Exact and heuristic approaches for the integrated block relocation and fleet allocation problem with soft precedence constraints

Optimization Letters https://doi.org/10.1007/s11590-026-02317-6 ORIGINAL PAPER Exact and heuristic approaches for the integrated block relocation and fleet allocation problem with soft precedence constraints Lucas Assunção1 · Giulia Caselli2 · Manuel Iori2 · Andréa Cynthia Santos1 · Rafael Castro de Andrade3 Received: 23 December 2025 / Accepted: 26 May 2026 © The Author(s) 2026 Abstract We introduce the integrated block relocation and fleet allocation problem with soft precedence constraints, which jointly determines the unloading sequence of items and their assignment to a heterogeneous fleet of capacity-limited vehicles. The new problem aims at maximizing the number of delivered items while minimizing violations of a given item precedence order. Items with different destinations cannot be allocated to the same vehicle. The problem finds applications in logistic operations in steel plants and container terminals, as well as in humanitarian supply operations in the context of natural or industrial disasters. We formalize the problem as a lexicographic bi-objective model, providing two compact integer linear programming formulations reflecting different modeling perspectives, two reformulations and a family of valid inequalities. The models incurring the best dual bounds are used as backbone for a Kernel Search heuristic exploiting problem-specific structural properties. Computational experiments on a benchmark derived from the block relocation literature show that the exact models solve most instances to optimality within one hour, while the heuristic provides high-quality solutions with short runtimes and near-optimal gaps, making it an effective and reproducible approach for larger instances. Keywords Scheduling · Soft precedence · Block relocation · Kernel search 1 Introduction Storage problems originate from a wide range of industrial and logistics applications. In steel production, slabs and coils are stored vertically in stacks distributed horizontally across large storage yards, where cranes can handle only the topmost items Extended author information available on the last page of the article L. Assunção et al. of each stack. Similarly, in maritime terminals, containers are stored in yard blocks and moved by cranes during unloading operations, with high costs and waiting times required to remove all containers stacked above the target ones [1]. Moreover, in berth allocation at transshipment ports, vessels must be assigned to specific berthing positions along the quay to optimize port throughput and reduce vessel waiting times [2]. In general, to optimize movements of large items in yards or warehouses in which the storage area is organized in stacks and items are put on top of each other in these stacks, it is desirable to minimize the number of item relocations required during the operations. In particular, in unloading operations, items often have to be taken out in a predefined sequence determined by production schedules or truck arrivals, but relocation operations are usually very time-consuming. The unloading problem in which a sequence of items must be unloaded while minimizing the total number of relocations is known as the Block Relocation Problem (BRP) [3]. The BRP has been extensively studied in the literature and applied in various contexts. In classical formulations, a sequence of outgoing item sets is often predefined, and items may be grouped according to destination or customer. The objective function typically focuses on minimizing relocations [4], without considering delivery truck characteristics and capacity constraints. In this work, we propose a new variant of the BRP, namely the integrated Block Relocation and Fleet Allocation Problem with Soft Precedence Constraints (BRASP), in which we consider a set of items stored in a stacked yard, sub-divided into available and unavailable for transport, and a heterogeneous fleet of vehicles with limited capacities, each assigned to load available items belonging to the same family based on destination. During the unloading of items of interest, any blocking item can be relocated to an unlimited storage area, each producing a single unproductive move. Unproductive moves are not effortless but may be unavoidable due to item sequencing, family grouping by destination, and the classification of items as available or unavailable. The primary objective is to maximize the number of items unloaded from the yard and assigned to vehicles, while the secondary objective is to minimize the number of unproductive moves. To the best of our knowledge, this is the first work to address the sequencing of items in yard unloading and their subsequent assignment/ loading to an outbound heterogeneous fleet with limited capacity in a combined way. This combination is not usual in the literature: routing and loading problems usually ignore unloading and stacking constraints in the yard [5], whereas most BRP variants do not consider truck capacity and assignment decisions. In practical situations, however, these two decisions are tightly interconnected. For instance, minimizing relocations independently may lead to vehicles departing partially empty. Alternatively, unnecessary relocations may occur because fleet constraints were ignored. Consequently, unloading sequencing should anticipate fleet allocation decisions, making a joint optimization approach desirable. Therefore, despite its practical relevance, the literature still presents a research gap regarding integrated inbound unloading and block relocation with outbound heterogeneous fleet allocation, particularly in terms of formulations, tailored solution methods, and benchmark instances. The contributions of this study are threefold. First, we introduce the BRASP, a new extension of the BRP that jointly considers yard unloading sequencing and fleet 13 Exact and heuristic approaches for the integrated block relocation and… allocation decisions. The problem is formulated as a lexicographic bi-objective optimization problem that prioritizes maximizing the number of items successfully delivered while minimizing unproductive relocations. Moreover, we define relocations as violations of soft precedence constraints, previously studied mainly in scheduling problems, integrating this concept within the block relocation literature. Second, we develop and analyze two alternative compact Integer Linear Programming (ILP) formulations. The first formulation extends classical BRP modeling approaches, while the second originally introduces pre-positioned vehicle slots, enabling the explicit integration of fleet allocation decisions. For each formulation, we provide a strengthened reformulation, based on a novel definition of binary decision variables associated with destination-based item partitions leading to stronger linear relaxations, and a family of valid inequalities. Third, we design a Kernel Search heuristic [6, 7] with initialization and bucket segmentation tai (...truncated)