A Branch-and-Price-and-Cut Algorithm for Heterogeneous Pickup and Delivery Problems with Configurable Vehicle Capacity

Abstract

This paper presents a mixed-integer programming model for a variant of the pickup and delivery problem with time windows. The fleet is assumed to be heterogeneous with a novel feature that allows the vehicles to be configured before service begins to handle various types of demand. The work was motivated by a daily route planning problem arising at a senior activity center. A fleet of configurable vans is available each day to transport participants to and from the center, as well as to secondary facilities for rehabilitative and medical treatment. The number of participants and support equipment that a van can accommodate depends on how it is configured. An exact method is introduced based on branch and price and cut. At each node in the search tree, the master problem is solved by column generation to find a lower bound. To improve the bound, subset-row inequalities are applied to the variables of the master problem. Columns are generated by solving the pricing subproblems with a labeling algorithm enhanced by new dominance conditions. Local search on the current set of columns is used to quickly find promising additions. Implementation details and ways to improve the performance of the overall procedure are discussed. Testing was done on a set of real instances as well as a set of randomly generated instances with up to 50 customer requests. The results show that optimal solutions are obtained in the majority of cases.

The fleet size and mix dial-a-ride problem with reconfigurable vehicle capacity

“…The assignment of configurations for a given set of routes is done in a feasibility check either by a heuristic or by solving a general assignment problem when the heuristic fails. A branch-and-price algorithm for this problem was also proposed in Qu and Bard (2015) for instances with up to 30 requests.…”

Thirty years of heterogeneous vehicle routing

“…Other extensions include stochastic demand (Teodorović et al, 1995), pickups and deliveries (Irnich, 2000;Qu and Bard, 2014), multi-trips (Prins, 2002;Seixas and Mendes, 2013), the use of external carriers (Chu, 2005;Potvin and Naud, 2011), backhauls (Belmecheri et al, 2013;Salhi et al, 2013), open routes (Li et al, 2012), overloads (Kritikos and Ioannou, 2013), site-dependencies (Nag et al, 1988;Chao et al, 1999), multi-vehicle task assignment (Franceschelli et al, 2013), green routing Koç et al, 2014), single and double container loads (Lai et al, 2013), two-dimensional loading (Leung et al, 2013;Dominguez et al, 2014), time-dependencies (AfsharNadjafi and Afshar-Nadjafi, 2014), multi-compartments (Wang et al, 2014), multiple stacks (Iori and RieraLedesma, 2015) and collection depot (Yao et al, 2015).…”

“…The solutions yielded cost savings from 30% to 40%. The same authors (Qu and Bard, 2014) later introduced the Pickup and Delivery FSMTW with Configurable Vehicle Capacity. They presented a mixed-integer programming model and a branch-and-price-and-cut algorithm.…”

A branch-and-cut algorithm for a realistic dial-a-ride problem

“…In many real-world applications of the DARP (Qu and Bard, 2014;Lim et al, in press), many requests have the same pickup or drop-off locations. In the data of the NEATS, according to our observation, a significant proportion of requests have the depot, which is a large local hospital, as their pickup or delivery points.…”

“…Treating these pickup or delivery points separately requires creating an artificial node for each of them, and hence complicates the model, as the number of decision variables and constraints in a model is usually proportional or even exponential to the number of nodes in the model. In the branch-and-price algorithm proposed for the PDP with configurable vehicle capacities, Qu and Bard (2014) impose the following two additional requirements on the operations on the requests with the same pickup or drop-off locations to speed up the algorithm: (i) clients who live together, who are being dropped off at the same location, and who have the same pickup/drop-off time windows should be routed together and (ii) clients in a vehicle who have the same destination should be dropped together. In our case, we adopt a similar way to simplify the operations on the requests whose pickup or delivery points correspond to the depot.…”