Sustainable Marine Structures

Predictive Motion Envelopes for Offshore Logistics via α-Cut Intervals

Suleiman Ibrahim Shelash Mohammad

Electronic Marketing and Social Media, Economic and Administrative Sciences, Zarqa University, Zarqa P.O. Box 132010, Jordan

Nijalingappa Yogeesh

Department of Mathematics, Government First Grade College, Tumkur 572101, India

Natarajan Raja

Department of Visual Communication, Sathyabama Institute of Science and Technology, Chennai 600001, India

Ziaulla

Department of Economics, Govt. First Grade College for Women, Kolara 563101, India

P. William

School of Engineering and Technology, Sanjivani University, Kopargaon 423603, India

Asokan Vasudevan

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

DOI: https://doi.org/10.36956/sms.v7i4.2434

Received: 8 July 2025 | Revised: 25 July 2025 | Accepted: 26 August 2025 | Published Online: 14 October 2025

---

Abstract

Offshore logistics operations must continuously balance safety, fuel efficiency, and emissions reduction while navigating under uncertain and highly variable sea states. To address this challenge, we present an α-cut interval framework in which environmental uncertainties, specifically wave height and wind speed, are modeled as fuzzy numbers. Their corresponding α-level intervals are systematically propagated through a discrete vessel dynamics model, focusing on surge and heave responses. This procedure generates families of nested motion envelopes that tighten monotonically with increasing α, thereby producing deterministic yet progressively refined safety bounds without relying on full probabilistic distributions. A case study off the Karnataka coast is used to demonstrate the approach for a 20 km offshore supply voyage. Route planning constrained by α-envelopes ensures adherence to vessel structural and stability limits while enabling optimized transit speed. Comparative evaluation indicates that, relative to standard interval analysis, α-cut propagation substantially reduces over-conservatism, while against Monte Carlo-based envelopes it achieves similar coverage with significantly lower computational effort. Sensitivity analyses further quantify the influence of α-grid resolution, membership-function design, and hydrodynamic coupling coefficients on envelope width, fuel use, and emissions. In the tested scenario, higher α levels allow up to ~15% reduction in worst-case energy consumption and nearly 10% reduction in CO₂ emissions, all while preserving safety margins. Overall, the proposed framework is transparent, computationally efficient, and easily integrable into digital-twin-enabled operational workflows, providing a practical and sustainable decision-support tool for adaptive offshore logistics planning.

Keywords: Fuzzy Uncertainty; Interval Propagation; α‑cut Methodology; Vessel Dynamics; Route Planning; Emis‑ sion Analysis

---

References

[1] Zadeh, L.A., 1965. Fuzzy sets. Information and Control. 8(3), 338–353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X

[2] Metropolis, N., Ulam, S., 1949. The Monte Carlo Method. Journal of the American Statistical Association. 44(247), 335–341. DOI: https://doi.org/10.1080/01621459.1949.10483310

[3] Moore, R.B., 1966. Interval Analysis. Prentice Hall: Englewood Cliff, NJ, USA. pp. 37–49.

[4] Klir, G.J., Folger, T.A., 1988. Fuzzy Sets, Uncertainty, and Information. Prentice Hall: Englewood Cliff, NJ, USA. pp. 120–138.

[5] Dubois, D., Prade, H., 1978. Possibility Theory. Revue d’Intelligence Artificielle. 12(2),1–25. (in French)

[6] Alefeld, G., Herzberger, J., 1983. Introduction to Interval Computations. Academic Press: New York, NY, USA. pp. 50–58.

[7] Hansen, E., Walster, G.W., 2003. Global Optimization Using Interval Analysis. CRC Press: Boca Raton, FL, USA. pp. 15–21. DOI: https://doi.org/10.1201/9780203026922

[8] Boyd, S., Vandenberghe, L., 2004. Convex Optimization, 1st ed. Cambridge University Press: Cambridge, UK. pp. 127–189. DOI: https://doi.org/10.1017/CBO9780511804441

[9] Khalil, H.K., 2002. Nonlinear Systems, 3rd ed. Prentice Hall: Englewood Cliff, NJ, USA. pp. 23–55.

[10] Goetschel, R., Vox, R., 1985. Fuzzy interval analysis. In: Dubois, D., Prade, H. (eds.). Uncertainty in Knowledge-Based Systems: Lecture Notes in Computer Science; Operations Research and Statistics. Springer: Cham, Switzerland. pp.117–128.

[11] Yager, R.L., Filev, D.P., 2000. Essentials of Fuzzy Modeling and Control. IEEE Press: Piscataway, NJ, USA. pp. 22–26.

[12] Stefanakos, Ch.N., Vanem, E., 2018. Nonstationary fuzzy forecasting of wind and wave climate in very long-term scales. Journal of Ocean Engineering and Science. 3(2), 144–155. DOI: https://doi.org/10.1016/j.joes.2018.04.001

[13] Jazlan, A., Sreeram, V., Shaker, H.R., et al., 2017. Frequency Interval Cross Gramians for Linear and Bilinear Systems. Asian Journal of Control. 19(1), 22–34. DOI: https://doi.org/10.1002/asjc.1330

[14] Evstigneev, N.M., Ryabkov, O.I., 2020. On the implementation of Taylor models on multiple graphics processing units for rigorous computations. In: International Conference on Parallel Computational Technologies. pp. 85–99. Springer International Publishing: Cham, Switzerland.

[15] Neumaier, A., 1991. Interval Methods for Systems of Equations, 1st ed. Cambridge University Press: Cambridge, UK. pp. 116–166. DOI: https://doi.org/10.1017/CBO9780511526473

[16] Hamann, D., Walz, N.-P., Fischer, A., et al., 2018. Fuzzy arithmetical stability analysis of uncertain machining systems. Mechanical Systems and Signal Processing. 98, 534–547. DOI: https://doi.org/10.1016/j.ymssp.2017.05.012

[17] Hu, K., Chen, F., Shao, Z., et al., 2023. Recursive Interval State Estimation Algorithm for Power System. In Proceedings of the 2023 IEEE Sustainable Power and Energy Conference (iSPEC), Chongqing, China, 28 November; pp. 1–5. DOI: https://doi.org/10.1109/iSPEC58282.2023.10403081

[18] Mohammad, A.A.S., Mohammad, S.I.S., Al-Daoud, K.I., et al., 2025. Digital ledger technology: A factor analysis of financial data management practices in the age of blockchain in Jordan. International Journal of Innovative Research and Scientific Studies. 8(2), 2567–2577. DOI: https://doi.org/10.53894/ijirss.v8i2.5737

[19] Esogbue, A.O., 1999. The computational complexity of some fuzzy dynamic programs. Computers & Mathematics with Applications. 37(11–12), 47–51. DOI: https://doi.org/10.1016/S0898-1221(99)00142-X

[20] Yaseen, H., Al-Adwan, A.S., Nofal, M., et al., 2023. Factors Influencing Cloud Computing Adoption Among SMEs: The Jordanian Context. Information Development. 39(2), 317–332. DOI: https://doi.org/10.1177/02666669211047916

[21] Johansson, F., 2020. Computing the Lambert W function in arbitrary-precision complex interval arithmetic. Numerical Algorithms. 83(1), 221–242.

[22] Mohammad, A.A.S., Mohammad, S.I.S., Al Daoud, K.I., et al., 2025. Optimizing the Value Chain for Perishable Agricultural Commodities: A Strategic Approach for Jordan. Research on World Agricultural Economy. 465–478. DOI: https://doi.org/10.36956/rwae.v6i1.1571

[23] Arici, S. S., Akyuz, E., & Arslan, O., 2020. Application of fuzzy bow-tie risk analysis to maritime transportation: The case of ship collision during the STS operation. Ocean Engineering, 217, 107960. https://doi.org/10.1016/j.oceaneng.2020.107960

[24] Nachawati, M., Brodsky, A., 2021. Mixed-Integer Constrained Grey-Box Optimization based on Dynamic Surrogate Models and Approximated Interval Analysis. In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems. Online Streaming, 4–7 February 2021; pp. 99–112. DOI: https://doi.org/10.5220/0010350100990112

[25] Mohammad, A.A., Shelash, S.I., Taher Saber, I., et al., 2025. Internal Audit Governance Factors and their effect on the Risk-Based Auditing Adoption of Commercial Banks in Jordan. Data and Metadata. 4, 464. DOI: http://dx.doi.org/10.56294/dm2025464

[26] India Meteorological Department, 2025. Offshore marine forecast: Mangalore. Available from: https://mausam.imd.gov.in/responsive/marine_forecast.php (cited 7 July 2025).

[27] Rahman, T., Kim, Y. S., Noh, M., et al., 2021. A study on the determinants of social media based learning in higher education. Educational Technology Research and Development. 69(2), 1325–1351.

[28] Konispoliatis, D.N., Mavrakos, S.A., 2024. Hydrodynamic Research of Marine Structures. Journal of Marine Science and Engineering. 12(11), 2049. DOI: https://doi.org/10.3390/jmse12112049

[29] Mohammad, A.A.S., Mohammad, S.I.S., Oraini, B.A., et al., 2025. Data security in digital accounting: A logistic regression analysis of risk factors. International Journal of Innovative Research and Scientific Studies. 8(1), 2699–2709. DOI: https://doi.org/10.53894/ijirss.v8i1.5044

[30] Li, H., Qian, L., Hong, M., Wang, X., & Guo, Z., 2023. An Efficient Maritime Route Planning Method Based on an Improved A* with an Adaptive Heuristic Function and Parallel Computing Structure. Applied Sciences, 13(19), 10873. https://doi.org/10.3390/app131910873

[31] Arican, O.H., 2025. Optimal tugboat placement in narrow straits for ship safety: a Fuzzy Delphi and P-Median approach, as exemplified by the Dardanelles Strait. Ships and Offshore Structures. 1–13. DOI: https://doi.org/10.1080/17445302.2025.2502862

[32] Rusu, E.V.-C., 2023. Harvesting Offshore Renewable Energy an Important Challenge for the European Coastal Environment. Sustainable Marine Structures. 5(1), 11–13. DOI: https://doi.org/10.36956/sms.v5i1.822

[33] Surendran, S., Lee, S.K., 2019. Application Of Fuzzy-Logic In Ship Manoeuvring In Confined Waters. Sustainable Marine Structures. 1(1), 11–20. DOI: https://doi.org/10.36956/sms.v1i1.2

[34] Aksel, M., Yagci, O., Valyrakis, M., et al., 2024. Flow Structures around a Sphere Attached to the Bottom of a Prismatic Sloshing Tank: Problem-oriented basic research. Sustainable Marine Structures. 42–63. DOI: https://doi.org/10.36956/sms.v6i2.1204

[35] Miller, K., Graham, C., 2025. Hydrogen Hub Potential in the Caribbean: Towards a Sustainable Future. Sustainable Marine Structures. 57(1), 2–65. DOI: https://doi.org/10.36956/sms.v7i1.1173

[36] Ksciuk, J., Kuhlemann, S., Tierney, K., et al., 2023. Uncertainty in maritime ship routing and scheduling: A Literature review. European Journal of Operational Research. 308(2), 499–524. DOI: https://doi.org/10.1016/j.ejor.2022.08.006

[37] Wang, Y., Qian, L., Hong, M., et al., 2024. Multi-Objective Route Planning Model for Ocean-Going Ships Based on Bidirectional A-Star Algorithm Considering Meteorological Risk and IMO Guidelines. Applied Sciences. 14(17), 8029. DOI: https://doi.org/10.3390/app14178029

[38] Fox, A.D., Corne, D.W., Mayorga Adame, C.G., et al., 2019. An Efficient Multi-Objective Optimization Method for Use in the Design of Marine Protected Area Networks. Frontiers in Marine Science. 6, 17. DOI: https://doi.org/10.3389/fmars.2019.00017