Approximation Algorithms for the Load-Balanced Capacitated Vehicle Routing Problem

We study the load-balanced capacitated vehicle routing problem (LBCVRP): the problem is to design a collection of tours for a fixed fleet of vehicles with capacity Q to distribute a supply from a single depot between a number of predefined clients, in a way that the total traveling cost is a minimum, and the vehicle loads are balanced. The unbalanced loads cause the decrease of distribution quality especially in business environments and flexibility in the logistics activities. The problem being NP-hard, we propose two approximation algorithms. When the demands are equal, we present a $$((1-\frac{1}{Q})\rho +\frac{3}{2})-$$ approximation algorithm that finds balanced loads. Here, $$\rho $$ is the approximation ratio for the known metric traveling salesman problem (TSP). This result leads to a $$2.5-\frac{1}{Q}$$ approximation ratio for the tree metrics since an optimal solution can be found for the TSP on a tree. We present an improved $$2-$$ approximation algorithm. When the demands are unequal, we focus on obtaining approximate solutions since finding balanced loads is NP-complete. We propose an algorithm that provides a $$4-$$ approximation for the balance of the loads. We assume a second approach to get around the difficulties of the feasibility. In this approach, we redefine and convert the problem into a multi-objective problem. The algorithm we propose has a 4 factor of approximation.