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.

我们研究了负载平衡的车辆通行路线问题（LBCVRP）：问题是为容量为Q的固定车队设计行程集合，以一种方式在多个预定义客户之间分配单个仓库的供应总行驶成本最低，车辆负载平衡。负载不平衡会导致分配质量下降，尤其是在商业环境中，并且物流活动的灵活性也会下降。这个问题是NP难的，我们提出了两种近似算法。当需求相等时，我们提出一个\（（（1- \ frac {1} {Q}）\ rho + \ frac {3} {2}）-\）近似算法，用于找到平衡负载。在这里，\（\ rho \）是已知度量旅行推销员问题（TSP）的近似比率。由于可以为树上的TSP找到最佳解决方案，因此该结果导致树度量的\（2.5- \ frac {1} {Q} \）近似比。我们提出一种改进的\（2- \）近似算法。当需求不相等时，我们将重点放在获得近似解上，因为找到平衡负载是NP完全的。我们提出一种算法，为负载平衡提供\（4- \）近似值。我们假设了第二种方法来解决可行性难题。在这种方法中，我们将问题重新定义并转换为多目标问题。我们提出的算法具有4个近似因子。