A cutting-plane algorithm for the Steiner team orienteering problem

Abstract

The Team Orienteering Problem (TOP) is an NP-hard routing problem in which a fleet of identical vehicles aims at collecting rewards (prizes) available at given locations, while satisfying restrictions on the travel times. In TOP, each location can be visited by at most one vehicle, and the goal is to maximize the total sum of rewards collected by the vehicles within a given time limit. In this paper, we propose a generalization of TOP, namely the Steiner Team Orienteering Problem (STOP). In STOP, we provide, additionally, a subset of mandatory locations. In this sense, STOP also aims at maximizing the total sum of rewards collected within the time limit, but, now, every mandatory location must be visited. In this work, we propose a new commoditybased formulation for STOP and use it within a cutting-plane scheme. The algorithm benefits from the compactness and strength of the proposed formulation and works by separating three families of valid inequalities, which consist of some general connectivity constraints, classical lifted cover inequalities based on dual bounds and a class of conflict cuts. To our knowledge, the last class of inequalities is also introduced in this work. A state-of-the-art branch-and-cut algorithm from the literature of TOP is adapted to STOP and used as baseline to evaluate the performance of the cutting-plane. Extensive computational experiments show the competitiveness of the new algorithm while solving several STOP and TOP instances. In particular, it is able to solve, in total, 14 more TOP instances than any other previous exact algorithm and finds eight new optimality certificates. With respect to the new STOP instances introduced in this work, our algorithm solves 30 more instances than the baseline.

Solving the team orienteering problem with nonidentical agents: A Lagrangian approach

“…The exact approaches for the conventional TOP (i.e., with identical agents) include the following: branch-and-price algorithm [6,21], branch-and-cut-and-price algorithm [21,26], and branch-and-cut algorithm [5,10,12]. Apart from these studies, exact approaches for some recent extensions of TOP, which also solve the conventional TOP as a special case, include the Steiner TOP using a cutting-plane approach [4], the TOP with overlaps using a branch-and-cut-and-price algorithm [25], and the multivisit TOP with precedence constraints using a kernel search framework and a branch-and-cut algorithm [18].…”

Coupling Feasibility Pump and Large Neighborhood Search to solve the Steiner team orienteering problem

“…STOP finds application, for instance, in devising the itinerary of the delivery of goods performed by shipping companies (Assunção and Mateus, 2019). Here, a reward value -which may rely on factors such as the urgency of the request and the customer priority -is associated with visiting each customer.…”

“…In this case, priority could me linked to the patients' needs. Similar applications arise in the planning of other sorts of technical visits (Tang and Miller-Hooks, 2005;Assunção and Mateus, 2019) STOP was only introduced quite recently by Assunção and Mateus (2019). In the work, a state-of-the-art branch-and-cut algorithm from the literature of TOP is adapted to STOP, and a cutting-plane algorithm is proposed.…”

“…The previous exact algorithms for TOP are based on cutting planes and column generation. In summary, they consist of branch-and-cut (Dang et al, 2013a;Bianchessi et al, 2018;Hanafi et al, 2020), branch-and-price (Boussier et al, 2007;Keshtkaran et al, 2016), branch-and-cut-and-price (Poggi et al, 2010;Keshtkaran et al, 2016) and cutting-plane algorithms (El-Hajj et al, 2016;Assunção and Mateus, 2019). Among them, we highlight the branch-and-cut of Hanafi et al (2020) as the current state-of-the-art for TOP, being able to solve to optimality five more instances than the previous state-of-the-art (Assunção and Mateus, 2019).…”

“…Given an MILP formulation for the problem addressed, FP works by sequentially and wisely rounding fractional solutions obtained from solving auxiliary problems based on the linear relaxation of the original MILP formulation. In this work, we take advantage from the compactness of the formulation presented by Assunção and Mateus (2019) to use it within the FP framework. In our experiments, we also consider a reinforced version of this formulation obtained from the addition of three classes of valid inequalities.…”