<?xml version="1.0" encoding="UTF-8"?><!DOCTYPE article PUBLIC "-//NLM//DTD Journal Publishing DTD v2.0 20040830//EN" "journalpublishing.dtd"><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" dtd-version="2.0" xml:lang="en" article-type="research-article"><front><journal-meta><journal-id journal-id-type="nlm-ta">Online J Public Health Inform</journal-id><journal-id journal-id-type="publisher-id">ojphi</journal-id><journal-id journal-id-type="index">45</journal-id><journal-title>Online Journal of Public Health Informatics</journal-title><abbrev-journal-title>Online J Public Health Inform</abbrev-journal-title><issn pub-type="epub">1947-2579</issn><publisher><publisher-name>JMIR Publications</publisher-name><publisher-loc>Toronto, Canada</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">v17i1e65039</article-id><article-id pub-id-type="doi">10.2196/65039</article-id><article-categories><subj-group subj-group-type="heading"><subject>Original Paper</subject></subj-group></article-categories><title-group><article-title>One Health Index Calculator for India Using Empirical Methods for Policy Stewardship: Development and Usability Study</article-title></title-group><contrib-group><contrib contrib-type="author" equal-contrib="yes"><name name-style="western"><surname>Meganathan</surname><given-names>Saveetha</given-names></name><degrees>PhD</degrees><xref ref-type="aff" rid="aff1"/><xref ref-type="fn" rid="equal-contrib1">*</xref></contrib><contrib contrib-type="author" equal-contrib="yes"><name name-style="western"><surname>Katiyar</surname><given-names>Arpit</given-names></name><degrees>MSc</degrees><xref ref-type="aff" rid="aff1"/><xref ref-type="fn" rid="equal-contrib1">*</xref></contrib><contrib contrib-type="author"><name name-style="western"><surname>Srivastava</surname><given-names>Esha</given-names></name><degrees>MPH</degrees><xref ref-type="aff" rid="aff1"/></contrib><contrib contrib-type="author" corresp="yes"><name name-style="western"><surname>Mishra</surname><given-names>Rakesh Kumar</given-names></name><degrees>PhD</degrees><xref ref-type="aff" rid="aff1"/></contrib></contrib-group><aff id="aff1"><institution>Tata Institute for Genetics and Society</institution><addr-line>Bengaluru</addr-line><country>India</country></aff><contrib-group><contrib contrib-type="editor"><name name-style="western"><surname>Mensah</surname><given-names>Edward</given-names></name></contrib></contrib-group><contrib-group><contrib contrib-type="reviewer"><name name-style="western"><surname>John</surname><given-names>Denny</given-names></name></contrib><contrib contrib-type="reviewer"><name name-style="western"><surname>Zhou</surname><given-names>Xiao-Nong</given-names></name></contrib></contrib-group><author-notes><corresp>Correspondence to Rakesh Kumar Mishra, PhD, Tata Institute for Genetics and Society, Bengaluru, 560065, India, 91 9441902188; <email>rakesh.mishra@tigs.res.in</email></corresp><fn fn-type="equal" id="equal-contrib1"><label>*</label><p>these authors contributed equally</p></fn></author-notes><pub-date pub-type="collection"><year>2025</year></pub-date><pub-date pub-type="epub"><day>25</day><month>6</month><year>2025</year></pub-date><volume>17</volume><elocation-id>e65039</elocation-id><history><date date-type="received"><day>02</day><month>08</month><year>2024</year></date><date date-type="rev-recd"><day>16</day><month>12</month><year>2024</year></date><date date-type="accepted"><day>19</day><month>04</month><year>2025</year></date></history><copyright-statement>&#x00A9; Saveetha Meganathan, Arpit Katiyar, Esha Srivastava, Rakesh Kumar Mishra. Originally published in the Online Journal of Public Health Informatics (<ext-link ext-link-type="uri" xlink:href="https://ojphi.jmir.org/">https://ojphi.jmir.org/</ext-link>), 25.6.2025. </copyright-statement><copyright-year>2025</copyright-year><license license-type="open-access" xlink:href="https://creativecommons.org/licenses/by/4.0/"><p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (<ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link>), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work, first published in the Online Journal of Public Health Informatics, is properly cited. The complete bibliographic information, a link to the original publication on <ext-link ext-link-type="uri" xlink:href="https://ojphi.jmir.org/">https://ojphi.jmir.org/</ext-link>, as well as this copyright and license information must be included.</p></license><self-uri xlink:type="simple" xlink:href="https://ojphi.jmir.org/2025/1/e65039"/><abstract><sec><title>Background</title><p><italic>One Health</italic> is a collaborative approach that can be used to evaluate and enhance the fields of human, animal, and environmental health and to emphasize their sectoral interconnectedness. Empirical evaluation of the <italic>One Health</italic> performance of a country in the form of an index, provides direction for actionable interventions such as targeted funding, prioritized resource allocation, rigorous data management, and evidence-based policy decisions. These efforts, along with public engagement and awareness on disease management; environmental degradation, and preparedness toward disease outbreaks, contribute to strengthening global health security. Thus, developing a One Health Index (OHI) calculator for India is a significant step toward evidence-based <italic>One Health</italic> governance in the context of low-and middle-income countries.</p></sec><sec><title>Objective</title><p>This study aimed to (1) develop a OHI Calculator for India using efficient and user-friendly weighting methods and demonstrate the calculation of the OHI; (2) develop India-specific datasets through secondary data collection from reliable data sources; and (3) determine data gaps for policy stewardship.</p></sec><sec sec-type="methods"><title>Methods</title><p>We proposed a OHI calculator to measure the OHI from an Indian context by adopting the Global One Health Index framework that comprises 3 categories: 13 key indicators, 57 indicators, and 216 subindicators. Secondary data collection was conducted to create a dataset for specific to India from reliable sources. For measuring OHI, we demonstrated two mathematical weighting methods: an efficient expert-based rating using fuzzy extent analysis and a modified entropy-based weightage method.</p></sec><sec sec-type="results"><title>Results</title><p>We demonstrate the step-by-step OHI calculation by determining indicator scores using both fuzzy extent analysis and modified entropy-based weightage method. Through secondary data collection an India-specific dataset was created using reliable sources. For the datasets from India, data for 156/216 subindicators were available, while that for the remaining 60 indicators were unavailable. Further, a pilot correlation analysis was performed between 20 indicator scores and relevant budget allocations for the years 2022&#x2010;2023, 2023&#x2010;2024, and 2024&#x2010;2025. It was found that increases in the budget allocation across consecutive years improved indicator scores or better performance and vice versa.</p></sec><sec sec-type="conclusions"><title>Conclusions</title><p>The demonstrated OHI calculator has the potential to serve as a governance tool while promoting data transparency and ethical data management. There is a need for a collaborative data federation approach to resolve data gaps, including incomplete, missing, or unavailable data. Further, the correlation analysis between budgetary allocation and performance of indicators provides empirical evidence for policymakers to improve intersectoral communication, multistakeholder engagement, concerted interventions, and informed policy decisions for resource allocation.</p></sec></abstract><kwd-group><kwd>one health</kwd><kwd>one health index calculator</kwd><kwd>fuzzy extent analysis</kwd><kwd>modified entropy-based weightage method</kwd><kwd>policy stewardship</kwd><kwd>India</kwd><kwd>low and middle income countries</kwd><kwd>LMIC</kwd><kwd>Index</kwd><kwd>indicator</kwd><kwd>secondary data collection</kwd></kwd-group></article-meta></front><body><sec id="s1" sec-type="intro"><title>Introduction</title><p><italic>One Health</italic> is a participatory, collaborative approach to enhancing the health of people, animals, and ecosystems over time. It recognizes the interdependence of the health of humans, domestic and wild animals, plants, and the larger environment. While health, food sources, water, energy, and the environment are broad topics with sector-specific concerns, cross-sectoral and cross-disciplinary collaboration helps protect human health, address health challenges such as the emergence of infectious diseases and antimicrobial resistance, ensure food safety, and promote the health and integrity of ecosystems. The <italic>One Health</italic> approach has the potential to address the complete spectrum of disease control, from prevention to detection, readiness, response, and management, while also contributing to global health security by considering the multisectoral interconnectedness and their mutual impact [<xref ref-type="bibr" rid="ref1">1</xref>].</p><p>The term &#x2019;One Health&#x2019; evolved during the outbreaks of SARS in 2003&#x2010;2004 and H5N1 influenza (ie, bird flu), highlighting the interconnectedness of human, animal, and environmental health. The &#x2019;Manhattan Principles&#x2019; underscored this link, recognizing the need for collaborative approaches in global disease prevention [<xref ref-type="bibr" rid="ref2">2</xref>]. These outbreaks emphasized the risks posed by unknown pathogens from wildlife, which led to the development of more effective alert and response systems. Global cooperation involving United Nations, World Health Organization, Food and Agriculture Organization, World Organization for Animal Health, United Nations Children&#x2019;s Fund, and the World Bank addressed the H5N1 outbreak, with the International Ministerial Conference on Avian and Pandemic Influenza playing a key role [<xref ref-type="bibr" rid="ref3">3</xref>]. The primary drivers of the emergence of novel zoonotic infectious diseases include human activities, changes to ecosystems, land use, agriculture intensification, urbanization, international travel, and trade over the past three decades. These diseases, predominantly originating in wildlife, pose significant public health risks. The <italic>One Health</italic> approach is pivotal for the prevention, monitoring, and surveillance of zoonotic and emerging infectious diseases, sustaining food security, and combating antimicrobial resistance, all of which affect human, animal, and environmental health. Subsequently, collaborative monitoring systems have become essential for effectively managing pandemics and outbreaks, given the multitude of health crises in the last decade [<xref ref-type="bibr" rid="ref3">3</xref>].</p><p>Developing a <italic>One Health</italic> Index (OHI) framework can contribute to the implementation of the <italic>One Health</italic> approach by focusing on intersectoral collaborations and the corresponding datasets. The value of such an index extends beyond compiling data; it has the potential to revolutionize our understanding, management, and response to the complex web of factors that influence global health. A OHI simplifies health data, making it accessible to diverse stakeholders including government functionaries, policymakers, health care providers and the public [<xref ref-type="bibr" rid="ref4">4</xref>]. It condenses a huge volume of information into a single quantifiable value. Government and health care organizations can use these health indices to make informed policy decisions regarding the allocation of resources and implementation of public health interventions [<xref ref-type="bibr" rid="ref5">5</xref>]. Further, these indices facilitate the monitoring of health trends in a given region or country, enabling the assessment of the effectiveness of public health campaigns and improvements in health care systems and policy frameworks [<xref ref-type="bibr" rid="ref6">6</xref>,<xref ref-type="bibr" rid="ref7">7</xref>].</p><p>In this context, the Global One Health Index (GOHI) serves as an empirical tool for the systematic assessment of <italic>One Health</italic> scores for more than 200 countries and territories worldwide [<xref ref-type="bibr" rid="ref8">8</xref>]. The GOHI framework consists of three major categories&#x2014;extrinsic driver index, core driver index, and intrinsic driver index&#x2014;which are further categorized into 13 key-indicators, 57 indicators, and 216 subindicators. This structure makes it possible to quantify the <italic>One Health</italic> Index for a country by mapping the multisectoral variables that contribute to the well-being of humans, animals, and the environment and their mutual interactions [<xref ref-type="bibr" rid="ref8">8</xref>]. Additionally, the GOHI framework accounts for the recruitment of 29 domain experts to attribute weightage to the indicators based on their sectoral experience, further demonstrating the need for cross-sectoral communication and concerted efforts to achieve better <italic>One Health</italic> outcomes for a country [<xref ref-type="bibr" rid="ref8">8</xref>].</p><p>In this study, we proposed to adapt and contextualize the <italic>One Health</italic> Index calculator for India based on the GOHI framework. We demonstrated the calculation of indicator scores using two mathematical weighting methods: (1) the efficient expert-based fuzzy extent analysis (FEA) method that requires recruitment of domain experts, and (2) the modified entropy-based weightage method (MEWM), which replaces expert-based weightage calculation with mathematical formula-based calculations. A country-specific database for India was developed through secondary data collection. The two weighting methods&#x2014;FEA and MEWM&#x2014;were demonstrated using the India-specific dataset to generate indicator scores, subject to data availability. In addition, we correlated sectoral budget allocations over the last two financial years with corresponding sectoral indicator scores. This analysis serves as an actionable pointer for policy makers, federal and state governments to support decision-making toward budgetary allocations.</p></sec><sec id="s2" sec-type="methods"><title>Methods</title><p>The OHI calculator for India is aimed to serve as a public health tool that addresses the need for multisectoral collaboration for data access and understanding the impact of sectoral performance, thereby improving the countrywide <italic>One Health</italic> performance. This calculator consists of the following steps:</p><sec id="s2-1"><title>Indicator Selection</title><p>To demonstrate the calculation of the OHI for India, we adopted three categories: 13 key indicators, 57 indicators, and 216 subindicators from the GOHI [<xref ref-type="bibr" rid="ref8">8</xref>] (<xref ref-type="supplementary-material" rid="app1">Multimedia Appendix 1</xref>).</p></sec><sec id="s2-2"><title>Database Building</title><p>For the demonstration of the OHI calculator for India, national-level data collection was conducted using secondary data sources such as the Press Information Bureau (Government of India), Department of Agriculture and Farmers Welfare (Government of India), World Bank, Food and Agriculture Organization of the United Nations, Yale Environmental Performance Index (Yale University), Our World in Data, The Global Economy, Statista, IndiaStat, Knoema, and other relevant databases (<xref ref-type="supplementary-material" rid="app2">Multimedia Appendix 2</xref>). Additionally, to correlate the sectoral budgets over the last two financial years with their respective indicator scores, the budget datasets were obtained from the Indian Union Budget documents for the financial years 2022&#x2010;2023, 2023&#x2010;2024, and 2024&#x2010;2025 (<xref ref-type="supplementary-material" rid="app3">Multimedia Appendix 3</xref>).</p></sec><sec id="s2-3"><title>Weight Determination</title><p>A sample proforma was developed to obtain expert ratings for various subindicators, indicators, key indicators, and categories. This proforma can be self-administered via email or used for in-person interviews, thereby providing efficient data collection from experts. It also allows the experts to provide additional information such as variables, data sources, and case studies (<xref ref-type="supplementary-material" rid="app4">Multimedia Appendix 4</xref>).</p></sec><sec id="s2-4"><title>Procedure to Calculate Expert Weights</title><p>There are two mathematical weightage methods for calculating the indicator scores: (1) fuzzy extent analysis (FEA)&#x2013;an efficient expert-based method and (2) Modified Entropy-Based Weightage Method (MEWM)&#x2013;a data-driven method.</p></sec><sec id="s2-5"><title>Fuzzy Extent Analysis (FEA)</title><p>The efficient expert-based FEA rating method used to calculate the indicator scores requires consultations with experts from diverse <italic>One Health</italic> sectors for ascertaining the priority or the weightage of different subindicators and indicators. The FEA is a multicriteria decision-making method that integrates both qualitative and quantitative approaches [<xref ref-type="bibr" rid="ref9">9</xref>]. Data collection for expert-based rating can be conducted using the provided proforma. A pairwise comparison was used for developing the proforma, that allowed the experts to rate the metrics [<xref ref-type="bibr" rid="ref10">10</xref>]. It is an effective way to gather opinions from many experienced experts, especially for complex decision-making problems involving multiple risks. The following is an example of how expert-based ratings are converted into weights. A pair of subindicators within an experts&#x2019; domain will be provided to them for pairwise comparison using linguistic ratings, which are then translated into an existing numerical scale of relative importance [<xref ref-type="bibr" rid="ref10">10</xref>]. Similarly, all possible combinations of subindicators will be provided to the expert for comparisons. The linguistic ratings obtained from an expert, such as equally important, moderately important, strongly important, and extremely important will be converted into numerical ratings. These numerical ratings are further converted into comparison matrix. Further, the consistency of the responses by the experts will be checked using the consistency ratio. After ascertaining the consistency of the responses, the next step will be &#x201C;fuzzification&#x201D; (ie, conversion of the numerical ratings to fuzzy numbers). To do this, we will use triangular fuzzy numbers (defined as a generalization of real numbers representing a set of possible values with weights, or membership functions) to calculate the weightage of different subindicators and indicators. Similarly, weights for the key indicators and categories may be obtained using this efficient expert-based weightage mechanism, or through a format similar to panel discussions which can be conducted to obtain the weights from the domain experts.</p></sec><sec id="s2-6"><title>Steps involved for OHI calculation using FEA</title><sec id="s2-6-1"><title>Step 1</title><list list-type="order"><list-item><p>Consult with domain experts from diverse sectors relevant to <italic>One Health</italic>.</p></list-item><list-item><p>Based on the systematic implementation of the semistructured interviews and a modified Delphi method, pairwise comparisons are made on the importance of each pair of parameters.</p></list-item><list-item><p>Given a hierarchy with n parameters, there will be n(n &#x2013; 1)/2 pairwise comparisons.</p></list-item><list-item><p>The comparisons are rated linguistically.</p></list-item><list-item><p>Linguistic variables are converted to numerical ratings using scale of relative importance [<xref ref-type="bibr" rid="ref11">11</xref>].</p></list-item></list></sec><sec id="s2-6-2"><title>Step 2</title><p>A comparison matrix will be established for each expert using the ratings from the proforma. The comparison matrix for the <inline-formula><mml:math id="ieqn1"><mml:mi> </mml:mi><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>expert is given by:</p><p><inline-formula><mml:math id="ieqn2"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msup><mml:mover><mml:mi>A</mml:mi><mml:mo>&#x223C;</mml:mo></mml:mover><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mstyle></mml:math></inline-formula> represented as:</p><p><inline-formula><mml:math id="ieqn3"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>=<inline-formula><mml:math id="ieqn4"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:mn>1</mml:mn></mml:mtd><mml:mtd><mml:mo>&#x22EF;</mml:mo></mml:mtd><mml:mtd><mml:msub><mml:mover><mml:mi>a</mml:mi><mml:mo>&#x223C;</mml:mo></mml:mover><mml:mrow><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>&#x22EE;</mml:mo></mml:mtd><mml:mtd><mml:mo>&#x22F1;</mml:mo></mml:mtd><mml:mtd><mml:mo>&#x22EE;</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mover><mml:mi>a</mml:mi><mml:mo>&#x223C;</mml:mo></mml:mover><mml:mrow><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>&#x22EF;</mml:mo></mml:mtd><mml:mtd><mml:mn>1</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula></p><p>The reciprocal matrix is denoted by <inline-formula><mml:math id="ieqn5"><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msup><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>.</p><p>The reciprocity properties are given by:</p><disp-formula id="E7"><mml:math id="eqn1"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mfrac><mml:mo>,</mml:mo><mml:mi mathvariant="normal">&#x2200;</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mstyle></mml:math></disp-formula></sec><sec id="s2-6-3"><title>Step 3</title><p>It crucial to maintain consistency for expert-based weightage process to derive the indicator scores. To achieve this, a Consistency Index (CI) was calculated to evaluate the consistency of the comparison matrix [<xref ref-type="bibr" rid="ref10">10</xref>]:</p><disp-formula id="E8"><mml:math id="eqn2"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>&#x03BB;</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:math></disp-formula><p>where <inline-formula><mml:math id="ieqn6"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>&#x03BB;</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mstyle></mml:math></inline-formula> is the maximum eigen value and <italic>n</italic> is the dimension of the comparison matrix [<xref ref-type="bibr" rid="ref9">9</xref>].</p><p>The consistency ratio is given by:</p><disp-formula id="E9"><mml:math id="eqn3"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>C</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:math></disp-formula><p>where, RI is the random consistency index which depends on <italic>n</italic> [<xref ref-type="bibr" rid="ref10">10</xref>].<graphic xlink:href="ojphi_v17i1e65039_fig01.png" position="float"/></p><p>If CR &#x003C;0.1, the judgements are considered consistent [<xref ref-type="bibr" rid="ref12">12</xref>].</p></sec><sec id="s2-6-4"><title>Step 4</title><list list-type="order"><list-item><p>Convert the ratings in the comparison matrix to triangular fuzzy number.</p></list-item><list-item><p>A fuzzy number serves as a tool to express values that are uncertain or imprecise particularly in the context of fuzzy set theory.</p></list-item><list-item><p>Unlike a value, it accommodates varying degrees of belongingness enabling it to account for the ambiguity often found in practical scenarios.</p></list-item><list-item><p>Fuzzy sets can compensate for the inconsistency and imprecisions in human judgements rather than random or stochastic ones.</p></list-item><list-item><p>Fuzzy numbers are depicted by a set of possible values having their own membership function ranging from 0 to 1.</p></list-item><list-item><p>A triangular fuzzy number is represented by [floor value, average value, ceiling value] ie, <inline-formula><mml:math id="ieqn7"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula> with the member function as:</p></list-item></list><disp-formula id="E18"><mml:math id="eqn4"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>&#x03BC;</mml:mi><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><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:mtable columnalign="center center" rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>l</mml:mi></mml:mrow></mml:mfrac><mml:mo>&#x2212;</mml:mo><mml:mfrac><mml:mi>l</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>l</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>&#x220A;</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfrac><mml:mo>&#x2212;</mml:mo><mml:mfrac><mml:mi>u</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>&#x220A;</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>o</mml:mi><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi><mml:mi>w</mml:mi><mml:mi>i</mml:mi><mml:mi>s</mml:mi><mml:mi>e</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo fence="true" stretchy="true" symmetric="true"/></mml:mrow></mml:mrow></mml:mstyle></mml:math></disp-formula><p><graphic xlink:href="ojphi_v17i1e65039_fig02.png" position="float"/>[<xref ref-type="bibr" rid="ref13">13</xref>].</p></sec><sec id="s2-6-5"><title>Step 5</title><p>To calculate weights from the comparison matrix, FEA is applied [<xref ref-type="bibr" rid="ref13">13</xref>].</p><p>In extent analysis, let <inline-formula><mml:math id="ieqn8"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula> be an object set and <inline-formula><mml:math id="ieqn9"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">G</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula> a goal set. For each object, extent analysis is performed for each goal <inline-formula><mml:math id="ieqn10"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>.</p><p>The value of fuzzy synthetic extent is given by:</p><disp-formula id="E10"><mml:math id="eqn5"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mo>&#x2297;</mml:mo><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mstyle></mml:mrow></mml:mstyle></mml:math></disp-formula><p>where,</p><disp-formula id="E11"><mml:math id="eqn6"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:math></disp-formula><p>We can calculate the ratings in the triangular fuzzy number format as shown above leading to a weight derived from these triangular weights.</p><p>A pairwise comparison of fuzzy weights needs to be performed and the computation of the degree of possibility of them being greater than the fuzzy weight will be obtained. The minimum of these possibilities is used as the overall score for each criterion <italic>i</italic>.</p><p>To compute the value of the fuzzy synthetic extent, <inline-formula><mml:math id="ieqn11"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> for the <inline-formula><mml:math id="ieqn12"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> object is as follows:</p><disp-formula id="E12"><mml:math id="eqn7"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mo>&#x2297;</mml:mo><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mstyle></mml:mrow></mml:mstyle></mml:math></disp-formula><p>where,</p><disp-formula id="E15"><mml:math id="eqn8"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:math></disp-formula><disp-formula id="equWL1"><mml:math id="eqn9"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><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:munderover><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mtext>&#x00A0;</mml:mtext><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mtext>&#x00A0;</mml:mtext><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:math></disp-formula></sec><sec id="s2-6-6"><title>Step 6</title><p>The fuzzy synthetic extent constitutes three values which will be averaged out to acquire a single value for the weights from them.</p><p>Weights = <inline-formula><mml:math id="ieqn13"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munderover><mml:mover><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>&#x223C;</mml:mo></mml:mover><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:mstyle></mml:math></inline-formula></p></sec><sec id="s2-6-7"><title>Step 7</title><p>Normalize each weight by dividing the individual weights by the sum of all weights.</p><disp-formula id="E13"><mml:math id="eqn10"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi mathvariant="normal">N</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">z</mml:mi><mml:mi mathvariant="normal">e</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mi mathvariant="normal">w</mml:mi><mml:mi mathvariant="normal">e</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">g</mml:mi><mml:mi mathvariant="normal">h</mml:mi><mml:mi mathvariant="normal">t</mml:mi><mml:msub><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mi mathvariant="normal">i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mstyle></mml:mrow></mml:mstyle></mml:math></disp-formula></sec><sec id="s2-6-8"><title>Step 8</title><p>If the ratings have been collected from multiple experts for the same indicator, either of these methods can be used to obtain optimized ratings:</p><sec id="s2-6-8-1"><title>Method 1</title><list list-type="order"><list-item><p>Calculate individual normalized weights using Steps 1-7 for every expert rating</p></list-item><list-item><p>Average the normalized weights to obtain optimized ratings for the indicators</p></list-item></list></sec><sec id="s2-6-8-2"><title>Method 2</title><list list-type="simple"><list-item><p>In continuation to Step 5, after calculating the fuzzy synthetic extent for every expert input individually,</p></list-item></list><list list-type="order"><list-item><p>Compute the degree of possibility of</p><list list-type="simple"><list-item><p><inline-formula><mml:math id="ieqn14"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula></p></list-item><list-item><p>where degree of possibility between two fuzzy synthetic extents is defined as:</p></list-item><list-item><p><inline-formula><mml:math id="ieqn15"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>s</mml:mi><mml:mi>u</mml:mi><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>&#x03BC;</mml:mi><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03BC;</mml:mi><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula></p></list-item><list-item><p><inline-formula><mml:math id="ieqn16"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mi>g</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2229;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03BC;</mml:mi><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula></p></list-item><list-item><p>where, <inline-formula><mml:math id="ieqn17"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>&#x03BC;</mml:mi><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>d</mml:mi><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:mtext>&#x00A0;</mml:mtext><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>&#x00A0;</mml:mtext><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mi>i</mml:mi><mml:mi>f</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2265;</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mi>o</mml:mi><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi><mml:mi>w</mml:mi><mml:mi>i</mml:mi><mml:mi>s</mml:mi><mml:mi>e</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo fence="true" stretchy="true" symmetric="true"/></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula></p></list-item><list-item><p>and&#x203A; d is the ordinate of the highest intersection point d between <inline-formula><mml:math id="ieqn18"><mml:msub><mml:mrow><mml:mi>&#x00B5;</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math id="ieqn19"><mml:msub><mml:mrow><mml:mi>&#x00B5;</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>.</p><p><graphic xlink:href="ojphi_v17i1e65039_fig03.png" position="float"/></p></list-item></list></list-item></list><list list-type="order"><list-item><p>Compute the degree of possibility for a convex fuzzy number to be &#x003E;k convex fuzzy numbers <inline-formula><mml:math id="ieqn20"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>(1, 2, &#x2026;, k)</p><list list-type="simple"><list-item><p><inline-formula><mml:math id="ieqn21"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:mi>V</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>S</mml:mi><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>S</mml:mi><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo stretchy="false">(</mml:mo><mml:mi>S</mml:mi><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>S</mml:mi><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>S</mml:mi><mml:mo>&#x2265;</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mstyle></mml:math></inline-formula></p></list-item></list></list-item><list-item><p>Compute the vector W&#x2019;.</p><list list-type="simple"><list-item><p><inline-formula><mml:math id="ieqn22"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msup><mml:mi>W</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mstyle></mml:math></inline-formula></p></list-item><list-item><p><inline-formula><mml:math id="ieqn23"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x2032;</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo movablelimits="true" form="prefix">min</mml:mo><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mo>&#x2265;</mml:mo><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula> Where</p></list-item><list-item><p>for i=1, 2, &#x2026;, k and j=1, 2, &#x2026;, k and i &#x2260; j</p></list-item><list-item><p>Normalized vector,  <inline-formula><mml:math id="ieqn24"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>A</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>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mstyle></mml:math></inline-formula></p></list-item><list-item><p>W is a nonfuzzy number calculated for each comparison matrix [<xref ref-type="bibr" rid="ref14">14</xref>],</p></list-item><list-item><p>Now, the weights have been calculated through FEA, thus using these weights and an appropriate weight accumulation formula, OHI can be calculated.</p></list-item></list></list-item></list></sec></sec></sec><sec id="s2-7"><title>Step 9</title><p>For the accumulation of the indicators and the weights, the following formula can be used:</p><disp-formula id="equWL4"><mml:math id="eqn11"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>i</mml:mi><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>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>w</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>w</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>100</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo fence="true" stretchy="true" symmetric="true"/></mml:mrow><mml:mo>&#x2217;</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:mstyle></mml:math></disp-formula><p>where <inline-formula><mml:math id="ieqn25"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> denotes the normalized score for the <inline-formula><mml:math id="ieqn26"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> sub indicator of <inline-formula><mml:math id="ieqn27"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> indicator; <inline-formula><mml:math id="ieqn28"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> denotes the original values for the <inline-formula><mml:math id="ieqn29"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> sub indicator of <inline-formula><mml:math id="ieqn30"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> indicator, <inline-formula><mml:math id="ieqn31"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> denotes the best value for the <inline-formula><mml:math id="ieqn32"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> sub indicator of the <inline-formula><mml:math id="ieqn33"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> indicator, <inline-formula><mml:math id="ieqn34"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> denotes the worst value for the <inline-formula><mml:math id="ieqn35"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> sub indicator of the <inline-formula><mml:math id="ieqn36"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup><mml:mi> </mml:mi></mml:math></inline-formula>indicator. In cases where no data is available for the sub-indicators, substitute <inline-formula><mml:math id="ieqn37"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> with 0.</p><p>The weighted sum of the scores can be given by:</p><p><inline-formula><mml:math id="ieqn38"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi>I</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:mi>S</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:msub><mml:mn>1</mml:mn><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mi>j</mml:mi><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>&#x2217;</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:msub><mml:mn>1</mml:mn><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mstyle></mml:math></inline-formula>,</p><p>where, m depicts the number of subindicators under the <inline-formula><mml:math id="ieqn39"><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> indicator, <inline-formula><mml:math id="ieqn40"><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> depicts the <inline-formula><mml:math id="ieqn41"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi> </mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>subindicator under the <inline-formula><mml:math id="ieqn42"><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> indicator, <inline-formula><mml:math id="ieqn43"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula> depicts the score of the <inline-formula><mml:math id="ieqn44"><mml:msup><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal"> </mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> sub indicator under <inline-formula><mml:math id="ieqn45"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> indicator; <inline-formula><mml:math id="ieqn46"><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula> depicts the weight of <inline-formula><mml:math id="ieqn47"><mml:msup><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal"> </mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> subindicator [<xref ref-type="bibr" rid="ref8">8</xref>].</p><p>The procedure is repeated stage-wise across sub-indicators, indicators, key indicators, and categories to calculate the OHI.</p></sec><sec id="s2-8"><title>Modified Entropy Based Weightage Method (MEWM)</title><p>This approach requires the indicators values for OHI calculation. Weightage mechanism is backed by the entropy-based weightage method for the indicator value and is a preferred method due to the ease of calculation and does not require expert-based rating.</p><sec id="s2-8-1"><title>Step 1</title><p>First, normalization (a systematic process of organizing data in a database to make it more flexible and cohesive) of the values of the sub indicators will be done using the following formula,</p><p><inline-formula><mml:math id="ieqn48"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>=<inline-formula><mml:math id="ieqn49"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi><mml:mtext>&#x00A0;</mml:mtext><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mo movablelimits="true" form="prefix">min</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:math></inline-formula></p><p>where max <inline-formula><mml:math id="ieqn50"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and min <inline-formula><mml:math id="ieqn51"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> are the maximum and minimum values among the alternatives for indicator <italic>j</italic> [<xref ref-type="bibr" rid="ref15">15</xref>] .</p></sec><sec id="s2-8-2"><title>Step 2</title><p>The entropy <inline-formula><mml:math id="ieqn52"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> of each subindicator <italic>i</italic> from the indicator <italic>j</italic>, the entropy <inline-formula><mml:math id="ieqn53"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> of each indicator was determined from the normalized values <inline-formula><mml:math id="ieqn54"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> as formulated:</p><disp-formula id="E14"><mml:math id="eqn12"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mfrac><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2217;</mml:mo><mml:mrow><mml:mi mathvariant="normal">l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">n</mml:mi></mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>ln</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mfrac><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></disp-formula><p>Where <inline-formula><mml:math id="ieqn55"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mstyle></mml:mrow></mml:mstyle></mml:math></inline-formula></p><p>For the cases where  <inline-formula><mml:math id="ieqn56"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>= 0or is not available, the entropy becomes not defined and, in those cases, substitute the entropy with 0.</p></sec><sec id="s2-8-3"><title>Step 3</title><p>For the applicability of the method, the weights <inline-formula><mml:math id="ieqn57"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> are computed as defined below:</p><p><inline-formula><mml:math id="ieqn58"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#x2212;</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:math></inline-formula> , <inline-formula><mml:math id="ieqn59"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mstyle></mml:math></inline-formula>, <inline-formula><mml:math id="ieqn60"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula></p></sec><sec id="s2-8-4"><title>Step 4</title><p>The above formula can be used at every step to calculate the relevant weights,</p><disp-formula id="equWL6"><mml:math id="eqn13"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x2217;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mtext>&#x00A0;</mml:mtext><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></disp-formula><p>where <inline-formula><mml:math id="ieqn61"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is the standardized value of the <inline-formula><mml:math id="ieqn62"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> sub-indicator for the <inline-formula><mml:math id="ieqn63"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi> </mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> state and <inline-formula><mml:math id="ieqn64"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> being the weight calculated for the <inline-formula><mml:math id="ieqn65"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> sub-indicator for the <inline-formula><mml:math id="ieqn66"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi> </mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> state; <inline-formula><mml:math id="ieqn67"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is the weighted indicator value.</p><disp-formula id="equWL7"><mml:math id="eqn14"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#x00A0;</mml:mtext><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munderover><mml:mo movablelimits="false">&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mstyle></mml:mrow></mml:mstyle></mml:math></disp-formula><p>where <inline-formula><mml:math id="ieqn68"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is the weighted indicator value for the <inline-formula><mml:math id="ieqn69"><mml:msup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> subindicator for the <inline-formula><mml:math id="ieqn70"><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi> </mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> state and <inline-formula><mml:math id="ieqn71"><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi> </mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is the index value of the indicator j using which, <inline-formula><mml:math id="ieqn72"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is calculated.</p><p>One can use this formula to arrive at the weighted value of the subindicators and then use those subindicator values to derive weights for the indicators and similarly repeating the process for all available data. These steps will transition stagewise in an agglomerative way from the scores of the sub indicators of a particular indicator to the OHI.</p></sec></sec><sec id="s2-9"><title>Ethical Considerations</title><p>A formal ethical review was waived. This study was not submitted for review as the Tata Institute for Genetics and Society (TIGS) adheres to Indian Council for Medical Research ethical guidelines, which allow exemption for minimal-risk studies using publicly available data. Further, our manuscript is solely based on secondary data from publicly available databases and these were appropriately cited to maintain research integrity.</p></sec></sec><sec id="s3" sec-type="results"><title>Results</title><sec id="s3-1"><title>Primary findings</title><p>Using secondary data sources, a country-specific dataset was developed for India. From this dataset, the indicator values for 156 of the 216 subindicators were gathered, while the remaining 60 subindicators lacked data (<xref ref-type="fig" rid="figure1">Figure 1</xref> and <xref ref-type="supplementary-material" rid="app5">Multimedia Appendix 5</xref>). This India-specific dataset reflects data gaps, that is, inconsistency in data availability and areas where there is absence of data requires planned interventions by governance systems. In some cases, due to the absence of historical data, the scores could not be computed.</p><fig position="float" id="figure1"><label>Figure 1.</label><caption><p>Data availability across indicators. Blue represents the number of subindicators for which the data is available and orange represents the number of subindicators for which the data is unavailable.</p></caption><graphic alt-version="no" mimetype="image" position="float" xlink:type="simple" xlink:href="ojphi_v17i1e65039_fig04.png"/></fig></sec><sec id="s3-2"><title>Pilot Analysis</title><p>Additionally, to evaluate the efficiency of weightage methods, a comparative analysis was conducted between FEA and MEWM. Scores were calculated for 23 indicators for which data for all or most sub-indicators were available (<xref ref-type="fig" rid="figure2">Figure 2</xref>). The findings demonstrate that the indicator scores obtained using the two weightage methods, FEA and MEWM for 23 indicators are significantly consistent and thereby render the two methods reliable. Further, to check the OHI calculator&#x2019;s applicability and utility in the Indian context, a correlation analysis was performed between 20 indicator scores and budget allocations for the years 2022&#x2010;2023, 2023&#x2010;2024, and 2024&#x2010;2025 (<xref ref-type="supplementary-material" rid="app3">Multimedia Appendix 3</xref>). The differences in budget allocations for each consecutive year were calculated, aiding in understanding the correlation between the indicator scores and the changes in budget allocations (<xref ref-type="fig" rid="figure3">Figure 3</xref> and <xref ref-type="fig" rid="figure4">Figure 4</xref>). <xref ref-type="table" rid="table1">Table 1</xref> shows that increases in budget allocations over consecutive years were associated with corresponding increase in indicator scores, and vice versa.</p><p>This emphasizes the importance of budgetary allocation in the performance of indicators, thereby impacting the overall performance of <italic>One Health</italic> in India. The OHI calculator has the potential to perform such nuanced correlation analysis; in this case, between budgetary allocation and performance of the indicators through their scores. This can serve as a valuable insight for policymakers and stakeholders alike in prioritizing sectoral interventions related to <italic>One Health</italic>.</p><fig position="float" id="figure2"><label>Figure 2.</label><caption><p>Comparison of indicator scores using FEA and MEWM. FEA: fuzzy extent analysis; MEWM: modified entropy-based weightage method.</p></caption><graphic alt-version="no" mimetype="image" position="float" xlink:type="simple" xlink:href="ojphi_v17i1e65039_fig05.png"/></fig><fig position="float" id="figure3"><label>Figure 3.</label><caption><p>Budgetary differences for the years 2023&#x2010;2024 and 2022&#x2010;2023 against the indicator scores for consecutive years of budget allocations using the FEA and MEWM. A positive correlation is observed for both methods, indicating that an increase in budget allocation positively impacts the indicator scores. FEA: fuzzy extent analysis; MEWM: modified entropy-based weightage method.</p></caption><graphic alt-version="no" mimetype="image" position="float" xlink:type="simple" xlink:href="ojphi_v17i1e65039_fig06.png"/></fig><fig position="float" id="figure4"><label>Figure 4.</label><caption><p>Budgetary differences for the years 2024&#x2010;2025 and 2023&#x2010;2024 against the indicator scores for consecutive years of budget allocations using the FEA and MEWM. A positive correlation is observed for both methods, indicating that an increase in budget allocation positively impacts the indicator scores. FAHP: fuzzy analytic hierarchy process; MEWM: modified entropy-based weightage method.</p></caption><graphic alt-version="no" mimetype="image" position="float" xlink:type="simple" xlink:href="ojphi_v17i1e65039_fig07.png"/></fig><table-wrap id="t1" position="float"><label>Table 1.</label><caption><p>Correlation between the increase in budget and indicator scores.</p></caption><table id="table1" frame="hsides" rules="groups"><thead><tr><td align="left" valign="bottom">Difference between years</td><td align="left" valign="bottom">Indicator scores (FEA)<sup><xref ref-type="table-fn" rid="table1fn1">a</xref></sup></td><td align="left" valign="bottom">Indicator scores (MEWM)<sup><xref ref-type="table-fn" rid="table1fn2">b</xref></sup></td></tr></thead><tbody><tr><td align="char" char="." valign="top">2024&#x2010;2025 and 2023&#x2010;2024</td><td align="char" char="." valign="top">0.353505892</td><td align="char" char="." valign="top">0.22310977</td></tr><tr><td align="char" char="." valign="top">2023&#x2010;2024 and 2022&#x2010;2023</td><td align="char" char="." valign="top">0.311369994</td><td align="char" char="." valign="top">0.20383558</td></tr></tbody></table><table-wrap-foot><fn id="table1fn1"><p><sup>a</sup>FEA: fuzzy extent analysis.</p></fn><fn id="table1fn2"><p><sup>b</sup>MEWM: modified entropy-based weightage method.</p></fn></table-wrap-foot></table-wrap></sec></sec><sec id="s4" sec-type="discussion"><title>Discussion</title><sec id="s4-1"><title>Summary</title><p>The OHI can be computed at various levels of governance in India, which is constituted of 28 states and 8 Union Territories. Its implementation depends on data availability, such as national, state, district or even block or village/panchayat levels. By calculating these values locally and then aggregating them, an accurate national value can be derived, which highlights demographic variations and provides a more precise measurement. For state-level calculations, population density can serve as a key weighting factor. Collaborative monitoring systems are now recognized as essential for effectively managing pandemics and outbreaks [<xref ref-type="bibr" rid="ref3">3</xref>]. Thus, developing a OHI calculator can contribute to improved implementation of policies using a participatory <italic>One Health</italic> approach. However, the value of such an index extends beyond compiling data; it has the potential to revolutionize our understanding, management, and actions concerning the complex web of factors that influence global health. The process of calculating a OHI for India sheds light on critical areas requiring systematic interventions, aided by policy decisions, particularly regarding resource allocation and strengthening of governance systems. This framework also promotes a collaborative data federation model to address data gaps&#x2014;such as incomplete data, lack of timely data, and the absence of appropriate data&#x2014;and correspondingly advocates for data transparency and ethical data management. Implementing a collaborative data federation model and maintaining consistent data collection will help address these challenges and establish a historical dataset for indexing indicators. This approach encourages intersectoral communication and multistakeholder engagement, garners interest from governance systems, and builds momentum toward improving poorly performing indicators, thereby achieving better <italic>One Health</italic> outcomes for the country. A study focusing on mitigating zoonotic disease risk using the economic approach stated that allocating budget and resources strategically at a higher level secures sufficient funding to manage diseases along the livestock value chain, leading to improvements in human health [<xref ref-type="bibr" rid="ref16">16</xref>]. As shown in <xref ref-type="table" rid="table1">Table 1</xref>, increasing the budget over consecutive years has led to higher indicator scores and vice versa, suggesting that budget increases for relevant ministries and departments contribute to improved indicator scores, thereby enhancing the OHI. Addressing the standardization of impact indicators and integrating new field knowledge are also essential.</p><p>In this study, we have provided a OHI calculator tool, demonstrating two weighting methods, FEA and MEWM, alongside the India-specific datasets that are accessible and efficient for use by multiple stakeholders such as government functionaries, policymakers, researchers, and institutes of disease surveillance and preparedness. While the scope of this study is to develop a OHI calculator for India, the objective is to provide a reliable, easy-to-use, and efficient tool that could be easily adapted by relevant sectoral stakeholders to compute empirical scores and plan informed and strategic interventions. We have also demonstrated OHI value calculations for India using both methods: the FEA method, which assigns equal weightage in the absence of expert weightage, yielded a score of 46.84, while the score obtained using the MEWM is 42.64. These values demonstrated that OHI calculation values fall within the range reported for the South Asia region, which is 35&#x2010;50 as published in 2022, using the GOHI framework [<xref ref-type="bibr" rid="ref8">8</xref>]. Further, it is also pertinent to conduct expert consultations to review the indicators adopted from the GOHI framework and to develop a list of indicators for India, which may be region-specific and locally adapted.</p><p>There are some limitations to be considered regarding the OHI calculator. For instance, the timeline for data availability is inconsistent across subindicators, and any gaps in data may impact the OHI score. Identifying reliable, validated data sources which is crucial for accuracy has been a challenge for many indicators. The selection of experts is a key factor, as their input determines the weightages assigned to different indicators. Any bias from the experts or the absence of a representative sample may result in a skewed OHI score. To avoid regional bias, it is crucial to ensure diverse representation from across the country for balanced input.</p></sec><sec id="s4-2"><title>Conclusion</title><p>The OHI calculator identifies key areas that require concerted action from <italic>One Health</italic>-centric stakeholders. During its development, data gaps and deficiencies for crucial <italic>One Health</italic> indicators were identified, suggesting that data federation and open data access from governance systems and research organizations working in the public interest should emerge as actionable agendas for collaborative efforts. In this context, the methodology and framework for calculating a single OHI value require multisectoral experts to come together to improve the OHI for India by recognizing the need for sectoral data. This in turn, leads to identifying disparities, targeting interventions, monitoring health trends, and implementing other strategic efforts aligned toward the health equity paradigm. Additionally, the importance of budgetary allocation for improving indicator scores that contribute to OHI, serves as a reminder to policymakers about the value of empirical and evidence-based decision making. The process of calculating an empirical OHI value demands consistent public engagement and awareness of the interconnected nature of the <italic>One Health</italic> approach. Ultimately, this leads to better preparedness for handling future pandemics, improved quality of life, and progress toward achieving the sustainable development goals.</p></sec></sec></body><back><ack><p>All authors declared that they had insufficient funding to support open access publication of this manuscript, including from affiliated organizations or institutions, funding agencies, or other organizations. 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KB"/></supplementary-material><supplementary-material id="app3"><label>Multimedia Appendix 3</label><p>Budgetary allocation sheet vs. indicator scores.</p><media xlink:href="ojphi_v17i1e65039_app3.xlsx" xlink:title="XLSX File, 13 KB"/></supplementary-material><supplementary-material id="app4"><label>Multimedia Appendix 4</label><p>Sample proforma for expert rating.</p><media xlink:href="ojphi_v17i1e65039_app4.docx" xlink:title="DOCX File, 55 KB"/></supplementary-material><supplementary-material id="app5"><label>Multimedia Appendix 5</label><p>List of subindicators for which data is unavailable.</p><media xlink:href="ojphi_v17i1e65039_app5.xlsx" xlink:title="XLSX File, 12 KB"/></supplementary-material></app-group></back></article>