None of the above studies, however, considered the open vehicle routing problem (OVRP) with a multiple capacitated heterogeneous fleet and multiple products with split deliveries together as in our study.
Open vehicle routing problems (OVRPs) have gained much attention recently since they represent a problem type that needs to be solved by many production companies.
The main distinction between OVRP and CVRP is that in CVRP each route is TSP which requires a Hamiltonian cycle , but in OVRP each route is a Hamiltonian path.
Improvements to the CW solution include proposed new parameters to the Clarke-Wright formulation composed of the nearest terminal k for solving multidepot VRP , deleting [c.sub.j, l] for solving OVRP , an estimate of the maximum savings value [s.sub.max], and a penalty multiplier a for solving VRP with backhauls , route shape [lambda] for solving CVRP [26, 27], weight [mu] for asymmetric solving CVRP , the customer demand v for solving CVRP , and the cosine value of polar coordinate angles of customers with the depot cos [[theta].sub.i, j] for solving CVRP .
According to (2), we also modified (4) by deleting for solving OVRP with the following equation:
In contrast, in OVRP each route has to construct for the open route (Hamiltonian path) by only assigning the first customer who starts at the depot.
In this paper, we adjust this procedure to deal with an OVRP by based on an operation of the genetic algorithm .
When the new chromosome which represents the new savings list is generated, it is calculated by the route merging procedure and the open-route construction procedure to produce a new OVRP solution.
The numerical experiment used five well-known data sets of Euclidean benchmarks (composed of 62 instances) of the OVRP consisting of Augerat et al.