In this manuscript, we delve into the realm of lattice ordered complex linear diophantine fuzzy soft set, which constitutes an invaluable extension to the existing Fuzzy set theories. Within this exploration, we investigate basic operations such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\oplus$\end{document}⊕ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\otimes$\end{document}⊗, together with their properties and theorems. This manuscript is more amenable in two ways, i.e., it enables real-life problems involving parametrization tool and applications with an existing order between the components of the parameter set based on the preference in the complex frame of reference. Adaptive cruise control (ACC) is a system designed for maintaining distance between two vehicles and to sustain a manually provided input speed. The purpose of cars with ACC is to avoid a collision that frequently happens nowadays, thereby improving road safety regulations amidst rising collision rates. The fundamental aim of this manuscript is to prefer an applicable car with ACC together with its latest model by defining a peculiar postulation of lattice ordered complex linear diophantine fuzzy soft set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\rm ( \widehat{LOCLDFSS} )$\end{document}(LOCLDFSS^). Emphasizing real-life applicability, we illustrate the effectiveness and validity of our suggested methodology in tackling current automotive safety concerns, providing useful guidance on reducing challenges related to contemporary driving conditions.

This paper addresses the concept of the vendor-managed inventory (VMI) problem, focusing on the replenishment policy and the vehicle routing problem (VRP) model. These components are integrated to tackle a three-echelon distribution issue comprising a single plant, multiple depots, and multiple retailers, with the primary objective of minimizing transportation and inventory costs within this complex distribution network. A three-phase methodology is proposed to optimize the entire supply chain, from the plant to the final retailer, and its performance is evaluated through computational experiments. This research is motivated by a real-life supply network, highlighting its practical relevance and applicability. To extend the capabilities of existing methods for solving the combined inventory and routing problem, an insertion heuristic is incorporated to enhance vehicle utilization, thereby reducing total costs. Computational results demonstrate the effectiveness of the improved algorithm, indicating that it is sufficiently robust for practical application. Significant cost savings can be achieved with the proposed approach, making it a valuable contribution to the field of supply chain optimization.

The eternal 1-secure set, strategically protects graphical structures against attacks by placing sentinels within dominating sets. This study adapts this concept to real-world scenarios, like emergency response systems, by addressing challenges such as roadblocks and traffic congestion. We introduce the secure set and the eternal 1-secure set for fuzzy graphs based on edge membership levels J. Our analysis examines the relationship between the security number of a fuzzy graph G at level J and the domination number of its spanning subgraph at level J, demonstrating the potential of the eternal 1-secure set to enhance security measures in dynamic environments.

Enterprise resource planning (ERP) is widely used to boost the total market power of businesses. The wrong selection is one of the key reasons why ERP installations fail. Due to the complexity of the business environment and the range of ERP systems, choosing an ERP system is a complex and time-consuming procedure. ERP alternatives may be assessed using several criteria, so the ERP selection process may be considered a multi-criteria decision-making (MCDM) problem. In this study, the rough best worst method (BWM) was used to determine criteria weights, while the newly developed rough integrated simple weighted sum product (WISP) was used to rank ERP alternatives. Results suggest that the SFT-4 coded software is regarded as the best option, followed by SFT-5, SFT-6, SFT-2, SFT-3, and SFT-1. Results of the newly developed rough WISP method are compared to those of existing rough techniques in the sensitivity analysis. The differences between them have been found to be negligible. The outcomes show how effectively developed rough BWM and WISP integrated method performs in terms of ERP selection with usability, accuracy, ease of use, and consistency. This study will help decision-makers in a context where ERP is implemented choose the best ERP software for different sectors.

The expansion of the open-pit exploitation of mineral raw materials, and especially the energy resources of fossil fuels, makes open-pit coal mines spatially dominant objects of large mining basins. Exploitation activities are accompanied by negative ecological impacts on the environment, which requires the integral planning, revitalization, reclamation, and rehabilitation of the disturbed area for human use in the post-exploitation period. The post-exploitation remediation and rehabilitation of open-pit mining areas and disposal sites, i.e., space disturbed by mining activities and accompanying facilities, are complex synthetic multidisciplinary multiphase engineering project tasks. In this paper, a hybrid fuzzy MCDM model (Multiple-Criteria Decision-Making) was developed for the selection of a reclamation solution for the Tamnava-West Field open-pit mine. IMF SWARA (Improved Fuzzy Stepwise Weight Assessment Ratio Analysis) was applied to define the weights of 12 criteria of different structures used in the evaluation of reclamation solutions. The Fuzzy ROV (Range of Value) method was applied to select the reclamation solution from a total of 11 solutions previously obtained using a process approach. The results of the hybrid IMF SWARA—Fuzzy ROV model show that forestry is the best solution for the Tamnava-West Field open-pit mine. After the results had been obtained, verification analyses of the proposed model were performed and the best stable proposed reclamation solution was determined.