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BRAIN Journal-An Efficient Combined Meta-Heuristic Algorithm for Solving the Traveling Salesman Problem-Figure 7. Execution plot, for instance Eil51 (left figure) and KroB100 (right figure)

The evolution of the best solution found by the proposed algorithm is plotted in Figure 7 during a typical execution when solving instance Eil51 and KroB100. In this figure, the horizontal and vertical axes show the number of iterations and gained values of the proposed algorithm respectively. Besides, there is a fast convergence toward the BKS at the beginning of the execution while in the rest of the search the evolution of the BKS is not that fast.

BRAIN Journal-An Efficient Combined Meta-Heuristic Algorithm for Solving the Traveling Salesman Problem-Figure 5. The process of MICALK for solving the TSP

Moreover, in order to prevent the ICA from getting trapped in stagnation, we used a local searching algorithm when the algorithm attained a better solution compared to previous iterations. In fact, the probability of finding better solutions near a good solution is relatively high. There exist many algorithms for the local search and they have of course their pros and cons. Since LinKernighan algorithm is simple and it is one of the most successful methods for generating optimal or near optimal solutions for the TSP, we have used it in this study. The main steps of MICALK are summarized in the pseudo-code given in Figure 5.

BRAIN Journal-An Efficient Combined Meta-Heuristic Algorithm for Solving the Traveling Salesman Problem-Figure 6. Some best routes found by the proposed algorithm

Figure 6 shows some of the best solutions searched by the proposed method. In this figure, the horizontal axis represents the x-axis with increasing positive values to the right and the vertical axis represents the y-axis with increasing positive values upward.

BRAIN Journal-An Efficient Combined Meta-Heuristic Algorithm for Solving the Traveling Salesman Problem-Figure 3. Flowchart of the ICA

At last, the most powerful empire will take the possession of other empires and will win the competition. In other words, imperialistic competition hopefully converges to a state in which there exists only one empire and its colonies are in the same position and have the same cost as the imperialist. Figure 3 shows the flowchart of the basic ICA.

BRAIN Journal-An Efficient Combined Meta-Heuristic Algorithm for Solving the Traveling Salesman Problem-Figure 1. The Initial Empires

The ICA is a novel global search strategy which uses imperialism and imperialistic competition process as a source of inspiration. This algorithm is based on the fact that in a real world, countries try to extend their power over other countries in order to use their resources and bolster their own government. The first step in ICA is to generate an initial population like other evolutionary algorithms. The population set includes a number of feasible solutions called a ‘country’, which corresponds to the term ‘chromosome’ in the GA method. These countries are of two types: colonies and imperialists that altogether form some empires. As it is shown in Figure 1 (Atashpaz Gargari & Lucas, 2007), bigger and stronger empires have more colonies than smaller and weaker ones.

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BRAIN Journal-An Efficient Combined Meta-Heuristic Algorithm for Solving the Traveling Salesman Problem-Figure 2. Eliminate the weakest colony of the weakest empire

After initial empires are formed, their colonies start moving toward their relevant imperialist country. This movement is a simple model of assimilation policy which was pursued by some of the imperialist states. If one of the colonies possesses more power than its relevant imperialist after this movement, they will exchange their positions. To begin the competition between empires, the total objective function of each empire should be calculated. It depends on the objective function of both an imperialist and its colonies. Imperialistic competition among these empires forms the basis of the proposed evolutionary algorithm. During this competition, weak empires collapse and powerful ones take the possession of their colonies - Figure 2 (Atashpaz Gargari & Lucas, 2007). The empire, which has lost all its colonies, will collapse.

BRAIN Journal-A Combination of Meta-heuristic and Heuristic Algorithms for the VRP, OVRP and VRP with Simultaneous Pickup and Delivery- Figure 4. The solution of C6 found by CEACO

Furthermore, the GA has not been able to find the best solutions in thirteen of the fourteen examples. Therefore, it is the weakest algorithm among the five presented algorithms. However, SS_ACO has been able to find better solutions than the GA and has come up with the best solution in 12 examples. Among remaining five algorithms, PSO has failed in improving the solutions in 10 examples and has come up with solutions similar to the ones found by GA. From the comparison between GAPSO and CEACO, it can be seen that GAPSO in six examples has been able to find better solutions than the proposed algorithm. However, the CEACO has found better solutions than this algorithm for one example. For example, the solution of C6 is shown in Figure 4 which is the best found solution until now by other algorithms.

BRAIN Journal-A Combination of Meta-heuristic and Heuristic Algorithms for the VRP, OVRP and VRP with Simultaneous Pickup and Delivery- Figure 3. The pseudo of CEACO

The idea here is that a better solution may have a better chance to find a global optimum. After each iteration, if the best solution is not changed during 20 times, the algorithm finished and the best found solution is reported. Otherwise, the algorithm goes to transition rule step. Figure 3 shows the pseudo code of the proposed algorithm.

BRAIN Journal-A Combination of Meta-heuristic and Heuristic Algorithms for the VRP, OVRP and VRP with Simultaneous Pickup and Delivery- Figure 1. The ACO for the TSP

Ant colony optimization (ACO) is one of the most popular meta-heuristic algorithms inspired by the behavior of real ants seeking a path between their colony and a source of food. For the first time, this algorithm was used to solve the traveling salesman problem (TSP) as shown in Figure 1

BRAIN Journal-A Combination of Meta-heuristic and Heuristic Algorithms for the VRP, OVRP and VRP with Simultaneous Pickup and Delivery- Figure 2. Insert (left), swap (middle) and 2-opt exchanges (right)

The literature on meta-heuristics indicates that a promising approach for obtaining highquality solutions is to couple a local search algorithm with a mechanism to generate initial solutions. So, after all ants have constructed their routes and before updating global pheromone, three types of local search schemes including 2- opt scheme, insert and swap moves are performed to further reduce the routes length (Figure 2). In insert algorithm, a customer is moved to another route but in swap algorithm a customer in a certain route is swapped with another customer from a different route. One of the most commonly encountered moves is the 2-opt which starts with a feasible tour and continues by omitting two arcs of the same route, which are not adjacent and then connects them again by another method in such a way that the new tour length is shorter. In multiple routes, two edges belong to different routes, which form a criss-cross, are selected and two new edges are replaced. It also should be noted that the new solution will be only accepted in state that first, the constraints are not violated specially about each vehicle’s capacity. It can be noted that there are several routes for connecting nodes and producing the tour again, but a state that satisfies the problem’s constraints is acceptable