Journal of Information & Computational Science 8: 5 (2011) 842–849 Available at http://www.joics.com
Optimal Aggregation of Interval Numbers Based on Genetic Algorithm in Group Decision ⋆ Bingyi Kang a,d , Yajuan Zhang a,b , Xinyang Deng a,e , Jiyi Wu b Xiaohong Sun c , Ya Li a , Yong Deng a,f,∗ a College
of Computer and Information Sciences, Southwest University, Chongqing 400715, China
b Hangzhou c College
Key Lab of E-Business and Information Security, Hangzhou Normal University Hangzhou 310036, China
of Food Science and Technology, Shanghai Ocean University, 999 Hucheng Huan Road Shanghai 201306, China
d Key
Subject Laboratory of National Defense for Radioactive Waste and Environmental Security Southwest University of Science and Technology, Mianyang 621010, China
e Key
Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, China f School
of Electronics and Information Technology, Shanghai Jiaotong University Shanghai 200240, China
Abstract A group of decision-makers may differ in their choice of alternatives while making a decision. As a result, how to aggregate opinions of different experts is still an open issue. Due to the uncertainty in the process of decision making, some mathematic tools such as fuzzy sets theory and evidence theory, are used to model uncertain information. In this paper, the experts’ opinions are represented by interval numbers. A new optimal aggregation method to combine interval data is proposed based on genetic algorithm. The new method can deal with experts’ opinions in an efficient manner. A numerical example on group decision making under uncertain environment is used to illustrate the efficiency of our proposed method. Keywords: Genetic Algorithm; Interval Number; Distance Measure; Group Decision Making
⋆ The work is partially supported by National Natural Science Foundation of China, Grant (No. 60874105, 60904099), Program for New Century Excellent Talents in University, Grant (No. NCET-08-0345), Shanghai Rising-Star Program Grant (No. 09QA1402900), Chongqing Natural Science Foundation, Grant (No. CSCT, 2010BA2003), Aviation Science Foundation, Grant (No. 20090557004), the Fundamental Research Funds for the Central Universities Grant (No. XDJK2010C030), National Defence Sciences Funding of Shanghai Jiao Tong University Grant (No. 11GFF-17), Doctor Funding of Southwest University Grant (No. SWU110021), Leading Academic Discipline Project of Shanghai Municipal Education Commission Grant (No. J50704), the Open Project Program of Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University Grant (No. 2009ICIP03), the Open Project Program of Hangzhou Key Lab of E-Business and Information Security, Hangzhou Normal University Grant (No. HZEB201001), Key Subject Laboratory of National defense for Radioactive Waste and Environmental Security Grant (No. 10zxnk08). ∗ Corresponding author. Email addresses:
[email protected],
[email protected] (Yong Deng).
1548–7741/ Copyright © 2011 Binary Information Press May 2011
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Introduction
In Multi-Criteria Decision Making (MCDM), each expert makes his judgement of the alternative based on certain criterion and provides his own rating or grading for that alternative. A decision maker may prefer imprecise representations to alleged exact estimates. It is because of the imprecision or vagueness inherent in the subjective assessment of the different decision makers that we use fuzzy set theory and interval analysis to arrive at a decision [1, 2]. Thus, finding a group consensus method of aggregating experts’ opinions to represent a common opinions is a very important issue under group decision making environments. Extensive researches have focused on aggregate subjective opinions via fuzzy sets theory and achieve fruitful results [4, 9]. As discussed above, experts may use fuzzy numbers or interval numbers to represent their opinions. An interval number can be thought as an extension of the concept of a real number and also as a subset of the real line R. As a coefficient, an interval signifies the extent of tolerance (or a region) that the parameter can possibly take. Hence, interval analysis is a simple and intuitive way to handle uncertainty information for complex decision problems. Decision makers or domain experts may choose interval numbers to represent their subjective opinions in some situations. Developing an efficient method to aggregate interval numbers is very important in group decision making. Though many relative works have been proposed [4-11], However, the use of interval number is not much attended as it merits. Intuitively, the aggregation of multiple imprecise information is a process of optimization to find a representation to take place of all the multiple imprecise information. Genetic Algorithm, one of the heuristic solution-search or optimization technique, originally motivated by the Darwinian principle of evolution through (genetic) selection [12], is widely used in science, engineering and industry. Hence, the idea of genetic algorithm is used to deal with the aggregation of different expert’s opinion with imprecise information. This paper is organized as follows. In section 2, The basic concept about interval numbers and the basic theory of genetic algorithms are introduced. The method of aggregating interval numbers is proposed in section 3. In section 4, a numerical example is shown to illustrate the procedure. Finally, a conclusion is made in section 5.
2
Preliminaries
In this section, we briefly introduce some basic definitions about interval numbers and genetic algorithm.
2.1
Interval Number
] [ Let R be the real number domain. A bounded interval xL , xU is called the interval number [2, 3]. In this paper, we assume that each expert constructs an interval number to represent their subjective estimate of the rating to a given criterion and alternative. For example, in a set-up having large number of personnel, when one deals with the problem of budget forecasting, wherein estimates of costs needed for the development of a new systems are required, the experts dealing with the system may give their estimates in terms of interval numbers. These experts may feel more comfortable if they are asked to give their cost estimates in terms of minimum cost, say
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′ ′ a1 ,
and maximum cost, ′ a′2 ,i.e. an interval number [a1 , a2 ]. Then the problem will be to find an overall cost estimate. The problem can be modeled as: Each expert Ei (i = 1, 2, . . . , n) constructs the interval number Oi = (ai , bi ) to represent his own opinion. Construct an aggregation function to combine these opinions Oi (i = 1, 2, . . . , n) to represent the common opinion O. Let F (R) be the set of interval numbers in R, and the distance between two interval numbers A(a1 , a2 ) and B(b1 , b2 ) be defined as [13]: ] [( ]}2 ) ) ∫ 1/2 {[( a1 + a2 b1 + b2 2 D (A, B) = + x (a2 − a1 ) − + x (b2 − b1 ) dx (1) 2 2 −1/2 The distance can also be calculated as follows [( ) ( )]2 a1 + a2 b1 + b2 1 2 − D (A, B) = + [(a2 − a1 ) + (b2 − b1 )]2 2 2 12
2.2
(2)
Genetic Algorithms
The simple genetic algorithm over populations defined as multisets P (t) = (a0 (t), a1 (t), ..., ar−1 (t)) consisting of r individual binary l−tuples ak (t) = (ak,0 (t), ak,1 (t), ..., ak,l−1 (t)) ∈ Ω with fitness values f (ak (t)). For the creation of offspring individual in each generation t random genetic operators like crossover χΩ and mutation µΩ are applied to parental individuals which are selected according to their fitness values as follows. The population P (0) is initialised appropriately, e.g. by randomly choosing individuals in Ω. %procedure : simple genetic algorithm ===================================== t := 0; while end of adaptation ̸= ture do f or k = 0 to r − 1 do select parental l − tuples b(t) and c(t) apply crossover χΩ and mutation µΩ ak (t + 1) := µΩ (χΩ (b(t), c(t))); evaluate f itness f (ak (t + 1)); end t := t + 1; end ===================================== (1) Selection For fitness-proportional selection each individual ak (t) in the population P (t) is selected with a probability pk (t) which is directly proportional to its fitness f (ak (t)) ≥ 0. The individual selection probability is then given by f (ak (t)) pk (t) = ∑r−1 j=0 f (aj (t))
(3)
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(2) Crossover The crossover operatorχΩ : Ω×Ω → Ω takes two selected individual l−tuples a = (a0 , ..., al−1 ) and b = (b0 , b1 , ..., bl−1 ) and randomly generates an offspring l−tuple c = (c0 , c1 , ..., cl−1 ) according to c = χΩ (a, b). For l − pointcrossover a position 1 ≤ λ ≤ l − 1 is chosen with uniform probability and the two l−tuples (a0 , ..., aλ−1 , bλ , ..., bl−1 ) and (b0 , ..., bλ−1 , aλ , ..., al−1 ) are generated one of which is randomly chosen as offspring c. The 1-point crossover operator is applied with crossover probability χ. In case of unif ormcrossover each bit of the offspring c is chosen with probability χ from the parental l−tuple a and with probability 1 − χ from the parental l−tuple b or vice versa. (3) Mutation In the simple genetic algorithm with binary l−tuples the bitwise mutation operator χΩ : Ω → Ω is defined by randomly flipping each bit of the individual l−tuple a = (a0 , ..., al−1 ) with small mutation probability µ. A typical value is µ ↔ 1l .
3
The Proposed Optimal Aggregation Method
The main idea of our proposed method can be shown as follows, this paper integrates the fuzzy numbers by applying the distance based on interval numbers to find a special interval number ([x1 , x2 ], a ≤ x1 ≤ x2 ≤ b) whose distance is the optimal value from all other normalized interval numbers with the searching method of Genetic Algorithms. The objective function f (x1 , x2 ) is constructed with the summation of all the distance between [x1 , x2 ] and each normalized interval number. Hence, the problem is mapped into find the optimal solution of f (x1 , x2 ) under the constraint a ≤ x1 < x2 ≤ b with the method of Genetic Algorithms. This process can be mathematically described by Eq. (4). n ∑ D2 (O, Oi ) min SDF (x1 , x2 ) = i=1 O = [x , x ] 1 2 a≤x ≤b 1 s.t. a ≤ x2 ≤ b x1 < x2
(4)
where Oi is the interval value of opinion of the ith expert. D2 (O, Oi ) is the interval distance between O and Oi . a and b is the boundary range of x1 and x2 . Now, the main procedure of aggregating interval numbers is introduced as follows: (1) Determination of the encoding method: The first step is to determine the encoding strategy, in other words, how to represent the data between a and b with the computer language. In this paper, the binary encoding strategy is adopted to represent the chromosome. The length of chromosome depends on the accuracy of encoding. Assume the domain of variable x1 is [a, b], and the accuracy of encoding prec is the digit after decimal point. The length of a binary string variable L can be calculated as follows: ( ) b−a +1] [ prec (5) L = max int log2
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[ b−a +1] where max int(x) means the minimum integer which is more than x, especially, L=log2prec , [ b−a +1] if log2prec is an integer in itself. (2) Determination of the decoding method: It is not very difficult to translate the binary string (bL−1 , ..., b0 ) to real number accordingly. First, the binary string (bL−1 , ..., b0 ) should be translate to decimal number x′ : (bL−1 , . . . , b0 )2 = (
L−1 ∑
bi × 2i )10 = x′
(6)
i=0 ′
Second, the decimal number x can be translate into the data x in the interval number [a, b] x = a + x′ ·
b−a 2L − 1
(7)
(3) Construction of the initial population: According to Eq. (5), the encoding length of chromosome for each variable can be obtained. Hence, the total length of chromosome for variable can be calculated by accumulating the length of two single long chromosome. For the method of generating population, the point position of each chromosome can be denoted as: { 1, ξi > 0.5 where ξi ∈ U (0, 1) (8) bi = 0, ξi ≤ 0.5 (4) Determination of the adaptive function and adaptive value: Generally, the adaptive function is designed according to the objective function f ∗ (x1 , x2 ), and the adaptive function is denoted as F (x1 , x2 ). In order to lay the foundation for calculating the selected probability of each individual behind, the optimalizing direction of adaptive function F (x1 , x2 ) should adapt to the incremental direction of adaptive value. Due to the value of objective function f ∗ (x1 , x2 ) > 0, the adaptive function F (x1 , x2 ) is defined as: F (x1 , x2 ) = e
−f ∗ (x1 ,x2 )
−
=e
n ∑
D2 (O,Oi )
i=1
(9)
(5) Determination of the selection criteria: In this paper, the proportional selection strategy of adaptive value is introduced, and the the proportion of every individual is defined as selected probability Pi . Assume the population whose scale is n as pop = {a1 , a2 , a3 , ..., an }, and the adaptive of ai as Fi , then the selected probability Pi is denoted as: Fi , i = 1, 2, 3, . . . , n Pi = ∑ n Fi
(10)
i=1
then the accumulative probability Qi of every chromosome is denoted as follows: Qi =
i ∑
Pj , j = 1, 2, . . .
(11)
j=1
After a random data r, r ∈ [0, 1] is generated, The selection of chromosome Ui for a new population can be selected if Qi−1 ≤ r ≤ Qi .
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(6) Determination of the genetic operators: Supposing we have a population pop(1) including four individuals described as follows: pop(1) = {< 1101011101001100011110 >, %%U1 < 1000011001010001000010 >, %%U2 < 0001100111010110000000 >, %%U3 < 0110101001101110010101 >}%%U4 After several Roulette Wheel testing, Assuming chromosome U2 occupying the most area of the whole circle and chromosome U3 occupying the least area of the whole circle. According to the selecting criteria, chromosome U2 is selected to make a reproduction, while chromosome U3 is fell into disuse. newpop(1) = {< 1101011101001100011110 >, %%U1 < 1000011001010001000010 >, %%U2 < 1000011001010001000010 >, %%U2 < 0110101001101110010101 >}%%U4 The crossover operator in this paper adopts the strategy of a single cutting crossover. This method considers the two flanks of the cutting into two substrings. Then the right substring should be exchanged with each other to get two new individuals. If the crossover probability Pc = 25%, it means that 25% of the chromosomes at average exchange each other. < 110101110
1001100011110 >
new :< 1101011101010001000010 >
crossover : < 100001100
1010001000010 >
new :< 1000011001001100011110 >
Mutation operator is to change some gene of chromosome with a tiny possibility. If the mutation probability Pm = 0.01, it means that 1% of all the genes are expected to mutate. (7) Determine the process parameter and terminal condition: Through the prior test, the crossover probability Pc is between 0.4 and 0.99, the mutation probability Pm is 0.0001 and 0.01, the scale of population is between 20 and 100, the terminated condition may be determined by the precise N iterated generation or determined by the minimum bias δ, which satisfies |f itnessmax − f itness∗ | ≤ δ
(12)
where f itnessmax is the maximum fitness value, and f itness∗ is the objective fitness value. If the judgement condition is not satisfied with the terminal condition, then goto step(4).
4
A Numerical Example
In this section, a numerical example is used to illustrate the procedure of our method. Three experts, E1 , E2 and E3 , give their interval number as follows: O1 = (5, 7); O1 = (5.5, 8); O1 = (6, 7.5)
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Assume the importance of each expert is same. First, We suppose that the integrated opinion of the three experts is represented with an interval numbers O = (x1 , x2 ), 4 ≤ x1 < x2 ≤ 9. So the distance between O and each expert Ei (i = 1, 2, 3) can be showed as follows according the distance of the interval numbers. f (x1 , x2 ) =
3 ∑
D2 (O, Oi ) = D2 (O, O1 ) + D2 (O, O2 ) + D2 (O, O3 )
i=1
= x21 +x1 x2 +x22 +7x1 − where 4 ≤ x1 < x2 ≤ 9
37 769 41 x1 − x2 + , 2 2 6
(13)
Hence, the integration of three interval numbers is mapped into solve the minimum of the formula f (x1 , x2 ) under the constraint of 4 ≤ x1 < x2 ≤ 9 Assuming the accuracy of encoding prec is 0.0001, the crossover probability pcro is 0.2, the mutation probability pmut is 0.05. First, the length of chromosome can be calculated as log2 [(9 − 4)/0.0001 + 1] = 16, Then the encoding length of x1 and x2 is 32. The adaptive function is 41 37 769 2 2 F (x1 , x2 ) = e−f (x1 ,x2 ) = e−(x1 +x1 x2 +x2 +7x1 − 2 x1 − 2 x2 + 6 ) . After 300 times repeat. we get the convergent result is (5.4949, 7.5023), the minimal value the objective function is 4.4315.
5
Conclusion
In multi-criteria decision making problems under group decision environment, there arise situations of conflict and agreement among opinions of expert. Hence, finding a group consensus to represent a common opinion is an important issue. This process can be modelled based on the idea of genetic algorithms searching for the optimal solution in the decision domain. In this paper, the expert opinions are represented with interval numbers due to the the subjectivity, vagueness and imprecision enter into the assessments of experts. After the simple process of encoding, decoding, determining the adaptive function and adaptive value, determining the genetic operator category, the optimal interval number (a general consensus opinion) can be determined in a efficient manner. The proposed method can be easily used in many group decision making problems.
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