Sustainable Marine Structures

Eco-Efficient Vessel Dynamics: An Interval Approach to Heave Response and Energy-Saving in Rough Seas

Suleiman Mohammad

Department of Electronic Marketing and Social Media, Economic and Administrative Sciences, Zarqa University, Zarqa 13132, Jordan

Faculty of Business and Communications, INTI International University, Nilai 71800, Malaysia

Markala Karthik

Department of Electrical and Electronics Engineering, SR University, Warangal 506371, India

Yogeesh Nijalingappa

Department of Electrical and Electronics Engineering, SR University, Warangal 506371, India

Department of Mathematics, Government First Grade College, Tumkur 572102, India

Hanan Jadallah

Department of Electronic Marketing and Social Media, Economic and Administrative Sciences, Zarqa University, Zarqa 13132, Jordan

Asokan Vasudevan

Faculty of Business and Communications, INTI International University, Nilai 71800, Malaysia

School of Management, Shinawatra University, Samkhok 12160, Thailand

Azizbek Matmuratov

Department of Pedagogical Sciences, Mamun University, Khiva 220900, Uzbekistan

Mashkhura Sultonova

Department of Pedagogy and Psychology, Urgench State University Named after Abu Raykhan Beruniy, Urgench 220100, Uzbekistan

DOI: https://doi.org/10.36956/sms.v8i1.2718

Received: 10 September 2025 | Revised: 9 October 2025 | Accepted: 23 December 2025 | Published Online: 16 January 2026

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Abstract

Maritime transport faces increasing pressure to reduce fuel consumption and emissions, yet vessel performance under variable sea states remains difficult to bound reliably. Traditional stochastic and data-driven models provide probabilistic forecasts but lack strict guarantees in extreme or out-of-sample conditions. This study develops a deterministic arithmetic-interval framework that replaces uncertain hydrodynamic parameters and wave forcing with bounded intervals. The vessel’s single-degree-of-freedom heave equation is reformulated as an interval differential equation, and existence and uniqueness of the resulting solution tube are established. Validated numerical techniques-interval Taylor expansions, Picard iteration, and adaptive subdivision-are used to compute tight heave envelopes. An interval energy metric integrates worst-case power demand over a voyage, and a branch-and-bound global optimizer selects control parameters (e.g., speed schedules) that minimize the upper-bound energy while satisfying seakeeping constraints. Two hypothetical Karnataka-coast scenarios (“calm” and “rough” seas) demonstrate the rigor and efficiency of the approach. Computed energy-consumption intervals exactly enclose corresponding Monte-Carlo extremes, confirming tightness without large sample sizes. Rough-sea conditions increase worst-case energy demand by approximately 75% despite negligible heave amplitudes at the micron scale. Sensitivity analysis shows that wave-amplitude uncertainty dominates energy variability, while vessel stiffness and damping have minimal influence. The proposed interval framework eliminates under-coverage of worst-case energy (0% missed extremes) and remains within 3–6% of the tightest Monte-Carlo 99% confidence bands, achieving comparable bound tightness with two orders of magnitude fewer model evaluations than CNN–BiLSTM–Attention and kernel-density-based predictors. Benchmarking against linear heave RAO predictions confirms hydrodynamic consistency. The approach provides decision-makers with mathematically guaranteed bounds, supporting targeted measurement, control, and sustainable maritime operations.

Keywords: Uncertainty Quantification; Marine Seakeeping; Validated Numerics; Branch-and-Bound Optimization; Computational Hydrodynamics; Sustainable Operations

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