SMCB-E-08102005-0551.R2 Leader Leaderr,and labeled as orgorg2 (AnStr1);otherwise they are generated by Annexing Strategy 2 (AnStr2).The subscript j in U(0,1)indicates that the random D.Evolutionary Operators for Organications number is generated anew for each value of /and AnStrl and In the real world situation,there is a severe competition AnStr2 are given by(13)and(14),respectively.Given that the among organizations,and the strong ones always annex the leader of orgpl is (x1,x2,...,x)and new members are r(,, weak ones.In OEA,the strength of an organization manifests in 52,…,5j户1,2，，N the fitness of the leader.So the purpose of each organization is In AnStr1,rj=1,2,...,N are determined by (13), to increase the leader's fitness as much as possible.To achieve B<X this purpose,each organization must interact with others.On the B>k, B=xx+ax(xx-yix) (13) basis of such interactions,three evolutionary operators are B.otherwise k=12,,n designed,that is,the splitting operator,the annexing operator, and the cooperating operator. where is a uniformly distributed random number over [0,1], Splitting operator:In human societies,an organization and is generated anew for each value of k. whose size is too large usually is split into several small In AnStr2,rj=1,2,...,N are determined by (14), organizations so that they can be easily managed.In OEA,if 4+×(-4)B(0,1)< most of the members belong to the same organization,the ,k=1,2,,n(14) otherwise evolutionary operations would be disabled.So when the size of an organization exceeds a limit,this organization must be split. where a and B are uniformly distributed random numbers over Let Maxos be the parameter controlling the maximum size of an [0,1],and B is generated anew for each value of k. organization,and Maxos1.If a parent organization,orgp After rj=1,2,...N are generated,,2,...,N are satisfies(12),then orgp will be split into two child organizations, determined by(15), orgel and org. r Dyi (orgMaxs)or(org Maxas)and (( ZM= (y,>r,)and{0,(0,)<exp(gy-y5)}(15) (12) otherwise where U(,)is a uniform random number generator,and No is As can be seen,when r is better thany gets into orge so as the number of organizations in the whole population.To keep to improve the quality of orge When r,is worse thany in order the randomicity and a small difference between the sizes of to maintain the diversity,gets into org with a probability.For org and org to members are first randomly more clarity.Fig.2 illustrates the operations in this operator. selected from orgp to form orge,and the remainder forms org2. orge has (M+N)members,where Member,i=1,2,...,M,come For more clarity,Fig.1 illustrates the operations in this operator. from orgpl and Member i=M+1,M+2,...,M+N are generated As can be seen,without loss of generality,let the ith member is by the leader of orgpl and the members of orgp2 together.After orge is generated,the leader is also selected.In this case,the the best,and Mbe a random integer between and leader is the ith member (the leader of orgpl),or the best Therefore,the ith member is the Leader of orgp Mmembers are member among Member,Memberv2,...,and MemberM-N. randomly selected from orgp to form orgel,and the other Cooperating operator:This operator realizes the lorgp-M members form orge2.After orgel and orge2 are cooperation between two organizations.The leaders of the generated,the best members in orge and org are selected to be organizations interact with each other to generate two new their leaders,that is,the jth member in orge and the kth member members.Then,in each organization,a member being worse in orge2 are selected. than the new member is replaced. Annexing operator:This operator realizes the competition Two parent organizations,orgp=1,x2,...,x and between two organizations.The better organization is the orgp=2,,),are randomly selected from the current winner,and the other is the loser.The winner will annex the generation.They will cooperate with each other to generate two loser to form a larger organization.The members of the winner child organizations,orge and org2.Let CSe(0,1)be a can directly go into the new organization while those of the loser predefined parameter.Then,if U(0,1)<CS,orgel and org2 are must die.But the members of the loser maybe still have useful generated by Cooperating Strategy 1(CoStr1);otherwise they information,so they interact with the leader of the winner to are generated by Cooperating Strategy 2 (CoStr2),where generate new members. CoStrl and CoStr2 are given by (16)and(17),respectively. Two parent organizations,orgp=2,and Given that the leader of orgpl is (x,x2..,),the leader of orgp2=1,2,...,yN),are randomly selected from the current orgp2 is (v1,y2,...,y),and two new members are q=(q,q2,.., generation.Without loss of generality,let orgplD orgp2.Thus,q)and r=(r,2,...m). orgp will annex orgp2 to generate a child organization,org=, In CoStrl,g and r are determined by (16), 22,....ZM Z,ZM2,....ZMN).Where i=1,2,...,M.Let ASE(0,1)be a predefined parameter.Then,if U(0,1)<AS, g4=0×+(1-X4,k=l,2,n (16) =M+1,M+2,...,M+N are generated by Annexing Strategy 1 =(1-a)xx+×ySMCB-E-08102005-0551.R2 4 1 2 Leader Leader org org
, and labeled as org1
org2. D. Evolutionary Operators for Organizations In the real world situation, there is a severe competition among organizations, and the strong ones always annex the weak ones. In OEA, the strength of an organization manifests in the fitness of the leader. So the purpose of each organization is to increase the leader’s fitness as much as possible. To achieve this purpose, each organization must interact with others. On the basis of such interactions, three evolutionary operators are designed, that is, the splitting operator, the annexing operator, and the cooperating operator. Splitting operator: In human societies, an organization whose size is too large usually is split into several small organizations so that they can be easily managed. In OEA, if most of the members belong to the same organization, the evolutionary operations would be disabled. So when the size of an organization exceeds a limit, this organization must be split. Let MaxOS be the parameter controlling the maximum size of an organization, and MaxOS>1. If a parent organization, orgp, satisfies (12), then orgp will be split into two child organizations, orgc1 and orgc2. ( ) ( ) () | | or | | and 0,1 { ( ) o } org OS OS N org Max org Max U >≤ < (12) where U(⋅, ⋅) is a uniform random number generator, and No is the number of organizations in the whole population. To keep the randomicity and a small difference between the sizes of orgc1 and orgc2, | | 3 p org to 2| | 3 p org members are first randomly selected from orgp to form orgc1, and the remainder forms orgc2. For more clarity, Fig.1 illustrates the operations in this operator. As can be seen, without loss of generality, let the ith member is the best, and M be a random integer between | | 3 p org and 2| | 3 p org . Therefore, the ith member is the Leader of orgp, M members are randomly selected from orgp to form orgc1, and the other |orgp-M| members form orgc2. After orgc1 and orgc2 are generated, the best members in orgc1 and orgc2 are selected to be their leaders, that is, the jth member in orgc1 and the kth member in orgc2 are selected. Annexing operator: This operator realizes the competition between two organizations. The better organization is the winner, and the other is the loser. The winner will annex the loser to form a larger organization. The members of the winner can directly go into the new organization while those of the loser must die. But the members of the loser maybe still have useful information, so they interact with the leader of the winner to generate new members. Two parent organizations, orgp1={x1, x2, …, xM} and orgp2={y1, y2, …, yN}, are randomly selected from the current generation. Without loss of generality, let orgp1
orgp2. Thus, orgp1 will annex orgp2 to generate a child organization, orgc={z1, z2, …, zM, zM+1, zM+2, …, zM+N}. Where zi=xi, i=1, 2, …, M. Let AS∈(0, 1) be a predefined parameter. Then, if Uj(0,1)<AS, zj, j=M+1, M+2, …, M+N are generated by Annexing Strategy 1 (AnStr1); otherwise they are generated by Annexing Strategy 2 (AnStr2). The subscript j in Uj(0,1) indicates that the random number is generated anew for each value of j, and AnStr1 and AnStr2 are given by (13) and (14), respectively. Given that the leader of orgp1 is (x1, x2, …, xn) and new members are rj=(rj,1, rj,2, …, rj,n), j=1, 2, …, N. In AnStr1, rj , j=1, 2, …, N are determined by (13), ( ) , , , 1, 2, ..., otherwise k kk k k k k jk jk k k k k x x x xy rx x k n β β α β β  <  =+× − = >   =  (13) where αk is a uniformly distributed random number over [0, 1], and is generated anew for each value of k. In AnStr2, rj , j=1, 2, …, N are determined by (14), ( ) () 1 , 0,1 , 1, 2, ..., otherwise k kk k n j k k x xx r kn x  +× − < α β = =   (14) where α and βk are uniformly distributed random numbers over [0, 1], and βk is generated anew for each value of k. After rj, j=1, 2, …, N are generated, zj+M, j=1, 2, …, N are determined by (15), ( ) { } ( ) ( ) and 0, 1 exp otherwise j jj F F jM j j j j j j j + U   = <−   

r ry z r yr r y y (15) As can be seen, when rj is better than yj, rj gets into orgc so as to improve the quality of orgc. When rj is worse than yj, in order to maintain the diversity, rj gets into orgc with a probability. For more clarity, Fig.2 illustrates the operations in this operator. orgc has (M+N) members, where Memberi, i=1, 2, …, M, come from orgp1 and Memberi, i=M+1, M+2, …, M+N are generated by the leader of orgp1 and the members of orgp2 together. After orgc is generated, the leader is also selected. In this case, the leader is the ith member (the leader of orgp1), or the best member among MemberM+1, MemberM+2, …, and MemberM+N. Cooperating operator: This operator realizes the cooperation between two organizations. The leaders of the organizations interact with each other to generate two new members. Then, in each organization, a member being worse than the new member is replaced. Two parent organizations, orgp1={x1, x2, …, xM} and orgp2={y1, y2, …, yN}, are randomly selected from the current generation. They will cooperate with each other to generate two child organizations, orgc1 and orgc2. Let CS∈(0, 1) be a predefined parameter. Then, if U(0, 1)<CS, orgc1 and orgc2 are generated by Cooperating Strategy 1 (CoStr1); otherwise they are generated by Cooperating Strategy 2 (CoStr2), where CoStr1 and CoStr2 are given by (16) and (17), respectively. Given that the leader of orgp1 is (x1, x2, …, xn), the leader of orgp2 is (y1, y2, …, yn), and two new members are q=(q1, q2, …, qn) and r=(r1, r2, …, rn). In CoStr1, q and r are determined by (16), ( ) ( ) 1 , 1, 2, ..., 1 k kk k k k k k kk qx y k n r xy α α α α  = × +− ×  =  =− ×+ ×  (16)