise comparison. Amer
et al. (2010) utilized IFSs to analyze the reliability of a large
system. In present study, therefore, IFS is introduced to pairwise comparison
matrix and refer to it as Intuitionistic Fuzzy Comparison Matrix (IFCM) or Intuitionistic
Fuzzy Preference Relation (IFPR) (Xu, 2007; Qian
and Feng, 2008). And then a new approach named IFAHP is proposed, in which
the eigenvector of IFCM is introduced to represent the priority of the compared
elements with respect to a criterion in the upper level of the hierarchy. The
outcome of a hierarchy, in other words, the priority of the alternatives with
respect to the object in the top level can be derived by synthesizing or multiplying
all the intuitionistic fuzzy matrices composed by the eigenvectors in the same
level orderly. Besides, its applications in decision support are also discussed
for prospective users or DMs.
Thus, the aim of this study was to develop a novel fuzzy AHP approach as a
solution of MCDM problems in intuitionistic fuzzy setting. Preliminaries such
as definition, product, eigenvectors and consistency of IFCMs are presented
hereinafter. The effectiveness of proposed methodology is requisite to clarify
as well, comparing to existing methods. That will be illustrated by two realistic
examples.
RECENT ADVANCES OF FUZZY AHP
Over the last twenty years several authors commented on shortcomings of pairwise
judgments and comparison scale of traditional AHP. Spontaneously, improved approaches
were worked out to focus on the denotative form of pairwise judgment matrix
and its resultant priority. To utilize fuzzy logic conveniently, the mutually
complement comparison matrix was used in fuzzy AHP (Kang
and Lee, 2007). Other more complex fuzzy theories were used and the comparison
matrix was extended. Intervalvalue fuzzy comparison matrix was used in (Chamodrakas
et al., 2010; Wang and Chin, 2006; Wang
and Chen, 2008). Triangular fuzzy comparison matrix was introduced and used
by Amiri (2010), Cakir (2008),
Chen and Wang (2010), Kaya and Kahraman
(2010) Sen and Cinar (2010) and Tang
(2009). And trapezoidal fuzzy comparison matrix emerged by Wang
and Chin (2006), Wang and Chin (2008), Fu
et al. (2008) and Huang et al. (2008).
Qian and Feng (2008) developed the concept of IFCM.
Meanwhile the methods of deriving the priority of a comparison matrix are mainly
listed as follow. The lambdamax method or fuzzy eigenvector method emerged
by Wang and Chin (2006), Chang et
al. (2009) Chang et al. (2008), Duran
and Aguilo (2008), Nepal et al. (2010), Huang
et al. (2008) and Csutora and Buckley (2001).
The geometric/arithmetic mean method can be found by Wang
and Chin (2008), Amiri (2010), Che
et al. (2010) Chen et al. (2008),
Chen and Wang (2010) and Kahraman
et al. (2009). And the extent analysis method was used by Bozbura
and Beskese (2007), Ertugrul and Karakasoglu (2009),
Heo et al. (2010), Kreng
and Wu (2007) and Secme et al. (2009) (which
is criticized by Wang et al. (2008)). Wang
et al. (2006) and Kang and Lee (2007) introduced
the least squares method and the entropy weight method, respectively. Further,
fuzzy preference programming method was used by Qian and
Feng (2008), Chamodrakas et al. (2010),
Cakir and Canbolat (2008) and Wang
and Chin (2008).
In the aspect of theory and technique of fuzzy AHP, it is focused on utilizing
fuzzy theory to overcome disadvantages of traditional AHP and new issues along
with it. Wang and Chin (2006) extend the lambdamax
method and proposed an eigenvector method to generate normalized interval, triangular
and trapezoidal fuzzy priority of comparison matrix. Situations where the eigenvector
method is inapplicable are also analyzed. A modified fuzzy logarithmic least
squares method which is formulated as a constrained nonlinear optimization model,
is suggested by Wang et al. (2006). Wang
et al. (2008) argued the shortcoming of the extent analysis method,
pointed out that untrue weight and wrong decision making may take place by the
extent analysis method. Furthermore, Wang and Chin (2008)
proposes a sound yet simple priority method for fuzzy AHP which utilizes a linear
goal programming model to derive normalized fuzzy weights for fuzzy pairwise
comparison matrices. Cakir (2008) focused on the issue
of rank reversal and suggested a fuzzy preference programming methodology to
prevent the order of preference. Wang and Chen (2008)
presented fuzzy linguistic preference relations method which is an easy and
practical way to provide a mechanism for improving consistency in fuzzy AHP
method.
Simultaneously, application of fuzzy AHP with different improvement of the
details is paid more attention. Kang and Lee (2007)
constructed an analytical approach under a fuzzy subjective judgment environment
to deal with uncertainty and to generate performance ranking of different priority
mixes for semiconductor fabrication. Kreng and Wu (2007)
evaluated three knowledge portal system development tools taking use of fuzzy
AHP in group decision setting. Similarly, Fu et al.
(2008) assessed the impact of market freedom on the adoption of thirdparty
electronic marketplace, while Chamodrakas et al.
(2010) discussed supplier selection in the same background using satisfying.
Kahraman et al. (2009), Huang
et al. (2008) and Gungor et al. (2009)
also talked over the multi criteria problem of selection from alternatives.
Duran and Aguilo (2008) proposed a method to computeraided
investment evaluation and justification of an advanced manufacturing system.
Nepal et al. (2010) utilized fuzzy AHP to prioritize
customer satisfaction attributes for automotive product development, considered
a broad range of strategic and tactical factors for determining the weights
which were then incorporated into target planning. Heo et
al. (2010) established the criteria and factors and assessed for renewable
energy dissemination program evaluation the importance of each factor using
fuzzy AHP. While Lin (2010) evaluated course website
quality with a fuzzy AHP approach to determine the relative weights linking
the criteria between high and low online learning experience groups.
Furthermore, scholars studied to combine fuzzy AHP with other decision making
methods based on advantages of both. The most familiar hybrid form is methods
aggregating fuzzy AHP and TOPSIS (Amiri, 2010; Dagdeviren
et al., 2009; Ertugrul and Karakasoglu, 2009;
On’t and Soner, 2008; Secme
et al., 2009; Torfi et al., 2010).
The framework of the aggregation could be come down to a pattern. Fuzzy AHP
is used to determine the relative weights of evaluation criteria and then TOPSIS
is used to rank the alternatives. Stirn (2006) integrated
fuzzy AHP with discrete dynamic programming approach to evaluate the conflicting
objectives to determine the optimal forest management decisions. Tang
(2009) budgeted allocation for an aerospace company with fuzzy AHP and artificial
neural network, respectively and then compared the two outcomes to help decisionmaking.
Chang et al. (2008) and Chen
and Wang (2010) proposed hybrid approaches combining fuzzy AHP with modified
Delphi approach. The former select unstable slicing machine to control wafer
slicing quality. While the later develop global business intelligence for information
service firms, respectively. The same idea for selection can be found by Hsu
et al. (2010). Besides, balanced scorecard method was associated
with fuzzy AHP to support decision making (Cebeci, 2009;
Lee et al., 2008), as well as data envelopment
analysis (Che et al., 2010), multidimensional
scaling analysis (Chen et al., 2008) and DempsterShafer
theory of evidence (Hua et al., 2008). Moreover,
Gumus (2009) presented a method which congregates fuzzy
AHP, TOPSIS and Delphi approach to evaluate hazardous waste transportation firms.
The aforementioned literatures in aspect of both theories and applications
expand the practicable scope of fuzzy AHP to a large extent. However, there
are still facets to be improved. Firstly, fuzzy number including interval, triangular
and trapezoidal fuzzy number consider merely degree of membership to a fuzzy
concept. For example, a triangular fuzzy number (7, 8, 9) can express a concept
“about 8”, while the degree of membership is default. By contraries,
theory of IFS consider degree of nonmembership besides membership with respect
to a fuzzy concept for each judgment, hence preference of judgments depicted
by IFS could include more information from DMs which may lead to more precise
decision making. That’s why we choose IFS in present study. In the proposed
approach, two problems caused by introducing IFS to comparison matrices are
consistency and priority of IFPR. Secondly, defuzzification in the process of
fuzzy AHP (Chang et al., 2009; Che
et al., 2010; Ertugrul and Karakasoglu, 2009;
Heo et al., 2010; Lin, 2010;
Fu et al., 2008) would cause loss of information
or even result in an untrue decision making. IFVs are used to express information
without defuzzification in the entire process of the proposed approach. Furthermore,
it is rational that if preference information takes the form of fuzzy number,
priorities of criteria or alternatives should take the form of fuzzy number
as well. Yet most of the existing fuzzy AHP approaches take the form of real
numbers to represent priorities. We propose an approach in which all information
for decision making take the form of IFVs. Lastly, fundamentals or illustrated
examples in some literatures mentioned above do spell out vividly the proposed
ways and means, whereas the lack of the intact and operable methodology may
limit the application for the perspective users.
PRELIMINARIES
IFS and its algorithm: Intuitionistic Fuzzy Sets (IFS) introduced by
Atanassov have been proven to be highly useful to deal with uncertainty and
vagueness which was characterized by a membership function and a nonmembership
function. Recently, many authors have applied the intuitionistic fuzzy set theory
to the field of decision making. Chen and Tan (1994)
and Hong and Choi (2000) presented some techniques for
handling multicriteria fuzzy decisionmaking problems based on vague set theory
Bustince and Burillo (1996) pointed out that vague sets
were IFS). They provided some functions to measure the degree of suitability
of each alternative with respect to a set of criteria presented by vague values.
Lin et al. (2007) proposed new method that allowed
the decisionmaker to assign the degrees of membership and nonmembership of
the criteria to the fuzzy concept “importance”. The concept of IFS
is as follows.
• 
Definition 1: (Atanassov, 1986).
Let X be an ordinary finite nonempty set. An IFS in X is an expression
A given by: 
where, u_{A}: X→[0, 1], v_{A}: X→[0, 1] with the condition: 0≤u_{A} (x)+v_{A} (x)≤1, for all x in X. The numbers u_{A} (x) and c_{A} (x) denote, respectively, the degree of the membership and the degree of the nonmembership of the element x in the set A. Denotation π_{A} (x) = 1u_{A} (x)v_{A} (x) represents the degree of hesitation or intuitionistic index or nondeterminacy of x to A. Therefore, for ordinary fuzzy sets the degree of hesitation π_{A} (x) = 0.
The ordered pair α_{A} (x) = (u_{α} (x), v_{α}
(x)) is referred to as an IFV (Xu, 2007), where, u_{α}
(x), v_{α} (x)ε[0, 1], π_{α} (x) and u_{α}
(x), v_{α} (x)≤1. Associated with the degree of hesitation,
an IFV can also be equivalently denoted by α (x) = (u_{α}
(x), v_{α} (x), π_{α} (x)) where u_{α}
(x), v_{α} (x), π_{α} (x)ε[0, 1] and u_{α}
(x), v_{α} (x), π_{α} (x) = 1. In the rest of
this study, IFV α = (u, v, π) is abbreviated as π = (u, v) when
no misunderstanding raises. Two useful operations on IFVs are as follows:
• 
Definition 2: (Xu, 2007). Let
a = (ú_{a}, v_{a}) and b = (u_{b}, v_{b})
be two IFVs, then: 
It is apparent that the results are also IFVs and both of the operations are commutative and associative as the following properties. If a = (u_{a}, u_{a}), b = u_{b}u_{b} and c = u_{c}u_{c} are three IFVs, then:
• 
Definition 3: (Qian and Feng, 2008).
X = (x_{1},…, _{n})^{T} with x_{1}
= (u_{i}, v_{i}) is said to be an IF vector if for all I
= 1,…, n satisfies: 
Especially, if an IF vector satisfies π_{i} = 1u_{i}v_{i} = 0 for all I = 1,…, n, then it is called certain IF vector. • 
Definition 4: (Qian and Feng, 2008).
If an IF vector x = (x_{1},…, x_{n})^{T} =
((u_{1}, v_{1}, π_{1}),…, (u_{n},
v_{n}, π_{n}))^{T} satisfies: 
then it is normalized; otherwise it is not normalized.
A normalized IF vector can represent the priority or relative weights of n
alternatives or criteria. In other words, if an IF vector x = (x_{1},…,
x_{n})^{T} is normalized, then there exist a vector
such that
If π_{j} = 0 for all j = 1,…, n, then the normalized certain
IF vector is a weight vector satisfying
IF matrix and its synthesis: The concept of matrix is extended to intuitionistic fuzzy setting for the purpose of representing decision information with intuitionistic fuzzy forms. • 
Definition 5: An IF matrix is a matrix denoted by M
= (M_{IJ})_{mxn} = ((u_{IJ}, v_{IJ}))_{mxn},
M_{IJ} in which is IFV, where, I = 1,…, m, j = 1,.…, n. 
Notice that the notion of IF matrix is just an extension of traditional real matrix. In present study, it is employed to represent priorities in a level of a hierarchical structure with respect to all elements in the above level. Thus columns of an IF matrix should be an IF vector so that it can represent priorities. On the other hand, a special IF matrix could be used to reflect preference relations as follow: • 
Definition 6: An square IF matrix M_{IJ} is
called an Intuitionistic Fuzzy Comparison Matrix (IFCM) or Intuitionistic
Fuzzy Preference Relation (IFPR) if ∀I, j = 1,..., n, such that: 
This kind of comparison matrix is called complementary matrix. If M_{IJ} = (0.6, 0.35), for example, it could be judged that 0.6 and 0.35 represent the certainty degree of which criterion I is preferred than criterion j and the certainty degree of that criterion j is important than criterion I, respectively, while 0.05 is interpreted as the uncertainty degree of which criterion I is preferred than criterion j, according to DMs. Especially, if an IFCM M satisfies π_{IJ} = 0 for all I, j = 1,…, n, then M is reduced to two fuzzy comparison matrix M_{1} = (u_{IJ}) and M_{2} = (v_{IJ}) which satisfy (M_{1})^{T} = M_{2}. Note that M_{IJ} and M_{JI} have the same hesitation. After defining basic concepts, an arithmetical algorithm is presented. • 
Definition 7: Let M, N be two IF matrices denoted by: 
a matrix
is entitled the product or synthesis of M multiply by N, denote by C = MqN, where: The arithmetical operation is selected here instead of fuzzy operations i.e., max and min as which leads to a loss of information obviously. It can be easily proved that the outcome of multiply is an IF matrix as well. • 
Theorem 1: If M = (M_{IJ})_{mxn}, N
= (N_{IJ})_{mxn} are two IF matrices, matrix C = (C_{IJ})_{mxn}
with: 
then Cis an IF matrix. The multiply of IF matrices will be used to both synthesize decision information in each level of a hierarchical structure (represented by an IF matrix) and derive priorities from IFCMs.
Consistency of IFCM: Consistency is essential in human thinking because
it enables ordering the world according to dominance. It is a necessary condition
for thinking about the world in a scientific way (Saaty, 1980).
Once judgments among a set of criteria or alternatives with respect to a certain
criteria or object are given, it is essential to make sure those judgment are
rational logically. In the following, the notion of consistent IFCM is presented.
• 
Definition 8: (Qian and Feng, 2008).
An IFCM M = (M_{IJ})_{mxn} = ((u_{IJ}, v_{IJ}))_{mxn}
is consistent, if there exists a certain IF vector x = (x_{1},...,
x_{n})^{T} with x_{1} = (u_{i}, v_{i})
such that: 
Based on Definition 4 and Definition 8, an approach is presented to check if an IFCM is consistent conveniently. • 
Theorem 2: M = (M_{IJ})_{mxn} = ((u_{IJ},
v_{IJ}))_{mxn} is a consistent IFCM, if and only if it satisfies
the following inequality constraints: 
Proof: If M is a consistent IFCM, then there exists a certain IF vector x = (x_{1},…, x_{n})^{T} with x_{1} = (u_{i}, v_{i}) such that: which means the vector x = (x_{1},…, x_{n})^{T} satisfies: Adding (16) by (17) leads to the following implied indirect inequalities: Since (18) holds for any k = 1,…, n, it follows that: Conversely, if (14) holds for ∀I, j, k, then:
By the condition of consistency of interval complementary comparison, there
exists
, where:
such that:
Suppose that
so u_{i}u_{u}+1≥2u_{ij} and v_{i}v_{u}+1≥2v_{ij}.
By Definition 8, M is a consistent IFCM.
The above Theorem 2 can be used to test whether or not an IFCM is consistent without solving any mathematical programming mode. It only requires simple algebraic operations. Because an IFCM is reciprocal in nature, only its lower or upper triangular need to be checked.
Rationally, it is not necessary to keep consistent everywhere. By contraries,
inconsistency is usually unavoidable, yet it is not useless indeed. Saaty
(1980) argued that inconsistency must be precisely one order of magnitude
less important than consistency, or simply 10% of the total concern with consistent
measurement. If it were larger it would disrupt consistent measurement and if
it were smaller it would make insignificant contribution to change in measurement.
We infer that satisfactory consistency is acceptable.
A programming model is proposed here to verify whether an IFCM is of satisfactory
consistency or not. First the Geometric Consistency Index (GCI) proposed by
Aguaron and Mereno (2003) which is similar to Consistency
Ratio (CR) proposed by Saaty, is introduced here.
• 
Definition 9: Aguaron and Mereno
(2003). Given a pairwise comparison matrix A = (a_{ij})_{nxn}
and the vector of priorities, ω = (ω_{1},..., ω_{n})^{T},
obtained by the Row Geometric Mean Method, the GCI can be defined as: 
where, e_{ij} = a_{ij} ω_{j}/ω_{i} is the error obtained when the ratio ω_{j}/ω_{i} is approximated by a_{ij} and I j = 1, 2,…, n
In addition, approximated thresholds for the GCI are proposed in (Aguaron
and Mereno, 2003). Qian et al. (2009), afterwards,
switch those thresholds to be available in the case of complementary comparison
matrix, namely, Complementary Geometric Consistency Index (CGCI) as shown in
Table 1 and the CGCI of a real complementary comparison matrix
could be defined as follows.
• 
Definition 10: Given a complementary pairwise comparison
matrix A = (a_{ij})_{nxn} and the vector of priorities,
ω = (ω_{1},…, ω_{n})^{T}, the
CGCI can be defined as: 
where, I j = 1, 2,…, n • 
Definition 11: An IFCM M = (M_{ij})_{nxn}
is of satisfactory consistency, if there exists a complementary pairwise
comparison matrix A = (a_{ij})_{nxn} with CGCI_{A}≤δ,
where δ is the approximated threshold in Table 1
and u_{ij}≤a_{ij}≤1v_{ij} 
Based on Definition 8 and 11, the relationship between consistency and satisfactory consistency of an IFCM is concluded as following theorem. • 
Theorem 3: If an IFCM M = (M_{ij})_{nxn}
is consistent, then it is of satisfactory consistency 
Further, Definition 11 gives a straightforward way to check if an IFCM is of satisfactory consistency. • 
Theorem 4: Let M = (M_{ij})_{nxn} =
((u_{ij}, v_{ij}))_{nxn} be an IFCM, M is of satisfactory
consistency if and only if there exist a vector of priorities ω = (ω_{1},…,
ω_{n})^{T} such that: 
where, u_{ij}≤a_{ij}≤1v_{ij} and δ is the approximated threshold in Table 1.
Then a programming model is established to verify the existence of vector ω:
Then the rule to make the judgment whether an IFCM is of satisfactory consistency could be come down to the following theorem. • 
Theorem 5: An IFCM M = (M_{ij})_{nxn}
is of satisfactory consistency if CGCI_{A}*≤δ, where, CGCI*
represent the optimal solution of programming model (24) 
Obviously, if δ = 0, then satisfactory consistency reduces to consistency. Thus consistency can be seen as a special case of satisfactory consistency. In addition, the vector ω = (ω_{1},…, ω_{n})^{T} derived by (24) is not appropriate to the priority reflected by an IFCM directly although the existence of ω ensures the satisfactory consistency of the IFCM. Because the optimal solution of (24) discards hesitation and uncertainty of the original IFCM, and which may lead to a loss of information. Similar to relative literatures, we used right eigenvector of IFCM to represent the priority instead.
Eigenvector of IFCM: If an IFCM is consistent or at least satisfactory
consistency, then it implies priority of DM in a level of hierarchical structure
with respect to a criterion of upper level. Motivated by the classical AHP (Saaty,
1980), the idea of right eigenvector is used to depict it. As hesitations
and uncertainties exist in an IFCM, they should be retained by the vector of
priority. Thus the eigenvector of an IFCM is defined by IF vector. First, the
definition is present as follows:
• 
Definition 12: Assume M is an nxn IFCM. If there exists
a nx1 IF vector: 
and an IFV λ = (u_{λ}, u_{λ}) which satisfying: then λ is named the eigenvalue of M, and x is the eigenvector of M correspondingly. Each IF matrix M can be split into two nonnegative matrices:
referred to as membership matrix and:
referred to as nonmembership matrix, and denote as M = ((u_{M}, v_{M})) for convenience. It is transparent that u_{M} and v_{M} are real number matrices but no longer complement matrices. • 
Theorem 6: Let M be an IFCM. If u_{λ}
v_{λ}, are the eigenvalue of u_{M}, v_{M} respectively,
u_{x} and v_{x} are the corresponding eigenvector of u_{M},
v_{M} respectively, i.e., u_{M}, u_{x} and v_{M},
v_{x}, then λ = u_{λ} v_{λ} is the
eigenvalue of M and the corresponding eigenvector of M can be denoted as
x = ku_{x}, 1lv_{x}, k and l in which are
real number such that 0≤ku_{x}≤lv_{x}≤1.
That is: 
Proof: Since 0≤ku_{x}≤lv_{x}≤1, therefore
we have 0≤ku_{x}≤lv_{x}≤1, then x = (ku_{x},
1lv_{x}) is IF vector. On the one hand:
On the other hand:
So, (26) is proofed. By Definition 12, the theorem is thus proofed completely.
Assume that are
the normalized principal eigenvectors of u_{M}, v_{M}, respectively,
which satisfying:
Then eigenvector of M can be denoted as
according to theorem 6. Substituting it to (8), we get:
In order to determine k and l, the following Linear Programming (LP)
model is constructed:
The meaning of maximizing δ is to make each weight interval as wide as possible, while the implication of maximizing k is to avoid k = 0. Note that not all IF comparison matrices can generate normalized eigenvector weights by LP model (28). • 
Example 1: Let M be the following IF comparison matrix: 
First, the consistency is checked by Theorem 2 and as a result, the matrix
is consistent. Then, we solve the LP models (28) after
and
are derived and get IF priority vector x = (x_{1},…, x_{n})^{T},
where:
The vector x is normalized according to Definition 4.
Comparisons of IFVs: In this study, all kinds of information are depicted
by IFVs. For example, priorities or weights are expressed by IF vectors in which
each entry is an IFV. IFVs need to be compared and further ranked in order to
rank alternatives or criteria. Following the idea of Wang
et al. (2005), the degree of preference of two IFVs based on probabilities
is defined as:
• 
Definition 13: Let a = (u_{a}, v_{a})
and b = u_{b}, v_{b} be two IFVs, the degree of preference
of a over b (a>b) is derived by: 
where, π_{a} = 1u_{a}v_{a}, π_{b} = 1u_{b}v_{b}. The degree of preference of b>a can be defined in the same way. Then two IFVs can be ranked by the following rules. • 
Definition 14: If P (a>b)>P (b>a), then a
is said to be superior to b to the degree of P (a>b), denoted by: 
if P (a>b)>P (b>a) = 0.5, then a is indifferent to b, denoted by b∼a; else if P (a>b)>P (b>a), then a is inferior to b to the degree of P (b>a), denoted by
• 
Example 2: Let a = (0.65, 0.3), b = (0.55, 0.25), then
P (a>b) = 0.6 and P (b>a) = 0.4, so according to Definition 14, a
is superior to b to the degree of 0.6, denoted by: 
For the purpose of the necessity of comparison of a serial IFVs of an IF vector hereinabove of this paper, a method based on Definitions 13 and 14 is introduced to achieve it. Let x_{i} = (u_{i}, v_{i}) π_{i} = 1u_{i}v_{i}, I = 1,…,n, where x_{i} is IFV defined by Definition 1. A ranking process is outlined below: • 
Step 1: Calculate the matrix of the degrees of preference: 
Where: Symbol ‘’ in P means that the value does not have to be calculated. • 
Step 2: Calculate the matrix of preference relation: 
Where:
• 
Step 3: Calculate the sum of the elements of
each row in the above matrix of preference relation and generate the final
aggregated ranking R = (r_{1},..., r_{n}). X_{i}
is ranked higher than x_{j} if and only if the sum for the ith row
is larger than that for the jth row 
It is worthy to note that this ranking method presents not only the order of n IFVs but also the degree of possibility of one IFV is superior to another. • 
Example 3: Ranking the order of normalized IF vector
derived by Example 1 
When utilizing the process above, the resultant preference order is: and the matrix of the degrees of preference P is: Notice that the probabilities of x_{4} preferring to x_{i} (I = 1, 2, 3, 5) are scarcely more than 0.5, so it isn’t very sure about the judgment x_{4} is superior to x_{i} (I = 1, 2, 3, 5) in fact. Therefore, the preference order isn’t sustained by merely probabilities, a modified process is suggested in this study to ranking the order of entries in an IF vector considering both probabilities and hesitations of them. The modified ranking process is as follows:
• 
Step 1: The same as Step 1 in the above process 
• 
Step 2: Calculate the sum of the elements of each row in the above
matrix P, denote by: 
and each entry in which divided by its hesitation π_{i}, respectively. The more superior x_{i}, the bigger value Sp_{i}/π_{i}. Denote the resultant index vector as I = (I_{1},…, I_{n})^{T}, where, I_{i} = Sp_{i}/π_{i} with I = 1,…, n. • 
Example 4: Ranking the order of normalized IF vector
derived by Example 1 by the modified ranking process 
The hesitations of x_{i} is presented in Example 1 and outcome of the modified process is (106.8181, 14.9740, 24.9558, 9.6312, 11.8220). Thus, the priority of IFCM in Example 1 is x_{5}>x_{1}>x_{3}>x_{2}>x_{4}. The modified ranking process considers probabilities and hesitations at the same time, it is clear that outcomes derived by which is more reasonable. However, it is nonsensical if π_{i} = 0. Note that this modified process is suitable when P (x_{i}>x_{j}) verges on 0.5 and hesitations are slightly bigger than 0, for other cases, the original ranking process is suitable. According to the fact that hesitations will magnify gradually along with the process of synthesis of IF matrices, the modified process is more adaptive to the proposed method in this study. IFAHP METHODOLOGY
A hierarchy is a powerful manner of classification used to order information
gained either from experience or from our own thinking. Thus, the complexity
of the world around us could be understood according to the order and distribution
of influences which make certain outcomes happen (Saaty
and Shih, 2009). Due to the confinement of the ability of expressing a judgment
accurately and the advantage of IFS in considering both degree of membership
and degree of nonmembership at the same time, IFS is introduced to the traditional
AHP.
Structuring hierarchies: Saaty and Shih (2009)
defined hierarchical structures by the notions of ordered sets and finite partially
ordered sets and suggested the following procedure for structuring hierarchies.
• 
Step 1: Define the goal or focus of the decision problem
at the top level. For instance, it could be a mission statement of an organization 
• 
Step 2: Break down the purpose into some supportive elements in
the first level below the goal. The elements on the first level should be
comparable and homogeneous or close in their possession of a common attribute.
For instance, a system can be broken down physically into subsystems, units,
subunits, components, etc 
• 
Step 3: Insert actors into a suitable level. The function of the
actors is similar to a filter that screens out some influences at the upper
levels. It might be more than one level of actors depending on the requirements 
• 
Step 4: Establish the bottom level for choice. The bottom level
of the hierarchy could be alternatives, actions, consequences, scenarios
or policies to be chosen 
• 
Step 5: Examine the hierarchic levels forward and backward. One
usually needs to check and revise the elements and even the levels, backward
and forward iteratively to ensure the consistency of the structure 
• 
Step 6: Check the validation of structures. Two guidelines to check
the structure are: 1) is the structure logical? and 2) is the structure
complete? 
The procedure of IFAHP: In present study, a panoramic methodology and
an operable procedure for the general and curbstone DMs is mainly focused on.
For the purpose, the proposed IFAHP approach is depicted step by step as follows.
• 
Step 1: Structuring hierarchies. DMs could structure
hierarchies following advices in section 4.1. Suppose there are n levels
in the hierarchical structure, in which the top level is named as the first
level and the last level or the alternatives level as the nth level, there
are n_{i} elements or criteria in the ith level, where the elements
are denoted by .
Note that n_{1} = 1 
• 
Step 2: Comparison. In order to give expression to a comparison
precisely, it is proposed in this paper to employ IFS to express all the
pairwise comparison in the hierarchies. Assume that comparison in level
I with respect to the jth criteria
in level I1, where I = 1,…, n, j = 1,…, n_{i1} is formed
to an IFCM .
Note that each entry (u, v) in
consists of two parts i.e., the certainty degree u of which a criterion
is preferred than another and the certainty degree v of that the latter
is important than the former, respectively, by two separate judgments such
that u+v≤1. In practice, DMs may not sure about translate judgments into
IFVs straightway. Therefore, two methods to accomplish successful and effectual
transformation of judgments are proposed when comparing
with with
respect to .
First, if a decision group is formed and x percents of the group prefer
while y percents of the group prefer ,
then the corresponding IFV of the judgments can be set as (u, v, π)
= (x/100, y/100, 1x/100y/100) and synthesize weights of members of the
group if needed. Second, if there is only one DM, he or she could estimate
the lower bound x and upper bound y of the degree or probability of
preferred to ,
then the corresponding IFV of the judgments can be derived as (u, v) = (x,
1y) 
• 
Step 3: Calculating eigenvectors. For each ,
if it is of satisfactory consistency, its weighting vector or eigenvectors
can be derived by the LP model (28) if it is of satisfactory consistent
according to Theorem 5. The eigenvectors of comparison in ith level with
respect to each ,
j = 1,..., n_{i1}, form a IF matrix as: 
• 
Step 4: Synthesis. The priority of elements
in the last level or alternatives with respect to general object is, immediately,
derived by x = M^{(n)}⊗…⊗ M^{(2)}. Obviously, x is an
IF vector and, usually, has been normalized (see example 5 and example 6) 
• 
Step 5: Ranking. Alternatives could be ranked according to entries
which are IFVS in x by methods in section 3.5 
Actually, the above process can deal with the case with incomplete information. If DMs have absolutely no idea about with one or some judgments of comparisons, just set the corresponding locations of the IFCM by IFV (0, 0) with 1 as its hesitation. The flow chart correspond to the above procedure is shown in Fig. 1.
Rating: There are two ways to create priorities of alternatives i.e.,
rank from comparisons which give relative measurements and from ratings which
give absolute measurements. When rating alternatives, they must be assumed to
be independent and rank should be preserved. While alternatives are usually
dependent with each other and rank may not always be preserved when comparing
(Saaty, 2006; Saaty and Sagir, 2009).
Due to the excellent performance of rating in aspect of rank preservation, rating
process in IF environments is addressed separately. The last level of a hierarchy
is always minute criteria rather than alternatives when rating, therefore priorities
of these criteria with respect to the goal of the hierarchy is represented by
x derived by Step 5 in Section 4.2.
Only one alternative is considered here with the hypothetic condition of that
e_{1},…, e_{n} are n criteria, entries in the normalized
IF vector x = (x_{1},…, x_{n})^{T} = ((u_{1},
v_{1}),…, (u_{n}, v_{n}))^{T} represent
the priority of all criteria. If the score of performance of the alternative
with respect to the ith criteria is denoted by an IFV ,
where I = 1,…, n, Then the alternative’s score is computed by:
Using the absolute or ratings method of the AHP, categories or standards are
usually established for each criterion.

Fig. 1: 
The flow chart of IFAHP 
Then alternatives are rated one at a time by selecting the appropriate category
under each criterion rather than compared against other alternatives. The standards
are prioritized for each criterion by making pairwise comparisons. The alternative’s
score is then computed by weighting the priority of the selected category by
the priority of the criterion and summing for all the criteria.
If there is only one alternative to be considered in this kind of rating problem and in which denotations of criteria and their priorities are the same as above. Suppose c_{i} categories are established for the criteria e_{i}, the utility of the category c_{ik} for criteria e_{i} is:
where, k = 1,…, c_{i} and I = 1,…, n. The alternative’s score is then calculated by:
EXAMPLES OF APPLICATION To illuminate the process and the validity of the proposed IFAHP in present study convectively, two examples are cited from literatures and data are used straightforwardly without any modification. Thus, it is easy to compare rank of alternatives with different versions of AHP. In order to give facilities for application of proposed method, the original data are necessary to transform into IFVs in both examples. Two examples: • 
Example 5: Supplier selection is a critical and demanding
task for companies that participate in electronic marketplaces to find suppliers
and to execute electronically their transactions. Chamodrakas
et al. (2010) aimed to suggest a fresh approach for decision
support enabling effective supplier selection processes in electronic marketplaces.
An evaluation method with two stages is consequently introduced: initial
screening of the suppliers and final supplier evaluation. Initial screening
is conducted through the enforcement of hard constraints on the selection
criteria. And final supplier evaluation is implemented through the application
of a modified variant of the fuzzy preference programming method. The approach
was demonstrated with the example of a hypothetical metal manufacturing
company that finds and selects suppliers in the environment of an electronic
marketplace. Here this example is figured out again with IFAHP. Its hierarchy
is presented in Fig. 2. Three main criteria are quality,
cost, delivery which form the second level of the hierarchy. The third level
of the hierarchy occupies the subcriteria defining the three criteria of
the second level. Only one subcriterion related to cost has been chosen,
namely cost reduction, two subcriteria related to quality, namely rejection
rate from quality control and supplier remedy of quality problems and two
subcriteria related to delivery, namely compliance with due date and compliance
with quantity. After initial screening, Supplier 3, Supplier 5 and Supplier
7 were moved out from original set of alternatives 
Since, the original data were presented by interval numbers with 19 scales which is suggested by Saaty, it is necessary to transform them to IFVs with a value range of 01. First, for an original interval comparison matrix:
it could be transformed into an interval comparison matrix:

Fig. 2: 
Hierarchical structure of example 5 

Fig. 3: 
Hierarchy of catering firm selection problem of example 6 
Xu (2002). Notice that ranges of each entry of T are
[0.1, 0.9]. Then these entries are transformed it to [0, 1] uniformly and still
denote as T. And then a corresponding IFCM is derived by
using the relationship between interval numbers and IFVs (Xu,
2007).
Follow the Steps 35 in Section 4.2, the IF priority and the index vector of 5 Suppliers derived by Matlab are, respectively, x = (x_{1},…, x_{n})^{T} with x_{1} = (0.0187, 0.1792), x_{2} = (0.0196, 0.1848), x_{3} = (0.0234, 0.1936), x_{4} = (0.0159, 0.1793), x_{5} = (0.0159, 0.1793). Apparently, x is normalized according to Definition 4 and the index vector: Hence, Supplier 4 is the best alternative followed by Supplier 2, 1, 6 and 8, while Supplier 6 (and 8) is superior that Supplier 1 in original literature. • 
Example 6: A big Turkish textile company wishes to
make a contract with one catering firm. Alternative Turkish catering firms
are Durusu, Mertol and Afiyetle. The goal is to select the best among the
three alternatives. The criteria to be considered are Hygiene (H), Quality
of meal (QM) and Quality of Service (QS) which involve 11 subcriteria,
i.e., Hygiene of Meal (HM), Hygiene of Service Personnel (HSP), Hygiene
of Service Vehicles (HSV), Variety of Meal (VM), Complementary meals in
a day (CoM), Calorie of meal (CaM), Taste of Meal (TM), Behaviour of Service
Personnel (BSP), Service Time (ST), Communication on Phone (CP) and Problem
Solving (PS) ability. Figure 3 shows the hierarchical
structure of the problem. A decisionmaking group consisting of the customers
of the catering firms and five experts is responsible for making comparisons
and constructing fuzzy comparison matrices. The information of judgments
is firstly shown by Kahraman et al. (2004)
and then by Wang et al. (2008) with a slight
change. The problem is computed with both methods and data are adopted from
the latter at first 
A prepared work needs to be completed due to the form of original data are presented by triangular fuzzy number. Let a = (l, m, r) be a triangular fuzzy number and α = (u, v) is a corresponding IFV calculated by u = l and v = 1r. It seems as if some information for decision making is discarded during the transformation. But this example aims at comparing the rank of alternatives with distinct methods. In the framework of proposed method, only two pieces of data are needed: membership function and nonmembership function. The effectiveness of IFAHP will be compared with other methods using only part of the given information. Then all original triangular fuzzy judgments could be switched to IFCMs so as to apply the proposed IFAHP method. Follow the steps in Section 4.2, IF priorities, namely, eigenvectors of three alternatives derived by Matlab are: x = (x_{1},…, x_{n})^{T} with x_{1} = (0.0596, 0.4331), x_{2} = (0.0546, 0.39), x_{3} = (0.0668, 0.4854), which is already normalized according to Definition 4 and the matrix of the degrees of preference P is: Then final index of the IF priority vector x is derived by the modified ranking process mentioned in Section 3.5 is:
All appearances, Afiyetle is the best alternatives followed by Durusu and Mertol
which is absolutely the same as in Wang et al. (2008).
Whereas, there is a rank reversal between Durusu and Mertol in Kahraman
et al. (2004). Moreover, nearly the opposite result is offered by
Wang et al. (2008) if the Extend Analysis Method
(EAM) proposed by Kahraman et al. (2004) is used.
Meanwhile, the proposed IFAHP is operated again with the original data presented
in Kahraman et al. (2004) and the following result
is obtained:
This accorded with the result of Kahraman et al.
(2004) nicely.
DISCUSSION
Example 5 is presented to illustrate the validity and capability of IFAHP
to deal with MCDM problems, while Example 6 is presented to compare three outcomes
derived by different approaches including IFAHP, the extend analysis method
and modified fuzzy logarithmic least squares method (MFLLSM) (Wang
et al., 2006). Comparing Example 5 with Chamodrakas
et al. (2010), it can be concluded that the proposed IFAHP could
offer a satisfying assessment of a MCDM problem and furthermore provide reliable
and believable decision making support for DMs. For the convenience of comparison
three methods, the local weight of three alternatives with respect to four subcriteria
of quality of service using data of Wang et al. (2008)
by different methods are shown in Table 14
for example. The following differences and indifference are summarized from
these Tables.
Table 1: 
Approximated thresholds for the CGCI 

Table 2: 
Local weights of three catering firms with respect to QS
by EAM 

Table 3: 
Local weights of three catering firms with respect to QS
by MFLLSM using IFVs 

Table 4: 
Local weights of three catering firms with respect to QS
by IFAHP using IFVs 

• 
All judgments expressed by triangular fuzzy numbers or IFVs
yet priorities derived by EAM are real numbers which may loss some information
of uncertainty and fuzziness. No defuzzification process exists in both
MFLLSM and IFAHP, so results by which are more reliable 
• 
EAM may assign a zero weight to a decision criterion or alternative,
leading to the criterion or alternative not to be considered in decision
analysis as can be seen in Table 2. Approximated thresholds
for the CGCI are presented in Table 1 
• 
Comparing Table 3 with Table 4, it
is found that hesitations of MFLLSM are smaller than that of IFAHP on average
which mainly because there is a most possible value in the middle of a triangular
fuzzy number 
• 
MFLLSM and IFAHP both adopt optimization model to derive priorities,
the former takes use of anonlinear optimization model yet the latter utilize
a linear optimization model (28) along with smaller degree of complexity
of calculation 
• 
MFLLSM obtains global fuzzy weights by solving two LP models and an equation
for each alternative, while IFAHP obtains global fuzzy weights by product
or synthesis of matrices constituting by priorities expressed by eigenvectors
of IFCMs 
Furthermore, the advantages and limitations of the proposed IFAHP are concluded as follows:
• 
Each comparison in IFAHP composes with two separate judgments
with the point of view of degree of membership and nonmembership which
allows arbitrary hesitation in interval [0, 1] in nature. So the method
could be applicable to different actual cases with different hesitations
for diverse comparisons. Whereas some existing methods define fixed hesitation
with a certain linguistic variable which may be only applicable for a certain
case 
• 
Prospective users and DMs could adopt the proposed method step by step
to deal with MCDM problems because of the detailed procedures presented
in this study. This enhances the scope of its application to some extent 
• 
In the entire process of IFAHP, all information such as comparisons,
eigenvalues and eigenvectors or priorities are expressed by IFVs which guarantees
no loss of information during procedures of calculation 
• 
However, the modified ranking process for IF vectors proposed in this
study is suitable for IFAHP merely but not fit the case when a hesitation
is convergent to zero. The original ranking process based only on the degree
of probability is available for the special case; nevertheless its validity
is not very satisfactory as shown in Example 3 
CONCLUSION
In view of the prominent advantage of IFS, a new IF AHP based on eigenvectors of IFCMs and their synthesis is proposed to deal with fuzzy uncertainties in MCDM problems. For the purpose of practical applications of the proposed methodology, some stepbystep procedures are presented in this study to facilitate idiographic operation of the potential users or DMs. At last two numerical examples are calculated to illustrate validity and correctness of IFAHP and a comparison with some existing approaches is further given. However, there are several further works worthy to study. First, the validity of IFAHP is confirmed by examples, yet whether the problem of rank reversal when alternatives are ranked from comparisons is essential for extensive application. Second, a decision support system or simply a toolbox based on Matlab is needful to be developed for prospective users so that they need only to construct a hierarchical model and give judgments of preferences and leave the task of calculating to the software. ACKNOWLEDGMENTS The authors acknowledge support of the National Natural Science Fund of China (71071161) and University Science Research Project of Jiangsu Province (11KJD630001).

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