A New Approach for Uncertain Multiobjective Programming Problem Based on P E Principle

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1 INFORMATION Volume xx, Number xx, pp ISSN c 21x International Information Institute A New Approach for Uncertain Multiobjective Programming Problem Based on P E Principle Zutong Wang 1, Jiansheng Guo 1, Mingfa Zheng 2, Youshe Yang 3 1 College of Material Management and Safety Engineering, Air Force Engineering University, Xi an, 7151, China 2 College of Science, Air Force Engineering University, Xi an, 7151, China 3 Department of Mathematics, College of Xingjing, Xi an, 71236, China *corresponding author: mingfazheng@126.com Abstract On the basis of the uncertainty theory, this paper is devoted to the uncertain multiobjective programming problem. Firstly, several principles are provided to define the relationship between uncertain variables; then a new approach is proposed for obtaining Pareto efficient solutions in uncertain multiobjective programming problem based on P E principle, which involves transforming the uncertain multiobjective problem into a problem with only one uncertain objective function, and its validity has been proved. Due to the complexity of this problem, it is very suitable for the use of genetic algorithm. Finally, a numerical example is presented to illustrate the novel approach proposed, and adopt the genetic algorithm to solve it. Keywords: Uncertain Programming; Uncertainty Theory; Multiobjective Programming; Genetic Algorithms 1 Introduction The deterministic multiobjective programming (MOP) problem has been widely studied by researchers in a variety of fields, where the most intensive development of the theory and methods can be found in the books by Kaisa (1999) and in the articles of Melih and Meral (29), Tarek (29). However, real world situations are often indeterministic; the decision-maker usually cannot obtain the values of some parameters at the moment he or she has to make the decision. With the great improvement of probability theory, the stochastic MOP methods have been widely used in many realworld decision making problems, including power systems planning (Teghem and Kunsch 1985), distributed energy resources planning (Arturo, Graham, and Stuart, 21), network design (Anthony, Juyoung, Seungjae, and Youngchan, 21), etc. Unfortunately, when the sample size is too small to estimate a probability distribution, the frequently used probability distribution is not always appropriate, especially when the information is vague; we have to invite some domain experts to evaluate their belief degree that each event will occur in this case. Such types of indeterminacy are called uncertainty. A lot of surveys showed that human beings usually overweight unlikely events, and the personal belief degree may have much larger variance than the real frequency (Liu, 212b). Liu (212a) declared that it is inappropriate to apply both probability theory and fuzzy set theory to uncertainty, because both theories may lead to counterintuitive results in this case. In order to deal with such kind of uncertain problem, Liu (29b) founded the uncertainty theory, which is a branch of mathematics based on normality, monotonicity, self-duality, and countable subadditivity axioms, as a mean of handling uncertainty that is due to imprecision rather than randomness. Obviously, the uncertain MOP models can lead to very large scale problems. Though Liu (27) first introduced the uncertainty theory into uncertain programming problem in 27, the literatures about uncertain MOP problem study are still very few. To the best of our knowledge, only Liu and Chen (212) put forward several definitions and models with respect to uncertain MOP problem in the previous work. The traditional solution method for uncertain MOP problem proposed by Liu and Chen (212) is to convert the uncertain MOP problem into a deterministic MOP problem first, and then solve the deterministic MOP problem, which can be referred to using the term multiobjective approach. However, when obtaining the equivalent deterministic problem using multiobjective approach, each objective function is transformed into deterministic function separately, so the possible existence of uncertain dependencies between objectives and the uncertainty nature of original uncertain MOP problem are not taken into account. In this paper, instead of transforming the uncertain MOP problem into a deterministic MOP problem directly, a new solution method is proposed to generate Pareto efficient solution of uncertain MOP problem, by considering the transformation of the uncertain MOP problem into an uncertain programming problem with only one uncertain objective function first, and then into its equivalent deterministic problem, which is called as uncertain approach. This approach 54

2 A NEW APPROACH FOR UNCERTAIN MULTIOBJECTIVE PROGRAMMING PROBLEM BASED ON P E PRINCIPLE will guarantee that the converted single uncertain objective function is still provided with the uncertain nature of original uncertain MOP problem. Personally speaking, the Pareto efficient solution in uncertain MOP problem should be defined on the uncertain objectives directly instead of the converted deterministic objectives, which will assure the uncertain nature of uncertain MOP problem. The symbol or is used to denote the relationship between uncertain objectives. For instance, f( x, ξ) f(x, ξ) means that the valuation of uncertain objective f( x, ξ) is lower than or equal to that of uncertain objective f(x, ξ), and f( x, ξ) f(x, ξ) means that the valuation of uncertain objective f( x, ξ) is strictly lower than that of uncertain objective f(x, ξ), where the valuation is a function defined under certain principles that determine the value of uncertain objectives. Due to different real-life problems call for different meanings of uncertain objectives valuation to satisfy its need in practical application, we have to propose corresponding principle P to define this valuation according to the real context of problem. For example, in the uncertain machine scheduling problem, since we want to minimize the makespan and cost in the long run, we would like to use the expected value of makespan and cost to define this valuation. In this paper, several consistent comparison methods are presented to define the relationship between uncertain variables, which use expected-value, variance, α optimistic value and α pessimistic value of uncertain variables respectively. Following these relationship definitions and the definition of Pareto efficient solution in uncertain MOP problem, several common principles in uncertain MOP problem are proposed, such as expected-value principle, minimum-variance principle, α optimistic value principle and α pessimistic value principle, which are denoted as P E, P V, P αsup and P αinf, respectively. As the expected value of uncertain variable is widely used in real-life problem, in this paper the P E principle of uncertain MOP problem is adopted to obtain P E Pareto efficient solutions. Based on the P E principle and the definition of P E Pareto efficient solution in uncertain MOP problem, it is proved that the optimal solution obtained using uncertain approach is P E Pareto efficient solution in original uncertain MOP problem, and it is shown that the solution obtained using uncertain approach is usually different from that obtained using multiobjective approach. Considering the complexity of the uncertain MOP problem, such as NP-hard with many local extremums, and that the results obtained in uncertain MOP problem are very sensitive to the solutions, evolutionary algorithms should be widely applied to the uncertain MOP problem for successful generation of solutions. Optimization based on genetic algorithms (GA) is known to be an efficient approach to solve classical multiobjective optimization problem under constraints (Fonseca C.M. and Fleming P.J., 1993), therefore, a new approach using GA is explored for obtaining Pareto efficient solutions in this paper. The paper is organized in the following manner. In Section 2, some useful definitions and properties about uncertain theory with application to uncertain MOP problem are introduced. In Section 3, a new approach to generate P E Pareto efficient solution of uncertain MOP problem based on the P E principle is proposed and its validity has been proved. In Section 4, a numerical example with many uncertain local minimums is provided. Finally, a brief summary is given and some open points are stated for future research work in Section 5. 2 Preliminaries Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ in L is called an event. A set function M from L to [, 1] is called an uncertain measure if it satisfies the following axioms (Liu, 27): Axiom 1. (Normality Axiom) M{Γ} = 1 for the universal set Γ. Axiom 2. (Duality Axiom) M{Λ} + M{Λ c } = 1 for any event Λ. Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ 1, Λ 2,, we have M{ Λ i } M{Λ i } Axiom 4. (Product Axiom) Let (Γ k, L k, M k ) be uncertainty spaces for k = 1, 2,. The product uncertain measure M is an uncertain measure satisfying M{ Λ k } = M k {Λ k } k=1 where Λ k are arbitrarily chosen events from L k for k = 1, 2,..., respectively. The triplet (Γ, L, M) is referred to as a uncertainty space (Liu, 27), in which an uncertain variable is defined as follows: Definition 1 (Liu, 27) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event. k=1 {ξ B} = {γ Γ ξ(γ) B} 55

3 ZUTONG WANG, JIANSHENG GUO, MINGFA ZHENG, YOUSHE YANG Definition 2 (Liu, 29b) The uncertain variables ξ 1, ξ 2,, ξ n are said to be independent if M{ n (ξ i B i )} = n M{ξ i B i } for any Borel sets B 1, B 2,, B n of real numbers. Definition 3 (Liu, 29a) The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ(x) = M{ξ x} for any real number x. Definition 4 (Liu, 211) Let ξ be an uncertain variable with regular uncertainty distribution Φ. Then the inverse function Φ 1 is called the inverse uncertainty distribution of ξ. Definition 5 (Liu, 27) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E[ξ] = M{ξ x}dr M{ξ x}dr provided that at least one of the two integrals is finite. Definition 6 (Liu, 27) Let ξ be an uncertain variable with finite expected value e. Then the variance of ξ is V [ξ] = E[(ξ e) 2 ]. Theorem 1 (Liu, 29b) Let ξ 1, ξ 2,, ξ n be uncertain variables, and f a real-valued measurable function. Then f(ξ 1, ξ 2,, ξ n ) is an uncertain variable. Theorem 2 (Liu, 211) Let ξ be an uncertain variable with regular uncertainty distribution Φ. If the expected value exists, then E[ξ] = Φ 1 (α)dα Theorem 3 (Liu, 211) Let ξ 1, ξ 2,, ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If the function f(x 1, x 2,, x n ) is strictly increasing with respect to x 1, x 2,, x m and strictly decreasing with respect to x m+1, x m+2,, x n, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain variable with inverse uncertainty distribution Ψ 1 (α)=f(φ 1 1 (α), Φ 1(α),, Φ 1 m (α), Φ 1 2 m+1 (1 α), Φ 1 3 A new solution approach uncertain approach m+2 (1 α),, Φ 1(1 α)). 3.1 Definitions of relationship between uncertain variables In order to define Pareto efficiency in uncertain MOP problem on the uncertain objectives directly, the definition of relationship between uncertain variables should be proposed first. Definition 7 Let ξ and η be two uncertain variables, we say ξ (or ) η if and only if P[ξ] (or <)P[η], where ξ η means that the valuation of ξ is lower than or equal to that of η, ξ η means that the valuation of ξ is strictly lower than that of η, and P denotes the principle used to define the valuation of uncertain variable. Remark 1. It is worth pointing out that for the comparison between uncertain variables, the relationship is defined under valuation of uncertain variables. Different real-life problems call for different meanings of valuation to satisfy its need in practical application, therefore, corresponding principle P should be proposed to define this valuation according to the real context of problem, where P is the generalization of meanings of this valuation. Obviously, the principle P used to define the valuation of uncertain objectives is of a great generality, it not only include the existing principles, but also those principles we have not constructed yet. However, the principle P can not be constructed at will, usually it is constructed according to the need of practical application or the context of specific problems. For example, in the uncertain machine scheduling problem, since we want to minimize the makespan T (x, ξ) in the long run, we would like to use the expected value of makespan E[T (x, ξ)] to define the relationship T ( x, ξ) T (x, ξ), that is, E[T ( x, ξ)] E[T (x, ξ)], we call it the expected-value principle P E, which will be defined in Definition 3.2. Definition 8 (Expected-value principle P E ) Let ξ and η be two uncertain variables, we say ξ (or ) η if and only if E[ξ] (or <)E[η], where E[ ] denotes the expected value of uncertain variable. Definition 9 (Minimum-variance principle P V ) Let ξ and η be two uncertain variables, we say ξ (or ) η if and only if V [ξ] (or <)V [η], where V [ ] denotes the variance of uncertain variable. Definition 1 (α optimistic value principle P αsup ) Let ξ and η be two uncertain variables, we say ξ (or ) η if and only if ξ sup (α) (or <) η sup (α) for a given confidence level α (, 1], where ξ sup (α) and η sup (α) denote the α optimistic value of uncertain variable ξ and η respectively. n 56

4 A NEW APPROACH FOR UNCERTAIN MULTIOBJECTIVE PROGRAMMING PROBLEM BASED ON P E PRINCIPLE Definition 11 (α pessimistic value principle P αinf ) Let ξ and η be two uncertain variables, we say ξ (or ) η if and only if ξ inf (α) (or <) η inf (α) for a given confidence level α (, 1], where ξ inf (α) and η inf (α) denote the α pessimistic value of uncertain variable ξ and η respectively. As the expected value of uncertain variable is widely used in real-life problem, in this paper the uncertain MOP problem is solved under P E principle to obtain P E Pareto efficient solutions, and it will find that the P E Pareto efficient solutions obtained using uncertain approach are usually different from that obtained using multiobjective approach. In the remainder of paper, the principle P is denoted as P E. In order to prove the validity of uncertain approach in uncertain MOP problem under P E principle, four theorems are presented as follows: Theorem 4 Let ξ and η be two uncertain variables with regular uncertainty distributions Φ and Ψ respectively, if ξ (or ) η, then for any real number λ >, we have λξ (or ) λη Proof 1 Since ξ (or ) η, according to P E principle, we have E[ξ] (or <)E[η] It follows from Theorem 2.2 that For any real number λ >, we can obtain that λ Φ 1 (α)dα (or <) Φ 1 (α)dα (or <)λ λφ 1 (α)dα (or <) Ψ 1 (α)dα Ψ 1 (α)dα λψ 1 (α)dα As E[λξ] = λφ 1 (α)dα and E[λη] = λψ 1 (α)dα, thus E[λξ] (or <)E[λη] that is to say, λξ (or ) λη. The theorem is proved. Theorem 5 Assume that ξ 1 and ξ 2 are two independent uncertain variables with regular uncertainty distributions Φ 1 and Φ 2, η 1 and η 2 are two independent uncertain variables with regular uncertainty distributions Ψ 1 and Ψ 2, if ξ 1 η 1, ξ 2 η 2, then we have ξ 1 + ξ 2 η 1 + η 2 Proof 2 Since ξ 1 η 1, ξ 2 η 2, according to P E principle, we have E[ξ 1 ] < E[η 1 ], E[ξ 2 ] E[η 2 ] It follows from Theorem 2.2 that Φ 1 1 (α)dα < Ψ 1 1 (α)dα Then we can obtain that Φ 1 Φ 1 1 (α) + Φ 1 According to Theorem 2.3, we can get that E[ξ 1 + ξ 2 ] = E[η 1 + η 2 ] = 2 (α)dα 2 (α)dα < Ψ 1 2 (α)dα Ψ 1 1 (α) + Ψ 1 2 (α)dα Φ 1 1 (α) + Φ 1 2 (α)dα Ψ 1 1 (α) + Ψ 1 2 (α)dα thus E[ξ 1 + ξ 2 ] < E[η 1 + η 2 ] that is to say, ξ 1 + ξ 2 η 1 + η 2. The theorem is proved. Theorem 6 Let ξ and η be two uncertain variables with regular uncertainty distributions Φ and Ψ respectively, if ξ (or ) η, and the lower bounds of ξ and η, ξ and η exist, then for t = min(ξ, η ), we have (ξ t ) 2 (or ) (η t ) 2 57

5 ZUTONG WANG, JIANSHENG GUO, MINGFA ZHENG, YOUSHE YANG Proof 3 Since ξ (or ) η, according to P E principle, we have E[ξ] (or <)E[η] It follows from Theorem 2.2 that Φ 1 (α)dα (or <) Ψ 1 (α)dα As ξ and η are the lower bounds of ξ and η respectively, they are the lower bounds of Φ 1 and Ψ 1 respectively. For t = min(ξ, η ), we can obtain that (Φ 1 (α) t ) 2 dα (or <) (Ψ 1 (α) t ) 2 dα For t = min(ξ, η ), (ξ t ) 2 and (η t ) 2 are strictly increasing with respect to ξ and η, according to Theorem 2.3, we can get that E[(ξ t ) 2 ] = (Φ 1 (α) t ) 2 dα and Thus E[(η t ) 2 ] = (Ψ 1 (α) t ) 2 dα E[(ξ t ) 2 ] (or <)E[(η t ) 2 ] that is to say, (ξ t ) 2 (or ) (η t ) 2. The theorem is proved. Theorem 7 Let ξ and η be two uncertain variables with regular uncertainty distributions Φ and Ψ respectively, if ξ (or ) η, and ξ and η exist, then we have ξ (or ) η Proof 4 Since ξ (or ) η, according to P E principle, we have E[ξ] (or <)E[η] It follows from Theorem 2.2 that and thus Φ 1 (α)dα (or <) Ψ 1 (α)dα Since ξ and η exist, Φ 1 and Ψ 1 exist, then we can obtain that Φ 1 (α)dα (or <) Ψ 1 (α)dα For x is a strictly increasing function, according to Theorem 2.3, we can get that E[ ξ] = Φ 1 (α)dα E[ η] = that is to say, ξ (or ) η. The theorem is proved. Ψ 1 (α)dα E[ ξ] (or <)E[ η] 3.2 Uncertain MOP problem under P E principle Let us consider the uncertain MOP problem as follows: min f(x, ξ) = (f 1(x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x, ξ n )) x R n (1) g i (x, η i ), i = 1, 2,, m where x R n is a vector of decision variables of the problem; ξ 1, ξ 2,, ξ p are independent uncertain vectors whose components are independent continuous uncertain variables, ξ i = (ξ i1, ξ i2,, ξ in ), and η 1, η 2,, η m are independent uncertain vectors whose components are independent continuous uncertain variables, η i = (η i1, η i2,, η in ), defined on the uncertainty space (Γ, L, M). Since the objectives are usually in conflict in MOP problem, there is no optimal solution that simultaneously minimizes all the objective functions. In this case, we have to introduce the concept of P E Pareto efficient solution in uncertain MOP problem under P E principle, which means that it is impossible to improve any one of objectives without sacrificing on one or more of the other objectives. Definition 12 A feasible solution x is said to be P E Pareto efficient to the uncertain MOP problem if there is no feasible solution x such that f k (x, ξ k ) f k (x, ξ k ), k = 1, 2,, p and f k (x, ξ k ) f k (x, ξ k ) for at least one index k. 58

6 A NEW APPROACH FOR UNCERTAIN MULTIOBJECTIVE PROGRAMMING PROBLEM BASED ON P E PRINCIPLE 3.3 Uncertain approach In order to obtain P E Pareto efficient solution in problem (1), we convert the uncertain MOP problem (1) into an uncertain single objective programming (SOP) problem using a real-valued measurable function F, that is min U(x, ξ) = F (f 1(x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x, ξ n )) x R n (2) M{g i (x, η i ) } α i, i = 1, 2,, m Note that since the uncertain constraints g i (x, η i ), i = 1, 2,, m, do not define a crisp feasible set, they have been converted into chance constraints hold with confidence levels α 1, α 2,, α m in problem (2), which is a crisp feasible set. Theorem 8 Let F be a real-valued measurable function, then F (f 1 (x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x, ξ n )) is an uncertain variable in problem (1). Proof 5 Note that for any x R n, f k (x, ξ k ) is an uncertain variable for k = 1, 2,, p, and (f 1 (x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x, ξ n )) is an uncertain vector. For any Borel set B of real numbers, since F is a measurable function, F 1 (B) is also a Borel set. Thus {F (f 1 (x, ξ 1 ),, f p (x, ξ n )) B} = {(f 1 (x, ξ 1 ),, f p (x, ξ n )) F 1 (B)} is an event for any Borel set B. Hence U(x, ξ)=f (f 1 (x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x, ξ n )) is an uncertain variable. This theorem guarantees the uncertain nature of uncertain MOP problem using uncertain approach. Before we solve problem (2), we need to define the optimal solution of it. Definition 13 A feasible solution x is called an optimal solution to the uncertain problem (2) if for any feasible solution x. U(x, ξ) U(x, ξ) Obviously, the optimal solution to problem (2) is also defined under the relationship between uncertain variables. Under P E principle the equivalent deterministic SOP problem can be obtained as follows min E[U(x, ξ)] = E[F (f 1(x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x, ξ n ))] x R n (3) M{g i (x, η i ) } α i, i = 1, 2,, m According to Definition 13 and the requirement of Pareto efficiency, we can draw a road map for this new solution method, that is, find a real-valued measurable function F first, try to transform the uncertain objective function vector into a single uncertain objective function, then obtain its equivalent single deterministic objective function under P E principle, finally prove that the optimal solution to uncertain SOP problem (2) x is still Pareto efficient to the original uncertain MOP problem (1). Since the order in uncertain approach is different from that in multiobjective approach, the results obtained using these two approaches are usually different. Theorem 9 Let ξ 1, ξ 2,, ξ p be independent uncertain vectors, and f 1, f 2,, f p measurable functions. Then f 1 (x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x are independent uncertain variables for x R n. Proof 6 For any Borel sets B 1, B 2,, B p of real numbers, and x R n, it follows from the definition of independence that M{ p f i (x, ξ i ) B i } = M{ p ξ i f 1 i (x, B i )} = p M{ξ i f 1 i (x, B i )} = p M{f i (x, ξ i ) B i } Then f 1 (x, ξ 1 ), f 2 (x, ξ 2 ),, f p (x, ξ p ) are independent uncertain variables for x R n. According to those well-known methods used in transforming the deterministic MOP problem into a deterministic SOP problem, the linear weighted method and ideal point method are considered in our new solution approach for uncertain MOP problem here, and then their validity with application to this new solution road map is proved, respectively. 59

7 ZUTONG WANG, JIANSHENG GUO, MINGFA ZHENG, YOUSHE YANG 3.4 Linear weighted method The first method we proposed is the linear weighted method, which converts the uncertain MOP problem (1) into an uncertain SOP problem by weighting the objective functions according to the importance of each objective, i.e., min U(x, ξ) = n x R λ j f j (x, ξ j ) M{g i (x, η i ) } α i, i = 1, 2,, m where λ Λ ++ = {λ = (λ 1,, λ p ) T λ j >, p λ j = 1}. Theorem 1 The optimal solution to uncertain SOP problem (4) under P E principle x must be P E Pareto efficient to the original uncertain MOP problem (1). Proof 7 Suppose that x is the optimal solution to uncertain SOP problem (4), but it isn t the P E Pareto efficient solution to the original uncertain MOP problem (1), by the Definition 3.6, there must exist some x such that f j ( x, ξ j ) f j (x, ξ j ), and f j ( x, ξ j ) f j (x, ξ j ) for at least one index j, (j = 1, 2,, p). Since λ Λ ++ = {λ = (λ 1,, λ p ) T λ j >, p λ j = 1}, it follows from Theorem 3.1, 3.2 and 3.6 that λ j f j ( x, ξ j ) λ j f j (x, ξ j ) that is to say, U( x, ξ) U(x, ξ). It follows from Definition 13 that x is not the optimal solution to uncertain SOP problem (4), which contradicts with the previous hypothesis that x is the optimal solution. Hence, x is P E Pareto efficient to the original uncertain MOP problem (1). The theorem is proved. 3.5 Ideal point method The second method we proposed is the ideal point method, which converts the uncertain MOP problem (1) into an uncertain SOP problem by minimizing the distance function from a solution as follows: min U(x, ξ) = (f x Rn j (x, ξ j ) fj ) 2 (5) M{g i (x, η i ) } α i, i = 1, 2,, m where fj denotes the lower bound of single objective f j(x, ξ j ) (j = 1, 2,, p) on feasible set without considering other objectives. Theorem 11 The optimal solution to uncertain SOP problem (5) under P E principle x must be Pareto efficient to the original uncertain MOP problem (1). Proof 8 Suppose that x is the optimal solution to uncertain SOP problem (5), but it isn t the P E Pareto efficient solution to the original uncertain MOP problem (1), by the Definition 3.6, there must exist some x such that f j ( x, ξ j ) f j (x, ξ j ), and f j ( x, ξ j ) f j (x, ξ j ) for at least one index j, (j = 1, 2,, p). Without any loss of generality, let us assume when j = j, f j ( x, ξ j ) f j (x, ξ j ), as fj is the lower bound of f j (x, ξ j ), according to Theorem 3.3 we can get that (f j ( x, ξ j ) fj ) 2 (f j (x, ξ j ) fj ) 2 When j j, according to Theorem 3.3 we can get that (f j ( x, ξ j ) fj ) 2 (f j (x, ξ j ) fj ) 2 According to Theorem 3.2, we can obtain that (f j ( x, ξ j ) fj ) 2 (f j (x, ξ j ) fj ) 2 It follows from Theorem 3.4 that (f j ( x, ξ j ) fj ) 2 (f j (x, ξ j ) fj ) 2 that is to say, U( x, ξ) U(x, ξ). It follows from Definition 13 that x is not the optimal solution to uncertain SOP problem (5), which contradicts with the previous hypothesis that x is the optimal solution. Hence, x is P E Pareto efficient to the original uncertain MOP problem (1).The theorem is proved. (4) 6

8 A NEW APPROACH FOR UNCERTAIN MULTIOBJECTIVE PROGRAMMING PROBLEM BASED ON P E PRINCIPLE Table 1: Control parameters adopted in the genetic algorithm Control parameters in genetic algorithm Number of generation 5 Number of initial population 2 Crossover probability.6 Mutation probability.3 Note that, in order to guarantee that the availability of Theorem 3.9, the lower bound of single objective f j (x, ξ j ) (j = 1, 2,, p) on the feasible set, fj must exist. Actually, in the real-life operation research problems, nearly all of optimization objectives are bounded, such as distance, cost, time, etc. Theorem 3.9 is also applicable to the situations as follows U(x, ξ) = ( (f j (x, ξ j ) fj ) q ) 1/q, or U(x, ξ) = ( λ j (f j (x, ξ j ) fj ) q ) 1/q, where λ Λ ++ = {λ = (λ 1,, λ p ) T λ j >, p λ j = 1}, q is integer greater than one. 4 An illustrative example Here, a numerical example is provided to illustrate the proposed new solution method, and the results obtained using this new approach are compared with the results obtained using the multiobjective approach. Since the linear weighted method in uncertain approach is as same as that in multiobjective approach when the uncertain parameters involved are independent, we only consider the ideal point method here. Assume that x 1, x 2 are nonnegative decision variables, ξ 1, ξ 2, ξ 3, ξ 4, ξ 5, ξ 6 are independent linear uncertain variables L(1,3),L(2,4),L(3,5),L(4,6),L(5,7),L(6,8), and η 1, η 2 are independent zigzag uncertain variables Z(1, 2, 3), Z(2, 3, 4), respectively. The problem under consideration is the following triple-objective programming problem involving uncertain variables both in the objective functions and the constraints, while the feasible set has been transformed into its equivalent deterministic set. min f 2 (x, ξ 3, ξ 4 ) = ξ 3 (x 1 +.5)(cos(2x 2 )+1) ξ 4 (x 2 +.5)(sin(2x 1 )+1)+35 x 1,x 2 sin min f 1 (x, ξ 1, ξ 2 ) = ξ 1 x 1,x 2 x x ξ x x 2 2 e 2 cos(2πx 1 ) + cos(2πx 2 ) 2 min f 3 (x, ξ 5, ξ 6 ) = x x ξ 5 (cos(2x 1 x 2 ) + 1) ξ 6 (sin(2x 1 x 2 ) + 1) x 1,x 2 M{(x 1 + η 1 ) 2 + (x 2 + η 2 ) 2 5}.9 x 1, x 2 (4.1) Given the complexity of this numerical example, we adopt the genetic algorithm to be the optimization algorithm. We denote (x 1, x 2 ) as an individual, and assume that if the individual is not in the feasible solution set, then its fitness is zero. In this paper, the control parameters in GA are given in Table 1. Now, let us consider the ideal point method in uncertain approach. Firstly, we need to obtain the minimum of every single objective function on the feasible set. As f 1 (x, ξ 1, ξ 2 ) is strictly increasing with respect to ξ 1, ξ 2, f 2 (x, ξ 3, ξ 4 ) is strictly decreasing with respect to ξ 3, ξ 4, f 3 (x, ξ 5, ξ 6 )is strictly increasing with respect to ξ 5, while strictly decreasing with respect to ξ 6, we set ξ 1, ξ 2, ξ 3, ξ 4, ξ 5, ξ 6 be real value 1, 2, 5, 6, 5, 8 respectively, and then get the lower bound of f 1, f 2, f 3 on the feasible set in turn. As a result, we obtain that the lower bound of f 1 (x, ξ 1, ξ 2 ), f 2 (x, ξ 3, ξ 4 ), f 3 (x, ξ 5, ξ 6 ) are , , respectively. Employing the ideal point method in uncertain approach, we can transform this uncertain triple-objective programming problem into an uncertain SOP problem as follows U(x, ξ) = (f ) 2 + (f 2 ( )) 2 + (f 3 ( )) 2 where f 1 denotes f 1 (x, ξ 1, ξ 2 ), f 2 denotes f 2 (x, ξ 3, ξ 4 ), f 3 denotes f 3 (x, ξ 5, ξ 6 ). The deterministic SOP problem under the P E principle can be obtained as follows 61

9 ZUTONG WANG, JIANSHENG GUO, MINGFA ZHENG, YOUSHE YANG Table 2: Results obtained using Uncertain approach VS Multiobjective approach in ideal point method Solutions Uncertain approach Multiobjective approach Obj Obj Obj Optimal solution (.523,2.443) (.515,2.419) min E[U(x, ξ)] = (f ) 2 +(f 2 ( )) 2 +(f 3 ( )) 2 x 1,x 2 (x 1 + Ψ 1 1 (.9))2 + (x 2 + Ψ 1 2 (.9))2 5 x 1, x 2 (4.2) where f 1 denotes f 1 (x, Φ 1 1 (α), Φ 1 2 (α)), f 2 denotes f 2 (x, Φ 1 3 (1 α), Φ 1 4 (1 α)), and f 3 denotes f 3 (x, Φ 1 5 (α), Φ 1 6 (1 α)). Using the ideal point method in multiobjective approach, we need to obtain the optimal expected values of each objective function without considering other objectives first, they are , , respectively. Then we can get the equivalent deterministic single objective programming problem as follows: min E[U(x)]= (E[f 1 ] ) 2 +(E[f 2 ] (2.5892)) 2 +(E[f 3 ] ( )) 2 x 1,x 2 (x 1 + Ψ 1 1 (.9))2 + (x 2 + Ψ 1 2 (.9))2 5 x 1, x 2 (4.3) where f 1 denotes f 1 (x, ξ 1, ξ 2 ), f 2 denotes f 2 (x, ξ 3, ξ 4 ), f 3 denotes f 3 (x, ξ 5, ξ 6 ). Following genetic algorithm procedure,the results obtained using these two approaches are shown in Table 2. As shown in the Table 2, the results obtained using the ideal point method in these two approaches are different, but both P E Pareto efficient. This is due to that when the uncertain variables in objective functions are independent, the uncertain approach emphasizes the uncertain nature of the original problem, while the multiobjective approach emphasizes the multiobjective nature of the original problem. 5 Conclusions The general purpose of this study is to propose a novel solution approach to generate Pareto efficient solution for uncertain multiobjective problem based on uncertainty theory. Our study shows that, the multiobjective approach is different from the uncertain approach under the ideal point method. Both of these two approaches can generate Pareto efficient solutions when the uncertain variables are independent, so we cannot assert which approach is more appropriate until use them to solve the uncertain multiobjective problem with dependent uncertain variables. In our view, to be studied in future, other principles of uncertain MOP problem should be proposed and considered, such as the Minimum-Variance Principle, Minimum-Entropy Principle, α-optimistic Value Principle, α-pessimistic Value Principle, Minimum-Uncertainty Measure Principle, etc. Acknowledgements This work was supported by Natural Science Foundation of Shaanxi Province under Grant No. S213JC References [1] Chen X, Gao J, Stability analysis of linear uncertain differential equations, Industrial Engineering & Management Systems, 12(213), 2-8. [2] Chen X. and Ralescu D.A., Liu process and uncertain calculus, Journal of Uncertainty Analysis and Applications, 1(213), Article 2. [3] Arturo A., Graham A., and Stuart G., Multi-objective planning of distributed energy resources: A review of the state-of-the-art, Renewable and Sustainable Energy Reviews, 14(21), [4] Anthony C., Juyoung K., Seungjae L., and Youngchan K., Stochastic multi-objective models for network design problem, Expert Systems with Applications, 37(21), [5]Fonseca C.M. and Fleming P.J., Genetic algorithms for multiobjective optimization: formulation,discussion,and generalization. The Fifth International Conference on Genetic Algorithms, 1993, [6] Kaisa M., Nonlinear Multi-objective Optimization, Kluwer Academic Publishers,

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