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)

<p>&nbsp;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&rsquo;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&rsquo;s constraints is acceptable</p>