Fuzzy multiple criteria decision making: Recent
developments ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
rfuller@abo.fi
Abstract
Multiple Criteria Decision Making (MCDM) shows signs of becoming a maturing field. There are
four quite distinct families of methods: (i) the outranking, (ii) the value and utility theory based, (iii)
the multiple objective programming, and (iv) group decision and negotiation theory based methods.
Fuzzy MCDM has basically been developed along the same lines, although with the help of fuzzy set
theory a number of innovations have been made possible; the most important methods are reviewed
and a novel approach - interdependence in MCDM - is introduced.
1
Introduction
Multiple Criteria Decision Making was introduced as a promising and important field of study in the
early 1970’es. Since then the number of contributions to theories and models, which could be used as
a basis for more systematic and rational decision making with multiple criteria, has continued to grow
at a steady rate. A number of surveys, cf e.g. Bana e Costa [2], show the vitality of the field and the
multitude of methods which have been developed. When Bellman and Zadeh, and a few years later
Zimmermann, introduced fuzzy sets into the field, they cleared the way for a new family of methods to
deal with problems which had been inaccessible to and unsolvable with standard MCDM techniques.
There are many variations on the theme MCDM depending upon the theoretical basis used for the modelling. Zeleny [135] shows that multiple criteria include both multiple attributes and multiple objectives,
and there are two major theoretical approaches built around multiple attribute utility theory (MAUT) and
multiple objective linear programming (MOLP), which have served as basis for a number of theoretical
variations. Bana e Costa and Vincke [3] argue that with MCDM the first contributions to a truly scientific
approach to decision making were made, but find fault with the objectives to carry this all the way as
we have to deal with human decision makers who can never reach the degree of consistency needed.
They introduce multiple criteria decision aid MCDA as a remedy; this approach can be given the aim
”to enhance the degree of conformity and coherence” in the decision processes carried out among (predominantly groups of) decision makers - this is done with a cross-adaptation of the value systems and
the objectives of those involved in the process. Even if there are some distinctions between MCDM and
MCDA the overall objective is the same: to help decision makers solve complex decision problems in a
systematic, consistent and more productive way.
There are four major families of methods in MCDM: (i) the outranking approach based on the pioneering
work by Bernard Roy, and implemented in the Electre and Promethee methods; (ii) the value and utility
theory approaches mainly started by Keeney and Raiffa, and then implemented in a number of methods;
∗
The final version of this paper appeared in: C. Carlsson and R. Fullér, Fuzzy multiple criteria decision making: Recent
developments, Fuzzy Sets and Systems, 78(1996) 139-153. doi: 10.1016/0165-0114(95)00165-4
1
a special method in this family is the Analytic Hierarchy Process (AHP) developed by Thomas L. Saaty
and then implemented in the Expert Choice software package; (iii) the largest group is the interactive
multiple objective programming approach with pioneering work done by P.L.Yu, Stanley Zionts, Milan
Zeleny, Ralph Steuer and a number of others; the MOLP family has been built around utility theorybased trade offs among objectives, with reference point techniques, ideal points, etc and the models have
had a number of features including stochastic and integer variables; one of the best interactive methods
available is the VIG software package developed by Pekka Korhonen; (iv) group decision and negotiation
theory introduced new ways to work explicitly with group dynamics and with differences in knowledge,
value systems and objectives among group members.
When fuzzy set theory was introduced into MCDM research the methods were basically developed along
the same lines. There are a number of very good surveys of fuzzy MCDM (cf [26, 49, 75, 89, 105] and
Ribeiro’s contribution in this issue), which is why we will not go into details here but just point to some
essential contributions. One of the good surveys is done by Chen and Hwang [26]: they make distinctions
between fuzzy ranking methods and fuzzy multiple attribute decision making methods, which contain all
the families (i)- (iv) listed above.
The first category contains a number of ways to find a ranking: degree of optimality (Baas-Kwakernaak,
Watson, Baldwin-Guild), Hamming distance (Yager, Kerre, Nakamura, Kolodziejczyk, -cuts Adamo,
Buckley-Chanas, Mabuchi), comparison function (Dubois-Prade, Tsukamoto, Delgado), fuzzy mean and
spread (Lee-Li), proportion to the ideal) McChahone, Zeleny), left and right scores (Jain, Chen, ChenHwang), centroid index (Yager, Murakami), area measurement (Yager), and linguistic ranking methods
(Efstathiou-Tong, Tong-Bonissone).
The second category is built around methods which utilize various ways to assess the relative importance of multiple attributes: fuzzy simple additive weighting methods (Baas-Kwakernaak, Kwaakernak,
Dubois-Prade, Chen-McInnis, Bonissone), analytic hierarchy process (Saaty, Laarhoven-Pedrycz, Buckley), fuzzy conjunctive / disjunctive methods (Dubois, Prade, Testemale), fuzzy outranking methods
(Roy, Sisko, Brans, Takeda), and maximin methods (Bellman-Zadeh, Yager).
The category with the most frequent contributions is, of course, fuzzy mathematical programming.
Inuiguchi et al [55] give a useful survey of recent developments in fuzzy programming in which they
work with the following families of applications: flexible programming (Tanaka, Zimmermann, SakawaYano), possibilistic programming (Tanaka, Tanaka-Asai, Dubois, Dubois-Prade), possibilistic linear
programming using fuzzy max (Dubois-Prade, Tanaka, Ramik-Rimanek, Rommelfanger, Luhandjula,
Inuiguchi-Kume), robust programming (Dubois-Prade, Negoita, Soyster), possibilistic programming
with fuzzy preference relations (Orlovski), possibilistic linear programming with fuzzy goals (Inuiguchi,
Tanaka, Buckley).
In order to introduce some of the key issues in fuzzy multiple criteria decison making we will work
through a number of examples with a novel approach we have recently introduced (cf Carlsson-Fullér
[20]), a method in which we allow the criteria to be interdependent. Then we will a give a brief overview
of the contributions to this issue and close with a fairly comprehensive list of recent publications on fuzzy
MCDM problems.
2
Decision-making with interdependent criteria
P.L. Yu explains that we have habitual ways of thinking, acting, judging and responding, which when
taken together form our habitual domain (HD) [134]. This domain is very nicely illustrated with the
following example ([134] page 560):
2
A retiring chairman wanted to select a successor from two finalists (A and B). The chairman
invited A and B to his farm, and gave each finalist an equally good horse. He pointed out the
course of the race and the rules saying, ”From this point whoever’s horse is slower reaching
the final point will be the new chairman”. This rule of horse racing was outside the habitual
ways of thinking of A and B. Both of them were puzzled and did not know what to do.
After a few minutes, A all of a sudden got a great idea. he jumped out of the constraint of
his HD. He quickly mounted B’s horse and rode as fast as possible, leaving his own horse
behind. When B realized what was going on, it was too late. A became the new chairman.
Part of the HD of multiple criteria decision-making is the intuitive assumption that all criteria are independent; this was initially introduced as a safeguard to get a feasible solution to a multiple criteria
problem, as there were no means available to deal with interdependence. Then, gradually, conflicts were
introduced as we came to realize that multiple goals or objectives almost by necessity represent conflicting interests [135, 126]. Here we will ”jump out of the constraints” of the HD of MCDM and leave out
the assumption of independent criteria.
Decision-making with interdependent multiple criteria is a surprisingly difficult task. If we have clearly
conflicting objectives there normally is no optimal solution which would simultaneously satisfy all the
criteria. On the other hand, if we have pair-wisely supportive objectives, such that the attainment of
one objective helps us to attain another objective, then we should exploit this property in order to find
effective optimal solutions.
In their classical text Theory of Games and Economic Behavior John von Neumann and Oskar Morgenstern (1947) described the problem with interdependence; in their outline of a social exchange economy
they discussed the case of two or more persons exchanging goods with each others (page 11):
. . . then the result for each one will depend in general not merely upon his own actions
but on those of the others as well. Thus each participant attempts to maximize a function
. . . of which he does not control all variables. This is certainly no maximum problem,
but a peculiar and disconcerting mixture of several conflicting maximum problems. Every
participant is guided by another principle and neither determines all variables which affect
his interest.
This kind of problem is nowhere dealt with in classical mathematics. We emphasize at the risk of being
pedantic that this is no conditional maximum problem, no problem of the calculus of variations, of
functional analysis, etc. It arises in full clarity, even in the most ”elementary” situations, e.g., when all
variables can assume only a finite number of values.
This interdependence is part of the economic theory and all market economies, but in most modelling
approaches in multiple criteria decision making there seems to be an implicit assumption that objectives
should be independent. This appears to be the case, if not earlier then at least at the moment when
we have to select some optimal compromise among a set of nondominated decision alternatives. Milan
Zeleny (1982) - and many others - recognizes one part of the interdependence (page 1),
Multiple and conflixting objectives, for example, ”minimize cost” and ”maximize the quality of service” are the real stuff of the decision maker’s or manager’s daily concerns. Such
problems are more complicated than the convenient assumptions of economics indicate. Improving achievement with respect to one objective can be accomplished only at the expense
of another.
but not the other part: objectives could support each others. We will in the following explore the consequences of allowing objectives to be interdependent.
3
In spite of the significant developements which have taken place in both the theory and the methodology
MCDM is still not an explicit part of managerial decision-making [136]. By not allowing interdependence multiple criteria problems are simplified beyond recognition and the solutions reached by the traditional algorithms have only marginal interest. Zeleny also points to other circumstances [136] which
have reduced the visibility and usefulness of MCDM: (i) time pressure reduces the number of criteria to
be considered; (ii) the more complete and precise the problem definition, the less criteria are needed; (iii)
autonomous decision makers are bound to use more criteria than those being controlled by a strict hierarchical decision system; (iv) isolation from the perturbations of changing environment reduces the need
for multiple criteria; (v) the more complete, comprehensive and integrated knowledge of the problem the
more criteria will be used - but partial, limited and non-integrated knowledge will significantly reduce
the number of criteria; and (vi) cultures and organisations focused on central planning and collective
decision-making rely on aggregation and the reduction of criteria in order to reach consensus.
Felix [44] presented a novel theory for multiple attribute decision making based on fuzzy relations between objectives, in which the interactive structure of objectives is inferred and represented explicitely.
With the following example in [45] he explains the need for a detailed automated reasoning about relationships between goals when we have to deal with nontrivial decision problems.
Example 1 Let us suppose that there is a decision maker who wants to earn money (goal 1) and to have
fun (goal 2) simultaneously, and the only way to earn money is to work. Then at least two situations are
possible:
Situation 1: The decision maker does not like to work. Therefore, while working he will not have fun.
The alternative working supports goal 1 but hinders goal 2.
Situation 2: The decision maker likes to work. Therefore, while working he will have fun. The alternative working supports both goal 1 and goal 2.
Relationships between two goals are defined using fuzzy inclusion and non-inclusion between the support
and hindering sets of the corresponding goals. Felix [45] also illustrates, with an example, that the
decision-making model based on relationships between goals can be used as a powerful MADM-method
for solving vector maximum problems.
In multiple objective linear programming (MOLP), application functions are established to measure the
degree of fulfillment of the decision maker’s requirements (achievement of goals, nearness to an ideal
point, satisfaction, etc.) on the objective functions (see e.g. [35, 137]) and are extensively used in the
process of finding ”good compromise” solutions.
In [20] we demonstrated that the use of interdependences among objectives of a MOLP in the definition
of the application functions provides for more correct solutions and faster convergence. Generalizing
the principle of application functions to fuzzy multiple objective programs (FMOP) with interdependent
objectives, in [20], we have defined a large family of application functions for FMOP and illustrated our
ideas by a simple three-objective program. Let us now discuss our approach to interdependent MCDM.
In [20] we have introduced interdependences among the objectives of a crisp multi-objective programming problem and developed a new method for finding a compromise solution by using explicitely the
interdependences among the objectives and combining the results of [17, 19, 35, 137].
Consider the following problem
max f1 (x), . . . , fk (x)
x∈X
(1)
where fi : Rn → R are objective functions, x ∈ Rn is the decision variable, and X is a subset of Rn
without any additional conditions for the moment.
4
Definition 1 Let fi and fj be two objective functions of (1). We say that
• fi supports fj on X (denoted by fi ↑ fj ) if fi (x0 ) ≥ fi (x) entails fj (x0 ) ≥ fj (x), for all
x0 , x ∈ X;
• fi is in conflict with fj on X (denoted by fi ↓ fj ) if fi (x0 ) ≥ fi (x) entails fj (x0 ) ≤ fj (x), for all
x0 , x ∈ X;
• fi and fj are independent on X, otherwise.
Figure 1: A typical example of conflict on R.
Figure 2: Supportive functions on R.
Let fi be an objective function of (1). Then we define the grade of interdependency, denoted by ∆(fi ),
of fi as
X
X
1, i = 1, . . . , k.
(2)
∆(fi ) =
1−
fi ↑fj ,i6=j
fi ↓fj
If ∆(fi ) is positive and large then fi supports a majority of the objectives, if ∆(fi ) is negative and large
then fi is in conflict with a majority of the objectives, if ∆(fi ) is positive and small then fi supports
more objectives than it hinders, and if ∆(fi ) is negative and small then fi hinders more objectives than
it supports. Finally, if ∆(fi ) = 0 then fi is independent from the others or supports the same number of
objectives as it hinders.
Following [137, 35] we introduce an application function
hi : R → [0, 1]
such that hi (t) measures the degree of fulfillment of the decision maker’s requirements about the ith objective by the value t. In other words, with the notation of Hi (x) = hi (f (x)), Hi (x) may be
considered as the degree of membership of x in the fuzzy set ”good solutions” for the i-th objective.
Then a ”good compromise solution” to (1) may be defined as an x ∈ X being ”as good as possible”
for the whole set of objectives. Taking into consideration the nature of Hi (.), i = 1, . . . k, it is quite
reasonable to look for such a kind of solution by means of the following auxiliary problem
max H1 (x), . . . , Hk (x)
(3)
x∈X
As max H1 (x), . . . , Hk (x) may be interpreted as a synthetical notation of a conjuction statement
(maximize jointly all objectives) and Hi (x) ∈ [0, 1], it is reasonable to use a t-norm T [108] to represent
the connective AND. In this way (3) turns into the single-objective problem
max T (H1 (x), . . . , Hk (x)).
x∈X
There exist several ways to introduce application functions [59]. Usually, the authors consider increasing
membership functions (the bigger is better) of the form

if t ≥ Mi

 1
vi (t) if mi < t < Mi
hi (t) =
(4)


0
if t ≤ mi
5
where mi := minx∈X fi (x) is the independent mimimum and Mi := maxx∈X fi (x) is the independent
maximum of the i-th criterion.
As it has been stated before, our idea is to use explicitely the interdependences in the solution method.
To do so, first we define Hi by

if fi (x) ≥ Mi

 1


Mi − fi (x)
Hi (x) =
if mi < fi (x) < Mi
1−

Mi − mi



0
if fi (x) ≤ mi
i.e. all membership functions are defined to be linear.
Figure 3: Linear membership function.
Then from (2) we compute ∆(fi ) for i = 1, . . . , k, and we change the shapes of Hi according to the
value of ∆(fi ) as follows
(1) If ∆(fi ) = 0 then we do not change the shape.
(2) If ∆(fi ) > 0 then instead of the linear membership function we use


1
if fi (x) ≥ Mi



!

1/(∆(f
)+1)
i

Mi − fi (x)
Hi (x, ∆(fi )) =
if mi < fi (x) < Mi
1−

M i − mi




 0
if fi (x) ≤ mi
(3) If ∆(fi ) < 0 then instead of the linear membership function we use

1
if fi (x) ≥ Mi


!|∆(fi )|+1


Mi − fi (x)
Hi (x, ∆(fi )) =
if mi < fi (x) < Mi
1−

Mi − m i



0
if fi (x) ≤ mi
Then we solve the following auxiliary problem
max T (H1 (x, ∆(f1 )), . . . , Hk (x, ∆(fk )))
x∈X
(5)
Let us suppose that we have a decision problem with many (k ≥ 7) objective functions (cf Example 2).
It is clear (due to the interdependences between the objectives), that we will find optimal compromise
solutions rather closer to the values of independent minima than maxima.
The basic idea of introducing this type of shape functions can be explained then as follows: if we manage
to increase the value of the i-th objective having a large positive ∆(fi ) then it entails the growth of the
majority of criteria (because it supports the majority of the objectives), so we are getting essentially
closer to the optimal value of the scalarizing function (because the losses on the other objectives are not
so big, due to their definition).
The efficiency of the obtained compromise solutions can be shown by using the results from [35].
Figure 4: Concave membership function.
6
Let us now consider a fuzzy version of (1)
max f˜1 (x), . . . , f˜k (x)
x∈X
(6)
where F(R) denotes the family of fuzzy numbers, f˜i : Rn → F(R) (i.e. a fuzzy-number-valued function) and X ⊂ Rn .
An application function for the FMOP of (6) is defined as
hi : F(R) → [0, 1]
such that hi (t̃) measures the degree of fulfillment of the decision maker’s requirements about the i-th
objective by the (fuzzy number) value t̃. In other words, with the notation of
Hi (x) = hi (f˜i (x)),
Hi (x) may be considered as the degree of membership of x in the fuzzy set ”good solutions” for the i-th
fuzzy objective.
To construct such application functions for FMOP problems is usually not an easy task. Suppose that
we have two reference points from F(R), denoted by m̃i and M̃i , which represent undesired and desired
levels for each objective function f˜i . We can now state (6) as follows: find an x∗ ∈ X such that f˜i (x∗ ) is
as close as possible to the desired point M̃i and as far as possible from the undisered point m̃i , for each
i = 1, . . . , k.
We suggest the use of the following family of application functions
1
1
Hi (x) = min 1 −
,
1 + D(m̃i , f˜i (x)) 1 + D(M̃i , f˜i (x))
or, more generally,
Hi (x) = T 1 −
1
1
,
1 + D(m̃i , f˜i (x)) 1 + D(M̃i , f˜i (x))
(7)
where T is a t-norm, D is a metric in F(R). It is clear that the bigger the value of Hi (x) the closer the
value of the i-th objective function to the desired level or/and further from the undesired level, and vica
versa the smaller the value of Hi (x) the closer its value to the undesired level or/and further from the
desired level.
Figure 5: Convex membership function.
In (7) the t-norm T measures the degree of satisfaction of two (conflicting) goals ”to be far from the
undesired point and to be close to the desired point”. The particular t-norm T should be chosen very
carefully, because it can occur that Hi (x) attends its maximal value at a point which is very far from
the undesired point, but not close enough to the desired point. For example, if T is the weak t-norm
(T (x, y) = min{x, y} if max{x, y} = 1 and T (x, y) = 0 otherwise) then Hi (x) positive if and only
if f˜i (x) = M̃i , i.e. we have managed to reach completely the desired point, which is rarely the case,
because M̃i is not necessarily in the range of f˜i . Another crucial point is the relative setting of the
desired and undesired points. If D(m̃i , M̃i ) is small then it is impossible to find an x∗ ∈ X satisfying
the condition ”f˜i (x) is close to M̃i and is far from m̃i ”.
Then, similarly to the crisp case, FMOP (6) turns into the single-objective problem
max T (H1 (x), . . . , Hk (x)).
x∈X
7
(8)
It is clear that the bigger the value of the objective function of problem (8) the closer the fuzzy functions
are to their desired levels.
Similarly to the crisp case, we shall modify the application functions, Hi , i = 1, . . . , k with respect to
the interdependences among the objectives of FMOP (6).
We will now define the interdependences with the help of their application functions.
Definition 2 Let f˜i and f˜j be two objective functions of (6), and let Hi and Hj be the associated application functions. We say that
(i) f˜i supports f˜j on X (denoted by f˜i ↑ f˜j ) if Hi (x0 ) ≥ Hi (x) entails Hj (x0 ) ≥ Hj (x) for all
x0 , x ∈ X;
(ii) f˜i is in conflict with f˜j on X (denoted by f˜i ↓ f˜j ) if Hi (x0 ) ≥ Hi (x) entails Hj (x0 ) ≤ Hj (x), for
all x0 , x ∈ X;
(iii) f˜i and f˜j are independent on X, otherwise.
Let f˜i be an objective function of (6) and let Hi be its application function. We define the grade of
interdependence, denoted by ∆(f˜i ), of f˜i as
X
X
∆(f˜i ) =
1−
1, i = 1, . . . , k.
(9)
Hi ↑Hj ,i6=j
Hi ↓Hj
Then similarly to the crisp case, if ∆(f˜i ) is positive and large then f˜i supports a majority of the objectives,
if ∆(f˜i ) is negative and large then f˜i is in conflict with a majority of the objectives, if ∆(f˜i ) is positive
and small then f˜i supports more objectives than it hinders, and if ∆(f˜i ) is negative and small then f˜i
hinders more objectives than it supports. Finally, if ∆(f˜i ) = 0 then f˜i is independent from the others or
supports the same number of objectives as it hinders.
It should be noted that interdependences among the fuzzy objectives of (6) strongly depend on the definition of their application functions. For example, if the application functions are defined in the sense
of (7) then by altering the desired or/and undesired levels for the i-th objective function, it can modify
∆(f˜i ).
We use explicitely the interdependences in the solution method. Namely, first we change the shape of Hi
according to the value of ∆(f˜i ) as follows:
if ∆(f˜i ) = 0 then we do not change the shape, i.e Hi (x, ∆(f˜i )) := Hi (x); if ∆(f˜i ) > 0 then instead of
Hi (x) we take
˜
Hi (x, ∆(f˜i )) := Hi (x)1/(∆(fi )+1) ,
finally, if ∆(f˜i ) < 0 then instead of Hi (x) we use
˜
Hi (x, ∆(f˜i )) := Hi (x)|∆(fi )|+1
Then we solve the single objective problem
max T (H1 (x, ∆(f˜1 )), . . . , Hk (x, ∆(f˜k )))
x∈X
(10)
As in the crisp case, if we manage to increase the value of the i-th objective having a large positive ∆(f˜i )
then follows that a majority of criteria will grow (because it supports a majority of the objectives); i.e.
we are getting essentially closer to the optimal value of the scalarizing function (because the losses on
the conflicting objectives are not so big, due to their definition).
8
It should be noted that we can use the above principles when the objective functions of both the crisp and
fuzzy MOP are known exactly. If we do not know exactly the behavoir of objectives then we should use
approximate reasoning methods to find good compromise solutions to FMCDM problems. The following
example illustrates this case.
Example 2 In corporate takeover negotiations the Buyer and the Seller have two conflicting objectives:
the Buyer wants the takeover price to be as low as possible c1 , but the Seller wants it to be as high as
possible c2 . There is, however, much more behind corporate takeovers. In a real case, in which two
Finnish companies were involved and finally merged, there were a number of more objectives which
could be identified and gradually formulated.
Buyer
c1
c3
c5
c7
Seller
aquisition price low
overall profits high
investments medium
total loans low
c2
c4
c6
c8
aquistion price high
cash inflow high
max corporate ROC
RD investments high
The Seller’s objectives c4 , c6 , and c8 all support his objective of getting a high aquisition price; nevertheless, the objectives (c4 , c6 ), (c6 , c8 ) and (c8 , c4 ) are all pairwise conflicting.
The Buyer’s objective c1 supports his objectives c3 , c5 and c7 . There is no conflict among his objectives,
but the objectives c3 and c7 support each others. There is also some interaction among the Seller’s and
the Buyer’s objectives, which partly explains why they are negotiating: c3 and c4 are supporting each
others, like c6 and c3 , but c5 and c8 are conflicting:
With the notation we introduced for the interdependence above, the takeover has the following objective
structure:
Buyer:
c2 ↑ c4 , c2 ↑ c6 , c2 ↑ c8 , c4 ↓ c6 , c6 ↓ c8 , c8 ↓ c4
Seller:
c1 ↑ c3 , c1 ↑ c5 , c1 ↑ c7 , c3 ↑ c7
Buyer/Seller:
c3 ↑ c4 , c6 ↑ c3 , c5 ↓ c8
It seems clear that it would be rather difficult to find a negotiated solution which would be simultaneously
optimal for all the objectives, as the conflicts seem to eliminate this possibility. It should, however, be
noted that the conflicts are fuzzy, as most of the objectives are given in a fuzzy form (high, medium,
low), which indicates that some other solution than a simultaneous optimum for all the objectives should
be attempted. There are two possibilities: (i) a negotiated compromise , based on trade-offs among
the conflicting objectives (this was carried out in an intuitive fashion in the real case), or (ii) alternate
optima for combinations of subsets of the objectives during a negotiated interval (this was also attempted
by representatives of the Seller, but without any success).
Buckley and Hayashi [12] introduced fuzzy genetic algorithms to (approximately) solve fuzzy optimization problems. Fuzzy genetic algorithms look like an interesting method of producing approximate
solutions to fuzzy optimization problems when the variables can be arbitrary discrete fuzzy subsets of
certain intervals.
3
Practical Applications
One of the earliest practical application of fuzzy multicriteria decison making is a commercial application
for the evaluation of the credit-worthiness of credit card applicants; this was developed eleven years ago
9
in Germany [138]. Nowdays one encounters more and more real applications of FMCDM. We will in
the following briefly outline four recent applications (cf [50, 57, 86, 89, 109, 131] for others):
• Evaluation of weapon systems
• A project maturity evaluation system implemented at Mercedes-Benz in Germany
• Technology transfer strategy selection in biotechnology
• Aggregation of market research data
Cheng and Mon [28] propose a new algorithm for evaluating weapon systems by the Analytical Hierarchy
Process (AHP) based on fuzzy scales. There are two basic problems in weapon system evaluation: the
objectives of the evaluations are generally multiple and generally in conflict, and the descriptions of
the weapon systems are usually linguistic and vague. The first problem can be solved by conventional
MCDM techniques, but in order to tackle the second problem we usually need FMCDM techniques.
The AHP is a very useful decision analysis tool in dealing with MCDM problems and has successfully
been applied to many actual decision areas. The systematic procedures used by Saaty’s AHP method
[95] results in a cardinal order, which can be used to select or rank alternatives. Cheng and Mon derive
a simple and general algorithm for fuzzy AHP by using triangular fuzzy numbers, α-cuts and interval
arithmetic. Triangular fuzzy numbers 1̃ to 9̃ are used to build a judgement matrix through pair-wise
comparison techniques. They estimate the fuzzy eigenvectors of the judgement matrix by using an ”index
of optimism” λ indicating the degree of satisfaction of the decision maker. The proposed technique is
illustrated with the selection of an anti-aircraft artillery system from several alternatives.
In [1] Altrock and Krause present a fuzzy multi-criteria decision-making system for optimizing the design process of truck components, such as gear boxes, axes or steering. For this optimization, it is
necessary to measure the maturity of the design process with a single parameter. The exisiting data
consists both of numerical data (objective criteria) which describe aspects such as the number of design
changes last month and qualitative data (subjective criteria) such as maturity of parts of a component.
After the single parameter describing the design maturity has been derived by the fuzzy data analysis
system, it is used to determine the optimum amount of design effort to be put in the project until completion. Optimality is defined by minimizing the total cost consisting of developement, warranty and
opportunity parts. The degree of maturity of the design process is derived from 10 input variables by
using Zimmermann’s γ-operator [138] for the aggregation process. Their hierarchically defined system
(using the commercial fuzzy logic design tool fuzzyTECH) is now in use at Mercedes-Benz in Germany.
Chang and Chen [25] discuss the potential application of FMCDM techniques the selection of to technology transfer strategies in the area of biotechnology management. The transfer of technology from its
source to the developement of commercial applications is a very complex process. It is clearly a multiperson multi-criteria problem in an ill-structured situation. One should make a careful analysis among
criteria, alternatives, weight, and decision makers before making a decision. If we want to use convential
crisp decision methods we will always have to find precise data. The assessments of alternatives on relation to various criteria, and the importance weights of these criteria, will have to rely on judgement or
approximation. The authors use linguistic variables and fuzzy numbers to aggregate the decision makers’ subjective assessments of the weighting of criteria. Their method is based on using data input for
computing the total index of optimism in a multi-person decision-making problem, instead of having a
decision maker to give the index of optimism independently. This new approach to using the index of
optimism reflects the pooled risk-bearing attitude of several decision makers. The index of optimism is
determined by the evaluation data conveyed by the decision makers at the beginning of the data input
stage. Finally, a novel method is introduced to rank the fuzzy appropriateness indices for choosing the
10
best technology transfer strategy.
The paper ends with a case study of the selection of a strategy for the transfer of hepatitis B vaccine
technology.
Forecasting consumer purchases of homes, cars, consumer electronics and appliances, and vacations is
of great importance to many sectors of the world economy. To address this concern, studies of consumer
preception of business conditions are continuously being conducted to help predict these purchases. In
[130] Yager et al. suggest a methodology for using information obtained by market surveys to predict
the values of other related (linguistic) variables of interest to market research analysts. Based upon a use
of Shannon’s entropy [61] the authors suggest a measure for calculating the relative predictive powers of
two linguistic variables. Yager’s OWA operators [126, 129] are used to carry out aggregation of single
concepts to form complex concepts. Finally, they select the best predictive model by finding the one
which has the minimum entropy.
With the help of a survey of respondents on their attitudes toward some present and future economic
conditions the authors illustrate the suggested mechanism. In this case study, respondents were asked
to rate each of five economic conditions as being either good, normal, or bad. These five economic
conditions were:
1,2. Business conditions now and six months from now.
3,4. The availability of jobs now and six months from now.
5. The family income six months from now.
A follow-up survey was conducted six months later to determine whether or not they had purchased a
house, a car, an electrical appliance or had taken a vacation in the preceding six months.
Using the data obtained from this survey Yager et al construct a model to best aggregate the respondents’
answers to the questions on economic conditions as a predictor of their purchasing a house, car etc.
As a result of the process they found, for example, that the best predictor of the home purchases was
consumers who rated three economic conditions as good, while the best predictor of car purchases was
consumers who rated only two economic conditions as good.
4
Future perspectives
In 1984 French [51] predicted a very pessimistic future for fuzzy decision-making:
It is now some sixteen years since Zadeh awakened interest in the concept of fuzzy sets and
over a decade since he and Bellman extended the analysis to decision-making in a fuzzy
environment. The intervening years have seen the development of a large and growing literature. Yet, despite the enormous amount of research into the theory and applications of
fuzzy sets, there are still some fundamental questions to be answered. It is my contention
that these cannot be resolved as favourably as the proponents of fuzzy mathematics suggest.
Moreover, I argue that the emphasis placed upon the modelling of imprecision is inappropriate to many of the applications suggested for the theory.
Fortunately, everything developed exactly contrary to French’s assessments, and the last ten years have
justified Gaines, Zadeh and Zimmermann’s visions [52] of the future possibilities for fuzzy decision
making.
11
Decision making in practice has shown that fuzzy logic allows decision making with estimated values inspite of incomplete information. It should be noted, however, that a decision may not be correct and can
be improved later when additional information is available. Of course, a complete lack of information
will not support any decision making using any form of logic. For difficult problems, conventional (nonfuzzy) methods are usually expensive and depend on mathematical approximations (e.g. linearization of
nonlinear problems), which may lead to poor performance. Under such circumstances, fuzzy systems
often outperform conventional MCDM methods.
A good example of a case where this happened is given by Munakata and Jani [77]:
Yamaichi Fuzzy Fund. This is a premier financial application for trading systems. It handles 65
industries and a majority of the stocks listed on Nikkei Dow and consists of approximately 800 fuzzy
rules. Rules are determined monthly by a group of experts and modified by senior business analysts as
necessary. The system was tested for two years, and its performance in terms of the return and growth
exceeds the Nikkei Avarage by over 20 %. While in testing, the system recommended ”sell” 18 days
before the Black Monday in 1987. The system went to commercial operations in 1988. All financial
analysts including Western analysts will agree that the rules for trading are all ”fuzzy”.
And it is just one example from 1500 applications of fuzzy systems listed in 1993 [77]...
5
About the papers
Let us now briefly summarize the contents of each paper from this special issue.
The paper Fuzzy multiple attribute decision-making: A review and new preference elicitation techniques
by R.A.Riberio provides an overview of the underlying concepts and theories of decision-making in a
fuzzy environment and the scope of this type of research. This is a well-written and up-to-date survey of
different kinds of methods for rating, comparison and ranking of alternatives. A new weighting technique
is introduced to elicit criteria importance. The paper ends with an example illustrating the proposed
weighting procedure.
Using fuzzy relations and interval valued fuzzy sets as the basic tool for modeling I.B. Turksen and T.
Bilgic in Interval valued strict preference with Zadeh triples introduce a new technique to model vague
preferences with the aim of making a choice at the final stage. They propose that the initial vagueness
in the weak preferences of a decision maker is represented by a fuzzy relation and that further constructs
from this concept introduce a higher order of vagueness which is represented by interval-valued fuzzy
sets. It is shown that conditions weaker than min-transitivity on the representation of initial vagueness
are necessary and sufficient for the alternatives to be partially ranked. Conditions for the existence of
nonfuzzy non-dominated alternatives are also explored.
The paper Scheduling as a fuzzy multiple criteria optimization problem by W.Slany is dealing with a
real-world scheduling problem with uncertain data and vague constraints of different importance, where
compromises between antagonistic criteria are also allowed. A new type of combination of fuzzy setbased constraints and iterative repair-based heuristics is introduced to model these scheduling problems.
Sensitivity analysis is performed by introducing a new consistency test for configuration changes.
The paper A fuzzy satisficing method for multiobjective linear optimal control problems by M.Sakawa,
M.Inuiguchi and T.Ikeda focuses on multiobjective linear control problems. To solve these problems,
they first discretize the time and then replace the system of differential equations with difference equations. Then they formulate approximate linear multi-objective programming problems by introducing
auxiliary variables. Assuming that the decision maker may have fuzzy goals for the objective functions
they determine the corresponding membership functions through interaction with the decision maker.
12
Finally, it is shown that the resulting problem can be reduced to a linear programming problem and that
the satisficing solution for the decision maker can be obtained with the standard simplex method of linear
programming.
In the paper Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test M.Inuiguchi and M.Sakawa extend the concept of efficient solutions
of conventional multi-objective linear programming problems to the case of fuzzy (possibilistic) coefficients by introducing possibly and necessarily efficient solution sets. These are defined as fuzzy sets
whose membership grades represent the possibility and necessity degrees to which the solution is efficient. A test for possible efficiency is presented, when a feasible solution is given. A necessary and
sufficient condition for the possible efficiency for the interval case is provided.
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Follow ups
The results of this paper have been improved and/or generalized in the following works.
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A18-c217 A. Hadi-Vencheh, M.N. Mokhtarian, A new fuzzy MCDM approach based on centroid of
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http://dx.doi.org/10.1016/j.eswa.2010.10.036
A18-c216 J. Ignatius et al, A multi-objective sensitivity approach to training providers’ evaluation and
quota allocation planning, INTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY
AND DECISION MAKING, 10(2011), issue 1, pp. 147-174. 2011
http://dx.doi.org/10.1142/S0219622011004269
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A18-c215 A. Choi, W. Woo, Multiple-Criteria Decision-Making Based On Probabilistic Estimation
With Contextual Information For Physiological Signal Monitoring, INTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY AND DECISION MAKING, 10(2011), number 1,
pp. 109-120. 2011
http://dx.doi.org/10.1142/S0219622011004245
A18-c214 Chi-Cheng Huang and Pin-Yu Chu, Using the fuzzy analytic network process for selecting technology R&D projects, INTERNATIONAL JOURNAL OF TECHNOLOGY MANAGEMENT, 53(2011), number 1, pp. 89-115. 2011
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A18-c213 Anjali Awasthi, S S Chauhan, S K Goyal, A multi-criteria decision making approach for location planning for urban distribution centers under uncertainty, MATHEMATICAL AND COMPUTER MODELLING, 53(2011), pp. 98-109. 2011
http://dx.doi.org/10.1016/j.mcm.2010.07.023
A18-c212 Kejiang Zhang; Gopal Acharia, Uncertainty propagation in environmental decision making
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A18-c211 Maninder Jeet Kaur, Moin Uddin, Harsh K Verma, Analysis of Decision Making Operation
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http://dx.doi.org/10.5120/861-1210
A18-c210 Dorit S Hochbaum, Asaf Levin, How to allocate review tasks for robust ranking, ACTA
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A18-c209 Rajender Kumar, Brahmjit Singh, Comparison of vertical handover mechanisms using generic
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NETWORKS, 2(2010), number 3, pp. 80-97 [doi 10.5121/ijngn.2010.2308]. 2010
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A18-c208 V. Peneva, I. Popchev, Fuzzy multi-criteria decision making algorithms, COMPTES RENDUS DE L’ACADEMIE BULGARE DES SCIENCES, 63(2010), Issue 7, pp. 979-992. 2010
A18-c207 Tony Prato, Sustaining Ecological Integrity with Respect to Climate Change: Adaptive Management Approach, ENVIRONMENTAL MANAGEMENT 45(2010), number 6, pp. 1344-1351.
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http://dx.doi.org/10.1007/s00267-010-9493-3
The framework employs ex post and ex ante evaluations of ecological integrity. The
ex post evaluation uses fuzzy logic (Barrett and Pattanaik 1989; Carlsson and Fuller
1996; Andriantiatsaholiniaina and others 2004; Prato 2005, 2009) to test hypotheses
about the vulnerability to losing ecological integrity in an historical period and the ex
ante evaluation determines the best CMA for alleviating potential adverse impacts of
climate change on ecosystem vulnerability to losing ecological integrity in a future
period. (page 1346)
22
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A18-c206 Kalyan Mitra, Validating AHP, fuzzy alpha cut and fuzzy preference programming method
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A18-c204 B. K. Mohanty and B. Aouni, Product selection in Internet business: a fuzzy approach, INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, 17(2010) 317-331. 2010
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A18-c202 Amir Sanayei, S. Farid Mousavi, A. Yazdankhah, Group decision making process for supplier
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Since 1970, multiple criteria decision making (MCDM) has been a promising and important field of study with many practical application (Carlsson and Fuller 1996). TraditionalMCDM methods have been extended to support the fuzzy decision making. Fuzzy
MCDM methods have found many practical applications in the real word (Carlsson and
Fuller 1996; Chen 2001). (page 220)
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A18-c198 Javier Munguia; Joaquim Lloveras; Sonia Llorens; Tahar Laoui, Development of an AIbased Rapid Manufacturing Advice System, INTERNATIONAL JOURNAL OF PRODUCTION
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A18-c197 ZHANG Tao; XU Yunyun; LI Zancheng; LIN Zhenrong, Fuzzy Interpolation Algorithm
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A18-c196 G. Uthra, R. Sattanathan, Confidence Analysis for Fuzzy Multicriteria Decision Making
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A18-c195 Amir Sanayei, Seyed Farid Mousavi, and Catherine Asadi Shahmirzadi, A Group Based
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http://www.waset.org/journals/waset/v52/v52-72.pdf
This approach helps decision-makers solve complex decision-making problems in a
systematic, consistent and productive way [A18] and has been widely applied to tackle
DM problems with multiple criteria and alternatives [27]. In short, fuzzy set theory
offers a mathematically precise way of modeling vague preferences for example when
it comes to setting weights of performance scores on criteria. Simply stated, fuzzy
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A18-c194 Emre Cevikcan, Selcuk Cebi and Ihsan Kaya, Fuzzy VIKOR and Fuzzy Axiomatic Design
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A18-c192 Gholam Ali Montazer, Hamed Qahri Saremi, Maryam Ramezani, Design a new mixed expert decision aiding system using fuzzy ELECTRE III method for vendor selection, EXPERT
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A18-c191 Min Guo; Jian-Bo Yang; Kwai-Sang Chin; Hong-Wei Wang; Xin-Bao Liu, Evidential Reasoning Approach for Multiattribute Decision Analysis Under Both Fuzzy and Interval Uncertainty,
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http://dx.doi.org/10.1109/TFUZZ.2008.928599
24
A18-c190 Mousumi Dutta, Zakir Husain An application of Multicriteria Decision Making to built heritage. The case of Calcutta, JOURNAL OF CULTURAL HERITAGE, 10(2009), pp. 237-243.
2009
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A18-c189 Tony Prato, Adaptive management of natural systems using fuzzy logic, ENVIRONMENTAL MODELLING & SOFTWARE, 24(2009), pp. 940-944. 2009
http://dx.doi.org/10.1016/j.envsoft.2009.01.007
A18-c188 Ta-Chung Chu, Yichen Lin, An extension to fuzzy MCDM, COMPUTERS AND MATHEMATICS WITH APPLICATIONS, 57(2009), pp. 445-454. 2009
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A18-c187 Tony Prat, Evaluating and managing wildlife impacts of climate change under uncertainty,
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A18-c186 Torbert, H.A., Krueger, E., Kurtener, D., Potter, K.N. Evaluation of Tillage Systems for Grain
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A18-c185 P. Kumar, P. Bauer, Progressive design methodology for complex engineering systems based
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A18-c184 Ni-Bin Chang, Ying-Hsi Chang, Ho-Wen Chen, Fair fund distribution for a municipal incinerator using GIS-based fuzzy analytic hierarchy process, JOURNAL OF ENVIRONMENTAL
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A18-c180 Chung-Tsen Tsao, Applying a fuzzy multiple criteria decision-making approach to the M &
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A18-c179 Chen Qin Lan, Guodong Jiang, Fuzzy AHP method based on the service industry, and elements of evaluation model and empirical study - A Case Study of Quanzhou, Fujian Tourism
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A18-c178 Robert I. John, Shang-Ming Zhou, Jonathan M. Garibaldi and Francisco Chiclana, Automated Group Decision Support Systems Under Uncertainty: Trends and Future Research, INTERNATIONAL JOURNAL OF COMPUTATIONAL INTELLIGENCE RESEARCH, 4(2008), pp.
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A18-c176 Mehrzad Samadi; Ali Afzali-Kusha, Dynamic power management with fuzzy decision support system, IEICE ELECTRONICS EXPRESS, 5(2008) , No. 19, pp. 789-795. 2008
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A18-c171 Fuzhan Nasiri, Gordon Huang, A fuzzy decision aid model for environmental performance
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A18-c135 Zhang Guoli; Geng Silver; Xie Hong; Li Yuanyuan, Multi-objective weighted fuzzy nonlinear programming, JOURNAL OF NORTH CHINA ELECTRIC POWER UNIVERSITY, 31(2004),
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multilevel model of fuzzy comprehensive evaluation for natural resources in agriculture, JOURNAL OF ZHEJIANG UNIVERSITY (AGRICULTURE & LIFE SCIENCES) 28: (2002), nuber
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A18-c118 Tan Ansheng, Optimum Seeking Model and Method of Fuzzy Multiobjective Group Decision
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A18-c115 Satyadas A, Harigopal U, Cassaigne NP Knowledge management tutorial: An editorial
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A18-c114 Zhou DN, Ma H, Turban E Journal quality assessment: An integrated subjective and objective
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A18-c113 Yu CS, Li HL An algorithm for generalized fuzzy binary linear programming problems
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A18-c108 Royes, G.F., Bastos, R.C. Political analysis using fuzzy MCDM Journal of Intelligent and
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A18-c76 Dale, M.B. Functional synonyms and environmental homologues: An empirical approach to
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A18-c105 A. Valls, V. Torra, Using classification as an aggregation tool in MCDM, FUZZY SETS AND
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A18-c104 Cuong BG, On group decision making under linguistic assessments INTERNATIONAL
JOURNAL OF UNCERTAINTY, FUZZINESS AND KNOWLEDGE-BASED SYSTEMS, 7 (4):
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A18-c103 Blasco F, Cuchillo-Ibanez E, Moron MA, et al. On the monotonicity of the compromise set
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A18-c81 Sohrab Khanmohammadi, Javad Jassbi, Electrical Power Scheduling in Emergency Conditions using a new Fuzzy Decision Making Procedure, 2010 IEEE International Conference on
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Fuzzy logic is based on fuzzy set theory in which every object has a grade of membership in various sets. Inputs are mapped to membership functions, or sets (fuzzification
process). Knowledge of a restricted domain is captured in the form of linguistic rules.
Relationships between two goals are defined using fuzzy inclusion and non-inclusion
between the supporting and hindering sets of the corresponding goals [A18]. (page
472)
A18-c76 Bui Cong Cuong, Dinh Tuan Long, Nguyen Thanh Huy, Pham Hong Phong, New Computing
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A18-c75 Guozheng Zhang, Research on Supplier Selection Based on Fuzzy Sets Group Decision, International Symposium on Computational Intelligence and Design, Changsha, Hunan, China. December 12-December 14, 2009, [ISBN 978-0-7695-3865-5], pp. 529-531. 2009
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A18-c74 Yong Shi, Shouyang Wang, Yi Peng, Jianping Li and Yong Zeng, Ecological Risk Assessment with MCDM of Some Invasive Alien Plants in China, in: Cutting-Edge Research Topics on
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A18-c70 E. Kornyshova, R. Deneckére, and C. Salinesi, Improving Software Development Processes
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A18-c69 James J. Buckley and Leonard J. Jowers, Fuzzy Multiobjective LP, in: Monte Carlo Methods
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A18-c68 Quan Li, A fuzzy neural network based multi-criteria decision making approach for outsourcing supplier evaluation, 3rd IEEE Conference on Industrial Electronics and Applications (ICIEA
2008), 3-5 June 2008, pp.192-196. 2008
http://dx.doi.org/10.1109/ICIEA.2008.4582505
As we can see, we have to deal with both quantitative and qualitative criteria during outsourcing supplier evaluation process, an effective decision making tool must be utilized
to fulfill our needs. Although several fuzzy MCDM methods such as multi-objective
fuzzy decision making (MOFDM), hierarchical weight decision making (HWDM),
PROMETHEE, etc. could be utilized to deal with such problem, the complexity of
such methods makes it harder to be undertaken [A18] (page 193)
A18-c67 Wen-Hsiang Lai; Pao-Long Chang; Ying-Chyi Chou, Fuzzy MCDM Approach to R&D
Project Evaluation in Taiwan’s Public Sectors, Portland International Conference on Management
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A18-c66 Tsung-Han Chang; Tien-Chin Wang, Fuzzy Preference Relation Based Multi-Criteria Decision Making Approach for WiMAX License Award, IEEE International Conference on Fuzzy
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A18-c65 Jie Lu; Xiaoguang Deng; Vroman, P.; Xianyi Zeng; Jun Ma; Guangquan Zhang, A fuzzy
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Carlsson and Fullér [A18] indicated four major families of methods in MCDM. One line
of the MCDM is multi-attributive decision-making (MADM) approach, which is based
on the use of fuzzy indicators and the minimum average weighted deviation method
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A18-c61 M.J. Beynon, Fuzzy Outranking Methods Including Fuzzy PROMETHEE, in: Jose Galindo
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A18-c60 Irina Georgescu, Concluding remarks, in: Fuzzy Choice Functions, Studies in Fuzziness and
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A18-c59 Hepu Deng, A Discriminative Analysis of Approaches to Ranking Fuzzy Numbers in Fuzzy
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A18-c57 Hamed Qahri Saremi, Gholam Ali Montazer, Website Structures Ranking: Applying Extended
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A18-c52 Feng Kong, Hong-yan Liu, A Hybrid Fuzzy LMS Neural Network Model for Determining
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A18-c51 Omar F. El-Gayar and Kanchana Tandekar, An IDSS for Regional Aquaculture Planning,
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A18-c37 Ashley Morris and Piotr Jankowski, Spatial Decision Making Using Fuzzy GIS, in: Frederick
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A18-c36 Zhu Dazhong, Han-Zhong, Application of Fuzzy Multi-Criteria Decision Making to the Development Sequence of New Products, In: First Symposium of the Taiwan Society of Operations
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A18-c35 P L Kunsch, Ph Fortemps, Evaluation by fuzzy rules of multicriteria valued preferences in
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A18-c34 Ta-Chung Chu and Tzu-Ming Chang Solving Fuzzy MCDM Using Fuzzy Weighted Average
Arithmetic, The 17th International Conference on Multiple Criteria Decision Making Whistler,
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A18-c33 Chi-Chun Lo and Ping Wang, Using Fuzzy Distance to Evaluate the Consensus of Group
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A18-c29 Martinovska, C. Agent-based emotional architecture for directing the adaptive robot behavior
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A18-c27 C. Mohan and S.K. Verma, Interactive algorithms using fuzzy concepts for solving mathematical models of real life optimization problems, in: J. L. Verdegay ed., Fuzzy Sets based Heuristics
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A18-c26 Wei Wang, Kim-Leng Poh, Fuzzy multicriteria decision making under attitude and confidence
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A18-c25 Bailey, David, Campbell, Duncan and Goonetilleke, Ashantha, An experiment with approximate reasoning in site selection using InfraPlanner’, in: Proceedings of the Conference: Australia
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A18-c24 Bailey, D., Goonetilleke, A., Campbell, D. Information analysis and dissemination for site
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A18-c23 Michelle R. Lavagna and Amalia Ercoli Finzi, Concurrent Processes within Preliminary Spacecraft Design: An Autonomous Decisional Support Based on Genetic Algorithms and Analytic Hierarchical Process, in Proceedings of the 17th International Symposium on Space Flight Dynamics,
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A18-c22 Lavagna, M., Finzi, A.E. Preliminary spacecraft design: Genetic algorithms and AHP to support the concurrent process approach 54th International Astronautical Congress of the International
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A18-c21 Liu, X.-W., Da, Q.-L., Chen, L.-H. A note on the interdependence of the objectives and their
entropy regularization solution International Conference on Machine Learning and Cybernetics, 5,
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A18-c20 Kuo, Y.-L., Yeh, C.-H., Chau, R. A validation procedure for fuzzy multiattribute decision
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A18-c19 P.M.L Chan, Y.F. Hu and R.E. Sheriff, Implementation of fuzzy multiple Objective decision
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A18-c18 Liu, X.-W. Fuzzy inference based aggregation method for multiobjective decision making
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A18-c17 Tao Wang; Yan-Ping Wang, A new algorithm for nonlinear mathematical programming based
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The involvement of different kinds of fuzziness in these problems is a matter which
also has received a great dealt of work since the early eighties, as it is very frequent that
decision makers have some lack of precision is stating some of the parameters involved
in the model [A18, . . . ]. (page 3)
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A18-c3 Federico Frattini, Vittorio Chiesa, Evaluation and Performance Measurement of Research and
Development: Techniques and Perspectives for Multi-Level Analysis, Edward Elgar Publishing,
Cheltenham, [ISBN 978 1 84720 948 1]. 2009
A18-c2 Jie Lu, Guangquan, Zhang, Da Ruan, Fengjie Wu, Multi-objective Group Decision Making:
Methods Software and Applications with Fuzzy Set Techniques, Series in Electrical and Computer
Engineering, Vol. 6, Imperial College Press, [ISBN 186094793X]. 2007
A18-c1 J. Smed and H. Hakonen, Algorithms and Networking for Computer Games, John Wiley &
Sons, New York, NY, USA, [ISBN 9780470018125], 2006.
in Ph.D. dissertations
• Patrick Meyer, Progressive Methods in Multiple Criteria Decision Analysis, Faculte de Droit, d’Economie
et de Finance, Universite du Luxembourg, 2007.
http://citeseerx.ist.psu.edu/
viewdoc/download?doi=10.1.1.124.9749&rep=rep1&type=pdf
• Erkki Patokorpi, ROLE OF ABDUCTIVE REASONING IN DIGITAL INTERACTION, Faculty of
Technology at Åbo Akademi University, December 2006.
• Elcin Kentel, Uncertainty Modeling Health Risk Assessment and Groundwater Resources Management, Georgia Institute of Technology, August 2006
http://hdl.handle.net/1853/11584
• Mao-Hua Yang, Nash-Stackelberg equilibrium solutions for linear multidivisional multilevel programming problems, STATE UNIVERSITY OF NEW YORK AT BUFFALO. 2005
http://www.acsu.buffalo.edu/˜bialas/public/pub/Papers/YangMHPhD05.pdf
• Fabian Bastin, Trust-Region Algorithms for Nonlinear Stochastic Programming and Mixed Logit
Models, FACULTES UNIVERSITAIRES NOTRE-DAME DE LA PAIX NAMUR, FACULTE
DES SCIENCES. 2004
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• Sudaryanto, A fuzzy multi-attribute decision making approach for the identification of the key sectors
of an economy: The case of Indonesia, RWTH Aachen Germany. 2003
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42
The main feature of this approach is that the imprecision inherent in the qualitative information can be formalized by applying fuzzy sets theory. The fuzzy-MCDM methods
have basically been developed along the same lines as conventional MCDM methods,
but are designed with the help of fuzzy set theory to deal specifically with MCDM problems containing fuzzy data [Zimmmermann, 1987, 1996], [Chen and Hwang, 1992],
[Carlsson and Fuller, 1996, p. 139]. The introduction of fuzzy set theory to the field of
decision making provides a consistent representation of qualitatively or linguistically
formulated knowledge in such a way that still allows the use of precise operators and
algorithms. (pages 168-169)
• A. Valls Mateu, CLUSDM: a multiple criteria decision making method for heterogeneous data sets,
Polytechnic University of Catalonia. 2002
http://www.tesisenred.net/TDX-0206103-205841
43