Complexity Analysis and Application of Gray Code-Based Genetic Algorithm for Container Loading Problem Optimization

Abstract

The Container Loading Problem involves arranging boxes of varying sizes and weights into constrained three-dimensional spaces. This study employs a Genetic Algorithm (GA) enhanced with Gray Code (GC) encoding for more stable and efficient optimization. GC reduces disruptions during mutation and crossover, ensuring smoother evolution. The proposed method was evaluated on fifty simulated datasets with randomly generated dimensions and weights. Complexity was analyzed theoretically and empirically by comparing execution times to various Complexity models, including linear, linear-logarithmic, quadratic, and cubic. Results revealed that the cubic model best represents the experimental data, emphasizing high computational demands, particularly during placement. The GC-GA achieved a superior <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${R}^{{2}}$</tex> value of 0.9769, outperforming the Original and Binary GA, thus highlighting enhanced performance and scalability. This confirms the effectiveness of integrating GC within GA frameworks for solving complex, high-dimensional optimization tasks like Container Loading.

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