Issue 28

B. Ye et alii, Frattura ed Integrità Strutturale, 28 (2014) 32-41 ; DOI: 10.3221/IGF-ESIS.28.04 37 1 1 1 1 1 1 1 [( , ),( , )] [( , ),( , )] [( , ),( , )] [( , ),( , )] [( , ),( , )] [ ( )] [ ( )] ( ) [ ( )] [ ( )] r r i r j r i r j k r i r j M r i r j r i r j j t t P t t t                  (12) In the process of creating solution, ants visit eachnode andupdate the pheromone using local pheromone along the path. 1 1 1 1 new old [( , ),( , )] [( , ),( , )] ( ) r r j r i r j            (13) where, ρ is the decay parameter of pheromone, 0< ρ < 1. Letbe Δ τ = τ 0 = C , τ 0 is the initial pheromone. After all ants walk through all nodes, the global pheromone is updated as: 1 1 1 1 1 1 1 [( , ),( , )] min gb [( , ),( , )] [( , ),( , )] ( ) ( ) if[( , ),( , )] global-best-tour ( ) otherwise r i r j r i r j r i r j f r i r j                         (14) where f min gb is the value of objective function at the beginning of iteration. In addition, for continuous domain object, stochastic search leads to low solution efficiency and solution result dispersion. This paper combines the stochastic search and the deterministic search. After each iterating, ants need tomove using the deterministic search strategy. Theywill then gradually move to the optimal solution. If the solution is worse at this node, the distance of the deterministic moving is longer. At time t , the deterministicmoving principle of node ( r , i ) is: 1 best , , , , best , , , if if t t r i r i r i r t r i t t r i r i r i r X X X X X X X X X          (15) min ( , ) , max min r i r r i r r f f X e f f     (16) where best r X is the optimal solution of parameter r at present, e is step size, f ( r , i ) is the minimum value finding in the process of passing through the node ( r , i ). When the feasible scheme is not obtained, min ( , ) r i r f f  . min r f is the minimum value of all nodes for parameter r , while max r f is themaximum value of all nodes for parameter r . By the nodes moving, nodes will gradually move to the optimal solution. The precision of search results is improved. Until to all r , , , t t r i r j X X e   , that is to say the distance of arbitrary two nodes of all parameters is less than e , which shows that the algorithm is convergence. At that time, the search range has focused near the optimal solution. The optimal solution of parameter r is best r X . The iterationwill stop. In additionwhen the cumulative times t of searchmoving is greater than the maximum moving times t max the algorithm will stop iteration. In the process of iteration, the deterministic search continuously modified the moving path. It will be helpful to overcome the disadvantage of result discretization and improve the global optimization capability. Therefore, the IACA can obtain the high precision results and need not to set too largeN. Its application to large-scale optimization problems can be easily implemented on parallel machines resulting in a significant reductionof required time. In common, the IACA is conducted according to the following procedure: Step 1Parameter initialization, parameter equal division ( N ). Step 2 m ants walk from the node “start”. According to Eq.10 and Eq.12, every ant individually computes the transition probability and creates the parameter scheme. Step 3Compute and save the object function value of feasible scheme, and save the optimal scheme at present. Step 4Update the pheromone according toEq.13 andEq.14 using the pathof the ant associatedwith the feasible scheme. Step 5Eachnode deterministicallymoves according toEq.16. Step 6 1 t t   , If to every parameter r, , , t t r i r j X X e   , or t ≥ t max , then output the optimal scheme. Otherwise, return to Step 2.