SMCB-E-08102005-0551.R2 and this method in Subsection IV.B. feasible solution wins; Reference [20]used the Pareto ranking to deal with 3)If both solutions are infeasible,the one with the smaller constraints.Although this method can reduce the number of sum of constraint violation is preferred. additional inputs required for constraints handling,the This technique is non problem-dependent,and no parameter computational time involved in nondominated sorting was needs to be tuned.In the following text,this technique is labeled increased.Reference [20]used four well-studied engineering as CHc. design problems to evaluate the performance of the proposed C.Organization Definition method,so a comparison is also made between OEA and SCA 20]in Subsection IV.D. In OEA.an organization consists of several members,and a There are many other constraints handling methods making member is a solution of the objective function.Since a UCOP use of different techniques,such as GENOCOP [21], can be viewed as a COP satisfying the constraints,we define GENOCOP II [22],GENOCOP III [23],microgenetic Member for both problems as follows, algorithm [24],varying fitness functions [25],homomorphous Definition 1:A Member is a real-valued vector that belongs mappings method [26],coevolutionary augmented Lagrangian to the search space,namely,Membere S.There are two method [27],the methods based on multiobjective algorithms measures,Memberf and Member',to evaluate the member's [28],[29],and so on.More detailed descriptions about quality.Member denotes the fitness and constraints handling in EAs were given in [30],[31]. [-f(Member)for UCOPs Member" -f(Member)for COPs with CHc (5) II.ORGANIZATIONAL EVOLUTIONARY ALGORITHM -(Member)for COPs with CHp Member denotes the sum of constraint violation and A.Problem Definition A UCOP can be formulated as solving the objective function for UCOPs minimize fx),1 ...,x)eS (1) Member ={0 for COPs with CHp where SCR"defines the search space which is an n-dimensional maxg(Member)for COPs with CHe space bounded by the parametric constraints x,sx,x,,i=1, (6) 2,,n.Thus,S=[s,,where=(s,玉，，x)and Thus,the purpose of OEA is maximizing Member.When comparing the qualities of two members,both Memberf and 天=（瓦，x32,,n) Member'should be considered.So the members are compared A COP can be formulated as solving the objective function according to Definition 2. minimize f(x),x=(x,x,,x)esF (2) Definition 2:If two members,Member and Memberz, where S is the same with that of(1),and the feasible region Fis satisfy (7)or(8)or(9),then Member is better than Member2, and labeled as Member Member,. F={x∈R"g,(x)≤0，j=1,2,…,m (3) (Member"=0)and(Member:=0)and(Member>Member) where gx),/=1,2,...,m are constraints. (7) B.Constraints Handling (Member'=o）and（Member,>0j (8) The main purpose of this paper is to propose OEA and (Member>0)and(Member >0)and(Member<Member) illustrate the effectiveness of OEA's search ability,so two kinds (9) of simple constraints handling techniques are used. Definition 2 realizes the three rules of CHc.By Definitions 1 The first technique uses a simplified version of the static penalty method [32],that is,COPs are transformed into UCOPs and 2,OEA can use a uniform framework to deal with both by adding a very simple penalty term in(4) UCOPs and COPs with the two different constraints handling techniques.On the basis of Definitions 1 and 2,an w(x)=f(x)+max(0.g(x) (4) Organization is defined as follows, where AeR is a penalty coefficient and needs to be tuned for Definition 3:An Organization,org,is a set of members,and different problems.In the following text,this technique is the best member is called Leader,which is labeled as Leader labeled as CHp. that is,an organization and the leader satisfy (10)and(11), The second technique has been widely used in EAs for COPs where lorg indicates the cardinality of org. [18],[33],[34],and has been also extended to multiobjective (org={Member,Member,,Member》and(org≠) optimization problems [35],[36],[37].This technique uses a (10) comparison mechanism based on the following rules, VMembere org,LeaderMember (11) 1)Between two feasible solutions,the one with the larger fitness value wins; When two organizations are compared,only the two leaders are 2)If one solution is feasible and the other is infeasible.the considered.Namely,org is better than org2 ifSMCB-E-08102005-0551.R2 3 and this method in Subsection IV.B. Reference [20] used the Pareto ranking to deal with constraints. Although this method can reduce the number of additional inputs required for constraints handling, the computational time involved in nondominated sorting was increased. Reference [20] used four well-studied engineering design problems to evaluate the performance of the proposed method, so a comparison is also made between OEA and SCA [20] in Subsection IV.D. There are many other constraints handling methods making use of different techniques, such as GENOCOP [21], GENOCOP II [22], GENOCOP III [23], microgenetic algorithm [24], varying fitness functions [25], homomorphous mappings method [26], coevolutionary augmented Lagrangian method [27], the methods based on multiobjective algorithms [28], [29], and so on. More detailed descriptions about constraints handling in EAs were given in [30], [31]. II. ORGANIZATIONAL EVOLUTIONARY ALGORITHM A. Problem Definition A UCOP can be formulated as solving the objective function minimize f(x), x=(x1, x2, …, xn)∈S (1) where S⊆Rn defines the search space which is an n-dimensional space bounded by the parametric constraints iii xxx ≤ ≤ , i=1, 2, …, n. Thus, S = [ ] x x , , where 1 2 ( , , ..., ) n x = xx x and 1 2 ( , , ..., ) n x = xx x . A COP can be formulated as solving the objective function minimize ( ), ( , , , ) 1 2 n f xx x x x = ∈
∩ S F (2) where S is the same with that of (1), and the feasible region F is { ( ) 0, 1, 2, , } n j F R =∈ ≤ = x x g j m
(3) where gj(x), j=1, 2, …, m are constraints. B. Constraints Handling The main purpose of this paper is to propose OEA and illustrate the effectiveness of OEA’s search ability, so two kinds of simple constraints handling techniques are used. The first technique uses a simplified version of the static penalty method [32], that is, COPs are transformed into UCOPs by adding a very simple penalty term in (4) { } 1 ( ) ( ) max 0, ( ) m j j ψ fA g = xx x = + ∑ (4) where A∈R is a penalty coefficient and needs to be tuned for different problems. In the following text, this technique is labeled as CHp. The second technique has been widely used in EAs for COPs [18], [33], [34], and has been also extended to multiobjective optimization problems [35], [36], [37]. This technique uses a comparison mechanism based on the following rules, 1) Between two feasible solutions, the one with the larger fitness value wins; 2) If one solution is feasible and the other is infeasible, the feasible solution wins; 3) If both solutions are infeasible, the one with the smaller sum of constraint violation is preferred. This technique is non problem-dependent, and no parameter needs to be tuned. In the following text, this technique is labeled as CHc. C. Organization Definition In OEA, an organization consists of several members, and a member is a solution of the objective function. Since a UCOP can be viewed as a COP satisfying the constraints, we define Member for both problems as follows, Definition 1: A Member is a real-valued vector that belongs to the search space, namely, Member∈S. There are two measures, MemberF and MemberV , to evaluate the member’s quality. MemberF denotes the fitness and ( ) for UCOPs ( ) for COPs with CHc ( ) for COPs with CHp F f f ψ −  = −  − Member Member Member Member (5) MemberV denotes the sum of constraint violation and { } 1 0 for UCOPs 0 for COPs with CHp max 0, ( ) for COPs with CHc V m j j g =    =   ∑ Member Member (6) Thus, the purpose of OEA is maximizing MemberF . When comparing the qualities of two members, both MemberF and MemberV should be considered. So the members are compared according to Definition 2. Definition 2: If two members, Member1 and Member2, satisfy (7) or (8) or (9), then Member1 is better than Member2, and labeled as Member Member 1 2
. 1 2 12 ( =0)and( 0)and( ) V V FF Member Member Member Member = > (7) ( )( ) 1 2 =0 and 0 V V Member Member > (8) 1 2 12 ( >0)and( 0)and( ) V V VV Member Member Member Member > < (9) Definition 2 realizes the three rules of CHc. By Definitions 1 and 2, OEA can use a uniform framework to deal with both UCOPs and COPs with the two different constraints handling techniques. On the basis of Definitions 1 and 2, an Organization is defined as follows, Definition 3: An Organization, org, is a set of members, and the best member is called Leader, which is labeled as Leaderorg, that is, an organization and the leader satisfy (10) and (11), where |org| indicates the cardinality of org. ( ) org org = ≠∅ {Member Member Member 1 2 || , , ..., and org } ( ) (10) , org ∀ ∈ Member Leader Member org ≺ (11) When two organizations are compared, only the two leaders are considered. Namely, org1 is better than org2 if