A Branch-and-Cut-and-Price algorithm for the Multi-trip Separate Pickup and Delivery Problem with Time Windows at Customers and Facilities

Andrea Bettinelli, Valentina Cacchiani, Teodor Gabriel Crainic, Daniele Vigo*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We study the Multi-trip Separate Pickup and Delivery Problem with Time Windows at Customers and Facilities (MT-PDTWCF), arising in two-tiered city logistics systems. The first tier refers to the transportation between the city distribution centers, in the outskirts of the city, and intermediate facilities, while the second tier refers to the transportation of goods between the intermediate facilities and the (pickup and delivery) customers. We focus on the second tier, and consider that customers and facilities have time windows in which they can be visited. Waiting is possible at waiting stations for free or at customers and facilities at a given cost or penalty. Therefore, it is relevant to coordinate the arrivals of vehicles at facilities and customers with the corresponding time windows. The MT-PDTWCF calls for determining minimum (fixed, routing and waiting) cost multi-trip routes, for a given fleet of vehicles, to service separately pickup and delivery customers, while taking into account vehicle capacity and time windows both at customers and facilities. We propose the first exact algorithm for MT-PDTWCF, namely a Branch-and-Cut-and-Price algorithm. It is based on column generation, where the pricing problem is solved by a bi-directional dynamic programming algorithm designed to cope with the features of the problem. Subset-row and rounded capacity inequalities are adapted to deal with MT-PDTWCF and inserted in the Branch-and-Cut-and-Price algorithm. The performance of the proposed algorithm is tested on benchmark instances with up to 200 customers, showing its effectiveness.

Original languageEnglish
Pages (from-to)824-839
Number of pages16
JournalEuropean Journal of Operational Research
Volume279
Issue number3
DOIs
Publication statusPublished - 16 Dec 2019

Keywords

  • Bi-directional dynamic programming
  • Branch-and-Cut-and-Price
  • Routing
  • Separate Pickup and Delivery Problem

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