The double traveling salesman problem with partial last-in-first-out loading constraints

In this paper, we introduce the Double Traveling Salesman Problem with Partial Last-In-First-Out Loading Constraints (DTSPPL), a pickup-and-delivery single-vehicle routing problem where all pickup operations must be performed before any delivery one because the pickup and delivery areas are geographically separated. The vehicle collects items in the pickup area and loads them into its container, a horizontal stack. After performing all pickup operations, the vehicle begins delivering the items in the delivery area. Loading and unloading operations must obey a partial Last-In-First-Out (LIFO) policy, i.e., a version of the LIFO policy that may be violated within a given reloading depth. The objective of the DTSPPL is to minimize the total cost, which involves the total distance traveled by the vehicle and the number of reloaded items due to violations of the standard LIFO policy. We formally describe the DTSPPL by means of two Integer Linear Programming (ILP) formulations, and propose a heuristic algorithm based on the Biased Random-Key Genetic Algorithm (BRKGA) to find high-quality solutions. The performance of the proposed solution approaches is assessed over a broad set of instances. Computational results have shown that both ILP formulations were able to solve only the smaller instances, whereas the BRKGA obtained better solutions for almost all instances, requiring shorter computational time.