# Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data

Authors: Chen Li; Gong Zeng-tai; Duan Gang
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

Fuzzy measures and fuzzy integrals have been successfully used in many real applications. How to determine fuzzy measures is a very difficult problem in these applications. Though there have existed some methodologies for solving this problem, such as genetic algorithms, gradient descent algorithms, neural networks, and particle swarm algorithm, it is hard to say which one is more appropriate and more feasible. Each method has its advantages. Most of the existed works can only deal with the data consisting of classic numbers which may arise limitations in practical applications. It is not reasonable to assume that all data are real data before we elicit them from practical data. Sometimes, fuzzy data may exist, such as in pharmacological, financial and sociological applications. Thus, we make an attempt to determine a more generalized type of general fuzzy measures from fuzzy data by means of genetic algorithms and Choquet integrals. In this paper, we make the first effort to define theσ-λ rules. Furthermore we define and characterize the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on σ-λ rules. In addition, we design a special genetic algorithm to determine a type of general fuzzy measures from fuzzy data.


---

## Body

## 1. Introduction

Fuzzy measures [1–4] and fuzzy integrals [5–9] have been applied successfully in multiattributes decision-making [10, 11], classification [12, 13], information fusion [14–18], nonlinear multiregression [19], feature selection [20, 21] and image processing. The reason of success is from the highly nonadditive and non-linear characteristics of fuzzy measures and fuzzy integrals. Fuzzy measure is the generalization of classical measure by using nonadditivity instead of additivity, which makes fuzzy measure be able to describe the importance of each individual information source (attribute or classifier) as well as the interaction [13], among them.The Choquet integral [22–26] with respect to fuzzy measure is often used in information fusion and data mining as a nonlinear aggregation tool. The nonadditivity of fuzzy measures can effectively describe the interaction among the contributions from each attribute toward some target. Some works have shown successful applications of the Choquet integral in nonlinear multiregressions, classifications, and decisionmakings [19, 25, 27–30], where the values of fuzzy measure are usually regarded as unknown parameter to be elicited from training data sets.Most of existed works can only deal with the data consisting of classic numbers which may arise limitations in practical applications. It is not reasonable to assume that all data are real data before we elicit them from practical data. Sometimes, fuzzy data may exist, such as in pharmacological, financial, and sociological applications. Genetic algorithm (GA) is a stochastic search method for optimization problems based on the mechanics of natural selection and natural genetics. GA has demonstrated considerable success in providing good solutions to many complex optimization problems and received more and more attentions during the past three decades. The advantage of GA just make it able to obtain the global optimal solution fairly. In addition, compared with the traditional methods, GA has the ability to avoid getting stuck at a local optimal solution, since GA search from a single point. Thus, we make an attempt to determine a more generalized type of general fuzzy measures from fuzzy data by means of genetic algorithms and Choquet integrals.The rest of this study is organized as follows. In Section2, the basic definitions of fuzzy measures based on σ-λ rules are reviewed. Section 3 briefly introduces the basic concepts on the Choquet integral of real-valued functions based on  rules and gives the operational schemes of its on discrete sets. In Section 4, we formulate the problems to be solved. Section 5 uses genetic algorithm optimization to determine fuzzy measures from real-valued data. Section 6, introduces the Choquet integral of interval-valued functions based on rules. Consequently, we use genetic algorithm optimization to determine fuzzy measures from interval-valued data. Section 7 discusses the Choquet integral of fuzzy number-valued functions based on  rules, and then uses genetic algorithm optimization to determine fuzzy measures from fuzzy number-valued data. Finally, conclusions are made in Section 8.


## 2. Fuzzy Measure Based onσ-λ Rules

Definition 1.
LetX be a nonempty set and 𝒜 a σ-algebra on the X. A set function μ is called a fuzzy measure based on σ-λ rules if
(1)μ(⋃i=1∞‍Ai)={1λ{∏i=1∞‍[1+λμ(Ai)]-1},λ≠0,∑i=1∞‍μ(Ai),λ=0,
where λ∈(-(1/supμ),∞)⋃‍{0}, {Ai}⊂𝒜, and Ai∩Aj=∅ for all i,j=1,2,… and i≠j.
Particularly, ifλ=0, then σ-λ rule is σ-additivity.Definition 2.
Let𝒜 be a σ-algebra on the X. μ is called Sugeno measure based on σ-λ rules if μ satisfies σ-λ rules and μ(X)=1. Briefly we denoted gλ.Remark 3.
In Definition1, if n=2, then
(2)μ(A∪B)={μ(A)+μ(B)+λμ(A)μ(B),λ≠0,μ(A)+μ(B),λ=0.Remark 4.
In Definition2, gλ is a classical probability measure if λ=0, and it can be represented by a wide range of nonadditive measure as long as we select proper parameters, many scholars think that it is a very important kind of nonadditive measure [31–33].Example 5.
Three workers,x1,x2, and x3, are engaged in producing the same kind of products; the efficiencies of every people are given as follows: μ(x1)=5,μ(x2)=6, and μ(x3)=8. Then we can get the joint efficiencies by use of σ-λ rules as shown in Table 1.Table 1
The values of set functionμ in Example 5.
Set
Value ofμ
E
1
=
{
x
1
}
5
E
2
=
{
x
2
}
6
E
3
=
{
x
1
,
x
2
}
11
+
30
λ
E
4
=
{
x
3
}
8
E
5
=
{
x
1
,
x
3
}
13
+
40
λ
E
6
=
{
x
2
,
x
3
}
14
+
48
λ
E
7
=
{
x
1
,
x
2
,
x
3
}
240
λ
2
+
118
λ
+
19Remark 6.
In Example5, xi can be viewed as a attribute, i=1,2,3, we can calculate the contribution of their joint attributes by use of σ-λ rules if we only know the contribution of individual attribute gλ(xi),i=1,2,3.Theorem 7.
Letgλ be a Sugeno measure based on σ-λ rules. If λ≥0, then gλ has the monotonicity.Proof.
LetE,F∈ℱ and E⊂F. Since F=E∪(F-E), this implies that
(3)gλ(F)=gλ(E∪(F-E))=gλ(E)+gλ(F-E)+λgλ(E)gλ(F-E),(4)gλ(F)≥gλ(E)
for λ≥0 and gλ≥0.
Due to the limitation of the classical measure, Sugeno, the Japanese scholar, presents set functions called fuzzy measures which use the monotonicity instead of the additivity. In practical applications, we often use regular fuzzy measure [32] on finite sets.Definition 8 (see [28]).
LetX be a finite set and 2X be the power set of X. Set function μ:2X→[0,1] is called a regular fuzzy measure defined on let 2X if the following conditions are satisfied:(1)
μ(∅)=0,μ(X)=1;(2)
ifE∈2X,G∈2X, and E⊂G, then μ(E)≤μ(G).Definition 9 (see [28]).
LetX be a finite set and 2X be the power set, of X. A fuzzy measure μ:2X→[0,1] is called a regular λ-fuzzy measure defined on let 2X if the following conditions are satisfied:(1)
μ(∅)=0,μ(X)=1;(2)
ifμ(A∪B)=μ(A)+μ(B)+λμ(A)μ(B), where A⊂X,B⊂X,A∩B=∅ and λ∈(-1,∞).Theorem 10.
Letgλ be a Sugeno measure based on σ-λ rules. Then gλ is a regular λ-fuzzy measure defined on 𝒜.Proof.
We could prove thatgλ(∅)=0. Otherwise for every λ≥0,A∈𝒜, we have
(5)gλ(A)=gλ(A∪∅)=gλ(A)+gλ(∅)+λgλ(A)gλ(∅),gλ(∅)=-λgλ(A)gλ(∅).
Since gλ(∅)≠0, we have λgλ(∅)=-1; it is a contradiction. Furthermore, we obtain gλ(X)=1, and gλ has the monotonicity by Definition 2 and Theorem 7.Denoting finite setX={x1,x2,…,xn}, the value gλi=gλ(xi) for all i=1,2,…,n is called measure density.Theorem 11.
The parameterλ of a regular Sugeno measure based on σ-λ rules is determined by the following equation:
(6)∏i=1n‍(1+λgλi)=1+λ.Proof.
We can prove the above theorem by Theorem10 and [32].If we know the values of Sugeno measure based onσ-λ on singleton sets, we can use Theorem 11 to obtain the values of λ and then use Definition 1 to obtain the values on the other sets. It implies that a Sugeno measure based on σ-λ can be determined by measure densities.Theorem 12 (see [28]).
If one knows the measure densitygλi on finite set X={x1,x2,…,xn}, then there is only one solution λ obtained from ∏i=1n(1+λgλi)=1+λ.


## 3. The Choquet Integrals of Real-Valued Function Based onσ-λ Rules

Definition 13 (see [34]).
A regular fuzzy number, denoted bya~, is a fuzzy subset of ℝ with membership function m:ℝ→[0,1] satisfying the following conditions.(RFN1) there exists at least one number a0∈ℝ such that a0=1;(RFN2) m(t) is nondecreasing on (-∞,a0] and nonincreasing on [a0,+∞);(RFN3) m(t) is upper semicontinuous; that is, limt→t0+=m(t0) if t0<a0 and limt→t0-=m(t0) if t0>a0;(RFN4)∫-∞∞‍m(t)dt<∞.
The set of all regular fuzzy numbers is denoted by𝒩.
Letf:X→(-∞,∞) be a measurable function with respect to 𝒜; that is, f satisfies the condition that
(7){x∣f(x)≥α}∈𝒜
for any α∈ℝ.From now on, we suppose that all functions defined onX appearing as an integrand of the Choquet integral in this paper are measurable.Definition 14 (see [22]).
Let(X,𝒜) be a measurable space, and let gλ be a Sugeno measure based on σ-λ rules on 𝒜. The Choquet integral of a real-valued function f:X→(-∞,+∞) is defined as
(8)(c)∫X‍fdgλ=∫-∞0‍[gλ(Fα)-gλ(X)]dα+∫0∞‍gλ(Fα)dα,
where
(9)Fα={x∣f(x)≥α}
for α∈(-∞,+∞), if both of Riemann integrals exist and at least one of them has finite value.
LetA∈𝒜, then Choquet integral of a nonnegative real-valued function f:X→(0,+∞) is defined as
(10)(c)∫A‍fdgλ=∫0∞‍gλ(A∩Fα)dα.Without loss of the generality, Yang et al. [34] have proposed a new scheme to calculate the value of a Choquet integral with a real-valued integrand.WhenX={x1,x2,…,xn}, for any function f:X→(-∞,+∞), both [gλ(Fα)-gλ(X)] and gλ(Fα) are functions of α with bounded variance; therefore, their Riemann integrals with respect to α exist and are finite. So, Choquet integral (c)∫‍fdgλ is well defined. To calculate the value of the Choquet integral of a given real-valued function f, usually the values of f, that is, f(x1),f(x2),…,f(xn), should be sorted in a nondecreasing order, so that f(x1′)≤f(x2′)≤⋯≤f(xn′), where {x1′,x2′,…,xn′} is a certain permutation of {x1,x2,…,xn}. Then, the value of the Choquet integral is obtained by
(11)(c)∫‍fdgλ=∑i=1n‍[f(xi′)-f(xi-1′)]·gλ({xi′,xi+1′,…,xn′}),
where f(x0′)=0.Example 15.
LetX={x1,x2,x3},gλ(x1)=0.1,gλ(x2)=0.2,gλ(x3)=0.3,f(x1)=0.3,f(x2)=0.7, and f(x3)=0.5. We can obtain ∏i=13(1+λgλi)=1+λby Theorem 11. Furthermore we get λ=3.1.
By Definition1, we can get the following results:
(12)gλ({x1,x2})=gλ(x1)+gλ(x2)+λgλ(x1)·gλ(x2)gλ({x1,x2})=0.1+0.2+3.1×0.1×0.2gλ({x1,x2})=0.362.
Similarly, gλ({x1,x3})=0.493,gλ({x2,x3})=0.593,  gλ({x1,x2,x3})=1.
By Definition14, we can get the Choquet integrals of f(x) with respect to gλ as follows:
(13)(c)∫‍fdgλ=(0.3-0)·1+(0.5-0.3)·gλ({x3,x2})+(0.7-0.5)·gλ(x2)=0.4586.Example 16.
LetX={x1,x2,x3,x4,x5},gλ(x1)=0.1,gλ(x2)=0.2,gλ(x3)=0.3,gλ(x4)=0.15, and gλ(x5)=0.175 (Table 2).
By Theorem11, we obtain that ∏i=15(1+λgλi)=1+λ, and with the aid of Mathematica software, we calculate that λ=0.218; furthermore, we get Choquet integrals which are shown in Table 3.Table 2
Data of Example16.
i
f
i
1
f
i
2
f
i
3
f
i
4
f
i
5
1
1
0.6
0.8
0.8
1
2
0.8
0.5
0.7
1
0.7
3
0.8
0.8
0.5
0.8
0.9
4
0.5
0.5
1
0.7
0.6
5
0.2
0.9
0.7
0.6
0.3
6
0.7
0.3
0.9
0.6
0.8
7
1
0.6
1
0.5
0.8
8
0.5
0.8
0.4
0.7
0.5
9
0.4
0.7
0.6
0.9
0.2
10
0.7
0.5
0.6
0.8
0.8Table 3
Results of Example16.
(
c
)
∫
f
1
d
g
λ
(
c
)
∫
‍
f
2
d
g
λ
(
c
)
∫
‍
f
3
d
g
λ
(
c
)
∫
‍
f
4
d
g
λ
(
c
)
∫
‍
f
5
d
g
λ
0.8
0.7
0.71
0.79
0.58
(
c
)
∫
‍
f
6
d
g
λ
(
c
)
∫
‍
f
7
d
g
λ
(
c
)
∫
‍
f
8
d
g
λ
(
c
)
∫
‍
f
9
d
g
λ
(
c
)
∫
‍
f
10
d
g
λ
0.67
0.79
0.56
0.56
0.65


## 4. Questions Description: Determine Fuzzy Measures

In this section, we formulate our problems to be solved.If we regard fuzzy integrals as multi-input single-output systems, we can obtain the Data through handling these systems. Suppose that we have several information sourcesx1,x2,…,xn,n≥2 and a given object y. Let X={x1,x2,…,xn}; we have the data with sample size m as shown in Table 4, where fij is the ith value of source xj and Ei is the ith value of object.Table 4
Data.
x
1
x
2
⋯
x
n
y
f
11
f
12
⋯
f
1
n
E
1
f
21
f
22
⋯
f
2
n
E
2
⋮
⋮
⋮
⋮
⋮
f
m
1
f
m
2
⋯
f
m
n
E
mWe hope to find a Sugeno measuregλ on measurable space (X,2X), such that Ei=(c)∫‍fidgλ, i=1,2,…,m, where function fi is defined by fi(xj)=fij,j=1,2,…,n for i=1,2,…,m.If such a Sugeno measuregλ does not exist, we hope to find the optimally approximate solution. This is just the inverse problem of synthetic evaluation. We can also use the least square method to transform the above problem to a constrained optimization problem. An optimization problem is described as follows:
(14)min(V)=1m∑i=1m‍(Ei-(c)∫‍fidgλ)2.A result ofmin(V)=0 also means that a precise solution is found.Here, we discuss this problem in three aspects. The first one is the values offij, and Ei(i=1,2,…,m,j=1,2,…,n) are a real-valued data. The second one is the values of fij, and Ei(i=1,2,…,m,j=1,2,…,n) are an interval-valued data. The last one is fij, and Ei(i=1,2,…,m,j=1,2,…,n) are a fuzzy-valued data.


## 5. Using Genetic Algorithm to Determine Fuzzy Measures from Real-Valued Data

In this section, we use genetic algorithm to determine fuzzy measures from real-valued data.


### 5.1. Genetic Algorithm (GA)

Genetic algorithm (GA) is a stochastic search method for optimization problems based on the mechanics of natural selection and natural genetics (i.e., survival of the fittest). GA has demonstrated considerable success in providing good solutions to many complex optimization problems and received more and more attentions during the past three decades. When the objective functions to be optimized in the optimization problems are multimodal or the search spaces are particularly irregular, algorithms need to be highly robust in order to avoid getting stuck at a local optimal solution. The advantage of GA just makes it able to obtain the global optimal solution fairly. In addition, GA does not require the specific mathematical analysis of optimization problems, which makes GA easily coded by users who are not necessarily good at mathematics and algorithms.


#### 5.1.1. The Decimal Coding

ChromosomeV is denoted by V=(a1,a2,…,an), where gene ai∈[0,1] for all i=1,2,…,n, and a1=gλ1,a2=gλ2,…, and an-1=gλn-1,an=λ.


#### 5.1.2. The Decoding

Find the formula(15)gλn=1λ(1+λ∏i=1n-1(1+λgλi)-1)
by gλi(i=1,2,…,n-1),λ, and Definition 1. Furthermore, the values of gλ(x1,x2,…, xk) for all k=1,2,…,n can be obtained by Definition 1. The encoding can guarantee to get a feasible solution. That is, the solution satisfies Definition 1, and it will not undermine the feasibility of the solution no matter what kind of genetic operation (crossover or mutation) be used to chromosome.


#### 5.1.3. The Arithmetic Crossover

Use the crossover probabilityPc to choose two parent chromosomes V1=(a11,a21,…,an1) and V2=(a12,a22,…,an2) and use the arithmetic crossover to get two offspring chromosomes V3 and V4:
(16)V3=αV1+(1-α)V2=(αa11+(1-α)a12,αa21+(1-α)a22,…,αan1+(1-α)an2)V4=(1-α)V1+αV2=((1-α)a11+αa12,(1-α)a21+αa22,…,(1-α)an1+αan2),
where α∈[0,1].


#### 5.1.4. The Nonuniform Mutation

Select parent chromosomeV=(a1,a2,…,an) according to the mutation probability Pm and take the mutation to ak for the random generation of k in [1,n]. Let
(17)ak′=ak+ϕ(t,akU-ak)
or
(18)ak′=ak-ϕ(t,ak-akL),
where akU and akL are the upper and lower bound of ak, respectively, and t is the number of generations. Function ϕ(t,b) is defined as follows:
(19)ϕ(t,b)=b·r·(1-tT)b,
where r is a random number of [0,1], T is the largest number of generations, and b is the parameter. Obviously, limn→Tϕ(t,b)=0.


#### 5.1.5. The Evaluation Function

Evaluation function is defined by(20)min(V)=1n∑i=1n‍(Ei-(c)∫‍fidgλ)2,
where (c)∫‍fidgλ is defined by Definition 14. Use the objective function as the evaluation function of a single chromosome.The genetic algorithm procedure is summarized as follows.Step 1.
Initialize pop size chromosomes randomly.Step 2.
Update the chromosomes by crossover and mutation operations.Step 3.
Calculate the evaluation function for all chromosomes.Step 4.
Select the chromosomes by spinning the roulette wheel.Step 5.
Repeat the Step 2 to Step 3 for a given number of cycles.Step 6.
Report the best chromosome as the optimal solution.


### 5.2. Examples and Results

Example 17.
A railway administration chooses 15 passengers randomly to evaluate the passenger train plan in the administration (Table5). Customer bases its overall scores on transfer times, in-train congestion, travel time, and ticket price, and also they have a score for each of four aspects. Let X={x1,x2,x3,x4,} be the attributes of transfer times, congestion, travel time, and ticket price, respectively. We want to know which attribute is the most important for passengers. Here, Pc=0.85, Pm=0.3, and the population size is 20.
We can obtain that the transfer times are the most important for passengers from Table6 and the convergence rate of Example 17 as shown in Figure 1.Table 5
Data of Example17 (the customer evaluation).
Customer
Transfer times
Congestion
Travel time
Ticket price
EvaluationEi
1
0
0
0.3
1
0.2
2
0.3
0.8
0.5
0.6
0.5
3
0.7
0.9
1
0.7
0.8
4
0.1
0
0.4
0.3
0.15
5
0.5
0.4
0.5
0.6
0.5
6
1
0.3
0.8
0.8
0.7
7
0.3
1
0
1
0.5
8
0.9
0.6
0.5
0.7
0.7
9
0.6
0.8
0.2
0.4
0.5
10
0.4
1
0.5
0
0.4
11
0.8
0.4
1
0.3
0.6
12
1
0.8
0.7
0.5
0.75
13
0.2
0.6
1
0.7
0.5
14
0
0.3
0.2
0.9
0.2
15
0.4
0.2
0.8
0
0.3Table 6
Results of Example17.
Set
λ
g
λ
Error
Number of generation
{
x
1
}
0.965
0.2969
0.0154
100
{
x
2
}
0.1842
{
x
3
}
0.1347
{
x
4
}
0.1529Figure 1
The convergence rate of Example17.


## 5.1. Genetic Algorithm (GA)

Genetic algorithm (GA) is a stochastic search method for optimization problems based on the mechanics of natural selection and natural genetics (i.e., survival of the fittest). GA has demonstrated considerable success in providing good solutions to many complex optimization problems and received more and more attentions during the past three decades. When the objective functions to be optimized in the optimization problems are multimodal or the search spaces are particularly irregular, algorithms need to be highly robust in order to avoid getting stuck at a local optimal solution. The advantage of GA just makes it able to obtain the global optimal solution fairly. In addition, GA does not require the specific mathematical analysis of optimization problems, which makes GA easily coded by users who are not necessarily good at mathematics and algorithms.


### 5.1.1. The Decimal Coding

ChromosomeV is denoted by V=(a1,a2,…,an), where gene ai∈[0,1] for all i=1,2,…,n, and a1=gλ1,a2=gλ2,…, and an-1=gλn-1,an=λ.


### 5.1.2. The Decoding

Find the formula(15)gλn=1λ(1+λ∏i=1n-1(1+λgλi)-1)
by gλi(i=1,2,…,n-1),λ, and Definition 1. Furthermore, the values of gλ(x1,x2,…, xk) for all k=1,2,…,n can be obtained by Definition 1. The encoding can guarantee to get a feasible solution. That is, the solution satisfies Definition 1, and it will not undermine the feasibility of the solution no matter what kind of genetic operation (crossover or mutation) be used to chromosome.


### 5.1.3. The Arithmetic Crossover

Use the crossover probabilityPc to choose two parent chromosomes V1=(a11,a21,…,an1) and V2=(a12,a22,…,an2) and use the arithmetic crossover to get two offspring chromosomes V3 and V4:
(16)V3=αV1+(1-α)V2=(αa11+(1-α)a12,αa21+(1-α)a22,…,αan1+(1-α)an2)V4=(1-α)V1+αV2=((1-α)a11+αa12,(1-α)a21+αa22,…,(1-α)an1+αan2),
where α∈[0,1].


### 5.1.4. The Nonuniform Mutation

Select parent chromosomeV=(a1,a2,…,an) according to the mutation probability Pm and take the mutation to ak for the random generation of k in [1,n]. Let
(17)ak′=ak+ϕ(t,akU-ak)
or
(18)ak′=ak-ϕ(t,ak-akL),
where akU and akL are the upper and lower bound of ak, respectively, and t is the number of generations. Function ϕ(t,b) is defined as follows:
(19)ϕ(t,b)=b·r·(1-tT)b,
where r is a random number of [0,1], T is the largest number of generations, and b is the parameter. Obviously, limn→Tϕ(t,b)=0.


### 5.1.5. The Evaluation Function

Evaluation function is defined by(20)min(V)=1n∑i=1n‍(Ei-(c)∫‍fidgλ)2,
where (c)∫‍fidgλ is defined by Definition 14. Use the objective function as the evaluation function of a single chromosome.The genetic algorithm procedure is summarized as follows.Step 1.
Initialize pop size chromosomes randomly.Step 2.
Update the chromosomes by crossover and mutation operations.Step 3.
Calculate the evaluation function for all chromosomes.Step 4.
Select the chromosomes by spinning the roulette wheel.Step 5.
Repeat the Step 2 to Step 3 for a given number of cycles.Step 6.
Report the best chromosome as the optimal solution.


## 5.1.1. The Decimal Coding

ChromosomeV is denoted by V=(a1,a2,…,an), where gene ai∈[0,1] for all i=1,2,…,n, and a1=gλ1,a2=gλ2,…, and an-1=gλn-1,an=λ.


## 5.1.2. The Decoding

Find the formula(15)gλn=1λ(1+λ∏i=1n-1(1+λgλi)-1)
by gλi(i=1,2,…,n-1),λ, and Definition 1. Furthermore, the values of gλ(x1,x2,…, xk) for all k=1,2,…,n can be obtained by Definition 1. The encoding can guarantee to get a feasible solution. That is, the solution satisfies Definition 1, and it will not undermine the feasibility of the solution no matter what kind of genetic operation (crossover or mutation) be used to chromosome.


## 5.1.3. The Arithmetic Crossover

Use the crossover probabilityPc to choose two parent chromosomes V1=(a11,a21,…,an1) and V2=(a12,a22,…,an2) and use the arithmetic crossover to get two offspring chromosomes V3 and V4:
(16)V3=αV1+(1-α)V2=(αa11+(1-α)a12,αa21+(1-α)a22,…,αan1+(1-α)an2)V4=(1-α)V1+αV2=((1-α)a11+αa12,(1-α)a21+αa22,…,(1-α)an1+αan2),
where α∈[0,1].


## 5.1.4. The Nonuniform Mutation

Select parent chromosomeV=(a1,a2,…,an) according to the mutation probability Pm and take the mutation to ak for the random generation of k in [1,n]. Let
(17)ak′=ak+ϕ(t,akU-ak)
or
(18)ak′=ak-ϕ(t,ak-akL),
where akU and akL are the upper and lower bound of ak, respectively, and t is the number of generations. Function ϕ(t,b) is defined as follows:
(19)ϕ(t,b)=b·r·(1-tT)b,
where r is a random number of [0,1], T is the largest number of generations, and b is the parameter. Obviously, limn→Tϕ(t,b)=0.


## 5.1.5. The Evaluation Function

Evaluation function is defined by(20)min(V)=1n∑i=1n‍(Ei-(c)∫‍fidgλ)2,
where (c)∫‍fidgλ is defined by Definition 14. Use the objective function as the evaluation function of a single chromosome.The genetic algorithm procedure is summarized as follows.Step 1.
Initialize pop size chromosomes randomly.Step 2.
Update the chromosomes by crossover and mutation operations.Step 3.
Calculate the evaluation function for all chromosomes.Step 4.
Select the chromosomes by spinning the roulette wheel.Step 5.
Repeat the Step 2 to Step 3 for a given number of cycles.Step 6.
Report the best chromosome as the optimal solution.


## 5.2. Examples and Results

Example 17.
A railway administration chooses 15 passengers randomly to evaluate the passenger train plan in the administration (Table5). Customer bases its overall scores on transfer times, in-train congestion, travel time, and ticket price, and also they have a score for each of four aspects. Let X={x1,x2,x3,x4,} be the attributes of transfer times, congestion, travel time, and ticket price, respectively. We want to know which attribute is the most important for passengers. Here, Pc=0.85, Pm=0.3, and the population size is 20.
We can obtain that the transfer times are the most important for passengers from Table6 and the convergence rate of Example 17 as shown in Figure 1.Table 5
Data of Example17 (the customer evaluation).
Customer
Transfer times
Congestion
Travel time
Ticket price
EvaluationEi
1
0
0
0.3
1
0.2
2
0.3
0.8
0.5
0.6
0.5
3
0.7
0.9
1
0.7
0.8
4
0.1
0
0.4
0.3
0.15
5
0.5
0.4
0.5
0.6
0.5
6
1
0.3
0.8
0.8
0.7
7
0.3
1
0
1
0.5
8
0.9
0.6
0.5
0.7
0.7
9
0.6
0.8
0.2
0.4
0.5
10
0.4
1
0.5
0
0.4
11
0.8
0.4
1
0.3
0.6
12
1
0.8
0.7
0.5
0.75
13
0.2
0.6
1
0.7
0.5
14
0
0.3
0.2
0.9
0.2
15
0.4
0.2
0.8
0
0.3Table 6
Results of Example17.
Set
λ
g
λ
Error
Number of generation
{
x
1
}
0.965
0.2969
0.0154
100
{
x
2
}
0.1842
{
x
3
}
0.1347
{
x
4
}
0.1529Figure 1
The convergence rate of Example17.


## 6. Using Genetic Algorithm to Determine Fuzzy Measures from Interval-Valued Data

The intervals are derived from many practical application problems, when instead of knowing the precise valuesx of some quantity X we know only the intervals [x_,x¯], in which X, ranges. Since the comparison of two values or quantities is the basic and most frequently used step in optimization, interval-valued function plays an important role in interval computation development.


### 6.1. The Choquet Integrals of Interval-Valued Function Based onσ-λ Rules

With the definitions of the preceding subsections and from Wu et al. [35], we assume that R+=[0,+∞), I(R+)={r¯:[r-,r+]⊂R+} is the set of interval numbers, and F¯(X) is the set of all interval numbers f.¯Interval numbers satisfy the following basic operations:(1)
r¯*p¯=[r-*p-,r+*p+](*+∨∧);(2)
k·r¯=[kr-,kr+],(k∈ℝ+);(3)
r¯≤p¯⇔r-≤p-,r+≤p+;(4)
d(r¯,p¯)=max{|r--p-|,|r+-p+|};(5)
ifd(r¯n,r¯)→0, then r¯n→r¯.Definition 18 (see [34]).
An interval-valued functionf¯:X→I(R+) is measurable if both f-(x) and f+(x) are measurable function of x, where f¯(x)=[f-(x),f+(x)], f-(x) is the left end point of interval f¯(x), and f+(x) is the right end point of interval f¯(x).Definition 19.
Let(X,𝒜,gλ) be a nonadditive measure space based on σ-λ rules f¯:X→I(R+) a measurable function in X and E∈𝒜. Then the Choquet integral of f¯ with respect to gλ is defined by
(21)(c)∫E‍f¯dgλ=:{(c)∫E‍f¯dgλ∣g∈Sf¯(x)}
if (c)∫E‍f¯dgλ is a closed interval on I(R+), where Sf¯(x)={g∣g:X→R+} is a measurable selection on f¯(x).Theorem 20.
Letf¯:X→I(R+) be a measurable interval-valued function on X, and let gλ be a Sugeno measure based on σ-λ rules on 𝒜. The Choquet integral of f¯ with respect to gλ is
(22)(c)∫‍f¯dgλ=[(c)∫‍f-dgλ,(c)∫‍f+dgλ],
where f-(x) is the left end point of interval f¯(x) and f+(x) is the right end point of interval f¯(x), for every x∈X.Proof.
We can prove the above theorem by Theorem10 and [34].Using the continuity and the monotonicity of the Choquet integral with the nonnegativity and the monotonicity of the fuzzy measures, we may obtain the following theorem.Theorem 21.
Letf¯:X→I(R+) be a measurable interval-valued function on X, let gλ be a sugeno measure based on σ-λ rules on 𝒜, and X={x1,x2,…,xn}. Then, the value of Choquet integral of f¯ with respect to gλ is obtained by
(23)(c)∫‍f¯dgλ=[∑i=1n‍[f-(xi′)-f-(xi-1′)]·gλ({xi′,xi+1′,…,xn′}),∑i=1n‍[f+(xi′′)-f+(xi-1′′)]·gλ({xi′′,xi+1′′,…,xn′′})∑i=1n‍],
where f-(x0′)=0, f+(x0′′)=0, the values of f- and f+, that is, f-(x1),f-(x2),…,f-(xn), and f+(x1),f+(x2),…,f+(xn) should be sorted in a nondecreasing order, so that f-(x1′)≤f-(x2′)≤⋯≤f-(xn′) and f+(x1′′)≤f+(x2′′)≤⋯≤f+(xn′′), respectively, and {x1′,x2′,…,xn′} and {x1′′,x2′′,…,xn′′} are a certain permutation of {x1,x2,…,xn}, respectively.


### 6.2. Using Genetic Algorithm to Determine Fuzzy Measures from Interval-Valued Data

In this subsection, we use genetic algorithm to determine fuzzy measures from interval-valued data.


#### 6.2.1. Genetic Algorithm (GA)

In this subsection, the all algorithms are the same to the Section5.1 but the evaluation function. Here, the valuation function is defined by
(24)Eval(V)=(1m∑i=1m‍min{((c)∫‍fi-dgλ-Ei-)2,((c)∫‍fi+dgλ-Ei+)2})1/2,
where (c)∫‍fi-dgλ and (c)∫‍fi+dgλ are defined by Definition 14 and Theorem 20.


#### 6.2.2. Examples and Results

Example 22.
If the evaluation information in Example17 is represented by the interval-valued fuzzy numbers as shown in Table 7, then we redetermine fuzzy measures from interval-valued data by using genetic optimization.
We can obtain that the transfer times are the most important for passengers from Table8 and the convergence rate of Example 22 as shown in Figure 2.Table 7
Data of Example22 (the customer evaluation).
Customer
Transfer times
Congestion
Travel time
Ticket price
EvaluationEi
1
[
0.00,0.00
]
[
0.00,0.00
]
[
0.25,0.35
]
[
1.00,1.00
]
[
0.15,0.25
]
2
[
0.25,0.35
]
[
0.15,0.25
]
[
0.45,0.55
]
[
0.55,0.65
]
[
0.45,0.55
]
3
[
0.65,0.75
]
[
0.85,0.95
]
[
1.00,1.00
]
[
0.65,0.75
]
[
0.75,0.85
]
4
[
0.05,0.15
]
[
0.00,0.00
]
[
0.35,0.45
]
[
0.25,0.35
]
[
0.10,0.20
]
5
[
0.45,0.55
]
[
0.35,0.45
]
[
0.45,0.55
]
[
0.55,0.65
]
[
0.45,0.55
]
6
[
1.00,1.00
]
[
0.25,0.35
]
[
0.75,0.85
]
[
0.75,0.85
]
[
0.65,0.75
]
7
[
0.25,0.35
]
[
1.00,1.00
]
[
0.00,0.00
]
[
1.00,1.00
]
[
0.45,0.55
]
8
[
0.85,0.95
]
[
0.55,0.65
]
[
0.45,0.55
]
[
0.65,0.75
]
[
0.65,0.75
]
9
[
0.55,0.65
]
[
0.75,0.85
]
[
0.15,0.25
]
[
0.35,0.45
]
[
0.45,0.55
]
10
[
0.35,0.45
]
[
1.00,1.00
]
[
0.45,0.55
]
[
0.00,0.00
]
[
0.35,0.45
]
11
[
0.75,0.85
]
[
0.35,0.45
]
[
1.00,1.00
]
[
0.25,0.35
]
[
0.55,0.65
]
12
[
1.00,1.00
]
[
0.75,0.85
]
[
0.65,0.75
]
[
0.45,0.55
]
[
0.70,0.80
]
13
[
0.15,0.25
]
[
0.55,0.65
]
[
1.00,1.00
]
[
0.65,0.75
]
[
0.45,0.55
]
14
[
0.00,0.00
]
[
0.25,0.35
]
[
0.15,0.25
]
[
0.85,0.95
]
[
0.15,0.25
]
15
[
0.35,0.45
]
[
0.15,0.25
]
[
0.75,0.85
]
[
0.00,0.00
]
[
0.25,0.35
]Table 8
Results of Example22.
Set
λ
g
λ
Error
Number of generation
{
x
1
}
0.937
0.3186
0.010
100
{
x
2
}
0.1819
{
x
3
}
0.1407
{
x
4
}
0.1344Figure 2
The convergence rate of Example22.


## 6.1. The Choquet Integrals of Interval-Valued Function Based onσ-λ Rules

With the definitions of the preceding subsections and from Wu et al. [35], we assume that R+=[0,+∞), I(R+)={r¯:[r-,r+]⊂R+} is the set of interval numbers, and F¯(X) is the set of all interval numbers f.¯Interval numbers satisfy the following basic operations:(1)
r¯*p¯=[r-*p-,r+*p+](*+∨∧);(2)
k·r¯=[kr-,kr+],(k∈ℝ+);(3)
r¯≤p¯⇔r-≤p-,r+≤p+;(4)
d(r¯,p¯)=max{|r--p-|,|r+-p+|};(5)
ifd(r¯n,r¯)→0, then r¯n→r¯.Definition 18 (see [34]).
An interval-valued functionf¯:X→I(R+) is measurable if both f-(x) and f+(x) are measurable function of x, where f¯(x)=[f-(x),f+(x)], f-(x) is the left end point of interval f¯(x), and f+(x) is the right end point of interval f¯(x).Definition 19.
Let(X,𝒜,gλ) be a nonadditive measure space based on σ-λ rules f¯:X→I(R+) a measurable function in X and E∈𝒜. Then the Choquet integral of f¯ with respect to gλ is defined by
(21)(c)∫E‍f¯dgλ=:{(c)∫E‍f¯dgλ∣g∈Sf¯(x)}
if (c)∫E‍f¯dgλ is a closed interval on I(R+), where Sf¯(x)={g∣g:X→R+} is a measurable selection on f¯(x).Theorem 20.
Letf¯:X→I(R+) be a measurable interval-valued function on X, and let gλ be a Sugeno measure based on σ-λ rules on 𝒜. The Choquet integral of f¯ with respect to gλ is
(22)(c)∫‍f¯dgλ=[(c)∫‍f-dgλ,(c)∫‍f+dgλ],
where f-(x) is the left end point of interval f¯(x) and f+(x) is the right end point of interval f¯(x), for every x∈X.Proof.
We can prove the above theorem by Theorem10 and [34].Using the continuity and the monotonicity of the Choquet integral with the nonnegativity and the monotonicity of the fuzzy measures, we may obtain the following theorem.Theorem 21.
Letf¯:X→I(R+) be a measurable interval-valued function on X, let gλ be a sugeno measure based on σ-λ rules on 𝒜, and X={x1,x2,…,xn}. Then, the value of Choquet integral of f¯ with respect to gλ is obtained by
(23)(c)∫‍f¯dgλ=[∑i=1n‍[f-(xi′)-f-(xi-1′)]·gλ({xi′,xi+1′,…,xn′}),∑i=1n‍[f+(xi′′)-f+(xi-1′′)]·gλ({xi′′,xi+1′′,…,xn′′})∑i=1n‍],
where f-(x0′)=0, f+(x0′′)=0, the values of f- and f+, that is, f-(x1),f-(x2),…,f-(xn), and f+(x1),f+(x2),…,f+(xn) should be sorted in a nondecreasing order, so that f-(x1′)≤f-(x2′)≤⋯≤f-(xn′) and f+(x1′′)≤f+(x2′′)≤⋯≤f+(xn′′), respectively, and {x1′,x2′,…,xn′} and {x1′′,x2′′,…,xn′′} are a certain permutation of {x1,x2,…,xn}, respectively.


## 6.2. Using Genetic Algorithm to Determine Fuzzy Measures from Interval-Valued Data

In this subsection, we use genetic algorithm to determine fuzzy measures from interval-valued data.


### 6.2.1. Genetic Algorithm (GA)

In this subsection, the all algorithms are the same to the Section5.1 but the evaluation function. Here, the valuation function is defined by
(24)Eval(V)=(1m∑i=1m‍min{((c)∫‍fi-dgλ-Ei-)2,((c)∫‍fi+dgλ-Ei+)2})1/2,
where (c)∫‍fi-dgλ and (c)∫‍fi+dgλ are defined by Definition 14 and Theorem 20.


### 6.2.2. Examples and Results

Example 22.
If the evaluation information in Example17 is represented by the interval-valued fuzzy numbers as shown in Table 7, then we redetermine fuzzy measures from interval-valued data by using genetic optimization.
We can obtain that the transfer times are the most important for passengers from Table8 and the convergence rate of Example 22 as shown in Figure 2.Table 7
Data of Example22 (the customer evaluation).
Customer
Transfer times
Congestion
Travel time
Ticket price
EvaluationEi
1
[
0.00,0.00
]
[
0.00,0.00
]
[
0.25,0.35
]
[
1.00,1.00
]
[
0.15,0.25
]
2
[
0.25,0.35
]
[
0.15,0.25
]
[
0.45,0.55
]
[
0.55,0.65
]
[
0.45,0.55
]
3
[
0.65,0.75
]
[
0.85,0.95
]
[
1.00,1.00
]
[
0.65,0.75
]
[
0.75,0.85
]
4
[
0.05,0.15
]
[
0.00,0.00
]
[
0.35,0.45
]
[
0.25,0.35
]
[
0.10,0.20
]
5
[
0.45,0.55
]
[
0.35,0.45
]
[
0.45,0.55
]
[
0.55,0.65
]
[
0.45,0.55
]
6
[
1.00,1.00
]
[
0.25,0.35
]
[
0.75,0.85
]
[
0.75,0.85
]
[
0.65,0.75
]
7
[
0.25,0.35
]
[
1.00,1.00
]
[
0.00,0.00
]
[
1.00,1.00
]
[
0.45,0.55
]
8
[
0.85,0.95
]
[
0.55,0.65
]
[
0.45,0.55
]
[
0.65,0.75
]
[
0.65,0.75
]
9
[
0.55,0.65
]
[
0.75,0.85
]
[
0.15,0.25
]
[
0.35,0.45
]
[
0.45,0.55
]
10
[
0.35,0.45
]
[
1.00,1.00
]
[
0.45,0.55
]
[
0.00,0.00
]
[
0.35,0.45
]
11
[
0.75,0.85
]
[
0.35,0.45
]
[
1.00,1.00
]
[
0.25,0.35
]
[
0.55,0.65
]
12
[
1.00,1.00
]
[
0.75,0.85
]
[
0.65,0.75
]
[
0.45,0.55
]
[
0.70,0.80
]
13
[
0.15,0.25
]
[
0.55,0.65
]
[
1.00,1.00
]
[
0.65,0.75
]
[
0.45,0.55
]
14
[
0.00,0.00
]
[
0.25,0.35
]
[
0.15,0.25
]
[
0.85,0.95
]
[
0.15,0.25
]
15
[
0.35,0.45
]
[
0.15,0.25
]
[
0.75,0.85
]
[
0.00,0.00
]
[
0.25,0.35
]Table 8
Results of Example22.
Set
λ
g
λ
Error
Number of generation
{
x
1
}
0.937
0.3186
0.010
100
{
x
2
}
0.1819
{
x
3
}
0.1407
{
x
4
}
0.1344Figure 2
The convergence rate of Example22.


## 6.2.1. Genetic Algorithm (GA)

In this subsection, the all algorithms are the same to the Section5.1 but the evaluation function. Here, the valuation function is defined by
(24)Eval(V)=(1m∑i=1m‍min{((c)∫‍fi-dgλ-Ei-)2,((c)∫‍fi+dgλ-Ei+)2})1/2,
where (c)∫‍fi-dgλ and (c)∫‍fi+dgλ are defined by Definition 14 and Theorem 20.


## 6.2.2. Examples and Results

Example 22.
If the evaluation information in Example17 is represented by the interval-valued fuzzy numbers as shown in Table 7, then we redetermine fuzzy measures from interval-valued data by using genetic optimization.
We can obtain that the transfer times are the most important for passengers from Table8 and the convergence rate of Example 22 as shown in Figure 2.Table 7
Data of Example22 (the customer evaluation).
Customer
Transfer times
Congestion
Travel time
Ticket price
EvaluationEi
1
[
0.00,0.00
]
[
0.00,0.00
]
[
0.25,0.35
]
[
1.00,1.00
]
[
0.15,0.25
]
2
[
0.25,0.35
]
[
0.15,0.25
]
[
0.45,0.55
]
[
0.55,0.65
]
[
0.45,0.55
]
3
[
0.65,0.75
]
[
0.85,0.95
]
[
1.00,1.00
]
[
0.65,0.75
]
[
0.75,0.85
]
4
[
0.05,0.15
]
[
0.00,0.00
]
[
0.35,0.45
]
[
0.25,0.35
]
[
0.10,0.20
]
5
[
0.45,0.55
]
[
0.35,0.45
]
[
0.45,0.55
]
[
0.55,0.65
]
[
0.45,0.55
]
6
[
1.00,1.00
]
[
0.25,0.35
]
[
0.75,0.85
]
[
0.75,0.85
]
[
0.65,0.75
]
7
[
0.25,0.35
]
[
1.00,1.00
]
[
0.00,0.00
]
[
1.00,1.00
]
[
0.45,0.55
]
8
[
0.85,0.95
]
[
0.55,0.65
]
[
0.45,0.55
]
[
0.65,0.75
]
[
0.65,0.75
]
9
[
0.55,0.65
]
[
0.75,0.85
]
[
0.15,0.25
]
[
0.35,0.45
]
[
0.45,0.55
]
10
[
0.35,0.45
]
[
1.00,1.00
]
[
0.45,0.55
]
[
0.00,0.00
]
[
0.35,0.45
]
11
[
0.75,0.85
]
[
0.35,0.45
]
[
1.00,1.00
]
[
0.25,0.35
]
[
0.55,0.65
]
12
[
1.00,1.00
]
[
0.75,0.85
]
[
0.65,0.75
]
[
0.45,0.55
]
[
0.70,0.80
]
13
[
0.15,0.25
]
[
0.55,0.65
]
[
1.00,1.00
]
[
0.65,0.75
]
[
0.45,0.55
]
14
[
0.00,0.00
]
[
0.25,0.35
]
[
0.15,0.25
]
[
0.85,0.95
]
[
0.15,0.25
]
15
[
0.35,0.45
]
[
0.15,0.25
]
[
0.75,0.85
]
[
0.00,0.00
]
[
0.25,0.35
]Table 8
Results of Example22.
Set
λ
g
λ
Error
Number of generation
{
x
1
}
0.937
0.3186
0.010
100
{
x
2
}
0.1819
{
x
3
}
0.1407
{
x
4
}
0.1344Figure 2
The convergence rate of Example22.


## 7. Using Genetic Algorithm to Determine Fuzzy Measures from Fuzzy-Valued Data

In many respects, fuzzy numbers depict the physical world more realistically than single-valued numbers. suppose, for example, service quality is an intangible asset of enterprises that is related to customers judgments about the overall quality of a firm. Fuzzy numbers are used in statistics, computer programming, engineering (especially communications), and experimental science. The concept takes into account the fact that all phenomena in the physical universe have a degree of inherent uncertainty.Definition 23 (see [33, 36, 37]).
Let(X,ℱ~) be a fuzzy measurable space, and let ℱ*(ℝ) be the class of all fuzzy subsets of ℝ. A fuzzy-valued function f~:X→ℱ*(ℝ) is called a measurable function if for every λ∈(0,1], its α-cut fα(x) is measurable, where
(25)fα(x)=(f~(x))α={r∣f~(x)(r)≥α}.Remark 24.
A measurable fuzzy-valued function is a especially measurable fuzzy set-value function.
Letf~:X→ℱ~,∀λ∈(0,1], and (f~(x))α=[(f~(x))α-,(f~(x))α+]. We will simply denote that
(26)fα-(x)=(f~(x))α-,fα+(x)=(f~(x))α+.
Obviously, fα-(x) and fα+(x) are real functions.
From now on, we suppose that all functions defined onX appearing as an integrand of the Choquet integral in this paper are measurable.
According to Theorem10, gλ is a signed fuzzy measure on 𝒜. Therefore, we may give the following definition referring to [34].
Fuzzy-valued functionf~:X→ℱ~ is said to be a C-integrally bounded, if there exists a Choquet integrable function h:X→R+ such that |x˙|≤h(t) for x˙∈[f~(t)]0.Definition 25.
Letf~:X→ℱ~ be a measurable fuzzy-valued function on X, and let gλ be a Sugeno measure based on σ-λ rules. Assume that f~ is C-integrally bounded. f~ is called Choquet integrable with respect to gλ if
(27){[(c)∫‍f~dμ]λ=:{(c)∫gdμ∣g∈Sfλ},0≤λ≤1}
confirms a unique fuzzy number a~∈ℱ~, which is denoted by (c)∫‍f~dμ=a~, where Sfλ={g:X→R,g∈fλ is a measurable selection of fλ}.The exact membership function of the Choquet integral with respect to Sugeno fuzzy measuregλ for fuzzy-valued integrand is rather difficult to be found. In a simpler but common case where X={x1,x2,…,xn} is finite, according to Definition 25, the calculation of the Choquet integral with a fuzzy-valued function comes down to that of the Choquet integral with an interval-valued function. Here, let us look at examples to show how to calculate the fuzzy-valued Choquet integral with respect to a Sugeno measure gλ.Example 26.
Letf~(x1)=[0,1,1],f~(x2)=[0.5,0.5,1.5], and f~(x3)=[0,1,2]. Here, a triangular fuzzy number is denoted by [al,a0,ar], where al,a0,ar∈ℝ and al≤a0≤ar. Set function gλ is a sugeno measure based on σ-λ rules. Function f~ is triangular fuzzy valued. The membership function of f~(x1),f~(x2), and f~(x3) are
(28)m1(t)=mf~(x1)(t)={t,t∈[0,1],0,t∉[0,1],(29)m2(t)=mf~(x2)(t)={1.5-t,t∈[0.5,1.5],0,t∉[0.5,1.5],(30)m3(t)=mf~(x3)(t)={t,t∈[0,1],2-t,t∈(1,2],0,t∉[0,2],
respectively. They are shown in Figure 1. The α-cut of f~ is represented by interval
(31)f~α(x1)=Mαf~(x1)={t∣mf~(x1)(t)≥α}=[α,1];f~α(x2)=Mαf~(x2)={t∣mf~(x2)(t)≥α}=[0.5,1.5-α];f~α(x3)=Mαf~(x3)={t∣mf~(x3)(t)≥α}=[α,2-α],whereα∈[0,1].
It is easy to get[f~α(x1)]l=α,[f~α(x2)]l=0.5,[f~α(x3)]l=α,[f~α(x1)]r=1,[f~α(x2)]r=1.5-α, and [f~α(x3)]r=2-α.
We conclude that
(
1
)  When 0≤α≤0.5, we get [f~α(x1)]l=f~α(x3)]l≤[f~α(x2)]l. Using Definition 14, we can let x1′=x1,x2′=x3,x3′=x2, and then
(32)[(c)∫‍f~αdgλ]l={[f(x1′)]l-[f(x0′)]l}[(c)∫‍f~αdgλ]l·gλ(x1′,x2′,x3′)+{[f(x2′)]l-[f(x1′)]l},gλ(x2′,x3′)+{[f(x3′)]l-[f(x2′)]l}·gλ(x3′)=[α-0]·1+[α-α]·0.5931+[0.5-α]·0.3=0.7α+0.15,
where [f(x0′)]l=0,[f(x0′)]r=0.
When0≤α≤0.5, we get [f~α(x1)]r≤[f~α(x2)]r≤[f~α(x3)]r. Using Definition 14 we can let x1′=x1,x2′=x2, and x3′=x3, and then
(33)[(c)∫‍f~αdgλ]r={[f(x1′)]r-[f(x0′)]r}[(c)∫‍f~αdgλ]r·gλ(x1′,x2′,x3′)[(c)∫‍f~αdgλ]r+{[f(x2′)]r-[f(x1′)]r},gλ(x2′,x3′)+{[f(x3′)]r-[f(x2′)]r}·gλ(x3′)=[1-0]·1+[1.5-α-1]·0.593+[2-α-1.5+α]·0.3=1.4465-0.593α.
That is, (c)∫‍f~αdgλ=[0.7α+0.15,1.45-0.592].
(
2
) When 0.5<α≤1, we get [f~α(x2)]l<[f~α(x3)]l=[f~α(x1)]l. Using Definition 14, we can let x1′=x2,x2′=x1, and x3′=x3, and then
(34)[(c)∫‍f~αdgλ]l={[f(x1′)]l-[f(x0′)]l}[(c)∫‍f~αdgλ]r·gλ(x1′,x2′,x3′)[(c)∫‍f~αdgλ]r+{[f(x2′)]l-[f(x1′)]l},gλ(x2′,x3′)+{[f(x3′)]l-[f(x2′)]l}·gλ(x3′)=0.5·1+[α-α]·0.3+[α-0.5]·0.493=0.493α+0.2535,
where [f(x0′)]l=0,[f(x0′)]r=0.
When0.5<α≤1, we get [f~α(x2)]r≤[f~α(x1)]r≤[f~α(x3)]r. Using Definition 14, we can let x1′=x2,x2′=x1, and x3′=x3, and then
(35)[(c)∫‍f~αdgλ]r={[f(x1′)]r-[f(x0′)]r}[(c)∫‍f~αdgλ]r·gλ(x1′,x2′,x3′)[(c)∫‍f~αdgλ]r+{[f(x2′)]r-[f(x1′)]r},gλ(x2′,x3′)+{[f(x3′)]r-[f(x2′)]r}·gλ(x3′)=[1.5-α]·1+[1-(1.5-α)]·0.4931+[(2-α)-1]·0.3=1.55-0.8α.
That is, (c)∫‍f~αdgλ=[0.493α+0.2535,1.55-0.8α].Example 27.
If the evaluation information in Example17 is represented by the triangular fuzzy number as shown in Table 9, then we redetermine fuzzy measures from fuzzy-valued data by using genetic algorithm.
Fist, we use the method as shown in Example26 to change the triangular fuzzy number of Table 9 into interval number as shown in Table 10; that is, the optimization of triangular fuzzy number comes down to that of the interval number.Table 9
Data of Example27 (The customer evaluation).
Customer
Transfer times
Congestion
Travel time
Ticket price
EvaluationEi
1
[
-
0.1,0.00,0.10
]
[
-
0.1,0.00,0.10
]
[
0.20,0.30,0.40
]
[
0.90,1.00,1.10
]
[
0.10,0.20,0.30
]
2
[
0.20,0.30,0.40
]
[
0.70,0.80,0.90
]
[
0.40,0.50,0.60
]
[
0.50,0.60,0.70
]
[
0.40,0.50,0.60
]
3
[
0.60,0.70,0.80
]
[
0.80,0.90,1.00
]
[
0.90,1.00,1.10
]
[
0.60,0.70,0.80
]
[
0.70,0.80,0.90
]
4
[
0.00,0.10,0.20
]
[
-
0.1,0.00,0.10
]
[
0.30,0.40,0.50
]
[
0.20,0.30,0.40
]
[
0.05,0.15,0.25
]
5
[
0.40,0.50,0.60
]
[
0.30,0.40,0.50
]
[
0.40,0.50,0.60
]
[
0.50,0.60,0.70
]
[
0.40,0.50,0.60
]Table 10
The triangular fuzzy number changed into the interval number.
Customer
Transfer times
Congestion
Travel time
1
[
0.1
α
-
0.1
,
-
0.1
α
+
0.1
]
[
0.1
α
-
0.1
,
-
0.1
α
+
0.1
]
[
0.1
α
+
0.2
,
-
0.1
α
+
0.4
]
2
[
0.1
α
+
0.2
,
-
0.1
α
+
0.4
]
[
0.1
α
+
0.7
,
-
0.1
α
+
0.9
]
[
0.1
α
+
0.4
,
-
0.1
α
+
0.6
]
3
[
0.1
α
+
0.6
,
-
0.1
α
+
0.8
]
[
0.1
α
+
0.8
,
-
0.1
α
+
1.0
]
[
0.1
α
+
0.9
,
-
0.1
α
+
1.1
]
4
[
0.1
α
,
-
0.1
α
+
0.2
]
[
0.1
α
-
0.1
,
-
0.1
α
+
0.1
]
[
0.1
α
+
0.3
,
-
0.1
α
+
0.5
]
5
[
0.1
α
+
0.4
,
-
0.1
α
+
0.6
]
[
0.1
α
+
0.3
,
-
0.1
α
+
0.5
]
[
0.1
α
+
0.4
,
-
0.1
α
+
0.6
]
Customer
Ticket price
EvaluationEi
1
[
0.1
α
+
0.9
,
-
0.1
α
+
1.1
]
[
0.1
α
+
0.1
,
-
0.1
α
+
0.3
]
2
[
0.1
α
+
0.5
,
-
0.1
α
+
0.7
]
[
0.1
α
+
0.4
,
-
0.1
α
+
0.6
]
3
[
0.1
α
+
0.6
,
-
0.1
α
+
0.8
]
[
0.1
α
+
0.7
,
-
0.1
α
+
0.9
]
4
[
0.1
α
+
0.2
,
-
0.1
α
+
0.4
]
[
0.1
α
+
0.05
,
-
0.1
α
+
0.25
]
5
[
0.1
α
+
0.5
,
-
0.1
α
+
0.7
]
[
0.1
α
+
0.4
,
-
0.1
α
+
0.6
]We can obtain that the transfer times are the most important for passengers from Table11 and the convergence rate of Example 27 as shown in Figure 3, which represents the convergence rate of Example 27, where the solid line represents the convergence rate of the target value about the best optimal solution, and dotted line represents the convergence rate of average value of the target value about all the solutions.Table 11
Results of Example27.
Set
λ
g
λ
Error
Number of generation
{
x
1
}
0.05
0.4292
0.0049
100
{
x
2
}
0.2428
{
x
3
}
0.1586
{
x
4
}
0.1524Figure 3
The convergence rate of Example27.


## 8. Conclusions and Remarks

In this paper, we have considered the genetic algorithm optimization for determining fuzzy measures from fuzzy data. We have gotten joint measures by use of single measure with the aid ofσ-λ rules. Then we have formulated our problems to be solved; that is, how to determine the fuzzy measures from fixed data. Furthermore, we have introduced the Choquet integral of interval-valued functions and then given the genetic algorithm optimization to determine fuzzy measures from interval-valued data. Finally we have discussed the Choquet integral of fuzzy number-valued functions. Consequently, we have given the genetic algorithm optimization to determine fuzzy measures from fuzzy number-valued data.



---

*Source: 542153-2013-12-03.xml*