The Capacitated Arc Routing Problem (CARP)

The Capacitated Arc Routing Problem (CARP) is a representative NP-Hard combinatorial optimization problem (Mourão et al. 2005; Mourgaya et al., 2007; Shang et al. 2014; Han et al. 2004), which has been widely applied in real life, such as: the path planning of municipal waste cleanup (Teixeira, et al. 2004; Rosa et al. 2002), the path planning of sprinkler (Handa et al. 2006), mail delivery planning (Lu et al., 2006; Araoz et al. 2006), salt spreader path planning (Muyldermans et al., 2002), school bus scheduling (Bektas et al. 2007), and winter snow removing routes(Mourão et al.2009; Campbell et al. 2000) et al. For these reasons, CARP has attracted widespread attention. However, the complexity of the CARP leads to the high discrete of the
However, we are often unable to solve the large-scale practical problems accurately due to the constraints of the computing resource (Shang et al. 2015). Thus, a variety of algorithms are given by researchers to find sub-optimal solution of the problem in a limited time (Shang et al. 2015). In general, there are two categories: heuristics and meta-heuristics. Among them, the representative heuristic algorithms: in 1983, Golden et al. (1983) proposed Augment-Merge algorithm and Path-Scanning algorithm (Golden and Wong 1981). In Path-Scanning algorithm, all tasks will be inserted into the loop according to some rules. After all task sides are inserted into the loop, the algorithm constructs a feasible solution which meets the capacity constraints. Ulusoy proposed Ulusoy-Split algorithm (1985) which is a representative algorithm used for segmentation. This algorithm adopts the dynamic programming method to obtain the optimal segmentation scheme in a given circuit. In 1992, Benavent et al. presented Cycle-Assignment algorithm et al. (Benavent et al. 1992). The Heuristic algorithms can obtain a suboptimal solution rapidly with less computational resources. Meanwhile, a good quality
is the weight( 0 or 1) of i-th task in h-th route, and stands for the edge between i-th vertex and i+1-th vertex in h-th route, and represents vehicle task sequence of the h-th loop. indicates the length of the task sequence of the loop. Equation (2) is the total cost of all the routes and constraints (3)-(4) guarantee that each task is served only once by one vehicle. Constraint (5) represents that the total cost served by one vehicle can not exceed its capacity.
3 ESMAENS for solving