Fuzzy Nonlinear Programming with Applications in Decision Making
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Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data
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 the rules. Furthermore we define and characterize the Choquet integrals of intervalvalued functions and fuzzynumbervalued functions based on rules. In addition, we design a special genetic algorithm to determine a type of general fuzzy measures from fuzzy data.
1. Introduction
Fuzzy measures [1–4] and fuzzy integrals [5–9] have been applied successfully in multiattributes decisionmaking [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 nonlinear 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 Section 2, the basic definitions of fuzzy measures based on  rules are reviewed. Section 3 briefly introduces the basic concepts on the Choquet integral of realvalued 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 realvalued data. Section 6, introduces the Choquet integral of intervalvalued functions based on rules. Consequently, we use genetic algorithm optimization to determine fuzzy measures from intervalvalued data. Section 7 discusses the Choquet integral of fuzzy numbervalued functions based on rules, and then uses genetic algorithm optimization to determine fuzzy measures from fuzzy numbervalued data. Finally, conclusions are made in Section 8.
2. Fuzzy Measure Based on  Rules
Definition 1. Let be a nonempty set and a algebra on the . A set function is called a fuzzy measure based on  rules if
where , , and for all and .
Particularly, if , then  rule is additivity.
Definition 2. Let be a algebra on the . is called Sugeno measure based on  rules if satisfies  rules and . Briefly we denoted .
Remark 3. In Definition 1, if , then
Remark 4. In Definition 2, is a classical probability measure if , 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, , and , are engaged in producing the same kind of products; the efficiencies of every people are given as follows: , and . Then we can get the joint efficiencies by use of  rules as shown in Table 1.

Remark 6. In Example 5, can be viewed as a attribute, , we can calculate the contribution of their joint attributes by use of  rules if we only know the contribution of individual attribute .
Theorem 7. Let be a Sugeno measure based on  rules. If , then has the monotonicity.
Proof. Let and . Since , this implies that
for and .
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]). Let be a finite set and be the power set of . Set function is called a regular fuzzy measure defined on let if the following conditions are satisfied:(1);(2)if , and , then .
Definition 9 (see [28]). Let be a finite set and be the power set, of . A fuzzy measure is called a regular fuzzy measure defined on let if the following conditions are satisfied:(1);(2)if , where and .
Theorem 10. Let be a Sugeno measure based on  rules. Then is a regular fuzzy measure defined on .
Proof. We could prove that . Otherwise for every , we have Since , we have ; it is a contradiction. Furthermore, we obtain , and has the monotonicity by Definition 2 and Theorem 7.
Denoting finite set , the value for all is called measure density.
Theorem 11. The parameter of a regular Sugeno measure based on  rules is determined by the following equation:
Proof. We can prove the above theorem by Theorem 10 and [32].
If we know the values of Sugeno measure based on  on singleton sets, we can use Theorem 11 to obtain the values of and then use Definition 1 to obtain the values on the other sets. It implies that a Sugeno measure based on  can be determined by measure densities.
Theorem 12 (see [28]). If one knows the measure density on finite set , then there is only one solution obtained from
3. The Choquet Integrals of RealValued Function Based on  Rules
Definition 13 (see [34]). A regular fuzzy number, denoted by , is a fuzzy subset of with membership function satisfying the following conditions. (RFN1) there exists at least one number such that ; (RFN2) m(t) is nondecreasing on and nonincreasing on ; (RFN3) m(t) is upper semicontinuous; that is, if and if ; (RFN4) .
The set of all regular fuzzy numbers is denoted by .
Let be a measurable function with respect to ; that is, satisfies the condition that
for any .
From now on, we suppose that all functions defined on appearing as an integrand of the Choquet integral in this paper are measurable.
Definition 14 (see [22]). Let be a measurable space, and let be a Sugeno measure based on  rules on . The Choquet integral of a realvalued function is defined as
where
for , if both of Riemann integrals exist and at least one of them has finite value.
Let , then Choquet integral of a nonnegative realvalued function is defined as
Without loss of the generality, Yang et al. [34] have proposed a new scheme to calculate the value of a Choquet integral with a realvalued integrand.
When , for any function , both and are functions of with bounded variance; therefore, their Riemann integrals with respect to exist and are finite. So, Choquet integral is well defined. To calculate the value of the Choquet integral of a given realvalued function , usually the values of , that is, , should be sorted in a nondecreasing order, so that , where is a certain permutation of . Then, the value of the Choquet integral is obtained by where .
Example 15. Let , and . We can obtain by Theorem 11. Furthermore we get .
By Definition 1, we can get the following results:
Similarly, ,, .
By Definition 14, we can get the Choquet integrals of with respect to as follows:
Example 16. Let , and (Table 2).
By Theorem 11, we obtain that and with the aid of Mathematica software, we calculate that ; furthermore, we get Choquet integrals which are shown in Table 3.


4. Questions Description: Determine Fuzzy Measures
In this section, we formulate our problems to be solved.
If we regard fuzzy integrals as multiinput singleoutput systems, we can obtain the Data through handling these systems. Suppose that we have several information sources and a given object . Let ; we have the data with sample size as shown in Table 4, where is the th value of source and is the th value of object.
We hope to find a Sugeno measure on measurable space , such that , , where function is defined by for .
If such a Sugeno measure 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:
A result of also means that a precise solution is found.
Here, we discuss this problem in three aspects. The first one is the values of , and are a realvalued data. The second one is the values of , and are an intervalvalued data. The last one is , and are a fuzzyvalued data.
5. Using Genetic Algorithm to Determine Fuzzy Measures from RealValued Data
In this section, we use genetic algorithm to determine fuzzy measures from realvalued 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
Chromosome is denoted by , where gene for all , and , and .
5.1.2. The Decoding
Find the formula by , and Definition 1. Furthermore, the values of , for all 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 probability to choose two parent chromosomes and and use the arithmetic crossover to get two offspring chromosomes and : where .
5.1.4. The Nonuniform Mutation
Select parent chromosome according to the mutation probability and take the mutation to for the random generation of in . Let or where and are the upper and lower bound of , respectively, and is the number of generations. Function is defined as follows: where is a random number of , is the largest number of generations, and is the parameter. Obviously, .
5.1.5. The Evaluation Function
Evaluation function is defined by where 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 (Table 5). Customer bases its overall scores on transfer times, intrain congestion, travel time, and ticket price, and also they have a score for each of four aspects. Let 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, , , and the population size is 20.
We can obtain that the transfer times are the most important for passengers from Table 6 and the convergence rate of Example 17 as shown in Figure 1.


6. Using Genetic Algorithm to Determine Fuzzy Measures from IntervalValued Data
The intervals are derived from many practical application problems, when instead of knowing the precise values of some quantity we know only the intervals , in which , ranges. Since the comparison of two values or quantities is the basic and most frequently used step in optimization, intervalvalued function plays an important role in interval computation development.
6.1. The Choquet Integrals of IntervalValued Function Based on Rules
With the definitions of the preceding subsections and from Wu et al. [35], we assume that , is the set of interval numbers, and is the set of all interval numbers
Interval numbers satisfy the following basic operations:(1);(2);(3);(4);(5)if , then .
Definition 18 (see [34]). An intervalvalued function is measurable if both and are measurable function of , where , is the left end point of interval , and is the right end point of interval .
Definition 19. Let be a nonadditive measure space based on  rules a measurable function in and . Then the Choquet integral of with respect to is defined by if is a closed interval on , where is a measurable selection on .
Theorem 20. Let be a measurable intervalvalued function on , and let be a Sugeno measure based on  rules on . The Choquet integral of with respect to is where is the left end point of interval and is the right end point of interval , for every .
Proof. We can prove the above theorem by Theorem 10 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. Let be a measurable intervalvalued function on , let be a sugeno measure based on  rules on , and . Then, the value of Choquet integral of with respect to is obtained by where , , the values of and , that is, , and should be sorted in a nondecreasing order, so that and , respectively, and and are a certain permutation of , respectively.
6.2. Using Genetic Algorithm to Determine Fuzzy Measures from IntervalValued Data
In this subsection, we use genetic algorithm to determine fuzzy measures from intervalvalued data.
6.2.1. Genetic Algorithm (GA)
In this subsection, the all algorithms are the same to the Section 5.1 but the evaluation function. Here, the valuation function is defined by where and are defined by Definition 14 and Theorem 20.
6.2.2. Examples and Results
Example 22. If the evaluation information in Example 17 is represented by the intervalvalued fuzzy numbers as shown in Table 7, then we redetermine fuzzy measures from intervalvalued data by using genetic optimization.
We can obtain that the transfer times are the most important for passengers from Table 8 and the convergence rate of Example 22 as shown in Figure 2.


7. Using Genetic Algorithm to Determine Fuzzy Measures from FuzzyValued Data
In many respects, fuzzy numbers depict the physical world more realistically than singlevalued 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 ) be a fuzzy measurable space, and let be the class of all fuzzy subsets of . A fuzzyvalued function is called a measurable function if for every , its cut is measurable, where
Remark 24. A measurable fuzzyvalued function is a especially measurable fuzzy setvalue function.
Let , and . We will simply denote that
Obviously, and are real functions.
From now on, we suppose that all functions defined on appearing as an integrand of the Choquet integral in this paper are measurable.
According to Theorem 10, is a signed fuzzy measure on . Therefore, we may give the following definition referring to [34].
Fuzzyvalued function is said to be a integrally bounded, if there exists a Choquet integrable function such that for .
Definition 25. Let be a measurable fuzzyvalued function on , and let be a Sugeno measure based on  rules. Assume that is integrally bounded. is called Choquet integrable with respect to if confirms a unique fuzzy number , which is denoted by , where is a measurable selection of .
The exact membership function of the Choquet integral with respect to Sugeno fuzzy measure for fuzzyvalued integrand is rather difficult to be found. In a simpler but common case where is finite, according to Definition 25, the calculation of the Choquet integral with a fuzzyvalued function comes down to that of the Choquet integral with an intervalvalued function. Here, let us look at examples to show how to calculate the fuzzyvalued Choquet integral with respect to a Sugeno measure .
Example 26. Let ,, and . Here, a triangular fuzzy number is denoted by , where and . Set function is a sugeno measure based on  rules. Function is triangular fuzzy valued. The membership function of , and are
respectively. They are shown in Figure 1. The cut of is represented by interval
It is easy to get , and .
We conclude that
When , we get . Using Definition 14, we can let , and then
where .
When , we get . Using Definition 14 we can let , and , and then