INFORMATION
SCIENCES
AN m a ~ m ~ A ~ A L
ELSEVIER
JOUP~AL
Journal of Information Sciences 107 (1998) 107- 126
Fuzzy decision making through trade-off
analysis between criteria 1
Jonathan Lee *, Jong-Yih Kuo
Department o/" Computer Science and lnfimnation Engineering, National Central University Chungli.
Taiwan, ROC
Received 1 January 1996: received in revised form 10 March 1997; accepted 29 April 1997
Abstract
In this paper, we propose a new approach in performing trade-off analysis between
multiple criteria, called criteria trade-off analysis (CTA). Conflicting degree and cooperative degree between any two criteria are first formulated, Relationships between criteria
are identified based upon their conflicting and cooperative degrees. Criteria are converted into their disjunctive normal form to obtain a uniform representation of the criteria,
and then arranged into a four-level hierarchical aggregation structure. A set of parameterized aggregation operator, fuzz), and / or, is selected to aggregate the judgements
for the alternatives. A compromise alternative, which is proved to satisfy pareto optimality, can thus be obtained through the aggregation of judgements based on the aggregation hierarchical structure. © 1998 Elsevier Science Inc. All rights reserved.
Keywords." Multicriteria decision making; Interdependent relationships; Trade-off
analysis; Hierarchical aggregation; Averaging operators
1. Introduction
M o s t o f the existing a p p r o a c h e s in multiple criteria decision m a k i n g
( M C D M ) consist o f two phases [1]: (1) the a g g r e g a t i o n o f the j u d g e m e n t s with
Corresponding author. E-mail: [email protected].
This research is supported by National Science Council (Taiwan, ROC) under grant NSC852213-E-008-005.
0020-0255/98/$19.00 © 1998 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 0 - 0 2 5 5 ( 9 7 ) 1 0 0 2 0 - 2
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J. Lee, J.-Y. Kuo /Journal o[ In/brmation Sciences 107 (1998) 107-126
respect to all goals and per decision alternative; and (2) the rank ordering of the
decision alternatives according to the aggregated judgements. However, as was
pointed out by Felix [2], a few of these approaches refer to the aspect of an explicit modeling of relationships between goals. In addition, Carlsson and
Fuller [3] advocated that much closer to MCDM in the real world than the traditional MCDM are the cases with interdependent criteria. However, current
relationship analysis approaches (e.g. fuzzy multiple objective programs
(FMOP) [3] and decision making based on relationship between goals
(DMRG) [2,4,5]) usually result in identifying relationships that are contradictory to each other. Furthermore, the aggregation operators selected in their aggregation procedures either to derive more than one alternative or fail to come
up with any.
Our work in judgements aggregation is motivated by Turksen's Interval Valued Fuzzy Set approach (IVFS) [6, 9], which is focused on providing an appropriate interpretation for a composite concept from a collection of simple
concepts. In IVFS, the intended meaning of a composite concept usually exhibits a range of vagueness through its IVFS, which, in our view, resulted from the
lack of relationship analysis between the simple concepts.
In this paper, we propose a new approach in performing trade-off analysis
between multiple criteria, called criteria trade-off analysis (CTA). 2 In developing CTA, we represent the criteria based on Zadeh's canonical form in testscore semantics [7]. Conflicting degree and cooperative degree between any
two criteria are first formulated. The trade-off among criteria is analyzed by identifying the relationship between criteria which could be either conflicting,
cooperative, irrelevant or counterbalance. A set of parameterized aggregation
operators, fuzz), and / or, is selected to aggregate the judgements for the alternatives. An extended hierarchical aggregation structure is proposed to establish
a four-level criteria hierarchy to facilitate judgements and criticalities aggregation through the fuzzy and / or operator. A compromise alternative, which is
proved to be pareto optimal, can be obtained through the aggregation of judgements based on the proposed aggregation structure.
By using a simple example in a hypothetical manner, we examine the proposed approach and two related approaches using the relationship analysis
technique adopted. We assume that there are five criteria and four alternatives,
in which the judgements of each criterion with respect to the alternatives are
listed in Table 1.
In the sequel, the proposed approach is fully discussed in Section 2. The issue of optimality in CTA is described in Section 3. In Section 4, we examine
2 We have also applied the idea of trade-off analysis in requirements engineering to represent
better and analyze imprecise requirements [9].
J. Lee, J.-Y. Kuo / Journal of ln/brmation Sciences 107 (1998) 107 126
109
Table 1
Judgements for each criterion w.r.t, alternatives
Ci
C,
C~
(~
C~
al
a2
a3
a4
0.85
0.20
0.55
0.00
0.75
0.50
0.00
0.55
O. 15
0.25
0.00
0.90
0.55
0.00
0.45
0.00
0.60
0.55
0.85
0.00
two relationship analysis techniques. Finally, the potential benefits of CTA are
given in Section 5.
2. CTA: Criteria trade-off analysis
To alleviate the problems with the existing relationship analysis approaches,
we propose an alternative approach in performing trade-off analysis for finding
a pareto optimal alternative. There are five steps involved in the proposed analysis technique (see Fig. 1) (1) to represent criteria using Zadeh's canonical form
in test-score semantics [7] (2) to examine the conflicting and cooperative degrees for any two criteria (3) to identify the relationships between criteria based
on the conflicting and cooperative degrees (4) to aggregate judgements with
respect to all criteria and per decision alternative and (5) to rank the order
of the decision alternatives according to the aggregated judgements. Each of
the steps is fully discussed below.
2.1. Criteria representation
Fuzzy decision making consists of a set of decision alternatives and a set of
fuzzy criteria. Suppose A is a non-empty and finite set of possible alternatives,
and A = { a l , . . . ,ah}. Each alternative consists of a collection of properties
whose degrees of judgement are to be evaluated by each criterion. Note that
all alternatives have the same number of properties, and that the number of
properties of an alternative is equivalent to that of criteria that are to evaluate
the alternative. Let C be a non-empty and finite set of criteria,
ck E C, k = 1 , . . . , I. Decision maker determines the feasible alternative with
the highest degree of judgement w.r.t, all relevant criteria.
Criteria specified by the user are usually imprecise in nature, therefore, we
propose the use of Zadeh's canonical form in test-score semantics [7] to represent the criteria. The basic idea is that a criterion cA- can be a boolean
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J. Lee, J.-E Kuo / Journal of Information Sciences 107 (1998) 107-126
criteria
r e p r e s e n t the criteria
using canonical form
compute the cooperative
and conflicting degrees
relative weights
among criteria
apply pairwise comparsion
for criticality o f criteria
identify the relationships
among criteria
transform criteria into DNF
establish the hierarchical
aggregation structure
aggregate the judgements
to compute the compromise
solution
rank ordering of the
decision alternatives
Fig. 1. An overview of CTA.
combination of fuzzy propositions, 3 that is, ck ~ X is F [7,8]. The possible alternative set may be viewed as the domain of variable set X, and F, acting as an
elastic constraint on X, restricts the possible judgements of a property which X
3 The transformation approach advocated by Turksen [9] arrives at a similar propositional
representation. Turksen attempts to convert knowledge expressed in natural language into
computable knowledge. Three transformations occur between four levels knowledge representation. The four levels of knowledge representation are linguistic, metalinguistic, propositional, and
computational. At computational expressions level, symbolic elements of sets are assigned numeric
values.
Z Lee, Z-l( Kuo / Journal oflnJormation Sciences 107 (1998) 107-126
111
can take in U. The Poss{X = ai} is equivalent to flF(ai) ( f o r short, ~ck(x)),
where Poss{X = ai} is the possibility that X takes the judgements of a property
of ai as its value. Thus,/tc~ (x) can be viewed as the degree of judgement of the
criterion ck to the alternative x.
2.2. Defining conflicting and cooperative degrees
Intuitively, two criteria are conflicting with each other if an increase in the
degree to which one criterion is satisfied often decreases the degree to which
another criterion is satisfied with respect to an alternative pair, that is, the
kt~k decreases between the two alternatives (called a conflicting decision alternative pair). On the other hand, two criteria are said to cooperate with each other
if an increase (or a decrease) in the degree to which one criterion is satisfied often increases (or decreases) the degree to which another criterion is satisfied,
that is, the #ok increases (or decreases) between the two alternatives (called a
cooperative decision alternative pair). Note that the third possibility is that
the ~t,.k remains unchanged between the two alternatives, which is called an irrelevant decision alternative pair. We formally define conflicting, cooperative
and irrelevant pairs below, followed by the formal definitions of conflicting
and cooperative degrees.
Definition 1 (Conflicting, cooperative and irrelevant pairs). Assume that c, c' are
two criteria, and A is a set of alternatives, Vai, aj C A, i :/; j. A set of conflicting
decision alternative pairs (for short, conflicting pairs) is defined as,
CF = {(a,,aj) l(~,.(ai) - #c(a/)) × (pd(a,) - pc,(aj)) < 0}.
A set of cooperative decision alternative pairs (for short, cooperative pairs) is
defined as,
CP =
{(a,,a;)l(l~c(a, )
- ,u~(aj))
x (/~,,,(a,) - ,uc,(a.j)) > 0 } .
A set of irrelevant decision alternative pairs (for short, irrelevant pairs) is defined as,
IR = {(ai,aj) l(p~(a~ ) - t~,(aj)) x (pc,(a~) - #~,(aj)) = 0}.
Hence, a collection of pairs of alternatives, Ap = { (ai, aj) ]Vai, aj E A, i ¢ j},
can be divided into three classes: conflicting, cooperative and irrelevant, in such
a way that
Ap=CFUCPUIR
and
CFNCPAIR=~b.
Definition 2 (Conflicting and cooperative degrees). Assume that c, c t are two
criteria, and Ap is a collection of pairs of alternatives. CF and CP denote
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,L Lee, J.-Y. Kuo / Journal of lnJormation Sciences 107 (1998) 10~126
conflicting and cooperative pairs, respectively. The conflicting degree between
two criteria, c and c I, is defined as
cf(c: CI) =
~I~,,.~,)~CF (I/~c(a,) --/~c(a/)l + Izt~.,(ai) --/~c,(aj)l)
E(ah,ak)CmP ( ~
Z~
_~_),,,(ah)Z~)T)
"
The cooperative degree between two criteria, c and c', is defined as:
+
-
cp(c,c') = ~(,,,.o~)cA,.(Igc(a',)
~c(a,)[
+
I+(aj)l)
~
-
~
D
2.3. Relationships between criteria
The relationships among criteria are crucial for adequate treatment of fuzzy
decision making, because they reflect the structure of interaction among the criteria and represent user's preference of the criteria. Together with information
about the criticality of criteria, the relationships among criteria can serve as a
guideline for selecting appropriate aggregation operators and for aggregating
judgements.
In the proposed framework, the relationship between any two criteria, say c
and c', can be classified into four types: irrelevant, counterbalance, conflicting
and cooperative. Two criteria are irrelevant if there is no conflicting and cooperative relationships between the criteria, that is, all alternative pairs are irrelevant pairs (i.e. cp -- cf = 0). In the case of counterbalance relationships, both
the conflicting and cooperative relationships coexist and their degrees are
equivalent (i.e. cp = cf = 0.5).
To further refine conflicting and cooperative relationships, we have identified three sub-categories: strong, moderate and weak. A relationship is said
to be conflicting if the conflicting degree between c and c' is greater than the
cooperative degree. On the other hand, if the cooperative degree is greater than
the conflicting degree, then c cooperates with c'. In the case that there are only
conflicting pairs (i.e. cp = 0), c is strongly conflicting with c'. Similarly, if there
are only cooperative pairs (i.e. cf = 0), then two criteria strongly cooperate with
each other.
The coexistence of conflicting, cooperative and irrelevant pairs (i.e.
cp + cf < 1) usually drops either the conflicting degree or the cooperative degree further compared with the existence of only conflicting and cooperative
pairs (i.e. cp + cf = 1). The former is called weak conflicting or weak cooperative, while the later is called moderate. The relationships among criteria are
summarized in Table 2.
Consider our example, after performing the analysis, the relationships
among the five criteria are summarized in Table 3. It is of interest to note that
the relationships identified in CTA are very different from those identified in
FMOP and D M R G . For example, the relationship between Cj and C3 in
.L Lee. J.- E Kuo / Journal ~f lnJormation Sciences 107 (/998) 107 126
113
~v
I
V
I
+
II
+
II II
Ii
II
II +
II
II
II +
A
V
I
+
A
I
[I
+
©
II
II
.<
[..,
'~
~
.'2 ~,
o
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114
J. Lee, J.-Y. Kuo I Journal of lnformation Sciences 107 (1998) 107-126
CTA is irrelevant, however, the relationships in FMOP are supportive and conflicting that are obviously contradictory to each other. The relationships found
in D M R G also suffer from the similar problem. We will explain this problem in
more detail in Section 4.
2.4. Hierarchical aggregation of criteria
In general, there are two main issues needed to be addressed in using aggregation operators for M C D M problem:
1. The variety of aggregation operators could make it difficult to determine
which one to use in a specific application [10]. Zimmermann [1] has outlined
eight general criteria: axiomatic strength, empirical fit, adaptability, numerical
efficiency, compensation, range of compensation, aggregating behavior, and required scale level of membership functions. Another related issue is that human
use of linguistic connectives, as shown in several studies [6,11], do not directly
correspond to any t-norm or t-conorm, which is an important factor to consider in selecting appropriate aggregation operators.
2. The averaging operator is symmetrical, monotonic, commutative and idempotent, but the property of associativity is not available [1]. The lack of associativity with respect to the averaging operator raises several important issues
on the extension of the operator. Several researchers such as Yager [12] and
Cutello and Motero [13] have proposed different imperatives in holding the definition together as elements are added to an aggregation.
To alleviate these problems, we propose an extension of the hierarchical aggregation structure advocated in [14], where criteria in each disjunct and conjunct are expanded to form a criteria hierarchy. The steps in establishing a
hierarchical structure for criteria aggregation are discussed below:
Convert criteria into disjunctive normal form: We assume that criteria specified by users are connected by linguistic connectives in natural language. To
take these connectives into account, we proposed the use of disjunctive normal
form (DNF) to obtain a uniform representation of the criteria. Criteria in D N F
can then be arranged based on an extension of the notion of the hierarchical
aggregation structure advocated in [15,14,16,13], where criteria in each disjunct
and conjunct are expanded to form a criteria hierarchy. A criteria hierarchy
can be established based on the notion of the criteria criticality and the positive
cooperative degree.
In fact, the criteria may carry different weights reflecting their degrees of criticality, where a weight is a non-negative real number. We have adopted Saaty's
pairwise comparison approach to the assignment of weights to criteria [17].
That is, the relative weights of each criteria pair are used to form a reciprocal
matrix, and the absolute weight of each criteria is obtained from the normalized eigenvector using eigenvalue method. A criterion c can thus be represented
Ji Lee, J.-l~ Kuo / Journal of lnjormation Sciences 107 (1998) 107 126
115
by a triple; (c,/tc(a), we), where/tc(a) is the degree of judgement of the criterion
c w.r.t, an alternative a, and wc denotes the criticality associated with the criterion c. In our example, we assume that the normalized eigenvector (i.e. the
absolute criticality) is [0.25, 0.2, 0.2, 0.1, 0.25], and the D N F for the five criteria
is: (C 3 /~ C4 /~ C5) k/ (el /~ C2).
Establish a criteria hierarchy: A criteria hierarchy of n levels is defined as a
tree, denoted asHk = (G,Ek). C, = {cl,...,Cm,}, k E { 1 , . . . , n } is a set of criteria that are connected by either conjunctive or disjunctive connectives. E denotes a collection of edges. Each edge is an ordered pair of criteria (ci,cj)
indicating the presence of a connection directed from a parent criterion ci to
one of its child criterion cj, wc, > wcj. Let { c i , . . . ,Ch} be a set of criteria at
kth level and c at the level of k + 1 in H. The edge between c, and c (i.e.
(ci, c)) is formed if cp(ci, c) = max{cp(cl, c ) , . . . , cp(ch, c)}. A sorting permutation S for a set of criteria with the same parent, C ~ = { c l , . . . , %} (C' C C), is
any permutation of C' that produces an ordered list S ( U ) = [c~(l),..., c~(t,l],
where cp(c, c~(i/) > cp(c, c~(j)) for all i < j. A criteria hierarchy can be established by applying Algorithm 1.
Algorithm 1 (Establish a criteria hierarchy)
1. Topmlown:
(a) Sort the criticalities for all criteria.
(b) Arrange criteria from top down in a descending order of the criticalities.
Criteria with the same criticality will be placed at the same level.
2. Top-level criteria:
(a) If there is only one criteria with the highest criticality, place it on the top
of the hierarchy.
(b) Else if there are more than one criterion with the highest criticality,
i. Compute the total cooperative degree for each criterion with the rest
of the criteria whose criticality is the same.
ii. Sort the total cooperative degrees computed in the previous step.
iii. Arrange criteria from left to right on the top in a descending order of
the total cooperative degrees.
iv. Add a virtual criterion on the top of those criteria.
3. Grouping criteria (between two adjacent levels):
(a) For criteria (other than criteria at the top level) with the same criticality,
compute all the cooperative degrees for each criterion at level i with every criteria at level i - 1.
(b) Given a criterion, ch, at level i, for every criterion, ck, at level i - 1, sort
the cooperative degree of ch and ck, obtained from the previous step.
Group the criteria whose cooperative degree is the highest under ch.
(c) Continue the previous step until all the criteria at level i have been considered.
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J. Lee, J.- Y. Kuo I Journal of InJbrmation Sciences 107 (1998) 107-126
4. Left-right (for each group):
(a) Given a criterion, ck, at level i - 1, for every criterion, ch, at level i, sort
the cooperative degrees of ch and ck.
(b) Arrange from left to right the criteria at level i in a descending order.
This step is important in the sense that the ordering established through the
hierarchy helps alleviate the associativity problem inherited in averaging operators, namely, a unique ordered list can thus be obtained [18].
Select aggregation operators: To select appropriate aggregation operators,
we propose the consideration of operators that can: (1) reflect the intended relationship between criteria, (2) match the human use of linguistic connectives,
(3) associate the criticality with each criterion, (4) fit the aggregation structure,
and (5) incorporate the notion of conflicting and cooperative degrees. We have
chosen fuzzy and and fuzzy or operators proposed in [14] due to the fact that
fuzzy and operator can be used within each conjunction, while fuzzy or can
be applied between each disjunction, and that the compensation between aggregated sets can be achieved by incorporating the conflicting and cooperative
degrees into the parameters in these two operators, which in turn reflects the
different types of relationship between criteria. In addition, the mathematical
structure of these operators is easy and can be handled efficiently [14]. In order
to match our aggregation structure better, these two operators are also adopted
for criticalities aggregation. We formally define these two operators below.
Definition 3 (Extended fuzzy and). Assume two criteria (c,#o(a),wc),
(c', #c,(a), wo,), A is a set of alternatives, Va ¢ A, and A;.nJ denotes the extended
fuzzy and operator.
(C,#c(a),wc) A~, ~ (c',#c,(a),wc,) is defined as
( c A c t, #and(#c(a),#c,(a)),Wand(Wc~Wc,)),
where
#and(#~(a),
Po'(a))
Wand(W, Wc') : ~and
= ~and min{#c(a), #~,(a)} +
min{w, wo,} +
(1 - 7and)(#c(a) + #c,(a))
(1 -- ~)and)(W -~- Wet)
2
n/and : (cf - cp + 1)/2 and ~)and E [0~ 1].
Definition 4 (Extended fuzzy or). Assume two criteria (c,#o(a),Wc),
(c', #o,(a), wo,), A is a set of alternatives, Va E A, and V~or denotes the extended
fuzzy or operator.
(c,#o(a),Wc) V~.or (c',#~(a),w'c,) is defined as
(c v c', Por(Pc(a), #o,(a)), Wor(Wo,wo'))
J. Lee, J.- Y. Kuo / Journal o f Information Sciences 107 (1998) 107 126
117
where
/~or(/~c(a), Pc,(a)) = 7or max{lLc(a), p,.,(a)} -+
Wor(W,W,.') = ?'or max{w,w,.,) +
7o,.=(cp--cf+l)/2
and
(1 -
,or)(w +
(1 - 7or)(l~c(a) + p~,(a))
2
w,,)
2
7o,.EI0,1].
In the case that two criteria are connected by and, there are two situations:
(1) if 7and equals to 1 (i.e. criteria are strongly conflicting), the fuzzy and operator reduces to min, and (2) if 7and equals to 0 (i.e. criteria are strongly cooperative), the operator becomes arithmetic mean. If the criteria are connected by
or, the fuzzy or operator yields max under the condition that ~,'or equals to 1;
whereas, the operator boils down to arithmetic mean if 7or equals to 0.
~'and = ~'or = 0.5 indicates that two criteria are either irrelevant or counterbalance. These are summarized in Table 4.
Aggregate criteria: To aggregate criteria in a criteria hierarchy Hk, there are
two steps involved: (1) to apply the traversing procedure of breadth first search
algorithm (denoted as z) to the criteria hierarchy Hk to form an ordered list,
that is, rt(Hk) = [eke/l/,..., Ckz(m)]; and (2) to utilize fuzzy and or fuzzy or operator recursively to aggregate the criteria in the list. Finally, a four-level hierarchical aggregation structure can thus be built (see Fig. 2), and the compromise
alternative can be derived:
1. Criteria hierarchies Hk = (G,E~.) ( G
{ck,,... ,cz,~), k = 1,... ,n) built
from disjuncts by applying Algorithm l are placed at the bottom of the hierarchical structure. Each criteria hierarchy Hk is converted into its ordered
list, i.e. ~(Hk) = [Ckrr(1),..., Ckzt(mx)1.
2. Fuzzy and operator is applied recursively to glue criteria in each ordered list,
[ck,(11,..., Ck~(m~)]obtained from the previous step to establish an aggregated
judgement (denoted as CHk) which is placed on the top of the criteria
hierarchy. The degree of the aggregated judgement w.r.t, an alternative a is
3. All the aggregated judgements, ell, (k = 1 , . . . , n), will be used to build an aggregated judgements hierarchy (denoted as H), in which the aggregated
judgements in the hierarchy are in turn combined using fuzzy or operator
recursively to form an overall aggregated judgement, CH. The degree of
the overall aggregated judgements w.r.t, an alternative a is
4. Suppose that there is a set of alternatives A = { a l , . . . , ah}, the degree of the
judgement for the compromise alternative a* is then defined as
=
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J. Lee, J.-Y. Kuo / J o u r n a l o f lnformation Sciences 107 (1998) 107 126
^
tt~
A
^
?-
~.
^
tt~
^
c5
^
^
~.
^
c5
II
tt~
c5
o
A
A
A
tt2.
c5
o
^
^
A
q2.
+
+
^
P.
0
o
.*Vo
J. Lee, J.-Y. Kuo / Journal of Information Sciences 107 (1998) 10~126
the overall aggregated judgement
119
(Ca) l
fuzzy-or
aggregated judgements
h i e r a r c h y (H)
I
I
aggregated
j u d g e m e n t s (CH1)
I
aggregated
judgements
....
fuzzy-and
(CHn) ]
fuzzy-and
c r i t e r i a h i e r a r c h y (H1)
criteria h i e r a r c h y (Hn)
Fig. 2. The proposed extendedhierarchicalaggregation structure.
Refer to our example (see Fig. 3), by applying Algorithm 1 for the disjunct
(c2, c3, c5), we can build a criteria hierarchy in which c5 is on the top of the hierarchy, and c2 on the leftmost position under c5 where c3 sits right next to c2 at
the same level due to the fact that the criticality of c5 is higher than that of c2
and c3, and that the cooperative degree between c5 and c2 is greater than that of
c5 and c3. Similarly, for the disjunct (cl, c4), cl is on the top of c4. The disjoint
aggregated sets A~.and(Cl, C4) and ATand(C5,C2,C3) can also be arranged into a hierarchy based on the criticalities computed using fuzzy and operator. Finally,
the overall aggregated judgement is obtained by applying fuzzy or operator. In
our example, after applying the hierarchical aggregation structure, the judgements for the alternatives are (al,O.12)(az,O.17)(a3,0.O1), and (a4,0.11).
Therefore, a2 will be selected as the compromise alternative.
V'Yor(A'Yand(C1,C4), A'~/and(C5,C2,C3))
t Vyo,
AT~°~(C,,C.)
^7..,(d5,C2,C3)
/
A~and(C 1,C4)
I Aya.d
C4
I
F
^7~n~(C~,C:,C~)
I ^~n~
C2
C3
Fig. 3. An example of the extendedhierarchicalaggregationstructure.
120
Z Lee, J.-Y. Kuo / Journal of lnJbrmation Sciences 107 (1998) 107-126
3. CTA and pareto optimality
To provide a justification for CTA, it is important to show that the compromise alternative derived by CTA is pareto optimal. Generally, an alternative a*
in the alternatives set A is pareto optimal if and only if there does not exist an
alternative a in A such that the judgements of all criteria w.r.t, a* are less than
or equal to those w.r.t, a (that is, Ci(a*) <~Ci(a) for all i) with strict inequality
for at least one criterion [19]. In other words, a pareto optimal solution is one
where any improvement of one criterion can be achieved only at the expense of
another. A pareto optimal solution is also called an efficient or non-dominated
solution.
To prove that the compromise alternative a* derived by CTA is pareto optimal, we first assume that the alternative is not pareto optimal. Based on the
definition of pareto optimality, there exist at least an alternative a such that the
judgements of all criteria w.r.t, a* are less than or equal to those w.r.t, a, which
in turn implies that after applying the aggregation operators in CTA, the overall aggregated judgement of a will be greater than that of a* (i.e.
Pc, (a) > #c,,(a*)), which contradicts the definition of the compromise alternative. Thus, a* is proved to satisfy pareto optimality. We formally prove the argument as follows.
Theorem 1. I f a* is the compromise alternative derived by CTA, then a* is a
pareto optimal solution.
Proof. Given that a* is the compromise alternative, a* E A. Based on the
definition of the compromise alternative, we know that
#c, (a*) = max{#c, ( a t ) , . . . , / % , (ah)}.
Assume that a* is not pareto optimal solution, then there exists an alternative a
such that #c,(a) ~> #c,(a*), i = 1 , . . . , l, with strict inequality for at least one i.
Thus, #c,, (a) > #,,, (a*), which contradicts the definition of the compromise alternative. Therefore, a* is proved to satisfy pareto optimality. []
4. Related work
In this section, we examine the relationship analysis technique adopted by
two fuzzy decision making approaches: 4 F M O P [3] and D M R G [2,4,5].
4 Note that AHP can also be classified under the relationship analysis approach, but the only
relationship consideredin AHP [17] is the "importance" relation.
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J. Lee, J.-Y. Kuo / Journal of lnJormation Sciences 107 (1998) 107 126
4.1. Interdependency relationships between objectives
Recently, Carlsson and Fuller [3] proposed an approach to FMOP with interdependency relationships among objectives, which is an extension of Carlsson's MOP [20] to fuzzy logic. The basic idea is to utilize these relationships to
modify the membership function of the so-called "good solution", denoted as
Hi. In FMOP, Hi(aj) is defined as hi(Ci(aj)), where hi(Ci(aj)) measures the degree of fulfillment of the alternative aj w.r.t, the ith objective by the value
Ci(aj). A "good compromise solution" may be defined as an alternative a being
"as good as possible" for the whole set of objectives A. It is quite reasonable to
look for such kind of solution by means of the following auxiliary problem;
max {HI ( a ) , . . . , Hk (a)}.
acA
For maxa~A{Hi ( a ) , . . . , Hk (a)}, which may be interpreted as a synthetical notation of a conjunction statement (maximize jointly all objectives) and
Hi(a) E I0.1], it is reasonable to use a t-norm T to represent the connective
AND.
The
problem
turns
into
the
single-objective
problem
max,~A T(HI (a),...,Hk(a)). In our example, we assume that each criterion Ci
represented as a fuzzy subset is treated as an objective function in FMOP.
In FMOP, three kinds of relationship are identified: supportive, conflicting,
and independent. The ith objective Ci is said to be supportive with another objective Ci if an increase of Q raises an increase of cj for all alternatives. Similarly, the ith objective is said to be conflicting with another objective if an
increase of Ci raises a decrease of Cj for all alternatives. If the relationship between two objectives is not conflicting or supportive, it is said to be independent. Consider our example, the relationships among objectives are
summarized in Table 5.
In order to change the shape of Hi, Carlsson introduced the notion of the
grade of interdependency, denoted by A (C~) of Ci, which is defined as the difference of the total number of supportive pairs of objective functions and that
of conflicting pairs of objective functions. If A(Ci) is positive and large, then Ci
supports a majority of the objectives. If/I (C~) is negative and large, then Ci is in
conflict with a majority of the objectives. Finally, if A(Ci)= 0, then C is
Table 5
Carlsson's relationships
C~
()
C5
(~
C~
Ci
Cz
C~
C4
C~
Null
Independent
Supportive, Conflicting
Independent
Independent
independent
Null
Independent
Independent
Independent
Conflicting
Independent
Null
Independent
Independent
Independent
Independent
Independent
Null
Independent
Independent
Independent
Independent
Independent
Null
122
J. Lee, J.-Y. Kuo I Journal of Information Sciences 107 (1998) 107-126
independent from the others or supports the same number of objectives as it
hinders. The Hi is then re-formulated as Hi(aj, A(C/)) to use explicitly the interdependence in the solution method. Thus, the FMOP can be viewed as solving
the following auxiliary problem:
maxaj~AT-- norm(Hl(aj, A(C1),...,
Hk(aj, A(Ck)))), where A is the set of alternatives. For our example, all the
A(C/)=0
except A ( C I ) = - I ,
and therefore the result is to
max(0.00,0.00,0.00,0.00), that is, no alternative will be selected.
Problems with F M O P can be summarized as follows:
• Ignoring the fact that other types of relationship may exist between the supportive and independent, or between the conflicting and independent relationship, usually results in oversimplifying the relationship. In this case,
only the independent relation is identified.
• In FMOP, it is possible that the interdependency relationships between two
objectives can be contradictory to each other. More specifically, both conflicting and supportive relationships can be derived at the same time if the
following condition hold,
if C i ( d ) > Ci(a) and Cj.(a') = Cj(a)
Va',a C A.
For example, the relationships between C3 and C1 are both supportive and
conflicting which are contradictory to each other.
• It should also be noticed that no alternative can be obtained under the condition that there exists a zero value for the fulfillment of each alternative
w.r.t, all objectives.
4.2. DMRG: Decision making based on relationships between goals
Felix et al. [2,4,5] propose an approach, called D M R G , to analyzing the relationships between goals, and to determine the final set of decision alternatives
according to the relationships. The D M R G defines a spectrum of relationships
to reflect the interactive structure of goals. The relationships between goals are
defined based on a fuzzy inclusion and a fuzzy non-inclusion between the support and distraction sets of the corresponding goals. The support set Sg represents which alternatives support the goal g and the degree of support. The
distraction set Dg describes which alternatives distract the goal g and the degree
of distraction. Note that the concept of the support set is similar to that of the
objective function advocated by Carlsson. The priority of goals is
(gl, 0.25), (gz, 0.2), (g3,0.2), (g4, 0.1), (gs, 0.25). The support set and distraction
set in our example are given as follows:
Sg, ={(al,0.85),(a2,0.50)},
Dg~ ={(a3,0.65),
Sg2 ={(a,,O.20),(a3,0.90),(a4,0.60)},
(a4,0.50)},
De2 ={(a2,0.5)},
J. Lee, J.-Y. Kuo / Journal of Information Sciences 107 (1998) 10~126
Sg3 = { ( a l , 0 . 5 5 ) , ( a 2 , 0 . 5 5 ) , ( a 3 , 0 . 5 5 ) , ( a 4 , 0 . 5 5 ) } ,
={(a2,0.15),(a,,0.SS)},
Se~ ={(a,,O.75),(a2,0.25),(a3,0.45)},
123
Dg3 ={ },
Dg4 ={(a,,0.75),
(a3,0.45)},
Dg 5 ={(a4,0.60)).
The fuzzy inclusions and fuzzy non-inclusions indicate the existence of relationships between goals. For example, the higher the degree of fuzzy inclusions
between the support sets of two goals, the more cooperative the relationship
between them. Similarly, the higher the degree of non-inclusions between the
support sets of two goals, the less cooperative the relationship. Felix identified
eight types of relationship: independent, assist, cooperate, analogous, hinder,
compete, trade-off and unspecified dependent. Table 6 shows the relationships
among goals for our example.
After the relationships are identified, D M R G will determine the focus of attention F which is the final decision set. In the first step for each pair of goals g~
and gj, their relationship is inferred and the local focus of attention, the fuzzy
set ~i, is determined using an appropriate local decision strategy. This strategy
recommends alternatives supporting the more important objective regardless of
their impact on the other objective. Refer to our example, since g4 competes
with gl, we can adopt a strategy that recommends alternatives supporting
the more important goal regardless of their impact on the other one, which will
result in F 2 = {al,a2}.
Secondly, the focus of attention F is determined by considering all fuzzy sets
~! and the priorities of all goals simultaneously by successive interacting, which
is to build a sequence of interactions among the different Fj. After the successive interacting process, the focus of attention in our example is
F = Ni,j=l,...A,icj 6 i ~- {al, a2}. If there is no decision common to all local focuses of attention, the least important local focus will be excluded out from the
successive interacting. The process iterates until either a satisfying, non-empty
interaction is found or there is no alternative satisfying the current goals, relationships and priorities.
Problems with D M R G are two fold:
1. In D M R G , it is very likely that there exists more than one relationship
between two criteria. Moreover, these relationships could be even contradictory to each other. For example (see Table 6), the relationships between g~ and g3
could be either assist, cooperate, analogous or hinder, in which hinder contradicts to the other three relationships.
2. To come up with strategies which are a must in D M R G to derive a decision alternative is too cumbersome to be practical. More specifically, if the relationships between two criteria contradict to each other, the strategies for
local focus attention are difficult to obtain. In the case that no decision alternative can be found, local focuses w.r.t, less important goals will be excluded
124
J. Lee, J.-E Kuo / Journal of In/brmation Sciences 107 (1998) 107-126
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Z Lee, J.-Y. Kuo/Journal of Information Sciences 107 (1998) 107-126
125
out, namely, not all goals will be considered in DMRG. It is also possible that
the final decision set consists of more than one alternative, which will further
complicate the situation in determining the best alternative.
To sum up, current relationship analysis approaches usually result in identifying relationships that are contradictory to each other. Moreover, the aggregation operators selected in their aggregation procedures either derive more
than one alternative or fail to come up with any.
5. Conclusion
We believe that it is important to incorporate relationship analysis into
M C D M in order to evolve a more realistic technique to support fuzzy decision
making. The similar idea is also shared by researchers such as Carlsson [3,8],
Felix [4] etc. CTA is spawned based on such belief. It is of interest to note that
the relationships between criteria are not fixed. The relationships can be changed due to a different set of alternatives.
CTA offers several important benefits:
• The relationship analysis in CTA makes it easy to identify a unique relationship between any two criteria (i.e. no contradiction will occur).
• CTA also facilitates the selection of an appropriate set of operators from a
large class of fuzzy set operators to incorporate the interdependent relationships to better reflect the real world situations.
• The ordering established through the extended hierarchical aggregation
structure helps alleviate the problem of associativity inherited in averaging
operators.
• The compromise alternative obtained by CTA is pareto optimal.
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