A Block-Scaled Grey Wolf Optimizer for Heterogeneous Decision Structures in Split Delivery Vehicle Routing

The Split Delivery Vehicle Routing Problem (SDVRP) necessitates coordinating route sequencing and order-splitting decisions, whose heterogeneous structure may constrain the efficacy of metaheuristics that update all decision variables under uniform search dynamics. This study introduces a Block-Scaled Grey Wolf Optimizer (BS-GWO) for a parameterized variant of the SDVRP, in which order allocation and cycle time are determined in an initial stage, while the primary optimization process focuses on operational vehicle-level splitting and route sequencing. Each candidate solution is represented by a continuous vector divided into two blocks: one corresponding to the visiting sequence and the other to the allocation of product units among vehicles. In contrast to the classical Grey Wolf Optimizer (GWO), the proposed BS-GWO incorporates differential block-wise scaling and maximum displacement control per block while maintaining the original α, β, and δ leadership structure. The results show that BS-GWO achieved more consistent solution quality than the classical GWO and delivered the best overall performance among the compared metaheuristics. These findings suggest that adapting the search dynamics to the internal structure of the solution vector enhances the ability to solve heterogeneous SDVRP representations within the considered formulation.