A Hybrid Ant Colony Optimization Approach for Solving ... - infomesr

The Online Journal on Power and Energy Engineering (OJPEE)

Vol. (1) – No. (4)

A Hybrid Ant Colony Optimization Approach for Solving Multiobjective Design Optimization of Air-Cored Solenoid Waiel F. Abd El Wahed a

A. A. Mousa

b

I. M. El-Desoky b

R. M. Rizk-Allah b,

a

Department of Operations Research, Faculty of Computers and Information, Minoufia University, Egypt. b Department of Basic Engineering Science, Faculty Of Engineering , Shebin El-Kom, Minoufia University, Egypt. Abstract-This paper presents a novel multiobjective design optimization problem. The multiobjective design problem can be cast in theses terms: maximize the inductance and minimize the volume of a coreless solenoid. In order to obtain a set of equivalent optimal solutions, a hybrid ant colony optimization approach has been implemented. The proposed approach differs from the traditional ones in its design of a multipheromone ant colony optimization (MACO) as well as the inclusion of steady state genetic algorithm (SSGA) and local search approach. From the results, it is found that the proposed approach is fast and efficient and hence it is useful for multiobjective design of an air-cored solenoid. KeywordsAnt colony optimization, optimization, Multiobjective. Pareto solutions. I.

Design

INTRODUCTION

The multiobjective or vector optimization is a very important research area in engineering studies because real world design problems require the optimization of a group of objectives. Thanks to the effort of scientists and engineers during the last two decades, particularly the last decade, a wealth of multiobjective optimizers have been developed, and some multiobjective optimization problems that could not be solved hitherto were successfully solved by using these optimizers [1, 2]. It is observed that most of the multiobjective optimization studies have been focused on the heuristic algorithms such as genetic (GA) or evolutionary algorithm (EA) [3], particle swarm optimization (PSO) method [4], simulated annealing algorithm (SA) [5] and tabu search method [6], are proposed and used successfully to solve typical electromagnetic design problems. To meet the ever increasing demands in the design problems, a new EA called ant colony optimization algorithm have all been used successfully to mimic the corresponding natural, or physical or social phenomena. Recently, Ant colony optimization (ACO) [7] is one of the most techniques for approximate optimization. The inspiring source of ACO algorithms are real ant colonies.This characteristic of real ant colonies is exploited in ACO algorithms in order to solve optimization problems [8,9 ].

Reference Number: W09-0018

This paper intends to present an optimal design of an aircored solenoid using hybrid ant colony optimization approach with two objective functions namely, maximize the inductance and minimize the volume. This methodology consists of two phases .The first one employs the heuristic search by MACO based on SSGA to obtain near optimal solution, while the other employs efficient local search to improve the solution quality of multiobjective optimization problem (MOP). Our approach has several characteristic features. Firstly, the proposed approach implements MACO as a heuristic search technique. Secondly, SSGA is employed for enhancing the ant search. Finally, a dynamic version of pattern search technique was adopted to improve the solution quality of design problems. The remainder of the paper is organized as follows. In Section II we describe some preliminaries on MOP. In section III we review the standard ACO metaheuristic. In section IV we present the proposed approach. Experimental results are given and discussed in Section V. Section VI indicates our conclusion and notes for future work. II.

PRELIMINARIES

A general MOP is expressed by [1]: Min F ( x )  ( f 1 ( x ), f 2 ( x ), ...., f Q ( x )) T s.t x  S , x  ( x 1 , x 2 , ...., x n ) Where (f 1 ( x ), f 2 ( x ),...., f Q ( x ))

are the Q

(1)

objectives

( x 1 , x 2 , ...., x n ) are the n optimization n parameters, and S  R is the solution or parameter space. functions,

Obtainable objective vectors, by  , so

F : S

  , S

F  x  x  S 

are denoted

is mapped by F onto  .

Because F ( x ) is a vector, there is no unique solution to this problem, instead, the concept of noninferiority (also called Pareto-optimality) must be used to characterize the objectives. Definition 1 (Dominance Criteria [1]). For a problem having more than one objective function (say, 1 2 f i , i  1, ..., Q , Q  1 ), any two solution x and x can

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have one of two possibilities, one dominates the other or none 1 dominates the other. A solution x is said to dominate the 2 other solution x , if both the following condition are true 1 1. The solution x is no worse (say the operator  denotes 2 worse and  denotes better) than x in all objectives, 1 2 or f i ( x )  f i ( x ) for all i  1, ...Q objectives. 1 2 2. The solution x is strictly better than x in at least one 1 2 f i ( x )  f i ( x ) for objective, or at least one i  {1, 2,..., Q } . 1 If any of the above condition is violated, the solution x dose 2 not dominates the solution x . * Definition 2 (Pareto optimal solution [1]). x is said to be a Pareto optimal solution of MOP if there exists no other * feasible x (i.e., x  S ) such that, f i ( x )  f i ( x ) for all * i  1, 2, . . ., Q and f j ( x )  f j ( x ) for at least one objective function f j .

Vol. (1) – No. (4)

solution, while the other phase employs efficient local search to improve the solution quality. A. Multiobjective Optimization via MACO Dealing with several objectives in ant colonies that use the principles of MACO necessitates to answer three questions: (1) how to globally update pheromone according to the performance of each solution based on each objective, where each colony having its own pheromone structure (2) how does a given ant locally selects a path, according to the visibility and the desirability, at a given step of the approach (3) how to build the Pareto front. The main steps of the MACO are summarized as follows Step1: Construct Q Colonies. In a multiobjective optimization problem, multiobjective functions F  (f 1 , f 2 ,.., f Q ), need to be optimized simultaneously, there does not necessarily existence a solution that is best with respect to all objectives because of incommensurability and confliction among objectives. For this step, the number of colonies is set to Q with its own pheromone structure, where Q  F

is the number of

objectives to optimize. Step 2: Initialization. First, pheromones trails are initialized to a given value  0 where  0 is the pheromone information in

Initialize the pheromone trail. while stopping criteria is not met do Position each ant in a starting node repeat for each ant do Construct a new solution using the current pheromone trail. end for until every ant has build a solution Evaluate the solutions constructed. Update the pheromone trail end while

the current iteration and Pareto set are initialized to an empty set. Implementing the multipheromone ant colony optimization for a certain problem requires a representation of n variables for each ant, with each variable i has a set of ni options(nodes) with their values l ij which we generate(a

Figure (1). Ant Colony Optimization

and j  1, 2,...., ni . The process starts by generating m ants' position (solutions) from the population which is generated randomly, thus each ant k , k 1, 2,..., m  , has a position

III. ACO METAHEURISTIC The ACO algorithms were inspired by the observation of real ant colonies [7]. An interesting behavior is how ants can find the shortest paths between food sources and their nest. While walking from a food source to the nest and vice-versa, ants deposit on the ground a substance called pheromone. Ants can smell pheromone and then they tend to choose, in probability, paths marked by strong pheromone concentrations. The pheromone trail allows the ants to find their way back to the food source (or to the nest). The scenario of ACO is illustrated in figure 1. IV. THE PROPOSED APPROACH In this section, we present a framework for the proposed approach that involves two phases. The first one employs the heuristic search by MACO to obtain approximate Pareto


fully

connected

pheromone

graph ( n , ni ) ),

concentrations

and

{  ij };

their

associated

where i  1, 2,..., n ,

with a selected value for each variable ( l ij i  1, 2, ..., n  j  1, 2, ..., ni ) according to the associated pheromone with this value. This process continues for each objective. Consequently, path of each ant was consisted of n nodes with a value l ij for each node. Step 3: Evaluation. The MACO parameterized by the number of ant colonies Q and the number of associated pheromone structures. All the colonies have the same number of ants. Each colony i , i  (1, 2,..., Q ) tries to optimize an objective considering the pheromone information associated for each colony, where each colony is determined knowing only the relevant part of a solution. This methodology

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Vol. (1) – No. (4)

enforces both colonies to search in different regions of the nondominated front.

Step 6: Nondominated Solutions. The set of nondominated solutions is stored in an archive. During the optimization search, this set, which represents the Pareto front, is updated. Step 4: Trail Update and Reward Solutions. When updating At each iteration, the current solutions obtained are compared pheromone trails, one has to decide on which of the to those stored in the Pareto archive; the dominated ones are constructed solutions laying pheromone. The quantity of removed and the nondominated ones are added to the set. pheromone laying on a component represents the past Step 7: Steady State Genetic Algorithm. Steady state genetic experience of the colony with respect to choosing this algorithm [10] was implemented in such way that, two component. Then, at each cycle every ant constructs a offspring are produced in each generation. Parents are solution, and pheromone trails are updated. Once all ants have selected to produce offspring and then a decision is made as constructed their solutions, pheromone trails are updated as to which individuals in the population to select for deletion to usually in equation 2: first, pheromone trails are reduced by a make room for the new offspring. A replacement/ deletion constant factor to simulate evaporation to prevent premature strategy defines which member of the population will be convergence; then, some pheromone is laid on components of replaced by the new offspring. the best solution. Accordingly, pheromone concentration (i) Selections: Selection determines which individuals of the q ( ij (t ), q  1, 2,.., Q & i  1, 2,.., n & j  1, 2,.., ni ) associate population will have all or some of their genetic material passed on to the next generation of individuals. The d with each possible route (variable value) is changed in a mechanism for selecting the parents is based on a tournament way to reinforce good solutions, the change in pheromone selection. Tournament selection operates by choosing some q individuals randomly from a population and selecting the best concentration  ij is expressed in equation 2. from this group to survive into the next generation. For example, pairs of parents ( x , y ) are randomly chosen from q  C / f q for Min f q , if theoption l ij is chosen by ant k  ij   (2) the initial population. Their fitness values are compared and a otherwise 0 copy of the better performing individual becomes part of the Where C is a constant. A possibility is to reward every nondominated solution of the mating pool. The tournament will be performed repeatedly until the mating pool is filled. That way, the worst performing current cycle as follows patent in the population will never be selected for inclusion in q q q  ib (t )  (1   ) ib (t  1)   ib , (3) the mating pool. Tournaments are held between pairs of individuals are the most common. In this way all parents b  1, 2,.., B & B  ni necessary for a reproduction operator are selected. q Where  ib (t ) the revised concentration of pheromone is (ii) Recombination through Crossover and Mutation: After selection has been carried out, then the mechanisms of q associated with option b  ni at iteration t ,  ib (t  1) is the crossover and mutation are applied to produce an offspring, the following subsection outlines these genetic operators. concentration of pheromone at the previous iteration (t  1) ;  Crossover: Once the parents are created, the crossover step q is carried out by replacing the current value with a new one  ib is change in pheromone concentration that can be which produced stochastically with a probability determined according to equation 6; and B is the size of proportional to the crossover probability. Suppose the reward solutions crossover probability set by the system is pc . Generating a Step 5: Solution Construction. Once the pheromone is updated after an iteration, the next iteration starts by changing random number r  [0,1] , the crossover operation could be the ants’ paths (i.e. associated variable values) in a manner carried out only if r  pc . Suppose x and y are two that respects pheromone concentration and also some parents and  is a random number (i.e.   [0,1] ). The heuristic preference. For each ant and for each dimension construct a new candidate group to replace the old one. As result of crossover operation x  and y  can be obtained such, an ant k will change the value for each variable by the following linear combination of x and y : according to the transition probability. The transition x    . x  (1   ). y q (5) probability is done for each colony ( p ij (t ), q  1, 2,.., Q ) as y   (1   ). x   . y expressed in the following equation.  Mutation: Once the, the crossover is performed, the mutation step is carried out by replacing the current value   q  with a new one which produced stochastically with a   ij (t ) j , h  ni  q  probability proportional to the mutation q p ij (t )   (4)   ih (t )  probability p m .Generating a random number r  [0,1] , the  h ni  0 otherwise mutation operation is implemented only


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if r  p m .Suppose x ( j ) will be transformed into x ( j ) after mutation as follows: x ( j )  L ( j )   . (U ( j )  L ( j )), j  1, 2,.., n (6) Where  is a random number (i.e.   [0,1] ). Here L and U are the lower and upper bounds respectively (iii) Replacement / deletion strategy: A widely used combination is to replace the worst individual only if the new individual is better [10]. In the paper, this strategy will be suggested that the individual will be deleted if it was dominated by the new offspring as in algorithm 1. The MACO based on steady state genetic algorithm will report the near optimal Pareto front for a given MOPs. Besides, the accuracy of found set of nondominated solutions may vary with the problem under consideration. In order to improve the solution quality of MOPs, we implement the local search technique as a dynamic version of pattern search technique.

solution   while solution not completed do component  Construct Policy (Pheromone Information)

solution  solution  component end while end for XQ

 XQ 

{solution }

end for X  X 1 X 2 ... X Q  for objective=1:Q do fQ  f (X )

,



t

Q   Q   Q

end for

Pareto  Pareto ( X ) , Archive  Archive  Pareto Evaporation(Pheromone Information ,  )

Algorithm 1: The Strategy of Deletion

Update trail (Pheromone Information, best-so-far (pareto))





pop i 1  pop i according to Genetic operators , i  i  1 end while Output  Archive elements Local search phase by adaptive pattern search

1. INPUT: pop , x 2. if  x  pop  x   x , then 3. pop  pop 4. else if  x  pop  x  x  , then

0 , load the Pareto solutions while t  T do for j  1 : n do t

5. pop  pop  {x } {x } 6. end if 7. Output: pop 

Pareto( i , j )1  Pareto( i , j )   (t ) , {generate new search point}

If

Pareto(i ,:)1  Pareto(i ,:) do Pareto( i , :)  Pareto( i , :)1 ; break while (replace the old solution by the new)

end if end for , t  t  1 end while, Output  Pareto

Figure (3). The pseudo code of the proposed approach Figure (2). Mechanism of dynamic pattern search in R

Heuristic search phase by MACO Set Parameters Initialize(all Pheromone Information)

i  0, pop i  random solution Archive   while i for

 Max Iteration do colony  1 : Q do

X Q (partialpath)    f Q   for each ant do


2

B. The Local Search The local search phase is implemented as a dynamic version of pattern search technique. Pattern search technique is a popular paradigm in Direct Search (DS) methods [11]. DS methods are evolutionary algorithms used to solve constrained optimization problems [12]. DS methods, as opposed to more standard optimization methods, are often called derivative-free as they do not require any information about the gradient or higher derivatives of the objective function to search for an optimal solution. Therefore direct search methods may very well be used to solve noncontinuous, nondifferentiable and multimodal (i.e. multiple local optima) optimization problems. This study examines the usefulness of a dynamic version of pattern search technique to improve the solution quality of MOPs. The search procedure

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looks for the best solution “near” another solution by repeatedly making small changes to a starting solution until no further improved solutions can be found. The local search is started by loading the Pareto solutions for a given MOPs. At iteration t , we have an iterate

x t  Pareto , where the changes on the values for each dimension (i  1, 2,.., n ) can be implemented as (1t T )  (t )  R (1  r ) (7) Where r is the random number in the range [0..1]; T is the maximum number of iterations; R is the search radius. Let e i , (i  1, 2, .., n ) , denote the standard unit basis vectors. We successively look points x   x t   (t ).e i , (i  1, 2, .., n ) ,

x  for

which

f j ( x  )  f j ( x t ) for

objective f j , j  Q .

If

we

find

at until at

we

the find

least

one

no

x  such

that f j ( x  )  f j ( x t ) , then x   x t . Then we update the Pareto solutions by non dominated ones and the dominated ones are removed. This situation is represented in figure 2 for 2 the case in R . Without loss of generality, the elements discussed above are synthesized to evolve the proposed approach. The pseudo code of the proposed algorithm is given in figure 3. Parameters Number of objective function Number of ants for colony 1 Number of ants for colony 2 Number of iteration

 

C



Application 2 500 500 10 0.5 1 100 10

pc

0.85

pm

0.02

Table 1: parameters of the proposed approach V. NUMERICAL RESULTS This section is devoted to the discussion of the experimental results. In order to validate the proposed algorithm, the application of the multiobjective design optimization of air-cored solenoid is solved. The parameter settings of the proposed approach for all runs are presented in Table 1.


Vol. (1) – No. (4)

Figure (4). cross section of the solenoid and design variables The multiobjective shape optimization of a coreless solenoid with rectangular cross section a  b and a mean radius c (see figure 4) is tackled [13]. If the electric current is uniformly distributed over the cross section, it can be shown that if the number of turns ( N ) of the solenoid is given, the inductance L [  ] can be approximated from L  31.49

2 2 a N b

a a (9 6 10 ) b b

This multiobjective design problem can then be formally defined in the following two terms: maximize the inductance L (a , b , c ) and minimize the volume V (a , b , c ) for the given 6 2 length k 1  10 m and k 2  10 m of the current carrying wire. In order to simplify the analysis, two variables, a and b , are considered. Correspondingly, the computation of L and V are simplified, respectively, to 2 k1  4 b max f 1  k 1k 2 a 96 5 2 b ab 31.49

s .t :

a 

k1 k 2 b

, min f 2 

2 a b 4

2 2 k1 k 2 k 1k 2  2  a b 

, a  [0, 0.1], b  [0, 0.3]

The searched Pareto front using the proposed algorithm is illustrated in figure 5. Clearly, the proposed algorithm produces a uniform sampling of the Pareto front for this application. From the previous result, it was concluded that integration of MACO and local search technique has improved the quality of the founded solutions, also it guarantee the faster converge to the Pareto optimal solution. MACO has provided the initial set (close to the Pareto set as possible) followed by local search technique to improve the quality of the solutions. However, because of its stochastic behavior, MACO may suffer from slow convergence. In order to reach a quick and closer result to Pareto optimal solution, and to improve the efficiency of the MACO, steady state genetic algorithm was implemented.

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REFERENCES [1]

Figure (5). Pareto front by the proposed approach VI. CONCLUSION This paper intends to present an optimal design of an aircored solenoid using hybrid ant colony optimization approach with two objective functions namely, maximize the inductance and minimize the volume. The numerical results reveal that the proposed algorithm can find and sample uniformly the Pareto solutions of multiobjective problems very efficiently and is thus very suitable for engineering multiobjective optimal problems. The following are the significant contributions. (a) The proposed approach differs from the traditional ones in its design of a multipheromone ant colony optimization. Several pheromone structures, where, one usually associates a different colony of ants with each different objective, each colony having its own pheromone structure. (b) Improving the multipheromone ant colony optimization by Integration with steady state genetic algorithm and local search approach to improve the quality of the solutions. (c) The numerical results reveal that the proposed approach can find Pareto solutions of multiobjective problems very efficiently and is thus very suitable for engineering multiobjective optimal problems, with no limitation in handing higher dimensional problems. (d) a dynamic version of pattern search technique was adopted to improve the solution quality of design problems (e) The proposed approach is efficient for solving nonconvex multiobjective optimization problems where multiple Pareto optimal solutions can be found in one simulation run. For future work, we intend to test the approach on more problems. Also, we would like to use our approach for more complex real-world applications.


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