Open Loop Layout Optimization: Feasible Path Planning and Exact Door-to-Door Distance Calculation

TL;DR

This work tackles the Open Loop Layout Problem (OLLP) of placing rectangular cells without overlap while minimizing flow-weighted door-to-door transportation costs under the $l_2$ Euclidean metric. It introduces a formal mathematical model that uses exact door-to-door distances and feasible paths, where doors act as pickup-delivery points and paths must avoid intersections or follow rectangle edges. A novel encoding-driven metaheuristic framework embeds exact path calculations, enabling efficient search despite NP-hardness; extensive experiments show the approach outperforms centroid-based and existing methods on standard benchmarks. The proposed method holds practical significance for manufacturing, warehousing, and facility design by enabling more realistic, cost-efficient layouts, with potential enhancements through surrogate modeling and parallelization for larger-scale problems.

Abstract

The Open Loop Layout Problem (OLLP) seeks to position rectangular cells of varying dimensions on a plane without overlap, minimizing transportation costs computed as the flow-weighted sum of pairwise distances between cells. A key challenge in OLLP is to compute accurate inter-cell distances along feasible paths that avoid rectangle intersections. Existing approaches approximate inter-cell distances using centroids, a simplification that can ignore physical constraints, resulting in infeasible layouts or underestimated distances. This study proposes the first mathematical model that incorporates exact door-to-door distances and feasible paths under the Euclidean metric, with cell doors acting as pickup and delivery points. Feasible paths between doors must either follow rectangle edges as corridors or take direct, unobstructed routes. To address the NP-hardness of the problem, we present a metaheuristic framework with a novel encoding scheme that embeds exact path calculations. Experiments on standard benchmark instances confirm that our approach consistently outperforms existing methods, delivering superior solution quality and practical applicability.

Figures (10)

• Figure 1: Examples of different layout configurations.
• Figure 2: Suboptimality in centroid-based methods
• Figure 3: Illustrations of (a)Overlapping and, (b)Intersection of a path and the edges.
• Figure 4: Exact feasible $l_2$ distances.
• Figure 5: Orientation of the $i^{th}$ rectangular box ($s_i$,$t_i$), described by $\lambda_{i_j}$.

Theorems & Definitions (1)

• Definition 1: The Orientation Function