An effective GRASP+VND metaheuristic for the k-Minimum Latency Problem

Abstract

Minimum Latency Problem (MLP) is a class of NP-hard combinatorial optimization problems which has many practical applications. In this paper, a general variant of MLP, also known as k-MLP is introduced. In k — MLP problem, the cost of objective function becomes the sum of waiting times at sites and k vehicles cover one of k routes. The goal is to find the order of customer visits that minimizes the sum of waiting time. The problem is a natural and practical extension of the k =1 case. In our work, we propose the first meta-heuristic algorithm which is mainly based on the principles of Greedy Randomized Adaptive Search Procedure (GRASP) and Variable Neighborhood Descent (VND) to solve the problem. The GRASP is used to build an initial solution which is good enough in construction phase. In a cooperative way, the VND is employed to generate diverse neighborhoods in improvement phase, therefore, it can prevent the search to escape from local optimal. In addition, we also introduce a new novel neighborhoods' structure in VND. In order to evaluate the performance of our algorithm, we also discuss an upper bound and a lower bound of the optimal solution. Extensive numerical experiments on benchmark instances show that our algorithm finds good-quality solutions fast, even for large instances which are up to 1200 customers.