The time-dependent capacitated profitable tour problem with time windows and precedence constraints

Abstract

We introduce the time-dependent capacitated profitable tour problem with time windows and precedence constraints. This problem concerns determining a tour and its departure time at the depot that maximizes the collected profit minus the total travel cost (measured by total travel time). To deal with road congestion, travel times are considered to be time-dependent. We develop a tailored labeling algorithm to find the optimal tour. Furthermore, we introduce dominance criteria to discard unpromising labels. Our computational results demonstrate that the algorithm is capable of solving instances with up to 150 locations (75 pickup and delivery requests) to optimality. Additionally, we present a restricted dynamic programing heuristic to improve the computation time. This heuristic does not guarantee optimality, but is able to find the optimal solution for 32 instances out of the 34 instances.

Traveling Salesman Problem with Optional Bonus Collection, Pickup Time and Passengers

“…N(i) representa o conjunto de localidades não visitadas que podem ser alcançadas a partir de i. A próxima localidade a ser inserida na rota da formigá e determinada por seleção estocástica baseada nas probabilidades de transição apresentadas na equação (43). Caso o sorteio tenha selecionado uma opção da equação (43) com k = 1 o vértice j será inserido na rota com o bônus coletado e caso seja pelo k = 2 o vértice j será inserido na rota sem que o bônus seja coletado.…”

Integer programming formulations for the time-dependent elementary shortest path problem with resource constraints

“…Similarly to Dabia et al [1] and Sun et al [6], we focus on the path duration instead of the makespan. Consequently, the vehicle is allowed to delay its departure from v s to reduce the duration of the route.…”

“…In this sense, to the best of our knowledge, most of the research has been devoted to variants of the TDVRP with the presence of time windows. This is the case of Dabia et al [1] for the capacitated TDVRP with time windows (TDVRP-TW) and Sun et al [6], which further considers the TDESPPRC with time windows and precedence constraints. In both cases, the problem is tackled using dynamic programming (DP).…”

“…In this sense, we build upon the works of Taccari [7] and Jepsen et al [4] on the ESPPRC with timeindependent travel times, where they consider ILP formulations to address the ESPPRC. We propose a strengthened version of the ILP formulation proposed in Sun et al [6], studied also by Montero et al [5] for the Time-Dependent Traveling Salesman Problem with Time Windows (TDTSP-TW). In addition, we present a tailored formulation which reduces the number of variables and constraints in the model by exploiting the structure of the travel time function.…”

Linear edge costs and labeling algorithms: The case of the time‐dependent vehicle routing problem with time windows

“…To tackle the pricing problem, which stands for a time dependent ESPPRC (TDESPPRC), they develop a tailored labeling algorithm that is able to solve 63% of the instances with 25 customers and 15% of the instances with 100 customers. In a follow‐up paper, Sun et al isolate the pricing problem for a DM‐TDVRPTW with pickups and deliveries and consider a time dependent tour problem with capacity, time windows, and precedence constraints. Spliet et al consider the time window assignment with time dependent travel times, for which they developed a BPC algorithm that is able to solve instances with up to 25 customers and 3 different demand scenarios.…”