INTRODUCTION
Multiattribute Decision Making (MADM) is concerned with the elucidation of
the levels of preference of decision alternatives, through judgments made over
a number of criteria. Many complex MADM problems are characterized with both
quantitative and qualitativeof preference of decision alternatives, through
judgments made over a number of criteria. Many complex MADM problems are characterized
with both quantitative and qualitativeattributes. For instance, the design
evaluation of an engineering product may require the simultaneous consideration
of several attributes such as cost, quality, safety, reliability, maintainability
and environmental impact; in selection of its suppliers, an organization needs
to take into account such attributes as quality, technical capability, supply
chain management, financial soundness, environmental and so on. Most of such
attributes are qualitative and could only be properly assessed using human judgments
which are subjective in nature and are inevitably associated with uncertainties
due to the human being's inability to provide complete judgment, or the lack
of information, or the vagueness of the meanings about attributes and their
assessments. For decades, many MADM methods have been developed, such as Analytical
Hierarchy Process (AHP) (Saaty, 1980), TOPSIS (Lan,
2009), ELECTRE, PROMETHEE, LINMAP (Jiang and Fan, 2005)
etc. for certain MADA and fuzzy multiattribute decision making under uncertainty
(Wang et al., 2011). AHP has been widely used
in many areas such as accounting (Webber et al.,
1997), assessment (Wu et al., 2010), programming
(Yang and Kuo, 2003), research and development management
(Liu and Tsai, 2007) and information management (Liu
and Shih, 2005). AHP can be applied under the precondition that the decision
maker can make pairwise comparison between decision alternatives. This prerequisite,
however, may not be satisfied in practice. For a practical MADM problem, information
about decision alternatives may be incomplete because of time pressure, lack
of data, intangible of some attributes (Kim and Ahn, 1997,
1999), limitation of attention, or limitations on information
processing capabilities (Kahneman et al., 1982),
etc. The DempsterShafer (DS) theory of evidence (Dempster,
1967; Shafer, 1976) models have both quantitative and
qualitative attributes with an appropriate framework. The power of the DS theory
in handling uncertainties has found wide applications in many areas such as
expert systems (Beynon et al., 2001), diagnosis
and reasoning (Jones et al., 2002), pattern classification
(Denoeux and Zouhal, 2001), information fusion (Telmoudi
and Chakhar, 2004), sort (Xu, 2012). In recent years,
there have been several attempts to use the DS theory of evidence for MADA (Yang
et al., 2006; Liu et al., 2011; Zhang
et al., 2012a). In many cases, the DS theory has been used as an
alternative approach to Bayes decision theory (Beynon et
al., 2000; Yager et al., 1994) incorporated
the DS theory with the AHP process. The method can not only model both quantitative
and qualitative attributes but also take advantage of the AHP to lower the number
of alternatives that fit the limited number of opinions given so far, with only
a few opinions stated.
In some uncertain decision problems with qualitative attributes, however, it
may be difficult to define assessment grades as independent crisp sets. It would
be more natural to define assessment grades using subjective and vague linguistic
terms which may overlap in their meanings. While intuitionistic fuzzy set has
been proven to be highly useful in dealing with uncertainty and vagueness, accordingly,
intuitionistic fuzzy set is a very suitable tool to be used to describe the
imprecise uncertain decision information (Zhang et al.,
2012b; Amer et al., 2010). Nevertheless,
the current DSAHP approach does not take into account vagueness or fuzzy uncertainty.
As such, there is a clear need to combine the DSAHP theory for handling both
types of uncertainties. The purpose of this paper is to investigate how to incorporate
the intuitionistic fuzzy information with the DSAHP method.
PRELIMINARY
The existing DSAHP approach: On the base of DempsterShafer theory
of evidence, the method first identifies all possible focal elements from the
decision matrix and then it calculates the basic probability assignment of each
focal element and the belief interval of each decision alternative. The AHP
approach is used to describe the MADM problem. Using AHP approach, the MADM
is decomposed into three levels. The first level is the MADM problem with incomplete
information, the second level is the decision attributes of the MADM problem
and the third level is focal elements identified from the decision matrix. Next,
we will describe the main steps of the DSAHP approach (Gong,
2007; Hua et al., 2008).
First, we identify focal elements from an incomplete decision matrix. Let Θ = {a_{1},…, a_{N}} be a collectively exhaustive and mutually exclusive set of decision alternatives, called the frame of discernment.
Given a decision matrix V = f (a_{i}, C_{i}) where,
f (a_{i}, C_{i}) is the evaluation of the decision alternative
a_{i} (i = 1, 2,…, N) under the jth attribute C_{j} (j
= 1, 2,…, M). The decision matrix V implies the body of evidence of the
MADM and a focal element can be defined from the decision matrix as follows:
then a_{i} and a_{k} belong to the same focal element.
According to above definition, we can obtain the focal elements of each attribute. Then, the hierarchical structure of DSAHP can be constructed. And similar to AHP, the weights of importance of decision attributes ω_{j}, j = 1, 2,…, M in DSAHP can be determined through pairwise comparison.
Second, construct belief interval of each decision alternative. Denote by A^{j}_{k} (j = 1, 2,…, M; k = 1, 2,…, t; t<2^{N}) the set that consists of all focal elements under decision attribute C_{j} and if a_{i}∈A^{j}_{k}, we can view ω_{j}f (a_{i}, C_{j}) as the decision maker’s preference on the focal element A^{j}_{k}, where, ω_{j} is the importance weight of decision attribute C_{j}. Denote by P(A^{i}_{k}) the decision maker’s preference on the focal element A^{j}_{k}, then p(A^{j}_{k}) = ω_{j}f(a_{i}, C_{j}). Because Θ is the frame of discernment which consists of all decision alternatives, we let the p(Θ). Then we can define the basic probability assignment (BPA) of each focal element as follows:
Applying the operator of combination in the DS theory, the BPA of each focal element considering all decision attributes can be obtained. Suppose A^{j1}_{k} and A^{j2}_{k} are two focal elements under decision attribute C_{j1} and C_{j2}, respectively, j_{1}, j_{2}∈{1, 2,…, M}, j_{1} ≠ j_{2}. Denote the intersection of A^{j1}_{k} and A^{j2}_{k} as E, then according to DS rule of combination, the BPA of E is defines as follows:
After obtaining BPA of each focal element considering all decision attributes, we can define its belief measure (Bel) and plausibility measure (Pls).
Denote by Bel({a_{i}}) and Pls ({a_{i}}) the exact support to a decision alternative a_{i} (i = 1, 2,…, N) and the possible support to a_{i}, respectively. The two values can be got as follows:
Using Bel({a_{i}}) and Pls ({a_{i}}), we obtain the belief
interval [Bel({a_{i}}), Pls ({a_{i}})] for all decision alternatives
of the MADA problem with incomplete information. Then in order to obtain the
preference relations among all decision alternatives, we need a mechanism to
generate the rank of decision alternatives based in their belief intervals.
There’re many available methods to rank the alternatives. Here we cite
the method described as follows.
We define the degree of preference of a_{i} over a_{k}, denoted by P(a_{i}>a_{k}) as follows:
with P(a_{i}>a_{k})∈[0, 1].
By applying the above definition, we define the preference relation between decision alternative as follows.
• 
Decision alternative a_{i} is said to be superior
to a_{k} (denoted by a_{i }a_{k}
) if P(a_{i}>a_{k})>0.5 
• 
Decision alternative a_{i} is said to be inferior to a_{k}
(denoted by a_{i} a_{k}
) if P(a_{i}>a_{k})<0.5 
• 
Decision alternative a_{i} is said to be indifferent to a_{k}
(denoted by a_{i}~a_{k} ) if P(a_{i}>a_{k})
= 0.5 
The above formulas consist of the main process of the DSAHP approach. After these steps, we can obtain the rank of all the decision alternatives.
The intuitionistic fuzzy set (IFS): An intuitionistic fuzzy set A on
a universe U is defined as an object of the following form (Hua
et al., 2008):
where, the functions μ_{A}: U→ [0, 1] and v_{A}: U→[0, 1] define the degree of membership and the degree of nonmembership of the element u∈U in A, respectively and for ∀u∈U, 0≤μ_{A} (u)+v_{A} (u)≤1.
For convenience, we call u = (u, v) an intuitionistic fuzzy number.
THE COMBINATION OF THE DSAHP METHOD WITH INTUITIONISTIC FUZZY INFORMATION
Just as the existing DSAHP approach, we first need to obtain a decision matrix..
Here we assume there are n evaluation grades to which all alternatives can be
assessed, denoted by H = {h_{1}, h_{2},…, h_{n}}
are mutually exclusive and collectively exhaustive. And we also need to define
the utility of each grade as u (h_{r}) with u (h_{r+1})>u(h_{r})
if it is assumed that the grade h_{r+1} is preferred to h_{r}.
And the assessment value is provided with intuitionistic fuzzy information.
Then we have that: if an alternative a_{i} is assessed on an attribute
C_{j} to a grade h_{r} with an intuitionistic fuzzy data, we
denote this by:
where, μ_{r,j} (a_{i}) is the degree of membership of an alternative a_{i} on the attribute C_{j} associated with the grade h_{r} and v_{r,j}(a_{i}) is the degree of nonmembership. For convenience, we use (μ_{r,j}, v_{r,j}) to represent the assessment (μ_{r,j} (a_{i}), v_{r,j}(a_{i})) in the following paragraphs. After N alternatives are all assessed on M attributes, we obtain the following decision matrix: D = (S(C_{j}(a_{i}))).
First, to every alternative assessed on each attribute, combine the assessment given to the different grades. With the utility of each grade, a comprehensive assessment value of alternative a_{i} on the attribute C_{j} can be got as follows:
The operational rules of intuitionistic fuzzy set were defined by Atanassov
(1986). Obviously, u(a_{i}, C_{j}) is still an intuitionistic
fuzzy data. As the process of assessment is often accompanied with incomplete
information, so we need to revise the results obtained using the expected utilities.
We denote
as the weight assigned to alternative a_{i} on the attribute C_{j},
where:
And δ^{ij}_{r} represents the counts of grades an alternative is assigned to δ^{ij}_{r} is about the following formula:
Using the weight above, we can revise u(a_{i}, C_{j}). We denote
(μ_{ij}, v_{ij}) =
u(a_{i}, C_{j}). From now onwards, we can finally obtain the
transformed decision matrix V = (μ_{i,j}, v_{i,j}) NxM.
Second, identify the focal element of each attribute. On the base of the transformed decision matrix V = (μ_{i,j}, v_{i,j}) NxM, we can define the following method. The decision alternatives which have the same assessment value or the same degree of influence on an attribute belong to the same focal element of the attribute. A focal element can be defined as follows:
Definition 1: For ∀ a_{i}, a_{k}∈Θ and a_{i} ≠ a_{k}, if μ_{ij} = μ_{kj} then a_{i} and a_{k} belong to the same focal element.
The next step, we will define the interval probability masses of each focal element. Denote by A^{j}_{k} (j = 1, 2,…, M; k = 1, 2,…, t; t<2^{N})the set that consists of all focal elements under decision attribute C_{j}. We can obtain the interval probability masses of A^{j}_{k}.
Definition 2: Considering the weight of C_{j}, for ∀a_{i}∈Θ,
∀A^{j}_{k}∈2^{Θ} if a_{i}∈A^{j}_{k},
then we define the interval probability masses as:
We denote it as:
As to the whole set Θ = {a_{1},…, a_{N}}, the followings are used to obtain its interval probability masses:
where, t<2^{N}.
Third, to obtain the interval probability masses of each focal element considering all decision attributes, we need to define the combination rule.
Definition 3: Let m_{j1} (A^{j2}_{k}), m_{j2} (A^{j1}_{k}) be two interval belief structure with interval probability masses m_{j1} (A^{j1}_{k})¯≤m_{j1} (A^{j1}_{k})≤m_{j1} (A^{j1}_{k})^{+} k = 1, 2,…, and m_{j2} (A^{j2}_{k})¯≤m_{j2} (A^{j2}_{k})≤m_{j2} (A^{j2}_{k})^{+}, l = 1, 2,…, t.
We define E as the intersection of two focal elements A^{j1}_{k} and A^{j2}_{k}. Their combination, denoted by m_{j1}⊕m_{j2} is also an interval belief structure defined by:
where, (m_{ji}⊕m_{j2})¯ (E) and (m_{ji}⊕m_{j2})^{+} (E) are, respectively the minimum and the maximum of the following optimization model:
When there are three or more attributes to be combined, we can use the combination rule recursively to obtain the final combined results. It should be pointed out that when using the combination rule recursively, we should do follow the next processes. We can first do the combination between two attributes and then combine the intersections of the two with the third attribute. This process will be stopped when all decision attributes are considered. These nonlinear programming models can be solved by some mathematical software such as LINGO. Similar models can also be constructed for m_{j}(Θ).
The last step is to construct belief interval of each decision alternative.
Definition 4: The belief measure (Bel) and the plausibility measure (Pls) of alternatives a_{j} are defined, respectively by:
Where:
Finally, we obtain the belief interval of all the decision alternatives which can be used to rank the alternatives. As for the ranking problem, there are a lot of methods provided by many scholars. Here we use the following formula:
If P(a_{i}>a_{k})>0.5 then a_{i }
a_{k}; else if P(a_{i}>a_{k})<0.5; else a_{i}
a_{k}.
NUMERICAL EXAMPLE
Here, we give an example on how to select a good project when the investment
is carried on. The problem is described as follows: there are four available
projects which consist of the decision alternative set Θ = {a_{1},
a_{2}, a_{3}, a_{4}} and there are four attributes related
to the investment problem: marketing opportunity (C_{1}), support degree
of policy (C_{2}), economy (C_{3}), technical capability (C_{4})
and the weight of importance of decision attributes are ω = {ω_{1},
ω_{2}, ω_{3}, ω_{4}}.
Table 1: 
Decision matrix of the project selection problem 

Table 2: 
Transformed decision matrix 

All alternatives can be assessed to five grades H = {h_{1}, h_{2},
h_{3}, h_{4}, h_{5}}, here we assume the expected utility
of each grades are:
u(h_{1}) = 0.2, u(h_{2}) = 0.4, u(h_{3})
= 0.6, u(h_{4}) = 0.8, u(h_{5}) = 1 
First of all, we should calculate the weight of importance. By comparing with each other and using AHP algorithm, we can obtain ω = (0.517, 0.168, 0.077 and 0.238). After we assessed every alternative on each attribute, we obtain the decision matrix as shown in Table 1.
With the original information, we can obtain the transformed matrix V = ((μ_{i,j}, v_{i,j})) as shown in Table 2.
According to definition 2, we calculate the interval probability masses of each focal element as follows:
Under the decision attribute C_{1}:
Under the decision attribute C_{2}:
Under the decision attribute C_{3}:
Under the decision attribute C_{4}:
According to definition 3 and 4, we can obtain the belief measure and the plausible measure of alternative a_{1} as follows:
Then according to Eq. 1, we have the final rank of the four
alternatives: a_{1 }
a_{1 }
a_{3 }
a_{4}, that is to say a_{4} is the best project to invest.
CONCLUSION
The DSAHP approach is a novel, flexible and systematic method. It can solve the problems directly based on its decision matrix. In this paper, we introduced something about intuitionistic fuzzy data and gave the basic steps of the DSAHP method. We used the expected utilities to transform the original decision matrix. And then we defined interval probability masses which is different from the original basic probability assignment. In fact, it is an interval BPA. With the interval probability masses, we used a nonlinear model to combine the focal element. In the end, we obtain the belief interval, where belief measure and plausible measure are both intervals. As we know the intuitionistic fuzzy information is used commonly in real assessment, so this combination is meaningful.
Further extension about DSAHP approach includes developing methods with information expressed in interval intuitionistic fuzzy values or other uncertain values.
ACKNOWLEDGMENTS
The authors acknowledge support of Chinese Humanities and Social Sciences Project of Ministry of Education (10YJC630269) and University Science Research Project of Jiangsu Province (11KJD630001).