Expected value of fuzzy variable and fuzzy expected value models ...

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 4, AUGUST 2002

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Expected Value of Fuzzy Variable and Fuzzy Expected Value Models Baoding Liu, Senior Member, IEEE, and Yian-Kui Liu

Abstract—This paper will present a novel concept of expected values of fuzzy variables, which is essentially a type of Choquet integral and coincides with that of random variables. In order to calculate the expected value of general fuzzy variable, a fuzzy simulation technique is also designed. Finally, we construct a spectrum of fuzzy expected value models, and integrate fuzzy simulation, neural network, and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models. Index Terms—Expected value, fuzzy programming, fuzzy simulation, genetic algorithm, neural network.

I. INTRODUCTION

design a hybrid intelligent algorithm to solve general expected value models and provide three numerical examples to illustrate the effectiveness of the algorithm. II. EXPECTED VALUE OPERATOR Let be a fuzzy variable with possibility distribution function . A fuzzy variable is said to be normal if there . In this paper, we exits a real number such that always assume that fuzzy variables involved are normal. Let be a real number. It is well known that the possibility of is defined by

S

INCE Zadeh’s pioneering work [24], possibility theory was being perfected and became a strong tool to deal with incomplete and uncertain situation [2], [5], [19], [21]. On the other hand, many researchers such as Zimmermann [25], Luhandjula [11], [12], Yazenin [22], [23], Sakawa [17], Inuiguchi and Ramík [4], Tanaka et al. [20] applied the theory successfully to optimization problems. As the development of more effective computer and the appearance of new algorithms such as genetic algorithm, simulated annealing and neural networks, many complex optimization problems can be solved by computers. Recently, Liu [8] laid a foundation for optimization theory in uncertain (stochastic, fuzzy, fuzzy random, etc.) environments and called such a theory uncertain programming, in which numerous models and hybrid intelligent algorithms are documented. Especially, there are two known classes of fuzzy programming, one is the fuzzy chance-constrained programming [6], [7], the other is fuzzy dependent-chance programming [9], [10]. In this paper, we will present a novel definition of expected value of fuzzy variable and propose a new class of fuzzy programming called fuzzy expected value models. The interested readers may also consult the related work [3], where fuzzy intervals were viewed as consonant random sets, and the mean value of a fuzzy number is thus defined as an interval based on Dempster and Shafer (D–S) Theory. The paper is organized as follows. Section II introduces the definition of expected value of fuzzy variable and presents some basic properties of expected value operator. Three types of fuzzy expected value models are formulated in Section III and a convexity theorem is also proved in this section. Finally, we will

(1) while the necessity of

is defined by (2)

and show the possibility and necessity degrees to what extent is not smaller than . , If we denote the support of by and are two particular fuzzy measures (see [16]) then , where is the power set of . In addition, defined on and are a pair of dual fuzzy measures in the sense that with is the complement of . Furthermore, based on possibility measure and necessity measure, we give the third index, called credibility measure, as follows: (3) . It is easy to check that satisfies the following for any conditions: , and ; i) implies for any . ii) is also a fuzzy measure defined on . BeThus, is self dual, i.e., for any sides, . Example 1: Suppose that is a triangular fuzzy variable , for any real number , we can calculate as follows:

Manuscript received April 30, 2001; revised November 7, 2001 and December 26, 2001. This work was supported by National Natural Science Foundation of China Grant 69804006, and the Sino-French Joint Laboratory for Research in Computer Science, Control and Applied Mathematics (LIAMA). The authors are with the Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). Publisher Item Identifier 10.1109/TFUZZ.2002.800692. 1063-6706/02$17.00 © 2002 IEEE

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Before we present the expected value of a fuzzy variable, we first recall the definition of Choquet integral. This kind of integral was first introduced in [1], and later was restudied in the field of fuzzy measure theory by some researchers such as Murofushi and Sugeno [13], [14], Murofushi, Sugeno and Machida [15], and Narukawa, Murofushi and Sugeno [16]. be a fuzzy measure space. The Choquet inLet tegral of a nonnegative measurable function with respect to a fuzzy measure is defined as

If is a finite fuzzy measure, then the Choquet integral of a measurable function with respect to is defined as

Remark 2: Just like the case of random variable (for example, Cauchy distributed variable), the expected value does not exist for some fuzzy variable. One of the referees of this paper provided the example if if whose expected value does not exist because it is of the form . Let be a normalized discrete fuzzy variable whose possibility distribution function is defined by if

(4)

if

, , and is the dual of in where . the sense that From the measure-theoretic interpretation of Choquet integral, it is usually regarded as the generalization of usual mathematical expectation. Therefore, motivated by the idea of Choquet integral, we present the following definition. Definition 1: Let be a normalized fuzzy variable. The upper , of is defined by expected value,

if We assume without loss of generality that It follows from (7) that the expected value of

. is: (8)

where the weights are given by

(5) while the lower expected value,

, of

is defined by for (6)

The expected value of

is defined as

and satisfy the fol-

lowing constraints: and

(7) , the expected When the right-hand side of (7) is of form value is not defined. Remark 1: If the fuzzy variable is replaced with a random is replaced with variable (whose density function is ) and (whose dual is itself), then we have the probability measure

which is exactly the expected value of random variable . This means that the representation of expected value of fuzzy variable is identical to that of random variable.

since is a normalized fuzzy variable. According to Definition 1, for triangular, trapezoidal, and normal fuzzy variables, we have the following results. , Example 2: If is a triangular fuzzy variable . If is then the expected value of is , then the expected a trapezoidal fuzzy variable . If is a normal fuzzy value of is variable with the possibility distribution function

then the expected value of is exactly . and be real-valued functions defined on , Let we say that and are comonotonic if for any (see [16]). In addition, if is a fuzzy vector, then and are also fuzzy variables. The following proposition gives some properties about their expected values. Theorem 1: Let be a fuzzy vector. The expected value operator has the following properties: , then ; i) if ; ii)

LIU AND LIU: EXPECTED VALUE OF FUZZY VARIABLE AND FUZZY EXPECTED VALUE MODELS

iii) if functions and are comonotonic, then for and , we have any nonnegative real numbers . Proof: Applying [16, Th. 2.4] to expected values of fuzzy and , assertions i) and iii) can be proved. variables Next, we prove assertion ii). By Definition 1, we have

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satisfy as many goals as possible in the order specified. In order to balance the multiple conflicting objectives, we may employ the following fuzzy expected value goal programming model:

subject to:

which complete the proof of assertion ii). Remark 3: For detailed expositions about Chouqet integral and its properties used but not provided in this paper, the readers may consult [13]–[16], and [18].

B. Convexity

III. EXPECTED VALUE MODELS A. General Formulations There are several possibilities to construct fuzzy programming models, for example, in [6], [7], [9], and [10], fuzzy chance-constrained programming and fuzzy dependent-chance programming have been developed. In this section, we will open a new door to formulate fuzzy decision problems, called fuzzy expected value models. In order to obtain the decision with maximum expected return, we may employ the following single-objective fuzzy expected value model:

subject to:

(9)

where and are decision vector and fuzzy vector, respectively, is the objective function, and are the constraint functions . for In many applications, a decision maker may want to optimize multiple objectives. A fuzzy expected value multiobjective programming model may be formulated as follows:

subject to:

A mathematical programming is called convex if it has both convex objective function and convex feasible set. For the fuzzy expected value model, if we add some conditions to objective function and constraint functions, then we have the following result on convexity. be an -ary fuzzy vector. Suppose Theorem 2: Let and that, for any fixed , the functions are convex with respect to , and for any and , the functions and [resp. given and for ] are comonotonic with respect to . Then the following fuzzy expected value model: (12)

subject to:

is a convex programming problem. Proof: By supposition of the theorem, for any fixed , the following inequality:

is valid for any

and

. By Theorem 1, we have

(10)

are objective functions for and are constraint functions for . In multiobjective decision-making problems, the decisionmaker may assign a target level for each goal and the key idea is to minimize the deviations (positive, negative, or both) from the target levels. In the real-world situation, the goals are achievable only at the expense of other goals and these goals are usually incompatible. Therefore, there is a need to establish a hierarchy of importance among these incompatible goals so as to

where

(11) is the preemptive priority factor which expresses the where for all ; relative importance of various goals and (resp. ) is the weighting factor corresponding to positive (resp. negative) deviation for goal with priority assigned; and are the goal constraints and real constraints, respectively, and ; and are the for positive deviations and negative deviations from the target of , respectively. goal for

which implies that the objective function is convex. Now let and be any two feasible solutions. By the convexity of , we have

for any given

and

. It follows from Theorem 1 that:

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which implies that is a feasible solution. Hence, the feasible set is convex. The proof of the theorem is complete. IV. HYBRID INTELLIGENT ALGORITHM In order to solve general fuzzy expected value models, the crux is to calculate uncertain functions such as (13) for any given decision vector . can be expressed as linear form If the fuzzy variable and all s are noninteractive triangular is (resp. trapezoidal, or normal) fuzzy variables, then also a triangular (resp. trapezoidal, or normal) fuzzy variable. can be calculated For this case, the expected value easily. In most cases, we cannot do so due to the complexity of the function . In the following, we suggest a fuzzy simulation to . estimate the value of

we can estimate the expected value (13) by the formula (14) provided that is sufficiently large. According to (14), for any given decision vector , we can by the following algoevaluate the expected value rithm.

Step 1. Sample points uniformly for . from the -cut Step 2. Calculate the values for . Step 3. Rearrange the subscript of and such that . Step 4. Calculate the weights according to formula (15) for Step 5. Calculate the expected value according to formula (14).

and

.

B. Uncertain Function Approximation

if

Thus far, for any given decision vector , we can use fuzzy in simulation to compute the uncertain function value (13). However, it is clearly a time-consuming process. It is known that an NN has the ability to approximate continuous function and the high speed of operation, thus we wish to train a feedforward NN to approximate the uncertain function in order to speed up the solution process. In this paper, we will train the feedforward NN by the popular backpropagation algorithm.

if

C. Hybrid Intelligent Algorithm

A. Fuzzy Simulation In order to obtain the expected value of a fuzzy variable, we design a fuzzy simulation for both discrete and continuous cases. Case I: Assume that is a function, and is a discrete fuzzy vector whose joint possibility distribution function is defined by

if where and

with are the possibility distribution function of for . for . We assume without Let (otherwise we may loss of generality that rearrange them to satisfy the condition), then the expected value is given by (14) where (15) for . Case II: Assume that is a continuous fuzzy vector with a possibility distribution function . Let be a sufficiently small positive number, and the -cut is a bounded subset of . Also, we assume in this paper that . Thus, we can sample points uniformly from the set of and denote them by . and for Assume that such that (otherwise we rearrange these numbers to satisfy the condition), then

Generally speaking, fuzzy expected value models are neither convex nor unimodal. Traditional algorithms are not applicable to such problems, for example, we cannot obtain the derivative of objective function due to the fact that the expected value is estimated by the fuzzy simulation. Thus, we have to apply heuristic algorithms to solving general fuzzy expected value models. In this paper, we integrate fuzzy simulations, NN and a genetic algorithm (GA) to produce a powerful hybrid intelligent algorithm. The procedure to solve general fuzzy expected value models is summarized as follows. Step 1) Generate training input–output data for uncertain functions by fuzzy simulations. Step 2) Train an NN to approximate the uncertain functions by the generated training data. Step 3) Initialize pop_size chromosomes in which the trained NN can be used to calculate the values of uncertain functions. Step 4) Update the chromosomes by crossover and mutation operations and the trained NN may be employed to check the feasibility of offsprings. Step 5) Calculate the objective values for all chromosomes by the trained NN. Step 6) Compute the fitness of each chromosome by rank-based evaluation function based on the objective values. Step 7) Select the chromosomes by spinning the roulette wheel.

LIU AND LIU: EXPECTED VALUE OF FUZZY VARIABLE AND FUZZY EXPECTED VALUE MODELS

Step 8) Repeat the fourth to seventh steps a given number of cycles. Step 9) Report the best chromosome as the optimal solution.

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A run of the hybrid intelligent algorithm (6000 sample points in simulation, 3000 data in NN, 1000 generations in GA) shows that the optimal solution is

V. NUMERICAL EXPERIMENTS In this section, we will give some numerical examples to illustrate the procedure of solving fuzzy expected value models by the hybrid intelligent algorithm. Example 1: Consider first the following fuzzy expected value model:

whose objective value is 3.4496. Example 3: Consider the following fuzzy expected value goal programming model: lexmin subject to:

subject to: where , and are triangular fuzzy variables ( 4, 2, 0), ( 2, 0, 2), and (0, 2, 4), respectively. In order to solve this model, as previously discussed, we first generate input–output data for the uncertain function

by fuzzy simulation. Then we train an NN (three input neurons representing decision variables, five hidden neurons, one output neuron representing objective function) to approximate the uncertain function . Lastly, the trained NN is embedded into a GA to produce a hybrid intelligent algorithm. A run of the hybrid intelligent algorithm (6000 sample points in simulation, 2000 data in NN, 400 generations in GA) shows that the optimal solution (here the optimal solution is, in fact, a satisfactory solution) is

where and are triangular fuzzy variables ( 3, 2, 1) and (1, 2, 3), respectively, and , are normal fuzzy variables whose possibility distributions are given as follows:

and

In this problem, we first generate input–output data for the , where uncertain function :

whose objective value is 3.0103. Example 2: We now consider another fuzzy expected value model subject to:

by fuzzy simulation. Then for each , the values of objective functions are calculated as follows:

and where is a triangular fuzzy variable (1, 2, 3), is a fuzzy , variable with possibility distribution and is a trapezoidal fuzzy variable (2, 3, 4, 8). In order to solve the model, we first employ the fuzzy simulation technique to generate input–output data for the uncertain , where function :

Then we use the training data to train an NN (three input neurons, eight hidden neurons, two output neurons) to approximate the uncertain function . Finally, the trained NN is embedded into a GA to produce a hybrid intelligent algorithm.

Using the training data, we train an NN (four input neurons, ten hidden neurons, three output neurons) to approximate the uncertain function . After that, we embed the trained NN into a GA to produce a hybrid intelligent algorithm. A run of the hybrid intelligent algorithm (5000 sample points in simulation, 2000 data in NN, 3000 generations in GA) shows that the optimal solution is

which satisfies the first two goals, but the third objective value is 0.8309.

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VI. CONCLUSION In this paper, we contributed to the research area of fuzzy optimization in the following four aspects: 1) we presented a new concept of expected value operator of fuzzy variable; 2) we designed a fuzzy simulation to estimate the expected value; 3) we constructed a new class of fuzzy programming—fuzzy expected value models—in addition to fuzzy chance-constrained programming and fuzzy dependent-chance programming; and 4) we integrated fuzzy simulation, neural networks, and genetic algorithm to produce a hybrid intelligent algorithm for solving the fuzzy expected value models. ACKNOWLEDGMENT The authors would like to thank the anonymous referees for their valuable comments and suggestions. REFERENCES [1] G. Choquet, “Theory of capacities,” Ann. Inst. Fourier, Grenoble, no. 5, pp. 131–295, 1955. [2] D. Dubois and H. Prade, Possibility Theory. New York: Plenum, 1988. [3] , “The mean value of a fuzzy number,” Fuzzy Sets Syst., vol. 24, pp. 279–300, 1987. [4] M. Inuiguchi and J. Ramík, “Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem,” Fuzzy Sets Syst., vol. 111, pp. 3–28, 2000. [5] G. J. Klir, “On fuzzy-set interpretation of possibility theory,” Fuzzy Sets Syst., vol. 108, pp. 263–273, 1999. [6] B. Liu and K. Iwamura, “Chance-constrained programming with fuzzy parameters,” Fuzzy Sets Syst., vol. 94, pp. 227–237, 1998. [7] , “A note on chance-constrained programming with fuzzy coefficients,” Fuzzy Sets Syst., vol. 100, pp. 229–233, 1998. [8] B. Liu, Uncertain Programming. New York: Wiley, 1999. [9] , “Dependent-chance programming with fuzzy decisions,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 354–360, June 1999. [10] , “Dependent-chance programming in fuzzy environments,” Fuzzy Sets Syst., vol. 109, pp. 97–106, 2000. [11] M. K. Luhandjula, “Fuzzy optimization: An appraisal,” Fuzzy Sets Syst., vol. 30, pp. 257–282, 1989. [12] , “Fuzziness and randomness in an optimization framework,” Fuzzy Sets Syst., vol. 77, pp. 291–297, 1996. [13] T. Murofushi and M. Sugeno, “An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure,” Fuzzy Sets Syst., vol. 29, pp. 201–227, 1989. [14] , “A theory of fuzzy measure representations, the Choquet integral, and null set,” J. Math. Anal. Appl., vol. 159, pp. 532–549, 1991. [15] T. Murofushi, M. Sugeno, and M. Machida, “Non-monotone fuzzy measure and the Choquet integral,” Fuzzy Sets Syst., vol. 64, pp. 73–86, 1994. [16] Y. Narukawa, T. Murofushi, and M. Sugeno, “Regular fuzzy measure and representation of comonotonically additive functional,” Fuzzy Sets Syst., vol. 112, pp. 177–186, 2000.

[17] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization. New York: Plenum, 1993. [18] D. Schmeidler, “Integral representation without additivity,” in Proc. Amer. Mathematical Society, vol. 97, 1986, pp. 255–261. [19] M. Spott, “A theory of possibility distributions,” Fuzzy Sets Syst., vol. 102, pp. 135–155, 1999. [20] H. Tanaka, P. Guo, and H. J. Zimmermann, “Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems,” Fuzzy Sets Syst., vol. 113, pp. 323–332, 2000. [21] R. R. Yager, “On the specificity of a possibility distribution,” Fuzzy Sets Syst., vol. 50, pp. 279–292, 1992. [22] A. V. Yazenin, “Fuzzy and stochastic programming,” Fuzzy Sets Syst., vol. 22, pp. 171–180, 1987. , “On the problem of possibilistic optimization,” Fuzzy Sets Syst., [23] vol. 81, pp. 133–140, 1996. [24] L. A. Zadeh, “Fuzzy set as a basis for a theory of possibility,” Fuzzy Sets Syst., vol. 1, pp. 3–28, 1978. [25] H. J. Zimmermann, “Applications of fuzzy set theory to mathematical programming,” Inform. Sci., vol. 36, pp. 29–58, 1985.

Baoding Liu (M’99–SM’00) graduated from the Department of Mathematics, Nankai University, Tianjin, China, and received the M.S. and Ph.D. degrees from the Institute of Systems Science, Chinese Academy of Sciences, Beijing, China, in 1986, 1989, and 1993, respectively. He has been a Full Professor at the Department of Mathematical Sciences, Tsinghua University, Beijing, China, since 1998. His current research interests include stochastic programming, fuzzy programming, uncertain systems, intelligent systems, and applications in inventory, scheduling, reliability, project management, and engineering design. He is the author of Uncertain Programming (New York:Wiley, 1999), Decision Criteria and Optimal Inventory Processes (Boston, MA: Kluwer, 1999), and Stochastic Programming and Fuzzy Programming (Beijing, China: Tsinghua Univ. Press, 1998). He has published over 60 papers in international conferences and premier journals. He is an Editorial Board member of the international journal Information, and Associate Editor of Fuzzy Optimization and Decision Making. Dr. Liu is currently serving as Associate Editor of the IEEE TRANSACTIONS ON FUZZY SYSTEMS.

Yian-Kui Liu received the B.S. and M.S. degrees from the Department of Mathematics, Hebei University, Baoding, China, in 1989 and 1992, respectively. Since 1992, he has been with the College of Mathematics and Computer, Hebei University, where he is an Associate Professor. He is currently a Research Fellow at the Uncertain Systems Laboratory, Department of Mathematical Sciences, Tsinghua University, Beijing, China. His research interests, previously within the areas of nonadditive measure theory and multivalued analysis, have extended to include theory of optimization under uncertainty and intelligent systems. He has published more than 20 papers in national and international conferences and premier journals.