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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400.</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content=" Vehicular Networks, Multi-Objective Optimization, Routing Stability, Communication Delay, Evolutionary Algorithms, LSTM, Dynamic Environment " lang="en" name="keywords"/> <base href="/html/2503.12050v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S1" title="In Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S2" title="In Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Related Work</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S2.SS1" title="In II Related Work ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-A</span> </span><span class="ltx_text ltx_font_italic">Traditional Routing Approaches</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S2.SS2" title="In II Related Work ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-B</span> </span><span class="ltx_text ltx_font_italic">Optimization-Based Routing Methods</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S2.SS3" title="In II Related Work ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-C</span> </span><span class="ltx_text ltx_font_italic">Hybrid and Predictive Approaches</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S3" title="In Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Mathematical Modeling</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S3.SS1" title="In III Mathematical Modeling ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-A</span> </span><span class="ltx_text ltx_font_italic">Objective Functions</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S3.SS2" title="In III Mathematical Modeling ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-B</span> </span><span class="ltx_text ltx_font_italic">Multi-Objective Optimization</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S3.SS3" title="In III Mathematical Modeling ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-C</span> </span><span class="ltx_text ltx_font_italic">Handling Dynamic Environment</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S4" title="In Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Proposed Algorithm</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S4.SS1" title="In IV Proposed Algorithm ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span> </span><span class="ltx_text ltx_font_italic">LSTM-Based Prediction Model for Dynamic Network Forecasting</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S4.SS2" title="In IV Proposed Algorithm ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span> </span><span class="ltx_text ltx_font_italic">Hierarchical Evolutionary Optimization Framework</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S5" title="In Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Simulation and Experiments</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S5.SS1" title="In V Simulation and Experiments ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">V-A</span> </span><span class="ltx_text ltx_font_italic">Simulation Scenario</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S5.SS2" title="In V Simulation and Experiments ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">V-B</span> </span><span class="ltx_text ltx_font_italic">Simulation Results and Analysis</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S6" title="In Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks <br class="ltx_break"/><span class="ltx_note ltx_role_thanks" id="id1.id1"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">thanks: </span>This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400.</span></span></span> </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">1<sup class="ltx_sup" id="id2.1.id1">st</sup> Zhiou Zhang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><span class="ltx_text ltx_font_italic" id="id3.2.id1"> School of Computer Science and Engineering</span> <br class="ltx_break"/><span class="ltx_text ltx_font_italic" id="id4.3.id2">University of New South Wales <br class="ltx_break"/></span>Sydney, Australia <br class="ltx_break"/>zhiou.zhang@student.unsw.edu.au </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">2<sup class="ltx_sup" id="id5.1.id1">nd</sup> Weian Guo </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><span class="ltx_text ltx_font_italic" id="id6.2.id1">Sino-German College of Applied Sciences</span> <br class="ltx_break"/><span class="ltx_text ltx_font_italic" id="id7.3.id2">Tongji University <br class="ltx_break"/></span>Shanghai, China <br class="ltx_break"/>guoweian@tongji.edu.cn </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">3<sup class="ltx_sup" id="id8.1.id1">rd</sup> Qin Zhang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><span class="ltx_text ltx_font_italic" id="id9.2.id1">Department of Information Engineering</span> <br class="ltx_break"/><span class="ltx_text ltx_font_italic" id="id10.3.id2">Fuzhou Polytechnic College <br class="ltx_break"/></span>Fuzhou, Fujian, China <br class="ltx_break"/>zqsrxh@163.com </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">4<sup class="ltx_sup" id="id11.1.id1">rd</sup> Haibin Lin </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><span class="ltx_text ltx_font_italic" id="id12.2.id1">School of Traffic Engineering</span> <br class="ltx_break"/><span class="ltx_text ltx_font_italic" id="id13.3.id2">Fuzhou Polytechnic College <br class="ltx_break"/></span>Fuzhou, Fujian, China <br class="ltx_break"/>644352201@qq.com </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">5<sup class="ltx_sup" id="id14.1.id1">rd</sup> Dongyang Li </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><span class="ltx_text ltx_font_italic" id="id15.2.id1">Sino-German College of Applied Sciences</span> <br class="ltx_break"/><span class="ltx_text ltx_font_italic" id="id16.3.id2">Tongji University <br class="ltx_break"/></span>Shanghai, China <br class="ltx_break"/>lidongyang0412@163.com </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id17.id1">Vehicular Ad Hoc Networks (VANETs) are a cornerstone of intelligent transportation systems, facilitating real-time communication between vehicles and infrastructure. However, the dynamic nature of VANETs introduces significant challenges in routing, especially in minimizing communication delay while ensuring route stability. This paper proposes a hierarchical evolutionary optimization framework for delay-constrained routing in vehicular networks. Leveraging multi-objective optimization, the framework balances delay and stability objectives and incorporates adaptive mechanisms like incremental route adjustments and LSTM-based predictive modeling. Simulation results confirm that the proposed framework maintains low delay and high stability, adapting effectively to frequent topology changes in dynamic vehicular environments.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> Vehicular Networks, Multi-Objective Optimization, Routing Stability, Communication Delay, Evolutionary Algorithms, LSTM, Dynamic Environment </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Vehicular networks, commonly referred to as Vehicular Ad Hoc Networks (VANETs), have become integral to intelligent transportation systems, allowing vehicles to communicate with each other and roadside infrastructure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib1" title="">1</a>]</cite>. This communication enhances road safety, improves traffic efficiency, and provides infotainment services. However, the dynamic nature of vehicular networks, characterized by frequent topology changes due to vehicle mobility, introduces significant challenges in maintaining reliable routing and communication <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib2" title="">2</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Routing in vehicular networks must address both communication delay and route stability. Minimizing communication delay is essential for timely data delivery, particularly in safety-critical applications <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib3" title="">3</a>]</cite>, while maximizing route stability is necessary to reduce frequent disruptions, which can increase overhead and degrade network performance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib4" title="">4</a>]</cite>. Balancing these conflicting objectives requires advanced optimization techniques capable of adapting to the dynamic and unpredictable conditions of vehicular networks <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib5" title="">5</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">In this paper, we propose a hierarchical evolutionary optimization framework for delay-constrained routing in vehicular networks. The framework employs a Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) to optimize two key objectives: minimizing communication delay and maximizing routing stability. Additionally, the framework incorporates mechanisms tailored to the dynamic nature of vehicular networks, including incremental route adjustments and predictive modeling with Long Short-Term Memory (LSTM) networks to anticipate vehicle movements.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">The primary aim of this framework is to provide an efficient and adaptive solution for vehicular network routing, achieving low communication delay and high route stability under frequent topology changes. The main contributions of this work are summarized as follows. We formulate a mathematical model for delay-constrained routing in vehicular networks, integrating both communication delay and route stability as optimization objectives. Building on this model, we develop a hierarchical evolutionary optimization framework using a Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) to address the multi-objective optimization problem. To further enhance adaptability and performance, our framework incorporates dynamic adaptation mechanisms, including incremental route adjustments and LSTM-based predictive modeling, enabling the proposed solution to respond effectively to the highly dynamic vehicular environment.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">The remainder of this paper is organized as follows. Section II presents the mathematical model for the routing problem. Section III introduces the proposed optimization framework, detailing the MOEA/D algorithm and local search strategies. Section IV elaborates on the adaptation mechanisms, including LSTM-based predictions and incremental adjustments. Section V presents simulation results that demonstrate the effectiveness of the framework. Finally, Section VI concludes the paper.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Related Work</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">Recent studies on vehicular network routing address challenges posed by dynamic topology, communication delay, and route stability, with approaches generally classified into traditional, optimization-based, and hybrid/predictive methods.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS1.4.1.1">II-A</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS1.5.2">Traditional Routing Approaches</span> </h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">Traditional routing methods include position-based and delay-tolerant networking (DTN) protocols. Position-based routing, exemplified by Geographic Position-Based Routing (GPSR) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib6" title="">6</a>]</cite>, uses geographic data to make routing decisions, enhancing scalability in dynamic environments. Stability-aware routing protocols consider link duration to increase robustness <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib12" title="">12</a>]</cite>, though frequent disconnections remain challenging in dense urban areas. DTN protocols rely on store-carry-forward strategies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib7" title="">7</a>]</cite>, which can be effective in sparse networks but often introduce high latency unsuitable for time-sensitive applications. Reactive protocols, such as Ad Hoc On-Demand Distance Vector (AODV) routing <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib8" title="">8</a>]</cite>, reduce overhead by establishing routes on demand, though they may incur delays in highly dynamic settings.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS2.4.1.1">II-B</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS2.5.2">Optimization-Based Routing Methods</span> </h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">Optimization-based approaches, particularly those using evolutionary algorithms, have been applied to improve routing adaptability in vehicular networks. Particle Swarm Optimization (PSO) and Genetic Algorithms (GA) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib15" title="">15</a>]</cite> show promise in optimizing paths for dynamic conditions, and multi-objective optimization frameworks address trade-offs between delay, energy, and stability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib10" title="">10</a>]</cite>. Multi-Objective Evolutionary Algorithms based on Decomposition (MOEA/D) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib9" title="">9</a>]</cite> enable flexibility and adaptability but may struggle with computational demands in highly dynamic environments.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS3.4.1.1">II-C</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS3.5.2">Hybrid and Predictive Approaches</span> </h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">Hybrid approaches that integrate optimization with predictive modeling have emerged as promising solutions for dynamic vehicular networks. Models such as Long Short-Term Memory (LSTM) networks and Markov chains predict vehicle mobility, allowing for proactive routing adjustments <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib13" title="">13</a>]</cite>. Combining predictive models with optimization algorithms effectively balances stability and delay, as hybrid methods integrating evolutionary algorithms with mobility predictions demonstrate improved stability in dynamic conditions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#bib.bib17" title="">17</a>]</cite>. This paper adopts a similar approach, integrating MOEA/D and predictive modeling, specifically with LSTM, to enhance route adaptability amid frequent topology changes.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.1">This research builds on existing studies by targeting both communication delay and route stability while incorporating dynamic adaptation mechanisms. Our proposed hierarchical optimization framework provides a balanced, adaptive solution for the challenges posed by vehicular network dynamics.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Mathematical Modeling</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">In this section, we develop a mathematical model to optimize routing in vehicular networks under dynamic environments, focusing on minimizing communication delay and maximizing routing stability.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS1.4.1.1">III-A</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS1.5.2">Objective Functions</span> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.2">In this dynamic vehicular network environment, the optimization problem involves two conflicting objectives: minimizing communication delay and maximizing routing stability. 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xref="S3.SS1.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">P(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_P ( italic_t )</annotation></semantics></math> as the decision variable representing the selected route at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_t</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.1">The objectives are formulated as follows:</p> </div> <div class="ltx_para" 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class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p3.1">where:</p> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.2"><span class="ltx_text ltx_font_bold" id="S3.SS1.p4.2.1">1. 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italic_t ) )</annotation></semantics></math>: The total communication delay along the dynamic path <math alttext="P(t)" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mrow id="S3.SS1.p4.2.m2.1.2" xref="S3.SS1.p4.2.m2.1.2.cmml"><mi id="S3.SS1.p4.2.m2.1.2.2" xref="S3.SS1.p4.2.m2.1.2.2.cmml">P</mi><mo id="S3.SS1.p4.2.m2.1.2.1" xref="S3.SS1.p4.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p4.2.m2.1.2.3.2" xref="S3.SS1.p4.2.m2.1.2.cmml"><mo id="S3.SS1.p4.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS1.p4.2.m2.1.2.cmml">(</mo><mi id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml">t</mi><mo id="S3.SS1.p4.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS1.p4.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><apply id="S3.SS1.p4.2.m2.1.2.cmml" xref="S3.SS1.p4.2.m2.1.2"><times id="S3.SS1.p4.2.m2.1.2.1.cmml" xref="S3.SS1.p4.2.m2.1.2.1"></times><ci id="S3.SS1.p4.2.m2.1.2.2.cmml" xref="S3.SS1.p4.2.m2.1.2.2">𝑃</ci><ci id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">P(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">italic_P ( italic_t )</annotation></semantics></math>, defined as:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="T(P(t))=\sum_{i=0}^{|P|-2}t_{i,i+1}(t)" class="ltx_Math" display="block" id="S3.E2.m1.6"><semantics id="S3.E2.m1.6a"><mrow id="S3.E2.m1.6.6" xref="S3.E2.m1.6.6.cmml"><mrow id="S3.E2.m1.6.6.1" xref="S3.E2.m1.6.6.1.cmml"><mi id="S3.E2.m1.6.6.1.3" xref="S3.E2.m1.6.6.1.3.cmml">T</mi><mo id="S3.E2.m1.6.6.1.2" xref="S3.E2.m1.6.6.1.2.cmml">⁢</mo><mrow id="S3.E2.m1.6.6.1.1.1" xref="S3.E2.m1.6.6.1.1.1.1.cmml"><mo 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xref="S3.E2.m1.6.6.3.1.2.3.1"></eq><ci id="S3.E2.m1.6.6.3.1.2.3.2.cmml" xref="S3.E2.m1.6.6.3.1.2.3.2">𝑖</ci><cn id="S3.E2.m1.6.6.3.1.2.3.3.cmml" type="integer" xref="S3.E2.m1.6.6.3.1.2.3.3">0</cn></apply></apply><apply id="S3.E2.m1.1.1.1.cmml" xref="S3.E2.m1.1.1.1"><minus id="S3.E2.m1.1.1.1.2.cmml" xref="S3.E2.m1.1.1.1.2"></minus><apply id="S3.E2.m1.1.1.1.3.1.cmml" xref="S3.E2.m1.1.1.1.3.2"><abs id="S3.E2.m1.1.1.1.3.1.1.cmml" xref="S3.E2.m1.1.1.1.3.2.1"></abs><ci id="S3.E2.m1.1.1.1.1.cmml" xref="S3.E2.m1.1.1.1.1">𝑃</ci></apply><cn id="S3.E2.m1.1.1.1.4.cmml" type="integer" xref="S3.E2.m1.1.1.1.4">2</cn></apply></apply><apply id="S3.E2.m1.6.6.3.2.cmml" xref="S3.E2.m1.6.6.3.2"><times id="S3.E2.m1.6.6.3.2.1.cmml" xref="S3.E2.m1.6.6.3.2.1"></times><apply id="S3.E2.m1.6.6.3.2.2.cmml" xref="S3.E2.m1.6.6.3.2.2"><csymbol cd="ambiguous" id="S3.E2.m1.6.6.3.2.2.1.cmml" xref="S3.E2.m1.6.6.3.2.2">subscript</csymbol><ci id="S3.E2.m1.6.6.3.2.2.2.cmml" xref="S3.E2.m1.6.6.3.2.2.2">𝑡</ci><list id="S3.E2.m1.3.3.2.3.cmml" xref="S3.E2.m1.3.3.2.2"><ci id="S3.E2.m1.2.2.1.1.cmml" xref="S3.E2.m1.2.2.1.1">𝑖</ci><apply id="S3.E2.m1.3.3.2.2.1.cmml" xref="S3.E2.m1.3.3.2.2.1"><plus id="S3.E2.m1.3.3.2.2.1.1.cmml" xref="S3.E2.m1.3.3.2.2.1.1"></plus><ci id="S3.E2.m1.3.3.2.2.1.2.cmml" xref="S3.E2.m1.3.3.2.2.1.2">𝑖</ci><cn id="S3.E2.m1.3.3.2.2.1.3.cmml" type="integer" xref="S3.E2.m1.3.3.2.2.1.3">1</cn></apply></list></apply><ci id="S3.E2.m1.5.5.cmml" xref="S3.E2.m1.5.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.6c">T(P(t))=\sum_{i=0}^{|P|-2}t_{i,i+1}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.6d">italic_T ( italic_P ( italic_t ) ) = ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT | italic_P | - 2 end_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p4.6">Here, <math alttext="t_{i,i+1}(t)" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m1.3"><semantics id="S3.SS1.p4.3.m1.3a"><mrow id="S3.SS1.p4.3.m1.3.4" xref="S3.SS1.p4.3.m1.3.4.cmml"><msub id="S3.SS1.p4.3.m1.3.4.2" xref="S3.SS1.p4.3.m1.3.4.2.cmml"><mi id="S3.SS1.p4.3.m1.3.4.2.2" xref="S3.SS1.p4.3.m1.3.4.2.2.cmml">t</mi><mrow id="S3.SS1.p4.3.m1.2.2.2.2" xref="S3.SS1.p4.3.m1.2.2.2.3.cmml"><mi id="S3.SS1.p4.3.m1.1.1.1.1" xref="S3.SS1.p4.3.m1.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p4.3.m1.2.2.2.2.2" xref="S3.SS1.p4.3.m1.2.2.2.3.cmml">,</mo><mrow id="S3.SS1.p4.3.m1.2.2.2.2.1" xref="S3.SS1.p4.3.m1.2.2.2.2.1.cmml"><mi id="S3.SS1.p4.3.m1.2.2.2.2.1.2" xref="S3.SS1.p4.3.m1.2.2.2.2.1.2.cmml">i</mi><mo id="S3.SS1.p4.3.m1.2.2.2.2.1.1" xref="S3.SS1.p4.3.m1.2.2.2.2.1.1.cmml">+</mo><mn id="S3.SS1.p4.3.m1.2.2.2.2.1.3" xref="S3.SS1.p4.3.m1.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><mo id="S3.SS1.p4.3.m1.3.4.1" xref="S3.SS1.p4.3.m1.3.4.1.cmml">⁢</mo><mrow id="S3.SS1.p4.3.m1.3.4.3.2" xref="S3.SS1.p4.3.m1.3.4.cmml"><mo id="S3.SS1.p4.3.m1.3.4.3.2.1" stretchy="false" xref="S3.SS1.p4.3.m1.3.4.cmml">(</mo><mi id="S3.SS1.p4.3.m1.3.3" xref="S3.SS1.p4.3.m1.3.3.cmml">t</mi><mo id="S3.SS1.p4.3.m1.3.4.3.2.2" stretchy="false" xref="S3.SS1.p4.3.m1.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m1.3b"><apply id="S3.SS1.p4.3.m1.3.4.cmml" xref="S3.SS1.p4.3.m1.3.4"><times id="S3.SS1.p4.3.m1.3.4.1.cmml" xref="S3.SS1.p4.3.m1.3.4.1"></times><apply id="S3.SS1.p4.3.m1.3.4.2.cmml" xref="S3.SS1.p4.3.m1.3.4.2"><csymbol cd="ambiguous" id="S3.SS1.p4.3.m1.3.4.2.1.cmml" xref="S3.SS1.p4.3.m1.3.4.2">subscript</csymbol><ci id="S3.SS1.p4.3.m1.3.4.2.2.cmml" xref="S3.SS1.p4.3.m1.3.4.2.2">𝑡</ci><list id="S3.SS1.p4.3.m1.2.2.2.3.cmml" xref="S3.SS1.p4.3.m1.2.2.2.2"><ci id="S3.SS1.p4.3.m1.1.1.1.1.cmml" xref="S3.SS1.p4.3.m1.1.1.1.1">𝑖</ci><apply id="S3.SS1.p4.3.m1.2.2.2.2.1.cmml" xref="S3.SS1.p4.3.m1.2.2.2.2.1"><plus id="S3.SS1.p4.3.m1.2.2.2.2.1.1.cmml" xref="S3.SS1.p4.3.m1.2.2.2.2.1.1"></plus><ci id="S3.SS1.p4.3.m1.2.2.2.2.1.2.cmml" xref="S3.SS1.p4.3.m1.2.2.2.2.1.2">𝑖</ci><cn id="S3.SS1.p4.3.m1.2.2.2.2.1.3.cmml" type="integer" xref="S3.SS1.p4.3.m1.2.2.2.2.1.3">1</cn></apply></list></apply><ci id="S3.SS1.p4.3.m1.3.3.cmml" xref="S3.SS1.p4.3.m1.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m1.3c">t_{i,i+1}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m1.3d">italic_t start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> denotes the delay between nodes <math alttext="i" class="ltx_Math" display="inline" id="S3.SS1.p4.4.m2.1"><semantics id="S3.SS1.p4.4.m2.1a"><mi id="S3.SS1.p4.4.m2.1.1" xref="S3.SS1.p4.4.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.4.m2.1b"><ci id="S3.SS1.p4.4.m2.1.1.cmml" xref="S3.SS1.p4.4.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.4.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.4.m2.1d">italic_i</annotation></semantics></math> and <math alttext="i+1" class="ltx_Math" display="inline" id="S3.SS1.p4.5.m3.1"><semantics id="S3.SS1.p4.5.m3.1a"><mrow id="S3.SS1.p4.5.m3.1.1" xref="S3.SS1.p4.5.m3.1.1.cmml"><mi id="S3.SS1.p4.5.m3.1.1.2" xref="S3.SS1.p4.5.m3.1.1.2.cmml">i</mi><mo id="S3.SS1.p4.5.m3.1.1.1" xref="S3.SS1.p4.5.m3.1.1.1.cmml">+</mo><mn id="S3.SS1.p4.5.m3.1.1.3" xref="S3.SS1.p4.5.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.5.m3.1b"><apply id="S3.SS1.p4.5.m3.1.1.cmml" xref="S3.SS1.p4.5.m3.1.1"><plus id="S3.SS1.p4.5.m3.1.1.1.cmml" xref="S3.SS1.p4.5.m3.1.1.1"></plus><ci id="S3.SS1.p4.5.m3.1.1.2.cmml" xref="S3.SS1.p4.5.m3.1.1.2">𝑖</ci><cn id="S3.SS1.p4.5.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p4.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.5.m3.1c">i+1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.5.m3.1d">italic_i + 1</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p4.6.m4.1"><semantics id="S3.SS1.p4.6.m4.1a"><mi id="S3.SS1.p4.6.m4.1.1" xref="S3.SS1.p4.6.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.6.m4.1b"><ci id="S3.SS1.p4.6.m4.1.1.cmml" xref="S3.SS1.p4.6.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.6.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.6.m4.1d">italic_t</annotation></semantics></math>, which varies due to factors such as distance, signal-to-noise ratio (SINR), and data rate in the dynamic environment.</p> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.2"><span class="ltx_text ltx_font_bold" id="S3.SS1.p5.2.1">2. Routing Stability Maximization</span> <math alttext="S(P(t))" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.2"><semantics id="S3.SS1.p5.1.m1.2a"><mrow id="S3.SS1.p5.1.m1.2.2" xref="S3.SS1.p5.1.m1.2.2.cmml"><mi id="S3.SS1.p5.1.m1.2.2.3" xref="S3.SS1.p5.1.m1.2.2.3.cmml">S</mi><mo id="S3.SS1.p5.1.m1.2.2.2" xref="S3.SS1.p5.1.m1.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.p5.1.m1.2.2.1.1" xref="S3.SS1.p5.1.m1.2.2.1.1.1.cmml"><mo id="S3.SS1.p5.1.m1.2.2.1.1.2" stretchy="false" xref="S3.SS1.p5.1.m1.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS1.p5.1.m1.2.2.1.1.1" xref="S3.SS1.p5.1.m1.2.2.1.1.1.cmml"><mi id="S3.SS1.p5.1.m1.2.2.1.1.1.2" xref="S3.SS1.p5.1.m1.2.2.1.1.1.2.cmml">P</mi><mo id="S3.SS1.p5.1.m1.2.2.1.1.1.1" xref="S3.SS1.p5.1.m1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.p5.1.m1.2.2.1.1.1.3.2" xref="S3.SS1.p5.1.m1.2.2.1.1.1.cmml"><mo id="S3.SS1.p5.1.m1.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.p5.1.m1.2.2.1.1.1.cmml">(</mo><mi id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml">t</mi><mo id="S3.SS1.p5.1.m1.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.SS1.p5.1.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p5.1.m1.2.2.1.1.3" stretchy="false" xref="S3.SS1.p5.1.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.1.m1.2b"><apply id="S3.SS1.p5.1.m1.2.2.cmml" xref="S3.SS1.p5.1.m1.2.2"><times id="S3.SS1.p5.1.m1.2.2.2.cmml" xref="S3.SS1.p5.1.m1.2.2.2"></times><ci id="S3.SS1.p5.1.m1.2.2.3.cmml" xref="S3.SS1.p5.1.m1.2.2.3">𝑆</ci><apply id="S3.SS1.p5.1.m1.2.2.1.1.1.cmml" xref="S3.SS1.p5.1.m1.2.2.1.1"><times id="S3.SS1.p5.1.m1.2.2.1.1.1.1.cmml" xref="S3.SS1.p5.1.m1.2.2.1.1.1.1"></times><ci id="S3.SS1.p5.1.m1.2.2.1.1.1.2.cmml" xref="S3.SS1.p5.1.m1.2.2.1.1.1.2">𝑃</ci><ci id="S3.SS1.p5.1.m1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.1.m1.2c">S(P(t))</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.1.m1.2d">italic_S ( italic_P ( italic_t ) )</annotation></semantics></math>: The stability of the dynamic path <math alttext="P(t)" class="ltx_Math" display="inline" id="S3.SS1.p5.2.m2.1"><semantics id="S3.SS1.p5.2.m2.1a"><mrow id="S3.SS1.p5.2.m2.1.2" xref="S3.SS1.p5.2.m2.1.2.cmml"><mi id="S3.SS1.p5.2.m2.1.2.2" xref="S3.SS1.p5.2.m2.1.2.2.cmml">P</mi><mo id="S3.SS1.p5.2.m2.1.2.1" xref="S3.SS1.p5.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p5.2.m2.1.2.3.2" xref="S3.SS1.p5.2.m2.1.2.cmml"><mo id="S3.SS1.p5.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS1.p5.2.m2.1.2.cmml">(</mo><mi id="S3.SS1.p5.2.m2.1.1" xref="S3.SS1.p5.2.m2.1.1.cmml">t</mi><mo id="S3.SS1.p5.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS1.p5.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.2.m2.1b"><apply id="S3.SS1.p5.2.m2.1.2.cmml" xref="S3.SS1.p5.2.m2.1.2"><times id="S3.SS1.p5.2.m2.1.2.1.cmml" xref="S3.SS1.p5.2.m2.1.2.1"></times><ci id="S3.SS1.p5.2.m2.1.2.2.cmml" xref="S3.SS1.p5.2.m2.1.2.2">𝑃</ci><ci id="S3.SS1.p5.2.m2.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.2.m2.1c">P(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.2.m2.1d">italic_P ( italic_t )</annotation></semantics></math>, defined as the minimum link stability across all hops:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S(P(t))=\min_{i=0,\dots,|P|-2}S_{i,i+1}(t)" class="ltx_Math" display="block" id="S3.E3.m1.9"><semantics id="S3.E3.m1.9a"><mrow id="S3.E3.m1.9.9" xref="S3.E3.m1.9.9.cmml"><mrow id="S3.E3.m1.9.9.1" xref="S3.E3.m1.9.9.1.cmml"><mi id="S3.E3.m1.9.9.1.3" xref="S3.E3.m1.9.9.1.3.cmml">S</mi><mo id="S3.E3.m1.9.9.1.2" xref="S3.E3.m1.9.9.1.2.cmml">⁢</mo><mrow id="S3.E3.m1.9.9.1.1.1" xref="S3.E3.m1.9.9.1.1.1.1.cmml"><mo 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id="S3.E3.m1.9c">S(P(t))=\min_{i=0,\dots,|P|-2}S_{i,i+1}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.9d">italic_S ( italic_P ( italic_t ) ) = roman_min start_POSTSUBSCRIPT italic_i = 0 , … , | italic_P | - 2 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p5.6">where <math alttext="S_{i,i+1}(t)" class="ltx_Math" display="inline" id="S3.SS1.p5.3.m1.3"><semantics id="S3.SS1.p5.3.m1.3a"><mrow id="S3.SS1.p5.3.m1.3.4" xref="S3.SS1.p5.3.m1.3.4.cmml"><msub id="S3.SS1.p5.3.m1.3.4.2" xref="S3.SS1.p5.3.m1.3.4.2.cmml"><mi id="S3.SS1.p5.3.m1.3.4.2.2" xref="S3.SS1.p5.3.m1.3.4.2.2.cmml">S</mi><mrow id="S3.SS1.p5.3.m1.2.2.2.2" 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encoding="application/x-tex" id="S3.SS1.p5.3.m1.3c">S_{i,i+1}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.3.m1.3d">italic_S start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> represents the stability of the link between nodes <math alttext="i" class="ltx_Math" display="inline" id="S3.SS1.p5.4.m2.1"><semantics id="S3.SS1.p5.4.m2.1a"><mi id="S3.SS1.p5.4.m2.1.1" xref="S3.SS1.p5.4.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.4.m2.1b"><ci id="S3.SS1.p5.4.m2.1.1.cmml" xref="S3.SS1.p5.4.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.4.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.4.m2.1d">italic_i</annotation></semantics></math> and <math alttext="i+1" class="ltx_Math" display="inline" id="S3.SS1.p5.5.m3.1"><semantics id="S3.SS1.p5.5.m3.1a"><mrow id="S3.SS1.p5.5.m3.1.1" xref="S3.SS1.p5.5.m3.1.1.cmml"><mi id="S3.SS1.p5.5.m3.1.1.2" xref="S3.SS1.p5.5.m3.1.1.2.cmml">i</mi><mo id="S3.SS1.p5.5.m3.1.1.1" xref="S3.SS1.p5.5.m3.1.1.1.cmml">+</mo><mn id="S3.SS1.p5.5.m3.1.1.3" xref="S3.SS1.p5.5.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.5.m3.1b"><apply id="S3.SS1.p5.5.m3.1.1.cmml" xref="S3.SS1.p5.5.m3.1.1"><plus id="S3.SS1.p5.5.m3.1.1.1.cmml" xref="S3.SS1.p5.5.m3.1.1.1"></plus><ci id="S3.SS1.p5.5.m3.1.1.2.cmml" xref="S3.SS1.p5.5.m3.1.1.2">𝑖</ci><cn id="S3.SS1.p5.5.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p5.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.5.m3.1c">i+1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.5.m3.1d">italic_i + 1</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p5.6.m4.1"><semantics id="S3.SS1.p5.6.m4.1a"><mi id="S3.SS1.p5.6.m4.1.1" xref="S3.SS1.p5.6.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.6.m4.1b"><ci id="S3.SS1.p5.6.m4.1.1.cmml" xref="S3.SS1.p5.6.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.6.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.6.m4.1d">italic_t</annotation></semantics></math>, influenced by factors such as relative velocity of nodes and link availability duration in a dynamic vehicular environment.</p> </div> <div class="ltx_para" id="S3.SS1.p6"> <p class="ltx_p" id="S3.SS1.p6.2">In this formulation, the objectives <math alttext="T(P(t))" class="ltx_Math" display="inline" id="S3.SS1.p6.1.m1.2"><semantics id="S3.SS1.p6.1.m1.2a"><mrow id="S3.SS1.p6.1.m1.2.2" xref="S3.SS1.p6.1.m1.2.2.cmml"><mi id="S3.SS1.p6.1.m1.2.2.3" xref="S3.SS1.p6.1.m1.2.2.3.cmml">T</mi><mo id="S3.SS1.p6.1.m1.2.2.2" xref="S3.SS1.p6.1.m1.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.p6.1.m1.2.2.1.1" xref="S3.SS1.p6.1.m1.2.2.1.1.1.cmml"><mo id="S3.SS1.p6.1.m1.2.2.1.1.2" stretchy="false" 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id="S3.SS1.p6.2.m2.2.2.1.2" xref="S3.SS1.p6.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.p6.2.m2.2.2.1.1.1" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml"><mo id="S3.SS1.p6.2.m2.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.p6.2.m2.2.2.1.1.1.1" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml"><mi id="S3.SS1.p6.2.m2.2.2.1.1.1.1.2" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.p6.2.m2.2.2.1.1.1.1.1" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.p6.2.m2.2.2.1.1.1.1.3.2" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml"><mo id="S3.SS1.p6.2.m2.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml">(</mo><mi id="S3.SS1.p6.2.m2.1.1" xref="S3.SS1.p6.2.m2.1.1.cmml">t</mi><mo id="S3.SS1.p6.2.m2.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p6.2.m2.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.2.m2.2b"><apply id="S3.SS1.p6.2.m2.2.2.cmml" xref="S3.SS1.p6.2.m2.2.2"><minus id="S3.SS1.p6.2.m2.2.2.2.cmml" xref="S3.SS1.p6.2.m2.2.2"></minus><apply id="S3.SS1.p6.2.m2.2.2.1.cmml" xref="S3.SS1.p6.2.m2.2.2.1"><times id="S3.SS1.p6.2.m2.2.2.1.2.cmml" xref="S3.SS1.p6.2.m2.2.2.1.2"></times><ci id="S3.SS1.p6.2.m2.2.2.1.3.cmml" xref="S3.SS1.p6.2.m2.2.2.1.3">𝑆</ci><apply id="S3.SS1.p6.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.p6.2.m2.2.2.1.1.1"><times id="S3.SS1.p6.2.m2.2.2.1.1.1.1.1.cmml" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.1"></times><ci id="S3.SS1.p6.2.m2.2.2.1.1.1.1.2.cmml" xref="S3.SS1.p6.2.m2.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.p6.2.m2.1.1.cmml" xref="S3.SS1.p6.2.m2.1.1">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.2.m2.2c">-S(P(t))</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.2.m2.2d">- italic_S ( italic_P ( italic_t ) )</annotation></semantics></math> are minimized together, reflecting their conflicting nature within the dynamic environment. This conflict necessitates a balanced solution, achieved here by employing MOEA/D, which decomposes the multi-objective problem into scalar subproblems, effectively adapting to the time-varying network conditions.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS2.4.1.1">III-B</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS2.5.2">Multi-Objective Optimization</span> </h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">In this work, we aim to optimize two conflicting objectives in vehicular network routing: minimizing communication delay and maximizing routing stability. These objectives often conflict because minimizing delay can require selecting faster, shorter paths that may have less stable links due to frequent topology changes, while maximizing stability may favor more reliable, but potentially longer paths with higher latency.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.2"><span class="ltx_text ltx_font_bold" id="S3.SS2.p2.2.1">1. Communication Delay Minimization:</span> The communication delay, denoted as <math alttext="T(P)" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mrow id="S3.SS2.p2.1.m1.1.2" xref="S3.SS2.p2.1.m1.1.2.cmml"><mi id="S3.SS2.p2.1.m1.1.2.2" xref="S3.SS2.p2.1.m1.1.2.2.cmml">T</mi><mo id="S3.SS2.p2.1.m1.1.2.1" xref="S3.SS2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p2.1.m1.1.2.3.2" xref="S3.SS2.p2.1.m1.1.2.cmml"><mo id="S3.SS2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S3.SS2.p2.1.m1.1.2.cmml">(</mo><mi id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml">P</mi><mo id="S3.SS2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S3.SS2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.2"><times id="S3.SS2.p2.1.m1.1.2.1.cmml" xref="S3.SS2.p2.1.m1.1.2.1"></times><ci id="S3.SS2.p2.1.m1.1.2.2.cmml" xref="S3.SS2.p2.1.m1.1.2.2">𝑇</ci><ci id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">T(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_T ( italic_P )</annotation></semantics></math>, represents the sum of delays over all hops along the path <math alttext="P" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mi id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><ci id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_P</annotation></semantics></math> from source to destination. Reducing this delay is crucial for time-sensitive applications, as low latency directly impacts the network’s responsiveness and reliability. However, paths with lower delay can be less stable, as they often involve dynamic connections prone to disruptions.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.2"><span class="ltx_text ltx_font_bold" id="S3.SS2.p3.2.1">2. Routing Stability Maximization:</span> Routing stability, denoted as <math alttext="S(P)" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mrow id="S3.SS2.p3.1.m1.1.2" xref="S3.SS2.p3.1.m1.1.2.cmml"><mi id="S3.SS2.p3.1.m1.1.2.2" xref="S3.SS2.p3.1.m1.1.2.2.cmml">S</mi><mo id="S3.SS2.p3.1.m1.1.2.1" xref="S3.SS2.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p3.1.m1.1.2.3.2" xref="S3.SS2.p3.1.m1.1.2.cmml"><mo id="S3.SS2.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S3.SS2.p3.1.m1.1.2.cmml">(</mo><mi id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml">P</mi><mo id="S3.SS2.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S3.SS2.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.2"><times id="S3.SS2.p3.1.m1.1.2.1.cmml" xref="S3.SS2.p3.1.m1.1.2.1"></times><ci id="S3.SS2.p3.1.m1.1.2.2.cmml" xref="S3.SS2.p3.1.m1.1.2.2">𝑆</ci><ci id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">S(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_S ( italic_P )</annotation></semantics></math>, is defined as the minimum stability of all links along the path <math alttext="P" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mi id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><ci id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_P</annotation></semantics></math>. A stable path reduces the frequency of route interruptions, which is beneficial for minimizing route recovery overhead and improving the reliability of the network. However, highly stable paths can sometimes involve longer or less direct routes, which increases overall communication delay.</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.1">These two objectives conflict because optimizing for delay generally requires shorter paths with potentially less stable links, while optimizing for stability typically favors paths with fewer disruptions, even if they are longer. This trade-off requires a balanced approach to avoid excessive latency while maintaining route reliability.</p> </div> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.1">To address these conflicts, we employ the Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), which does not rely on a single combined objective function but rather decomposes the problem into multiple scalar subproblems. Each subproblem corresponds to a unique combination of delay and stability objectives, allowing the algorithm to explore and maintain a diverse set of solutions that offer various trade-offs between delay and stability.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS3.4.1.1">III-C</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS3.5.2">Handling Dynamic Environment</span> </h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">In a dynamic vehicular network, frequent changes in network topology occur due to vehicle movement. To address this, the optimization framework incorporates the following mechanisms:</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1"><span class="ltx_text ltx_font_bold" id="S3.SS3.p2.1.1">1. Incremental Adjustment</span>: When network topology changes, only the affected individuals in the population are adjusted, rather than re-optimizing the entire path from scratch. This approach reduces computational overhead and allows the algorithm to quickly adapt to changes.</p> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.1"><span class="ltx_text ltx_font_bold" id="S3.SS3.p3.1.1">2. Prediction Model</span>: A prediction model, such as an LSTM network, is used to forecast vehicle movements. By predicting future positions of vehicles, the algorithm can proactively adjust routing paths to maintain low delay and high stability.</p> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.1">These mechanisms enable the optimization framework to respond effectively to changes in the vehicular network, thereby maintaining optimal routing performance in real time.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Proposed Algorithm</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">This section presents the hierarchical evolutionary optimization framework designed to address the dynamic and delay-sensitive nature of routing in vehicular networks. The proposed algorithm aims to minimize communication delay and maximize route stability through a Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D). To further enhance adaptability in a dynamic environment, the algorithm incorporates incremental adjustment and predictive modeling mechanisms.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS1.4.1.1">IV-A</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS1.5.2">LSTM-Based Prediction Model for Dynamic Network Forecasting</span> </h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">In highly dynamic vehicular networks, node positions and link stability fluctuate frequently due to rapid vehicle movement. To address these fluctuations, we employ a Long Short-Term Memory (LSTM)-based prediction model that leverages historical data to forecast future states, enhancing routing stability and reducing communication delay.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.7">The input to the LSTM model is a time-series sequence of historical data for each node, denoted as <math alttext="X_{t-w:t}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><msub id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">X</mi><mrow id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml"><mrow id="S4.SS1.p2.1.m1.1.1.3.2" xref="S4.SS1.p2.1.m1.1.1.3.2.cmml"><mi id="S4.SS1.p2.1.m1.1.1.3.2.2" xref="S4.SS1.p2.1.m1.1.1.3.2.2.cmml">t</mi><mo id="S4.SS1.p2.1.m1.1.1.3.2.1" xref="S4.SS1.p2.1.m1.1.1.3.2.1.cmml">−</mo><mi id="S4.SS1.p2.1.m1.1.1.3.2.3" xref="S4.SS1.p2.1.m1.1.1.3.2.3.cmml">w</mi></mrow><mo id="S4.SS1.p2.1.m1.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p2.1.m1.1.1.3.1.cmml">:</mo><mi id="S4.SS1.p2.1.m1.1.1.3.3" xref="S4.SS1.p2.1.m1.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝑋</ci><apply id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3"><ci id="S4.SS1.p2.1.m1.1.1.3.1.cmml" xref="S4.SS1.p2.1.m1.1.1.3.1">:</ci><apply id="S4.SS1.p2.1.m1.1.1.3.2.cmml" xref="S4.SS1.p2.1.m1.1.1.3.2"><minus id="S4.SS1.p2.1.m1.1.1.3.2.1.cmml" xref="S4.SS1.p2.1.m1.1.1.3.2.1"></minus><ci id="S4.SS1.p2.1.m1.1.1.3.2.2.cmml" xref="S4.SS1.p2.1.m1.1.1.3.2.2">𝑡</ci><ci id="S4.SS1.p2.1.m1.1.1.3.2.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3.2.3">𝑤</ci></apply><ci id="S4.SS1.p2.1.m1.1.1.3.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">X_{t-w:t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_X start_POSTSUBSCRIPT italic_t - italic_w : italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="w" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_w</annotation></semantics></math> represents the window size, capturing past node positions or link stability values. 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During real-time deployment, the LSTM model receives the latest <math alttext="w" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mi id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><ci id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_w</annotation></semantics></math>-length sequence <math alttext="X_{i,t-w:t}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.2"><semantics id="S4.SS1.p2.7.m7.2a"><msub id="S4.SS1.p2.7.m7.2.3" xref="S4.SS1.p2.7.m7.2.3.cmml"><mi id="S4.SS1.p2.7.m7.2.3.2" xref="S4.SS1.p2.7.m7.2.3.2.cmml">X</mi><mrow id="S4.SS1.p2.7.m7.2.2.2" xref="S4.SS1.p2.7.m7.2.2.2.cmml"><mrow id="S4.SS1.p2.7.m7.2.2.2.2.1" xref="S4.SS1.p2.7.m7.2.2.2.2.2.cmml"><mi id="S4.SS1.p2.7.m7.1.1.1.1" xref="S4.SS1.p2.7.m7.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p2.7.m7.2.2.2.2.1.2" xref="S4.SS1.p2.7.m7.2.2.2.2.2.cmml">,</mo><mrow id="S4.SS1.p2.7.m7.2.2.2.2.1.1" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1.cmml"><mi id="S4.SS1.p2.7.m7.2.2.2.2.1.1.2" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1.2.cmml">t</mi><mo id="S4.SS1.p2.7.m7.2.2.2.2.1.1.1" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1.1.cmml">−</mo><mi id="S4.SS1.p2.7.m7.2.2.2.2.1.1.3" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1.3.cmml">w</mi></mrow></mrow><mo id="S4.SS1.p2.7.m7.2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p2.7.m7.2.2.2.3.cmml">:</mo><mi id="S4.SS1.p2.7.m7.2.2.2.4" xref="S4.SS1.p2.7.m7.2.2.2.4.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.2b"><apply id="S4.SS1.p2.7.m7.2.3.cmml" xref="S4.SS1.p2.7.m7.2.3"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m7.2.3.1.cmml" xref="S4.SS1.p2.7.m7.2.3">subscript</csymbol><ci id="S4.SS1.p2.7.m7.2.3.2.cmml" xref="S4.SS1.p2.7.m7.2.3.2">𝑋</ci><apply id="S4.SS1.p2.7.m7.2.2.2.cmml" xref="S4.SS1.p2.7.m7.2.2.2"><ci id="S4.SS1.p2.7.m7.2.2.2.3.cmml" xref="S4.SS1.p2.7.m7.2.2.2.3">:</ci><list id="S4.SS1.p2.7.m7.2.2.2.2.2.cmml" xref="S4.SS1.p2.7.m7.2.2.2.2.1"><ci id="S4.SS1.p2.7.m7.1.1.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1.1.1">𝑖</ci><apply id="S4.SS1.p2.7.m7.2.2.2.2.1.1.cmml" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1"><minus id="S4.SS1.p2.7.m7.2.2.2.2.1.1.1.cmml" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1.1"></minus><ci id="S4.SS1.p2.7.m7.2.2.2.2.1.1.2.cmml" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1.2">𝑡</ci><ci id="S4.SS1.p2.7.m7.2.2.2.2.1.1.3.cmml" xref="S4.SS1.p2.7.m7.2.2.2.2.1.1.3">𝑤</ci></apply></list><ci id="S4.SS1.p2.7.m7.2.2.2.4.cmml" xref="S4.SS1.p2.7.m7.2.2.2.4">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.2c">X_{i,t-w:t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.2d">italic_X start_POSTSUBSCRIPT italic_i , italic_t - italic_w : italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and provides proactive insights into network dynamics, allowing the routing algorithm to select stable, low-latency paths. This real-time prediction improves both network reliability and routing efficiency.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.3">The LSTM architecture comprises <math alttext="L" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mi id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">italic_L</annotation></semantics></math> LSTM layers followed by a dense output layer, capturing temporal dependencies in node positions and link stability. Training minimizes a loss function, such as Mean Squared Error (MSE), by comparing predicted outputs <math alttext="\hat{X}_{i,t+1}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.2"><semantics id="S4.SS1.p3.2.m2.2a"><msub id="S4.SS1.p3.2.m2.2.3" xref="S4.SS1.p3.2.m2.2.3.cmml"><mover accent="true" id="S4.SS1.p3.2.m2.2.3.2" xref="S4.SS1.p3.2.m2.2.3.2.cmml"><mi id="S4.SS1.p3.2.m2.2.3.2.2" xref="S4.SS1.p3.2.m2.2.3.2.2.cmml">X</mi><mo id="S4.SS1.p3.2.m2.2.3.2.1" xref="S4.SS1.p3.2.m2.2.3.2.1.cmml">^</mo></mover><mrow id="S4.SS1.p3.2.m2.2.2.2.2" xref="S4.SS1.p3.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p3.2.m2.1.1.1.1" xref="S4.SS1.p3.2.m2.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.2.m2.2.2.2.2.2" xref="S4.SS1.p3.2.m2.2.2.2.3.cmml">,</mo><mrow id="S4.SS1.p3.2.m2.2.2.2.2.1" xref="S4.SS1.p3.2.m2.2.2.2.2.1.cmml"><mi id="S4.SS1.p3.2.m2.2.2.2.2.1.2" xref="S4.SS1.p3.2.m2.2.2.2.2.1.2.cmml">t</mi><mo id="S4.SS1.p3.2.m2.2.2.2.2.1.1" xref="S4.SS1.p3.2.m2.2.2.2.2.1.1.cmml">+</mo><mn id="S4.SS1.p3.2.m2.2.2.2.2.1.3" xref="S4.SS1.p3.2.m2.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.2b"><apply id="S4.SS1.p3.2.m2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.2.3.1.cmml" xref="S4.SS1.p3.2.m2.2.3">subscript</csymbol><apply id="S4.SS1.p3.2.m2.2.3.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2"><ci id="S4.SS1.p3.2.m2.2.3.2.1.cmml" xref="S4.SS1.p3.2.m2.2.3.2.1">^</ci><ci id="S4.SS1.p3.2.m2.2.3.2.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2.2">𝑋</ci></apply><list id="S4.SS1.p3.2.m2.2.2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2"><ci id="S4.SS1.p3.2.m2.1.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1.1.1">𝑖</ci><apply id="S4.SS1.p3.2.m2.2.2.2.2.1.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2.1"><plus id="S4.SS1.p3.2.m2.2.2.2.2.1.1.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2.1.1"></plus><ci id="S4.SS1.p3.2.m2.2.2.2.2.1.2.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2.1.2">𝑡</ci><cn id="S4.SS1.p3.2.m2.2.2.2.2.1.3.cmml" type="integer" xref="S4.SS1.p3.2.m2.2.2.2.2.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.2c">\hat{X}_{i,t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.2d">over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_i , italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to actual states <math alttext="X_{i,t+1}" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.2"><semantics id="S4.SS1.p3.3.m3.2a"><msub id="S4.SS1.p3.3.m3.2.3" xref="S4.SS1.p3.3.m3.2.3.cmml"><mi id="S4.SS1.p3.3.m3.2.3.2" xref="S4.SS1.p3.3.m3.2.3.2.cmml">X</mi><mrow id="S4.SS1.p3.3.m3.2.2.2.2" xref="S4.SS1.p3.3.m3.2.2.2.3.cmml"><mi id="S4.SS1.p3.3.m3.1.1.1.1" xref="S4.SS1.p3.3.m3.1.1.1.1.cmml">i</mi><mo id="S4.SS1.p3.3.m3.2.2.2.2.2" xref="S4.SS1.p3.3.m3.2.2.2.3.cmml">,</mo><mrow id="S4.SS1.p3.3.m3.2.2.2.2.1" xref="S4.SS1.p3.3.m3.2.2.2.2.1.cmml"><mi id="S4.SS1.p3.3.m3.2.2.2.2.1.2" xref="S4.SS1.p3.3.m3.2.2.2.2.1.2.cmml">t</mi><mo id="S4.SS1.p3.3.m3.2.2.2.2.1.1" xref="S4.SS1.p3.3.m3.2.2.2.2.1.1.cmml">+</mo><mn id="S4.SS1.p3.3.m3.2.2.2.2.1.3" xref="S4.SS1.p3.3.m3.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.2b"><apply id="S4.SS1.p3.3.m3.2.3.cmml" xref="S4.SS1.p3.3.m3.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m3.2.3.1.cmml" xref="S4.SS1.p3.3.m3.2.3">subscript</csymbol><ci id="S4.SS1.p3.3.m3.2.3.2.cmml" xref="S4.SS1.p3.3.m3.2.3.2">𝑋</ci><list id="S4.SS1.p3.3.m3.2.2.2.3.cmml" xref="S4.SS1.p3.3.m3.2.2.2.2"><ci id="S4.SS1.p3.3.m3.1.1.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1.1.1">𝑖</ci><apply id="S4.SS1.p3.3.m3.2.2.2.2.1.cmml" xref="S4.SS1.p3.3.m3.2.2.2.2.1"><plus id="S4.SS1.p3.3.m3.2.2.2.2.1.1.cmml" xref="S4.SS1.p3.3.m3.2.2.2.2.1.1"></plus><ci id="S4.SS1.p3.3.m3.2.2.2.2.1.2.cmml" xref="S4.SS1.p3.3.m3.2.2.2.2.1.2">𝑡</ci><cn id="S4.SS1.p3.3.m3.2.2.2.2.1.3.cmml" type="integer" xref="S4.SS1.p3.3.m3.2.2.2.2.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.2c">X_{i,t+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.2d">italic_X start_POSTSUBSCRIPT italic_i , italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Training is optimized through Backpropagation Through Time (BPTT), allowing the model to learn long-term data dependencies. The LSTM training and deployment process is outlined in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#alg1" title="Algorithm 1 ‣ IV-A LSTM-Based Prediction Model for Dynamic Network Forecasting ‣ IV Proposed Algorithm ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.1">This LSTM-based prediction component enables the hierarchical evolutionary optimization framework to maintain optimal routing performance by anticipating topology changes, effectively achieving the dual objectives of low communication delay and high stability in dynamic vehicular networks.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.2.1.1">Algorithm 1</span> </span> LSTM-Based Prediction Model for Dynamic Network Forecasting</figcaption> <div class="ltx_listing ltx_listing" id="alg1.3"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l1.2">Input:</span> Historical position data <math alttext="X_{t-w:t}" class="ltx_Math" display="inline" id="alg1.l1.m1.1"><semantics id="alg1.l1.m1.1a"><msub id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml"><mi id="alg1.l1.m1.1.1.2" xref="alg1.l1.m1.1.1.2.cmml">X</mi><mrow id="alg1.l1.m1.1.1.3" xref="alg1.l1.m1.1.1.3.cmml"><mrow id="alg1.l1.m1.1.1.3.2" 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class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l2.2">Output:</span> Predicted position or link stability state <math alttext="\hat{X}_{t+1}" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><msub id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mover accent="true" id="alg1.l2.m1.1.1.2" xref="alg1.l2.m1.1.1.2.cmml"><mi id="alg1.l2.m1.1.1.2.2" xref="alg1.l2.m1.1.1.2.2.cmml">X</mi><mo id="alg1.l2.m1.1.1.2.1" xref="alg1.l2.m1.1.1.2.1.cmml">^</mo></mover><mrow id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml"><mi id="alg1.l2.m1.1.1.3.2" xref="alg1.l2.m1.1.1.3.2.cmml">t</mi><mo id="alg1.l2.m1.1.1.3.1" xref="alg1.l2.m1.1.1.3.1.cmml">+</mo><mn id="alg1.l2.m1.1.1.3.3" xref="alg1.l2.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply 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ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.1.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l3.2">Initialize</span> LSTM model with weight parameters </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span>Define model architecture with <math alttext="L" class="ltx_Math" display="inline" id="alg1.l4.m1.1"><semantics id="alg1.l4.m1.1a"><mi id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.1b"><ci id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.1d">italic_L</annotation></semantics></math> LSTM layers and a dense output layer </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l5.1.1.1" style="font-size:80%;">5:</span></span>Define loss function <math alttext="L(y,\hat{y})" class="ltx_Math" display="inline" id="alg1.l5.m1.2"><semantics id="alg1.l5.m1.2a"><mrow id="alg1.l5.m1.2.3" xref="alg1.l5.m1.2.3.cmml"><mi id="alg1.l5.m1.2.3.2" xref="alg1.l5.m1.2.3.2.cmml">L</mi><mo id="alg1.l5.m1.2.3.1" xref="alg1.l5.m1.2.3.1.cmml">⁢</mo><mrow id="alg1.l5.m1.2.3.3.2" xref="alg1.l5.m1.2.3.3.1.cmml"><mo id="alg1.l5.m1.2.3.3.2.1" stretchy="false" xref="alg1.l5.m1.2.3.3.1.cmml">(</mo><mi id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml">y</mi><mo id="alg1.l5.m1.2.3.3.2.2" xref="alg1.l5.m1.2.3.3.1.cmml">,</mo><mover accent="true" id="alg1.l5.m1.2.2" xref="alg1.l5.m1.2.2.cmml"><mi id="alg1.l5.m1.2.2.2" xref="alg1.l5.m1.2.2.2.cmml">y</mi><mo id="alg1.l5.m1.2.2.1" xref="alg1.l5.m1.2.2.1.cmml">^</mo></mover><mo id="alg1.l5.m1.2.3.3.2.3" stretchy="false" xref="alg1.l5.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.2b"><apply id="alg1.l5.m1.2.3.cmml" xref="alg1.l5.m1.2.3"><times id="alg1.l5.m1.2.3.1.cmml" xref="alg1.l5.m1.2.3.1"></times><ci id="alg1.l5.m1.2.3.2.cmml" xref="alg1.l5.m1.2.3.2">𝐿</ci><interval closure="open" id="alg1.l5.m1.2.3.3.1.cmml" xref="alg1.l5.m1.2.3.3.2"><ci id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1">𝑦</ci><apply id="alg1.l5.m1.2.2.cmml" xref="alg1.l5.m1.2.2"><ci id="alg1.l5.m1.2.2.1.cmml" xref="alg1.l5.m1.2.2.1">^</ci><ci id="alg1.l5.m1.2.2.2.cmml" xref="alg1.l5.m1.2.2.2">𝑦</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.2c">L(y,\hat{y})</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.2d">italic_L ( italic_y , over^ start_ARG italic_y end_ARG )</annotation></semantics></math> and optimization algorithm </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l6.2">Step 1: Train LSTM on Historical Data</span> </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l7.1.1.1" style="font-size:80%;">7:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l7.2">for</span> each training epoch <span class="ltx_text ltx_font_bold" id="alg1.l7.3">do</span> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l8.1.1.1" style="font-size:80%;">8:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l8.2">for</span> each node <math alttext="i" class="ltx_Math" display="inline" id="alg1.l8.m1.1"><semantics id="alg1.l8.m1.1a"><mi id="alg1.l8.m1.1.1" xref="alg1.l8.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.1b"><ci id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m1.1d">italic_i</annotation></semantics></math> in training data <span class="ltx_text ltx_font_bold" id="alg1.l8.3">do</span> </div> <div class="ltx_listingline" id="alg1.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l9.1.1.1" style="font-size:80%;">9:</span></span> Retrieve position sequence <math alttext="X_{i,t-w:t}" class="ltx_Math" display="inline" id="alg1.l9.m1.2"><semantics id="alg1.l9.m1.2a"><msub id="alg1.l9.m1.2.3" xref="alg1.l9.m1.2.3.cmml"><mi id="alg1.l9.m1.2.3.2" xref="alg1.l9.m1.2.3.2.cmml">X</mi><mrow id="alg1.l9.m1.2.2.2" xref="alg1.l9.m1.2.2.2.cmml"><mrow id="alg1.l9.m1.2.2.2.2.1" xref="alg1.l9.m1.2.2.2.2.2.cmml"><mi id="alg1.l9.m1.1.1.1.1" xref="alg1.l9.m1.1.1.1.1.cmml">i</mi><mo id="alg1.l9.m1.2.2.2.2.1.2" xref="alg1.l9.m1.2.2.2.2.2.cmml">,</mo><mrow 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xref="alg1.l9.m1.1.1.1.1">𝑖</ci><apply id="alg1.l9.m1.2.2.2.2.1.1.cmml" xref="alg1.l9.m1.2.2.2.2.1.1"><minus id="alg1.l9.m1.2.2.2.2.1.1.1.cmml" xref="alg1.l9.m1.2.2.2.2.1.1.1"></minus><ci id="alg1.l9.m1.2.2.2.2.1.1.2.cmml" xref="alg1.l9.m1.2.2.2.2.1.1.2">𝑡</ci><ci id="alg1.l9.m1.2.2.2.2.1.1.3.cmml" xref="alg1.l9.m1.2.2.2.2.1.1.3">𝑤</ci></apply></list><ci id="alg1.l9.m1.2.2.2.4.cmml" xref="alg1.l9.m1.2.2.2.4">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m1.2c">X_{i,t-w:t}</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m1.2d">italic_X start_POSTSUBSCRIPT italic_i , italic_t - italic_w : italic_t end_POSTSUBSCRIPT</annotation></semantics></math> for node <math alttext="i" class="ltx_Math" display="inline" id="alg1.l9.m2.1"><semantics id="alg1.l9.m2.1a"><mi id="alg1.l9.m2.1.1" xref="alg1.l9.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="alg1.l9.m2.1b"><ci id="alg1.l9.m2.1.1.cmml" xref="alg1.l9.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m2.1d">italic_i</annotation></semantics></math> up to time <math alttext="t" class="ltx_Math" display="inline" id="alg1.l9.m3.1"><semantics id="alg1.l9.m3.1a"><mi id="alg1.l9.m3.1.1" xref="alg1.l9.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="alg1.l9.m3.1b"><ci id="alg1.l9.m3.1.1.cmml" xref="alg1.l9.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m3.1d">italic_t</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l10.1.1.1" style="font-size:80%;">10:</span></span> Feed sequence <math alttext="X_{i,t-w:t}" class="ltx_Math" display="inline" id="alg1.l10.m1.2"><semantics id="alg1.l10.m1.2a"><msub id="alg1.l10.m1.2.3" xref="alg1.l10.m1.2.3.cmml"><mi id="alg1.l10.m1.2.3.2" xref="alg1.l10.m1.2.3.2.cmml">X</mi><mrow id="alg1.l10.m1.2.2.2" xref="alg1.l10.m1.2.2.2.cmml"><mrow id="alg1.l10.m1.2.2.2.2.1" xref="alg1.l10.m1.2.2.2.2.2.cmml"><mi id="alg1.l10.m1.1.1.1.1" xref="alg1.l10.m1.1.1.1.1.cmml">i</mi><mo id="alg1.l10.m1.2.2.2.2.1.2" xref="alg1.l10.m1.2.2.2.2.2.cmml">,</mo><mrow id="alg1.l10.m1.2.2.2.2.1.1" xref="alg1.l10.m1.2.2.2.2.1.1.cmml"><mi id="alg1.l10.m1.2.2.2.2.1.1.2" xref="alg1.l10.m1.2.2.2.2.1.1.2.cmml">t</mi><mo id="alg1.l10.m1.2.2.2.2.1.1.1" xref="alg1.l10.m1.2.2.2.2.1.1.1.cmml">−</mo><mi id="alg1.l10.m1.2.2.2.2.1.1.3" xref="alg1.l10.m1.2.2.2.2.1.1.3.cmml">w</mi></mrow></mrow><mo id="alg1.l10.m1.2.2.2.3" lspace="0.278em" rspace="0.278em" xref="alg1.l10.m1.2.2.2.3.cmml">:</mo><mi id="alg1.l10.m1.2.2.2.4" xref="alg1.l10.m1.2.2.2.4.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.2b"><apply 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encoding="application/x-llamapun" id="alg1.l10.m1.2d">italic_X start_POSTSUBSCRIPT italic_i , italic_t - italic_w : italic_t end_POSTSUBSCRIPT</annotation></semantics></math> into LSTM model </div> <div class="ltx_listingline" id="alg1.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l11.1.1.1" style="font-size:80%;">11:</span></span> Output prediction <math alttext="\hat{X}_{i,t+1}" class="ltx_Math" display="inline" id="alg1.l11.m1.2"><semantics id="alg1.l11.m1.2a"><msub id="alg1.l11.m1.2.3" xref="alg1.l11.m1.2.3.cmml"><mover accent="true" id="alg1.l11.m1.2.3.2" xref="alg1.l11.m1.2.3.2.cmml"><mi id="alg1.l11.m1.2.3.2.2" xref="alg1.l11.m1.2.3.2.2.cmml">X</mi><mo id="alg1.l11.m1.2.3.2.1" xref="alg1.l11.m1.2.3.2.1.cmml">^</mo></mover><mrow id="alg1.l11.m1.2.2.2.2" xref="alg1.l11.m1.2.2.2.3.cmml"><mi id="alg1.l11.m1.1.1.1.1" xref="alg1.l11.m1.1.1.1.1.cmml">i</mi><mo id="alg1.l11.m1.2.2.2.2.2" xref="alg1.l11.m1.2.2.2.3.cmml">,</mo><mrow 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id="alg1.l11.m1.2.2.2.2.1.2.cmml" xref="alg1.l11.m1.2.2.2.2.1.2">𝑡</ci><cn id="alg1.l11.m1.2.2.2.2.1.3.cmml" type="integer" xref="alg1.l11.m1.2.2.2.2.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m1.2c">\hat{X}_{i,t+1}</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.m1.2d">over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_i , italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> for position or link state at time <math alttext="t+1" class="ltx_Math" display="inline" id="alg1.l11.m2.1"><semantics id="alg1.l11.m2.1a"><mrow id="alg1.l11.m2.1.1" xref="alg1.l11.m2.1.1.cmml"><mi id="alg1.l11.m2.1.1.2" xref="alg1.l11.m2.1.1.2.cmml">t</mi><mo id="alg1.l11.m2.1.1.1" xref="alg1.l11.m2.1.1.1.cmml">+</mo><mn id="alg1.l11.m2.1.1.3" xref="alg1.l11.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l11.m2.1b"><apply id="alg1.l11.m2.1.1.cmml" xref="alg1.l11.m2.1.1"><plus id="alg1.l11.m2.1.1.1.cmml" xref="alg1.l11.m2.1.1.1"></plus><ci id="alg1.l11.m2.1.1.2.cmml" xref="alg1.l11.m2.1.1.2">𝑡</ci><cn id="alg1.l11.m2.1.1.3.cmml" type="integer" xref="alg1.l11.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m2.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.m2.1d">italic_t + 1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l12"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l12.1.1.1" style="font-size:80%;">12:</span></span> Compute loss <math alttext="L(X_{i,t+1},\hat{X}_{i,t+1})" class="ltx_Math" display="inline" id="alg1.l12.m1.6"><semantics id="alg1.l12.m1.6a"><mrow id="alg1.l12.m1.6.6" xref="alg1.l12.m1.6.6.cmml"><mi id="alg1.l12.m1.6.6.4" xref="alg1.l12.m1.6.6.4.cmml">L</mi><mo id="alg1.l12.m1.6.6.3" xref="alg1.l12.m1.6.6.3.cmml">⁢</mo><mrow id="alg1.l12.m1.6.6.2.2" xref="alg1.l12.m1.6.6.2.3.cmml"><mo id="alg1.l12.m1.6.6.2.2.3" stretchy="false" xref="alg1.l12.m1.6.6.2.3.cmml">(</mo><msub id="alg1.l12.m1.5.5.1.1.1" xref="alg1.l12.m1.5.5.1.1.1.cmml"><mi id="alg1.l12.m1.5.5.1.1.1.2" xref="alg1.l12.m1.5.5.1.1.1.2.cmml">X</mi><mrow id="alg1.l12.m1.2.2.2.2" xref="alg1.l12.m1.2.2.2.3.cmml"><mi id="alg1.l12.m1.1.1.1.1" xref="alg1.l12.m1.1.1.1.1.cmml">i</mi><mo id="alg1.l12.m1.2.2.2.2.2" xref="alg1.l12.m1.2.2.2.3.cmml">,</mo><mrow id="alg1.l12.m1.2.2.2.2.1" xref="alg1.l12.m1.2.2.2.2.1.cmml"><mi id="alg1.l12.m1.2.2.2.2.1.2" xref="alg1.l12.m1.2.2.2.2.1.2.cmml">t</mi><mo id="alg1.l12.m1.2.2.2.2.1.1" xref="alg1.l12.m1.2.2.2.2.1.1.cmml">+</mo><mn id="alg1.l12.m1.2.2.2.2.1.3" xref="alg1.l12.m1.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><mo id="alg1.l12.m1.6.6.2.2.4" xref="alg1.l12.m1.6.6.2.3.cmml">,</mo><msub id="alg1.l12.m1.6.6.2.2.2" xref="alg1.l12.m1.6.6.2.2.2.cmml"><mover accent="true" id="alg1.l12.m1.6.6.2.2.2.2" xref="alg1.l12.m1.6.6.2.2.2.2.cmml"><mi id="alg1.l12.m1.6.6.2.2.2.2.2" xref="alg1.l12.m1.6.6.2.2.2.2.2.cmml">X</mi><mo id="alg1.l12.m1.6.6.2.2.2.2.1" xref="alg1.l12.m1.6.6.2.2.2.2.1.cmml">^</mo></mover><mrow id="alg1.l12.m1.4.4.2.2" xref="alg1.l12.m1.4.4.2.3.cmml"><mi id="alg1.l12.m1.3.3.1.1" xref="alg1.l12.m1.3.3.1.1.cmml">i</mi><mo id="alg1.l12.m1.4.4.2.2.2" xref="alg1.l12.m1.4.4.2.3.cmml">,</mo><mrow id="alg1.l12.m1.4.4.2.2.1" xref="alg1.l12.m1.4.4.2.2.1.cmml"><mi id="alg1.l12.m1.4.4.2.2.1.2" xref="alg1.l12.m1.4.4.2.2.1.2.cmml">t</mi><mo id="alg1.l12.m1.4.4.2.2.1.1" xref="alg1.l12.m1.4.4.2.2.1.1.cmml">+</mo><mn id="alg1.l12.m1.4.4.2.2.1.3" xref="alg1.l12.m1.4.4.2.2.1.3.cmml">1</mn></mrow></mrow></msub><mo id="alg1.l12.m1.6.6.2.2.5" stretchy="false" xref="alg1.l12.m1.6.6.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l12.m1.6b"><apply id="alg1.l12.m1.6.6.cmml" xref="alg1.l12.m1.6.6"><times id="alg1.l12.m1.6.6.3.cmml" xref="alg1.l12.m1.6.6.3"></times><ci id="alg1.l12.m1.6.6.4.cmml" xref="alg1.l12.m1.6.6.4">𝐿</ci><interval closure="open" id="alg1.l12.m1.6.6.2.3.cmml" xref="alg1.l12.m1.6.6.2.2"><apply id="alg1.l12.m1.5.5.1.1.1.cmml" xref="alg1.l12.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="alg1.l12.m1.5.5.1.1.1.1.cmml" xref="alg1.l12.m1.5.5.1.1.1">subscript</csymbol><ci id="alg1.l12.m1.5.5.1.1.1.2.cmml" xref="alg1.l12.m1.5.5.1.1.1.2">𝑋</ci><list id="alg1.l12.m1.2.2.2.3.cmml" xref="alg1.l12.m1.2.2.2.2"><ci id="alg1.l12.m1.1.1.1.1.cmml" xref="alg1.l12.m1.1.1.1.1">𝑖</ci><apply id="alg1.l12.m1.2.2.2.2.1.cmml" xref="alg1.l12.m1.2.2.2.2.1"><plus id="alg1.l12.m1.2.2.2.2.1.1.cmml" xref="alg1.l12.m1.2.2.2.2.1.1"></plus><ci id="alg1.l12.m1.2.2.2.2.1.2.cmml" xref="alg1.l12.m1.2.2.2.2.1.2">𝑡</ci><cn id="alg1.l12.m1.2.2.2.2.1.3.cmml" type="integer" xref="alg1.l12.m1.2.2.2.2.1.3">1</cn></apply></list></apply><apply id="alg1.l12.m1.6.6.2.2.2.cmml" xref="alg1.l12.m1.6.6.2.2.2"><csymbol cd="ambiguous" id="alg1.l12.m1.6.6.2.2.2.1.cmml" xref="alg1.l12.m1.6.6.2.2.2">subscript</csymbol><apply id="alg1.l12.m1.6.6.2.2.2.2.cmml" xref="alg1.l12.m1.6.6.2.2.2.2"><ci id="alg1.l12.m1.6.6.2.2.2.2.1.cmml" xref="alg1.l12.m1.6.6.2.2.2.2.1">^</ci><ci id="alg1.l12.m1.6.6.2.2.2.2.2.cmml" xref="alg1.l12.m1.6.6.2.2.2.2.2">𝑋</ci></apply><list id="alg1.l12.m1.4.4.2.3.cmml" xref="alg1.l12.m1.4.4.2.2"><ci id="alg1.l12.m1.3.3.1.1.cmml" xref="alg1.l12.m1.3.3.1.1">𝑖</ci><apply id="alg1.l12.m1.4.4.2.2.1.cmml" xref="alg1.l12.m1.4.4.2.2.1"><plus id="alg1.l12.m1.4.4.2.2.1.1.cmml" xref="alg1.l12.m1.4.4.2.2.1.1"></plus><ci id="alg1.l12.m1.4.4.2.2.1.2.cmml" xref="alg1.l12.m1.4.4.2.2.1.2">𝑡</ci><cn id="alg1.l12.m1.4.4.2.2.1.3.cmml" type="integer" xref="alg1.l12.m1.4.4.2.2.1.3">1</cn></apply></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l12.m1.6c">L(X_{i,t+1},\hat{X}_{i,t+1})</annotation><annotation encoding="application/x-llamapun" id="alg1.l12.m1.6d">italic_L ( italic_X start_POSTSUBSCRIPT italic_i , italic_t + 1 end_POSTSUBSCRIPT , over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_i , italic_t + 1 end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l13"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l13.1.1.1" style="font-size:80%;">13:</span></span> Update model weights using backpropagation through time (BPTT) </div> <div class="ltx_listingline" id="alg1.l14"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l14.1.1.1" style="font-size:80%;">14:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l14.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l14.3">for</span> </div> <div class="ltx_listingline" id="alg1.l15"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l15.1.1.1" style="font-size:80%;">15:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l15.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l15.3">for</span> </div> <div class="ltx_listingline" id="alg1.l16"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l16.1.1.1" style="font-size:80%;">16:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l16.2">Step 2: Predict Next State in Real-Time</span> </div> <div class="ltx_listingline" id="alg1.l17"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l17.1.1.1" style="font-size:80%;">17:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l17.2">for</span> each node <math alttext="i" class="ltx_Math" display="inline" id="alg1.l17.m1.1"><semantics id="alg1.l17.m1.1a"><mi id="alg1.l17.m1.1.1" xref="alg1.l17.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="alg1.l17.m1.1b"><ci id="alg1.l17.m1.1.1.cmml" xref="alg1.l17.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l17.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="alg1.l17.m1.1d">italic_i</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="alg1.l17.m2.1"><semantics id="alg1.l17.m2.1a"><mi id="alg1.l17.m2.1.1" xref="alg1.l17.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="alg1.l17.m2.1b"><ci id="alg1.l17.m2.1.1.cmml" xref="alg1.l17.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l17.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="alg1.l17.m2.1d">italic_t</annotation></semantics></math> in deployment <span class="ltx_text ltx_font_bold" id="alg1.l17.3">do</span> </div> <div class="ltx_listingline" id="alg1.l18"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l18.1.1.1" style="font-size:80%;">18:</span></span> Retrieve latest position or link state sequence <math alttext="X_{i,t-w:t}" class="ltx_Math" display="inline" id="alg1.l18.m1.2"><semantics id="alg1.l18.m1.2a"><msub id="alg1.l18.m1.2.3" xref="alg1.l18.m1.2.3.cmml"><mi id="alg1.l18.m1.2.3.2" xref="alg1.l18.m1.2.3.2.cmml">X</mi><mrow id="alg1.l18.m1.2.2.2" xref="alg1.l18.m1.2.2.2.cmml"><mrow id="alg1.l18.m1.2.2.2.2.1" xref="alg1.l18.m1.2.2.2.2.2.cmml"><mi id="alg1.l18.m1.1.1.1.1" xref="alg1.l18.m1.1.1.1.1.cmml">i</mi><mo id="alg1.l18.m1.2.2.2.2.1.2" xref="alg1.l18.m1.2.2.2.2.2.cmml">,</mo><mrow id="alg1.l18.m1.2.2.2.2.1.1" xref="alg1.l18.m1.2.2.2.2.1.1.cmml"><mi id="alg1.l18.m1.2.2.2.2.1.1.2" xref="alg1.l18.m1.2.2.2.2.1.1.2.cmml">t</mi><mo id="alg1.l18.m1.2.2.2.2.1.1.1" xref="alg1.l18.m1.2.2.2.2.1.1.1.cmml">−</mo><mi id="alg1.l18.m1.2.2.2.2.1.1.3" xref="alg1.l18.m1.2.2.2.2.1.1.3.cmml">w</mi></mrow></mrow><mo id="alg1.l18.m1.2.2.2.3" lspace="0.278em" rspace="0.278em" xref="alg1.l18.m1.2.2.2.3.cmml">:</mo><mi id="alg1.l18.m1.2.2.2.4" xref="alg1.l18.m1.2.2.2.4.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l18.m1.2b"><apply id="alg1.l18.m1.2.3.cmml" xref="alg1.l18.m1.2.3"><csymbol cd="ambiguous" id="alg1.l18.m1.2.3.1.cmml" xref="alg1.l18.m1.2.3">subscript</csymbol><ci id="alg1.l18.m1.2.3.2.cmml" xref="alg1.l18.m1.2.3.2">𝑋</ci><apply id="alg1.l18.m1.2.2.2.cmml" xref="alg1.l18.m1.2.2.2"><ci id="alg1.l18.m1.2.2.2.3.cmml" xref="alg1.l18.m1.2.2.2.3">:</ci><list id="alg1.l18.m1.2.2.2.2.2.cmml" xref="alg1.l18.m1.2.2.2.2.1"><ci id="alg1.l18.m1.1.1.1.1.cmml" xref="alg1.l18.m1.1.1.1.1">𝑖</ci><apply id="alg1.l18.m1.2.2.2.2.1.1.cmml" xref="alg1.l18.m1.2.2.2.2.1.1"><minus id="alg1.l18.m1.2.2.2.2.1.1.1.cmml" xref="alg1.l18.m1.2.2.2.2.1.1.1"></minus><ci id="alg1.l18.m1.2.2.2.2.1.1.2.cmml" xref="alg1.l18.m1.2.2.2.2.1.1.2">𝑡</ci><ci id="alg1.l18.m1.2.2.2.2.1.1.3.cmml" xref="alg1.l18.m1.2.2.2.2.1.1.3">𝑤</ci></apply></list><ci id="alg1.l18.m1.2.2.2.4.cmml" xref="alg1.l18.m1.2.2.2.4">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l18.m1.2c">X_{i,t-w:t}</annotation><annotation encoding="application/x-llamapun" id="alg1.l18.m1.2d">italic_X start_POSTSUBSCRIPT italic_i , italic_t - italic_w : italic_t end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l19"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l19.1.1.1" style="font-size:80%;">19:</span></span> Feed sequence <math alttext="X_{i,t-w:t}" class="ltx_Math" display="inline" id="alg1.l19.m1.2"><semantics id="alg1.l19.m1.2a"><msub id="alg1.l19.m1.2.3" xref="alg1.l19.m1.2.3.cmml"><mi id="alg1.l19.m1.2.3.2" xref="alg1.l19.m1.2.3.2.cmml">X</mi><mrow id="alg1.l19.m1.2.2.2" xref="alg1.l19.m1.2.2.2.cmml"><mrow id="alg1.l19.m1.2.2.2.2.1" xref="alg1.l19.m1.2.2.2.2.2.cmml"><mi id="alg1.l19.m1.1.1.1.1" xref="alg1.l19.m1.1.1.1.1.cmml">i</mi><mo id="alg1.l19.m1.2.2.2.2.1.2" xref="alg1.l19.m1.2.2.2.2.2.cmml">,</mo><mrow id="alg1.l19.m1.2.2.2.2.1.1" xref="alg1.l19.m1.2.2.2.2.1.1.cmml"><mi id="alg1.l19.m1.2.2.2.2.1.1.2" xref="alg1.l19.m1.2.2.2.2.1.1.2.cmml">t</mi><mo id="alg1.l19.m1.2.2.2.2.1.1.1" xref="alg1.l19.m1.2.2.2.2.1.1.1.cmml">−</mo><mi id="alg1.l19.m1.2.2.2.2.1.1.3" xref="alg1.l19.m1.2.2.2.2.1.1.3.cmml">w</mi></mrow></mrow><mo id="alg1.l19.m1.2.2.2.3" lspace="0.278em" rspace="0.278em" xref="alg1.l19.m1.2.2.2.3.cmml">:</mo><mi id="alg1.l19.m1.2.2.2.4" xref="alg1.l19.m1.2.2.2.4.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l19.m1.2b"><apply id="alg1.l19.m1.2.3.cmml" xref="alg1.l19.m1.2.3"><csymbol cd="ambiguous" id="alg1.l19.m1.2.3.1.cmml" xref="alg1.l19.m1.2.3">subscript</csymbol><ci id="alg1.l19.m1.2.3.2.cmml" xref="alg1.l19.m1.2.3.2">𝑋</ci><apply id="alg1.l19.m1.2.2.2.cmml" xref="alg1.l19.m1.2.2.2"><ci id="alg1.l19.m1.2.2.2.3.cmml" xref="alg1.l19.m1.2.2.2.3">:</ci><list id="alg1.l19.m1.2.2.2.2.2.cmml" xref="alg1.l19.m1.2.2.2.2.1"><ci id="alg1.l19.m1.1.1.1.1.cmml" xref="alg1.l19.m1.1.1.1.1">𝑖</ci><apply id="alg1.l19.m1.2.2.2.2.1.1.cmml" xref="alg1.l19.m1.2.2.2.2.1.1"><minus id="alg1.l19.m1.2.2.2.2.1.1.1.cmml" xref="alg1.l19.m1.2.2.2.2.1.1.1"></minus><ci id="alg1.l19.m1.2.2.2.2.1.1.2.cmml" xref="alg1.l19.m1.2.2.2.2.1.1.2">𝑡</ci><ci id="alg1.l19.m1.2.2.2.2.1.1.3.cmml" xref="alg1.l19.m1.2.2.2.2.1.1.3">𝑤</ci></apply></list><ci id="alg1.l19.m1.2.2.2.4.cmml" xref="alg1.l19.m1.2.2.2.4">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l19.m1.2c">X_{i,t-w:t}</annotation><annotation encoding="application/x-llamapun" id="alg1.l19.m1.2d">italic_X start_POSTSUBSCRIPT italic_i , italic_t - italic_w : italic_t end_POSTSUBSCRIPT</annotation></semantics></math> into trained LSTM model </div> <div class="ltx_listingline" id="alg1.l20"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l20.1.1.1" style="font-size:80%;">20:</span></span> Output prediction <math alttext="\hat{X}_{i,t+1}" class="ltx_Math" display="inline" id="alg1.l20.m1.2"><semantics id="alg1.l20.m1.2a"><msub id="alg1.l20.m1.2.3" xref="alg1.l20.m1.2.3.cmml"><mover accent="true" id="alg1.l20.m1.2.3.2" xref="alg1.l20.m1.2.3.2.cmml"><mi id="alg1.l20.m1.2.3.2.2" xref="alg1.l20.m1.2.3.2.2.cmml">X</mi><mo id="alg1.l20.m1.2.3.2.1" xref="alg1.l20.m1.2.3.2.1.cmml">^</mo></mover><mrow id="alg1.l20.m1.2.2.2.2" xref="alg1.l20.m1.2.2.2.3.cmml"><mi id="alg1.l20.m1.1.1.1.1" xref="alg1.l20.m1.1.1.1.1.cmml">i</mi><mo id="alg1.l20.m1.2.2.2.2.2" xref="alg1.l20.m1.2.2.2.3.cmml">,</mo><mrow id="alg1.l20.m1.2.2.2.2.1" xref="alg1.l20.m1.2.2.2.2.1.cmml"><mi id="alg1.l20.m1.2.2.2.2.1.2" xref="alg1.l20.m1.2.2.2.2.1.2.cmml">t</mi><mo id="alg1.l20.m1.2.2.2.2.1.1" xref="alg1.l20.m1.2.2.2.2.1.1.cmml">+</mo><mn id="alg1.l20.m1.2.2.2.2.1.3" xref="alg1.l20.m1.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l20.m1.2b"><apply id="alg1.l20.m1.2.3.cmml" xref="alg1.l20.m1.2.3"><csymbol cd="ambiguous" id="alg1.l20.m1.2.3.1.cmml" xref="alg1.l20.m1.2.3">subscript</csymbol><apply id="alg1.l20.m1.2.3.2.cmml" xref="alg1.l20.m1.2.3.2"><ci id="alg1.l20.m1.2.3.2.1.cmml" xref="alg1.l20.m1.2.3.2.1">^</ci><ci id="alg1.l20.m1.2.3.2.2.cmml" xref="alg1.l20.m1.2.3.2.2">𝑋</ci></apply><list id="alg1.l20.m1.2.2.2.3.cmml" xref="alg1.l20.m1.2.2.2.2"><ci id="alg1.l20.m1.1.1.1.1.cmml" xref="alg1.l20.m1.1.1.1.1">𝑖</ci><apply id="alg1.l20.m1.2.2.2.2.1.cmml" xref="alg1.l20.m1.2.2.2.2.1"><plus id="alg1.l20.m1.2.2.2.2.1.1.cmml" xref="alg1.l20.m1.2.2.2.2.1.1"></plus><ci id="alg1.l20.m1.2.2.2.2.1.2.cmml" xref="alg1.l20.m1.2.2.2.2.1.2">𝑡</ci><cn id="alg1.l20.m1.2.2.2.2.1.3.cmml" type="integer" xref="alg1.l20.m1.2.2.2.2.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l20.m1.2c">\hat{X}_{i,t+1}</annotation><annotation encoding="application/x-llamapun" id="alg1.l20.m1.2d">over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_i , italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> for position or link state at time <math alttext="t+1" class="ltx_Math" display="inline" id="alg1.l20.m2.1"><semantics id="alg1.l20.m2.1a"><mrow id="alg1.l20.m2.1.1" xref="alg1.l20.m2.1.1.cmml"><mi id="alg1.l20.m2.1.1.2" xref="alg1.l20.m2.1.1.2.cmml">t</mi><mo id="alg1.l20.m2.1.1.1" xref="alg1.l20.m2.1.1.1.cmml">+</mo><mn id="alg1.l20.m2.1.1.3" xref="alg1.l20.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l20.m2.1b"><apply id="alg1.l20.m2.1.1.cmml" xref="alg1.l20.m2.1.1"><plus id="alg1.l20.m2.1.1.1.cmml" xref="alg1.l20.m2.1.1.1"></plus><ci id="alg1.l20.m2.1.1.2.cmml" xref="alg1.l20.m2.1.1.2">𝑡</ci><cn id="alg1.l20.m2.1.1.3.cmml" type="integer" xref="alg1.l20.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l20.m2.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="alg1.l20.m2.1d">italic_t + 1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l21"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l21.1.1.1" style="font-size:80%;">21:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l21.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l21.3">for</span> </div> <div class="ltx_listingline" id="alg1.l22"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l22.1.1.1" style="font-size:80%;">22:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l22.2">Return</span> predicted states <math alttext="\hat{X}_{t+1}" class="ltx_Math" display="inline" id="alg1.l22.m1.1"><semantics id="alg1.l22.m1.1a"><msub id="alg1.l22.m1.1.1" xref="alg1.l22.m1.1.1.cmml"><mover accent="true" id="alg1.l22.m1.1.1.2" xref="alg1.l22.m1.1.1.2.cmml"><mi id="alg1.l22.m1.1.1.2.2" xref="alg1.l22.m1.1.1.2.2.cmml">X</mi><mo id="alg1.l22.m1.1.1.2.1" xref="alg1.l22.m1.1.1.2.1.cmml">^</mo></mover><mrow id="alg1.l22.m1.1.1.3" xref="alg1.l22.m1.1.1.3.cmml"><mi id="alg1.l22.m1.1.1.3.2" xref="alg1.l22.m1.1.1.3.2.cmml">t</mi><mo id="alg1.l22.m1.1.1.3.1" xref="alg1.l22.m1.1.1.3.1.cmml">+</mo><mn id="alg1.l22.m1.1.1.3.3" xref="alg1.l22.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l22.m1.1b"><apply id="alg1.l22.m1.1.1.cmml" xref="alg1.l22.m1.1.1"><csymbol cd="ambiguous" id="alg1.l22.m1.1.1.1.cmml" xref="alg1.l22.m1.1.1">subscript</csymbol><apply id="alg1.l22.m1.1.1.2.cmml" xref="alg1.l22.m1.1.1.2"><ci id="alg1.l22.m1.1.1.2.1.cmml" xref="alg1.l22.m1.1.1.2.1">^</ci><ci id="alg1.l22.m1.1.1.2.2.cmml" xref="alg1.l22.m1.1.1.2.2">𝑋</ci></apply><apply id="alg1.l22.m1.1.1.3.cmml" xref="alg1.l22.m1.1.1.3"><plus id="alg1.l22.m1.1.1.3.1.cmml" xref="alg1.l22.m1.1.1.3.1"></plus><ci id="alg1.l22.m1.1.1.3.2.cmml" xref="alg1.l22.m1.1.1.3.2">𝑡</ci><cn id="alg1.l22.m1.1.1.3.3.cmml" type="integer" xref="alg1.l22.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l22.m1.1c">\hat{X}_{t+1}</annotation><annotation encoding="application/x-llamapun" id="alg1.l22.m1.1d">over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> for all nodes, used in routing decision-making </div> </div> </figure> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS2.4.1.1">IV-B</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS2.5.2">Hierarchical Evolutionary Optimization Framework</span> </h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">The proposed framework uses a population-based evolutionary approach, where candidate paths are evaluated based on delay and stability objectives. By decomposing the multi-objective problem into scalar subproblems with distinct weight vectors, MOEA/D enables exploration of diverse trade-offs between delay and stability. To efficiently adapt to network changes, the framework employs incremental adjustment for affected routes and utilizes a predictive model to anticipate node movements. The proposed algorithm is summarized in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#alg2" title="Algorithm 2 ‣ IV-B Hierarchical Evolutionary Optimization Framework ‣ IV Proposed Algorithm ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg2.2.1.1">Algorithm 2</span> </span> Hierarchical Evolutionary Optimization Framework for Delay-Constrained Routing in Vehicular Networks</figcaption> <div class="ltx_listing ltx_listing" id="alg2.3"> <div class="ltx_listingline" id="alg2.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l1.2">Input:</span> Vehicular network topology <math alttext="G" class="ltx_Math" display="inline" id="alg2.l1.m1.1"><semantics id="alg2.l1.m1.1a"><mi id="alg2.l1.m1.1.1" xref="alg2.l1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="alg2.l1.m1.1b"><ci id="alg2.l1.m1.1.1.cmml" xref="alg2.l1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m1.1d">italic_G</annotation></semantics></math>, source node <math alttext="s" class="ltx_Math" display="inline" id="alg2.l1.m2.1"><semantics id="alg2.l1.m2.1a"><mi id="alg2.l1.m2.1.1" xref="alg2.l1.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="alg2.l1.m2.1b"><ci id="alg2.l1.m2.1.1.cmml" xref="alg2.l1.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m2.1d">italic_s</annotation></semantics></math>, destination node <math alttext="d" class="ltx_Math" display="inline" id="alg2.l1.m3.1"><semantics id="alg2.l1.m3.1a"><mi id="alg2.l1.m3.1.1" xref="alg2.l1.m3.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="alg2.l1.m3.1b"><ci id="alg2.l1.m3.1.1.cmml" xref="alg2.l1.m3.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m3.1c">d</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m3.1d">italic_d</annotation></semantics></math>, population size <math alttext="N" class="ltx_Math" display="inline" id="alg2.l1.m4.1"><semantics id="alg2.l1.m4.1a"><mi id="alg2.l1.m4.1.1" xref="alg2.l1.m4.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="alg2.l1.m4.1b"><ci id="alg2.l1.m4.1.1.cmml" xref="alg2.l1.m4.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m4.1c">N</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m4.1d">italic_N</annotation></semantics></math>, maximum generations <math alttext="T" class="ltx_Math" display="inline" id="alg2.l1.m5.1"><semantics id="alg2.l1.m5.1a"><mi id="alg2.l1.m5.1.1" xref="alg2.l1.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="alg2.l1.m5.1b"><ci id="alg2.l1.m5.1.1.cmml" xref="alg2.l1.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m5.1d">italic_T</annotation></semantics></math>, weight vectors <math alttext="\lambda_{1},\lambda_{2},\dots,\lambda_{N}" class="ltx_Math" display="inline" id="alg2.l1.m6.4"><semantics id="alg2.l1.m6.4a"><mrow id="alg2.l1.m6.4.4.3" xref="alg2.l1.m6.4.4.4.cmml"><msub id="alg2.l1.m6.2.2.1.1" xref="alg2.l1.m6.2.2.1.1.cmml"><mi id="alg2.l1.m6.2.2.1.1.2" xref="alg2.l1.m6.2.2.1.1.2.cmml">λ</mi><mn id="alg2.l1.m6.2.2.1.1.3" xref="alg2.l1.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="alg2.l1.m6.4.4.3.4" xref="alg2.l1.m6.4.4.4.cmml">,</mo><msub id="alg2.l1.m6.3.3.2.2" xref="alg2.l1.m6.3.3.2.2.cmml"><mi id="alg2.l1.m6.3.3.2.2.2" xref="alg2.l1.m6.3.3.2.2.2.cmml">λ</mi><mn id="alg2.l1.m6.3.3.2.2.3" xref="alg2.l1.m6.3.3.2.2.3.cmml">2</mn></msub><mo id="alg2.l1.m6.4.4.3.5" xref="alg2.l1.m6.4.4.4.cmml">,</mo><mi id="alg2.l1.m6.1.1" mathvariant="normal" xref="alg2.l1.m6.1.1.cmml">…</mi><mo id="alg2.l1.m6.4.4.3.6" xref="alg2.l1.m6.4.4.4.cmml">,</mo><msub id="alg2.l1.m6.4.4.3.3" xref="alg2.l1.m6.4.4.3.3.cmml"><mi id="alg2.l1.m6.4.4.3.3.2" xref="alg2.l1.m6.4.4.3.3.2.cmml">λ</mi><mi id="alg2.l1.m6.4.4.3.3.3" xref="alg2.l1.m6.4.4.3.3.3.cmml">N</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m6.4b"><list id="alg2.l1.m6.4.4.4.cmml" xref="alg2.l1.m6.4.4.3"><apply id="alg2.l1.m6.2.2.1.1.cmml" xref="alg2.l1.m6.2.2.1.1"><csymbol cd="ambiguous" id="alg2.l1.m6.2.2.1.1.1.cmml" xref="alg2.l1.m6.2.2.1.1">subscript</csymbol><ci id="alg2.l1.m6.2.2.1.1.2.cmml" xref="alg2.l1.m6.2.2.1.1.2">𝜆</ci><cn id="alg2.l1.m6.2.2.1.1.3.cmml" type="integer" xref="alg2.l1.m6.2.2.1.1.3">1</cn></apply><apply id="alg2.l1.m6.3.3.2.2.cmml" xref="alg2.l1.m6.3.3.2.2"><csymbol cd="ambiguous" id="alg2.l1.m6.3.3.2.2.1.cmml" xref="alg2.l1.m6.3.3.2.2">subscript</csymbol><ci id="alg2.l1.m6.3.3.2.2.2.cmml" xref="alg2.l1.m6.3.3.2.2.2">𝜆</ci><cn id="alg2.l1.m6.3.3.2.2.3.cmml" type="integer" xref="alg2.l1.m6.3.3.2.2.3">2</cn></apply><ci id="alg2.l1.m6.1.1.cmml" xref="alg2.l1.m6.1.1">…</ci><apply id="alg2.l1.m6.4.4.3.3.cmml" xref="alg2.l1.m6.4.4.3.3"><csymbol cd="ambiguous" id="alg2.l1.m6.4.4.3.3.1.cmml" xref="alg2.l1.m6.4.4.3.3">subscript</csymbol><ci id="alg2.l1.m6.4.4.3.3.2.cmml" xref="alg2.l1.m6.4.4.3.3.2">𝜆</ci><ci id="alg2.l1.m6.4.4.3.3.3.cmml" xref="alg2.l1.m6.4.4.3.3.3">𝑁</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m6.4c">\lambda_{1},\lambda_{2},\dots,\lambda_{N}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m6.4d">italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_λ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l2.1.1.1" style="font-size:80%;">2:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l2.2">Output:</span> Optimal routing path <math alttext="P^{*}" class="ltx_Math" display="inline" id="alg2.l2.m1.1"><semantics id="alg2.l2.m1.1a"><msup id="alg2.l2.m1.1.1" xref="alg2.l2.m1.1.1.cmml"><mi id="alg2.l2.m1.1.1.2" xref="alg2.l2.m1.1.1.2.cmml">P</mi><mo id="alg2.l2.m1.1.1.3" xref="alg2.l2.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="alg2.l2.m1.1b"><apply id="alg2.l2.m1.1.1.cmml" xref="alg2.l2.m1.1.1"><csymbol cd="ambiguous" id="alg2.l2.m1.1.1.1.cmml" xref="alg2.l2.m1.1.1">superscript</csymbol><ci id="alg2.l2.m1.1.1.2.cmml" xref="alg2.l2.m1.1.1.2">𝑃</ci><times id="alg2.l2.m1.1.1.3.cmml" xref="alg2.l2.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m1.1c">P^{*}</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m1.1d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l3.1.1.1" style="font-size:80%;">3:</span></span>Initialize population <math alttext="P" class="ltx_Math" display="inline" id="alg2.l3.m1.1"><semantics id="alg2.l3.m1.1a"><mi id="alg2.l3.m1.1.1" xref="alg2.l3.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="alg2.l3.m1.1b"><ci id="alg2.l3.m1.1.1.cmml" xref="alg2.l3.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m1.1d">italic_P</annotation></semantics></math> with <math alttext="N" class="ltx_Math" display="inline" id="alg2.l3.m2.1"><semantics id="alg2.l3.m2.1a"><mi id="alg2.l3.m2.1.1" xref="alg2.l3.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="alg2.l3.m2.1b"><ci id="alg2.l3.m2.1.1.cmml" xref="alg2.l3.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m2.1d">italic_N</annotation></semantics></math> individuals (candidate paths from <math alttext="s" class="ltx_Math" display="inline" id="alg2.l3.m3.1"><semantics id="alg2.l3.m3.1a"><mi id="alg2.l3.m3.1.1" xref="alg2.l3.m3.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="alg2.l3.m3.1b"><ci id="alg2.l3.m3.1.1.cmml" xref="alg2.l3.m3.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m3.1c">s</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m3.1d">italic_s</annotation></semantics></math> to <math alttext="d" class="ltx_Math" display="inline" id="alg2.l3.m4.1"><semantics id="alg2.l3.m4.1a"><mi id="alg2.l3.m4.1.1" xref="alg2.l3.m4.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="alg2.l3.m4.1b"><ci id="alg2.l3.m4.1.1.cmml" xref="alg2.l3.m4.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m4.1c">d</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m4.1d">italic_d</annotation></semantics></math>) using a heuristic strategy </div> <div class="ltx_listingline" id="alg2.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l4.1.1.1" style="font-size:80%;">4:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l4.2">for</span> <math alttext="t=1" class="ltx_Math" display="inline" id="alg2.l4.m1.1"><semantics id="alg2.l4.m1.1a"><mrow id="alg2.l4.m1.1.1" xref="alg2.l4.m1.1.1.cmml"><mi id="alg2.l4.m1.1.1.2" xref="alg2.l4.m1.1.1.2.cmml">t</mi><mo id="alg2.l4.m1.1.1.1" xref="alg2.l4.m1.1.1.1.cmml">=</mo><mn id="alg2.l4.m1.1.1.3" xref="alg2.l4.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l4.m1.1b"><apply id="alg2.l4.m1.1.1.cmml" xref="alg2.l4.m1.1.1"><eq id="alg2.l4.m1.1.1.1.cmml" xref="alg2.l4.m1.1.1.1"></eq><ci id="alg2.l4.m1.1.1.2.cmml" xref="alg2.l4.m1.1.1.2">𝑡</ci><cn id="alg2.l4.m1.1.1.3.cmml" type="integer" xref="alg2.l4.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m1.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m1.1d">italic_t = 1</annotation></semantics></math> to <math alttext="T" class="ltx_Math" display="inline" id="alg2.l4.m2.1"><semantics id="alg2.l4.m2.1a"><mi id="alg2.l4.m2.1.1" xref="alg2.l4.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="alg2.l4.m2.1b"><ci id="alg2.l4.m2.1.1.cmml" xref="alg2.l4.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m2.1d">italic_T</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l4.3">do</span> </div> <div class="ltx_listingline" id="alg2.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l5.1.1.1" style="font-size:80%;">5:</span></span>Evaluate the objectives for each individual <math alttext="P_{i}" class="ltx_Math" display="inline" id="alg2.l5.m1.1"><semantics id="alg2.l5.m1.1a"><msub id="alg2.l5.m1.1.1" xref="alg2.l5.m1.1.1.cmml"><mi id="alg2.l5.m1.1.1.2" xref="alg2.l5.m1.1.1.2.cmml">P</mi><mi id="alg2.l5.m1.1.1.3" xref="alg2.l5.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l5.m1.1b"><apply id="alg2.l5.m1.1.1.cmml" xref="alg2.l5.m1.1.1"><csymbol cd="ambiguous" id="alg2.l5.m1.1.1.1.cmml" xref="alg2.l5.m1.1.1">subscript</csymbol><ci id="alg2.l5.m1.1.1.2.cmml" xref="alg2.l5.m1.1.1.2">𝑃</ci><ci id="alg2.l5.m1.1.1.3.cmml" xref="alg2.l5.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m1.1c">P_{i}</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m1.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="P" class="ltx_Math" display="inline" id="alg2.l5.m2.1"><semantics id="alg2.l5.m2.1a"><mi id="alg2.l5.m2.1.1" xref="alg2.l5.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="alg2.l5.m2.1b"><ci id="alg2.l5.m2.1.1.cmml" xref="alg2.l5.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m2.1d">italic_P</annotation></semantics></math>: <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f_{1}(P_{i})=T(P_{i}(t)),\quad f_{2}(P_{i})=-S(P_{i}(t))" class="ltx_Math" display="block" id="S4.E4.m1.4"><semantics id="S4.E4.m1.4a"><mrow id="S4.E4.m1.4.4.2" xref="S4.E4.m1.4.4.3.cmml"><mrow id="S4.E4.m1.3.3.1.1" xref="S4.E4.m1.3.3.1.1.cmml"><mrow id="S4.E4.m1.3.3.1.1.1" xref="S4.E4.m1.3.3.1.1.1.cmml"><msub id="S4.E4.m1.3.3.1.1.1.3" xref="S4.E4.m1.3.3.1.1.1.3.cmml"><mi id="S4.E4.m1.3.3.1.1.1.3.2" xref="S4.E4.m1.3.3.1.1.1.3.2.cmml">f</mi><mn id="S4.E4.m1.3.3.1.1.1.3.3" xref="S4.E4.m1.3.3.1.1.1.3.3.cmml">1</mn></msub><mo id="S4.E4.m1.3.3.1.1.1.2" xref="S4.E4.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.E4.m1.3.3.1.1.1.1.1" xref="S4.E4.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S4.E4.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.E4.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E4.m1.3.3.1.1.1.1.1.1" xref="S4.E4.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S4.E4.m1.3.3.1.1.1.1.1.1.2" xref="S4.E4.m1.3.3.1.1.1.1.1.1.2.cmml">P</mi><mi id="S4.E4.m1.3.3.1.1.1.1.1.1.3" xref="S4.E4.m1.3.3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.E4.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.E4.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E4.m1.3.3.1.1.3" xref="S4.E4.m1.3.3.1.1.3.cmml">=</mo><mrow id="S4.E4.m1.3.3.1.1.2" xref="S4.E4.m1.3.3.1.1.2.cmml"><mi id="S4.E4.m1.3.3.1.1.2.3" xref="S4.E4.m1.3.3.1.1.2.3.cmml">T</mi><mo id="S4.E4.m1.3.3.1.1.2.2" xref="S4.E4.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="S4.E4.m1.3.3.1.1.2.1.1" xref="S4.E4.m1.3.3.1.1.2.1.1.1.cmml"><mo id="S4.E4.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="S4.E4.m1.3.3.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.E4.m1.3.3.1.1.2.1.1.1" xref="S4.E4.m1.3.3.1.1.2.1.1.1.cmml"><msub id="S4.E4.m1.3.3.1.1.2.1.1.1.2" xref="S4.E4.m1.3.3.1.1.2.1.1.1.2.cmml"><mi 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rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> </div> <div class="ltx_listingline" id="alg2.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l6.1.1.1" style="font-size:80%;">6:</span></span>Each subproblem <math alttext="j" class="ltx_Math" display="inline" id="alg2.l6.m1.1"><semantics id="alg2.l6.m1.1a"><mi id="alg2.l6.m1.1.1" xref="alg2.l6.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="alg2.l6.m1.1b"><ci id="alg2.l6.m1.1.1.cmml" xref="alg2.l6.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l6.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="alg2.l6.m1.1d">italic_j</annotation></semantics></math> is solved by minimizing the scalarized objective: <table class="ltx_equation ltx_eqn_table" id="S4.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> </div> <div class="ltx_listingline" id="alg2.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l7.1.1.1" style="font-size:80%;">7:</span></span>Select parents based on neighborhood solutions, apply crossover and mutation to generate offspring <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="alg2.l7.m1.1"><semantics id="alg2.l7.m1.1a"><msup id="alg2.l7.m1.1.1" xref="alg2.l7.m1.1.1.cmml"><mi id="alg2.l7.m1.1.1.2" xref="alg2.l7.m1.1.1.2.cmml">P</mi><mo id="alg2.l7.m1.1.1.3" xref="alg2.l7.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="alg2.l7.m1.1b"><apply id="alg2.l7.m1.1.1.cmml" xref="alg2.l7.m1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m1.1.1.1.cmml" xref="alg2.l7.m1.1.1">superscript</csymbol><ci id="alg2.l7.m1.1.1.2.cmml" xref="alg2.l7.m1.1.1.2">𝑃</ci><ci id="alg2.l7.m1.1.1.3.cmml" xref="alg2.l7.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l7.m1.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="alg2.l7.m1.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l8.1.1.1" style="font-size:80%;">8:</span></span>Perform a local search on selected individuals to refine convergence </div> <div class="ltx_listingline" id="alg2.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l9.1.1.1" style="font-size:80%;">9:</span></span>Merge the parent and offspring populations and select the top <math alttext="N" class="ltx_Math" display="inline" id="alg2.l9.m1.1"><semantics id="alg2.l9.m1.1a"><mi id="alg2.l9.m1.1.1" xref="alg2.l9.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="alg2.l9.m1.1b"><ci id="alg2.l9.m1.1.1.cmml" xref="alg2.l9.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l9.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="alg2.l9.m1.1d">italic_N</annotation></semantics></math> individuals based on objective values for the next generation </div> <div class="ltx_listingline" id="alg2.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l10.1.1.1" style="font-size:80%;">10:</span></span> <span class="ltx_text ltx_font_bold" id="alg2.l10.2">if</span> network topology changes during iteration <span class="ltx_text ltx_font_bold" id="alg2.l10.3">then</span> </div> <div class="ltx_listingline" id="alg2.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l11.1.1.1" style="font-size:80%;">11:</span></span>Adjust only affected individuals in <math alttext="P" class="ltx_Math" display="inline" id="alg2.l11.m1.1"><semantics id="alg2.l11.m1.1a"><mi id="alg2.l11.m1.1.1" xref="alg2.l11.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="alg2.l11.m1.1b"><ci id="alg2.l11.m1.1.1.cmml" xref="alg2.l11.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l11.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="alg2.l11.m1.1d">italic_P</annotation></semantics></math> rather than re-optimizing the entire population </div> <div class="ltx_listingline" id="alg2.l12"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l12.1.1.1" style="font-size:80%;">12:</span></span>Use predictive models (e.g., LSTM in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#alg1" title="Algorithm 1 ‣ IV-A LSTM-Based Prediction Model for Dynamic Network Forecasting ‣ IV Proposed Algorithm ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_tag">1</span></a>) to forecast vehicle movements and adjust affected routes to maintain stability and low delay </div> <div class="ltx_listingline" id="alg2.l13"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l13.1.1.1" style="font-size:80%;">13:</span></span> <span class="ltx_text ltx_font_bold" id="alg2.l13.2">end</span> <span class="ltx_text ltx_font_bold" id="alg2.l13.3">if</span> </div> <div class="ltx_listingline" id="alg2.l14"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l14.1.1.1" style="font-size:80%;">14:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l14.2">end</span> <span class="ltx_text ltx_font_bold" id="alg2.l14.3">for</span> </div> <div class="ltx_listingline" id="alg2.l15"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l15.1.1.1" style="font-size:80%;">15:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l15.2">Return</span> the best individual <math alttext="P^{*}" class="ltx_Math" display="inline" id="alg2.l15.m1.1"><semantics id="alg2.l15.m1.1a"><msup id="alg2.l15.m1.1.1" xref="alg2.l15.m1.1.1.cmml"><mi id="alg2.l15.m1.1.1.2" xref="alg2.l15.m1.1.1.2.cmml">P</mi><mo id="alg2.l15.m1.1.1.3" xref="alg2.l15.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="alg2.l15.m1.1b"><apply id="alg2.l15.m1.1.1.cmml" xref="alg2.l15.m1.1.1"><csymbol cd="ambiguous" id="alg2.l15.m1.1.1.1.cmml" xref="alg2.l15.m1.1.1">superscript</csymbol><ci id="alg2.l15.m1.1.1.2.cmml" xref="alg2.l15.m1.1.1.2">𝑃</ci><times id="alg2.l15.m1.1.1.3.cmml" xref="alg2.l15.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l15.m1.1c">P^{*}</annotation><annotation encoding="application/x-llamapun" id="alg2.l15.m1.1d">italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> based on final population evaluation </div> </div> </figure> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">The proposed algorithm employs a structured methodology to address routing stability and delay in a dynamic vehicular network environment. The framework consists of the following key components:</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.3"><span class="ltx_text ltx_font_bold" id="S4.SS2.p3.3.1">1. Objective Evaluation:</span> Each individual path <math alttext="P_{i}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><msub id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2" xref="S4.SS2.p3.1.m1.1.1.2.cmml">P</mi><mi id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p3.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.1.2">𝑃</ci><ci id="S4.SS2.p3.1.m1.1.1.3.cmml" xref="S4.SS2.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">P_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is evaluated based on delay <math alttext="T(P_{i})" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><mrow id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml"><mi id="S4.SS2.p3.2.m2.1.1.3" xref="S4.SS2.p3.2.m2.1.1.3.cmml">T</mi><mo id="S4.SS2.p3.2.m2.1.1.2" xref="S4.SS2.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.2.m2.1.1.1.1" xref="S4.SS2.p3.2.m2.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S4.SS2.p3.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.2.m2.1.1.1.1.1" xref="S4.SS2.p3.2.m2.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.2.m2.1.1.1.1.1.2" xref="S4.SS2.p3.2.m2.1.1.1.1.1.2.cmml">P</mi><mi id="S4.SS2.p3.2.m2.1.1.1.1.1.3" xref="S4.SS2.p3.2.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS2.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S4.SS2.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><apply id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1"><times id="S4.SS2.p3.2.m2.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1.2"></times><ci id="S4.SS2.p3.2.m2.1.1.3.cmml" xref="S4.SS2.p3.2.m2.1.1.3">𝑇</ci><apply id="S4.SS2.p3.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1.1.2">𝑃</ci><ci id="S4.SS2.p3.2.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">T(P_{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">italic_T ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and stability <math alttext="S(P_{i})" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><mrow id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml"><mi id="S4.SS2.p3.3.m3.1.1.3" xref="S4.SS2.p3.3.m3.1.1.3.cmml">S</mi><mo id="S4.SS2.p3.3.m3.1.1.2" xref="S4.SS2.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p3.3.m3.1.1.1.1" xref="S4.SS2.p3.3.m3.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.3.m3.1.1.1.1.2" stretchy="false" xref="S4.SS2.p3.3.m3.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.3.m3.1.1.1.1.1" xref="S4.SS2.p3.3.m3.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.3.m3.1.1.1.1.1.2" xref="S4.SS2.p3.3.m3.1.1.1.1.1.2.cmml">P</mi><mi id="S4.SS2.p3.3.m3.1.1.1.1.1.3" xref="S4.SS2.p3.3.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS2.p3.3.m3.1.1.1.1.3" stretchy="false" xref="S4.SS2.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m3.1b"><apply id="S4.SS2.p3.3.m3.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1"><times id="S4.SS2.p3.3.m3.1.1.2.cmml" xref="S4.SS2.p3.3.m3.1.1.2"></times><ci id="S4.SS2.p3.3.m3.1.1.3.cmml" xref="S4.SS2.p3.3.m3.1.1.3">𝑆</ci><apply id="S4.SS2.p3.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.3.m3.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.3.m3.1.1.1.1.1.2">𝑃</ci><ci id="S4.SS2.p3.3.m3.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.3.m3.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m3.1c">S(P_{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m3.1d">italic_S ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> objectives. This step integrates real-time network conditions, allowing continuous recalculations of these metrics throughout each generation, ensuring responsiveness to environmental changes.</p> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.1"><span class="ltx_text ltx_font_bold" id="S4.SS2.p4.1.1">2. MOEA/D-Based Decomposition:</span> The multi-objective optimization problem is decomposed into a series of scalar subproblems, each with a unique weight vector <math alttext="\lambda_{j}" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><msub id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.2.cmml">λ</mi><mi id="S4.SS2.p4.1.m1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.1.m1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p4.1.m1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.2">𝜆</ci><ci id="S4.SS2.p4.1.m1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">\lambda_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. This decomposition enables the algorithm to explore a diverse set of routing solutions, effectively balancing between delay minimization and stability maximization.</p> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.1"><span class="ltx_text ltx_font_bold" id="S4.SS2.p5.1.1">3. Selection, Crossover, and Mutation:</span> The algorithm applies evolutionary operators to generate offspring solutions, promoting comprehensive exploration within the solution space. Parent selection is influenced by neighborhood solutions to reinforce local exploitation, while crossover and mutation introduce essential variability, supporting adaptation to dynamic network topologies.</p> </div> <div class="ltx_para" id="S4.SS2.p6"> <p class="ltx_p" id="S4.SS2.p6.1"><span class="ltx_text ltx_font_bold" id="S4.SS2.p6.1.1">4. Local Search Enhancement:</span> To expedite convergence, a local search process refines selected individuals, facilitating rapid movement towards high-quality solutions and enhancing overall optimization efficiency.</p> </div> <div class="ltx_para" id="S4.SS2.p7"> <p class="ltx_p" id="S4.SS2.p7.1"><span class="ltx_text ltx_font_bold" id="S4.SS2.p7.1.1">5. Handling Network Dynamics:</span></p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.1"><span class="ltx_text ltx_font_bold" id="S4.I1.i1.p1.1.1">Incremental Adjustment:</span> In response to topology changes, only the affected individuals in the population are adjusted, rather than re-evaluating the entire population, conserving computational resources.</p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.1"><span class="ltx_text ltx_font_bold" id="S4.I1.i2.p1.1.1">Predictive Modeling:</span> The framework incorporates predictive models, particularly Long Short-Term Memory (LSTM) networks, to forecast node movements. This proactive adjustment helps maintain stable routes and reduces communication delay, enabling the algorithm to adapt efficiently to frequently changing network environments.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S4.SS2.p8"> <p class="ltx_p" id="S4.SS2.p8.1">Through this structured approach, the proposed algorithm optimally balances delay and stability objectives, achieving efficient routing performance in vehicular networks, even under dynamic conditions.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Simulation and Experiments</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">To evaluate the proposed framework, we simulate an emergency response scenario in an intelligent transportation system, where low-delay, high-stability routing is essential for emergency vehicles (e.g., ambulances or fire trucks) navigating through a dynamic urban network. The system leverages MOEA/D for multi-objective optimization, minimizing communication delay and maximizing route stability, while an LSTM model predicts future node positions and link stability, enabling proactive routing adjustments. Historical data is used to train the LSTM model, which subsequently guides real-time path selection by forecasting potential link disruptions. This approach ensures the emergency vehicle can dynamically adapt to network changes, maintaining an optimal route that balances speed and reliability.</p> </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S5.SS1.4.1.1">V-A</span> </span><span class="ltx_text ltx_font_italic" id="S5.SS1.5.2">Simulation Scenario</span> </h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.1">To validate the effectiveness of the proposed framework, we design a simulation environment representing a dynamic urban vehicular network focused on emergency response routing. In this scenario, an emergency vehicle is tasked with reaching a designated destination in a city grid network, navigating through highly dynamic traffic conditions typical of urban environments.</p> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.1">The network consists of a predefined number of nodes representing intersections and vehicles, with links representing potential routes between them. Nodes and links experience frequent changes in connectivity due to varying traffic density, vehicle speed, and movement patterns, replicating real-world conditions. The emergency vehicle’s objective is to minimize overall travel delay while maximizing route stability, crucial for reliable and timely arrival at the destination.</p> </div> <div class="ltx_para" id="S5.SS1.p3"> <p class="ltx_p" id="S5.SS1.p3.1">The MOEA/D algorithm performs multi-objective optimization, dynamically selecting the optimal path based on communication delay and predicted link stability. To enhance adaptability, an LSTM model trained on historical traffic data predicts future node positions and link states, allowing the framework to proactively adjust routing decisions in response to predicted changes. Key parameters for the simulation include:</p> <ul class="ltx_itemize" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.2"><span class="ltx_text ltx_font_bold" id="S5.I1.i1.p1.2.1">Network Size and Density</span>: The urban grid network consists of <math alttext="N" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><mi id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><ci id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_N</annotation></semantics></math> intersections and <math alttext="M" class="ltx_Math" display="inline" id="S5.I1.i1.p1.2.m2.1"><semantics id="S5.I1.i1.p1.2.m2.1a"><mi id="S5.I1.i1.p1.2.m2.1.1" xref="S5.I1.i1.p1.2.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.2.m2.1b"><ci id="S5.I1.i1.p1.2.m2.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.2.m2.1c">M</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.2.m2.1d">italic_M</annotation></semantics></math> vehicles, with varying densities to assess the framework’s adaptability under different traffic conditions.</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.1"><span class="ltx_text ltx_font_bold" id="S5.I1.i2.p1.1.1">Communication Delay and Link Stability Metrics</span>: Delay between nodes is based on real-time traffic and distance, while link stability is affected by vehicle speed and relative positions.</p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.1"><span class="ltx_text ltx_font_bold" id="S5.I1.i3.p1.1.1">Prediction Window Size for LSTM</span>: The LSTM model utilizes a window of historical data to forecast short-term link stability and node positions, facilitating proactive routing.</p> </div> </li> </ul> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S5.SS2.4.1.1">V-B</span> </span><span class="ltx_text ltx_font_italic" id="S5.SS2.5.2">Simulation Results and Analysis</span> </h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.1">In the experiment, a desktop computer with an Intel 13790F CPU, 32GB of memory, and an RTX 4070ti GPU running Windows OS was used. The programming language was MATLAB. We first conducted the LSTM network training, where the batch size was set to 10, and the maximum number of epochs was 20. The training process involved approximately 9000 iterations, and the training results are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S5.F1" title="Figure 1 ‣ V-B Simulation Results and Analysis ‣ V Simulation and Experiments ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_tag">1</span></a>. The training error is presented in a logarithmic scale. The training process took approximately 20 seconds to complete.</p> </div> <figure class="ltx_figure" id="S5.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="623" id="S5.F1.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Training error of the LSTM model</figcaption> </figure> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.1">The simulation evaluates the framework’s performance by measuring travel time, routing stability, and adaptation speed to dynamic changes, highlighting its effectiveness in maintaining low-delay, high-stability paths under challenging conditions. Among all solutions, after 25 experiments, the average travel time and network stability score across all individuals over 25 runs were 2.13 seconds and 0.72, respectively. A randomly selected Pareto front from the solution set is presented in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12050v1#S5.F2" title="Figure 2 ‣ V-B Simulation Results and Analysis ‣ V Simulation and Experiments ‣ Hierarchical Evolutionary Optimization with Predictive Modeling for Stable Delay-Constrained Routing in Vehicular Networks This work is supported by the National Natural Science Foundation of China under Grant Number 62273263, 72171172 and 71771176; Shanghai Municipal Science and Technology Major Project (2022-5-YB-09); Natural Science Foundation of Shanghai under Grant Number 23ZR1465400."><span class="ltx_text ltx_ref_tag">2</span></a>. From the figure, we can observe that MOEA/D effectively finds the Pareto optimal solutions balancing travel time and network stability. Furthermore, the Pareto front has a good distribution, providing a strong basis for decision-making.</p> </div> <figure class="ltx_figure" id="S5.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="623" id="S5.F2.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Pareto Sets with Travel Times and Stability Score</figcaption> </figure> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span><span class="ltx_text ltx_font_smallcaps" id="S6.1.1">Conclusion</span> </h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">In this paper, we presented a hierarchical evolutionary optimization framework tailored for delay-constrained routing in dynamic vehicular networks. By leveraging the Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), the framework effectively balances two key objectives: minimizing communication delay and maximizing routing stability. To address the inherent challenges of frequent network topology changes, we incorporated both incremental adjustments for affected routes and predictive modeling techniques, including LSTM networks, to forecast vehicle movements and proactively optimize routing paths.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.1">Our simulation results demonstrate that the proposed framework achieves low-delay, high-stability routing under dynamic conditions, which is crucial for applications like emergency response in intelligent transportation systems. The framework’s use of MOEA/D ensures diverse Pareto optimal solutions, offering a robust foundation for decision-making. Additionally, the adaptability of the LSTM-based prediction model enhances the framework’s capability to handle real-time changes in network topology, reducing the need for continuous re-optimization.</p> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">Overall, the proposed framework provides an effective solution for routing in vehicular networks, demonstrating significant improvements in routing performance through optimized travel time and stability. Future work could explore further integration of other predictive models to refine stability assessments and adapt the framework for more complex and varied vehicular scenarios.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> J. Zhu and K. 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