A unified matheuristic for solving multi-constrained traveling salesman problems with profits

Abstract

In this paper, we address a rich Traveling Salesman Problem with Profits encountered in several real-life cases. We propose a unified solution approach based on variable neighborhood search. Our approach combines several removal and insertion routing neighborhoods and efficient constraint checking procedures. The loading problem related to the use of a multi-compartment vehicle is addressed carefully. Two loading neighborhoods based on the solution of mathematical programs are proposed to intensify the search. They interact with the routing neighborhoods as it is commonly done in matheuristics. The performance of the proposed matheuristic is assessed on various instances proposed for the Orienteering Problem and the Orienteering Problem with Time Window including up to 288 customers. The computational results show that the proposed matheuristic is very competitive compared with the state-of-the-art methods. To better evaluate its performance, we generate a new testbed including instances with various attributes. Extensive computational experiments on the new testbed confirm the efficiency of the matheuristic. A sensitivity analysis highlights which components of the matheuristic contribute most to the solution quality.

Introduction

Routing problems where profits are associated with the visits of customers are extensively studied in the combinatorial optimization literature. Many papers and book chapters discuss these problems, since many industrial applications are modeled according to this general framework [e.g., Laporte and Martello (1990); Gendreau et al. (1998a); Fischetti et al. (2007); Balas (2007)]. Some surveys appeared over the last decade review variants and applications of these problems [see e.g., Feillet et al. (2005); Vansteenwegen et al. (2011); Archetti et al. (2013)].

Traveling Salesman Problems (TSP) with Profits are single-vehicle routing problems with two conflicting objectives. One consists in maximizing the total collected profit while the other aims to reduce the total route cost. Depending on the definition of the objective function, three classes of TSP with profits can be distinguished (Feillet et al. 2005). When both criteria are combined linearly in the objective function, the problem is the so-called Profitable Tour Problem (PTP) introduced by Dell’Amico et al. (1995). When the total distance cost is upper bounded and the profit is maximized, the problem is referred to as the Orienteering Problem (OP) introduced by Tsiligirides (1984) or the selective traveling salesman problem. When the objective is to minimize the distance costs and the profit collected must exceed a preset lower bound, the problem is called the Prize Collecting TSP (PCTSP). The PCTSP is originally defined by Balas (1989) by penalizing the unvisited vertices in the objective function. In this paper, an extended variant of the PTP including multiple constraints is addressed. The OP and the Orienteering Problem with Time Windows (OPTW) are considered to assess the efficiency of the proposed approach.

As stated in Lahyani et al. (2015b), a Rich Vehicle Routing (RVRP) Problem is a problem reflecting the complexities of a real-life context by combining various challenges faced daily. In this paper, we are interested in solving a Rich variant of the PTP with a maximum route duration, referred to as (RPTP). The proposed RPTP enriches the basic PTP in many ways. It may be considered as a time constrained capacitated profitable tour problem with multiple products and incompatibility constraints. Dealing with several complicated constraints encountered in common practical situations makes this problem more challenging. Particularly, we examine a variant of the PTP arising when a multi-compartment vehicle is involved. The use of such vehicles is relevant in several practical situations dealing with incompatible products, (e.g., to deliver dry, refrigerated and frozen food (Derigs et al. 2011), to collect olive oil (Lahyani et al. 2015a). However, to the best of our knowledge, there is no work studying TSPs with multiple compartments.

In the RPTP, the request of a customer is composed of demands for different products. A profit is associated with the demand for each product. A customer may be satisfied partially by delivering one or more products of its placed request. Since feasible tours are limited in time and capacity, the vehicle might not visit all the customers. The vehicle has different compartments with different capacities. A key feature is that some products are incompatible and must be kept separated during transportation. There are also incompatibility relations between some products and some compartments. Last, a time window and a service time are associated with each customer. Waiting times at the customer site are permitted but penalized in the objective function. The total cost of a tour is the total profit minus the distance cost and the cost for the total waiting time.

Formally, the RPTP considered can be defined as follows. Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be an undirected complete graph where $\mathcal{V}=\{0,\dots ,n\}$ is the vertex set and $\mathcal{E}=\{(i,j):i,j\in \mathcal{V},i\ne j\}$ is the edge set. Feasible tours correspond to cycles including the depot, vertex 0. Vertices $i\in {\mathcal{V}}^{\prime }=\{1,\dots ,n\}$ correspond to potential customers. Let $d_{ij}$ and $t_{ij}$ denote the non-negative distance cost and the travel time associated with edge $(i,j)\in \mathcal{E}$. We set $d_{ij}=\infty$ if the edge $(i,j)\notin \mathcal{E}$. With each customer $i\in {\mathcal{V}}^{\prime }$ are associated a hard time window $[e_i,l_i]$, within which the deliveries of i take place, and a service time $s_i$. In the case of early arrival at customer i, a cost is incurred corresponding to the waiting time until $e_i$. Time windows are also associated with the depot, and they correspond to the opening hours. There is no service time at the depot. For the sake of simplicity, the service times are included in the travel time. Service times are then disregarded in the remainder of this paper. There is a set $p\in \mathcal{P}=\{1,\dots ,P\}$ of products. Each customer $i\in {\mathcal{V}}^{\prime }$ can place several orders, each referring to one single product p. We denote by $o_i^p\in \mathcal{O}$ the order placed by customer i for product p. With each order $o_i^p$ we associate a demand $q_i^p$. A positive integer profit $g_i^p$ is associated with each $q_i^p\ne 0$. There is at least one order for each product type $p\in \mathcal{P}$ and the delivery of any product must not be split. We consider one vehicle which can visit a subset of customers within a given time limit $T^{\max }$. The vehicle has a capacity Q with compartments $w\in \mathcal{W}=\{1,\dots ,W\}$. Each compartment w has a capacity $Q_w$ and is equipped with a debit meter. The set $\mathcal{IP}\subseteq \mathcal{P}\times \mathcal{P}$ denotes the incompatibility relation between products. $(p,q)\in \mathcal{IP}$ means that products p and q must not be carried together in the same compartment. The set $\mathcal{IPC}\subseteq \mathcal{P}\times \mathcal{W}$ defines incompatibilities between products and compartments, forbidding product p to be transported in compartment w. The objective function consists of minimizing the total costs including the total profit, the total distance cost and the total waiting time. These three terms are weighted by parameters depending on the problem addressed.

In this paper, we tackle a challenging problem routing problem with complicated loading restrictions. To the best of our knowledge, we identify only one article devoted to a routing problem with vehicles with compartments where loadings satisfying incompatibilities restrictions between products and products and compartments are considered (Pirkwieser 2012). The authors propose first-fit, best-fit and best-fit decreasing heuristics as well as a constraint programming algorithm to solve the problem of assigning products to compartments considering potential incompatibilities. Previous works (e.g., Muyldermans and Pang 2010; Fallahi et al. 2008) consider a simple case with two compartments and two products, each being dedicated to one compartment. There is no assignment problem, and the loading problem reduces to knapsack problems. Following Pirkwieser (2012), we denote the loading sub-problem as the Compartment Assignment Problem (CAP).

Considering the literature devoted to routing problems, some attempts have been made recently to propose unified models and algorithms tackling different classes of vehicle routing problems (e.g., Pisinger and Ropke 2007; Subramanian 2012; Vidal et al. 2014). Some studies describe optimization algorithms for multi-constrained OP with m vehicles, referred to as Team Orienteering Problem, (e.g., Garcia et al. 2010; Tricoire et al. 2010; Souffriau et al. 2013). However, no previous work has been dedicated to rich variants of TSP with profits. There are few studies dealing with basic extended variants of TSP with profits, e.g., the capacitated PTP (CPTP) (e.g., Archetti et al. 2009; Jepsen 2011), or the OPTW (e.g., Righini and Salani 2006; Tricoire et al. 2010; Labadie et al. 2011). In this paper, we present a unified solution approach for a large class of TSP with profits ranging from academic problems to multi-attribute problems with no need for customization. The main contributions of this paper are the following. We define a multi-constrained TSP with profits which is a generalization of the PTP, the CPTP, the PTP with time windows, the OP and the OPTW. We model the CAP with incompatible products. We develop a unified matheuristic combining routing and loading neighborhoods. The proposed solution approach, referred to as VNS*, addresses a large set of instances from the OP and the OPTW literature following a generic parameters tuning. Last, we design a new testbed for the multi-constrained PTP with compartments, which may be useful for future multi-compartment routing studies.

This paper is organized as follows. Section 2 describes the matheuristic approach and presents the detailed issues related to the constructive heuristic, the routing neighborhoods, the CAP solution and the route feasibility check procedures. Computational results and an extensive sensitivity analysis on a large class of problems are reported in Sect. 3. Section 4 concludes and discusses some future research guidelines.