- Email: [email protected]
On the fusion of multiple multi-criteria aggregation functions with focus on the fusion of OWA aggregations
Journal Pre-proof On the fusion of multiple multi-criteria aggregation functions with focus on the fusion of OWA aggregations Ronald R. Yager
PII: DOI: Reference:
S0950-7051(19)30545-3 https://doi.org/10.1016/j.knosys.2019.105216 KNOSYS 105216
To appear in:
Knowledge-Based Systems
Received date : 16 April 2019 Revised date : 4 November 2019 Accepted date : 7 November 2019 Please cite this article as: R.R. Yager, On the fusion of multiple multi-criteria aggregation functions with focus on the fusion of OWA aggregations, Knowledge-Based Systems (2019), doi: https://doi.org/10.1016/j.knosys.2019.105216. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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On the Fusion of Multiple Multi-Criteria Aggregation Functions with Focus on the Fusion of OWA Aggregations Ronald R. Yager
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Machine Intelligence Institute Iona College
New Rochelle, NY 10801
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[email protected]
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Technical Report #MII-3929R3
1
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Abstract We discuss the concepts of a fuzzy measure and the Choquet integral with respect to a fuzzy measure. We introduce aggregation functions and discus their role in fusing fuzzy measures. We describe the formulation of multi-criteria decision-making using fuzzy measures and explain
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how the measure is used to capture the relationship between the criteria. We look at some examples of multi-criteria decision-making using a measure-based approach.
Two notable
examples are the OWA aggregation of the criteria and a propositional logic expression of the relationship between criteria. We next look at the fusion of multiple multi-criteria decision functions. The major objective of this paper is to focus on the problem of the fusion of multiple OWA aggregation functions. A significant benefit of the methodology developed here is that it
many different aggregation imperatives.
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is based solely on the use of fuzzy measures, this allows us to represent in a unified fashion
1. Introduction
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Keywords: Fuzzy Measure, Multi-Criteria, Aggregation, OWA, Choquet Integral
Many decision problems can be formulated as multi-criteria problems [1-5]. Typically in
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these problems we have a set of objectives or criteria we desire to satisfy a set of alternatives from among which we must choose one that best satisfies these objectives. In addition we may have some quantification of how well each of the alternatives satisfies each of the individual criteria. Only in the special case where one of the alternatives is better than, dominates, all of the other alternatives in its satisfaction to all the criteria is there a best alternative. In the real world this is a very rare situation. In order to more generally enable a choice in this multi-objective
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environment we must provide some basis for associating with each alternative a scalar value based on its combined satisfaction to the individual objectives or criteria.
Having these
associated scalar values makes it easy to compare these different alternatives. This scalar value is referred to as a multi-criteria decision function. The rational for how to formulate this multicriteria decision function is implicit in the responsible decision makers understanding of the relationship between the individual criteria satisfactions. Many different kinds of relationships 2
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between the criteria are possible. Here we focus on the use of a measure fuzzy over the set of criteria as a way of representing/modeling the decision maker’s expressed relationship between criteria.
We show that the Choquet integral with respect to the measure over the set of
criteria satisfactions provides a multi-criteria decision function. We look at some special cases One case, is the
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of this fuzzy measure representations of the relationship between the criteria
representation in terms of a measure of a relationship between criteria initially expressed in a well-formed formula in the language of propositional logic [6]. Another case is the measure representation of the OWA aggregation [7, 8] of criteria satisfactions. We also look at the fusion of multiple measure modeled multi-criteria decision functions. Here we particularly focus on the fusion of multiple OWA type decision functions. As special case we provide for an OWA type
2. Fuzzy Measures and Integrals
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aggregation of multiple OWA type decision functions.
Assume Z = {z1, …, zn} is a set of objects. A fuzzy measure over Z, or more simply a
following properties [9] 1) () = 0
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2) (Z) = 1
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measure, is a set mapping, it maps subsets of Z into the unit interval, : 2Z → [0, 1], that has the
3) (A) (B) if B A
associates with each subset A Z a number (A) [0, 1] called the measure of A. If 1 and 2 are two measures on Z such that for any B Z we have 1(B) 2(B) then we say 1 2.
We say 𝜇̂ is the dual of if 𝜇̂ (A) = 1 - (𝐴̅) for all A. It is easy to show that 𝜇̂ is a
not 𝐴̅.
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measure on Z. If 1 2 then 𝜇̂ 1 𝜇̂ 2. We note that 𝜇̂ (A) is the complement of the measure of We call a measure self-dual if (A) = 𝜇̂ (A).
We see that if is self-dual then
(A) = 1 - (𝐴̅) and hence (A) + (𝐴̅) = 1 = (Z). Consider the set function 𝜇̃ defined so that 𝜇̃(A) = a measure: 𝜇̃() = 0, 𝜇̃(X) = 1 and 𝜇̃(A) 𝜇̃(B) if A B. Observation: For any , 𝜇̃ is self-dual. 3
̂ (𝐴) 𝜇(𝐴) + 𝜇 2
. It is easy to show that 𝜇̃ is
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Proof: 𝜇̃̂ = 1 -
̂ (𝐴)) (𝜇(𝐴) + 𝜇 2
=1–
(𝜇(𝐴) +1 − 𝜇(𝐴)) 2
=
𝜇(𝐴) 2
1
𝜇(𝐴)
2
2
+ -
=
̂ (𝐴) 𝜇(𝐴) + 𝜇 2
.
Thus 𝜇̃ associates with any measure a self-dual version Assume g is a mathematical structure, an object, a function, a property or an attribute, that has an associated value for each z Z. Here we shall generically denote this associated
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value as. In this work we shall, unless otherwise indicated, assume [0, 1]. Furthermore to be more consistent with the notational convention used in mathematics we shall at times denote as g(z).
One fundamental use of measures is to define integrals. An integral of g over the Z with respect to is a weighted aggregation of the collection { for i = 1 to n} where the weights are determined via the measure on Z. While there are many possible definitions of integrals of g over Z with respect to a very popular one is the Choquet integral [10, 11],
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denoted Choq(g) or Choq(, …, ). The Choquet integral is defined so that Choq(g) = ∑𝑛𝑗=1(𝜇(𝐻𝑗 ) - (Hj-1)) where is an index function so that is jth largest value of and
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Hj = {z(k), k = 1 to j}. With a little algebra we can show
Choq(g) = ∑𝑛𝑗=1 𝜇(𝐻𝑗 )( - ) + Another notable example of integral of g over Z with respect to is the Sugeno integral
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[10, 11]. In this case
𝑛
Sug(g) = 𝑀𝑎𝑥 [(Hj) )] 𝑗=1
More generally in [10] the author’s suggest a general class of integrals can be obtained using a strict copula C [12, 13]. In this case Cop(g) = ∑𝑛𝑗=1(𝐶(, (Hj)) - (C(, (Hj))) + We note here if C is the product copula, C(ab) = ab we get the Choquet integral.
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We recall a strict copula C is a binary function C: [0, 1] [0, 1] → [0, 1] having the properties
1) For any a [0, 1], C(a, 0) = C(0, a) = 0 2) For any a [0, 1] = C(1, a) = C(a, 1) = a 3) For all a1 a2 and b1 b2 where at least one of the “≤” is a “<” we have C(a2, b2) – C(a2, b1) – C(a1, b2) + C(a1, b1) > 0 4
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A notable copula is the Fréchet copula, C(a, b) = M(a, b) + W(a,
) + (1 – ( + )) (a, b)
where M(a, b ) = Min(a, b), W(a, b) = Max((a + b) – 1, 0), (a, b) = a b and with + < 1.
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3. Aggregation Functions An important class of functions that we shall have considerable interest in are aggregation functions [11, 14, 15].
Definition: An aggregation function is a function of n > 1 arguments, f: [0, 1]n → [0, 1], that has the following properties 1) f(0, 0, …, 0) = 0 2) f(1, 1, …, 1) = 1
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3) f(a1, …, an) f(b1, …, bn) if ai bi for all i
Properties 1 and 2 are called boundary conditions and property 3 is called the monotonicity condition.
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There are three interesting classes of aggregation functions [11].
A conjunctive
aggregation function is defined by the additional property f(a1, …, an) Min(a1, …, an)
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A disjunctive aggregation function is defined by the additional property f(a1, …, an) Max(a1, …, an) An averaging or mean like aggregation function is defined by the additional property Min[a1, …, an] f(a1, …, an) Max[a1, …, an] We now observe some notable examples of aggregation functions. Conjunctive:
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Min: f(a1, .., an) = Mini[ai]
Product: f(a1, ..., an) = ∏𝑛𝑖=1 𝑎𝑖
Disjunctive
Max: f(a1, …, an) = Maxi[ai] Disjunctive Sum: f(a1, …, an) = Min[1, ∑𝑛𝑖=1 𝑎𝑖 ]
Mean: 5
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Max, Min and Median 1
Arithmetic mean: f(a1, …, an) = ∑𝑛𝑖=1 𝑎𝑖 𝑛
1
Geometric mean: f(a1, …, an) = (∏𝑛𝑖=1 𝑎𝑖 )𝑛 𝑛 1 𝑎𝑖
∑𝑛 𝑖=1
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Harmonic mean: f(a1, …, an) =
Assume g is a function on Z, g: Z → [0, 1], where we denote g(zi) = ai. If is a measure on Z then the Choq(g) = Choq(g(z1), …, g(zn)) is a mean type aggregation function. In this case we can express this in aggregation function form as
ChoqC(g) = Choq(g(z1), …, g(zn)) = (g(z1), …, g(zn)) = f(a1, …, an) We now note some properties that can be associated with an aggregation function.
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Definition: An aggregation function f is called idempotent if with all ai = a, f(a1, …, an) = a. Definition: An aggregation function f is called symmetric or anonymous if the indexing of the arguments does not matter: f(a1, …, an) = f(a(1), …, a(n)) for any permutation . Definition: A function N: [0, 1] → [0, 1] is called a strong negation if monotonic decision
2) N(N(a)) = a
involutive
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1) N(a) < N(b) if a > b
The standard strong negation is N(x) = 1 – x
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Definition: Assume N: [0, 1] → [0, 1] is a strong negation and f: [0, 1]n → [0, 1] is an aggregation function. The aggregation function 𝑓̂ defined so that 𝑓̂(a1, …, an) = N(f(N(a1), N(a2), …, N(an))) is called the N-dual of f.
A fundamental theorem for fuzzy measures makes use of aggregation functions [16], Theorem: Assume 1, …, q are a collection of fuzzy measures on the space Z. If I = [0, 1] and
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if Agg is an aggregation function, Agg: Iq → I, then the set function defined so that (A) = Agg(1(A), …, q(A)) for all subsets A of Z is a fuzzy measure on Z. Proof: 1) () = Agg(1(), 2(), …, q()) = Agg(0, …, 0) = 0 2) (X) = Agg(1(X), …, q(X)) = Agg(1, …, 1) = 1 3) Assume A and B are two subsets of Z such that A B. Then (B) = Agg(1(B), …, q(B)) Agg(1(A), …, q(A)) (A). 6
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We shall refer to the preceding as the Fundamental Theorem on the Aggregation of Fuzzy Measures, FTAM. We shall use the notation = Agg(1, …, q) to indicate that is defined so that (A) = Agg(1(A), …, q(A)) for all subsets A of Z.
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We see that the FTAM provides a very general method for providing a measure from a combination of other measures.
Definition: A standard monotonic function is a mapping : [0, 1] → [0, 1] having the properties 1) (0) = 0 2) (1) = 1 3) (a) (b) if a b
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The standard monotonic function also provides a method for obtaining a measure from another measure.
Theorem: Assume is a measure on Z and is a standard monotonic function, the set function defined such that 1(A) = ((A)) is a measure on Z.
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We easily see that if 1, …, q are a collection of measures on Z and 1, …, q are a collection of standard monotonic functions, i:[0, 1] → [0, 1]. Then the set function defined so that for any subset A of Z
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(A) = Agg(1(1(A)), 2(2(A)), …, q(q(A))) is a measure on Z.
4. Multi-Criteria Decision Making
Many problems can be modeled in the following manner. We have a collection of criteria C = {C1, …, Cn} and we wish to satisfy some aggregation of the criteria in C. In
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addition we have a collection of alternatives X = {x1, …, xm} which can be used to attain our goal. Furthermore, for each alternative we have a value Cj(xk) [0, 1], the degree to which criteria Cj is satisfied by alternative xk. One commonly used approach is that for each alternative we combine, using an appropriate aggregation function, its satisfaction to that different criteria to obtain a score function S(xk) for xk, thus S(xk) = Agg(C1(xk), …, Cn(xk)). Here then we can use the S(xk) to decide what action to take. 7
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Here we require that Agg is mean type aggregation function, Mini[Ci(xk)] S(xk) Maxi[Ci(xk)]. A very general formulation for this mean type of aggregation, Agg(C1(x), C2(x), …, Cn(x)), can be obtained with the aid of a fuzzy measure on the space C of criteria and the use of an
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appropriate integral [17]. Here we shall use the Choquet integral. The measure in this case of aggregation of multiple criteria shall be interpreted such that for any subset A of C, (A) is the importance weight contributed by the satisfaction of the subset of criteria in A. So here S(x) = Agg(C1(x), …, Cn(x)) = Choq(C1(x), …, Cn(x)) where
Choq(C1(x), …, Cn(x)) = ∑𝑛𝑗=1(𝜇(𝐻𝑗 ) − 𝜇(𝐻𝑗−1 )) 𝐶𝜌(𝑗) (𝑥)
In the above is an index function defined so that C(j)(x) is the jth largest of the Ci(x) and
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Hj = {C(k) for k = 1 to j}, it is the subset of criteria in C with the j largest satisfactions by x. If we denote (Hj) - (Hj-1) = wj then wj 0, since (Hj) (Hj - 1) from the monotonicity of .
Furthermore we see ∑𝑛𝑗=1 𝑤𝑗 = ∑𝑛𝑗=1(𝜇(𝐻𝑗 ) − 𝜇(𝐻𝑗−1 )) = 𝜇(𝐻𝑛 ) −
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𝜇(𝐻0 ) = 1 since (Hn) = (C) = 1 and (H0) = () = 0. Thus here S(x) = Choq(C1(x), …, Cn(x)) = ∑𝑛𝑗=1 𝑤𝑗 𝐶𝜌(𝑗) (𝑥), is a weighted average of Ci(x). Here the weights are determined by the measure which is a
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reflection of the associated importances of the subsets of criteria. Furthermore Min[Ci(x)] jwjC(j)(x) Maxi[Ci(x)] and hence S(x) is a mean type aggregation of the Ci(x).
We note that if C(j)(x) = 1 for j = 1 to j* and C(j)(x) = 0 for j = j* + 1 to n then S(x) = (Hj*). Here Hj* = {C(j) for j = 1 to j*},
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5. Modeling Multi-Criteria Aggregation Using Measures Let us look at this for some notable examples of . The most common type of measure is one in which ({Ci}) = i and (A) = ∑𝐶𝑖∈𝐴 𝛼𝑖 . Here each Ci in A directly makes a contribution of i to the importance of the subset A. Here we observe that (Hj) = ∑𝑗𝑘=1 𝛼𝜌(𝑘) and (Hj - 1) = 𝑗−1
∑𝑘=1 𝛼𝜌(𝑘) . In this case wj = (Hj) - (Hj - 1) = (j). From this we see that
S(x) = ∑𝑛𝑗=1 𝛼𝜌(𝑗) C(j)(x) = ∑𝑛𝑖=1 𝛼𝑖 Ci(x) 8
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It is simply the weighted average of the Ci(x) where the weight associated with Ci is i. Consider the measure * such that *(C) = 1 and *(A) = 0 if A C. Here we need all the criteria for any contribution. In this case (Hj) = 1 for j = n and (Hj) = 0 for j n. In this case with S(x) = ∑𝑛𝑗=1(𝜇(𝐻𝑗 ) − 𝜇(𝐻𝑗−1 )) 𝐶𝜌(𝑗) (𝑥) = ∑𝑛𝑗=1 𝑤𝑗 𝐶𝜌(𝑗) (𝑥)we see that wn = 1 and
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wj = 0 for j n and hence S(x) = C(n)(x) = Mini[Ci(x)]. It is interesting to note here that since (A) = 0 for A C then for each Ci we have ({Ci}) = 0. Thus each criterion individually makes no contribution to the importance. Consider the measure * such that *() = 0 and (A) = 1 for A . In this case with (Hj) = 1 for j 0 and (Hj) = 0 for j = 0 we see that w1 = 1 and wj = 0 for j n. Hence S(x) = jwjC(j)(x) = C(1)(x) = Maxi[Ci(x)]
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We observe here that for each Ci, (Ci) = 1 and thus each Ci makes an importance contribution of one, but there is no addition.
Consider the situation where is a cardinality-based measure [18, 19]. Here we have a collection of parameters rj 0 for j = 0 to n with rj + 1 > rj and r0 = 0 and rn = 1 and (A) = r|A|,
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where |A| is the cardinality of A. Since Hj has cardinality j then (Hj) = rj. Thus here wj = (Hj) - (Hj - 1) = rj – rj - 1. In this case S(x) = ∑𝑛𝑗=1 𝑤𝑗 𝐶𝜌 (𝑗) (𝑥). There is essentially an OWA aggregation of Ci(x) where the OWA weights wj = rj – rj - 1. Further we observe that rj = 𝑗
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∑𝑘=1 𝑤𝑘 . Thus we see that if we want an OWA aggregation with weights wk for k = 1 to n we
can model it as a cardinality based measure with rj = ∑𝑗𝑘=1 𝑤𝑘 . We observe that here for all Cj, ({Cj}) = r1. Thus r1 is the importance of all individual criteria. We further observe that wj = rj – rj -1 is the amount of importance we gain by going from Hj - 1 to Hj. The median aggregation function can be modeled using a cardinality-based measure. Here we must consider two situations n odd and n even. If n is odd we let 𝑛+1
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rj = 0 for j = 0 to rj = 1 for j =
2
− 1
𝑛+1 2
to n
Since (Hj) = rj and wj = (Hj) - (Hj - 1) we see that wj = 0 for j = 1 to wj = 1 for j = wj = 0 for j =
𝑛+1 2
− 1
𝑛+1 2 𝑛+1 2
+ 1 to n 9
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Here we see S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x) = 𝐶𝜌(𝑛+1) (𝑥), which is the median. 2
If n is even we let 𝑛
rj = 0 for j = 0 to - 1 2
rj = 0.5 for j =
𝑛 2
𝑛
rj = 1 for j = + 1 to n
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2
With (Hj) = rj and wj = (Hj) - (Hj - 1) we see 𝑛
wj = 0 for j = 1 to - 1 2
wj = 0.5 j =
𝑛 2 𝑛
wj = 0.5 = for j = + 1 2
𝑛
wj = 0 for j = + 2 to n 2
Here we get 1
1
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S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x) = C 𝜌(𝑛) (x) + 𝐶𝜌(𝑛+2) (x) 2
2
2
2
Let us look at the situation for possibility measures. Here we have associated with each
𝐶𝑖 ∈𝐴
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Ci a value ({Ci}) = i, where at least one i = 1 and where for any subset A C we have (A) = 𝑀𝑎𝑥 [𝛽𝑖 ]. Here the importance contribution of a collection A of criteria is equal to maximal importance of any criteria in the set A. In this case with Hj = {C(k) for k = 1 to j} we have (Hj) = 𝑀𝑎𝑥[𝛽𝜌(𝑘) ] and (Hj - 1) = 𝑀𝑎𝑥[𝛽𝜌(𝑘) ]. Here with wj = (Hj) - (Hj - 1). We have 𝑘= 1 𝑡𝑜 𝑗
𝑘= 1 𝑡𝑜 𝑗−1
wj = 𝑀𝑎𝑥[𝛽𝜌(𝑘) ] -𝑀𝑎𝑥[𝛽𝜌(𝑘) ]. Using this we have S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(k)(x) 𝑘= 1 𝑡𝑜 𝑗
𝑘= 1 𝑡𝑜 𝑗−1
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Here we see that if (j) ≤ 𝑀𝑎𝑥[𝛽𝜌(𝑘) ]) then wj = 0 and if (j) 𝑀𝑎𝑥[𝛽𝜌(𝑘) ]then
wj = (j) -𝑀𝑎𝑥[𝛽𝜌(𝑘) ].
𝑘= 1 𝑡𝑜 𝑗−1
𝑘= 1 𝑡𝑜 𝑗−1
𝑘= 1 𝑡𝑜 𝑗−1
Closely related is a measure based on a t-conorm S [20]. Here S: [0, 1]2 → [0, 1] where 1) S(a, b) = S(b, a)
2) S(a, b, c) = S(a, S(b, c)) = S(S(a, b), c)
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3) S(a1, b1) S(a2, b2) if a1 a2 and b1 b2
4) S(a, 0) = a Here we also have ({Ci}) = i, with at least one i = 1 and (A) = 𝑆[𝛽𝑖 ]. For this measure (Hj) =
𝑆
𝑘= 1 𝑡𝑜 𝑗
[(k)] and hence wj =
𝐶𝑖 ∈𝐴
𝑆
𝑘= 1 𝑡𝑜 𝑗
[(k)] -
𝑆
[(k)]. We note Max is a
𝑘= 1 𝑡𝑜 𝑗−1
special case of this. Another special case is where S(a, b) = a + b – ab, this is called the probabilistic sum. Another example of this S(a, b) = Min(1, a + b), this called the bounded sum 10
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For this aggregation t-conorm Here we see that if
𝑆
𝑘= 1 𝑡𝑜 𝑗
𝑗−1 ∑𝑘=1 𝛽𝜌(𝑘) ≥
𝑗
[(k)] = Min[1, ∑𝑘=1 𝛽𝜌(𝑘) ].
𝑗−1 ∑ 1 then wj = 0 and ∑𝑗−1 𝑘=1 𝛽𝜌(𝑘) 1 then wj = (j) (1 - 𝑘= 𝛽𝜌(𝑘) )
Another interesting class of measures are the Boolean or binary measures. Here the measures have (A) {0, 1}, in this case (Hj) {0, 1}. Since (C) = 1 and () = 0 and
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(Hj) (Hj-1) we see that there exists some j* such (Hj*) = 1 and (Hj*-1) = 0 and hence wj* = (Hj*) - (Hj*-1) = 1 while all other wj = 0. For this measure S(x) = 𝐶𝜌(𝑗∗) (𝑥) . Thus we see that S(x) {Ck(x), for k = 1 to n}, here S(x) is always equal to the satisfaction of one of the criteria, there is no aggregation.
An interesting case of a cardinality-based measure is one which j = 0 for j = 0 to j* – 1 and j = 1* for j = j* to n it is a binary measure. In this case since (Hj) = j we see that wj = 0 for j = 1 to j* – 1 wj = 1 for j = j*
Here we see that S(x) = 𝐶𝜌(𝑗∗) (𝑥)
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wj = 0 for j = j* + 1 to n
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wj = (Hj) - (Hj - 1) is such that
Another class of aggregation functions are based on a granularization of C [21]. Here we let Ri for i = 1 to q be collection of a subsets of C called granules. We have no requirements on
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the Ri other than that they are non-empty. In particular they need not be disjoint. Conceptually we can view each granule Ri as a class or type of criteria. Associated with each Ri is an importance weight i it can contribute. Here we require each i [0, 1] and ∑𝑞𝑖=1 𝛼𝑖 = 1. Using these Ri we can construct a measure on C, called a plausibility measure, such that (A) = ∑𝑞𝑖=1 𝐺𝑖 (𝐴) 𝛼𝑖 for any subset A C where Gi(A) = 1 if A Ri and Gi(A) = 0 if A Ri = . We note here that (Hj) = ∑𝑞𝑖=1 𝐺𝑖 (𝐻𝑗 ) 𝛼𝑖 where Hj = {C(k) for k = 1 to j}. We
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see that since wj = (Hj) – (Hj - 1) = ∑𝑞𝑖=1(𝐺𝑖 (𝐻𝑗 ) − 𝐺𝑖 (𝐻𝑗−1 ))𝛼𝑖 and S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x). Closely related to the plausibility measure is a belief measure b that can also be used to generate an aggregation function. Here for any subset A C, b(A) = ∑𝑞𝑖=1 𝐺∗𝑖 (𝐴) 𝛼𝑖 where 𝐺𝑖∗ (𝐴) = 1 if Ri A and 𝐺𝑖∗ (𝐴) = 0if Ri A. For this measure b(Hj) = ∑𝑞𝑖=1 𝐺∗𝑖 (𝐻𝑗 ) 𝛼𝑖. Another related measure useful for generating an aggregation function is the pignistic ̃ 𝑖 (𝐴) 𝛼𝑖 measure. Here again pig(A) = ∑𝑞𝑖=1 𝐺
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|𝑅 ∩𝐴| where 𝐺̃𝑖 (A) = 𝑖 |𝑅𝑖 |
Two other interesting measures that can be constructed using the granularization Ri for i = 1 to q. However, here instead of associating with each Ri a weighted i [0, 1] where ∑𝑞𝑖=1 𝛼𝑖 = 1we associate with each Ri a weight i [0, 1] and require that 𝑀𝑎𝑥 [i] = 1, here at 𝑖=1 𝑡𝑜 𝑞
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least one of the i = 1. Using this granularization we define the measure 1 on C such that for any subset A C we have 1(A) = 𝑀𝑎𝑥 [Gi(A) i]. Using the above can also define the 𝑖=1 𝑡𝑜 𝑞
measure 2 on C such that for any A C we have that 2(A) = 𝑀𝑎𝑥 [𝐺𝑖∗ (A) i] 𝑖= 1 𝑡𝑜 𝑞
Another relationship between criteria is a prioritization. Here we suggested a formulation for a fuzzy measure that can be used to implement a priority relationship between the criteria. Assume C = {C1, …, Cn} are prioritized so that C1 > C2, …, > Cn. The fundamental idea of
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prioritization is that a lack of satisfaction to a higher priority criterion is not easily compensated by satisfaction to a lesser priority criterion. Here we shall model a measure that captures this idea. Let Lr = {Ck| for k = 1 to r} where r = 1 to n and let L0 = . Here we associate with each Lr a value Vr [0. 1] so that Vr Vr - 1 and Vn = 1.
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We now define a measure on C so that for any subset A we have (A) = 𝑀𝑎𝑥 [VrGr(A)] where Gr(A) = 1 if Lr A and Gr(A) = 0 if Lr A. Here we see 𝑟= 1 𝑡𝑜 𝑛 (A) = Vr* where Lr* is the largest Lr contained in A.
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Here again S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x). In this case again wj = (Hj) - (Hj - 1). Here (Hj) = 𝑀𝑎𝑥 [VrGr(H(j))] and here wj = ((Hj) - (Hj - 1)). 𝑟= 1 𝑡𝑜 𝑛
6. Representing a Logical Expression of Requirements in a Measure Formulation
Again assume C = {C1, …, Cn} are our criteria of interest. Often people can express
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their desired requirements with respect to these multiple criteria in natural language statement which in turn can be expressed in the form of a well-formed formula, WFF, in the language of propositional logic using the set of criteria C, the logic operators (and), (or), (not) and → (if … then) [6].
Examples of this are:
I desire C1 and C2.
C1 C2
I desire C4 or C2 and C3.
C4 (C2 and C3) 12
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C1 → (C2 C6)
I desire if C1 then C2 and not C6.
In the following discussion we shall use the term “literal” to indicate a criteria, Ck, or its negation, Ck. A logical formula is in disjunctive normal form, DNF, if it is a disjunction, sequence of or’s, consisting of one or more disjuncts, where each disjunct is a conjunction, an
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anding or one or more literals. Here then DNF = D1 D2 D3 ... Dm where each disjunct Dk is of the form
Dk = L1 L2 L3 ... L 𝑛𝑘
Where each literal, Li is an element from C or the negation of an element from C, such as C5. The following is a statement in DNF.
C1 (C8 C5) C2 (C5 C3) (C6 C7 C8).
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We shall say a DNF is pure if none of the literals involve a negation, all the literals in any Dk are simply elements from C.
As shown in Mendelson [6] any WFF from propositional logic can be equivalently
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expressed in DNF. We note two WFF are equivalent if their evaluated truth-values are the same for all allocations of truth-values to constituent elements Our objective here is to convert any WFF, P, expressing the aggregation of criteria into a
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measure over the appropriate set of criteria C. Our first step would be to convert P into its equivalent DNF using the ideas described in Mendelson [6]. Since our interest is not primarily in logic manipulation here we shall assume this step has been performed and our point of departure here is a statement in DNF.
We shall first consider the case where the associated DNF is pure, there are no negation type literals. Let DNF = D1 D2 D3 …, Dm , here each Dk is a conjunction of elements
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from C. We now associate with this DNF a measure on C using the following algorithm. 1. Associate with each Dk a subset Sk of C such that Sk is the set of literals in Dk. 2. For any subset A C define Gk(A) so that Gk(A) = 1 if Sk A Gk(A) = 0 if Sk A 3. For any subset A of C define (A) = Maxk[Gk(A)] 13
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Note: We can easily see that is a measure - () = 0, (X) = 1 if B A then (B) (A). Note: is a binary measure, (A) {0, 1} for all A Note: In a logic world in which all the criteria in any subset A where (A) = 1 have the logic value TRUE then the logical evaluation of P and its equivalent DNF is TRUE
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We now consider the situation where some of the literals involve the negation, we have terms like C3 in our DNF. Here we first replace in our DNF any negative type literal, Ci by a new non-negative criteria 𝐶̃𝑖 .
Furthermore 𝐶̃𝑖 is defined such that for any alternative x,
𝐶̃𝑖 (x) = 1 - Ci(x). Here we let 𝐶̃ = C {all 𝐶̃𝑖 }. We now replace our original DNF by an ̃ . We now express 𝐷𝑁𝐹 ̃ as associated DNF,𝐷𝑁𝐹 ̃= 𝐷 ̃ 𝐷 ̃ 𝐷 ̃ …. 𝐷 ̃ 𝐷𝑁𝐹 1 2 3 m
̃ is composed of where each negation type literal, Ci in Dk is replaced by𝐶̃𝑖 . Here each 𝐷 k
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̃a elements from 𝐶̃ , all of which are non-negative type literals. We now associate with 𝐷𝑁𝐹 measure 𝜇̃ on 𝐶̃ using the following algorithm
̃𝑘 a subset 𝑆̃𝑘 of 𝐶̃ such that 𝑆̃𝑘 is the set of literals in 𝐷 ̃𝑘 . 1. Associate with each 𝐷
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2. For any subset 𝐴̃ 𝐶̃ define Gk(𝐴̃) so that Gk(𝐴̃) = 1 if 𝑆̃𝑘 𝐴̃
Gk(𝐴̃) = 0 if 𝑆̃𝑘 𝐴̃
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3. For any subset 𝐴̃ of 𝐶̃ define 𝜇̃(𝐴̃) = Maxk[Gk(𝐴̃)] Example: Here C = {C1, C2. C3} and DNF = C1 (C2 C3).
Here S1 = {C1} and
S2 = {C2, C3}. We now list all the subsets of C. A1 = {} = , A2 = {C1}, A3 = {C2}, A4 = {C3}, A5 = {C1, C2}, A6 = {C1, C3}, A7 = {C2, C3}, A8 = C = {C1, C2. C3} We obtain G1 and G2.
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G1(A1) = 0, G1(A2) = 1, G1(A3) = 0, G1(A4) = 0, G1(A5) = 0, G1(A6) = 1, G1(A7) = 0, G1(A8) = 1
G2(A1) = 0, G2(A2) = 0, G2(A3) = 0, G2(A4) = 0, G2(A5) = 0, G2(A6) = 0, G2(A7) = 1, G2(A8) = 1.
With (A) = Maxk[Gk(A)] we get (A1) = 0, (A2) = 1, (A3) = 0, (A4) = 0, (A5) = 1, (A6) = 1, (A7) = 1, (A8) = 14
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1 If C2(x) > C3(x) > C1(x) then H0 = = A1, H1 = {C2} = A3, H2 = {C2, C3} = A7 and H3 = C = A8 and in this case S(x) = ((H1) - (H0))C2(x) + ((H2) - (H1))C3(x) + ((H3) - (H2))C1(x)
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S(x) = ((A3) - ())C2(x) + ((H7) - (A3))C3(x) + ((H7) - (H8))C1(x) = C3(x)
7. Fusion of Multi-Agent Multi-Criteria Decision Functions Assume C = {C1, C2,
Ci, …, Cn] is a collection of criteria of interest in a decision
problem. In addition assume we have m agents each of which has a preferred multi-criteria decision function expressed in terms of a measure k on C. Thus here we have k for k = 1 to m
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each of which is a measure on C. Our interest here is to fuse, appropriately aggregate, these individual multi-criteria decision function to obtain a fused group multi-criteria decision function. Thus our objective is to obtain + = Agg(1, 2, 3, …, m) where Agg is an aggregation function modeling the type of fusion we desire between the collection of k’s. Here
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we assume the meta-aggregation function is provided by an external decision maker. As we have earlier indicated if we define + so that for each subset A C we let +(A) = Agg(1(A), 2(A), 3(A), …, m(A))
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then + is a measure on C. If our desired imperative for Agg is to find + so that it satisfies all the k then, +(A) = Mink[k(A)]. If our desired imperative for Agg is to find + so that it satisfies at least one of the k then, +(A) = Maxk[k(A)]. If our desire is to find + so that it is a weighted average of the k then +(A) = ∑𝑚 𝑘=1 𝑤𝑘 k(A). Very general formulation for + can be obtained using the Choquet integral for the aggregation operator, this will give us a mean type aggregation. Assume D = {1, 2, …, m}
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are the set of measures on C corresponding to the individual agents multi-criteria decision functions. Let be a measure on D corresponding to the desired aggregation imperative of the individual agents k’s. Here for any subset B of measures from D, (B) is the importance weight, in the formulation of +, associated with the set of agents whose k’s are in B. It is with the measure we capture the type of aggregation of the k’s we desire in the formulation of the group multi-criteria decision function +. We now define our group aggregation measure +, a 15
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measure on C, so that for any subset A of C we have +(A) = Choq((1(A), 2(A), 3(A), …, m(A)). Here we shall let A be an index function so that 𝜌𝐴 (𝑗) (A) is the jth largest k(A). Thus if A(j) = 3 then 3(A) is the jth largest k(A). Furthermore if for = 1 to m we let 𝑀𝜏𝐴 be a
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collection of subsets of D where𝑀𝜏𝐴 = { 𝜌𝐴 (𝑘) for k = 1 to } then 𝐴 𝐴 +(A) = ∑𝑚 𝑗=1(𝜆(𝑀𝜏 ) - (𝑀𝜏−1 )) 𝜌𝐴 (𝑗) (A).
We note here that +(A) is a mean type aggregation, that is for each A C we have Mink[k(A)] +(A) Maxk[k(A)]. Note: Since for any alternative x,
S(x) = Choq 𝜇+ (C1(x), …, Cn(x)) = ∑𝑛𝑗=1(𝜇+ (𝐻𝑗 ) - +(Hj - 1))C(j)(x). where C(j)(x) is the jth largest of the Ci(x) and Hj = {C(k) for k = 1 to j}, then we just need
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calculate +(A) for only n + 1 subsets of C, the Hj for j = 0 to n.
In the preceding we considered the problem of aggregating multi-agent multi-criteria decision functions to obtain a group multi-criteria decision function. In providing a solution to
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this problem we assumed all of the measures being fused were defined over the same space of criteria C. Now we shall generalize this multi-agent fusion to the case where each of the individual measure’s k are defined over a set Ck of criteria which are not necessarily the same.
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In preparation for addressing this problem we provide a method for extending a measure. Assume X and Y are two sets and Z = X Y and let be a measure on X. We are now interested in obtaining an extension of to Z denoted E. Assume A is a subset of Z, A Z, we now define the extended measure of , E, so that E(A) = (A X). We now show that the set function E on Z is a measure on Z.
E() = ( X) = () = 0
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E(Z) = (Z X) = (X) = 1
Monotonicity: Assume A B Z then E(B) = (B X) (A X) E(A)
Thus we see that E is a measure on Z. We now show that E is an extension of from X to Z. Consider any subset A of X, A X, here E(A) = (A X) = (A). We now show that elements not in X have E measure 16
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zero. Assume B X, here B X = hence E(B) = (B X) = () = 0. We now consider the unification of multiple measures. Assume for k = 1 to m that each k is a measure on the set of criteria Ck. We assume no special relationship between the Ck. 𝑘 Under unification we shall make all k measures on the same space. Here we let C = ⋃𝑚 𝑘=1 𝐶 .
the k measures on the same set of criteria, C.
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We now extend each k to be a measure 𝜇𝑘𝐸 on C. We have now unified all the k by making all
We now turn to the problem of fusing a set D = {k for k = 1 to m} of multi-agent multi-criteria aggregation functions defined over possibly different sets of criteria, Ck where we assume our multi-agent aggregation agenda can be expressed via a measure on the space D. Our procedure for accomplishing this is the following: 𝑘 1. Obtain the unified of criteria C = ⋃𝑚 𝑘=1 𝐶 .
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2. Extend each to be a measure 𝜇𝑘𝐸 on C.
3. Fuse these 𝜇𝑘𝐸 using the Choquet integral and the measure to give us a group measure + on C. Here for any subset A of C we have +(A) = Choq(𝜇𝑘𝐸 (A) for k = 1 to m).
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Note: Again we just need calculate +(Hj) where j = 0 to Card (C)
8. Fusion of OWA Aggregators
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Our main concern here is with fusion of multiple OWA multi-criteria decision functions. Here we shall first review the OWA based multi-criteria aggregation process from a perspective useful for our task. Assume C = {Ci for i = 1 to n} is a collection of criteria and Ci(x) [0, 1] is the satisfaction of criteria Ci by alternative x. The OWA operator can be used to provide a method for obtaining the overall satisfaction, S(x), by alternative x to the collection of criteria C. As originally introduced by Yager [8] the OWA operator is defined by a set of weights, wj for
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j = 1 to n where each wj [0, 1] and ∑𝑛𝑗=1 𝑤𝑗 = 1 and an ordering based aggregation. As suggested by Yager [8] the OWA aggregation of the Ci(x) is S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x) where (j) is the index of the jth largest criteria satisfaction. Thus if (j) = L then CL(x) is the jth largest of 𝑤1 𝑤2 the Ci(x). Here we shall denote W = [ ] and refer to it as the OWA weighting vector. By 𝑤𝑛 choosing different weights in W we can implement different types of aggregation of the Ci(x). 17
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For example if w1 = 1 and all other wj = 0 then S(x) = Maxi[Ci(x)], if wn = 1 and all other wj = 0 1
1
𝑛
𝑛
then S(x) = Mini[Ci(x)]. In the case where all wj = then S(x) = ∑𝑛𝑗=1 𝐶𝜌(𝑗) (x). Many different types of aggregation can implement by appropriate choice of weights W. The value of S(x) depends on both the OWA weighting vector W and the criteria satisfactions, the Ci(x). The
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OWA operator is known to be a mean type aggregation function, Mini(Ci(x)) OWA(C1(x), …, Cn(x)) Maxi(Ci(x)).
In [22] we introduced a characterizing property of an OWA aggregation called the 𝑛−𝑗 attitudinal character, AC, which depends solely on W, AC(W) = ∑𝑛𝑗=1 𝑤𝑗 ( 𝑛−1 ). It can be shown
that AC(W) [0, 1] and for the Max aggregation AC(W) = 1, for the simple average aggregation AC(W) = 0.5 and for the Min aggregation AC(W) = 0. Generally the larger AC(W) the larger
(
The attitudinal character is essentially the OWA aggregation of the argument set
𝑛−1 𝑛−2 𝑛−3
𝑛−𝑛
𝑛−1 𝑛−1 𝑛−1
𝑛−1
,
,
, …,
)
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S(x).
We recall that a cardinality-based measure on the set C of criteria is defined in terms of set of parameters rj for j = 0 to n where rj rj - 1, r0 = 0 and rn = 1. Furthermore for any subset A
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of C we have (A) = r|A|. By appropriate choice of the parameters rj, a cardinality-based measure can be used in coordination with the Choquet to implement a given OWA aggregation of the Ci(x). Here we let r0 = 0 and for j = 1 to n we define rj = ∑𝑗𝑘=1 𝑤𝑘 where wk are the 1
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weights of the OWA aggregation being modeled. Here we see that wj = rj – rj - 1. We note that 𝑟
if all wj = then rj = . 𝑛
𝑛
Using the above-defined cardinality based measure we see Choq(C1(x), C2(x), …, Cn(x)) = ∑𝑛𝑘=1(𝜇(𝐻𝑗 ) - (Hj - 1))C(j)(x) Where C(j)(x) is the jth most satisfied criteria under x and Hj = {C(j) for k = 1 to j}, the j most satisfied criteria under x. Since H has j elements and is a cardinality based measure with
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parameters rj then (Hj) = rj. From this we get Choq(C1(x), C2(x), …, Cn(x)) = ∑𝑛𝑖=1(𝜇(𝐻𝑗 ) - (Hj - 1))C(j)(x) = ∑𝑛𝑖=1(𝑟𝑗 - rj 1))C(j)(x)
Choq(C1(x), C2(x), …., Cn(x)) = ∑𝑛𝑖=1 𝑤𝑗 C(j)(x) = S(x) Thus we see that any OWA aggregation can be formulated in terms of an appropriated cardinality-based measure on the space C of criteria. 18
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9. On the OWA Aggregation of OWA Aggregators Assume C = {Ci for i = 1 to n} are a set of criteria. Let Ci(x) indicate the satisfaction of criteria Ci by alternative x.
Assume (j) is the index of the criteria with the jth largest
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satisfaction by x, its value is C(j)(x). Here j = 1 to n. Assume D = {k for k = 1 to m} are our different OWA aggregations, we have m of these different OWA aggregations. Here w kj is the jth OWA weight for the kth OWA aggregation. We note that ∑𝑛𝑗=1 𝑤𝑘𝑗 = 1 for each of these k. Here the OWA aggregated value under k is Sk(x) = ∑𝑛𝑗=1 𝑤𝑘𝑗 C(j)(x).
We now obtain the simple weighted aggregation of these n OWA aggregations where k is the weighted associated with k. In this case 𝑆̃(x) = ∑𝑚 𝑘=1 𝛽𝑘 Sk(x). This is essentially an OWA
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aggregation 𝜇̃ of the C(j)(x) where the n OWA weights are 𝑤 ̃𝑗 = ∑𝑚 𝑘=1 𝛽𝑘 wkj for j = 1 to n. Here 𝑆̃(x) = ∑𝑛𝑗=1(∑𝑚 𝑘=1 𝛽𝑘 wkj)C(j)(x).
We next consider an OWA Aggregation of the Sk(x) with OWA weights t. Here t is
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the weight associated with the tth largest Sk(x). We note here the m weights, t for t = 1 to m, sum to one. Here we let be an index function so that (k) in the ordered position of Sk(x). Thus if (k) = t then k has the tth largest OWA aggregated value under x. Using this we see the
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𝑚 𝑛 OWA aggregation of the Sk(x) is 𝑆⃡(x) = ∑𝑚 𝑘=1 𝛼𝑣(𝑘) Sk(x) = ∑𝑘=1 𝛼𝑣(𝑘) (∑𝑗=1 𝑤𝑘𝑗 C(j)(x)). We see
𝑆⃡(x) is essentially an OWA aggregation of the C(j)(x) which is based on an OWA aggregation using the cardinality-based measure 𝜇 ⃡ where the OWA weights are 𝑤 ⃡ 𝑗 = ∑𝑚 𝑘=1 𝛼𝑣(𝑘) wkj. Thus ⃡(x) we need (j), the ⃡ 𝑗 C(j)(x). It is important to emphasize that to calculate 𝑆 𝑆⃡(x) = ∑𝑛𝑗=1 𝑤 ordering of Ci(x) and the v(k) the ordering of the Sk(x) which depends on (j), the ordering of the Ci(x).
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Reiterating the OWA aggregation of a collection of m OWA operators of C criteria is also OWA operator of the C criteria with OWA weights 𝑤 ⃡ 𝑗 = ∑𝑚 𝑘=1 𝛼𝑣(𝑘) wkj where wkj are the OWA weights of the constituent operators being aggregated and t are the OWA weights of the aggregating OWA operator and v(k) is the position of the kth operator being aggregated with respect to alternative x. More precisely since 𝑤 ⃡ 𝑗 depends on the alternative x it should be denoted as 𝑤 ⃡ 𝑗 (x). Also since v(k) depends on the alternative x we should denote it as vx(k). 19
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In the following we provide an example of the OWA aggregation of OWA multi-criteria aggregators. Example: Assume we have three criteria, C = {C1, C2, C3}. The satisfaction of these criteria by alternative x is C1(x) = 0, C2(x) = 0.5 and C3(x) = 1. In this case the function is such that (1)
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= 3, (2) = 2 and (3) = 1. Here we have three OWA operators we want to aggregate: 1, 2 and 3. The weighting vectors wkj for these operators are: 𝑘 = 1 𝑘 = 2 𝑘 = 3 𝐶𝜌(𝑗) (𝑥) 𝑤1 0.2 1 0.333 𝐶𝜌(1) (𝑥) = 1 𝑤2 0.5 0 0.333 𝐶𝜌(2) (𝑥) = 0.5 𝑤3 0.3 0 0.333 𝐶𝜌(3) (𝑥) = 0
Since for each OWA operator we have Sk(x) = ∑3𝑗−1 𝑤𝑗𝑘 C(j)(x) then S1(x) = (0.2)(1) + (0.5)(0.5) + (0.3)(0) = 0.2 + 0.25 = 0.45
S3(x) = (0.333)(1 + 0.5 + 0) = 0.5
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S2(x) = (1)(1) + (0)(0.5) + (0)(0) = 1 + 0 + 0 = 1
S2(x) > S3(x) > S1(x) and hence v(1) = 3 v(3) = 1 v(3) = 2
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Below we have v(k) the ordered position of OWA aggregation k, Sk (x). We see that
1 = 0.6 2 = 0.3 3 = 0.1
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We assume the aggregation of the k is implemented by an OWA operator with weights
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Using this we have 𝑆⃡(x) = ∑3𝑘=1 𝛼𝑣(𝑘) Sk(x) = 0.6(1) + 0.3(0.5) + 0.1(0.45) = 0.795 As we previously indicated this OWA aggregation of 1, 2 and 3 results in an OWA operator 𝜇 ⃡ on C with weights 𝑤 ⃡𝑗 0.1 0.6 𝑤1 0.2 1 𝑤2 0.5 0 𝑤3 0.3 0
= ∑𝑚 𝑘=1 𝛼𝑣(𝑘) wkj. 0.3 0.333 0.333 0.333
From the table above we easily get 20
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𝑤 ⃡ 1 = (0.1)(0.2) + (0.6)(1) + (0.3(0.333) = 0.72 𝑤 ⃡ 2 = (0.1)(0.5) + (.6)(0) + (0.3(0.333) = 0.15 𝑤 ⃡ 3 = (0.1)(0.3) + (.6)(0) + (0.3(0.333) = 0.13 We now calculate the OWA aggregation of the Ci(x) using the OWA operator 𝜇 ⃡ with the weights
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𝑤 ⃡ 𝑗 here we get ⃡ 𝑗 C(j)(x) = (0.72)(1) + (0.15)(0.5) + (0.13)(0) = 0.795 𝑆̆(x) = ∑3𝑗=1 𝑤
which is the same value as 𝑆(x).
10. Conclusion
Our particular interest in this paper was in investigating the fusion of multiple OWA aggregation
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functions. Our approach made use of a measure based approach to the formulation and fusion of multi-criteria aggregation functions. First we showed how the OWA operator can be formulated using a cardinality-based measure over the relevant criteria. Thus any OWA operator can be simply represented as cardinality-based measure. A second result we used was the Fundamental
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Theorem on the Aggregation of Fuzzy Measures, the FTAM, which provided a methodology for aggregating measures under various imperatives. By combining our ability to represent the OWA aggregation function as a cardinality-based measure with the FTAM we provided a
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methodology for fusing multiple OWA aggregation functions under different aggregation imperatives. A significant feature of the methodology developed here is that it is based solely on the use of fuzzy measures.
11. References [2] [3] [4] [5] [6] [7]
Zopounidis, C. and Doumpos, M., "Multiple Criteria Decision Making: Applications in Management and Engineering," Springer: Heidelberg, 2017. Greco, S., Ehrgott, M., and Figueira, J. R., Multiple Criteria Decision Analysis: State of the Art Surveys. Heidelberg: Springer, 2016. Al-Shammari, M. and Masri, H., "Multiple Criteria Decision Making in Finance, Insurance and Investment," Springer; Heidelberg, 2015. Ventre, A. G. S., Maturo, A., Hoskova-Mayerova, S., and Kacprzyk, J., "Multicriteria and Multiagent Decision Making with Applications to Economics and Social Sciences," Springer, Heidelberg, 2013. Mateo, J., Multi Criteria Analysis in the Renewable Energy Industry (Green Energy and Technology). London: Springer, 2012. Mendelson, E., Introduction to Mathematical Logic. New York: D. Van Nostrand, 1964. Yager, R. R., Kacprzyk, J., and Beliakov, G., Recent Developments in the Ordered
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[1]
21
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[15] [16] [17] [18] [19] [20] [21] [22]
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[13] [14]
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[12]
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[11]
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[9] [10]
Weighted Averaging Operators: Theory and Practice. Berlin: Springer, 2011. Yager, R. R., "On ordered weighted averaging aggregation operators in multi-criteria decision making," IEEE Transactions on Systems, Man and Cybernetics 18, pp. 183-190, 1988. Wang, Z. Y. and Klir, G. J., Fuzzy Measure Theory. New York: Plenum Press, 1992. Klement, E. P., Mesiar, R., and Pap, E., "A universal integral as common frame for Choquet and Sugeno," IEEE Transaction of Fuzzy Systems 18, pp. 178-187, 2010. Beliakov, G., Pradera, A., and Calvo, T., Aggregation Functions: A Guide for Practitioners. Heidelberg: Springer, 2007. Durante, F. and Sempi, C., "Copula theory: an introduction," in Copula Theory and Its Applications, P. Jaworski, F. Durante, W. Hardle and T. Rychlik (eds.), Springer, Berlin, pp. 3-31, 2010. Nelsen, R. B., An Introduction to Copulas. New York: Springer-Verlag, 1999. Yager, R. R., "Intelligent aggregation and time series smoothing," In Time Series Analysis, Modeling and Applications, Pedrycz, W. and Chen, S. M. (Eds), Springer, Heidelberg, pp. 53-75, 2013. Merigó, J. M. and Yager, R. R., "Generalized moving averages, distance measures and OWA operators," International Journal of Fuzziness, Uncertainty and Knowledge-Based Systems 21, pp. 533-559, 2013. Yager, R. R., "A measure based approach to the fusion of possibilistic and probabilistic uncertainty," Fuzzy Optimization and Decision Making 10, pp. 91-113, 2011. Yager, R. R., "Criteria aggregations functions using fuzzy measures and the Choquet integral," International Journal of Fuzzy Systems 1, pp. 96-112, 1999. Yager, R. R., "Satisfying uncertain targets using measure generalized Dempster-Shafer belief structures " Knowledge Based Systems 142, pp. 1-6, 2018. Yager, R. R., "Uncertainty modeling using fuzzy measures," Knowledge-Based Systems 92, pp. 1-8, 2016. Weber, S., "Decomposable measures and integrals for Archimedean t-conorms," Journal of Mathematical Analysis and Applications 101, pp. 114-138, 1984. Pedrycz, W., "Granular Computing," CRC Press, Boca Raton, Fl, 2013. Yager, R. R., "On the cardinality index and attitudinal character of fuzzy measures," International Journal of General Systems 31, pp. 303-329, 2002.
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[8]
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*Conflict of Interest Form
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No Conflict Exists
PII: DOI: Reference:
S0950-7051(19)30545-3 https://doi.org/10.1016/j.knosys.2019.105216 KNOSYS 105216
To appear in:
Knowledge-Based Systems
Received date : 16 April 2019 Revised date : 4 November 2019 Accepted date : 7 November 2019 Please cite this article as: R.R. Yager, On the fusion of multiple multi-criteria aggregation functions with focus on the fusion of OWA aggregations, Knowledge-Based Systems (2019), doi: https://doi.org/10.1016/j.knosys.2019.105216. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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On the Fusion of Multiple Multi-Criteria Aggregation Functions with Focus on the Fusion of OWA Aggregations Ronald R. Yager
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Machine Intelligence Institute Iona College
New Rochelle, NY 10801
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[email protected]
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Technical Report #MII-3929R3
1
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Abstract We discuss the concepts of a fuzzy measure and the Choquet integral with respect to a fuzzy measure. We introduce aggregation functions and discus their role in fusing fuzzy measures. We describe the formulation of multi-criteria decision-making using fuzzy measures and explain
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how the measure is used to capture the relationship between the criteria. We look at some examples of multi-criteria decision-making using a measure-based approach.
Two notable
examples are the OWA aggregation of the criteria and a propositional logic expression of the relationship between criteria. We next look at the fusion of multiple multi-criteria decision functions. The major objective of this paper is to focus on the problem of the fusion of multiple OWA aggregation functions. A significant benefit of the methodology developed here is that it
many different aggregation imperatives.
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is based solely on the use of fuzzy measures, this allows us to represent in a unified fashion
1. Introduction
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Keywords: Fuzzy Measure, Multi-Criteria, Aggregation, OWA, Choquet Integral
Many decision problems can be formulated as multi-criteria problems [1-5]. Typically in
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these problems we have a set of objectives or criteria we desire to satisfy a set of alternatives from among which we must choose one that best satisfies these objectives. In addition we may have some quantification of how well each of the alternatives satisfies each of the individual criteria. Only in the special case where one of the alternatives is better than, dominates, all of the other alternatives in its satisfaction to all the criteria is there a best alternative. In the real world this is a very rare situation. In order to more generally enable a choice in this multi-objective
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environment we must provide some basis for associating with each alternative a scalar value based on its combined satisfaction to the individual objectives or criteria.
Having these
associated scalar values makes it easy to compare these different alternatives. This scalar value is referred to as a multi-criteria decision function. The rational for how to formulate this multicriteria decision function is implicit in the responsible decision makers understanding of the relationship between the individual criteria satisfactions. Many different kinds of relationships 2
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between the criteria are possible. Here we focus on the use of a measure fuzzy over the set of criteria as a way of representing/modeling the decision maker’s expressed relationship between criteria.
We show that the Choquet integral with respect to the measure over the set of
criteria satisfactions provides a multi-criteria decision function. We look at some special cases One case, is the
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of this fuzzy measure representations of the relationship between the criteria
representation in terms of a measure of a relationship between criteria initially expressed in a well-formed formula in the language of propositional logic [6]. Another case is the measure representation of the OWA aggregation [7, 8] of criteria satisfactions. We also look at the fusion of multiple measure modeled multi-criteria decision functions. Here we particularly focus on the fusion of multiple OWA type decision functions. As special case we provide for an OWA type
2. Fuzzy Measures and Integrals
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aggregation of multiple OWA type decision functions.
Assume Z = {z1, …, zn} is a set of objects. A fuzzy measure over Z, or more simply a
following properties [9] 1) () = 0
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2) (Z) = 1
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measure, is a set mapping, it maps subsets of Z into the unit interval, : 2Z → [0, 1], that has the
3) (A) (B) if B A
associates with each subset A Z a number (A) [0, 1] called the measure of A. If 1 and 2 are two measures on Z such that for any B Z we have 1(B) 2(B) then we say 1 2.
We say 𝜇̂ is the dual of if 𝜇̂ (A) = 1 - (𝐴̅) for all A. It is easy to show that 𝜇̂ is a
not 𝐴̅.
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measure on Z. If 1 2 then 𝜇̂ 1 𝜇̂ 2. We note that 𝜇̂ (A) is the complement of the measure of We call a measure self-dual if (A) = 𝜇̂ (A).
We see that if is self-dual then
(A) = 1 - (𝐴̅) and hence (A) + (𝐴̅) = 1 = (Z). Consider the set function 𝜇̃ defined so that 𝜇̃(A) = a measure: 𝜇̃() = 0, 𝜇̃(X) = 1 and 𝜇̃(A) 𝜇̃(B) if A B. Observation: For any , 𝜇̃ is self-dual. 3
̂ (𝐴) 𝜇(𝐴) + 𝜇 2
. It is easy to show that 𝜇̃ is
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Proof: 𝜇̃̂ = 1 -
̂ (𝐴)) (𝜇(𝐴) + 𝜇 2
=1–
(𝜇(𝐴) +1 − 𝜇(𝐴)) 2
=
𝜇(𝐴) 2
1
𝜇(𝐴)
2
2
+ -
=
̂ (𝐴) 𝜇(𝐴) + 𝜇 2
.
Thus 𝜇̃ associates with any measure a self-dual version Assume g is a mathematical structure, an object, a function, a property or an attribute, that has an associated value for each z Z. Here we shall generically denote this associated
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value as
One fundamental use of measures is to define integrals. An integral of g over the Z with respect to is a weighted aggregation of the collection {
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denoted Choq(g) or Choq(
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Hj = {z(k), k = 1 to j}. With a little algebra we can show
Choq(g) = ∑𝑛𝑗=1 𝜇(𝐻𝑗 )(
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[10, 11]. In this case
𝑛
Sug(g) = 𝑀𝑎𝑥 [(Hj)
More generally in [10] the author’s suggest a general class of integrals can be obtained using a strict copula C [12, 13]. In this case Cop(g) = ∑𝑛𝑗=1(𝐶(
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We recall a strict copula C is a binary function C: [0, 1] [0, 1] → [0, 1] having the properties
1) For any a [0, 1], C(a, 0) = C(0, a) = 0 2) For any a [0, 1] = C(1, a) = C(a, 1) = a 3) For all a1 a2 and b1 b2 where at least one of the “≤” is a “<” we have C(a2, b2) – C(a2, b1) – C(a1, b2) + C(a1, b1) > 0 4
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A notable copula is the Fréchet copula, C(a, b) = M(a, b) + W(a,
) + (1 – ( + )) (a, b)
where M(a, b ) = Min(a, b), W(a, b) = Max((a + b) – 1, 0), (a, b) = a b and with + < 1.
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3. Aggregation Functions An important class of functions that we shall have considerable interest in are aggregation functions [11, 14, 15].
Definition: An aggregation function is a function of n > 1 arguments, f: [0, 1]n → [0, 1], that has the following properties 1) f(0, 0, …, 0) = 0 2) f(1, 1, …, 1) = 1
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3) f(a1, …, an) f(b1, …, bn) if ai bi for all i
Properties 1 and 2 are called boundary conditions and property 3 is called the monotonicity condition.
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There are three interesting classes of aggregation functions [11].
A conjunctive
aggregation function is defined by the additional property f(a1, …, an) Min(a1, …, an)
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A disjunctive aggregation function is defined by the additional property f(a1, …, an) Max(a1, …, an) An averaging or mean like aggregation function is defined by the additional property Min[a1, …, an] f(a1, …, an) Max[a1, …, an] We now observe some notable examples of aggregation functions. Conjunctive:
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Min: f(a1, .., an) = Mini[ai]
Product: f(a1, ..., an) = ∏𝑛𝑖=1 𝑎𝑖
Disjunctive
Max: f(a1, …, an) = Maxi[ai] Disjunctive Sum: f(a1, …, an) = Min[1, ∑𝑛𝑖=1 𝑎𝑖 ]
Mean: 5
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Max, Min and Median 1
Arithmetic mean: f(a1, …, an) = ∑𝑛𝑖=1 𝑎𝑖 𝑛
1
Geometric mean: f(a1, …, an) = (∏𝑛𝑖=1 𝑎𝑖 )𝑛 𝑛 1 𝑎𝑖
∑𝑛 𝑖=1
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Harmonic mean: f(a1, …, an) =
Assume g is a function on Z, g: Z → [0, 1], where we denote g(zi) = ai. If is a measure on Z then the Choq(g) = Choq(g(z1), …, g(zn)) is a mean type aggregation function. In this case we can express this in aggregation function form as
ChoqC(g) = Choq(g(z1), …, g(zn)) = (g(z1), …, g(zn)) = f(a1, …, an) We now note some properties that can be associated with an aggregation function.
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Definition: An aggregation function f is called idempotent if with all ai = a, f(a1, …, an) = a. Definition: An aggregation function f is called symmetric or anonymous if the indexing of the arguments does not matter: f(a1, …, an) = f(a(1), …, a(n)) for any permutation . Definition: A function N: [0, 1] → [0, 1] is called a strong negation if monotonic decision
2) N(N(a)) = a
involutive
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1) N(a) < N(b) if a > b
The standard strong negation is N(x) = 1 – x
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Definition: Assume N: [0, 1] → [0, 1] is a strong negation and f: [0, 1]n → [0, 1] is an aggregation function. The aggregation function 𝑓̂ defined so that 𝑓̂(a1, …, an) = N(f(N(a1), N(a2), …, N(an))) is called the N-dual of f.
A fundamental theorem for fuzzy measures makes use of aggregation functions [16], Theorem: Assume 1, …, q are a collection of fuzzy measures on the space Z. If I = [0, 1] and
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if Agg is an aggregation function, Agg: Iq → I, then the set function defined so that (A) = Agg(1(A), …, q(A)) for all subsets A of Z is a fuzzy measure on Z. Proof: 1) () = Agg(1(), 2(), …, q()) = Agg(0, …, 0) = 0 2) (X) = Agg(1(X), …, q(X)) = Agg(1, …, 1) = 1 3) Assume A and B are two subsets of Z such that A B. Then (B) = Agg(1(B), …, q(B)) Agg(1(A), …, q(A)) (A). 6
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We shall refer to the preceding as the Fundamental Theorem on the Aggregation of Fuzzy Measures, FTAM. We shall use the notation = Agg(1, …, q) to indicate that is defined so that (A) = Agg(1(A), …, q(A)) for all subsets A of Z.
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We see that the FTAM provides a very general method for providing a measure from a combination of other measures.
Definition: A standard monotonic function is a mapping : [0, 1] → [0, 1] having the properties 1) (0) = 0 2) (1) = 1 3) (a) (b) if a b
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The standard monotonic function also provides a method for obtaining a measure from another measure.
Theorem: Assume is a measure on Z and is a standard monotonic function, the set function defined such that 1(A) = ((A)) is a measure on Z.
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We easily see that if 1, …, q are a collection of measures on Z and 1, …, q are a collection of standard monotonic functions, i:[0, 1] → [0, 1]. Then the set function defined so that for any subset A of Z
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(A) = Agg(1(1(A)), 2(2(A)), …, q(q(A))) is a measure on Z.
4. Multi-Criteria Decision Making
Many problems can be modeled in the following manner. We have a collection of criteria C = {C1, …, Cn} and we wish to satisfy some aggregation of the criteria in C. In
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addition we have a collection of alternatives X = {x1, …, xm} which can be used to attain our goal. Furthermore, for each alternative we have a value Cj(xk) [0, 1], the degree to which criteria Cj is satisfied by alternative xk. One commonly used approach is that for each alternative we combine, using an appropriate aggregation function, its satisfaction to that different criteria to obtain a score function S(xk) for xk, thus S(xk) = Agg(C1(xk), …, Cn(xk)). Here then we can use the S(xk) to decide what action to take. 7
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Here we require that Agg is mean type aggregation function, Mini[Ci(xk)] S(xk) Maxi[Ci(xk)]. A very general formulation for this mean type of aggregation, Agg(C1(x), C2(x), …, Cn(x)), can be obtained with the aid of a fuzzy measure on the space C of criteria and the use of an
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appropriate integral [17]. Here we shall use the Choquet integral. The measure in this case of aggregation of multiple criteria shall be interpreted such that for any subset A of C, (A) is the importance weight contributed by the satisfaction of the subset of criteria in A. So here S(x) = Agg(C1(x), …, Cn(x)) = Choq(C1(x), …, Cn(x)) where
Choq(C1(x), …, Cn(x)) = ∑𝑛𝑗=1(𝜇(𝐻𝑗 ) − 𝜇(𝐻𝑗−1 )) 𝐶𝜌(𝑗) (𝑥)
In the above is an index function defined so that C(j)(x) is the jth largest of the Ci(x) and
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Hj = {C(k) for k = 1 to j}, it is the subset of criteria in C with the j largest satisfactions by x. If we denote (Hj) - (Hj-1) = wj then wj 0, since (Hj) (Hj - 1) from the monotonicity of .
Furthermore we see ∑𝑛𝑗=1 𝑤𝑗 = ∑𝑛𝑗=1(𝜇(𝐻𝑗 ) − 𝜇(𝐻𝑗−1 )) = 𝜇(𝐻𝑛 ) −
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𝜇(𝐻0 ) = 1 since (Hn) = (C) = 1 and (H0) = () = 0. Thus here S(x) = Choq(C1(x), …, Cn(x)) = ∑𝑛𝑗=1 𝑤𝑗 𝐶𝜌(𝑗) (𝑥), is a weighted average of Ci(x). Here the weights are determined by the measure which is a
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reflection of the associated importances of the subsets of criteria. Furthermore Min[Ci(x)] jwjC(j)(x) Maxi[Ci(x)] and hence S(x) is a mean type aggregation of the Ci(x).
We note that if C(j)(x) = 1 for j = 1 to j* and C(j)(x) = 0 for j = j* + 1 to n then S(x) = (Hj*). Here Hj* = {C(j) for j = 1 to j*},
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5. Modeling Multi-Criteria Aggregation Using Measures Let us look at this for some notable examples of . The most common type of measure is one in which ({Ci}) = i and (A) = ∑𝐶𝑖∈𝐴 𝛼𝑖 . Here each Ci in A directly makes a contribution of i to the importance of the subset A. Here we observe that (Hj) = ∑𝑗𝑘=1 𝛼𝜌(𝑘) and (Hj - 1) = 𝑗−1
∑𝑘=1 𝛼𝜌(𝑘) . In this case wj = (Hj) - (Hj - 1) = (j). From this we see that
S(x) = ∑𝑛𝑗=1 𝛼𝜌(𝑗) C(j)(x) = ∑𝑛𝑖=1 𝛼𝑖 Ci(x) 8
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It is simply the weighted average of the Ci(x) where the weight associated with Ci is i. Consider the measure * such that *(C) = 1 and *(A) = 0 if A C. Here we need all the criteria for any contribution. In this case (Hj) = 1 for j = n and (Hj) = 0 for j n. In this case with S(x) = ∑𝑛𝑗=1(𝜇(𝐻𝑗 ) − 𝜇(𝐻𝑗−1 )) 𝐶𝜌(𝑗) (𝑥) = ∑𝑛𝑗=1 𝑤𝑗 𝐶𝜌(𝑗) (𝑥)we see that wn = 1 and
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wj = 0 for j n and hence S(x) = C(n)(x) = Mini[Ci(x)]. It is interesting to note here that since (A) = 0 for A C then for each Ci we have ({Ci}) = 0. Thus each criterion individually makes no contribution to the importance. Consider the measure * such that *() = 0 and (A) = 1 for A . In this case with (Hj) = 1 for j 0 and (Hj) = 0 for j = 0 we see that w1 = 1 and wj = 0 for j n. Hence S(x) = jwjC(j)(x) = C(1)(x) = Maxi[Ci(x)]
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We observe here that for each Ci, (Ci) = 1 and thus each Ci makes an importance contribution of one, but there is no addition.
Consider the situation where is a cardinality-based measure [18, 19]. Here we have a collection of parameters rj 0 for j = 0 to n with rj + 1 > rj and r0 = 0 and rn = 1 and (A) = r|A|,
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where |A| is the cardinality of A. Since Hj has cardinality j then (Hj) = rj. Thus here wj = (Hj) - (Hj - 1) = rj – rj - 1. In this case S(x) = ∑𝑛𝑗=1 𝑤𝑗 𝐶𝜌 (𝑗) (𝑥). There is essentially an OWA aggregation of Ci(x) where the OWA weights wj = rj – rj - 1. Further we observe that rj = 𝑗
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∑𝑘=1 𝑤𝑘 . Thus we see that if we want an OWA aggregation with weights wk for k = 1 to n we
can model it as a cardinality based measure with rj = ∑𝑗𝑘=1 𝑤𝑘 . We observe that here for all Cj, ({Cj}) = r1. Thus r1 is the importance of all individual criteria. We further observe that wj = rj – rj -1 is the amount of importance we gain by going from Hj - 1 to Hj. The median aggregation function can be modeled using a cardinality-based measure. Here we must consider two situations n odd and n even. If n is odd we let 𝑛+1
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rj = 0 for j = 0 to rj = 1 for j =
2
− 1
𝑛+1 2
to n
Since (Hj) = rj and wj = (Hj) - (Hj - 1) we see that wj = 0 for j = 1 to wj = 1 for j = wj = 0 for j =
𝑛+1 2
− 1
𝑛+1 2 𝑛+1 2
+ 1 to n 9
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Here we see S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x) = 𝐶𝜌(𝑛+1) (𝑥), which is the median. 2
If n is even we let 𝑛
rj = 0 for j = 0 to - 1 2
rj = 0.5 for j =
𝑛 2
𝑛
rj = 1 for j = + 1 to n
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2
With (Hj) = rj and wj = (Hj) - (Hj - 1) we see 𝑛
wj = 0 for j = 1 to - 1 2
wj = 0.5 j =
𝑛 2 𝑛
wj = 0.5 = for j = + 1 2
𝑛
wj = 0 for j = + 2 to n 2
Here we get 1
1
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S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x) = C 𝜌(𝑛) (x) + 𝐶𝜌(𝑛+2) (x) 2
2
2
2
Let us look at the situation for possibility measures. Here we have associated with each
𝐶𝑖 ∈𝐴
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Ci a value ({Ci}) = i, where at least one i = 1 and where for any subset A C we have (A) = 𝑀𝑎𝑥 [𝛽𝑖 ]. Here the importance contribution of a collection A of criteria is equal to maximal importance of any criteria in the set A. In this case with Hj = {C(k) for k = 1 to j} we have (Hj) = 𝑀𝑎𝑥[𝛽𝜌(𝑘) ] and (Hj - 1) = 𝑀𝑎𝑥[𝛽𝜌(𝑘) ]. Here with wj = (Hj) - (Hj - 1). We have 𝑘= 1 𝑡𝑜 𝑗
𝑘= 1 𝑡𝑜 𝑗−1
wj = 𝑀𝑎𝑥[𝛽𝜌(𝑘) ] -𝑀𝑎𝑥[𝛽𝜌(𝑘) ]. Using this we have S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(k)(x) 𝑘= 1 𝑡𝑜 𝑗
𝑘= 1 𝑡𝑜 𝑗−1
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Here we see that if (j) ≤ 𝑀𝑎𝑥[𝛽𝜌(𝑘) ]) then wj = 0 and if (j) 𝑀𝑎𝑥[𝛽𝜌(𝑘) ]then
wj = (j) -𝑀𝑎𝑥[𝛽𝜌(𝑘) ].
𝑘= 1 𝑡𝑜 𝑗−1
𝑘= 1 𝑡𝑜 𝑗−1
𝑘= 1 𝑡𝑜 𝑗−1
Closely related is a measure based on a t-conorm S [20]. Here S: [0, 1]2 → [0, 1] where 1) S(a, b) = S(b, a)
2) S(a, b, c) = S(a, S(b, c)) = S(S(a, b), c)
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3) S(a1, b1) S(a2, b2) if a1 a2 and b1 b2
4) S(a, 0) = a Here we also have ({Ci}) = i, with at least one i = 1 and (A) = 𝑆[𝛽𝑖 ]. For this measure (Hj) =
𝑆
𝑘= 1 𝑡𝑜 𝑗
[(k)] and hence wj =
𝐶𝑖 ∈𝐴
𝑆
𝑘= 1 𝑡𝑜 𝑗
[(k)] -
𝑆
[(k)]. We note Max is a
𝑘= 1 𝑡𝑜 𝑗−1
special case of this. Another special case is where S(a, b) = a + b – ab, this is called the probabilistic sum. Another example of this S(a, b) = Min(1, a + b), this called the bounded sum 10
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For this aggregation t-conorm Here we see that if
𝑆
𝑘= 1 𝑡𝑜 𝑗
𝑗−1 ∑𝑘=1 𝛽𝜌(𝑘) ≥
𝑗
[(k)] = Min[1, ∑𝑘=1 𝛽𝜌(𝑘) ].
𝑗−1 ∑ 1 then wj = 0 and ∑𝑗−1 𝑘=1 𝛽𝜌(𝑘) 1 then wj = (j) (1 - 𝑘= 𝛽𝜌(𝑘) )
Another interesting class of measures are the Boolean or binary measures. Here the measures have (A) {0, 1}, in this case (Hj) {0, 1}. Since (C) = 1 and () = 0 and
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(Hj) (Hj-1) we see that there exists some j* such (Hj*) = 1 and (Hj*-1) = 0 and hence wj* = (Hj*) - (Hj*-1) = 1 while all other wj = 0. For this measure S(x) = 𝐶𝜌(𝑗∗) (𝑥) . Thus we see that S(x) {Ck(x), for k = 1 to n}, here S(x) is always equal to the satisfaction of one of the criteria, there is no aggregation.
An interesting case of a cardinality-based measure is one which j = 0 for j = 0 to j* – 1 and j = 1* for j = j* to n it is a binary measure. In this case since (Hj) = j we see that wj = 0 for j = 1 to j* – 1 wj = 1 for j = j*
Here we see that S(x) = 𝐶𝜌(𝑗∗) (𝑥)
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wj = 0 for j = j* + 1 to n
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wj = (Hj) - (Hj - 1) is such that
Another class of aggregation functions are based on a granularization of C [21]. Here we let Ri for i = 1 to q be collection of a subsets of C called granules. We have no requirements on
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the Ri other than that they are non-empty. In particular they need not be disjoint. Conceptually we can view each granule Ri as a class or type of criteria. Associated with each Ri is an importance weight i it can contribute. Here we require each i [0, 1] and ∑𝑞𝑖=1 𝛼𝑖 = 1. Using these Ri we can construct a measure on C, called a plausibility measure, such that (A) = ∑𝑞𝑖=1 𝐺𝑖 (𝐴) 𝛼𝑖 for any subset A C where Gi(A) = 1 if A Ri and Gi(A) = 0 if A Ri = . We note here that (Hj) = ∑𝑞𝑖=1 𝐺𝑖 (𝐻𝑗 ) 𝛼𝑖 where Hj = {C(k) for k = 1 to j}. We
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see that since wj = (Hj) – (Hj - 1) = ∑𝑞𝑖=1(𝐺𝑖 (𝐻𝑗 ) − 𝐺𝑖 (𝐻𝑗−1 ))𝛼𝑖 and S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x). Closely related to the plausibility measure is a belief measure b that can also be used to generate an aggregation function. Here for any subset A C, b(A) = ∑𝑞𝑖=1 𝐺∗𝑖 (𝐴) 𝛼𝑖 where 𝐺𝑖∗ (𝐴) = 1 if Ri A and 𝐺𝑖∗ (𝐴) = 0if Ri A. For this measure b(Hj) = ∑𝑞𝑖=1 𝐺∗𝑖 (𝐻𝑗 ) 𝛼𝑖. Another related measure useful for generating an aggregation function is the pignistic ̃ 𝑖 (𝐴) 𝛼𝑖 measure. Here again pig(A) = ∑𝑞𝑖=1 𝐺
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|𝑅 ∩𝐴| where 𝐺̃𝑖 (A) = 𝑖 |𝑅𝑖 |
Two other interesting measures that can be constructed using the granularization Ri for i = 1 to q. However, here instead of associating with each Ri a weighted i [0, 1] where ∑𝑞𝑖=1 𝛼𝑖 = 1we associate with each Ri a weight i [0, 1] and require that 𝑀𝑎𝑥 [i] = 1, here at 𝑖=1 𝑡𝑜 𝑞
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least one of the i = 1. Using this granularization we define the measure 1 on C such that for any subset A C we have 1(A) = 𝑀𝑎𝑥 [Gi(A) i]. Using the above can also define the 𝑖=1 𝑡𝑜 𝑞
measure 2 on C such that for any A C we have that 2(A) = 𝑀𝑎𝑥 [𝐺𝑖∗ (A) i] 𝑖= 1 𝑡𝑜 𝑞
Another relationship between criteria is a prioritization. Here we suggested a formulation for a fuzzy measure that can be used to implement a priority relationship between the criteria. Assume C = {C1, …, Cn} are prioritized so that C1 > C2, …, > Cn. The fundamental idea of
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prioritization is that a lack of satisfaction to a higher priority criterion is not easily compensated by satisfaction to a lesser priority criterion. Here we shall model a measure that captures this idea. Let Lr = {Ck| for k = 1 to r} where r = 1 to n and let L0 = . Here we associate with each Lr a value Vr [0. 1] so that Vr Vr - 1 and Vn = 1.
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We now define a measure on C so that for any subset A we have (A) = 𝑀𝑎𝑥 [VrGr(A)] where Gr(A) = 1 if Lr A and Gr(A) = 0 if Lr A. Here we see 𝑟= 1 𝑡𝑜 𝑛 (A) = Vr* where Lr* is the largest Lr contained in A.
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Here again S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x). In this case again wj = (Hj) - (Hj - 1). Here (Hj) = 𝑀𝑎𝑥 [VrGr(H(j))] and here wj = ((Hj) - (Hj - 1)). 𝑟= 1 𝑡𝑜 𝑛
6. Representing a Logical Expression of Requirements in a Measure Formulation
Again assume C = {C1, …, Cn} are our criteria of interest. Often people can express
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their desired requirements with respect to these multiple criteria in natural language statement which in turn can be expressed in the form of a well-formed formula, WFF, in the language of propositional logic using the set of criteria C, the logic operators (and), (or), (not) and → (if … then) [6].
Examples of this are:
I desire C1 and C2.
C1 C2
I desire C4 or C2 and C3.
C4 (C2 and C3) 12
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C1 → (C2 C6)
I desire if C1 then C2 and not C6.
In the following discussion we shall use the term “literal” to indicate a criteria, Ck, or its negation, Ck. A logical formula is in disjunctive normal form, DNF, if it is a disjunction, sequence of or’s, consisting of one or more disjuncts, where each disjunct is a conjunction, an
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anding or one or more literals. Here then DNF = D1 D2 D3 ... Dm where each disjunct Dk is of the form
Dk = L1 L2 L3 ... L 𝑛𝑘
Where each literal, Li is an element from C or the negation of an element from C, such as C5. The following is a statement in DNF.
C1 (C8 C5) C2 (C5 C3) (C6 C7 C8).
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We shall say a DNF is pure if none of the literals involve a negation, all the literals in any Dk are simply elements from C.
As shown in Mendelson [6] any WFF from propositional logic can be equivalently
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expressed in DNF. We note two WFF are equivalent if their evaluated truth-values are the same for all allocations of truth-values to constituent elements Our objective here is to convert any WFF, P, expressing the aggregation of criteria into a
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measure over the appropriate set of criteria C. Our first step would be to convert P into its equivalent DNF using the ideas described in Mendelson [6]. Since our interest is not primarily in logic manipulation here we shall assume this step has been performed and our point of departure here is a statement in DNF.
We shall first consider the case where the associated DNF is pure, there are no negation type literals. Let DNF = D1 D2 D3 …, Dm , here each Dk is a conjunction of elements
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from C. We now associate with this DNF a measure on C using the following algorithm. 1. Associate with each Dk a subset Sk of C such that Sk is the set of literals in Dk. 2. For any subset A C define Gk(A) so that Gk(A) = 1 if Sk A Gk(A) = 0 if Sk A 3. For any subset A of C define (A) = Maxk[Gk(A)] 13
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Note: We can easily see that is a measure - () = 0, (X) = 1 if B A then (B) (A). Note: is a binary measure, (A) {0, 1} for all A Note: In a logic world in which all the criteria in any subset A where (A) = 1 have the logic value TRUE then the logical evaluation of P and its equivalent DNF is TRUE
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We now consider the situation where some of the literals involve the negation, we have terms like C3 in our DNF. Here we first replace in our DNF any negative type literal, Ci by a new non-negative criteria 𝐶̃𝑖 .
Furthermore 𝐶̃𝑖 is defined such that for any alternative x,
𝐶̃𝑖 (x) = 1 - Ci(x). Here we let 𝐶̃ = C {all 𝐶̃𝑖 }. We now replace our original DNF by an ̃ . We now express 𝐷𝑁𝐹 ̃ as associated DNF,𝐷𝑁𝐹 ̃= 𝐷 ̃ 𝐷 ̃ 𝐷 ̃ …. 𝐷 ̃ 𝐷𝑁𝐹 1 2 3 m
̃ is composed of where each negation type literal, Ci in Dk is replaced by𝐶̃𝑖 . Here each 𝐷 k
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̃a elements from 𝐶̃ , all of which are non-negative type literals. We now associate with 𝐷𝑁𝐹 measure 𝜇̃ on 𝐶̃ using the following algorithm
̃𝑘 a subset 𝑆̃𝑘 of 𝐶̃ such that 𝑆̃𝑘 is the set of literals in 𝐷 ̃𝑘 . 1. Associate with each 𝐷
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2. For any subset 𝐴̃ 𝐶̃ define Gk(𝐴̃) so that Gk(𝐴̃) = 1 if 𝑆̃𝑘 𝐴̃
Gk(𝐴̃) = 0 if 𝑆̃𝑘 𝐴̃
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3. For any subset 𝐴̃ of 𝐶̃ define 𝜇̃(𝐴̃) = Maxk[Gk(𝐴̃)] Example: Here C = {C1, C2. C3} and DNF = C1 (C2 C3).
Here S1 = {C1} and
S2 = {C2, C3}. We now list all the subsets of C. A1 = {} = , A2 = {C1}, A3 = {C2}, A4 = {C3}, A5 = {C1, C2}, A6 = {C1, C3}, A7 = {C2, C3}, A8 = C = {C1, C2. C3} We obtain G1 and G2.
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G1(A1) = 0, G1(A2) = 1, G1(A3) = 0, G1(A4) = 0, G1(A5) = 0, G1(A6) = 1, G1(A7) = 0, G1(A8) = 1
G2(A1) = 0, G2(A2) = 0, G2(A3) = 0, G2(A4) = 0, G2(A5) = 0, G2(A6) = 0, G2(A7) = 1, G2(A8) = 1.
With (A) = Maxk[Gk(A)] we get (A1) = 0, (A2) = 1, (A3) = 0, (A4) = 0, (A5) = 1, (A6) = 1, (A7) = 1, (A8) = 14
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1 If C2(x) > C3(x) > C1(x) then H0 = = A1, H1 = {C2} = A3, H2 = {C2, C3} = A7 and H3 = C = A8 and in this case S(x) = ((H1) - (H0))C2(x) + ((H2) - (H1))C3(x) + ((H3) - (H2))C1(x)
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S(x) = ((A3) - ())C2(x) + ((H7) - (A3))C3(x) + ((H7) - (H8))C1(x) = C3(x)
7. Fusion of Multi-Agent Multi-Criteria Decision Functions Assume C = {C1, C2,
Ci, …, Cn] is a collection of criteria of interest in a decision
problem. In addition assume we have m agents each of which has a preferred multi-criteria decision function expressed in terms of a measure k on C. Thus here we have k for k = 1 to m
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each of which is a measure on C. Our interest here is to fuse, appropriately aggregate, these individual multi-criteria decision function to obtain a fused group multi-criteria decision function. Thus our objective is to obtain + = Agg(1, 2, 3, …, m) where Agg is an aggregation function modeling the type of fusion we desire between the collection of k’s. Here
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we assume the meta-aggregation function is provided by an external decision maker. As we have earlier indicated if we define + so that for each subset A C we let +(A) = Agg(1(A), 2(A), 3(A), …, m(A))
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then + is a measure on C. If our desired imperative for Agg is to find + so that it satisfies all the k then, +(A) = Mink[k(A)]. If our desired imperative for Agg is to find + so that it satisfies at least one of the k then, +(A) = Maxk[k(A)]. If our desire is to find + so that it is a weighted average of the k then +(A) = ∑𝑚 𝑘=1 𝑤𝑘 k(A). Very general formulation for + can be obtained using the Choquet integral for the aggregation operator, this will give us a mean type aggregation. Assume D = {1, 2, …, m}
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are the set of measures on C corresponding to the individual agents multi-criteria decision functions. Let be a measure on D corresponding to the desired aggregation imperative of the individual agents k’s. Here for any subset B of measures from D, (B) is the importance weight, in the formulation of +, associated with the set of agents whose k’s are in B. It is with the measure we capture the type of aggregation of the k’s we desire in the formulation of the group multi-criteria decision function +. We now define our group aggregation measure +, a 15
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measure on C, so that for any subset A of C we have +(A) = Choq((1(A), 2(A), 3(A), …, m(A)). Here we shall let A be an index function so that 𝜌𝐴 (𝑗) (A) is the jth largest k(A). Thus if A(j) = 3 then 3(A) is the jth largest k(A). Furthermore if for = 1 to m we let 𝑀𝜏𝐴 be a
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collection of subsets of D where𝑀𝜏𝐴 = { 𝜌𝐴 (𝑘) for k = 1 to } then 𝐴 𝐴 +(A) = ∑𝑚 𝑗=1(𝜆(𝑀𝜏 ) - (𝑀𝜏−1 )) 𝜌𝐴 (𝑗) (A).
We note here that +(A) is a mean type aggregation, that is for each A C we have Mink[k(A)] +(A) Maxk[k(A)]. Note: Since for any alternative x,
S(x) = Choq 𝜇+ (C1(x), …, Cn(x)) = ∑𝑛𝑗=1(𝜇+ (𝐻𝑗 ) - +(Hj - 1))C(j)(x). where C(j)(x) is the jth largest of the Ci(x) and Hj = {C(k) for k = 1 to j}, then we just need
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calculate +(A) for only n + 1 subsets of C, the Hj for j = 0 to n.
In the preceding we considered the problem of aggregating multi-agent multi-criteria decision functions to obtain a group multi-criteria decision function. In providing a solution to
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this problem we assumed all of the measures being fused were defined over the same space of criteria C. Now we shall generalize this multi-agent fusion to the case where each of the individual measure’s k are defined over a set Ck of criteria which are not necessarily the same.
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In preparation for addressing this problem we provide a method for extending a measure. Assume X and Y are two sets and Z = X Y and let be a measure on X. We are now interested in obtaining an extension of to Z denoted E. Assume A is a subset of Z, A Z, we now define the extended measure of , E, so that E(A) = (A X). We now show that the set function E on Z is a measure on Z.
E() = ( X) = () = 0
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E(Z) = (Z X) = (X) = 1
Monotonicity: Assume A B Z then E(B) = (B X) (A X) E(A)
Thus we see that E is a measure on Z. We now show that E is an extension of from X to Z. Consider any subset A of X, A X, here E(A) = (A X) = (A). We now show that elements not in X have E measure 16
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zero. Assume B X, here B X = hence E(B) = (B X) = () = 0. We now consider the unification of multiple measures. Assume for k = 1 to m that each k is a measure on the set of criteria Ck. We assume no special relationship between the Ck. 𝑘 Under unification we shall make all k measures on the same space. Here we let C = ⋃𝑚 𝑘=1 𝐶 .
the k measures on the same set of criteria, C.
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We now extend each k to be a measure 𝜇𝑘𝐸 on C. We have now unified all the k by making all
We now turn to the problem of fusing a set D = {k for k = 1 to m} of multi-agent multi-criteria aggregation functions defined over possibly different sets of criteria, Ck where we assume our multi-agent aggregation agenda can be expressed via a measure on the space D. Our procedure for accomplishing this is the following: 𝑘 1. Obtain the unified of criteria C = ⋃𝑚 𝑘=1 𝐶 .
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2. Extend each to be a measure 𝜇𝑘𝐸 on C.
3. Fuse these 𝜇𝑘𝐸 using the Choquet integral and the measure to give us a group measure + on C. Here for any subset A of C we have +(A) = Choq(𝜇𝑘𝐸 (A) for k = 1 to m).
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Note: Again we just need calculate +(Hj) where j = 0 to Card (C)
8. Fusion of OWA Aggregators
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Our main concern here is with fusion of multiple OWA multi-criteria decision functions. Here we shall first review the OWA based multi-criteria aggregation process from a perspective useful for our task. Assume C = {Ci for i = 1 to n} is a collection of criteria and Ci(x) [0, 1] is the satisfaction of criteria Ci by alternative x. The OWA operator can be used to provide a method for obtaining the overall satisfaction, S(x), by alternative x to the collection of criteria C. As originally introduced by Yager [8] the OWA operator is defined by a set of weights, wj for
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j = 1 to n where each wj [0, 1] and ∑𝑛𝑗=1 𝑤𝑗 = 1 and an ordering based aggregation. As suggested by Yager [8] the OWA aggregation of the Ci(x) is S(x) = ∑𝑛𝑗=1 𝑤𝑗 C(j)(x) where (j) is the index of the jth largest criteria satisfaction. Thus if (j) = L then CL(x) is the jth largest of 𝑤1 𝑤2 the Ci(x). Here we shall denote W = [ ] and refer to it as the OWA weighting vector. By 𝑤𝑛 choosing different weights in W we can implement different types of aggregation of the Ci(x). 17
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For example if w1 = 1 and all other wj = 0 then S(x) = Maxi[Ci(x)], if wn = 1 and all other wj = 0 1
1
𝑛
𝑛
then S(x) = Mini[Ci(x)]. In the case where all wj = then S(x) = ∑𝑛𝑗=1 𝐶𝜌(𝑗) (x). Many different types of aggregation can implement by appropriate choice of weights W. The value of S(x) depends on both the OWA weighting vector W and the criteria satisfactions, the Ci(x). The
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OWA operator is known to be a mean type aggregation function, Mini(Ci(x)) OWA(C1(x), …, Cn(x)) Maxi(Ci(x)).
In [22] we introduced a characterizing property of an OWA aggregation called the 𝑛−𝑗 attitudinal character, AC, which depends solely on W, AC(W) = ∑𝑛𝑗=1 𝑤𝑗 ( 𝑛−1 ). It can be shown
that AC(W) [0, 1] and for the Max aggregation AC(W) = 1, for the simple average aggregation AC(W) = 0.5 and for the Min aggregation AC(W) = 0. Generally the larger AC(W) the larger
(
The attitudinal character is essentially the OWA aggregation of the argument set
𝑛−1 𝑛−2 𝑛−3
𝑛−𝑛
𝑛−1 𝑛−1 𝑛−1
𝑛−1
,
,
, …,
)
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S(x).
We recall that a cardinality-based measure on the set C of criteria is defined in terms of set of parameters rj for j = 0 to n where rj rj - 1, r0 = 0 and rn = 1. Furthermore for any subset A
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of C we have (A) = r|A|. By appropriate choice of the parameters rj, a cardinality-based measure can be used in coordination with the Choquet to implement a given OWA aggregation of the Ci(x). Here we let r0 = 0 and for j = 1 to n we define rj = ∑𝑗𝑘=1 𝑤𝑘 where wk are the 1
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weights of the OWA aggregation being modeled. Here we see that wj = rj – rj - 1. We note that 𝑟
if all wj = then rj = . 𝑛
𝑛
Using the above-defined cardinality based measure we see Choq(C1(x), C2(x), …, Cn(x)) = ∑𝑛𝑘=1(𝜇(𝐻𝑗 ) - (Hj - 1))C(j)(x) Where C(j)(x) is the jth most satisfied criteria under x and Hj = {C(j) for k = 1 to j}, the j most satisfied criteria under x. Since H has j elements and is a cardinality based measure with
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parameters rj then (Hj) = rj. From this we get Choq(C1(x), C2(x), …, Cn(x)) = ∑𝑛𝑖=1(𝜇(𝐻𝑗 ) - (Hj - 1))C(j)(x) = ∑𝑛𝑖=1(𝑟𝑗 - rj 1))C(j)(x)
Choq(C1(x), C2(x), …., Cn(x)) = ∑𝑛𝑖=1 𝑤𝑗 C(j)(x) = S(x) Thus we see that any OWA aggregation can be formulated in terms of an appropriated cardinality-based measure on the space C of criteria. 18
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9. On the OWA Aggregation of OWA Aggregators Assume C = {Ci for i = 1 to n} are a set of criteria. Let Ci(x) indicate the satisfaction of criteria Ci by alternative x.
Assume (j) is the index of the criteria with the jth largest
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satisfaction by x, its value is C(j)(x). Here j = 1 to n. Assume D = {k for k = 1 to m} are our different OWA aggregations, we have m of these different OWA aggregations. Here w kj is the jth OWA weight for the kth OWA aggregation. We note that ∑𝑛𝑗=1 𝑤𝑘𝑗 = 1 for each of these k. Here the OWA aggregated value under k is Sk(x) = ∑𝑛𝑗=1 𝑤𝑘𝑗 C(j)(x).
We now obtain the simple weighted aggregation of these n OWA aggregations where k is the weighted associated with k. In this case 𝑆̃(x) = ∑𝑚 𝑘=1 𝛽𝑘 Sk(x). This is essentially an OWA
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aggregation 𝜇̃ of the C(j)(x) where the n OWA weights are 𝑤 ̃𝑗 = ∑𝑚 𝑘=1 𝛽𝑘 wkj for j = 1 to n. Here 𝑆̃(x) = ∑𝑛𝑗=1(∑𝑚 𝑘=1 𝛽𝑘 wkj)C(j)(x).
We next consider an OWA Aggregation of the Sk(x) with OWA weights t. Here t is
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the weight associated with the tth largest Sk(x). We note here the m weights, t for t = 1 to m, sum to one. Here we let be an index function so that (k) in the ordered position of Sk(x). Thus if (k) = t then k has the tth largest OWA aggregated value under x. Using this we see the
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𝑚 𝑛 OWA aggregation of the Sk(x) is 𝑆⃡(x) = ∑𝑚 𝑘=1 𝛼𝑣(𝑘) Sk(x) = ∑𝑘=1 𝛼𝑣(𝑘) (∑𝑗=1 𝑤𝑘𝑗 C(j)(x)). We see
𝑆⃡(x) is essentially an OWA aggregation of the C(j)(x) which is based on an OWA aggregation using the cardinality-based measure 𝜇 ⃡ where the OWA weights are 𝑤 ⃡ 𝑗 = ∑𝑚 𝑘=1 𝛼𝑣(𝑘) wkj. Thus ⃡(x) we need (j), the ⃡ 𝑗 C(j)(x). It is important to emphasize that to calculate 𝑆 𝑆⃡(x) = ∑𝑛𝑗=1 𝑤 ordering of Ci(x) and the v(k) the ordering of the Sk(x) which depends on (j), the ordering of the Ci(x).
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Reiterating the OWA aggregation of a collection of m OWA operators of C criteria is also OWA operator of the C criteria with OWA weights 𝑤 ⃡ 𝑗 = ∑𝑚 𝑘=1 𝛼𝑣(𝑘) wkj where wkj are the OWA weights of the constituent operators being aggregated and t are the OWA weights of the aggregating OWA operator and v(k) is the position of the kth operator being aggregated with respect to alternative x. More precisely since 𝑤 ⃡ 𝑗 depends on the alternative x it should be denoted as 𝑤 ⃡ 𝑗 (x). Also since v(k) depends on the alternative x we should denote it as vx(k). 19
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In the following we provide an example of the OWA aggregation of OWA multi-criteria aggregators. Example: Assume we have three criteria, C = {C1, C2, C3}. The satisfaction of these criteria by alternative x is C1(x) = 0, C2(x) = 0.5 and C3(x) = 1. In this case the function is such that (1)
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= 3, (2) = 2 and (3) = 1. Here we have three OWA operators we want to aggregate: 1, 2 and 3. The weighting vectors wkj for these operators are: 𝑘 = 1 𝑘 = 2 𝑘 = 3 𝐶𝜌(𝑗) (𝑥) 𝑤1 0.2 1 0.333 𝐶𝜌(1) (𝑥) = 1 𝑤2 0.5 0 0.333 𝐶𝜌(2) (𝑥) = 0.5 𝑤3 0.3 0 0.333 𝐶𝜌(3) (𝑥) = 0
Since for each OWA operator we have Sk(x) = ∑3𝑗−1 𝑤𝑗𝑘 C(j)(x) then S1(x) = (0.2)(1) + (0.5)(0.5) + (0.3)(0) = 0.2 + 0.25 = 0.45
S3(x) = (0.333)(1 + 0.5 + 0) = 0.5
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S2(x) = (1)(1) + (0)(0.5) + (0)(0) = 1 + 0 + 0 = 1
S2(x) > S3(x) > S1(x) and hence v(1) = 3 v(3) = 1 v(3) = 2
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Below we have v(k) the ordered position of OWA aggregation k, Sk (x). We see that
1 = 0.6 2 = 0.3 3 = 0.1
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We assume the aggregation of the k is implemented by an OWA operator with weights
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Using this we have 𝑆⃡(x) = ∑3𝑘=1 𝛼𝑣(𝑘) Sk(x) = 0.6(1) + 0.3(0.5) + 0.1(0.45) = 0.795 As we previously indicated this OWA aggregation of 1, 2 and 3 results in an OWA operator 𝜇 ⃡ on C with weights 𝑤 ⃡𝑗 0.1 0.6 𝑤1 0.2 1 𝑤2 0.5 0 𝑤3 0.3 0
= ∑𝑚 𝑘=1 𝛼𝑣(𝑘) wkj. 0.3 0.333 0.333 0.333
From the table above we easily get 20
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𝑤 ⃡ 1 = (0.1)(0.2) + (0.6)(1) + (0.3(0.333) = 0.72 𝑤 ⃡ 2 = (0.1)(0.5) + (.6)(0) + (0.3(0.333) = 0.15 𝑤 ⃡ 3 = (0.1)(0.3) + (.6)(0) + (0.3(0.333) = 0.13 We now calculate the OWA aggregation of the Ci(x) using the OWA operator 𝜇 ⃡ with the weights
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𝑤 ⃡ 𝑗 here we get ⃡ 𝑗 C(j)(x) = (0.72)(1) + (0.15)(0.5) + (0.13)(0) = 0.795 𝑆̆(x) = ∑3𝑗=1 𝑤
which is the same value as 𝑆(x).
10. Conclusion
Our particular interest in this paper was in investigating the fusion of multiple OWA aggregation
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functions. Our approach made use of a measure based approach to the formulation and fusion of multi-criteria aggregation functions. First we showed how the OWA operator can be formulated using a cardinality-based measure over the relevant criteria. Thus any OWA operator can be simply represented as cardinality-based measure. A second result we used was the Fundamental
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Theorem on the Aggregation of Fuzzy Measures, the FTAM, which provided a methodology for aggregating measures under various imperatives. By combining our ability to represent the OWA aggregation function as a cardinality-based measure with the FTAM we provided a
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methodology for fusing multiple OWA aggregation functions under different aggregation imperatives. A significant feature of the methodology developed here is that it is based solely on the use of fuzzy measures.
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Zopounidis, C. and Doumpos, M., "Multiple Criteria Decision Making: Applications in Management and Engineering," Springer: Heidelberg, 2017. Greco, S., Ehrgott, M., and Figueira, J. R., Multiple Criteria Decision Analysis: State of the Art Surveys. Heidelberg: Springer, 2016. Al-Shammari, M. and Masri, H., "Multiple Criteria Decision Making in Finance, Insurance and Investment," Springer; Heidelberg, 2015. Ventre, A. G. S., Maturo, A., Hoskova-Mayerova, S., and Kacprzyk, J., "Multicriteria and Multiagent Decision Making with Applications to Economics and Social Sciences," Springer, Heidelberg, 2013. Mateo, J., Multi Criteria Analysis in the Renewable Energy Industry (Green Energy and Technology). London: Springer, 2012. Mendelson, E., Introduction to Mathematical Logic. New York: D. Van Nostrand, 1964. Yager, R. R., Kacprzyk, J., and Beliakov, G., Recent Developments in the Ordered
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Weighted Averaging Operators: Theory and Practice. Berlin: Springer, 2011. Yager, R. R., "On ordered weighted averaging aggregation operators in multi-criteria decision making," IEEE Transactions on Systems, Man and Cybernetics 18, pp. 183-190, 1988. Wang, Z. Y. and Klir, G. J., Fuzzy Measure Theory. New York: Plenum Press, 1992. Klement, E. P., Mesiar, R., and Pap, E., "A universal integral as common frame for Choquet and Sugeno," IEEE Transaction of Fuzzy Systems 18, pp. 178-187, 2010. Beliakov, G., Pradera, A., and Calvo, T., Aggregation Functions: A Guide for Practitioners. Heidelberg: Springer, 2007. Durante, F. and Sempi, C., "Copula theory: an introduction," in Copula Theory and Its Applications, P. Jaworski, F. Durante, W. Hardle and T. Rychlik (eds.), Springer, Berlin, pp. 3-31, 2010. Nelsen, R. B., An Introduction to Copulas. New York: Springer-Verlag, 1999. Yager, R. R., "Intelligent aggregation and time series smoothing," In Time Series Analysis, Modeling and Applications, Pedrycz, W. and Chen, S. M. (Eds), Springer, Heidelberg, pp. 53-75, 2013. Merigó, J. M. and Yager, R. R., "Generalized moving averages, distance measures and OWA operators," International Journal of Fuzziness, Uncertainty and Knowledge-Based Systems 21, pp. 533-559, 2013. Yager, R. R., "A measure based approach to the fusion of possibilistic and probabilistic uncertainty," Fuzzy Optimization and Decision Making 10, pp. 91-113, 2011. Yager, R. R., "Criteria aggregations functions using fuzzy measures and the Choquet integral," International Journal of Fuzzy Systems 1, pp. 96-112, 1999. Yager, R. R., "Satisfying uncertain targets using measure generalized Dempster-Shafer belief structures " Knowledge Based Systems 142, pp. 1-6, 2018. Yager, R. R., "Uncertainty modeling using fuzzy measures," Knowledge-Based Systems 92, pp. 1-8, 2016. Weber, S., "Decomposable measures and integrals for Archimedean t-conorms," Journal of Mathematical Analysis and Applications 101, pp. 114-138, 1984. Pedrycz, W., "Granular Computing," CRC Press, Boca Raton, Fl, 2013. Yager, R. R., "On the cardinality index and attitudinal character of fuzzy measures," International Journal of General Systems 31, pp. 303-329, 2002.
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*Conflict of Interest Form
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No Conflict Exists