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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Sustain. Food Syst.</journal-id>
<journal-title>Frontiers in Sustainable Food Systems</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Sustain. Food Syst.</abbrev-journal-title>
<issn pub-type="epub">2571-581X</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.3389/fsufs.2022.1007915</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Sustainable Food Systems</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Assessing the efficiency level of the &#x0201C;poverty alleviation through agriculture project&#x0201D;: A case study of fixed observation points in China</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name><surname>Yan</surname> <given-names>Shuya</given-names></name>
<xref ref-type="aff" rid="aff1"><sup>1</sup></xref>
</contrib>
<contrib contrib-type="author" corresp="yes">
<name><surname>Li</surname> <given-names>Lipeng</given-names></name>
<xref ref-type="aff" rid="aff1"><sup>1</sup></xref>
<xref ref-type="corresp" rid="c001"><sup>&#x0002A;</sup></xref>
<uri xlink:href="http://loop.frontiersin.org/people/1982293/overview"/>
</contrib>
<contrib contrib-type="author">
<name><surname>Sarkar</surname> <given-names>Apurbo</given-names></name>
<xref ref-type="aff" rid="aff2"><sup>2</sup></xref>
<uri xlink:href="http://loop.frontiersin.org/people/1779042/overview"/>
</contrib>
<contrib contrib-type="author">
<name><surname>Yang</surname> <given-names>Guotao</given-names></name>
<xref ref-type="aff" rid="aff1"><sup>1</sup></xref>
</contrib>
</contrib-group>
<aff id="aff1"><sup>1</sup><institution>School of Economics and Management, Ningxia University</institution>, <addr-line>Yinchuan, Ningxia</addr-line>, <country>China</country></aff>
<aff id="aff2"><sup>2</sup><institution>College of Economics and Management, Northwest A&#x00026;F University</institution>, <addr-line>Xianyang</addr-line>, <country>China</country></aff>
<author-notes>
<fn fn-type="edited-by"><p>Edited by: Roberta Selvaggi, University of Catania, Italy</p></fn>
<fn fn-type="edited-by"><p>Reviewed by: Filippo Sgroi, University of Palermo, Italy; Justice Gameli Djokoto, Central University, Ghana; Sahya Maulu, Nanjing Agricultural University, China</p></fn>
<corresp id="c001">&#x0002A;Correspondence: Lipeng Li <email>feixiang2022&#x00040;nxu.edu.cn</email></corresp>
<fn fn-type="other" id="fn001"><p>This article was submitted to Social Movements, Institutions and Governance, a section of the journal Frontiers in Sustainable Food Systems</p></fn></author-notes>
<pub-date pub-type="epub">
<day>26</day>
<month>09</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="collection">
<year>2022</year>
</pub-date>
<volume>6</volume>
<elocation-id>1007915</elocation-id>
<history>
<date date-type="received">
<day>31</day>
<month>07</month>
<year>2022</year>
</date>
<date date-type="accepted">
<day>07</day>
<month>09</month>
<year>2022</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#x000A9; 2022 Yan, Li, Sarkar and Yang.</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>Yan, Li, Sarkar and Yang</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/"><p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p></license>
</permissions>
<abstract>
<p>In the modern era, development organizations and governments worldwide are undertaking various policies and projects to eradicate poverty. However, there is a lack of evidence that can trigger the efficiency level of those. Based on the survey data of the Ministry of Agriculture and Rural Affairs, which was acquired at rural fixed observation points across 31 provinces of China from 2012 to 2016, the study evaluates the overall efficiency, stage-specific efficiency, and indicator-based efficiency of &#x0201C;Poverty alleviation through agriculture projects of China&#x0201D;. First of all, the entire process of agricultural poverty relief is divided into two stages: (i) agricultural production and (ii) social governance. Accordingly, the study proposes a two-stage theoretical analysis framework for agricultural poverty relief and decomposes the mechanisms; it also discusses the potential for improved efficiency levels in both agricultural production efficiency and social governance efficiency. Therefore, we utilize the two-stage dynamic data envelopment analysis (DEA) model to outline the findings. The outcomes showed the efficiency level of the projects can play an important role in addressing rural poverty in China. This study&#x00027;s major findings are summarized as follows: (i) the overall efficiency of the projects tends to be stable undauntedly. While agricultural production efficiency is the major cause and social governance efficiency in the second stage has been a minor cause for maintaining a relatively lower level of overall efficiency. (ii) There is significant room for improving the efficiency of certain input indicators (including total labor force, productive fixed assets, and education attainment of rural labor) and intermediate variables (i.e., income gap of village households). However, limited room has been found for certain output indicators (including the total output of grain, the poverty elimination index, and an aggregate index of social harmony). Thus, in China, poverty alleviation projects should be revitalized and targeted instead of concentrated. It is required to advance a long-term structure for rural poverty and promote the smooth transition of poverty alleviation projects and working criteria. Moreover, the government should strengthen the top-level design for addressing the relative poverty problem and incorporate it into the rural revitalization strategy.</p></abstract>
<kwd-group>
<kwd>poverty alleviation</kwd>
<kwd>agriculture</kwd>
<kwd>production</kwd>
<kwd>social governance</kwd>
<kwd>efficiency</kwd>
<kwd>dynamic data envelopment analysis</kwd>
</kwd-group>
<contract-sponsor id="cn001">National Natural Science Foundation of China<named-content content-type="fundref-id">10.13039/501100001809</named-content></contract-sponsor>
<counts>
<fig-count count="5"/>
<table-count count="3"/>
<equation-count count="12"/>
<ref-count count="71"/>
<page-count count="14"/>
<word-count count="8715"/>
</counts>
</article-meta>
</front>
<body>
<sec sec-type="intro" id="s1">
<title>Introduction</title>
<p>The notion of poverty is primarily regarded as an obvious deprivation of sustainable development goals set by the united nations, especially for emerging nations it has been causing significant and persistent burdens (Adamkovi&#x0010D; and Marton&#x0010D;ik, <xref ref-type="bibr" rid="B1">2017</xref>; Maulu et al., <xref ref-type="bibr" rid="B45">2021</xref>). In economics, it is considered a worldwide socioeconomic phenomenon that is generally studied from macroeconomic aspects and often involves multi-dimensional concepts (Bradshaw, <xref ref-type="bibr" rid="B8">2007</xref>; Zhou and Che, <xref ref-type="bibr" rid="B68">2021</xref>). In recent trends of poverty-related literature, there is a clearer understanding of the reality that market-driven economic expansion does not have to be incompatible with the reduction of poverty (Liu, <xref ref-type="bibr" rid="B41">2018</xref>; Leng et al., <xref ref-type="bibr" rid="B36">2021</xref>). In general, improving chances for sustainable and financially rewarding work (such as through entrepreneurship and laboring) is the most essential approach for socioeconomic development to assist the poor (Peng et al., <xref ref-type="bibr" rid="B50">2021</xref>; Shen and Li, <xref ref-type="bibr" rid="B54">2022</xref>). Conversely, several initiatives such as easy mortgage and financing options, improve risk-taking mechanisms and machinery supports which eventually can rectify the overall growth by availing allocates efficient resources allocation (Maja Gavrilovic, <xref ref-type="bibr" rid="B43">2016</xref>; Guo and Liu, <xref ref-type="bibr" rid="B26">2021</xref>). In this thrives poverty alleviation through agriculture becomes a prominent concept for the government, academia, and international and regional development organizations. Therefore, organizations like FAO and World Bank are predominantly highlighting the significance of agricultural interventions toward poverty alleviation (Charles, <xref ref-type="bibr" rid="B11">2019</xref>; World Bank, <xref ref-type="bibr" rid="B62">2021</xref>). According to the recent reports of the World Bank and Nations (Dah and Bassolet, <xref ref-type="bibr" rid="B16">2021</xref>; World Bank and Nations, <xref ref-type="bibr" rid="B63">2021</xref>), the rapid outward-oriented agricultural expansion has been linked to significant decreases in widespread poverty in several nations, particularly in East and Southeast Asia. According to Ellis and Freeman (Ellis and Freeman, <xref ref-type="bibr" rid="B20">2005</xref>), the growth in agriculture further stimulates demand for other goods, acting as a growth multiplier in the rural economy and eventually reducing overall rural poverty. As a result, agricultural intervention is a critical first step toward alleviating regional poverty as a source of socioeconomic development (Shen, <xref ref-type="bibr" rid="B53">2022</xref>). This is particularly so if the traded goods sector is more labor-intensive than the non-traded goods sector and if exports are more labor-intensive than import substitutes (assuming, of course, that the workers have some basic education and skills). Since China&#x00027;s reform and opening up, its per capita gross domestic product has continued to grow, the standard of living has greatly improved, and the absolute poverty-stricken population in the countryside has decreased significantly (Cheng et al., <xref ref-type="bibr" rid="B15">2018</xref>; Miller et al., <xref ref-type="bibr" rid="B46">2021</xref>). According to current poverty standards, China&#x00027;s rural poor population was 770 million in 1978, with a poverty incidence of 97.5%, but decreased to merely 5.51 million in 2019, with a poverty incidence of only 0.6% (Dong et al., <xref ref-type="bibr" rid="B18">2021</xref>). President Xi Jinping announced China&#x00027;s victory against absolute poverty at the national commendation conference held in Beijing, on 25 February 2021 (Eryong and Xiuping, <xref ref-type="bibr" rid="B21">2018</xref>). In the 20th National Congress of the Chinese Communist Party, President Xi committed that, the rural poor population will be lifted out of poverty by 2030, and existing poverty-stricken counties will all be delisted, thus eliminating overall regional poverty (Dong, <xref ref-type="bibr" rid="B17">2022</xref>).</p>
<p>Various studies of poverty-stricken areas across the globe showed that a significant poverty level has a greater impact on unemployment and eventually hinders the development policies taken by the government (Bouwman et al., <xref ref-type="bibr" rid="B7">2021</xref>; Zhu et al., <xref ref-type="bibr" rid="B70">2022</xref>). Therefore, the relationship between agricultural interventions and availing better livelihoods for farmers and employment opportunities for rural households has been long examined worldwide (for example Hurst et al., <xref ref-type="bibr" rid="B30">2005</xref>; Knickel et al., <xref ref-type="bibr" rid="B34">2009</xref>; Martens et al., <xref ref-type="bibr" rid="B44">2020</xref>). In a recent study, Wang et al. (<xref ref-type="bibr" rid="B59">2020</xref>) pointed out the existing need to direct attention toward the currently neglected issue of measuring efficiencies of poverty alleviation policies and projects, which has generally been overlooked in favor of assessing poverty from a solely economical or even macro economical perspective (see e.g., Kaidi et al., <xref ref-type="bibr" rid="B31">2019</xref>; Ali et al., <xref ref-type="bibr" rid="B3">2020</xref>; Zameer et al., <xref ref-type="bibr" rid="B67">2020</xref>). Sole observations of economic expansion have been impeded in many situations, whereas inequalities have remained constant or to some extent even exacerbated (Bardhan, <xref ref-type="bibr" rid="B5">1996</xref>; Liu et al., <xref ref-type="bibr" rid="B39">2021a</xref>). Seemingly, various initiatives have been working to alleviate the tremendous limitations encountered by the poor, and enhancing their conditions can also aid macroeconomic expansion (Macfadyen et al., <xref ref-type="bibr" rid="B42">2002</xref>; Mujuru et al., <xref ref-type="bibr" rid="B47">2022</xref>). The situation may often contradict the conventional theory of economics with the equity-efficiency trade-off, as the efficiency rectifies different allocation costs, such as lower monetary incentives and overall competence (Kerr, <xref ref-type="bibr" rid="B33">2002</xref>; Wongnaa et al., <xref ref-type="bibr" rid="B61">2019</xref>). However, the impacts of governmental poverty relief programs on poverty alleviation have long been explored globally. For example, Fabiyi and Akande (Fabiyi and Akande, <xref ref-type="bibr" rid="B22">2015</xref>) explored Nigerian rural women and found agricultural interventions can substantially help for fostering women empowerment and poverty relief. In a study of Bangladeshi rural farmers, Wei et al. (<xref ref-type="bibr" rid="B60">2021</xref>) indicated that the targeted poverty alleviation and food security programs administrated by the government significantly changed the socioeconomic conditions of rural households and help substantially to eliminate them from extreme poverty lines. Seemingly, Alema&#x000F1; et al. (<xref ref-type="bibr" rid="B2">2019</xref>) depicted the major transitional effects of &#x0201C;poverty reduction through seaweed cultivation&#x0201D; among Latin American rural farmers.</p>
<p>Moreover, after the critical tasks of poverty elimination are completed, China&#x00027;s poverty status will change remarkably (Li et al., <xref ref-type="bibr" rid="B38">2018</xref>). The key to poverty relief lies in agricultural and rural economic growth, thus attaching importance to the agricultural sector is a prerequisite for continuous poverty relief (Wu et al., <xref ref-type="bibr" rid="B65">2018</xref>). Thus, poverty-relief projects will be refocused to address relative poverty and will be carried out in a normal rather than centralized manner. Hence, it is necessary to develop a long-term mechanism for reducing relative poverty and promoting the smooth transformation of the poverty relief strategy and work system (Wu et al., <xref ref-type="bibr" rid="B64">2019</xref>; Bai et al., <xref ref-type="bibr" rid="B4">2021</xref>). The Chinese government is also taking several policies such as China declaring that &#x0201C;the National Strategic Plan on Rural Revitalization <italic>Via</italic> Agriculture&#x0201D; will be prolonged till 2030. President and General Secretary of the Communist Party of China, Xi Jinping, gave important instructions to the National Spring Agricultural Production Work conference held in February 2020: &#x0201C;The more we face risks and challenges, the more we need to stabilize agriculture, and the more we need to ensure the security of food and important subsidiary toward poverty alleviation&#x0201D;. Those highlights the integrity of China&#x00027;s long-term agenda toward the reform of agricultural development in quality, efficiency, and impetus and improving the innovativeness, competitiveness, and total factor productivity of agriculture continuously. Thus, it depicted the importance of examining whether the poverty alleviation projects efficiently help reduce poverty sustainably. However, the existing studies mainly explored the impacts of poverty alleviation and the linkage of core agriculture factors such as income stability, production factors, factor and resources endowments using different indicators in an isolated manner [Such as Cai and Xia (<xref ref-type="bibr" rid="B10">2018</xref>), Gassner et al. (<xref ref-type="bibr" rid="B24">2019</xref>) and Bird et al. (<xref ref-type="bibr" rid="B6">2022</xref>)]. While these factors are substantially interrelated and demand an integrated assessment within a single framework (Maulu et al., <xref ref-type="bibr" rid="B45">2021</xref>; Sgroi and Marino, <xref ref-type="bibr" rid="B52">2022</xref>).</p>
<p>Therefore, the prime goal of the study is to evaluate the efficiency level of China&#x00027;s poverty alleviation through agriculture projects in terms of agricultural production efficiency and social governance efficiency. We explore the probable efficiency improvement opportunities and summarize the experience and practical views of the projects. More specifically, the study evaluates the overall efficiency, stage-specific efficiency, and indicator-based efficiency of agricultural poverty relief in China. Seemingly, we redefined the social governance efficiency according to the local prospects, education attainment, and income gap are used as input variables, and the aggregate index of social harmony is used as an output variable. The empirical framework of the study is supported by the data collected by the Ministry of Agriculture and Rural Affairs, acquired at rural fixed observation points across 31 provinces of China from 2012 to 2016.</p>
<p>The major contributions of the study are as follows. First, the entire process of agricultural poverty relief is divided into the agricultural production and social governance stages. A novel rural revitalization model triggering the overall efficiency and stage-specific efficiency of poverty alleviation through agriculture projects has been proposed and tested based on village-level data. This would be the major contribution of the study. Second, a two-stage dynamic SBM-DEA model with undesirable output is applied to the study of agricultural production efficiency and social governance efficiency which would provide a more comprehensive outcome. Third, to the best of our knowledge, the study will serve as the first attempt to integrate several crucial factors of poverty alleviation such as social governance, education attainment, disposable income, and income gap in a single framework. The study will be crucial for governmental bodies and development partner organizations as it eventually provides in-depth policy suggestions for promoting the comprehensive revitalization of China&#x00027;s rural areas and fosters references for rural development in developing countries.</p>
</sec>
<sec sec-type="materials and methods" id="s2">
<title>Materials and methods</title>
<sec>
<title>The theoretical and analytical framework</title>
<p>In this study, the entire process of agricultural poverty relief is divided into two stages, an agricultural production stage and a social governance stage. Whereas agricultural production efficiency is evaluated in the first stage, and social governance efficiency is evaluated in the second stage. Overall efficiency is equal to the weighted average of both. In the first stage, the total labor force, sown area of farm crops, and productive fixed assets are used as input variables, and the total output of grain is an output variable as suggested by Thongdara et al. (<xref ref-type="bibr" rid="B55">2012</xref>) and Liu et al. (<xref ref-type="bibr" rid="B40">2021b</xref>). While according to the study by Ellis and Freeman (<xref ref-type="bibr" rid="B19">2004</xref>), disposable income per village household and the income gap of village households are used as intermediate variables. At the social governance stage, education attainment of rural labor is used as an input variable and the aggregate index of social harmony and the poverty elimination index are used as output variables, as suggested by Huang et al. (<xref ref-type="bibr" rid="B28">2021a</xref>). The study used an aggregate index of social harmony as an undesirable output variable because the degree of social harmony can explore the external impacts of poverty more comprehensively as suggested by Pan et al. (<xref ref-type="bibr" rid="B49">2021b</xref>). Moreover, Social governance efficiency in the second stage covers two aspects: (1) poverty elimination effectiveness in poor areas, measured in terms of non-poor population; and (2) social harmony and stability, measured in terms of the aggregate index of social harmony. Moreover, productive fixed assets are not only an input variable in the first stage but also an inter-period variable for the two-stage dynamic SBM-DEA model, to connect different periods. <xref ref-type="fig" rid="F1">Figure 1</xref> shows the theoretical analysis framework for two-stage agricultural poverty relief.</p>
<fig id="F1" position="float">
<label>Figure 1</label>
<caption><p>Theoretical analysis framework for two-stage agricultural poverty relief.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fsufs-06-1007915-g0001.tif"/>
</fig>
</sec>
<sec>
<title>Methodology</title>
<p>The study utilized Data Envelopment Analysis (DEA) for crafting its findings. The DEA method is a linear programming model which is popularly used in social sciences and expressed as a ratio of output to input (Pan et al., <xref ref-type="bibr" rid="B48">2021a</xref>). The study utilized a combination of a two-stage dynamic Slacks-Based Measure (SBM) model for performing the Data Envelopment Analysis (DEA). That is based on panel data that connects the linear programming problems of each period and does not require parameter estimation in the calculation process (Kao, <xref ref-type="bibr" rid="B32">2018</xref>; Kuang et al., <xref ref-type="bibr" rid="B35">2020</xref>). The primary reasons behind choosing the methodology are as follows: the conventional DEA model is used to measure the relative efficiency of multi-input and multi-output decision-making units (DMUs) (Charnes et al., <xref ref-type="bibr" rid="B12">1978</xref>). However, it only considers input and output but omits the internal structure of DMUs (Chen et al., <xref ref-type="bibr" rid="B14">2010</xref>). Existing studies [such as Tone and Tsutsui (<xref ref-type="bibr" rid="B57">2010</xref>, <xref ref-type="bibr" rid="B58">2014</xref>), and Guo et al. (<xref ref-type="bibr" rid="B25">2017</xref>)] criticized the issue of internal structure and are widely referred to as a &#x0201C;black box,&#x0201D; and possessed a certain level of difficulty to understand the critical factors when the framework incorporates input and output characteristics. In social sciences and behavioral studies, each of many problems or events comprises multiple stages or departments, which are linked to each other and have specific input and output variables, as well as linking variables interlinked with other stages or departments. Therefore, it is necessary to construct a multi-stage or multi-department network for assessing the situations. F&#x000E4;re et al. (<xref ref-type="bibr" rid="B23">2007</xref>) were the first who use network DEA to address the &#x0201C;black box&#x0201D; problem in the production process, which was ignored by the traditional DEA model. Tone and Tsutsui (<xref ref-type="bibr" rid="B56">2009</xref>) proposed a weighted SBM network and DEA model to explore the linkage between different stages or departments of the DMU, where the stages or departments were regarded as sub-DMUs and the SBM model was used to find the optimal solution.</p>
<p>Although network DEA has resolved the &#x0201C;black box&#x0201D; problem, it is restricted to only one stage, therefore Tone and Tsutsui (<xref ref-type="bibr" rid="B57">2010</xref>) proposed the dynamic SBM (DSBM) model, in which carry-over was used as a variable of linkage between different stages. Based on dynamic DEA combined with network DEA, Tone and Tsutsui (<xref ref-type="bibr" rid="B58">2014</xref>) evaluated the overall and internal variation of DMUs in the long term. This combined model can not only measure the overall efficiency of a DMU across the whole observation period but also perform further analysis and observe the dynamic variation in the DMU&#x00027;s overall and stage-specific efficiency. Chen et al. (<xref ref-type="bibr" rid="B13">2019</xref>) amended the dynamic network DEA model proposed by Tone and Tsutsui (<xref ref-type="bibr" rid="B57">2010</xref>), specifically considering undesirable output and referring to the amended model as the improved dynamic network model with undesirable output. Therefore, the improved two-stage dynamic SBM-DEA model has been selected in the study. <xref ref-type="fig" rid="F2">Figure 2</xref> shows the two-stage dynamic DEA model framework.</p>
<fig id="F2" position="float">
<label>Figure 2</label>
<caption><p>Two-stage dynamic SBM-DEA model frameworks with consideration of undesirable output.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fsufs-06-1007915-g0002.tif"/>
</fig>
<p>The selection of the model is mainly based on the following points: first, the theoretical analysis framework of the study divides the research problem into two stages (i) agricultural production and (ii) social governance. The traditional DEA model omits the internal structure of the model and cannot decompose and measure the total efficiency. Therefore, we use a two-stage dynamic SBM model to decompose and carry the research process more robustly as suggested by Huang et al. (<xref ref-type="bibr" rid="B29">2021b</xref>). Second, the static model based on cross-sectional data cannot see the dynamic efficiency changes between DMUs. This paper introduces a dynamic model to analyze the efficiency changes between periods. Third, there are often undesirable output indicators in most forms of measuring efficiency analysis (Zhuang et al., <xref ref-type="bibr" rid="B71">2021</xref>; Sgroi, <xref ref-type="bibr" rid="B51">2022</xref>). Therefore, the study considers undesired outputs in the core model to provide comprehensive efficiency outcomes. The empirical analysis of the study follows four steps. First, we analyze the overall results from the perspective of comprehensive efficiency and compare the efficiency differences between DMUs. Second, the efficiency of agricultural production and social governance has been measured, and then eventually measured the total efficiency of each stage. Third, the study compares and analyzes the dynamic changes in the efficiency of each year&#x00027;s decision-making units. Finally, we analyze the efficiency difference between the various indicators and draw research conclusions based on the empirical analysis.</p>
</sec>
<sec>
<title>Data sources</title>
<p>Considering that there may be a non-linear relationship between some variables and food imports, and the economic threshold model can more accurately explore this relationship between variables, the study adopted the theory and practice of &#x0201C;Threshold Auto-regression&#x0201D; proposed by Hansen (<xref ref-type="bibr" rid="B27">2011</xref>) and built the following single threshold panel model to explore whether each variable has a threshold value. <xref ref-type="table" rid="T1">Table 1</xref> depicts the descriptive statistics of input and output variables.</p>
<table-wrap position="float" id="T1">
<label>Table 1</label>
<caption><p>Descriptive statistics of input and output variables.</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th valign="top" align="left"><bold>Input/output variable</bold></th>
<th valign="top" align="left"><bold>Unit</bold></th>
<th valign="top" align="center"><bold>Mean</bold></th>
<th valign="top" align="center"><bold>Minimum</bold></th>
<th valign="top" align="center"><bold>Maximum</bold></th>
<th valign="top" align="center"><bold>Standard deviation</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">Total labor force</td>
<td valign="top" align="left">Person</td>
<td valign="top" align="center">1,376.07</td>
<td valign="top" align="center">238</td>
<td valign="top" align="center">4,355</td>
<td valign="top" align="center">961.47</td>
</tr>
<tr>
<td valign="top" align="left">Productive fixed assets</td>
<td valign="top" align="left">10,000 yuan</td>
<td valign="top" align="center">752.80</td>
<td valign="top" align="center">12.42</td>
<td valign="top" align="center">30,872.24</td>
<td valign="top" align="center">3,041.56</td>
</tr>
<tr>
<td valign="top" align="left">The sown area of farm crops</td>
<td valign="top" align="left">Mu</td>
<td valign="top" align="center">6,029.09</td>
<td valign="top" align="center">278</td>
<td valign="top" align="center">34,000</td>
<td valign="top" align="center">6,452.10</td>
</tr>
<tr>
<td valign="top" align="left">The total output of grain</td>
<td valign="top" align="left">Ton</td>
<td valign="top" align="center">7,240.36</td>
<td valign="top" align="center">295</td>
<td valign="top" align="center">83,407.05</td>
<td valign="top" align="center">10,452.97</td>
</tr>
<tr>
<td valign="top" align="left">Disposable income per village household</td>
<td valign="top" align="left">Yuan</td>
<td valign="top" align="center">31,733.01</td>
<td valign="top" align="center">8,416.89</td>
<td valign="top" align="center">82,955.13</td>
<td valign="top" align="center">11,645.33</td>
</tr>
<tr>
<td valign="top" align="left">Income gap of village households</td>
<td valign="top" align="left">/</td>
<td valign="top" align="center">6.16</td>
<td valign="top" align="center">1.49</td>
<td valign="top" align="center">19.74</td>
<td valign="top" align="center">3.65</td>
</tr>
<tr>
<td valign="top" align="left">Educational attainment of rural labor</td>
<td valign="top" align="left">Year</td>
<td valign="top" align="center">8.06</td>
<td valign="top" align="center">3</td>
<td valign="top" align="center">10</td>
<td valign="top" align="center">1.10</td>
</tr>
<tr>
<td valign="top" align="left">Poverty elimination index</td>
<td valign="top" align="left">Person</td>
<td valign="top" align="center">2,429.82</td>
<td valign="top" align="center">11</td>
<td valign="top" align="center">7,805</td>
<td valign="top" align="center">1,689.38</td>
</tr>
<tr>
<td valign="top" align="left">Aggregate index of social harmony</td>
<td valign="top" align="left">/</td>
<td valign="top" align="center">23.45</td>
<td valign="top" align="center">1</td>
<td valign="top" align="center">463</td>
<td valign="top" align="center">44.71</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
<sec>
<title>Measurement model</title>
<p>The study assumes that there are <italic>n DMUs</italic> (<italic>j</italic> &#x0003D; 1, &#x02026;, <italic>n</italic>) and each DMU has <italic>k</italic> (<italic>k</italic> &#x0003D; 1, &#x02026;, <italic>K</italic>) stages and <italic>T</italic> (<italic>t</italic> &#x0003D; 1, &#x02026;, <italic>T</italic>) periods. For each DMU, the t<sup>th</sup> period and (t&#x0002B;1)<sup>th</sup> period are linked through carry-over. Assuming that <italic>m</italic><sub><italic>k</italic></sub> and <italic>r</italic><sub><italic>k</italic></sub> denotes the input and output of each stage (<italic>K</italic>), denotes the variable of the link from the <italic>k</italic><sup>th</sup> stage to <italic>h</italic><sup>th</sup> stage, and <italic>L</italic><sub><italic>hk</italic></sub> denotes the number of stages from the <italic>k</italic><sup>th</sup> stage to the <italic>h</italic><sup>th</sup> stage. The input, output, and intermediate indicators, and the inter-period link indicator (carry-over) are defined as follows.</p>
<sec>
<title>Input and output indicators</title>
<p>Firstly, <inline-formula><mml:math id="M1"><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x02208;</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x0002B;</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><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:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>n</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</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:mo>;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> denotes the input <italic>i</italic> of DMU<sub>j</sub> at the k<sup>th</sup> stage of the t<sup>th</sup> period; <inline-formula><mml:math id="M2"><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x02208;</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x0002B;</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><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:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>n</mml:mi><mml:mo>;</mml:mo><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</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:mo>;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> denotes the desirable output of at the k<sup>th</sup> stage of the t<sup>th</sup> period; <inline-formula><mml:math id="M3"><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x02208;</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x0002B;</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>(<italic>r</italic> &#x0003D; 1, &#x02026;<italic>r</italic><sub>2<italic>k</italic></sub>; <italic>j</italic> &#x0003D; 1, 2, &#x02026;<italic>n</italic>; <italic>k</italic> &#x0003D; 1, &#x02026;, <italic>K</italic>; <italic>t</italic> &#x0003D; 1, &#x02026;<italic>T</italic>) denotes the undesirable output of DMU<sub>j</sub> at the k<sup>th</sup> stage of the t<sup>th</sup> period.</p>
</sec>
<sec>
<title>Intermediate indicators (links)</title>
<p>First,<inline-formula><mml:math id="M4"><mml:msubsup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x02208;</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x0002B;</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><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:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>n</mml:mi><mml:mo>;</mml:mo><mml:mi>l</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:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mi>t</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:mi>T</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> denotes the inter-period link (carry-over) of DMU<sub>j</sub> at the k<sup>th</sup> stage of the t<sup>th</sup> period; <italic>L</italic><sub><italic>k</italic></sub> denotes the number of inter-period link indicators at the k<sup>th</sup> stage. As mentioned above, there are four types of carry-over. In this study, advisable carry-over is selected, thus the number of inter-period link indicators at the k<sup>th</sup> stage is equal to the advisable carry-over quantity at the k<sup>th</sup> stage <italic>ngood</italic><sub><italic>k</italic></sub>.</p>
<p>Second, <inline-formula><mml:math id="M5"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></inline-formula> (<italic>t</italic> &#x0003D; 1, &#x02026;, <italic>T</italic>) (the production possibility set) is defined as follows:</p>
<disp-formula id="E1"><mml:math id="M6"><mml:mtable columnalign='left'><mml:mtr><mml:mtd><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>x</mml:mtext></mml:mstyle><mml:mi>k</mml:mi><mml:mi>t</mml:mi></mml:msubsup><mml:mo>&#x02265;</mml:mo><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>x</mml:mtext></mml:mstyle><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:msubsup><mml:mi>&#x003BB;</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup></mml:mrow></mml:mstyle><mml:mtext>&#x000A0;(</mml:mtext><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mtext>)</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle><mml:mrow><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:mo>&#x02264;</mml:mo><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:msubsup><mml:mi>&#x003BB;</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup></mml:mrow></mml:mstyle><mml:mtext>&#x000A0;(</mml:mtext><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mtext>)</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle><mml:mrow><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:mo>&#x02264;</mml:mo><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:msubsup><mml:mi>&#x003BB;</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup></mml:mrow></mml:mstyle><mml:mtext>&#x000A0;(</mml:mtext><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mtext>)</mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p><inline-formula><mml:math id="M7"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x02211;</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>n</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">(</mml:mtext></mml:mstyle><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">)</mml:mtext></mml:mstyle></mml:math></inline-formula> <inline-formula><mml:math id="M8"><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x02211;</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>n</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">(</mml:mtext></mml:mstyle><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">)</mml:mtext></mml:mstyle></mml:math></inline-formula> (carry-over of the t<sup>th</sup> period) <inline-formula><mml:math id="M11"><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x02211;</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>n</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">(</mml:mtext></mml:mstyle><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">)</mml:mtext></mml:mstyle></mml:math></inline-formula> (carry-over of the (t&#x0002B;1)<sup>th</sup> period) <inline-formula><mml:math id="M12"><mml:msubsup><mml:mrow><mml:mo>&#x02211;</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>n</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">(</mml:mtext></mml:mstyle><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">),</mml:mtext></mml:mstyle><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x02265;</mml:mo><mml:mn>0</mml:mn><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">(</mml:mtext></mml:mstyle><mml:mo>&#x02200;</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">)</mml:mtext></mml:mstyle></mml:math></inline-formula> (constant returns to scale).</p>
<p>Where <inline-formula><mml:math id="M13"><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>&#x02208;</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x0002B;</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> denotes the weight vector at the Stage of <italic>k</italic>(&#x02200;<italic>k</italic>) the period <italic>t</italic>(&#x02200;<italic>t</italic>).<inline-formula><mml:math id="M14"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mo>&#x003BB;</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> is used as a variable to evaluate the overall efficiency of DMU<sub>&#x000B0;</sub> (o = 1,&#x02026;.,n) &#x02208; <italic>P</italic><sup><italic>t</italic></sup>. The input and output constraints are expressed as follows:</p>
<disp-formula id="E2"><label>(1)</label><mml:math id="M15"><mml:mrow><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>x</mml:mtext></mml:mstyle><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>X</mml:mtext></mml:mstyle><mml:mi>k</mml:mi><mml:mi>t</mml:mi></mml:msubsup><mml:msubsup><mml:mo>&#x003BB;</mml:mo><mml:mi>k</mml:mi><mml:mi>t</mml:mi></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>s</mml:mtext></mml:mstyle><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x02212;</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy='false'>(</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy='false'>)</mml:mo></mml:mrow></mml:math></disp-formula>
<disp-formula id="E3"><label>(2)</label><mml:math id="M16"><mml:mrow><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>Y</mml:mtext></mml:mstyle><mml:mrow><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mo>&#x003BB;</mml:mo></mml:mstyle><mml:mi>k</mml:mi><mml:mi>t</mml:mi></mml:msubsup><mml:mo>&#x02212;</mml:mo><mml:msubsup><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>s</mml:mtext></mml:mstyle><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy='false'>(</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy='false'>)</mml:mo></mml:mrow></mml:math></disp-formula>
<disp-formula id="E4"><label>(3)</label><mml:math id="M17"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>Y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mo>&#x003BB;</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:msubsup><mml:mtext>&#x000A0;</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E5"><label>(4)</label><mml:math id="M18"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:mi>e</mml:mi><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mtext>&#x000A0;</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E6"><label>(5)</label><mml:math id="M19"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mo>&#x003BB;</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x02265;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>&#x000A0;</mml:mtext><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:msubsup><mml:mo>&#x02265;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>&#x000A0;</mml:mtext><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo></mml:mrow></mml:msubsup><mml:mo>&#x02265;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>&#x000A0;</mml:mtext><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:msubsup><mml:mo>&#x02265;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>&#x000A0;</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>Where, <inline-formula><mml:math id="M20"><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>X</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x02208;</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>&#x000D7;</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M21"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>Y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x02208;</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>&#x000D7;</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M22"><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>Y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant='bold' mathsize='normal'><mml:mtext>y</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>k</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x02208;</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>&#x000D7;</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> and are input and output matrices; <inline-formula><mml:math id="M23"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>o</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M24"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula> and <inline-formula><mml:math id="M25"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>k</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula> are matrices of input, desirable output, and undesirable output, respectively. The intermediate variables as input constraints are as follows:</p>
<disp-formula id="E7"><label>(6)</label><mml:math id="M26"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>Z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mo>&#x003BB;</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mtext>&#x000A0;</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>k</mml:mi><mml:mi>i</mml:mi><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>Where <inline-formula><mml:math id="M27"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>Z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x02208;</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>&#x000D7;</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>; <inline-formula><mml:math id="M28"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x02208;</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup></mml:math></inline-formula> is a non-negative reduced value. The advisable carry-over constraints are as follows:</p>
<disp-formula id="E8"><label>(7)</label><mml:math id="M29"><mml:mrow><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:msub><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy='false'>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mo stretchy='false'>(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy='false'>)</mml:mo><mml:mo stretchy='false'>)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mstyle><mml:msubsup><mml:mi>&#x003BB;</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:msub><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy='false'>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mo stretchy='false'>(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy='false'>)</mml:mo><mml:mo stretchy='false'>)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mstyle><mml:msubsup><mml:mi>&#x003BB;</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mtext>&#x000A0;</mml:mtext><mml:mo stretchy='false'>(</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mo>&#x02200;</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo>;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x02026;</mml:mo><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>&#x02212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy='false'>)</mml:mo></mml:mrow></mml:math></disp-formula>
<disp-formula id="E9"><mml:math id="M31"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>&#x02211;</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>n</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mtext>&#x000A0;</mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E10"><label>(8)</label><mml:math id="M32"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E11"><label>(9)</label><mml:math id="M33"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x02265;</mml:mo><mml:mn>0</mml:mn><mml:mtext>&#x000A0;</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>&#x02200;</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mo>&#x02200;</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>Where, <inline-formula><mml:math id="M34"><mml:msubsup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula> is a reduced value. The objective function is as follows:</p>
<disp-formula id="E12"><label>(10)</label><mml:math id="M9"><mml:mrow><mml:msup><mml:mi>&#x003B8;</mml:mi><mml:mo>&#x0002A;</mml:mo></mml:msup><mml:mtext>=</mml:mtext><mml:mi>min</mml:mi><mml:mfrac><mml:mrow><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:msubsup><mml:mrow><mml:msup><mml:mi>W</mml:mi><mml:mi>t</mml:mi></mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:msup><mml:mi>w</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x02212;</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>k</mml:mi><mml:mi>i</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:mfrac><mml:mrow><mml:msubsup><mml:mi>s</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x02212;</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:msubsup><mml:mi>x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msubsup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mstyle displaystyle='true'><mml:msubsup><mml:mo>&#x02211;</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy='false'>(</mml:mo><mml:mi>k</mml:mi><mml:mi>h</mml:mi><mml:mo 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stretchy='false'>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mo stretchy='false'>(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy='false'>)</mml:mo><mml:mo stretchy='false'>)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula>
</sec>
</sec>
</sec>
<sec sec-type="results" id="s3">
<title>Results</title>
<sec>
<title>Theoretical and analytical framework analyzing the overall efficiency of agricultural poverty relief</title>
<p><xref ref-type="table" rid="T2">Table 2</xref> describes the overall efficiency and stage-specific efficiency from 2012 to 2016. Among the 32 villages, the overall efficiency of Yunnan01 was 1, demonstrating the highest overall efficiency. The overall efficiency of 13 villages (e.g., Heilongjiang06) was higher than 0.7, indicating good performance overall, while the overall efficiency of 11 villages (e.g., Shaanxi01) was lower than 0.5 [specifically, the overall efficiency of Shaanxi01 is the lowest (0.3421)]. In general, overall efficiency is substandard for most villages from 2012 to 2016 and differs significantly across the 32 villages.</p>
<table-wrap position="float" id="T2">
<label>Table 2</label>
<caption><p>Overall efficiency and stage-specific efficiency from 2012 to 2016.</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th valign="top" align="left"><bold>DMU</bold></th>
<th valign="top" align="center"><bold>Stage 1</bold></th>
<th valign="top" align="center"><bold>Stage 2</bold></th>
<th valign="top" align="center"><bold>Overall efficiency</bold></th>
<th valign="top" align="left"><bold>DMU</bold></th>
<th valign="top" align="center"><bold>Stage 1</bold></th>
<th valign="top" align="center"><bold>Stage 2</bold></th>
<th valign="top" align="center"><bold>Overall efficiency</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">Yunnan01</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="left">Anhui02</td>
<td valign="top" align="center">0.5146</td>
<td valign="top" align="center">0.6583</td>
<td valign="top" align="center">0.5865</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang06</td>
<td valign="top" align="center">0.9885</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9942</td>
<td valign="top" align="left">Hubei01</td>
<td valign="top" align="center">0.7478</td>
<td valign="top" align="center">0.4161</td>
<td valign="top" align="center">0.5819</td>
</tr>
<tr>
<td valign="top" align="left">Jilin03</td>
<td valign="top" align="center">0.9756</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9878</td>
<td valign="top" align="left">Gansu02</td>
<td valign="top" align="center">0.5192</td>
<td valign="top" align="center">0.6135</td>
<td valign="top" align="center">0.5664</td>
</tr>
<tr>
<td valign="top" align="left">Guangdong01</td>
<td valign="top" align="center">0.8786</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9393</td>
<td valign="top" align="left">Heilongjiang02</td>
<td valign="top" align="center">0.5203</td>
<td valign="top" align="center">0.5410</td>
<td valign="top" align="center">0.5307</td>
</tr>
<tr>
<td valign="top" align="left">Jilin01</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7962</td>
<td valign="top" align="center">0.8981</td>
<td valign="top" align="left">Gansu01</td>
<td valign="top" align="center">0.5900</td>
<td valign="top" align="center">0.4610</td>
<td valign="top" align="center">0.5255</td>
</tr>
<tr>
<td valign="top" align="left">Zhejiang02</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7782</td>
<td valign="top" align="center">0.8891</td>
<td valign="top" align="left">Henan01</td>
<td valign="top" align="center">0.4513</td>
<td valign="top" align="center">0.5476</td>
<td valign="top" align="center">0.4995</td>
</tr>
<tr>
<td valign="top" align="left">Jilin02</td>
<td valign="top" align="center">0.9601</td>
<td valign="top" align="center">0.7906</td>
<td valign="top" align="center">0.8754</td>
<td valign="top" align="left">Anhui05</td>
<td valign="top" align="center">0.2093</td>
<td valign="top" align="center">0.7778</td>
<td valign="top" align="center">0.4935</td>
</tr>
<tr>
<td valign="top" align="left">Zhejiang01</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.5886</td>
<td valign="top" align="center">0.7943</td>
<td valign="top" align="left">Anhui07</td>
<td valign="top" align="center">0.2819</td>
<td valign="top" align="center">0.6939</td>
<td valign="top" align="center">0.4879</td>
</tr>
<tr>
<td valign="top" align="left">Shaanxi03</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.5211</td>
<td valign="top" align="center">0.7606</td>
<td valign="top" align="left">Fujian03</td>
<td valign="top" align="center">0.2420</td>
<td valign="top" align="center">0.7092</td>
<td valign="top" align="center">0.4756</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang04</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.5188</td>
<td valign="top" align="center">0.7594</td>
<td valign="top" align="left">Anhui04</td>
<td valign="top" align="center">0.2670</td>
<td valign="top" align="center">0.6834</td>
<td valign="top" align="center">0.4752</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang03</td>
<td valign="top" align="center">0.9253</td>
<td valign="top" align="center">0.5828</td>
<td valign="top" align="center">0.7540</td>
<td valign="top" align="left">Xinjiang01</td>
<td valign="top" align="center">0.2869</td>
<td valign="top" align="center">0.6283</td>
<td valign="top" align="center">0.4576</td>
</tr>
<tr>
<td valign="top" align="left">Fujian01</td>
<td valign="top" align="center">0.6121</td>
<td valign="top" align="center">0.8786</td>
<td valign="top" align="center">0.7454</td>
<td valign="top" align="left">Guangxi01</td>
<td valign="top" align="center">0.3316</td>
<td valign="top" align="center">0.5727</td>
<td valign="top" align="center">0.4522</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang05</td>
<td valign="top" align="center">0.6924</td>
<td valign="top" align="center">0.7551</td>
<td valign="top" align="center">0.7238</td>
<td valign="top" align="left">Guizhou02</td>
<td valign="top" align="center">0.2620</td>
<td valign="top" align="center">0.6105</td>
<td valign="top" align="center">0.4362</td>
</tr>
<tr>
<td valign="top" align="left">Fujian02</td>
<td valign="top" align="center">0.6582</td>
<td valign="top" align="center">0.6285</td>
<td valign="top" align="center">0.6434</td>
<td valign="top" align="left">Shaanxi02</td>
<td valign="top" align="center">0.4812</td>
<td valign="top" align="center">0.3536</td>
<td valign="top" align="center">0.4174</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang01</td>
<td valign="top" align="center">0.5196</td>
<td valign="top" align="center">0.7071</td>
<td valign="top" align="center">0.6134</td>
<td valign="top" align="left">Guizhou01</td>
<td valign="top" align="center">0.2787</td>
<td valign="top" align="center">0.4672</td>
<td valign="top" align="center">0.3729</td>
</tr>
<tr>
<td valign="top" align="left">Anhui06</td>
<td valign="top" align="center">0.8135</td>
<td valign="top" align="center">0.3648</td>
<td valign="top" align="center">0.5892</td>
<td valign="top" align="left">Shaanxi01</td>
<td valign="top" align="center">0.1195</td>
<td valign="top" align="center">0.5648</td>
<td valign="top" align="center">0.3421</td>
</tr>
</tbody>
</table>
</table-wrap>
<p><xref ref-type="fig" rid="F3">Figure 3</xref> shows the overall efficiency of agricultural poverty relief in 32 villages from 2012 to 2016. There are seven villages whose overall efficiency is higher than 1 in at least 2 years; specifically, the overall efficiency of Yunnan01 was always 1 over 5 consecutive years. Over the 5 years, the overall efficiency of 14 villages (e.g., Jilin01) tended to rise to undulate, and the overall efficiency of 18 villages (e.g., Heilongjiang05) tended to decline to undulate. The overall efficiency of four villages (e.g., Shaanxi01) was relatively low, indicating no obvious effectiveness in agricultural poverty relief. In general, the overall efficiency of agricultural poverty relief tends to be stable and undulate from 2012 to 2016, implying the significant potential for improvement.</p>
<fig id="F3" position="float">
<label>Figure 3</label>
<caption><p>Overall efficiency from 2012 to 2016.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fsufs-06-1007915-g0003.tif"/>
</fig>
</sec>
<sec>
<title>Analyzing agricultural production efficiency and social governance efficiency in different years</title>
<p>In this study, the overall efficiency of agricultural poverty relief is further decomposed into efficiency in two stages: agricultural production efficiency and social governance efficiency. In the first stage, the agricultural production efficiency of six villages (e.g., Heilongjiang04) all reach the efficiency frontiers; in the second stage, the social governance efficiency of four villages (e.g., Guangdong01) reaches the efficiency frontiers. In the first stage, there were 19 villages (e.g., Shaanxi02) whose agricultural production efficiency was lower than 0.5; in the second stage, there were five villages (e.g., Guizhou01) whose social governance efficiency was lower than 0.5. The low overall efficiency of eight villages (e.g., Xinjiang01) is mainly due to their low agricultural production efficiency at the first stage; the low overall efficiency of five villages (e.g., Guizhou01) is mainly due to their low social governance efficiency at the second stage. Overall, there is a slight difference in efficiency between the two stages. Specifically, the efficiency values in the second stage were slightly higher than those in the first stage.</p>
<p>Here, <xref ref-type="fig" rid="F4">Figures 4</xref>, <xref ref-type="fig" rid="F5">5</xref> show the agricultural production efficiency and social governance efficiency of 32 villages, respectively, from 2012 to 2016. The agricultural production efficiency of ten villages (e.g., Anhui06) tended to decline and undulate, that of Xinjiang01 continues to decline, and that of six villages (e.g., Anhui07) was always lower than 0.4 over the 5 years, showing no obvious improvement across the years. The social governance efficiency of eight villages (e.g., Jilin01) tended to increase undulated, that of ten villages (e.g., Heilongjiang05) tended to decline undulated, and that of Shaanxi02 was always lower than 0.5 over the 5 years. Overall, agricultural production efficiency in the first stage is the primary cause of the low overall efficiency of agricultural poverty relief; social governance efficiency in the second stage is a minor cause. In the whole process of agricultural poverty relief, there is significant potential for improvement in both agricultural production efficiency at the first stage and social governance efficiency in the second stage, particularly at the first stage. Moreover, agricultural production efficiency at the first stage does not vary significantly across different years, whereas social governance efficiency at the second stage does.</p>
<fig id="F4" position="float">
<label>Figure 4</label>
<caption><p>Agricultural production efficiency at the first stage, from 2012 to 2016.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fsufs-06-1007915-g0004.tif"/>
</fig>
<fig id="F5" position="float">
<label>Figure 5</label>
<caption><p>Social governance efficiency in the second stage, from 2012 to 2016.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fsufs-06-1007915-g0005.tif"/>
</fig>
</sec>
<sec>
<title>Comparison between agricultural production efficiency and social governance efficiency</title>
<p>For the 32 villages, the average agricultural production efficiency and average social governance efficiency are respectively 0.6290 and 0.6628. To further analyze the potential for improvement in agricultural production and social governance efficiency, the specific values of the 32 villages were compared with the average values (as described in <xref ref-type="table" rid="T3">Table 3</xref>).</p>
<table-wrap position="float" id="T3">
<label>Table 3</label>
<caption><p>Average efficiency of input and output indicators in 5 years (2012 to 2016).</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th/>
<th valign="top" align="center" colspan="4" style="border-bottom: thin solid #000000;"><bold>Input variable</bold></th>
<th valign="top" align="center" colspan="2" style="border-bottom: thin solid #000000;"><bold>Intermediate variable</bold></th>
<th valign="top" align="center" colspan="3" style="border-bottom: thin solid #000000;"><bold>Output variable</bold></th>
</tr>
<tr>
<th valign="top" align="left"><bold>DMU</bold></th>
<th valign="top" align="center"><bold>Labor force</bold></th>
<th valign="top" align="center"><bold>Sown area</bold></th>
<th valign="top" align="center"><bold>Educational attainment</bold></th>
<th valign="top" align="center"><bold>Fixed assets</bold></th>
<th valign="top" align="center"><bold>Disposable income per village household</bold></th>
<th valign="top" align="center"><bold>Income gap</bold></th>
<th valign="top" align="center"><bold>Total output of grain</bold></th>
<th valign="top" align="center"><bold>Non-poor population</bold></th>
<th valign="top" align="center"><bold>Public order</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">Yunnan01</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang06</td>
<td valign="top" align="center">0.9804</td>
<td valign="top" align="center">0.9956</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9860</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
</tr>
<tr>
<td valign="top" align="left">Jilin03</td>
<td valign="top" align="center">0.9935</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9449</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
</tr>
<tr>
<td valign="top" align="left">Zhejiang02</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9053</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7176</td>
<td valign="top" align="center">0.9780</td>
</tr>
<tr>
<td valign="top" align="left">Jilin02</td>
<td valign="top" align="center">0.8958</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8631</td>
<td valign="top" align="center">0.9963</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9942</td>
<td valign="top" align="center">0.9803</td>
<td valign="top" align="center">0.7698</td>
<td valign="top" align="center">0.8207</td>
</tr>
<tr>
<td valign="top" align="left">Jilin01</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9286</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7662</td>
<td valign="top" align="center">0.9554</td>
</tr>
<tr>
<td valign="top" align="left">Guangdong01</td>
<td valign="top" align="center">0.8521</td>
<td valign="top" align="center">0.9675</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8485</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
</tr>
<tr>
<td valign="top" align="left">Shaanxi03</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8020</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.4534</td>
<td valign="top" align="center">0.8449</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang03</td>
<td valign="top" align="center">0.9583</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.6066</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8814</td>
<td valign="top" align="center">0.5850</td>
<td valign="top" align="center">0.7430</td>
</tr>
<tr>
<td valign="top" align="left">Zhejiang01</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9816</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.5180</td>
<td valign="top" align="center">0.9711</td>
</tr>
<tr>
<td valign="top" align="left">Henan01</td>
<td valign="top" align="center">0.6423</td>
<td valign="top" align="center">0.8274</td>
<td valign="top" align="center">0.7618</td>
<td valign="top" align="center">0.5259</td>
<td valign="top" align="center">0.9402</td>
<td valign="top" align="center">0.4063</td>
<td valign="top" align="center">0.6941</td>
<td valign="top" align="center">0.8300</td>
<td valign="top" align="center">0.7006</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang01</td>
<td valign="top" align="center">0.6080</td>
<td valign="top" align="center">0.8557</td>
<td valign="top" align="center">0.8066</td>
<td valign="top" align="center">0.8692</td>
<td valign="top" align="center">0.9285</td>
<td valign="top" align="center">0.8424</td>
<td valign="top" align="center">0.4447</td>
<td valign="top" align="center">0.7981</td>
<td valign="top" align="center">0.8004</td>
</tr>
<tr>
<td valign="top" align="left">Fujian02</td>
<td valign="top" align="center">0.6135</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7884</td>
<td valign="top" align="center">0.9360</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.5876</td>
<td valign="top" align="center">0.7246</td>
<td valign="top" align="center">0.3781</td>
</tr>
<tr>
<td valign="top" align="left">Fujian01</td>
<td valign="top" align="center">0.4975</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9255</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9908</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.5208</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7288</td>
</tr>
<tr>
<td valign="top" align="left">Anhui06</td>
<td valign="top" align="center">0.9611</td>
<td valign="top" align="center">0.7626</td>
<td valign="top" align="center">0.7776</td>
<td valign="top" align="center">0.9729</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8980</td>
<td valign="top" align="center">0.7240</td>
<td valign="top" align="center">0.3188</td>
<td valign="top" align="center">0.8684</td>
</tr>
<tr>
<td valign="top" align="left">Hubei01</td>
<td valign="top" align="center">0.8128</td>
<td valign="top" align="center">0.9120</td>
<td valign="top" align="center">0.5164</td>
<td valign="top" align="center">0.9713</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8949</td>
<td valign="top" align="center">0.7585</td>
<td valign="top" align="center">0.4280</td>
<td valign="top" align="center">0.7098</td>
</tr>
<tr>
<td valign="top" align="left">Gansu01</td>
<td valign="top" align="center">0.6420</td>
<td valign="top" align="center">0.9412</td>
<td valign="top" align="center">0.4848</td>
<td valign="top" align="center">0.8791</td>
<td valign="top" align="center">0.9394</td>
<td valign="top" align="center">0.6232</td>
<td valign="top" align="center">0.4908</td>
<td valign="top" align="center">0.5197</td>
<td valign="top" align="center">0.8671</td>
</tr>
<tr>
<td valign="top" align="left">Gansu02</td>
<td valign="top" align="center">0.7025</td>
<td valign="top" align="center">0.8429</td>
<td valign="top" align="center">0.7727</td>
<td valign="top" align="center">0.8980</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7862</td>
<td valign="top" align="center">0.3248</td>
<td valign="top" align="center">0.6070</td>
<td valign="top" align="center">0.9954</td>
</tr>
<tr>
<td valign="top" align="left">Anhui02</td>
<td valign="top" align="center">0.4909</td>
<td valign="top" align="center">0.7904</td>
<td valign="top" align="center">0.7418</td>
<td valign="top" align="center">0.9557</td>
<td valign="top" align="center">0.9238</td>
<td valign="top" align="center">0.6095</td>
<td valign="top" align="center">0.5356</td>
<td valign="top" align="center">0.8362</td>
<td valign="top" align="center">0.9395</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang02</td>
<td valign="top" align="center">0.3802</td>
<td valign="top" align="center">0.9785</td>
<td valign="top" align="center">0.5825</td>
<td valign="top" align="center">0.9334</td>
<td valign="top" align="center">0.9576</td>
<td valign="top" align="center">0.6129</td>
<td valign="top" align="center">0.3596</td>
<td valign="top" align="center">0.9924</td>
<td valign="top" align="center">0.3356</td>
</tr>
<tr>
<td valign="top" align="left">Guizhou01</td>
<td valign="top" align="center">0.6404</td>
<td valign="top" align="center">0.8641</td>
<td valign="top" align="center">0.6202</td>
<td valign="top" align="center">0.4798</td>
<td valign="top" align="center">0.9532</td>
<td valign="top" align="center">0.5450</td>
<td valign="top" align="center">0.2689</td>
<td valign="top" align="center">0.6481</td>
<td valign="top" align="center">0.4550</td>
</tr>
<tr>
<td valign="top" align="left">Anhui07</td>
<td valign="top" align="center">0.3395</td>
<td valign="top" align="center">0.8980</td>
<td valign="top" align="center">0.7932</td>
<td valign="top" align="center">0.6957</td>
<td valign="top" align="center">0.9212</td>
<td valign="top" align="center">0.6903</td>
<td valign="top" align="center">0.2294</td>
<td valign="top" align="center">0.9548</td>
<td valign="top" align="center">0.7263</td>
</tr>
<tr>
<td valign="top" align="left">Shaanxi02</td>
<td valign="top" align="center">0.8286</td>
<td valign="top" align="center">0.9869</td>
<td valign="top" align="center">0.7289</td>
<td valign="top" align="center">0.9485</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.4591</td>
<td valign="top" align="center">0.2038</td>
<td valign="top" align="center">0.3711</td>
<td valign="top" align="center">0.8518</td>
</tr>
<tr>
<td valign="top" align="left">Guangxi01</td>
<td valign="top" align="center">0.2230</td>
<td valign="top" align="center">0.9388</td>
<td valign="top" align="center">0.5217</td>
<td valign="top" align="center">0.7908</td>
<td valign="top" align="center">0.9841</td>
<td valign="top" align="center">0.7068</td>
<td valign="top" align="center">0.2846</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.3990</td>
</tr>
<tr>
<td valign="top" align="left">Guizhou02</td>
<td valign="top" align="center">0.4143</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.4868</td>
<td valign="top" align="center">0.6385</td>
<td valign="top" align="center">0.9605</td>
<td valign="top" align="center">0.9369</td>
<td valign="top" align="center">0.2118</td>
<td valign="top" align="center">0.9732</td>
<td valign="top" align="center">0.4295</td>
</tr>
<tr>
<td valign="top" align="left">Anhui04</td>
<td valign="top" align="center">0.2344</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.6547</td>
<td valign="top" align="center">0.6849</td>
<td valign="top" align="center">0.9675</td>
<td valign="top" align="center">0.7346</td>
<td valign="top" align="center">0.1925</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.6696</td>
</tr>
<tr>
<td valign="top" align="left">Xinjiang01</td>
<td valign="top" align="center">0.5971</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.6305</td>
<td valign="top" align="center">0.7248</td>
<td valign="top" align="center">0.9893</td>
<td valign="top" align="center">0.8651</td>
<td valign="top" align="center">0.1480</td>
<td valign="top" align="center">0.9230</td>
<td valign="top" align="center">0.4212</td>
</tr>
<tr>
<td valign="top" align="left">Fujian03</td>
<td valign="top" align="center">0.4244</td>
<td valign="top" align="center">0.9352</td>
<td valign="top" align="center">0.8826</td>
<td valign="top" align="center">0.8481</td>
<td valign="top" align="center">0.9623</td>
<td valign="top" align="center">0.8657</td>
<td valign="top" align="center">0.1277</td>
<td valign="top" align="center">0.9196</td>
<td valign="top" align="center">0.5138</td>
</tr>
<tr>
<td valign="top" align="left">Anhui05</td>
<td valign="top" align="center">0.2882</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8494</td>
<td valign="top" align="center">0.7801</td>
<td valign="top" align="center">0.8648</td>
<td valign="top" align="center">0.7196</td>
<td valign="top" align="center">0.1174</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9398</td>
</tr>
<tr>
<td valign="top" align="left">Shaanxi01</td>
<td valign="top" align="center">0.4986</td>
<td valign="top" align="center">0.8257</td>
<td valign="top" align="center">0.5883</td>
<td valign="top" align="center">0.3986</td>
<td valign="top" align="center">0.9430</td>
<td valign="top" align="center">0.6097</td>
<td valign="top" align="center">0.1234</td>
<td valign="top" align="center">0.8626</td>
<td valign="top" align="center">0.6925</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang04</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.6405</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.7237</td>
<td valign="top" align="center">0.5613</td>
</tr>
<tr>
<td valign="top" align="left">Heilongjiang05</td>
<td valign="top" align="center">0.7579</td>
<td valign="top" align="center">0.5442</td>
<td valign="top" align="center">0.8616</td>
<td valign="top" align="center">0.8911</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.9707</td>
<td valign="top" align="center">1.0000</td>
<td valign="top" align="center">0.8007</td>
<td valign="top" align="center">0.9651</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
<sec>
<title>Analyzing the efficiency of input and output indicators</title>
<p><xref ref-type="table" rid="T3">Table 3</xref> describes the average efficiency of the input, intermediate, and output variables of the 32 villages. Regarding input variables, there are six villages whose efficiency of the total labor force is 1, reaching efficiency frontiers; the remaining 26 villages all have the potential for improvement. There are 16 villages (e.g., Guangxi01) whose efficiency of the total labor force is below the average (0.6962); specifically, Guangxi04 has the lowest efficiency of the total labor force (0.2230). There is labor redundancy in agricultural production, thus it is necessary to reduce the agricultural labor force to improve agricultural production efficiency. Regarding the sown areas of farm crops, there are 15 villages (e.g., Yunnan01) that reach efficiency frontiers, and the remaining 17 villages all have some potential for improvement; overall, the efficiency of sown areas of farm crops is relatively high. There are 11 villages (e.g., Yunnan01) whose efficiency of productive fixed assets is 1, reaching efficiency frontiers. The remaining 21 villages all have some potential for improvement. Specifically, it is necessary to raise the utilization rate of productive fixed assets, thus improving agricultural production efficiency. Regarding the educational attainment of rural labor, there are four villages (e.g., Zhejiang02) that reach efficiency frontiers, and the 28 remaining villages have the potential for improvement to varying degrees. The average educational attainment of rural laborers is 8.06. Regional social governance can be improved by increasing the average educational attainment of rural laborers.</p>
<p>Regarding output variables, there are seven villages (e.g., Yunnan01) whose total output of grain reaches efficiency frontiers; the remaining 25 villages have the potential for improvement to varying degrees. There are 17 villages (e.g., Anhui05) with below-average efficiency in the total output of grain; Anhui05 also has the lowest efficiency in the total output of labor (0.1174). Agricultural production efficiency can be improved to some extent by increasing the output of grain. Regarding the poverty elimination effect, eight villages (e.g., Yunnan01) reach efficiency frontiers, and the remaining 24 villages have potential for improvement to varying degrees. Regarding undesirable output variables, there are four villages (i.e., Yunnan01, Heilongjiang06, Jilin03, and Guangdong01) whose aggregate index of social harmony reaches efficiency frontiers, and the remaining 28 villages have room for improvement to varying degrees. Moreover, there are 15 villages (e.g., Heilongjiang02) with below-average efficiency in the aggregate index of social harmony. The efficiency of the two output variables in the second stage shows that remarkable achievements have been made in poverty relief and social harmony and stability, but the aggregate index of social harmony has more potential for improvement.</p>
<p>Regarding disposable income per village household, 17 villages reach efficiency frontiers, while the remaining 15 villages have the potential for improvement to varying degrees. Regarding the income gap of village households, 12 villages reached efficiency frontiers; the remaining 20 villages have some potential for improvement. For the 32 villages, the efficiency of disposable income per village household is appreciable, while the efficiency of the income gap of village households has significant potential for improvement. Thus, to improve social governance efficiency, it is necessary to reduce the income gap of village households. <xref ref-type="table" rid="T3">Table 3</xref> Average efficiency of input and output indicators in 5 years (2012 to 2016).</p>
</sec>
</sec>
<sec sec-type="discussion" id="s4">
<title>Discussion</title>
<p>It is beneficial to draw some emphasis on the outcomes produced by the study and synthesize those with the existing literature. However, China&#x00027;s success in reducing poverty across the board is unevenly dispersed throughout the country (Zameer et al., <xref ref-type="bibr" rid="B67">2020</xref>). Like most other emerging countries Chinese poverty alleviation may possess uneven development trends and the efficiency level is also random which is mostly characterized by factor endowment, distributions of labor and natural resources, local governments support, and sociodemographic notions. Interestingly, the potential success of alleviating poverty through agriculture projects is dependent upon the simultaneous development of several interrelated mechanisms from production factors <italic>via</italic> labor and resource allocations through household income and gaps of income to social harmony aggressions and managing undesired outcomes (Ellis and Freeman, <xref ref-type="bibr" rid="B19">2004</xref>; Thongdara et al., <xref ref-type="bibr" rid="B55">2012</xref>; Huang et al., <xref ref-type="bibr" rid="B28">2021a</xref>). Therefore, we divided the whole process of agricultural poverty reduction into two stages, namely the agricultural production stage and the social governance stage, and include the desirable and undesirable outcomes in an integrated manner.</p>
<p>In general comprehensive efficiency value of China&#x00027;s agricultural poverty reduction shows a relatively stable trend in the fluctuations. The agricultural production efficiency value has a large improvement space, and the social governance efficiency value has a small space for improvement. The outcome is parallel with the study of Zhou et al. (<xref ref-type="bibr" rid="B69">2019</xref>) and Peng et al. (<xref ref-type="bibr" rid="B50">2021</xref>). The study believes that in the first-stage value of agricultural production efficiency is significantly lower than the value of poverty reduction efficiency in the second stage, and there is still much room for improvement in agricultural production efficiency. In the study of China&#x00027;s Targeted Poverty Alleviation Policies from 2013 to 2019, Li and Li (<xref ref-type="bibr" rid="B37">2021</xref>) also found similar findings. The study found in the future, due to the rapid development of the agricultural scale, mechanization, and smart agriculture in China, there will be a lot of room for improvement in agricultural production efficiency. After evaluating the notion of anti-poverty policy efficiencies and the problem of agricultural poverty reduction in China based on the Chinese rural statistical data of 28 provinces, municipalities, and autonomous regions from 2013 to 2017, Yang et al. (<xref ref-type="bibr" rid="B66">2021</xref>) also outlined parallel assumptions. Interestingly, Zameer et al. (<xref ref-type="bibr" rid="B67">2020</xref>) found China&#x00027;s poverty alleviation trends have much room for increasing greater social harmony and social obligation is relatively weak. The current study also found limited room for probable expansions concerning the output of grain, the poverty elimination index, and an aggregate index of social harmony which is slightly different from some studies. The study found grain productivity will lead to an increase in farmers&#x00027; disposable income and reduce the income gaps which eventually fosters better performance of poverty alleviation. While ins a study of poverty relief and grassland dilapidation in Inner Mongolia Briske et al. (<xref ref-type="bibr" rid="B9">2015</xref>) had quite different outcomes as they outline maximization of livestock revenue reduces household income and creates a burden for the household to get rid of poverty.</p>
</sec>
<sec sec-type="conclusions" id="s5">
<title>Conclusion</title>
<p>Like most other developing countries Chinese government is also fostering various poverty alleviation projects. However, among them &#x0201C;Poverty alleviation through agriculture projects&#x0201D; has drawn much more focal attention toward policy dimensions and empirical setup. While the existing literature mainly lacks an evaluation of the level of efficiencies of these particular types of projects. Moreover, how and to what extent the projects foster and perform in alleviating poverty at the village level has not been fully grasped by other studies. Therefore, the study has been designed to fulfill those gaps by using the data from rural fixed observation points across 31 provinces of China from 2012 to 2016. For doing so, we have adopted a two-stage dynamic Slacks-Based Measure (SBM) model for performing the Data Envelopment Analysis (DEA). The study depicted the following outcomes: the overall efficiency of agricultural poverty relief across the 32 villages tended to be stable and undulate. Agricultural production efficiency in the first stage is the primary cause of the low overall efficiency of agricultural poverty relief, and social governance efficiency in the second stage is its secondary cause. Throughout the entire process of agricultural poverty relief, there is significant potential for improvement in both agricultural production efficiency at the first stage and social governance efficiency in the second stage, particularly at the first stage. Agricultural poverty relief provides an important pathway to addressing rural poverty in China. Specifically, we can achieve the goal of agricultural poverty relief by improving the conditions of agricultural production and operation and enhancing the potential of rural social governance at the grassroots level. It is necessary to improve agricultural production efficiency in particular. Among the 32 villages, agricultural production efficiency in the first stage differs significantly from social governance efficiency in the second stage. Village-based agricultural production efficiency does not vary significantly across different years, whereas village-based social governance efficiency does.</p>
<p>Regarding specific indicators, the efficiency of two input indicators (total labor force and productive fixed assets) at the first stage is poor, implying the significant potential for improvement; though the efficiency value of sown areas of farm crops is appreciable. Therefore, it is necessary to reduce the labor force of agricultural production, raise the utilization rate of productive fixed assets, develop modern agriculture, and improve agricultural production efficiency. Among the input indicators in the second stage, the efficiency of educational attainment of rural labor is poor, implying the significant potential for improvement. Therefore, it is necessary to improve the educational attainment of rural labor, thus facilitating poverty relief and social harmony. Among the output indicators in the second stage, the efficiency of the total output of grain has some potential for improvement. A high-yield, high-quality, and low-consumption agricultural production system can be created by improving agricultural production efficiency in the first stage. The analysis of the efficiency of output indicators at the second stage shows that remarkable achievements have been made in poverty relief and social harmony and stability, but the aggregate index of social harmony has more potential for improvement. The analysis of intermediate indicators shows that the efficiency of disposable income per village household is appreciable, while the efficiency of the income gap of village households has significant potential for improvement. To improve social governance efficiency, it is necessary to reduce the income gap of village households. The income gap affects not only the progress in poverty relief but also rural social harmony.</p>
</sec>
<sec sec-type="data-availability" id="s6">
<title>Data availability statement</title>
<p>The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.</p>
</sec>
<sec id="s7">
<title>Ethics statement</title>
<p>Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. Written informed consent for participation was not required for this study in accordance with the national legislation and the institutional requirements.</p>
</sec>
<sec id="s8">
<title>Author contributions</title>
<p>Conceptualization and methodology: SY and AS. Software and resources: AS. Formal analysis and visualization: GY. Data curation: SY. Writing&#x02014;original draft preparation, investigation, supervision, and funding acquisition: LL. Writing&#x02014;review and editing and validation: SY, AS, and LL. Project administration: GY and AS. All authors have read and agreed to the published version of the manuscript.</p>
</sec>
<sec sec-type="funding-information" id="s9">
<title>Funding</title>
<p>This research was funded by the National Natural Science Foundation of China (Grant Nos. 71763022, 72263027, and 71963028) and Ningxia Natural Science Foundation (Grant No. 2022AAC03027).</p>
</sec>
<sec sec-type="COI-statement" id="conf1">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="disclaimer" id="s10">
<title>Publisher&#x00027;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
</body>
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