Set Function Representations of Alternatives’
Relative Features
Eiichiro Takahagi
Senshu University, Kawasaki 2148580, Japan
[email protected]
Abstract. In a weighted sum model such as the Analytic Hierarchy Process, a set function value is constructed from the weights of the model.
In this paper, we use relative individual scores to propose a set function that shows the features of an alternative. The set function value
for the alternative is calculated by averaging the values of the set function representation of the weights generated when the alternative has
the highest comprehensive score. By interpreting the functions, we can
understand the features of an alternative. We discuss the properties of
the set functions, and extend to Choquet integral models.
Keywords: alternatives, multi-criteria decision-making, set functions,
fuzzy measures, Choquet integral, Shapley value, Möbius transformation.
1
Introduction
In multi-criteria decision-making models, such as the Analytic Hierarchy Process [3] and Choquet integral models [2][4], it is useful to display the features
of an alternative. The figures are determined from the relative evaluation scores
among alternatives. There are some works to explain of a decision such as [6],
[8], and [9]. Especially in [9], explanations are generated by analyzing whether
the decision would be taken by changing the values of weights. In this paper, by
changing the values of weights, for an alternative, we show which set of criteria
is important differnt conditions on the comparison of weights and the figure of
the degree. The aim of this paper is to define the figures using set functions, and
analyze the set function properties.
Table 1. Example 1 (Two Criteria and Four Alternatives Model)
Alternatives Criterion 1 (C1) Criterion 2 (C2)
A1
0.35
0.15
0.05
0.50
A2
0.24
0.34
A3
0.36
0.01
A4
A. Laurent et al. (Eds.): IPMU 2014, Part I, CCIS 442, pp. 246–255, 2014.
c Springer International Publishing Switzerland 2014
Set Function Representations of Alternatives’ Relative Features
247
Table 1 is an example of a two-criteria (C1 and C2) and four-alternatives (A1,
. . ., A4) multi-criteria decision-making model. In the weighted sum model, if a
decision maker (DM) sets greater importance on C1, then the alternative A1 is
selected, because it has a large score (0.35) for C1 and a modest score (0.15)
for C2. If the DM sets extremely high importance on C1, then A4 is selected,
because the A4 score for C1 is a little higher than that of A1. A2 is selected
if the DM sets greater importance on C2, and A3 is selected if C1 and C2 are
considered to be equally important.
Using these relations among alternatives scores, we propose the set function
representation EX,j (A) of alternative j. Figure 2 illustrates the result of the
proposed set function.
In [10], we proposed the concept of set functions. In this paper, we propose a
uniform generating method, compare this with the random generating method,
and extend the concept to Choquet integral models.
2
Notation
Let Y = {1, . . . , n} be the set of criteria (n: number of criteria), and m be the
number of alternatives. xji ∈ [0, 1], i = 1, . . . , n, j = 1, . . . , m) is the individual
score of the i-th criterion and j-th alternative. The xji have the property of
strong commensurability. X is the matrix representation of xji . In this paper, X
is given in advance, such as by pairwise
comparisons.
w k = (w1k , . . . , wnk ), wik ∈ [0, 1], i wik = 1 gives the weight of each criterion.
As we calculate various cases by varying w, we use the superscript k.
The comprehensive score yjk is calculated as the weighted sum
yjk =
n
i=1
(wik xji ).
(1)
A fuzzy measure μ is defined as
μ : 2Y → [0, 1] where μ(∅) = 0 and μ(A) ≥ μ(B) if A ⊇ B.
The comprehensive score
yjk =
n
i=1
yjk
(2)
calculated by Choquet integral is
[xjσ(i) − xjσ(i+1) ]μ({σ(1), . . . , σ(i)})
(3)
where σ(i) is the permutation on Y , that is, xjσ(1) ≥ . . . ≥ xjσ(n) , Y =
{σ(1), . . . , σ(n)}, σ(n + 1) = n + 1, and xjn+1 = 0. The Shapley value of μ [1][5] is
φi (μ) ≡
γn (S)[μ(S) − μ(S \ {i})] where γn (S) =
S⊆Y
(n− | S |)!(| S | −1)!
(4)
n!
and the Möbius transformation [7] is
ν(A) ≡
(−1)|A\B| μ(B) and μ(A) =
ν(B), ∀A ∈ 2Y .
B⊆A
B⊆A
(5)
248
3
3.1
E. Takahagi
Set Function Representations (Weighted Sum Cases)
Set Functions
EX,j (A) shows the degree to which alternative j is the best alternative when
criteria in A have the large weights. If EX,j (A) has a high score, we interpret
that alternative j is usually selected when wi , ∀i ∈ A are large values. For some
X, EX,j (A) is defined as
EX,j : 2Y → R+ , EX,j (∅) = 0, j = 1, . . . , m.
(6)
When n = 2, Y = {1, 2}, EX,j is constituted of EX,j (∅), EX,j ({1}), EX,j ({2}),
and EX,j ({1, 2}). EX,j ({1}) shows the degree to which alternative j has the highest comprehensive score when only w1 is assigned a greater score. EX,j ({1, 2})
shows the degree to which alternative j has the highest comprehensive score when
both w1 and w2 are assigned greater values, that is, cases close to w1 = w2 .
3.2
Set Function for wk
E k (A) shows the reconstituted weight of set A and is calculated by rank depended method such as the Choquet integrals. For example, if n = 3 and w2k ≥
w3k ≥ w1k , then E k ({2}) = w2k − w3k , E k ({2, 3}) = w3k − w1k , and E k ({1, 2, 3}) =
w1k .
Definition 1 (Set Function for w). For some wk , the set function E k is
k
k
− wσ(i+1)
],
E k ({σ(1), . . . , σ(i)}) = [wσ(i)
(7)
k
k
(i = 1, . . . , n) where σ(i) is the permutation on Y , that is, wσ(1)
≥ . . . ≥ wσ(n)
,
k
Y = {σ(1), . . . , σ(n)}, σ(n + 1) = n + 1, and wn+1 = 0.Values that are not asn
signed by eq. (7) are zero, that is, E k (A) = 0, ∀A ∈ (2Y \ i=1 {{σ(1), . . . , σ(i)}}).
For A ∈ {{σ(1)}, {σ(1), σ(2)}, . . . , {σ(1), . . . , σ(n)}}, the E k (A) can be active,
E k (A) > 0. ({σ(1)}, {σ(1), σ(2)}, . . . , {σ(1), . . . , σ(n)})) is the maximum chain
of Y .
Theorem 1. For any wk and i,
E k (B) = wik ,
(8)
[| A | E k (A)] = 1.
(9)
Bi
A∈2Y
Proof. Eq. (8) is trivial.
=
A∈2Y
n
i=1
[| A | E k (A)] =
n
[| {σ(1), . . . , σ(i)} | E k ({σ(1), . . . , σ(i)})]
i=1
k
k
[i(wσ(i)
− wσ(i+1)
)] =
n
i=1
k
wσ(i)
= 1.
Set Function Representations of Alternatives’ Relative Features
249
For example, if w k = (0.7, 0.3), then E k ({1}) = 0.7 − 0.3 = 0.4, E k ({2}) = 0,
E ({1, 2}) = 0.3, and E k ({1}) + E k ({2}) + 2E k ({1, 2}) = 1.
k
3.3
Set Functions for the Alternative j
For weights w k , if the alternative j is selected, the set function E k (A), ∀A belongs
∗
. qjk
to alternative j. By generating multiple w k , k = 1, . . . , K, we calculate EX,j
is a flag for when alternative j is selected for w k . If two or more alternatives are
selected for w k , qjk is assigned on a pro-rata basis, that is,
H k = {j | yjk ≥ ylk , ∀l = 1, . . . , m},
K
1/ | H k | if j ∈ H k
qjk .
qjk =
, Qj =
0
otherwise
k=1
(10)
(11)
m
Obviously, j qjk = 1∀k and j=1 Qj = K.
As the selection of the best alternative is depended on weights and individual
∗
that shows the relative features of alternative j
scores, the set function EX,j
is defined as the average value of E k (A) when alternative j is selected. In this
paper, we calculate the set function using simulations.
+
∗
∗
). EX,j
and EX,j
are defined as
Definition 2 (EX,j
+
EX,j
(A) =
∗
EX,j
(A)
K
[qjk E k (A)],
k=1
+
=EX,j
(A)/Qj , ∀A
(12)
∈ 2X .
(13)
Let d > 0 be the simulation number (positive integer). In this method, we use
wi ∈ {0, 1/d, 2/d, . . . , d/d}.
Definition 3 (Uniform Generating Method). Ω U is the set of all weights
wk for the simulation.
Ω U = {w | wi ∈ {0, 1/d, 2/d, . . . , d/d} where
wi = 1 }
(14)
We number the elements of Ω U , that is, Ω U = {w1 , . . . , wK }.
k
Definition 4 (Random Generating Method). w i is assigned by uniform random numbers in (0, 1). wik is given by wik =
w k
i k ∀k,
iw i
and Ω R = {w1 , . . . , wK }.
Y
are
The average
standard deviation of A ∈ 2
and
K
K
M
k
SD
M
EX (A) ≡ [ k=1 E (A)]/K and EX (A) ≡ [ k=1 (E k (A) − EX
(A))2 /K]0.5 .
∗
Figure 1 illustrates the calculation process of EX,j (A) in the uniform generating method, example 1, and d = 100. For each w1 and w2 = 1 − w1 , E k (A) are
∗
calculated from eq. (7). For w1 ∈ [0, 0.457], A2 is selected, therefore EX,2
({1})
k
∗
is the average value of E ({1}) for w1 ∈ [0, 0.457], that is, 0.567, EX,2 ({2}) = 0,
∗
and EX,2
({1, 2}) = 0.2165.
250
E. Takahagi
1
k
E ({1})
Ϯ͘ϱ
k
E ({2})
0.9
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0.7
k
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0.3
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0
A2
0
A3
A1
0.5
A4
1 w1
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΂ϭ΃͕ͲϬ͘ϳϳ
ϭ
Ͳϭ͘ϱ
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΂Ϯ΃͕ͲϬ͘ϳϳ
ϰ
΂ϭ͕Ϯ΃͕Ͳϭ͘ϱϬ
ͲϮ͘Ϭ
W
Fig. 2. EX,j
of Example 1
Fig. 1. Calculation Process
3.4
΂ϭ΃͕ͲϬ͘ϰϲ
΂Ϯ΃͕ͲϬ͘ϳϳ
Ͳϭ͘Ϭ
Interpretation and Properties of the Set Functions
Theorem 2. For any j where Qj > 0,
∗
[| A | EX,j
(A)] = 1
(15)
A∈2Y
Proof.
∗
| A | EX,j
(A) =
A∈2Y
=
K
[qjk
k=1
+
| A | EX,j
(A)/Qj =
A∈2Y
A∈2Y
(| A | (E k (A)))]/
|A|
qjk =
k=1
K
k=1
qjk /
K
(qjk E k (A))/
k=1
A∈2Y
K
K
K
qjk
k=1
qjk = 1
k=1
∗
(A) shows the selectability of alternative j, because
From the theorem, EX,j
is the average value when alternative j is selected. As the average and
W
(A) as
standard deviation of E k (A) differ according to | A |, we define EX,j
∗
(A)
EX,j
Definition 5
W
EX,j
(A) ≡
∗
M
EX,j
(A) − EX
(A)
SD (A)
EX
.
(16)
W
Figure 2 is EX,j
(A) from example 1. The figure shows the relative future
performance of each alternative. For example, A1 is selected when E k ({1}) has
a high score, and is not selected when E k ({2}) has a high score. A3 is selected
when E k ({1, 2}) has a high score, and is not selected when either E k ({1}) or
E k ({2}) has a high score.
Set Function Representations of Alternatives’ Relative Features
251
Theorem 3. For any A,
m
W
Qj EX,j
(A) = 0.
(17)
j=1
Proof.
m
W
Qj EX,j
(A) =
j=1
m
=
j=1
m
+
M
[EX,j
(A)/Qj − EX
(A)]Qj
SD (A)
EX
j=1
+
M
[EX,j
(A) − EX
(A)Qj ]
=
SD (A)
EX
W
EX,j
(A)
EX,j (A) as
[
K
m M
qjk E k (A)] − KEX
(A)
j=1 k=1
SD (A)
EX
=0
can be used to compare alternatives. For this purpose, we define
Definition 6
+
EX,j
(A) ≡ EX,j
(A)/K,
∀A ∈ 2Y
(18)
Theorem 4
m j=1 A∈2Y
[| A | EX,j
(A)] = 1
(19)
Proof
m [| A |
j=1 A∈2Y
1 =
K
Y
A∈2
m
j=1
EX,j
(A)]
[| A |
=[
+
EX,j
(A)]
m j=1 A∈2Y
+
| A | EX,j
(A)]/K
K
1 K
=
[| A | E k (A)] =
=1
K
K
Y
k=1 A∈2
Figure 3 shows a comparison between the uniform and random generating
methods. Both graphs have the same trend—if E k ({1}) is high, A1 is usually
selected, and if E k ({1, 2}) is high, A2 or A3 are usually selected.
3.5
Shapley Value and Möbius Transformation
As set functions are closely related to fuzzy measure theories, we use the Möbius
transformation and discuss the resulting properties.
Definition 7. The average importance of alternative j and criterion i is defined
as
=
EX,j (B).
(20)
SX,ji
Bi
252
E. Takahagi
Ϭ͘ϯ
Ϭ͘ϯ
΂Ϯ΃͕Ϭ͘Ϯϱ
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΂Ϯ΃͕Ϭ͘ϭϵ
Ϭ͘Ϯ
΂ϭ΃͕Ϭ͘ϭϳ
Ϯ
ϭ
΂ϭ͕Ϯ΃͕Ϭ͘ϭϬ
Ϭ͘ϭ
Ϯ
΂ϭ΃͕Ϭ͘ϭϯ
ϯ
ϭ
ϰ
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΂ϭ͕Ϯ΃͕Ϭ͘Ϭϴ
΂ϭ͕Ϯ΃͕Ϭ͘Ϭϲ
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΂ϭ΃͕Ϭ͘ϬϬ
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΂ϭ͕Ϯ΃͕Ϭ͘ϬϬ
΂ϭ΃͕Ϭ͘ϬϮ
΂Ϯ΃͕Ϭ͘ϬϬ
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΂ϭ͕Ϯ΃͕Ϭ͘ϭϮ
ϯ
΂ϭ΃͕Ϭ͘Ϭϯ
ϰ
΂ϭ΃͕Ϭ͘Ϭϯ
΂ϭ͕Ϯ΃͕Ϭ͘ϬϬ
΂Ϯ΃͕Ϭ͘ϬϬ
΂Ϯ΃͕Ϭ͘ϬϬ
΂Ϯ΃͕Ϭ͘ϬϬ
΂ϭ΃͕Ϭ͘ϬϬ
΂Ϯ΃͕Ϭ͘ϬϬ
Ϭ͘Ϭ
΂Ϯ΃͕Ϭ͘ϬϬ
Fig. 3. Set Function Representation of Example 1 (Weighted Sum)
From the definition,
SX,ji
=[
K
qjk wik ]/K.
(21)
k=1
Definition 8 (Cumulative Set Function). The Cumulative Set Function
GX,j for EX,j
is defined as
GX,j (A) =
B⊆A
[| B | EX,j
(B)].
(22)
Eq. (22) is the inverse Möbius transformation of [| B | EX,j
(B)]. As [| B |
≥ 0∀B, GX,j are monotone and super-additive fuzzy measures. As the
(B))/ | B |, SX,ji
is
Shapley value of the fuzzy measure GX,j is Bi (| B | EX,j
the Shapley value of criterion i when alternative j is selected.
(B)]
EX,j
Theorem 5. GX,j and SX,ji
have the following properties:
SX,ji
m n
j=1 i=1
m
j=1
K
1 k k
=
qj wi
K
(23)
k=1
SX,ji
=1
SX,ji
=
(24)
K
K
1
, ∀i if
w1k = . . . =
wnk
n
GX,j (Y ) = Qj /K
k=1
(25)
k=1
(26)
Set Function Representations of Alternatives’ Relative Features
253
Proof. From Theorem 1,
SX,ji
=
m n
j=1 i=1
As
Bi
EX,j
(B) =
SX,ji
=
K
K
1 k k
1 k k
qj E (B) =
qj wi .
K
K
k=1 Bi
k=1
m K
K
n
n
1 k
1 k k
qj wi =
wi = 1.
K
K
j=1 i=1
i=1
k=1
k=1
K
n
K
1 k
1
1 k
wi = from the assumption of eq. (25) and
wi = 1,
K
n
K
i=1
k=1
k=1
m
j=1
SX,ji
=
K m
K
1 k k
1 k
1
qj wi =
wi = .
K
K
n
j=1
k=1
k=1
From Theorem 1,
GX,j (Y ) =
[| B | E (B)] =
[| B |
B∈2Y
k
k=1
K
B∈2Y
k
qjk E k (B)]
=
qjk
k=1
K
=
Qj
.
K
SX,ji
is the average weight i when alternative j is selected and the sum of i and
j is 1. GX,j (Y ) shows the frequency with which alternative j is selected.
4
Set Function Representations (Choquet Integral Cases)
In the Choquet integral cases, we divide a fuzzy measure feature into weights
and interaction degrees. In this model, we use the normal fuzzy measure, that
is μ(Y ) = 1. The weight of a fuzzy measure μk is the Shapley value of the
fuzzy measure, and the interaction degree of a fuzzy measure μk is the Möbius
transformation ν k of μk without singletons.
Definition 9 (Weights and Interaction Degrees).
wik ≡ φi (μk ),
0
k
M (A) ≡
ν k (A)
(27)
if | A |≤ 1
.
otherwise
(28)
Analogous to the weighted sum cases, by generating lots of μk , we analyze
the average set functions when the alternative is selected.
Definition 10 (Uniform Generating Method μ). Ω F is the set of all fuzzy
measures μk for the simulation,
Ω F = {μ | μ(A) ∈ {0, 1/d, 2/d, . . . , d/d}, ∀A ∈ 2Y \ {∅, Y }
where μ is a fuzzy measure }.
(29)
254
E. Takahagi
Let K =| Ω F |, and let qjk and Qj be as defined in eqs. (11) , where the yjk are
∗
W
, EX,j
, and EX,j
are calculated in the same way as
calculated by eq. (3). EX,j
for the weighted sum cases.
Definition 11.
∗
MX,j
(A) ≡
k
[qjk M k (A)]/Qj ,
C
MX,j
(A) ≡
∗
[| B | MX,j
(B)]
(30)
B⊆A
k=1
∗
C
MX,j
(A) shows the interaction degree among the elements of A. MX,j
(A) shows
∗
C
the cumulative set function of MX,j (A). If MX,j (A) is positive, alternative j is
usually selected when the fuzzy measure μk has super-additivity among A.
5
Numerical Examples
In Table 2, X denotes the model of example 2, where n = 4 and m = 5. Table
2 gives the set function representations of the weighted sum model using the
uniform generating method, d = 100, and K = 176, 851. As Q1 = 0, A1 is
not selected for any weights. As EX,3
({3}) has the highest value, A3 is usually
k
({2}) = 0.01 is
selected when E ({3}) is a high value. Despite x32 = 0.4, EX,3
5
k
fairly low, because x2 = 0.44. When E ({2}) has a high value, A5 is selected.
Table 2. Set Function Representation of Example 2 (Weighted Sum)
j
A1
A2
A3
A4
A5
j
1
2
3
4
5
{1}
0.03
0.00
0.04
0.00
{2} {1, 2}
0.00
0.01
0.00
0.05
0.00
0.00
0.01
0.01
C1
0.18
0.44
0.01
0.35
0.02
X
C2 C3
0.1 0.2
0.01 0.04
0.4 0.45
0.05 0.11
0.44 0.2
C4
0.12
0.03
0.05
0.5
0.3
{3} {1, 3} {2, 3} {1, 2, 3}
0.00
0.06
0.00
0.00
0.00
0.01
0.01
0.00
0.00
0.02
0.00
0.00
0.00
0.02
0.00
0.00
Qj
0
10,553
53,425
75,605
37,269
EX,j
SX,ji
C1 C2 C3 C4
0.04
0.05
0.13
0.03
0.01
0.08
0.06
0.11
0.01
0.14
0.08
0.03
0.00
0.03
0.16
0.05
{4} {1, 4} {2, 4} {1, 2, 4} {3, 4} {1, 3, 4} {2, 3, 4}
0.00
0.00
0.06
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.01
0.01
0.00
0.01
0.01
0.00
0.00
0.00
0.02
0.00
0.00
0.01
0.00
0.01
Y
0.00
0.02
0.03
0.01
Table 3 gives results for the Choquet integral case using the uniform generating method, where d = 10 and K = 25.683 × 109 . Unlike the weighted
sum model, A1 is selected when super-additivity exists between C1, C2, and C3
C
C
({1, 2, 3}) = 2.55), or Y (MX,1
(Y ) = 4.26). A2 is usually selected when
(MX,1
k
W
E ({1}) has a high value (EX,2 ({1} = 3.66) and/or when {1, 2}, {1, 3} and/or
C
sub-additivity exists between C1, C2, and C3 (MX,2
({1, 2, 3}) = −3.02).
Set Function Representations of Alternatives’ Relative Features
255
Table 3. Set Function Representation of Example 2 (Choquet Integral)
W
EX,j
j
1
2
3
4
5
0.92
3.66
-0.27
0.38
-0.31
j
1
2
3
4
5
Qj
0.080 × 109
0.130 × 109
7.554 × 109
10.227 × 109
7.692 × 109
6
{1}
{2} {1, 2}
-0.39
-0.41
0.13
-0.37
0.38
0.39
0.64
-0.02
-0.07
0.11
{3} {1, 3} {2, 3} {1,2,3}
-0.04
-0.40
0.54
-0.29
-0.13
1.99
0.86
0.12
0.04
-0.21
-0.32
-0.33
0.43
-0.32
0.01
0.92
0.43
0.42
-0.28
-0.06
{4} {1, 4} {2, 4} {1,2,4} {3, 4} {1,3,4} {2, 3,4}
-0.40
-0.41
-0.36
0.35
-0.10
-0.23 -0.33
-0.11 -0.33
-0.32 -0.18
0.44 -0.11
-0.27 0.34
C
MX,j
-0.26
-0.10
-0.33
0.21
0.05
-0.31
-0.33
-0.03
0.04
-0.02
0.07
-0.06
-0.24
0.40
-0.30
-0.43
-0.43
0.16
-0.29
0.24
{1, 2} {1, 3} {2, 3} {1, 2, 3} {1, 4} {2, 4} {1, 2, 4} {3, 4} {1, 3, 4} {2, 3, 4}
0.41
-0.85
0.08
0.04
-0.13
1.02
-0.93
-0.07
0.02
0.05
-0.37
-0.18
0.71
-0.18
-0.46
2.55
-3.02
0.03
0.12
-0.17
-0.49
-0.84
-0.12
0.27
-0.21
-0.21
0.20
0.03
-0.23
0.28
2.32
-2.43
0.28
-0.28
0.12
-0.02
0.27
-0.01
-0.15
0.21
2.46
-2.73
-0.16
-0.25
0.51
-0.22
0.57
0.08
-0.56
0.67
Y
-0.46
-0.88
-0.01
-0.05
0.09
Y
4.26
-2.67
-0.07
-0.29
0.46
Conclusion
We have defined set functions that show the relative features of alternatives. There
is room for further research into the interpretation of these set functions. In the
uniform generating method, the simulation number K increases exponentially with
n or d, especially in Choquet integral cases. In Choquet integral cases, it is hard to
W
∗
(A) and MX,j
(A) when
give an intuitive explanation of the difference between EX,j
| A |≥ 2.
References
1. Shapley, L.: A value for n-person games. Contribution Theory of Games, II. Annals
of Mathematics Studies 28, 307–317 (1953)
2. Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1954)
3. Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill (1980)
4. Murofushi, T., Sugeno, M.: A theory of fuzzy measure: Representation, the Choquet
integral and null sets. J. Math. Anal. Appl. 159, 532–549 (1991)
5. Murofushi, T., Soneda, S.: Techniques for reading fuzzy measures (III): interaction
index. In: 9th Fuzzy System Symposium, Sapporo, Japan, pp. 693–696 (1993) (in
Japanese)
6. Klein, D.A.: Decision analytic intelligent systems: automated explanation and
knowledge acquisition. Lawrence Erlbaum Associates, Psychology Press (1994)
7. Fujimoto, K., Murofushi, T.: Some Characterizations of the Systems Represented by
Choquet and Multi-Linear Functionals through the use of Möbius Inversion. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5, 547–561
(1997)
8. Montmain, J., Mauris, G., Akharraz, A.: Elucidation and decisional risk in a multi
criteria decision based on a choquet integral aggregation: A cybernetic framework.
International Journal of Multi-Criteria Decision Analysis 13, 239–258 (2005)
9. Labreuche, C.: A general framework for explaining the results of a multi-attribute
preference model. Artificial Intelligence 175, 1410–1448 (2011)
10. Takahagi, E.: Feature Analysis of Alternatives: Set Function Representations and
Interpretations. Journal of Japan Society for Fuzzy Theory and Intelligent Informatics 26, 827–833 (2013) (in Japanese)