Neutrosophic Z-Number Framework for Intelligent Multi-Objective
Solid Transportation Systems
Muhammad Kamran1,7,8,∗, Anns Uzair2, Muhammad Tahir3, Muhammad Farman4,5,6,∗,
Ixtiyarov Farxod7, Mohamed Hafez8,9
1Research Institute of Business Analytics and SCM, College of Management, Shenzhen University, China
2Department of Mathematics, Govt College University Faisalabad
3Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan, 29050,
KPK, Pakistan
4Faculty of Arts and Sciences, Department of Mathematics, Near East University, Nicosia, 99010, Turkey
5Jadara University Research Center, Jadara University, Irbid, Jordan
6Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
7Center for Reseaech and Innovation, Asia International University, Yangibod MFY, G’ijduvon street, House
74, Bukhara, Uzbekistan
8Faculty of Engineering and Quantity Surviving, INTI International University Colleges, Nilai, Malaysia
9Faculty of Mangement, Shinawatra, Pathum Thani, Thailand
Emails: kamrankfueit@gmail.com; UzairAnns@gmail.com; TahirdMuhamma@gmail.com;
FarmanMuhammad@gmail.com; FarxodIxtiyarov@gmail.com; HafezMohamed@gmail.com
Abstract
Transportation optimization remains a critical challenge in international businesses, particularly given the in-
herent uncertainties of supply chain networks. This paper proposes a novel machine learning-based model for
solving multi-objective, multi-item solid transportation problems that fundamentally advances beyond exist-
ing fuzzy and neutrosophic approaches. Our key innovation lies in the synergistic integration of neutrosophic
Z-numbers (NZNs) with adaptive machine learning techniques, creating a framework that simultaneously cap-
tures value vagueness, information reliability, and dynamic uncertainty patterns capabilities absent in con-
ventional fuzzy transportation models. Unlike traditional fuzzy methods that treat all uncertainty uniformly,
our NZN representation provides a three-dimensional structure incorporating truth, indeterminacy, and falsity
measures, each with associated reliability metrics. This enriched uncertainty modeling enables three ground
breaking advancements over existing approaches: (1) a neural scoring system that autonomously learns opti-
mal NZN comparison functions from historical decision patterns, overcoming the limitations of static aggre-
gation operators in fuzzy systems; (2) LSTM networks that jointly forecast demand values and their reliability
evolution under uncertainty; and (3) reinforcement learning optimizers that dynamically balance economic
efficiency with information quality in routing decisions. Computational experiments demonstrate superior
performance compared to six established baseline methods, including traditional fuzzy, intuitionistic fuzzy,
neutrosophic, and pure machine learning approaches. Our hybrid framework achieves a 23.4% reduction in
transportation costs and 35.4% improvement in uncertainty handling compared to conventional fuzzy trans-
portation models, with statistically significant improvements (p < 0.001) across all evaluation metrics. By
coupling the theoretical rigor of neutrosophic mathematics with the adaptive power of machine learning, this
study provides businesses with a transformative decision-support system for transportation planning under
real-world uncertainty conditions.
Keywords: Machine Learning; Neutrosophic Z-Numbers; Supply Chain Optimization; Cost Optimization;
Sustainability