are copied in the archive and the dominated individuals already present are deleted. If the number of individuals in the archive exceeds a given number, a clustering technique is performed to reduce the archive. The performance of every individual is then updated before doing the selection while using the two groups. To finish, one applies the genetic operators of crossing and mutation [3].

ALGORITHM 1: SPEA ALGORITHM
Step 1: Generate random initial population P and create the empty external set of non-dominated individuals P0.

Step 2: Evaluate objective functions for each individual in P.

Step 3: Copy non-dominated members of P to Pa. Step 4: Remove solutions within Pa which are covered by any other member of Pa.

Step 5: If the number of externally kept non-dominated solutions exceeds a given size, then use the clustering technique.

Step 6: Calculate the fitness of each individual in P, as well as, in Pa.

Step 7: Select individuals from P U Pa, until the mating pool is filled.

Step 8: Apply recombination and mutation to members of the mating pool in order to create a new population P.

Step 9: If a maximum number of generations is reached, then stop, else go to Step 2.

## Iii. Tabu Search

Tabu search (TS) is a metaheuristic procedure introduced by Glover [4]. It is used for a local search heuristic algorithm to explore the space of solutions beyond the simple local optimum. It has been applied to a wide range of practical optimization applications, producing an accelerated growth of Tabu search algorithms in recent years.

 A tabu list is a set of solutions determined by historical information from the last iterations of the algorithm. At each iteration, given the current solution x and its corresponding neighborhood N(x), the procedure moves to the solution in the neighborhood N(x) that mostly improves the objective function. However, moves that lead to solutions on the tabu list are forbidden or are tabu. If there are no improving moves, TS 
chooses the move which at least changes the objective function value. The tabu list avoids returning to the local optimum from which the procedure has recently escaped. A basic element of the tabu search is the aspiration criterion, which determines when a move is admissible despite being on the tabu list. One termination criterion for the tabu procedure is a limit in the number of consecutive moves for which no improvement occurs.

ALGORITHM 2: TABU SEARCH ALGORITHM
Step 1: Choose an initial solution in S (search space). Set i * = i and k = 0 Step 2: Apply k = k+1 and generate a subset of solutions in N(i,k) so that: The tabu movements are not chosen The aspiration criterion a(i,m ) is applied N(i,k) is the neighborhood of the current solution i at iteration k. Step 3: Choose the best solution i' among N(i,k), then apply i = better i' Step 4: If f(i) <= f(i *), then apply i * = i Step 5: Update the list T and aspiration criterion.

Step 6: If a stop condition is reached, then stop. Otherwise, return to Step 2.

IV. COOPERATIVE MODEL: EVOLUTIONARY ALGORITHMTABU SEARCH FOR THE MULTI-OBJECTIVE 
OPTIMIZATION(COMOEATS):
Our goal is to exploit the advantages of EA and TS to 

![0_image_0.png](0_image_0.png)

achieve a level-headed research that leads to a uniform distribution which covers the totality of the pareto optimal solutions. This search avoids the premature convergence and the omission of promising solutions at the same time.

 Fig.1. shows two possible distributions where the solutions given are limited on a short part of the front and the choice of decider is reduced. To overcome this problem, we are developing a new hybrid model in order to find an unexplored zone to which the search is directed (using Diversificator tabu search). After that an intensificator tabu search is applied in Fig. 3.