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<front>
<journal-meta>
<journal-id journal-id-type="pmc">CMC</journal-id>
<journal-id journal-id-type="nlm-ta">CMC</journal-id>
<journal-id journal-id-type="publisher-id">CMC</journal-id>
<journal-title-group>
<journal-title>Computers, Materials &#x0026; Continua</journal-title>
</journal-title-group>
<issn pub-type="epub">1546-2226</issn>
<issn pub-type="ppub">1546-2218</issn>
<publisher>
<publisher-name>Tech Science Press</publisher-name>
<publisher-loc>USA</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">31962</article-id>
<article-id pub-id-type="doi">10.32604/cmc.2023.031962</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Article</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>Metaheuristics-based Clustering with Routing Technique for Lifetime Maximization in Vehicular Networks</article-title>
<alt-title alt-title-type="left-running-head">Metaheuristics-based Clustering with Routing Technique for Lifetime Maximization in Vehicular Networks</alt-title>
<alt-title alt-title-type="right-running-head">Metaheuristics-based Clustering with Routing Technique for Lifetime Maximization in Vehicular Networks</alt-title>
</title-group>
<contrib-group content-type="authors">
<contrib id="author-1" contrib-type="author" corresp="yes">
<name name-style="western"><surname>Muthukrishnan</surname><given-names>P.</given-names></name><email>muthueng0@gmail.com</email></contrib>
<contrib id="author-2" contrib-type="author">
<name name-style="western"><surname>Muthu Kannan</surname><given-names>P.</given-names></name></contrib>
<aff id="aff-1"><institution>Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University</institution>, <addr-line>Saveetha Nagar, Thandalam, Chennai, 602105, Tamilnadu</addr-line>, <country>India</country></aff>
</contrib-group>
<author-notes>
<corresp id="cor1"><label>&#x002A;</label>Corresponding Author: P. Muthukrishnan. Email: <email>muthueng0@gmail.com</email></corresp>
</author-notes>
<pub-date pub-type="epub" date-type="pub" iso-8601-date="2022-08-16"><day>16</day>
<month>08</month>
<year>2022</year></pub-date>
<volume>74</volume>
<issue>1</issue>
<fpage>1107</fpage>
<lpage>1122</lpage>
<history>
<date date-type="received"><day>01</day><month>5</month><year>2022</year></date>
<date date-type="accepted"><day>08</day><month>6</month><year>2022</year></date>
</history>
<permissions>
<copyright-statement>&#x00A9; 2023 Muthukrishnan and Muthu Kannan</copyright-statement>
<copyright-year>2023</copyright-year>
<copyright-holder>Muthukrishnan and Muthu Kannan</copyright-holder>
<license xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>This work is licensed under a <ext-link ext-link-type="uri" xlink:type="simple" xlink:href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</ext-link>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</license-p>
</license>
</permissions>
<self-uri content-type="pdf" xlink:href="TSP_CMC_31962.pdf"></self-uri>
<abstract>
<p>Recently, vehicular ad hoc networks (VANETs) finds applicability in different domains such as security, rescue operations, intelligent transportation systems (ITS), etc. VANET has unique features like high mobility, limited mobility patterns, adequate topology modifications, and wireless communication. Despite the benefits of VANET, scalability is a challenging issue which could be addressed by the use of cluster-based routing techniques. It enables the vehicles to perform intercluster communication via chosen CHs and optimal routes. The main drawback of VANET network is the network unsteadiness that results in minimum lifetime. In order to avoid reduced network lifetime in VANET, this paper presents an enhanced metaheuristics based clustering with multihop routing technique for lifetime maximization (EMCMHR-LM) in VANET. The presented EMCMHR-LM model involves the procedure of arranging clusters, cluster head (CH) selection, and route selection appropriate for VANETs. The presented EMCMHR-LM model uses slime mold optimization based clustering (SMO-C) technique to group the vehicles into clusters. Besides, an enhanced wild horse optimization based multihop routing (EWHO-MHR) protocol by the optimization of network parameters. The presented EMCMHR-LM model is simulated using Network Simulator (NS3) tool and the simulation outcomes reported the enhanced performance of the proposed EMCMHR-LM technique over the other models.</p>
</abstract>
<kwd-group kwd-group-type="author">
<kwd>Scalability</kwd>
<kwd>VANET</kwd>
<kwd>clustering</kwd>
<kwd>multihop routing</kwd>
<kwd>metaheuristics</kwd>
<kwd>route selection</kwd>
<kwd>fitness function</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="s1"><label>1</label><title>Introduction</title>
<p>The Vehicular Ad hoc Networks (VANETs) are regarded in the subsection of the conventional structure of the Mobile Adhoc Network system (MANETs) [<xref ref-type="bibr" rid="ref-1">1</xref>]. The important character of VANET indicates the telecommunication nodes mention the vehicles which were granted with critical &#x2018;on-board unit&#x2019; apparatus, transportable in limited roads or lanes, and interacting with one another for swapping messages via the Vehicle to Vehicle (V2V) protocol of transmission, that describes the extent amongst the vehicles and standard roadside accessible points [<xref ref-type="bibr" rid="ref-2">2</xref>]. This extent explanation further inculcates the wireless network and mobile substructure, in those cases where V2V form of information transfer is done [<xref ref-type="bibr" rid="ref-3">3</xref>]. Even though there comes numerous resemblance among VANETs and MANETs, the particular features of the vehicle motion models cause VANETs to have higher difficulties and issues [<xref ref-type="bibr" rid="ref-4">4</xref>]. For example, the higher mobility of vehicles creates recurrent differences in the network topology in VANETs, paving the way to have complications in routing table maintenance and worst node connectivity [<xref ref-type="bibr" rid="ref-5">5</xref>]. The vehicle could just move in a single direction and might be met with emergency accidents or traffic jams, such situations would make the transmission position among vehicle nodes to be very complex. Over several decades, several authors make much effort and contributing to the model of routing protocol for VANETs and suggested distinct routing methods [<xref ref-type="bibr" rid="ref-6">6</xref>]. Few are categorized into broadcast-related, topology-related, position-related, infrastructure-related, and cluster-related. By comparing, location-related routing denotes a hopeful scheme in vehicular circumstances as it assists geographical location data of every vehicle to provide routing [<xref ref-type="bibr" rid="ref-7">7</xref>]. Whenever a source node initiates to transmit packets towards the destination node, the route path cannot be fixed due to the relay node is computed from the position information of itself, neighbors, and the destination. Nodes do not sustain the routing table and it does not accomplish the global path amongst destination and source [<xref ref-type="bibr" rid="ref-8">8</xref>]. This routing scheme could implement recurrent frequent topology variations in a better way. But, such location-oriented routing protocols contain some restrictions. The node&#x2019;s connectivity is not considered in the routing method that is amplified in insufficient cases.</p>
<p>For addressing such issue, numerous new routing methods were suggested. Few utilize Road Side Units (RSU) for making statistics on the road situations, like road connectivity traffic volume, and so on. In general terms, the prevailing routing schemes are divided into a couple of groups depending on their route-finding procedure and substructure [<xref ref-type="bibr" rid="ref-9">9</xref>]. But every group has particular restrictions in several aspects. Particularly, linear routing schemes or the non-hierarchical executions degrade in very dense network systems in relation to transmission and computational cost. Therefore, cluster related routing protocols are presented to optimal manage vehicles to groups depending on their similarity, targeting to minimize the computational overhead through choosing a maximum sum of clusters [<xref ref-type="bibr" rid="ref-10">10</xref>]. Distinct protocols leverage several variables for the selection of cluster head (CH) and cluster members in the literature, like velocity, travel time, ID, link reliability, route confidence level, degree of neighbors connectivity, average link lifetime, propagation delay, average relative velocity, total quantity of vehicles to pursue, and so on.</p>
<p>This paper presents an enhanced metaheuristics based clustering with multihop routing technique for lifetime maximization (EMCMHR-LM) in VANET. The presented EMCMHR-LM model involves the procedure of arranging clusters, cluster head (CH) selection, and route selection appropriate for VANETs. The presented EMCMHR-LM model uses slime moldoptimization based clustering (SMO-C) technique to group the vehicles into clusters. Besides, an enhanced wild horse optimization based multihop routing (EWHO-MHR) protocol by the optimization of network parameters. The presented EMCMHR-LM model is simulated using Network Simulator (NS3) tool.</p>
</sec>
<sec id="s2"><label>2</label><title>Related Works</title>
<p>ChandrenMuniyandi&#x00A0;et&#x00A0;al.&#x00A0;[<xref ref-type="bibr" rid="ref-11">11</xref>] presented an enhanced harmony search optimization (EHSO) system which assumes the outline of the OLSR variables by pairing 2 phases, a course for optimization take place by the EHSO methodology depends on implanting 2 famous selection techniques in its storage, like, tournament selection roulette and wheel selection. In [<xref ref-type="bibr" rid="ref-12">12</xref>], a hybrid clustering system is suggested that complies with the context related geographic related clustering techniques. The hybrid clustering minimizes the traffic and controls overhead in the networks. Secondly, the destination aware routing protocol is suggested for inter clustering routing that enhances the complete packet delivery ratio and minimizes the endwise postponement. In [<xref ref-type="bibr" rid="ref-13">13</xref>], a clustering routing protocol, termed QMM-VANET, that assumes Quality of Service (QoS) needs, the distrust value variables, and mobility restraints, is suggested. This protocol indicates a dependable and standard cluster and rises the durability and connectedness at the time of transmissions. Javadpour&#x00A0;et&#x00A0;al.&#x00A0;[<xref ref-type="bibr" rid="ref-14">14</xref>] decrease the impacts of unexpected issues on the best track for replacing the broken link or path. A QoSR having Particle Swarm Optimization (QoSR-PSO) is utilized in this work to enhance QoSs in vehicular ad hoc network systems. By shaping the attitude of a set of particles, particle swarm optimization methods discover the superior feasible resolution to the issues. Ardakani [<xref ref-type="bibr" rid="ref-15">15</xref>] developed a cluster related routing protocol for VANET using an innovative addressing scheme in that every node receives an address respective to their mobility paradigm. Hamming distance approach is utilized after making the separation of the network in an address centric mode. Sangeetha&#x00A0;et&#x00A0;al.&#x00A0;[<xref ref-type="bibr" rid="ref-16">16</xref>] suggested an unclear genetic optimization approach for VANETs. The node clustering, base station movement, and cluster head selection is depends on unclear sense. Moreover, genetic method is utilized for finding the superior feasible routing route.</p>
<p>The researchers in [<xref ref-type="bibr" rid="ref-17">17</xref>] suggested a cluster related method for enhancing data communication. In this algorithm, an original technique is recommended for selecting the CH. The CH selection method is carried out in 3 stages, initial stage is normal CH selection, next stage is cluster leaving and final stage is cluster merging. Afterward selecting the CH, the routing path must be accomplished on the basis of the link quality that can be measured with the help of the link life period. In [<xref ref-type="bibr" rid="ref-18">18</xref>], a hierarchical method was devised for managing huge VANETs by offering superior mobility management. The creation of manifold overlay clusters from huge VANETs leveraging k-means method is main feature of this technique. Moreover, an addressing infrastructure was launched by 2 data registers.</p>
</sec>
<sec id="s3"><label>3</label><title>The Proposed Model</title>
<p>In this study, a new EMCMHR-LM model was enhanced to resolve scalability issues and increase lifetime of VANET. The presented EMCMHR-LM model involves the procedure of arranging clusters, CH selection, and route selection appropriate for VANETs. At the initial stage, the presented EMCMHR-LM model uses SMO-C technique to group the vehicles into clusters. In addition, the WHO-MHR protocol has been developed by the optimization of network parameters. <xref ref-type="fig" rid="fig-1">Fig. 1</xref> depicts the overall process of EMCMHR-LM technique.</p>
<fig id="fig-1"><label>Figure 1</label><caption><title>Overall process of EMCMHR-LM technique</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-1.png"/></fig>
<sec id="s3_1"><label>3.1</label><title>Overview of SMO Algorithm</title>
<p>The SMO algorithm has been developed depending upon the oscillation mode of slime mould in nature. Consider <italic>N</italic> slime mould is existing in the searching space with lower boundary <inline-formula id="ieqn-1"><mml:math id="mml-ieqn-1"><mml:mrow><mml:mo>(</mml:mo><mml:mi>L</mml:mi><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and upper boundary <inline-formula id="ieqn-2"><mml:math id="mml-ieqn-2"><mml:mrow><mml:mo>(</mml:mo><mml:mi>U</mml:mi><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. Next, <inline-formula id="ieqn-3"><mml:math id="mml-ieqn-3"><mml:mi>i</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-4"><mml:math id="mml-ieqn-4"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> slime moulds located in <inline-formula id="ieqn-5"><mml:math id="mml-ieqn-5"><mml:mi>d</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-6"><mml:math id="mml-ieqn-6"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> dimension are formulated by <inline-formula id="ieqn-7"><mml:math id="mml-ieqn-7"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="normal">&#x2200;</mml:mi><mml:mi>i</mml:mi><mml:mo>&#x2208;</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>, and fitness (odor) of <inline-formula id="ieqn-8"><mml:math id="mml-ieqn-8"><mml:mi>i</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-9"><mml:math id="mml-ieqn-9"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> slime is characterized by <inline-formula id="ieqn-10"><mml:math id="mml-ieqn-10"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="normal">&#x2200;</mml:mi><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>]</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></inline-formula> Hence, the fitness and position of <italic>N</italic> slime mould at <italic>t</italic> time is formulated by using <xref ref-type="disp-formula" rid="eqn-1">Eq. (1)</xref>:
<disp-formula id="eqn-1"><label>(1)</label><mml:math id="mml-eqn-1" display="block"><mml:mi>X</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mo>&#x22EF;</mml:mo></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mo>&#x22EF;</mml:mo></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>&#x22EE;</mml:mo></mml:mtd><mml:mtd><mml:mo>&#x22EE;</mml:mo></mml:mtd><mml:mtd><mml:mo>&#x22EE;</mml:mo></mml:mtd><mml:mtd><mml:mo>&#x22EE;</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mo>&#x22EF;</mml:mo></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mtable rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>&#x22EE;</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo>]</mml:mo></mml:mrow></mml:math></disp-formula>
<disp-formula id="eqn-2"><label>(2)</label><mml:math id="mml-eqn-2" display="block"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>The location of slime mould for the second iteration <inline-formula id="ieqn-11"><mml:math id="mml-ieqn-11"><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> in SMO algorithm can be upgraded as follows [<xref ref-type="bibr" rid="ref-19">19</xref>]:
<disp-formula id="eqn-3"><label>(3)</label><mml:math id="mml-eqn-3" display="block"><mml:mtable columnalign='left'><mml:mtr><mml:mtd><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo> <mml:mrow><mml:mtable columnalign='left'><mml:mtr columnalign='left'><mml:mtd columnalign='left'><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>W</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign='left'><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2265;</mml:mo><mml:mi>&#x03B4;</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:msub><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x003C;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign='left'><mml:mtd columnalign='left'><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign='left'><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2265;</mml:mo><mml:mi>&#x03B4;</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:msub><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign='left'><mml:mtd columnalign='left'><mml:mrow><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:mtext>&#x2009;</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>U</mml:mi><mml:mi>B</mml:mi><mml:mtext>---</mml:mtext><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign='left'><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mi>&#x03B4;</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow> </mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;</mml:mtext><mml:mo>&#x2200;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2208;</mml:mo><mml:mrow><mml:mo>[</mml:mo> <mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow> <mml:mo>]</mml:mo></mml:mrow><mml:mtext>&#x00A0;&#x00A0;</mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-3">Eq. (3)</xref>, The local optimal individual for the existing iteration is represented as <inline-formula id="ieqn-12"><mml:math id="mml-ieqn-12"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, randomly pooled slime mould from current population is denoted by <inline-formula id="ieqn-13"><mml:math id="mml-ieqn-13"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-14"><mml:math id="mml-ieqn-14"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, <inline-formula id="ieqn-15"><mml:math id="mml-ieqn-15"><mml:mi>W</mml:mi></mml:math></inline-formula>indicates the weight factor, <inline-formula id="ieqn-16"><mml:math id="mml-ieqn-16"><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-17"><mml:math id="mml-ieqn-17"><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> denotes the arbitrary velocity. The <inline-formula id="ieqn-18"><mml:math id="mml-ieqn-18"><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-19"><mml:math id="mml-ieqn-19"><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> are the two arbitrary values lie within <inline-formula id="ieqn-20"><mml:math id="mml-ieqn-20"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>. <inline-formula id="ieqn-21"><mml:math id="mml-ieqn-21"><mml:mi>&#x03B4;</mml:mi></mml:math></inline-formula>represent the probability of slime mould that initialized to an arbitrary searching position that is set to 0.03, <inline-formula id="ieqn-22"><mml:math id="mml-ieqn-22"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>indicates the thresholding value of <inline-formula id="ieqn-23"><mml:math id="mml-ieqn-23"><mml:mi>i</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-24"><mml:math id="mml-ieqn-24"><mml:mspace width="thickmathspace" /><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> slime moulds that assist in choosing the slime mould location with the optimal individuals as follows:
<disp-formula id="eqn-4"><label>(4)</label><mml:math id="mml-eqn-4" display="block"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mspace width="thickmathspace" /></mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo>|</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="normal">&#x2200;</mml:mi><mml:mi>i</mml:mi><mml:mo>&#x2208;</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-4">Eq. (4)</xref>, <inline-formula id="ieqn-25"><mml:math id="mml-ieqn-25"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>represent the fitness value of <inline-formula id="ieqn-26"><mml:math id="mml-ieqn-26"><mml:mi>i</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-27"><mml:math id="mml-ieqn-27"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> slime moulds <inline-formula id="ieqn-28"><mml:math id="mml-ieqn-28"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, and the global optimal fitness value <inline-formula id="ieqn-29"><mml:math id="mml-ieqn-29"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is assessed by the following equation of the global optimal location <inline-formula id="ieqn-30"><mml:math id="mml-ieqn-30"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></inline-formula>
<disp-formula id="eqn-5"><label>(5)</label><mml:math id="mml-eqn-5" display="block"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>Next, the weight <italic>W</italic> for <italic>N</italic> slime moulds in an existing iteration <italic>t</italic> can be described by the <xref ref-type="disp-formula" rid="eqn-6">Eq. (6)</xref>:
<disp-formula id="eqn-6"><label>(6)</label><mml:math id="mml-eqn-6" display="block"><mml:mi>W</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext mathvariant="italic">SortIn</mml:mtext></mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mml:mtr><mml:mtd><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mspace width="thickmathspace" /><mml:mo>&#x22C5;</mml:mo><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mfrac><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mfrac><mml:mi>N</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>1</mml:mn><mml:mspace width="thickmathspace" /><mml:mstyle displaystyle="false" scriptlevel="0"><mml:mtext>---</mml:mtext></mml:mstyle><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mo>.</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mfrac><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mfrac><mml:mi>N</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle><mml:mo>&#x003C;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mi>N</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo fence="true" stretchy="true" symmetric="true"></mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>Let rand be an arbitrary value lies within <inline-formula id="ieqn-31"><mml:math id="mml-ieqn-31"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> indicates the local best fitness and the local worst fitness value is <inline-formula id="ieqn-32"><mml:math id="mml-ieqn-32"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></inline-formula> The <inline-formula id="ieqn-33"><mml:math id="mml-ieqn-33"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-34"><mml:math id="mml-ieqn-34"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is defined from the fitness value <inline-formula id="ieqn-35"><mml:math id="mml-ieqn-35"><mml:mi>f</mml:mi></mml:math></inline-formula>. For a minimization problem, sort the fitness value in ascending order as:
<disp-formula id="eqn-7"><label>(7)</label><mml:math id="mml-eqn-7" display="block"><mml:mrow><mml:mo>[</mml:mo><mml:mi>S</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mrow><mml:mtext mathvariant="italic">SortIn</mml:mtext></mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>f</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>The local best individual <inline-formula id="ieqn-36"><mml:math id="mml-ieqn-36"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and local best fitness <inline-formula id="ieqn-37"><mml:math id="mml-ieqn-37"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> are extracted by:
<disp-formula id="eqn-8"><label>(8)</label><mml:math id="mml-eqn-8" display="block"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>S</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula>
<disp-formula id="eqn-9"><label>(9)</label><mml:math id="mml-eqn-9" display="block"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext mathvariant="italic">SortIn</mml:mtext></mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>The local worst fitness <inline-formula id="ieqn-38"><mml:math id="mml-ieqn-38"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is extracted by:
<disp-formula id="eqn-10"><label>(10)</label><mml:math id="mml-eqn-10" display="block"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>S</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>The <inline-formula id="ieqn-39"><mml:math id="mml-ieqn-39"><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-40"><mml:math id="mml-ieqn-40"><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> indicates the arbitrary velocity selected from the uniform distribution <inline-formula id="ieqn-41"><mml:math id="mml-ieqn-41"><mml:mrow><mml:mo>[</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>b</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula id="ieqn-42"><mml:math id="mml-ieqn-42"><mml:mrow><mml:mo>[</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>c</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>. The <italic>b</italic> and <italic>c</italic> for the iteration <italic>t</italic> are selected by:
<disp-formula id="eqn-11"><label>(11)</label><mml:math id="mml-eqn-11" display="block"><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mspace width="thickmathspace" /><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mspace width="thickmathspace" /></mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula>
<disp-formula id="eqn-12"><label>(12)</label><mml:math id="mml-eqn-12" display="block"><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mfrac><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:mfrac></mml:math></disp-formula>where the maximal iteration refers to <inline-formula id="ieqn-43"><mml:math id="mml-ieqn-43"><mml:mi>T</mml:mi><mml:mo>.</mml:mo><mml:mspace width="thickmathspace" /></mml:math></inline-formula></p>
</sec>
<sec id="s3_2"><label>3.2</label><title>Process Involved in SMO-C Technique</title>
<p>The SMO-C algorithm obtains a fitness function using three input variables namely energy for CH selection, trust level, and distance to neighbors.</p>
<p>Distance to neighbor: It is applicable for selecting CH with minimal distance amongst neighboring nodes. In the intra-cluster transmission technique, energy consumption of sensors to CH transmission. After the neighbor node distance is minimized, subsequently the energy of intracluster transmission is minimalized correspondingly.</p>
<p>Objective 1: Minimize
<disp-formula id="eqn-13"><label>(13)</label><mml:math id="mml-eqn-13" display="block"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mfrac><mml:mn>1</mml:mn><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msubsup><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>s</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>Trust factor (TF): Initially, each node is specified that TF is one. The TF value is reduced through anomalous prediction method while the node process the abnormal tasks and such nodes are named malicious nodes.</p>
<p>Objective 2: Maximize
<disp-formula id="eqn-14"><label>(14)</label><mml:math id="mml-eqn-14" display="block"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mfrac><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>Energy: It is amount of energy expended as <inline-formula id="ieqn-44"><mml:math id="mml-ieqn-44"><mml:mi>C</mml:mi><mml:mi>H</mml:mi></mml:math></inline-formula> to RE of <inline-formula id="ieqn-45"><mml:math id="mml-ieqn-45"><mml:mi>C</mml:mi><mml:mi>H</mml:mi></mml:math></inline-formula>. When a CH consumes less energy as sensing, process, and also transmission process with high RE is gathered of low energy ratio. Therefore, lower as energy ratio, the CH selection develops further feasible.</p>
<p>Objective 3: Minimize
<disp-formula id="eqn-15"><label>(15)</label><mml:math id="mml-eqn-15" display="block"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mfrac><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:math></disp-formula></p>
<p>In the proposed SMO-C method, it is crucial for decreasing the linear integration of objective function. Thus, the possible energy function of SMO-C algorithm is implemented by:
<disp-formula id="eqn-16"><label>(16)</label><mml:math id="mml-eqn-16" display="block"><mml:mrow><mml:mtext mathvariant="italic">Minimize</mml:mtext></mml:mrow><mml:mspace width="thickmathspace" /><mml:mrow><mml:mtext mathvariant="italic">Potiential</mml:mtext></mml:mrow><mml:mspace width="thickmathspace" /><mml:mrow><mml:mtext mathvariant="italic">energy</mml:mtext></mml:mrow><mml:mspace width="thickmathspace" /><mml:mrow><mml:mtext mathvariant="italic">function</mml:mtext></mml:mrow><mml:mo>=</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x00D7;</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>&#x00D7;</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>&#x00D7;</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:math></disp-formula></p>
<p>Here <inline-formula id="ieqn-46"><mml:math id="mml-ieqn-46"><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>A</mml:mi><mml:mi>l</mml:mi><mml:mi>s</mml:mi><mml:mi>o</mml:mi><mml:mspace width="thickmathspace" /><mml:mn>0</mml:mn><mml:mo>&#x003C;</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mn>1.</mml:mn></mml:math></inline-formula></p>
</sec>
<sec id="s3_3"><label>3.3</label><title>Steps Involved in EWHO-MHR Technique</title>
<p>In this study, the EWHO-MHR technique has been introduced to select optimal routes [<xref ref-type="bibr" rid="ref-20">20</xref>,<xref ref-type="bibr" rid="ref-21">21</xref>]. The WHO method is a Meta-heuristic swarm-based approach stimulated by the social behavior of horses [<xref ref-type="bibr" rid="ref-20">20</xref>], namely mating, grazing, domination, and leadership hierarchy. The WHO approach comprises five discrete steps, as follows: when <italic>G</italic> groups and <italic>N</italic> individuals exist, the number of leaders is <italic>G</italic> and the number of non-leaders (foals and mares) is <inline-formula id="ieqn-47"><mml:math id="mml-ieqn-47"><mml:mi>N</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>G</mml:mi></mml:math></inline-formula>. The proportion of stallions is determined as <inline-formula id="ieqn-48"><mml:math id="mml-ieqn-48"><mml:mi>P</mml:mi><mml:mi>S</mml:mi></mml:math></inline-formula>, that is <inline-formula id="ieqn-49"><mml:math id="mml-ieqn-49"><mml:mi>G</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:math></inline-formula>. As previously mentioned, most foals spent a lifetime grazing nearby their group. To simulate the grazing stage, we consider that the stallion location existed in the grazing area center. To enable other individuals to move, the following formula is used. <xref ref-type="fig" rid="fig-2">Fig. 2</xref> depicts the flowchart of WHO technique.
<disp-formula id="eqn-17"><label>(17)</label><mml:math id="mml-eqn-17" display="block"><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>Z</mml:mi><mml:mi>cos</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03C0;</mml:mi><mml:mi>R</mml:mi><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-17">Eq. (17)</xref>, position of the <inline-formula id="ieqn-50"><mml:math id="mml-ieqn-50"><mml:mi>i</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-51"><mml:math id="mml-ieqn-51"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> group member and stallion in the <inline-formula id="ieqn-52"><mml:math id="mml-ieqn-52"><mml:mi>j</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-53"><mml:math id="mml-ieqn-53"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> group can be represented as <inline-formula id="ieqn-54"><mml:math id="mml-ieqn-54"><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> and <inline-formula id="ieqn-55"><mml:math id="mml-ieqn-55"><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, correspondingly, <inline-formula id="ieqn-56"><mml:math id="mml-ieqn-56"><mml:mi>R</mml:mi></mml:math></inline-formula>refers to random number that lies within the range of [<inline-formula id="ieqn-57"><mml:math id="mml-ieqn-57"><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>, 2], and <italic>Z</italic> indicates an adoptive variable calculated as follows:
<disp-formula id="eqn-18"><label>(18)</label><mml:math id="mml-eqn-18" display="block"><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy="false">&#x2192;</mml:mo></mml:mover><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mi>T</mml:mi><mml:mi>D</mml:mi><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>I</mml:mi><mml:mi>D</mml:mi><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>P</mml:mi><mml:mo>==</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>Z</mml:mi><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mn>2</mml:mn><mml:mi mathvariant="normal">&#x0398;</mml:mi><mml:mi>I</mml:mi><mml:mi>D</mml:mi><mml:mi>X</mml:mi><mml:mo>+</mml:mo><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy="false">&#x2192;</mml:mo></mml:mover><mml:mn>3</mml:mn><mml:mi mathvariant="normal">&#x0398;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>&#x223C;</mml:mo><mml:mi>I</mml:mi><mml:mi>D</mml:mi><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>Let <italic>P</italic> be a vector comprising <inline-formula id="ieqn-58"><mml:math id="mml-ieqn-58"><mml:mn>0</mml:mn></mml:math></inline-formula> and 1, and their dimension equivalents to the dimensional problem, <inline-formula id="ieqn-59"><mml:math id="mml-ieqn-59"><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy="false">&#x2192;</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:math></inline-formula> and <inline-formula id="ieqn-60"><mml:math id="mml-ieqn-60"><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy="false">&#x2192;</mml:mo></mml:mover><mml:mn>3</mml:mn></mml:math></inline-formula> represent arbitrary vector ranges from <inline-formula id="ieqn-61"><mml:math id="mml-ieqn-61"><mml:mn>0</mml:mn></mml:math></inline-formula> and 1, and <inline-formula id="ieqn-62"><mml:math id="mml-ieqn-62"><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> denotes a random number ranging from <inline-formula id="ieqn-63"><mml:math id="mml-ieqn-63"><mml:mn>0</mml:mn></mml:math></inline-formula> and 1. <inline-formula id="ieqn-64"><mml:math id="mml-ieqn-64"><mml:mi>T</mml:mi><mml:mi>D</mml:mi><mml:mi>R</mml:mi></mml:math></inline-formula> signifies a linear diminishing variable calculated as follows.
<fig id="fig-2"><label>Figure 2</label><caption><title>Flowchart of WHO technique</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-2.png"/></fig>
<disp-formula id="eqn-19"><label>(19)</label><mml:math id="mml-eqn-19" display="block"><mml:mi>T</mml:mi><mml:mi>D</mml:mi><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mfrac><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:mfrac></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-19">Eq. (19)</xref> <inline-formula id="ieqn-65"><mml:math id="mml-ieqn-65"><mml:mi>t</mml:mi></mml:math></inline-formula>and <italic>T</italic> denote the present and maximal iterations, correspondingly. As previously mentioned, the unique behavior of horses compared to other animals is splitting foals from the original group prior to the mating and reaching puberty. The following formula is used to simulate the behavior of mating between horses:
<disp-formula id="eqn-20"><label>(20)</label><mml:math id="mml-eqn-20" display="block"><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Crossover</mml:mtext></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>i</mml:mi><mml:mrow><mml:mi mathvariant="normal">&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mi>j</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi></mml:math></disp-formula>
<disp-formula id="ueqn-1">
<mml:math id="mml-ueqn-1" display="block"><mml:mrow><mml:mtext mathvariant="italic">Crossover</mml:mtext></mml:mrow><mml:mo>=</mml:mo><mml:mi>M</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-20">Eq. (20)</xref>, the position of horse <italic>p</italic> in group can be represented by <inline-formula id="ieqn-66"><mml:math id="mml-ieqn-66"><mml:mi>k</mml:mi><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, i.e., formed by position of horse <italic>z</italic> in group <italic>j</italic> and horse <italic>q</italic> in group <inline-formula id="ieqn-67"><mml:math id="mml-ieqn-67"><mml:mi>i</mml:mi></mml:math></inline-formula>. In the simple WHO, the possibility of crossover is fixed as a constant called <inline-formula id="ieqn-68"><mml:math id="mml-ieqn-68"><mml:mi>P</mml:mi><mml:mi>C</mml:mi></mml:math></inline-formula>. The group leader (stallion) leads other group members to a convenient location (waterhole). Also, Group leader (stallion) competes for the waterhole, which leads the dominant group to use the waterhole initially. To simulate this behavior, the following formula is used:
<disp-formula id="eqn-21"><label>(21)</label><mml:math id="mml-eqn-21" display="block"><mml:msub><mml:mover><mml:mrow><mml:mtext mathvariant="italic">Stallion</mml:mtext></mml:mrow><mml:mo accent="false">&#x00AF;</mml:mo></mml:mover><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mml:mtr><mml:mtd><mml:mn>2</mml:mn><mml:mi>Z</mml:mi><mml:mi>cos</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03C0;</mml:mi><mml:mi>R</mml:mi><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>W</mml:mi><mml:mi>H</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>W</mml:mi><mml:mi>H</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mo>&#x003E;</mml:mo><mml:mn>0.5</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>2</mml:mn><mml:mi>Z</mml:mi><mml:mi>cos</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03C0;</mml:mi><mml:mi>R</mml:mi><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>W</mml:mi><mml:mi>H</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>W</mml:mi><mml:mi>H</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mn>0.5</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo fence="true" stretchy="true" symmetric="true"></mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-21">Eq. (21)</xref>, the candidate position and the current leadership position in the <inline-formula id="ieqn-69"><mml:math id="mml-ieqn-69"><mml:mi>j</mml:mi><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> group are denoted as <inline-formula id="ieqn-70"><mml:math id="mml-ieqn-70"><mml:msub><mml:mover><mml:mrow><mml:mtext mathvariant="italic">Stallion</mml:mtext></mml:mrow><mml:mo accent="false">&#x00AF;</mml:mo></mml:mover><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-71"><mml:math id="mml-ieqn-71"><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, correspondingly, WH stands for the location of waterhole. In the beginning, leaders are designated in a random fashion. Next, leaders are designated on the basis of fitness value. The following formula is used to simulate the exchange between leader positions and other individuals:
<disp-formula id="eqn-22"><label>(22)</label><mml:math id="mml-eqn-22" display="block"><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x003C;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2265;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo fence="true" stretchy="true" symmetric="true"></mml:mo></mml:mrow></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-22">Eq. (22)</xref>, the fitness values of foal and stallion are characterized by <inline-formula id="ieqn-72"><mml:math id="mml-ieqn-72"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula id="ieqn-73"><mml:math id="mml-ieqn-73"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, correspondingly. The dynamic weight approach makes use of EWHO algorithm and is beneficial to finding a global optimum solution once they are brought into algorithm [<xref ref-type="bibr" rid="ref-21">21</xref>]. Consequently, to assist the stallion to find the best waterhole, a dynamic weight is added to the waterhole in the initial formulation of <xref ref-type="disp-formula" rid="eqn-21">Eq. (21)</xref>. In the following, the modified formula and weight are measured:
<disp-formula id="eqn-23"><label>(23)</label><mml:math id="mml-eqn-23" display="block"><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mml:mtr><mml:mtd><mml:mi>w</mml:mi><mml:mo movablelimits="true" form="prefix">min</mml:mo><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>w</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mi>w</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mspace width="thickmathspace" /></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x00D7;</mml:mo><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mfrac><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo movablelimits="true" form="prefix">min</mml:mo></mml:mrow><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>a</mml:mi><mml:mi>v</mml:mi><mml:mi>g</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo movablelimits="true" form="prefix">min</mml:mo></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>a</mml:mi><mml:mi>v</mml:mi><mml:mi>g</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>w</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mspace width="thickmathspace" /><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mspace width="thickmathspace" /><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mo>&#x003E;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>a</mml:mi><mml:mi>v</mml:mi><mml:mi>g</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo fence="true" stretchy="true" symmetric="true"></mml:mo></mml:mrow></mml:math></disp-formula>
<disp-formula id="eqn-24"><label>(24)</label><mml:math id="mml-eqn-24" display="block"><mml:msub><mml:mover><mml:mrow><mml:mtext mathvariant="italic">Stallion</mml:mtext></mml:mrow><mml:mo accent="false">&#x00AF;</mml:mo></mml:mover><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>Z</mml:mi><mml:mi>cos</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03C0;</mml:mi><mml:mi>R</mml:mi><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>W</mml:mi><mml:mi>H</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mtext mathvariant="italic">Stallio</mml:mtext></mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>w</mml:mi><mml:mo>&#x00D7;</mml:mo><mml:mi>W</mml:mi><mml:mi>H</mml:mi></mml:math></disp-formula></p>
<p>From the above equations, the upper and lower boundary values are denoted as <inline-formula id="ieqn-74"><mml:math id="mml-ieqn-74"><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mo movablelimits="true" form="prefix">min</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-75"><mml:math id="mml-ieqn-75"><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula> correspondingly, the fitness value of the current stallion at <inline-formula id="ieqn-76"><mml:math id="mml-ieqn-76"><mml:mi>t</mml:mi></mml:math></inline-formula>-<inline-formula id="ieqn-77"><mml:math id="mml-ieqn-77"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> iteration is indicated by<inline-formula id="ieqn-78"><mml:math id="mml-ieqn-78"><mml:mspace width="thickmathspace" /><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, the average fitness value of each stallion is denoted as<inline-formula id="ieqn-79"><mml:math id="mml-ieqn-79"><mml:mspace width="thickmathspace" /><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mi>a</mml:mi><mml:mi>v</mml:mi><mml:mi>g</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, and the minimum fitness value of the population is represented as <inline-formula id="ieqn-80"><mml:math id="mml-ieqn-80"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo movablelimits="true" form="prefix">min</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>.</p>
<p>In order to define a set of routes utilizing the EWHO-MHR model, the given function is applied to define the subsequent hop to Base Station (BS) as follows:
<disp-formula id="eqn-25"><label>(25)</label><mml:math id="mml-eqn-25" display="block"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mrow><mml:mi>f</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mtext>&#xA0;</mml:mtext><mml:mi>w</mml:mi><mml:mi>h</mml:mi><mml:mi>i</mml:mi><mml:mi>c</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo fence="false" stretchy="false">|</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mfrac><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mrow><mml:msub><mml:mi>i</mml:mi><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo fence="false" stretchy="false">|</mml:mo><mml:mtext>&#xA0;</mml:mtext><mml:mrow><mml:mi>i</mml:mi><mml:mi>s</mml:mi><mml:mtext>&#xA0;</mml:mtext><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>m</mml:mi><mml:mi>u</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:msub><mml:mi mathvariant="normal">&#x2200;</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mn>1</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mi>k</mml:mi></mml:math></disp-formula></p>
<p>The aim is to define a set of routes in CH to BS using an Fitness Function (FF) comprising 2 variables such as distance and energy. At first, the Residual Energy (RE) of next-hop node is established and the nodes with the highest energy are provided as relay nodes. For data transmission, the source node transmits to relay node that further transfer to BS through inter CH. Hence, the node with greater RE is provided as next-hop node. The <inline-formula id="ieqn-81"><mml:math id="mml-ieqn-81"><mml:mi>f</mml:mi><mml:mn>1</mml:mn></mml:math></inline-formula> initial sub-objectives are provided as follows:
<disp-formula id="eqn-26"><label>(26)</label><mml:math id="mml-eqn-26" display="block"><mml:mi>f</mml:mi><mml:mn>1</mml:mn><mml:mo>=</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mi>H</mml:mi></mml:mrow></mml:msub></mml:math></disp-formula></p>
<p>In <xref ref-type="disp-formula" rid="eqn-22">Eq. (22)</xref>, Euclidean distance can be employed to determine the distance between CH to BS. The reduction of energy dissipation is based largely on the broadcasting distance. By using a minimal distance, the energy is retained considerable. When the distance is enhanced, additional amount of energy is consumed. Consequently, the nodes with lower distances are desired for a relay node. Therefore, the next sub-objectives using distance is <inline-formula id="ieqn-82"><mml:math id="mml-ieqn-82"><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:math></inline-formula> can be formulated by:
<disp-formula id="eqn-27"><label>(27)</label><mml:math id="mml-eqn-27" display="block"><mml:mi>f</mml:mi><mml:mn>2</mml:mn><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow></mml:mrow></mml:msubsup><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>s</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>s</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mi>H</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mi>S</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:math></disp-formula></p>
<p>The abovementioned sub-objectives are regarded as to FF as given under where <inline-formula id="ieqn-83"><mml:math id="mml-ieqn-83"><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula id="ieqn-84"><mml:math id="mml-ieqn-84"><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> represent the weight allotted for each sub-objective.
<disp-formula id="eqn-28"><label>(28)</label><mml:math id="mml-eqn-28" display="block"><mml:mrow><mml:mtext mathvariant="italic">Fitness</mml:mtext></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>f</mml:mi><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>f</mml:mi><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="thickmathspace" /><mml:mrow><mml:mtext mathvariant="italic">where</mml:mtext></mml:mrow><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>;</mml:mo></mml:math></disp-formula></p>
</sec>
</sec>
<sec id="s4"><label>4</label><title>Experimental Validation</title>
<p>The proposed model is simulated using NS3 tool. The performance validation of the EMCMHR-LM model is tested using distinct aspects. <xref ref-type="table" rid="table-1">Tab. 1</xref> and <xref ref-type="fig" rid="fig-3">Fig. 3</xref> report a detailed packet delivery ratio (PDR) examination of the EMCMHR-LM model with recent models under varying vehicle counts [<xref ref-type="bibr" rid="ref-22">22</xref>&#x2013;<xref ref-type="bibr" rid="ref-24">24</xref>]. The experimental values implied that the EMCMHR-LM model has resulted in maximum&#x00A0;PDR values. For instance, with 50 vehicles per km<sup>2</sup>, the EMCMHR-LM model has obtained increased PDR of 20.90&#x0025; whereas DACRS, CBCLR, AODV-CV, CBR, and DBR models have obtained reduced PDR of 14.69&#x0025;, 14&#x0025;, 16.30&#x0025;, 14.46&#x0025;, and 10.78&#x0025; respectively. Meanwhile, with 75 vehicles per km<sup>2</sup>, the EMCMHR-LM model has reached maximum PDR of 59.56&#x0025; whereas DACRS, CBCLR, AODV-CV, CBR, and DBR models have accomplished minimal PDR of 56.11&#x0025;, 50.82&#x0025;, 37.93&#x0025;, 30.34&#x0025;, and 28.73&#x0025; respectively.</p>
<table-wrap id="table-1"><label>Table 1</label><caption><title>PDR analysis of EMCMHR-LM technique with existing algorithms under varying vehicle count</title></caption>
<table frame="hsides">
<colgroup>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
</colgroup>
<thead>
<tr>
<th align="center" colspan="7">Packet delivery ratio (&#x0025;)</th>
</tr>
<tr>
<th align="left">No. of vehicles per <inline-formula id="ieqn-85"><mml:math id="mml-ieqn-85"><mml:mi>K</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></th>
<th align="left">EMCMHR-LM</th>
<th align="left">DACRS model</th>
<th align="left">CBCLR model</th>
<th align="left">AODV-CV ALGORITHM</th>
<th align="left">CBR Algorithm</th>
<th align="left">DBR Algorithm</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">50</td>
<td align="left">20.90</td>
<td align="left">14.69</td>
<td align="left">14.00</td>
<td align="left">16.30</td>
<td align="left">14.46</td>
<td align="left">10.78</td>
</tr>
<tr>
<td align="left">55</td>
<td align="left">29.19</td>
<td align="left">26.43</td>
<td align="left">21.82</td>
<td align="left">21.36</td>
<td align="left">18.14</td>
<td align="left">13.54</td>
</tr>
<tr>
<td align="left">60</td>
<td align="left">38.85</td>
<td align="left">31.72</td>
<td align="left">29.42</td>
<td align="left">24.81</td>
<td align="left">20.67</td>
<td align="left">14.23</td>
</tr>
<tr>
<td align="left">65</td>
<td align="left">45.75</td>
<td align="left">39.77</td>
<td align="left">34.71</td>
<td align="left">27.58</td>
<td align="left">23.20</td>
<td align="left">20.67</td>
</tr>
<tr>
<td align="left">70</td>
<td align="left">52.20</td>
<td align="left">46.67</td>
<td align="left">42.99</td>
<td align="left">36.32</td>
<td align="left">27.12</td>
<td align="left">22.74</td>
</tr>
<tr>
<td align="left">75</td>
<td align="left">59.56</td>
<td align="left">56.11</td>
<td align="left">50.82</td>
<td align="left">37.93</td>
<td align="left">30.34</td>
<td align="left">28.73</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="fig-3"><label>Figure 3</label><caption><title>PDR analysis of EMCMHR-LM technique under varying vehicle count</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-3.png"/></fig>
<p><xref ref-type="table" rid="table-2">Tab. 2</xref> and <xref ref-type="fig" rid="fig-4">Fig. 4</xref> define a detailed throughput (THPT) inspection of the EMCMHR-LM approach with recent models under varying vehicle counts. The experimental values implied that the EMCMHR-LM system has resulted in maximal THPT values. For instance, with 50 vehicles per km<sup>2</sup>, the EMCMHR-LM algorithm has obtained higher THPT of 32 whereas DACRS, CBCLR, AODV-CV, CBR, and DBR systems have obtained decreased THPT of 30, 27, 25, 27, and 23 correspondingly. In the meantime, with 75 vehicles per km<sup>2</sup>, the EMCMHR-LM algorithm has reached maximal THPT of 188 whereas DACRS, CBCLR, AODV-CV, CBR, and DBR methodologies have accomplished minimal THPT of 178, 169, 141, 121, and 112 correspondingly.</p>
<table-wrap id="table-2"><label>Table 2</label><caption><title>Throughput analysis of EMCMHR-LM approach with existing algorithms under varying vehicle count</title></caption>
<table frame="hsides">
<colgroup>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
</colgroup>
<thead>
<tr>
<th align="center" colspan="7">Throughput</th>
</tr>
<tr>
<th align="left">No. of vehicles per <inline-formula id="ieqn-86"><mml:math id="mml-ieqn-86"><mml:mi>K</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></th>
<th align="left">EMCMHR-LM</th>
<th align="left">DACRS model</th>
<th align="left">CBCLR model</th>
<th align="left">AODV-CV Algorithm</th>
<th align="left">CBR Algorithm</th>
<th align="left">DBR Algorithm</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">50</td>
<td align="left">32</td>
<td align="left">30</td>
<td align="left">27</td>
<td align="left">25</td>
<td align="left">27</td>
<td align="left">23</td>
</tr>
<tr>
<td align="left">55</td>
<td align="left">126</td>
<td align="left">112</td>
<td align="left">100</td>
<td align="left">93</td>
<td align="left">91</td>
<td align="left">70</td>
</tr>
<tr>
<td align="left">60</td>
<td align="left">150</td>
<td align="left">135</td>
<td align="left">124</td>
<td align="left">85</td>
<td align="left">83</td>
<td align="left">73</td>
</tr>
<tr>
<td align="left">65</td>
<td align="left">165</td>
<td align="left">154</td>
<td align="left">143</td>
<td align="left">93</td>
<td align="left">101</td>
<td align="left">80</td>
</tr>
<tr>
<td align="left">70</td>
<td align="left">186</td>
<td align="left">171</td>
<td align="left">152</td>
<td align="left">119</td>
<td align="left">111</td>
<td align="left">96</td>
</tr>
<tr>
<td align="left">75</td>
<td align="left">188</td>
<td align="left">178</td>
<td align="left">169</td>
<td align="left">141</td>
<td align="left">121</td>
<td align="left">112</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="fig-4"><label>Figure 4</label><caption><title>Throughput analysis of EMCMHR-LM algorithm under varying vehicle count</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-4.png"/></fig>
<p>A detailed cluster overhead (COH) inspection of the EMCMHR-LM model with recent models is made in <xref ref-type="table" rid="table-3">Tab. 3</xref> and <xref ref-type="fig" rid="fig-5">Fig. 5</xref>. The experimental outcomes portrayed that the EMCMHR-LM model has obtained lower COH over other models. For instance, with 50 vehicles per km<sup>2</sup>, the EMCMHR-LM model has resulted in least COH of 13 whereas DACRS, CBCLR, AODV-CV, CBR, and DBR models have demonstrated increased COH of 22, 26, 31, 31, and 45 respectively.</p>
<table-wrap id="table-3"><label>Table 3</label><caption><title>COH analysis of EMCMHR-LM technique with existing algorithms under varying vehicle count</title></caption>
<table frame="hsides">
<colgroup>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
</colgroup>
<thead>
<tr>
<th align="center" colspan="7">Cluster overhead</th>
</tr>
<tr>
<th align="left">No. of vehicles per <inline-formula id="ieqn-87"><mml:math id="mml-ieqn-87"><mml:mi mathvariant="bold-italic">K</mml:mi><mml:msup><mml:mi mathvariant="bold-italic">m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></th>
<th align="left">EMCMHR-LM</th>
<th align="left">DACRS model</th>
<th align="left">CBCLR model</th>
<th align="left">AODV-CV Algorithm</th>
<th align="left">CBR Algorithm</th>
<th align="left">DBR Algorithm</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">50</td>
<td align="left">32</td>
<td align="left">30</td>
<td align="left">27</td>
<td align="left">25</td>
<td align="left">27</td>
<td align="left">23</td>
</tr>
<tr>
<td align="left">55</td>
<td align="left">126</td>
<td align="left">112</td>
<td align="left">100</td>
<td align="left">93</td>
<td align="left">91</td>
<td align="left">70</td>
</tr>
<tr>
<td align="left">60</td>
<td align="left">150</td>
<td align="left">135</td>
<td align="left">124</td>
<td align="left">85</td>
<td align="left">83</td>
<td align="left">73</td>
</tr>
<tr>
<td align="left">65</td>
<td align="left">165</td>
<td align="left">154</td>
<td align="left">143</td>
<td align="left">93</td>
<td align="left">101</td>
<td align="left">80</td>
</tr>
<tr>
<td align="left">70</td>
<td align="left">186</td>
<td align="left">171</td>
<td align="left">152</td>
<td align="left">119</td>
<td align="left">111</td>
<td align="left">96</td>
</tr>
<tr>
<td align="left">75</td>
<td align="left">188</td>
<td align="left">178</td>
<td align="left">169</td>
<td align="left">141</td>
<td align="left">121</td>
<td align="left">112</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="fig-5"><label>Figure 5</label><caption><title>COH analysis of EMCMHR-LM algorithm under varying vehicle count</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-5.png"/></fig>
<p>Also, with 75 vehicles per km<sup>2</sup>, the EMCMHR-LM system has resulted in least COH of 77 whereas DACRS, CBCLR, AODV-CV, CBR, and DBR techniques have outperformed higher COH of 109, 132, 140, 152, and 210 correspondingly. <xref ref-type="table" rid="table-4">Tab. 4</xref> and <xref ref-type="fig" rid="fig-6">Fig. 6</xref> determines a detailed coverage examination of the EMCMHR-LM approach with recent methods under varying vehicle count. The experimental values referred that the EMCMHR-LM method has resulted in higher coverage values. For instance, with 50 vehicles per km<sup>2</sup>, the EMCMHR-LM methodology has obtained increased coverage of 28.74&#x0025; whereas DACRS, CBCLR, AODV-CV, CBR, and DBR systems have obtained decreased coverage of 21.39&#x0025;, 19.55&#x0025;, 15.87&#x0025;, 15.87&#x0025;, and 8.52&#x0025; correspondingly. Besides, with 75 vehicles per km<sup>2</sup>, the EMCMHR-LM algorithm has reached increased coverage of 93.43&#x0025; whereas DACRS, CBCLR, AODV-CV, CBR, and DBR approaches have accomplished minimal coverage of 89.02&#x0025;, 80.94&#x0025;, 82.41&#x0025;, 70.28&#x0025;, and 60.72&#x0025; correspondingly.</p>
<table-wrap id="table-4"><label>Table 4</label><caption><title>Coverage analysis of EMCMHR-LM technique with existing algorithms under varying vehicle count</title></caption>
<table frame="hsides">
<colgroup>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
</colgroup>
<thead>
<tr>
<th align="center" colspan="7">Coverage (&#x0025;)</th>
</tr>
<tr>
<th align="left">No. of vehicles per <inline-formula id="ieqn-88"><mml:math id="mml-ieqn-88"><mml:mi>K</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></th>
<th align="left">EMCMHR-LM</th>
<th align="left">DACRS model</th>
<th align="left">CBCLR model</th>
<th align="left">AODV-CV Algorithm</th>
<th align="left">CBR Algorithm</th>
<th align="left">DBR Algorithm</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">50</td>
<td align="left">28.74</td>
<td align="left">21.39</td>
<td align="left">19.55</td>
<td align="left">15.87</td>
<td align="left">15.87</td>
<td align="left">8.52</td>
</tr>
<tr>
<td align="left">55</td>
<td align="left">55.94</td>
<td align="left">47.85</td>
<td align="left">30.21</td>
<td align="left">26.53</td>
<td align="left">24.33</td>
<td align="left">17.71</td>
</tr>
<tr>
<td align="left">60</td>
<td align="left">65.86</td>
<td align="left">57.04</td>
<td align="left">50.43</td>
<td align="left">26.53</td>
<td align="left">39.03</td>
<td align="left">29.47</td>
</tr>
<tr>
<td align="left">65</td>
<td align="left">75.79</td>
<td align="left">66.23</td>
<td align="left">55.94</td>
<td align="left">39.77</td>
<td align="left">60.72</td>
<td align="left">39.40</td>
</tr>
<tr>
<td align="left">70</td>
<td align="left">89.02</td>
<td align="left">79.83</td>
<td align="left">68.81</td>
<td align="left">66.60</td>
<td align="left">58.88</td>
<td align="left">53.37</td>
</tr>
<tr>
<td align="left">75</td>
<td align="left">93.43</td>
<td align="left">89.02</td>
<td align="left">80.94</td>
<td align="left">82.41</td>
<td align="left">70.28</td>
<td align="left">60.72</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="fig-6"><label>Figure 6</label><caption><title>Coverage analysis of EMCMHR-LM technique under varying vehicle count</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-6.png"/></fig>
<p>A detailed delivery delay (DD) analysis of the EMCMHR-LM technique with recent methods is made in <xref ref-type="table" rid="table-5">Tab. 5</xref> and <xref ref-type="fig" rid="fig-7">Fig. 7</xref>. The experimental outcomes outperformed that the EMCMHR-LM approach has obtained lower DD over other methods. For instance, with 50 vehicles per km<sup>2</sup>, the EMCMHR-LM approach has resulted in minimal DD of 15&#x2005;s whereas DACRS, CBCLR, AODV-CV, CBR, and DBR algorithms have demonstrated superior DD of 19, 21, 17, 19 and 22&#x2005;s correspondingly. Besides, with 75 vehicles per km<sup>2</sup>, the EMCMHR-LM algorithm has resulted in lower DD of 15&#x2005;s whereas DACRS, CBCLR, AODV-CV, CBR, and DBR models have depicted higher DD of 19, 21, 17, 19 and 22&#x2005;s correspondingly.</p>
<table-wrap id="table-5"><label>Table 5</label><caption><title>DD analysis of EMCMHR-LM technique with existing algorithms under varying vehicle count</title></caption>
<table frame="hsides">
<colgroup>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
</colgroup>
<thead>
<tr>
<th align="center" colspan="7">Delivery delay (sec)</th>
</tr>
<tr>
<th align="left">No. of vehicles per <inline-formula id="ieqn-89"><mml:math id="mml-ieqn-89"><mml:mi>K</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></th>
<th align="left">EMCMHR-LM</th>
<th align="left">DACRS Model</th>
<th align="left">CBCLR Model</th>
<th align="left">AODV-CV Algorithm</th>
<th align="left">CBR Algorithm</th>
<th align="left">DBR Algorithm</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">50</td>
<td align="left">15</td>
<td align="left">19</td>
<td align="left">21</td>
<td align="left">17</td>
<td align="left">19</td>
<td align="left">22</td>
</tr>
<tr>
<td align="left">55</td>
<td align="left">11</td>
<td align="left">14</td>
<td align="left">18</td>
<td align="left">15</td>
<td align="left">18</td>
<td align="left">19</td>
</tr>
<tr>
<td align="left">60</td>
<td align="left">8</td>
<td align="left">11</td>
<td align="left">16</td>
<td align="left">14</td>
<td align="left">16</td>
<td align="left">17</td>
</tr>
<tr>
<td align="left">65</td>
<td align="left">6</td>
<td align="left">9</td>
<td align="left">13</td>
<td align="left">12</td>
<td align="left">15</td>
<td align="left">15</td>
</tr>
<tr>
<td align="left">70</td>
<td align="left">5</td>
<td align="left">7</td>
<td align="left">10</td>
<td align="left">10</td>
<td align="left">13</td>
<td align="left">12</td>
</tr>
<tr>
<td align="left">75</td>
<td align="left">15</td>
<td align="left">19</td>
<td align="left">21</td>
<td align="left">17</td>
<td align="left">19</td>
<td align="left">22</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="fig-7"><label>Figure 7</label><caption><title>DD analysis of EMCMHR-LM algorithm under varying vehicle count</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-7.png"/></fig>
<p>A detailed average number of collisions (ANOC) investigation of the EMCMHR-LM methodology with recent methods is made in <xref ref-type="table" rid="table-6">Tab. 6</xref> and <xref ref-type="fig" rid="fig-8">Fig. 8</xref>. The experimental outcomes exhibited that the EMCMHR-LM model has obtained lesser ANOC over other models.</p>
<table-wrap id="table-6"><label>Table 6</label><caption><title>ANOC analysis of EMCMHR-LM technique with existing algorithms under varying vehicle count</title></caption>
<table frame="hsides">
<colgroup>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
<col align="left"/>
</colgroup>
<thead>
<tr>
<th align="center" colspan="7">Average no. of collisions</th>
</tr>
<tr>
<th align="left">No. of vehicles per <inline-formula id="ieqn-90"><mml:math id="mml-ieqn-90"><mml:mi>K</mml:mi><mml:msup><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></th>
<th align="left">EMCMHR-LM</th>
<th align="left">DACRS model</th>
<th align="left">CBCLR model</th>
<th align="left">AODV-CV Algorithm</th>
<th align="left">CBR Algorithm</th>
<th align="left">DBR Algorithm</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">50</td>
<td align="left">2</td>
<td align="left">3</td>
<td align="left">4</td>
<td align="left">5</td>
<td align="left">6</td>
<td align="left">8</td>
</tr>
<tr>
<td align="left">55</td>
<td align="left">2</td>
<td align="left">4</td>
<td align="left">6</td>
<td align="left">7</td>
<td align="left">8</td>
<td align="left">10</td>
</tr>
<tr>
<td align="left">60</td>
<td align="left">4</td>
<td align="left">6</td>
<td align="left">10</td>
<td align="left">11</td>
<td align="left">12</td>
<td align="left">13</td>
</tr>
<tr>
<td align="left">65</td>
<td align="left">5</td>
<td align="left">7</td>
<td align="left">11</td>
<td align="left">13</td>
<td align="left">14</td>
<td align="left">16</td>
</tr>
<tr>
<td align="left">70</td>
<td align="left">6</td>
<td align="left">8</td>
<td align="left">11</td>
<td align="left">12</td>
<td align="left">14</td>
<td align="left">17</td>
</tr>
<tr>
<td align="left">75</td>
<td align="left">8</td>
<td align="left">10</td>
<td align="left">12</td>
<td align="left">12</td>
<td align="left">15</td>
<td align="left">18</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="fig-8"><label>Figure 8</label><caption><title>ANOC analysis of EMCMHR-LM technique under varying vehicle count</title></caption><graphic mimetype="image" mime-subtype="png" xlink:href="CMC_31962-fig-8.png"/></fig>
<p>For instance, with 50 vehicles per km<sup>2</sup>, the EMCMHR-LM technique has resulted in minimal ANOC of 2 whereas DACRS, CBCLR, AODV-CV, CBR, and DBR systems have demonstrated increased ANOC of 3, 4, 5, 6, and 8 respectively. Followed by, with 75 vehicles per km<sup>2</sup>, the EMCMHR-LM approach has resulted in minimum ANOC of 8 whereas DACRS, CBCLR, AODV-CV, CBR, and DBR algorithms have depicted superior ANOC of 10, 12, 12, 15, and 18 correspondingly. From the detailed experimental analysis, it is apparent that the EMCMHR-LM model has shown better results than other models.</p>
</sec>
<sec id="s5"><label>5</label><title>Conclusion</title>
<p>In this study, a new EMCMHR-LM model was enhanced to resolve scalability issues and increase lifetime of VANET. The presented EMCMHR-LM model involves the procedure of arranging clusters, CH selection, and route selection appropriate for VANETs. At the initial stage, the presented EMCMHR-LM model uses SMO-C technique to group the vehicles into clusters. In addition, the EWHO-MHR protocol has been developed by the optimization of network parameters. The presented EMCMHR-LM model is simulated using NS3 tool and the simulation outcomes reported the advanced performance of the proposed EMCMHR-LM technique over the other models. Thus, the EMCMHR-LM model can be utilized for effectual network performance in VANET. In future, privacy preserving blockchain enabled secure data transmission protocol can be designed for VANET.</p>
</sec>
</body>
<back>
<fn-group>
<fn fn-type="other"><p><bold>Funding Statement:</bold> The authors received no specific funding for this study.</p></fn>
<fn fn-type="conflict"><p><bold>Conflicts of Interest:</bold> The authors declare that they have no conflicts of interest to report regarding the present study.</p></fn>
</fn-group>
<ref-list content-type="authoryear">
<title>References</title>
<ref id="ref-1"><label>[1]</label><mixed-citation publication-type="conf-proc"><person-group person-group-type="author"><string-name><given-names>M.</given-names> <surname>Elhoseny</surname></string-name> and <string-name><given-names>K.</given-names> <surname>Shankar</surname></string-name></person-group>, &#x201C;<article-title>Energy efficient optimal routing for communication in VANETs via clustering model</article-title>,&#x201D; in <conf-name>Proc. Emerging Technologies for Connected Internet of Vehicles and Intelligent Transportation System Networks</conf-name>, Switzerland, <publisher-name>Springer, Cham</publisher-name>, pp. <fpage>1</fpage>&#x2013;<lpage>14</lpage>, <year>2020</year>.</mixed-citation></ref>
<ref id="ref-2"><label>[2]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>B.</given-names> <surname>Singh</surname></string-name>, <string-name><given-names>P.</given-names> <surname>Kavitha</surname></string-name>, <string-name><given-names>R.</given-names> <surname>Regin</surname></string-name>, <string-name><given-names>K.</given-names> <surname>Praghash</surname></string-name>, <string-name><given-names>S.</given-names> <surname>Sujatha</surname></string-name> <etal>et al.,</etal></person-group> &#x201C;<article-title>Optimized node clustering based on received signal strength with particle ordered-filter routing used in VANET</article-title>,&#x201D; <source>Webology</source>, vol. <volume>17</volume>, no. <issue>2</issue>, pp. <fpage>1</fpage>&#x2013;<lpage>12</lpage>, <year>2020</year>.</mixed-citation></ref>
<ref id="ref-3"><label>[3]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>Z.</given-names> <surname>Khan</surname></string-name>, <string-name><given-names>P.</given-names> <surname>Fan</surname></string-name>, <string-name><given-names>S.</given-names> <surname>Fang</surname></string-name> and <string-name><given-names>F.</given-names> <surname>Abbas</surname></string-name></person-group>, &#x201C;<article-title>An unsupervised cluster-based VANET-oriented evolving graph (CVoEG) model and associated reliable routing scheme</article-title>,&#x201D; <source>IEEE Transactions on Intelligent Transportation Systems</source>, vol. <volume>20</volume>, no. <issue>10</issue>, pp. <fpage>3844</fpage>&#x2013;<lpage>3859</lpage>, <year>2019</year>.</mixed-citation></ref>
<ref id="ref-4"><label>[4]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>W.</given-names> <surname>Qi</surname></string-name>, <string-name><given-names>B.</given-names> <surname>Landfeldt</surname></string-name>, <string-name><given-names>Q.</given-names> <surname>Song</surname></string-name>, <string-name><given-names>L.</given-names> <surname>Guo</surname></string-name> and <string-name><given-names>A.</given-names> <surname>Jamalipour</surname></string-name></person-group>, &#x201C;<article-title>Traffic differentiated clustering routing in DSRC and C-V2X hybrid vehicular networks</article-title>,&#x201D; <source>IEEE Transactions on Vehicular Technology</source>, vol. <volume>69</volume>, no. <issue>7</issue>, pp. <fpage>7723</fpage>&#x2013;<lpage>7734</lpage>, <year>2020</year>.</mixed-citation></ref>
<ref id="ref-5"><label>[5]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>R.</given-names> <surname>Kaur</surname></string-name>, <string-name><given-names>R. K.</given-names> <surname>Ramachandran</surname></string-name>, <string-name><given-names>R.</given-names> <surname>Doss</surname></string-name> and <string-name><given-names>L.</given-names> <surname>Pan</surname></string-name></person-group>, &#x201C;<article-title>The importance of selecting clustering parameters in VANETs: A survey</article-title>,&#x201D; <source>Computer Science Review</source>, vol. <volume>40</volume>, pp. <fpage>100392</fpage>, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-6"><label>[6]</label><mixed-citation publication-type="conf-proc"><person-group person-group-type="author"><string-name><given-names>S.</given-names> <surname>Yogarayan</surname></string-name>, <string-name><given-names>S. F. A.</given-names> <surname>Razak</surname></string-name>, <string-name><given-names>A.</given-names> <surname>Azman</surname></string-name>, <string-name><given-names>M. F. A.</given-names> <surname>Abdullah</surname></string-name>, <string-name><surname>S. Z. Ibrahim</surname></string-name> <etal>et al.,</etal></person-group> &#x201C;<article-title>A review of routing protocols for vehicular ad-hoc networks (VANETs)</article-title>,&#x201D; in <conf-name>Proc. 8th Int. Conf. on Information and Communication Technology (ICoICT)</conf-name>, <publisher-name>IEEE</publisher-name>, <conf-loc>Yogyakarta, Indonesia</conf-loc>, pp. <fpage>1</fpage>&#x2013;<lpage>7</lpage>, <year>2020</year>.</mixed-citation></ref>
<ref id="ref-7"><label>[7]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>U. A.</given-names> <surname>Khan</surname></string-name> and <string-name><given-names>S. S.</given-names> <surname>Lee</surname></string-name></person-group>, &#x201C;<article-title>Multi-layer problems and solutions in VANETs: A review</article-title>,&#x201D; <source>Electronics</source>, vol. <volume>8</volume>, no. <issue>2</issue>, pp. <fpage>204</fpage>, <year>2019</year>.</mixed-citation></ref>
<ref id="ref-8"><label>[8]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>A.</given-names> <surname>Katiyar</surname></string-name>, <string-name><given-names>D.</given-names> <surname>Singh</surname></string-name> and <string-name><given-names>R. S.</given-names> <surname>Yadav</surname></string-name></person-group>, &#x201C;<article-title>State-of-the-art approach to clustering protocols in VANET: A survey</article-title>,&#x201D; <source>Wireless Networks</source>, vol. <volume>26</volume>, no. <issue>7</issue>, pp. <fpage>5307</fpage>&#x2013;<lpage>5336</lpage>, <year>2020</year>.</mixed-citation></ref>
<ref id="ref-9"><label>[9]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>K.</given-names> <surname>Kandali</surname></string-name>, <string-name><given-names>L.</given-names> <surname>Bennis</surname></string-name> and <string-name><given-names>H.</given-names> <surname>Bennis</surname></string-name></person-group>, &#x201C;<article-title>A new hybrid routing protocol using a modified K-means clustering algorithm and continuous hopfield network for VANET</article-title>,&#x201D; <source>IEEE Access</source>, vol. <volume>9</volume>, pp. <fpage>47169</fpage>&#x2013;<lpage>47183</lpage>, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-10"><label>[10]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>J.</given-names> <surname>Cheng</surname></string-name>, <string-name><given-names>G.</given-names> <surname>Yuan</surname></string-name>, <string-name><given-names>M.</given-names> <surname>Zhou</surname></string-name>, <string-name><given-names>S.</given-names> <surname>Gao</surname></string-name>, <string-name><given-names>Z.</given-names> <surname>Huang</surname></string-name> <etal>et al.,</etal></person-group> &#x201C;<article-title>A Connectivity-prediction-based dynamic clustering model for VANET in an urban scene</article-title>,&#x201D; <source>IEEE Internet of Things Journal</source>, vol. <volume>7</volume>, no. <issue>9</issue>, pp. <fpage>8410</fpage>&#x2013;<lpage>8418</lpage>, <year>2020</year>.</mixed-citation></ref>
<ref id="ref-11"><label>[11]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>R.</given-names> <surname>Chandren Muniyandi</surname></string-name>, <string-name><given-names>M. K.</given-names> <surname>Hasan</surname></string-name>, <string-name><given-names>M. R.</given-names> <surname>Hammoodi</surname></string-name> and <string-name><given-names>A.</given-names> <surname>Maroosi</surname></string-name></person-group>, &#x201C;<article-title>An improved harmony search algorithm for proactive routing protocol in VANET</article-title>,&#x201D; <source>Journal of Advanced Transportation</source>, vol. <volume>2021</volume>, pp. 1&#x2013;14, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-12"><label>[12]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>K.</given-names> <surname>Aravindhan</surname></string-name> and <string-name><given-names>C.</given-names> <surname>Dhas</surname></string-name></person-group>, &#x201C;<article-title>Destination-aware context-based routing protocol with hybrid soft computing cluster algorithm for VANET</article-title>,&#x201D; <source>Soft Computing</source>, vol. <volume>23</volume>, no. <issue>8</issue>, pp. <fpage>2499</fpage>&#x2013;<lpage>2507</lpage>, <year>2019</year>.</mixed-citation></ref>
<ref id="ref-13"><label>[13]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>H.</given-names> <surname>Fatemidokht</surname></string-name> and <string-name><given-names>M. K.</given-names> <surname>Rafsanjani</surname></string-name></person-group>, &#x201C;<article-title>QMM-VANET: An efficient clustering algorithm based on QoS and monitoring of malicious vehicles in vehicular ad hoc networks</article-title>,&#x201D; <source>Journal of Systems and Software</source>, vol. <volume>165</volume>, pp. <fpage>110561</fpage>, <year>2020</year>.</mixed-citation></ref>
<ref id="ref-14"><label>[14]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>A.</given-names> <surname>Javadpour</surname></string-name>, <string-name><given-names>S.</given-names> <surname>Rezaei</surname></string-name>, <string-name><given-names>A. K.</given-names> <surname>Sangaiah</surname></string-name>, <string-name><given-names>A.</given-names> <surname>Slowik</surname></string-name> and <string-name><given-names>S.</given-names> <surname>Mahmoodi Khaniabadi</surname></string-name></person-group>, &#x201C;<article-title>Enhancement in quality of routing service using metaheuristic PSO algorithm in VANET networks</article-title>,&#x201D; <source>Soft Computing</source>, Early Access, pp. <fpage>1</fpage>&#x2013;<lpage>12</lpage>, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-15"><label>[15]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>S. P.</given-names> <surname>Ardakani</surname></string-name></person-group>, &#x201C;<article-title>ACR: A cluster-based routing protocol for VANET</article-title>,&#x201D; <source>International Journal of Wireless &#x0026; Mobile Networks (IJWMN)</source>, vol. <volume>10</volume>, no. <issue>2</issue>, pp. <fpage>39</fpage>&#x2013;<lpage>53</lpage>, <year>2018</year>.</mixed-citation></ref>
<ref id="ref-16"><label>[16]</label><mixed-citation publication-type="conf-proc"><person-group person-group-type="author"><string-name><given-names>M.</given-names> <surname>Sangeetha</surname></string-name>, <string-name><given-names>T.</given-names> <surname>Vijayan</surname></string-name>, <string-name><given-names>N. A.</given-names> <surname>Unnikrishnan</surname></string-name>, <string-name><given-names>G.</given-names> <surname>Kalanandhini</surname></string-name> and <string-name><given-names>R. N.</given-names> <surname>Shaw</surname></string-name></person-group>, &#x201C;<article-title>Genetic fuzzy based VANET routing algorithm for better efficiency</article-title>,&#x201D; in <conf-name>Proc. Int. Conf. on Electrical and Electronics Engineering</conf-name>, <conf-loc>Springer, Singapore</conf-loc>, pp. <fpage>669</fpage>&#x2013;<lpage>677</lpage>, <year>2022</year>.</mixed-citation></ref>
<ref id="ref-17"><label>[17]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>V. G.</given-names> <surname>Krishnan</surname></string-name> and <string-name><given-names>N. S.</given-names> <surname>Ram</surname></string-name></person-group>, &#x201C;<article-title>A novel approach to select cluster head and optimising the data transmission in VANET</article-title>,&#x201D; <source>International Journal of Advanced Intelligence Paradigms</source>, vol. <volume>21</volume>, no. <issue>3&#x2013;4</issue>, pp. <fpage>235</fpage>&#x2013;<lpage>246</lpage>, <year>2022</year>.</mixed-citation></ref>
<ref id="ref-18"><label>[18]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>P. K.</given-names> <surname>Pandey</surname></string-name>, <string-name><given-names>V.</given-names> <surname>Kansal</surname></string-name> and <string-name><given-names>A.</given-names> <surname>Swaroop</surname></string-name></person-group>, &#x201C;<article-title>OCSR: Overlapped cluster-based scalable routing approach for vehicular ad hoc networks (VANETs)</article-title>,&#x201D; <source>Wireless Communications and Mobile Computing</source>, vol. <volume>2022</volume>, pp. <fpage>1</fpage>&#x2013;<lpage>18</lpage>, <year>2022</year>.</mixed-citation></ref>
<ref id="ref-19"><label>[19]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>R.</given-names> <surname>Zheng</surname></string-name>, <string-name><given-names>H.</given-names> <surname>Jia</surname></string-name>, <string-name><given-names>L.</given-names> <surname>Abualigah</surname></string-name>, <string-name><given-names>Q.</given-names> <surname>Liu</surname></string-name> and <string-name><given-names>S.</given-names> <surname>Wang</surname></string-name></person-group>, &#x201C;<article-title>Deep ensemble of slime mold algorithm and arithmetic optimization algorithm for global optimization</article-title>,&#x201D; <source>Processes</source>, vol. <volume>9</volume>, no. <issue>10</issue>, pp. <fpage>1774</fpage>, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-20"><label>[20]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>I.</given-names> <surname>Naruei</surname></string-name> and <string-name><given-names>F.</given-names> <surname>Keynia</surname></string-name></person-group>, &#x201C;<article-title>Wild horse optimizer: A new meta-heuristic algorithm for solving engineering optimization problems</article-title>,&#x201D; <source>Engineering with Computers</source>, Early Access, pp. <fpage>1</fpage>&#x2013;<lpage>32</lpage>, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-21"><label>[21]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>R.</given-names> <surname>Zheng</surname></string-name>, <string-name><given-names>A. G.</given-names> <surname>Hussien</surname></string-name>, <string-name><given-names>H. M.</given-names> <surname>Jia</surname></string-name>, <string-name><given-names>L.</given-names> <surname>Abualigah</surname></string-name>, <string-name><given-names>S.</given-names> <surname>Wang</surname></string-name> <etal>et al.,</etal></person-group> &#x201C;<article-title>An improved wild horse optimizer for solving optimization problems</article-title>,&#x201D; <source>Mathematics</source>, vol. <volume>10</volume>, no. <issue>8</issue>, pp. <fpage>1311</fpage>&#x2013;<lpage>1324</lpage>, <year>2022</year>.</mixed-citation></ref>
<ref id="ref-22"><label>[22]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>B.</given-names> <surname>Elira</surname></string-name>, <string-name><given-names>K. P.</given-names> <surname>Keerthana</surname></string-name> and <string-name><given-names>K.</given-names> <surname>Balaji</surname></string-name></person-group>, &#x201C;<article-title>Clustering scheme and destination aware context based routing protocol for VANET</article-title>,&#x201D; <source>International Journal of Intelligent Networks</source>, vol. <volume>2</volume>, pp. <fpage>148</fpage>&#x2013;<lpage>155</lpage>, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-23"><label>[23]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>W.</given-names> <surname>Sun</surname></string-name>, <string-name><given-names>G. C.</given-names> <surname>Zhang</surname></string-name>, <string-name><given-names>X. R.</given-names> <surname>Zhang</surname></string-name>, <string-name><given-names>X.</given-names> <surname>Zhang</surname></string-name> and <string-name><given-names>N. N.</given-names> <surname>Ge</surname></string-name></person-group>, &#x201C;<article-title>Fine-grained vehicle type classification using lightweight convolutional neural network with feature optimization and joint learning strategy</article-title>,&#x201D; <source>Multimedia Tools and Applications</source>, vol. <volume>80</volume>, no. <issue>20</issue>, pp. <fpage>30803</fpage>&#x2013;<lpage>30816</lpage>, <year>2021</year>.</mixed-citation></ref>
<ref id="ref-24"><label>[24]</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><string-name><given-names>W.</given-names> <surname>Sun</surname></string-name>, <string-name><given-names>X.</given-names> <surname>Chen</surname></string-name>, <string-name><given-names>X. R.</given-names> <surname>Zhang</surname></string-name>, <string-name><given-names>G. Z.</given-names> <surname>Dai</surname></string-name>, <string-name><given-names>P. S.</given-names> <surname>Chang</surname></string-name> <etal>et al.,</etal></person-group> &#x201C;<article-title>A Multi-feature learning model with enhanced local attention for vehicle re-identification</article-title>,&#x201D; <source>Computers, Materials &#x0026; Continua</source>, vol. <volume>69</volume>, no. <issue>3</issue>, pp. <fpage>3549</fpage>&#x2013;<lpage>3560</lpage>, <year>2021</year>.</mixed-citation></ref>
</ref-list>
</back>
</article>