\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \fancyhead[CE]{A. SOHRABI, J. GERAMI, M. M. MOZAFFARI} \fancyhead[CO]{A NOVEL INVERSE DEA-R MODEL FOR INPUTS/OUTPUTS...} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\noindent Journal of Mathematical Extension \\ Journal Pre-proof}\\ ISSN: 1735-8299\\ URL: http://www.ijmex.com\\ ‎\vspace*{9mm} ‎ \begin{center} {\Large \bf A Novel Inverse DEA-R Model for Inputs/Outputs estimation\\} \let\thefootnote\relax\footnote{\scriptsize Received: May 2021; Accepted: October 2021} {\bf A. Sohrabi}\vspace*{-2mm}\\ \vspace{2mm} {\small Kerman Branch, Islamic Azad University} \vspace{2mm} {\bf J. Gerami$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\ \vspace{2mm} {\small Shiraz Branch, Islamic Azad University} \vspace{2mm} {\bf M. M. Mozaffari}\vspace*{-2mm}\\ \vspace{2mm} {\small Shiraz Branch, Islamic Azad University} \vspace{2mm} \end{center} \vspace{4mm} {\footnotesize \begin{quotation} {\noindent \bf Abstract.} In this paper, we propose inverse data envelopment analysis (DEA) models in the presence of ratio data. We present the inputs/output estimation process based on ratio based DEA (DEA-R) models. We first present a multiple objective linear programming (MOLP) model to determine the level of inputs based on the perturbed outputs, assuming that the relative efficiency of the under evaluation decision making unit (DMU) preserve. We also present the relationship between the Pareto solutions of the proposed MOLP model and the optimal level of inputs and outputs of the new DMU. We presented criterion models to determine the efficiency of the new DMU in the inputs/output estimation process based on inverse DEA-R models in the presence of ratio data. We showed that in the presence of ratio data the selection of criterion model can be important, in order to we provide a new criterion model in the inputs/output estimation process in the presence of ratio data, and so on the amount of calculations is reduced. We have shown that the results for the new criterion model are the same as the existing criterion model presented in the paper. In order to show the validity of the proposed approach in the inputs/output estimation process based on the inverse DEA-R models, we provide an application of our models in a real life for a set of data regarding to medical centers in Taiwan and finally we present the research results. \end{quotation} \begin{quotation} \noindent{\bf AMS Subject Classification:} MSC code1; MSC code 2; more \noindent{\bf Keywords and Phrases:} Data envelopment analysis; Multiple objective linear programming; Input/output estimation; Ratio data; Criterion model; Inverse DEA. \end{quotation}} \section{Introduction} \label{intro} % It is advised to give each section and subsection a unique label. Traditional DEA models determine the efficiency of DMUs based on their inputs and outputs. However, in inverse DEA, the efficiency of the DMU is predetermined by the decision-maker (DM), and based on this score of efficiency, the optimal level of inputs or outputs are determined. This amount of efficiency that is predetermined by the DM is called the target efficiency. The concept of inverse DEA was first present by Wei et al. [45] and then by Yan et al. [46] developed on the issue of resource allocation. Hadi-Vencheh and Foroughi [26] proposed a generalized inverse DEA mode based on the model of Wei et al. [45]. They showed that some special cases of the inverse DEA model proposed by Wei et al. [45] may fail in some situations and then they revised these failures. Lertworasirikul et al. [32] considered the issue of inverse DEA by considering two different strategies. In the first strategy, by determining the specific level of efficiency for each unit under evaluation DMU, they determined the best possible level of inputs corresponding to a given level of outputs. In the second strategy, again considering a specific level for the efficiency for the under evaluation DMU, they determined the best possible level of outputs corresponding to a level of given inputs and presented their models as resource allocation models. But the early models that they presented were nonlinear models. Due to the problems in solving nonlinear models, they presented new their inverse DEA model in the form of MOLP model. In the following, Ghiyasi [25] points out the drawbacks of Lertworasirikul et al. [32] and then revised the use of MOLP in the proposed inverse DEA model considering the variable return to scale technology (VRS) (Banker, Charnes and Cooper [10]). Gattoufi et al. [20] presented a new model of inverse DEA on mergers and acquisitions to estimate the optimal level of inputs and outputs for the merged entity for a given target efficiency value. Amin and Oukil [9] proposed a Flexible target setting in mergers using inverse data envelopment analysis. Amin et al. [5,6] presented a general model on mergers and acquisitions. They presented a generalized firm restructuring in two scenarios in the form of consolidation or a split. They considering a set of DMUs that called pre-restructuring DMUs, they produced a set of new DMUs that called post-restructuring DMUs, and the level of inputs and outputs from post-restructuring DMUs are determined based on the level of inputs and outputs of pre-restructuring DMUs also the efficiency scores of post-restructuring DMUs are predetermined by the DM as target efficiency scores. Emrouznejad et al. [16] proposed a new application of inverse DEA in environmental efficiency to determine the optimal allocation of CO2 emissions reduction by Chinese manufacturing industries. Wegener and Amin [41] suggested an inverse DEA model for minimizing greenhouse gas emissions with an application in oil and gas. Other applications of inverse DEA including an application in resource allocation (Ghiyasi [23, 25]), new product target setting given expected changes of production frontier (Lim [33]), inter-temporal dependence (Jahanshahloo et al. [29]), revenue setting problems of chain stores, inverse DEA models based on cost and revenue efciency (Ghiyasi [24]), application of the inverse DEA to sensitivity analysis of DMUs (Eyni et al. [18]). Amin and Al-Muharrami [3] addresses the model of inverse DEA in the mergers and acquisitions of firms with negative data. Amin et al. [4] suggested a combined inverse DEA and goal programming approach for target setting of mergers as allows DM to incorporate their preferences. Emrouznejad and Yang [15] presented a literature review of DEA and inverse DEA. Amin and Ibn Boamah [7] proposed a new model of inverse DEA for estimating potential merger gains based on cost efficiency and used the proposed approach in the Canadian banking industry. Amin and Ibn Boamah [8] presented an inverse DEA approach for the two-stage network in the US banking sector. DEA and Multiple Objective Linear Programming (MOLP) have similar structure, and a some of literature can be found on the integration of these two types of methods. The application of an interactive decision-making technique that incorporates both DEA and MOLP is applied to allow the DM to estimate the target set of input levels from a set of alternative points on the efficiency frontier, given the available output levels of a DMU. The DEA method is employed to assess the past performances of DMUs and then MOLP to plan future targets by the DM. The effective combination of DEA and MOLP with the consideration of decision maker’s preferences is useful in decision theory. Hosseinzadeh Lotfi et al. [28] established an equivalence model between DEA and MOLP, then they used the Zionts-Wallenius method to reflecting the decision maker’s preferences in the process of efficiency assessment. Yang et al. [47] developed three equivalence models between the output-oriental dual DEA model and MOLP, including the super-ideal point model, the ideal point model, and the shortest distance model. These models can support the trade-off analysis for setting target values by the DM, However, they only consider the output-oriented dual DEA model, which is a radial model that focuses more on output increase. Malekmohammadi et al. [34] proposed a target model by solving only one rather than n mathematical programming problem, addressing both input reduction and output increase simultaneously. But, these techniques require prior preference knowledge from the DM, and it could be subjective and uncertain due to the multiple attributes and conflicting objectives in most cases. To handle the data uncertainty and target setting according to the decision maker’s preferences, an interactive robust DEA model was proposed by Sadjadi et al. [40] with the combination of DEA and step method in MOLP. In following, Keshavarz and Toloo [30] proposed efficiency status of a feasible solution in the multi-objective integer linear programming problems as a DEA methodology. Ehrgott, Hasannasab, and Raith [17] developed a multi-objective optimization approach to compute the efficient frontier in data envelopment analysis. Agasisti, Munda, and Hippe [2] proposed a measuring the efficiency of European education systems by combining DEA and multiple-criteria evaluation. In the real world, there are many cases in which data are ratio and the ratio of input data to output data or vice versa is important to the DM or input/output data is presented in the form of ratio or percentage data. Traditional DEA models can no longer be used to evaluate the efficiency of DMUs if the ratio of input to output or vice versa is important to the DM, or if the input and output data are ratio data. We need to develop DEA models and in this situation we use DEA-R models. In general, we divide ratio data into three categories as follows. The first category includes ratio data in which the input and output data of ratio numbers are in the form of a fraction and the numerator and denominator corresponding to these fractions are known, but the DM can use this ratio data in the form of decimal numbers in the model. In this case, the data are used in both absolute and ratio forms in the efficiency evaluation model. In the presence of ratio data, the principle of convexity in underlying assumptions of the production possibility set (PPS) is not established in DEA. Among the articles that have been presented in this category to deal with ratio data in DEA models, the following articles can be mentioned. Emrouznejad and Amin [14], Hatami-Marbini and Toloo [27], Khoshnevis and Teirlinck [31]. The above articles modified DEA models to evaluate efficiency in the presence of ratio data. In this category, the numerator and denominator corresponding to these fractions corresponding to the ratio data are known, but the nature of the data is ratio. The second category includes ratio data in which the ratio data are in the form of a fraction and the numerator and denominator corresponding to these fractional numbers may not be available and we have ratio numbers only available as decimal numbers or percentages. It is known and we must use these decimal numbers as ratio data in the model. From a series of articles that modified DEA models and change the underlying assumptions of the PPS in constant and variable return to scale technologies in the presence of ratio data. These papers provide new DEA models to calculate efficiency in the presence of absolute and ratio data. Among the articles that have been presented in this category to deal with ratio data in DEA models, the following articles can be mentioned. Olesen, Petersen and Podinovski [38, 39]. The third category includes ratio data in which the ratio data are in the form of a fraction and the numerator and denominator corresponding to these fractional numbers are important for the DM and the DM cannot use these fractions as decimal numbers in the model. These ratio data are in the form of the ratio of components input to components output or vice versa. These models were initially presented as ratio analysis models (Fernandez-Castro and Smith [19]). In these models, we must use the ratio of inputs to outputs and vice versa in the model. Among the articles that have been presented in this category to deal with ratio data in DEA models, the following articles can be mentioned. Fernandez-Castro and Smith [19], Despic, Despic, and Paradi [13], Wei et al. [42, 43, 44], Mozaffari et al. [35], Mozaffari, Gerami and Jablonsky [36], Gerami et al. [21, 22], Mozaffari et al. [37]. Despic, Despic and Paradi [13] presented DEA-R models by combining ratio analysis and DEA models. They proposed DEA-R models in the output orientation to calculate the efficiency of DMUs in the presence of ratio data as the ratio of components output to components input. Wei et al. [42, 43, 44] examined DEA-R models in the input orientation. They showed that by using DEA-R models in the input orientation, we can avoid the available problems in of traditional DEA models such as efficiency underestimation and pseudo-inefficiency. They showed that DEA-R models in the input orientation have higher efficiency scores than their corresponding scores from CCR models in the input orientation. Mozaffari et al. [37] used DEA-R models to evaluate cost and revenue efficiency. Gerami et al. [22] used DEA-R models to evaluate the efficiency of the hospital supply chain in the presence of ratio data. Gerami, Mozaffari, and Wanke [21] proposed DEA-R models to evaluate the efficiency of two-stage network structures in the presence of ratio data. In the first category of ratio data, we can refer to the ratio of the number of research projects presented by some professors in a course to the total number of professors in a university, and these ratios are important for the DM. DM used of the numerator and denominator corresponding ratio data in the model. In the second category of ratio data can be referred to the percentage of successful operations performed to the total number of operations performed in a hospital during a treatment period. But the DM only uses the decimal form of this data and this data is only available in the form of decimal numbers. In the third category of data, we can refer to some concepts in economics such as immediate and current profit, or the number of patients treated to the total number of patients admitted to a hospital during a treatment period. It should be noted that in the third category of data, we use the input and output data of each of the DMUs directly in the model and put this data as the form of the ratio of components input to components output or vice versa in the model, but what is important is that we assume that the input and output data are definite and their ratio is important for the DM. In this paper, we use the ratio data in the third category and assume that the input and output data are definite numbers and their ratio is important for the DM, and we put this data as the form of the ratio of components input to components output in the models and do not use the fractional or decimal form of these numbers. It can be said that the main contribution of the article is as follows. In this paper, we examine one of the most important issues in DAE, namely inverse DEA, and estimate inputs and outputs if some of the input and output components change and the DM wants create a new DMU with a relative efficiency score that predetermined and is equal to relative efficiency score of the initial unit. In the process of estimating the level of inputs and outputs, we can choose two different strategies in inverse DEA models in the presence of ratio data. In the first strategy, by determining the specific level of efficiency for each unit under evaluation DMU, they determined the best possible level of inputs corresponding to a given level of outputs. In the second strategy, again considering a specific level for the efficiency for the under evaluation DMU, they determined the best possible level of outputs corresponding to a level of given inputs. That is, if we want the efficiency of the DMU to remain unchanged, we determine the optimal level of input or output based on DEA-R models. We obtain the necessary and sufficient conditions for inverse DEA-R in the input orientation models. As we know, one of the important issues in inverse DEA is the selection of a criterion model for comparing the efficiency scores of DMU before and after the process of estimating inputs and outputs. In this paper, we first develop inverse DEA models in the presence of ratio data, and by providing a suitable criterion model in the presence of ratio data, we show that we can significantly reduce the computations and thus show that the new criterion model presented have the same results as the previous criterion models. Finally, we provide a case study to examine the validity of the proposed models. The remainder of the paper unfolds as follows. In the Section 2, we examine DEA-R models in the input and output orientations and present the relationship between these models and traditional DEA models. Section 3 proposes the inverse DEA-R models and the inputs/output estimation process based on inverse DEA-R models in the presence of ratio data, in following, we present the criterion models for evaluating the efficiency of the new units created. Section 4 provides a numerical example, in this way, we illustrate the inputs/output estimation process based on inverse DEA-R models in the presence of ratio data. Section 5 provides a real world data empirical investigation and shows the applicability and potential use of the proposed models, we present an application of the proposed approach related to medical centers in Taiwan and at the end, we present the results of the research. \section{Ratio-Based DEA Models} \label{sec:2} \noindent Suppose we have $n$ decision units as ${DMU}_j=\left(x_j,y_j\right),$ $j=1,\dots ,n$. The input and output vectors corresponding to ${DMU}_j,$ $j=1,\dots ,n$, as $x_j\mathrm{=}\left(x_{1j}{,\dots ,x}_{mj}\right)$ and${\ y}_j\mathrm{=}\left(y_{1j}{,\dots ,y}_{sj}\right)$. We suppose that\\ $x_{ij}>0$, $y_{rj}>0$, $i=1,\dots ,m,$ $r=1,\dots ,s$, $j=1,\dots ,n$. Suppose the ratios $\frac{x_j}{y_j},$ $j=1,\dots ,n$, in the input orientation and the ratios $\frac{y_j}{x_j}$, $ j=1,\dots ,n,$ in the output orientation are defined. Suppose, we consider the multiples corresponding to the ratios $\frac{x_{ij}}{y_{rj}},$ $j=1,\dots ,n$, $i=1,\dots ,m,$ $r=1,\dots ,s,$ as $w_{ir}$. Fernandez-Castro and Smith [19] proposed ratio analysis model in the input orientation as follows. \begin{equation}\begin{array}{cc} \hspace{-7cm}\min\hspace{0.2cm}\displaystyle\sum_{i=1}^{m}\sum_{r=1}^{s}w_{ir}(\frac{x_{io}}{y_{ro}})\\ \hspace{-3.6cm}s.t.\quad \displaystyle\sum_{i=1}^{m}\sum_{r=1}^{s}w_{ir}(\frac{x_{ij}}{y_{rj}})\geq 1,\hspace{0.25cm} j=1,\cdots ,n,\\ \hspace{0.35cm} w_{ir}\geq 0,\hspace{0.3cm} i=1,\cdots ,m,\hspace{0.3cm} r=1,\cdots ,s. \\ \end{array} \end{equation} \noindent We consider the variable corresponding to first constraint in Model $(1)$ as ${\widehat{\mu }}_{j}$, $j=1,\dots ,n$. The dual Model $(1)$ is as follows. \begin{equation}\begin{array}{cc} \hspace{-8.5cm}\max\hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\widehat{\mu }}_{j}\\ \hspace{-1.1cm}s.t.\quad \displaystyle\sum_{j=1}^{n} {\widehat{\mu }}_{j}(\frac{x_{ij}}{y_{rj}})\leq (\frac{x_{io}}{y_{ro}}),\hspace{0.3cm} i=1,\cdots ,m,\hspace{0.3cm} r=1,\cdots ,s,\\ \hspace{-3.6cm} {\widehat{\mu }}_{j}\geq 0,\hspace{0.3cm} j=1,\cdots ,n. \\ \end{array} \end{equation} \noindent By considering $\sum^n_{j=1}{{\widehat{\mu }}_j}=t$, ${\mu }_j=\frac{{\widehat{\mu }}_j}{t}$ and placing ${\theta }_R\mathrm{=}\frac{1}{t}$ from the optimization point of the model $(2)$ is converted as follows. \begin{equation}\begin{array}{cc} \hspace{-10.5cm}\min \hspace{0.1cm}{\theta }_{R}\\ \hspace{-2.1cm}s.t.\hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\mu }_{j}(\frac{x_{ij}}{y_{rj}})\leq {\theta }_{R}(\frac{x_{io}}{y_{ro}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s,\\ \hspace{-4.2cm} \displaystyle\sum_{j=1}^{n}{\mu }_{j}=1,\hspace{0.2cm} {\mu }_{j}\geq 0,\hspace{0.2cm} j=1,\cdots ,n. \\ \end{array} \end{equation} Model $(3)$ is called the DEA-R model in the input orientation in the envelopment form. Model $(3)$ by Wei et al. [42, 43, 44] and Mozaffari et al. [36] were also studied. We now examine the relationship between the above Model $(1)$ to traditional DEA model.\\ \noindent \textbf{Theorem 1.} \textit{Model $(1)$ is equivalent to the CCR multiplier model in the input orientation.}\\ \noindent \textbf{Proof}: If we define $\widehat{{DMU}_j}\mathrm{=}\left(\frac{x_{ij}}{y_{rj}},1\right),\ \ \ j=1,\dots ,n,\ \ \ \ i=1,\dots ,m,\ \ \ r=1,\dots ,s.$ Then we have $n$ DMUs with one output and $m+s$ input. Considering the multiples corresponding to the input components of the new units as $w_{ir}\ge 0,\ \ i=1,\dots ,m,\ \ r=1,\dots ,\ s$, and the multiples corresponding to the output component of the new units as $u_1$. Then model $(1)$ is converted as follows. \begin{equation}\begin{array}{cc} \hspace{-7cm}\min\hspace{0.2cm}\displaystyle\sum_{i=1}^{m}\sum_{r=1}^{s}w_{ir}(\frac{x_{io}}{y_{ro}})\\ \hspace{-2.7cm}s.t.\quad \displaystyle\sum_{i=1}^{m}\sum_{r=1}^{s}w_{ir}(\frac{x_{ij}}{y_{rj}})-1u_{1}\geq 1,\hspace{0.25cm} j=1,\cdots ,n,\\ \hspace{-3.35cm} 1u_{1}=1,\hspace{0.3cm} u_{1}\geq 0,\\ \hspace{0.15cm} w_{ir}\geq 0,\hspace{0.3cm} i=1,\cdots ,m,\hspace{0.3cm} r=1,\cdots ,s. \\ \end{array} \end{equation} \noindent That model $(4)$ is CCR multiplier model in the input orientation (Charnes, Cooper and Rhodes [11]) in evaluation $\widehat{{DMU}_o}\mathrm{=}\left(\frac{x_{io}}{y_{ro}},1\right),\ \ \ j=1,\dots ,n,\ \ \ \ \\ i=1,\dots ,m,\ \ \ r=1,\dots ,s.$ If the set of DMUs be as $\widehat{{DMU}_j}\mathrm{=}\left(\frac{x_{ij}}{y_{rj}},1\right),\ \ \\\ j=1,\dots ,n,\ \ \ \ i=1,\dots ,m,\ \ \ r=1,\dots ,s$, which completes the proof.$\blacksquare $\\ \noindent Similarly, suppose, we consider the multiples corresponding to the ratios$\\\ \frac{y_{rj}}{x_{ij}},$ $j=1,\dots ,n$,$\ i=1,\dots ,m,$ $r=1,\dots ,s,$ as $w_{ir}$. Fernandez-Castro and Smith [19] proposed ratio analysis model in the output orientation as follows. \begin{equation}\begin{array}{cc} \hspace{-7cm}\min\hspace{0.2cm}\displaystyle\sum_{i=1}^{m}\sum_{r=1}^{s}u_{ir}(\frac{y_{ro}}{x_{io}})\\ \hspace{-3.6cm}s.t.\quad \displaystyle\sum_{i=1}^{m}\sum_{r=1}^{s}u_{ir}(\frac{y_{rj}}{x_{ij}})\leq 1,\hspace{0.25cm} j=1,\cdots ,n,\\ \hspace{0.35cm} u_{ir}\geq 0,\hspace{0.3cm} i=1,\cdots ,m,\hspace{0.3cm} r=1,\cdots ,s. \\ \end{array} \end{equation} \noindent We consider the variable corresponding to first constraint in Model $(5)$ as ${\widehat{\lambda }}_j\ ,\ \ j=1,\dots ,n$. The dual Model $(5)$ is as follows. \begin{equation}\begin{array}{cc} \hspace{-8.5cm}\max\hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\widehat{\lambda }}_{j}\\ \hspace{-1.1cm}s.t.\quad \displaystyle\sum_{j=1}^{n} {\widehat{\lambda }}_{j}(\frac{y_{rj}}{x_{ij}})\leq(\frac{y_{ro}}{x_{io}}),\hspace{0.3cm} r=1,\cdots ,s,\hspace{0.3cm} i=1,\cdots ,m,\\ \hspace{-3.6cm} {\widehat{\lambda }}_{j}\geq 0,\hspace{0.3cm} j=1,\cdots ,n. \\ \end{array} \end{equation} \noindent By considering $\sum^n_{j=1}{\widehat{\lambda }}_{j}=t,$\hspace{0.1cm}$ {\lambda }_j=\frac{{\widehat{\lambda }}_j}{t}$, and placing ${\varphi }_R\mathrm{=}\frac{1}{t}$, from the optimization point, the Model $(5)$ is converted as follows. \begin{equation}\begin{array}{cc} \hspace{-10.5cm}\max \hspace{0.1cm}{\varphi }_{R}\\ \hspace{-2.1cm}s.t.\hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\lambda}_{j}(\frac{y_{rj}}{x_{ij}})\leq{\varphi }_{R}(\frac{y_{ro}}{x_{io}}),\hspace{0.2cm} r=1,\cdots ,s,\hspace{0.2cm} i=1,\cdots ,m,\\ \hspace{-4.2cm} \displaystyle\sum_{j=1}^{n}{\lambda }_{j}=1,\hspace{0.2cm} {\lambda }_{j}\geq 0,\hspace{0.2cm} j=1,\cdots ,n. \\ \end{array} \end{equation} \noindent Model $(7)$ is called the DEA-R model in the output orientation in the envelopment form. Model $(7)$ by Despic, Despic and Paradi [13] were also studied. We now examine the relationship between the above model $(5)$ to traditional DEA model.\\ \noindent \textbf{Theorem 2.} \textit{Model $(5)$ is equivalent to the CCR multiplier model in the output orientation.}\\ \noindent \textbf{Proof}: If we define $\check{{DMU}_j}\mathrm{=}\left(1,\frac{y_{rj}}{x_{ij}}\right),\ \ \ j=1,\dots ,n,\ \ \ \ i=1,\dots ,m,\ \ \ r=1,\dots ,s$. Then we have $n$ DMUs with one input and $m+s$ output. Considering the multiples corresponding to the output components of the new units as $u_{ir}\ge 0,\ \ i=1,\dots ,m,\ r=1,\dots ,\ s$, and the multiples corresponding to the input component of the new units as $v_1$. Then Model $(5)$ is converted as follows. \begin{equation}\begin{array}{cc} \hspace{-7cm}\max\hspace{0.2cm}\displaystyle\sum_{r=1}^{s}\sum_{i=1}^{m}u_{ir}(\frac{y_{ro}}{x_{io}})\\ \hspace{-2.7cm}s.t.\quad \displaystyle\sum_{i=1}^{m}\sum_{r=1}^{s}u_{ir}(\frac{y_{rj}}{x_{ij}})-1v_{1}\leq 1,\hspace{0.25cm} j=1,\cdots ,n,\\ \hspace{-3.35cm} 1v_{1}=1,\hspace{0.3cm} v_{1}\geq 0,\\ \hspace{0.15cm} u_{ir}\geq 0,\hspace{0.3cm} i=1,\cdots ,m,\hspace{0.3cm} r=1,\cdots ,s. \\ \end{array} \end{equation} \noindent That model $(8)$ is CCR multiplier model in the output orientation (Charnes, Cooper and Rhodes [11]) in evaluation $\check{{DMU}_o}\mathrm{=}\left(1,\frac{y_{ro}}{x_{io}}\right),\ \ \ j=1,\dots ,n,\ \ \ \ i=1,\dots ,m,\ \ \ r=1,\dots ,s$, if we consider the set of DMUs as $\check{{DMU}_j}\mathrm{=}\left(1,\frac{y_{rj}}{x_{ij}}\right),\ \ \ j=1,\dots ,n,\ \ \ \ i=1,\dots ,m,\ \ r=1,\dots ,s$, which completes the proof.$\blacksquare $ \section{The Inputs/Output Estimation Process Based on Inverse DEA-R} \noindent In this section, we present inverse DEA-R models in the presence of ratio data in the input orientation. In this regard, we present inputs/output estimation process based on inverse DEA-R models. We provide criterion models to evaluate the efficiency of new units. In other words, we need to find the new input level of under evaluation DMU that guarantees unchanged relative efficiency for this DMU. \noindent Suppose we have $n$ DMUs as ${DMU}_j=\left(x_j,y_j\right),\ \ j=1,\dots ,n$ that each DMU consume input vector $x_j\mathrm{=}\left(x_{1j}{,\dots ,x}_{mj}\right)$ to product output vector${\ y}_j\mathrm{=}\left(y_{1j}{,\dots ,y}_{sj}\right)$. We suppose that $x_{ij}>0,\ \ y_{rj}>0,\ \ i=1,\dots ,m,$ $r=1,\dots ,s,\ \mathrm{\ \ }j=1,\dots ,n$. Suppose the ratios $\frac{x_{ij}}{y_{rj}},$ $j=1,\dots ,n$, are the ratio ith input component to rth output component of\noindent ${DMU}_j=\left(x_j,y_j\right),\ \ j=1,\dots ,n$. We show under evaluation DMU as ${DMU}_o=\left(x_o,y_o\right)$, also, assume ${DMU}_o$ perturbs its output level into ${\eta }^o=y_o+\Delta y_o$, $\Delta y_o\ge -y_o$, $\Delta y_o\in R$. Now we want to know how much we need to change the input level of ${DMU}_o=\left(x_o,y_o\right)$, so that the relative efficiency of this unit remains unchanged. In other words, we first perturbs the output level of ${DMU}_o=\left(x_o,y_o\right)$ to a certain extent, and then, we must determine the input level of the new DMU namely ${\gamma }^o=x_o+\Delta x_o$, $\Delta x_o\ge -x_o$, $\Delta x_o\in R$, in such a way that the relative efficiency of the new unit is equal to the relative efficiency of ${DMU}_o=\left(x_o,y_o\right)$. We proposed the following MOLP model in the inverse DEA-R and in the presence of ratio data to determine ${\gamma }^o=x_o+\Delta x_o$, $\Delta x_o\ge -x_o$, $\Delta x_o\in R$, as follows. \begin{equation}\begin{array}{cc} \hspace{-8cm}\min \hspace{0.1cm}(\gamma_{1},\gamma_{2},...,\gamma_{m})\\ \hspace{-1.4cm}s.t.\hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\mu }_{j}(\frac{x_{ij}}{y_{rj}})\leq {\theta }_{R}^{I}(\frac{\gamma_{i}}{y_{ro}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s,\\ \hspace{-3.4cm} \displaystyle\sum_{j=1}^{n}{\mu }_{j}=1,\hspace{0.2cm} {\mu }_{j}\geq 0,\hspace{0.2cm} j=1,\cdots ,n. \end{array} \end{equation} \textbf{Definition 1. }${DMU}_o=\left(x_o,y_o\right)$ is called a weak efficient solution in evaluation with model $(3)$ if the optimal value of model $(3)$ is equal to one.\\ \textbf{Definition 2. }Suppose $\left(\mu ,\gamma \right)$ that $\gamma=(\gamma_{1},\gamma_{2},...,\gamma_{m})$ and\\ $\mu=(\mu_{1},\mu_{2},...,\mu_{n})$ are a feasible solution of model Model $(9)$. If there does not exist a feasible solution $(\bar{\mu},\bar{\gamma})$ of model $(9)$ such that $\bar{\gamma}\leq \gamma$ will be a weakly efficient solution of model $(9)$.\\ \noindent \textbf{Theorem 3.} The Model $(9)$ is a feasible.\\ \noindent \textbf{Proof}: We can show that ${\mu }_o=1$, ${\mu }_j=0$, $j=1,\dots ,n,\ j\neq o$, ${\theta }^I_R=1$, ${\gamma }_i=x_{io}$, ${\eta }^o_r=y_{ro},\ i=1,\dots ,m,\ \ r=1,\dots ,s,$ is a feasible for a Model $(9)$, and the proof is complete. $\mathrm{\blacksquare }$\\ \noindent A weak efficient solution of Model $(9)$ are as new input values from ${DMU}_o=\left(x_o,y_o\right)$ for a disturbed output level ${\eta }^o=y_o+\Delta y_o$, $\Delta y_o\ge -y_o$, $\Delta y_o\in R$, to preserve relative efficiency of \noindent ${DMU}_o=\left(x_o,y_o\right)$ after the output changes. At first, to check the relative efficiency of the new unit namely ${DMU}^{new}_o=\left(x_o+\Delta x_o,y_o+\Delta y_o\right)=\left({\gamma }^o,{\eta }^o\right)$, we present the following criterion model. \begin{equation}\begin{array}{cc} \hspace{-10.5cm}\min \hspace{0.1cm}{\theta }_{R}\\ \hspace{-0.1cm}s.t.\hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\mu }_{j}(\frac{x_{ij}}{y_{rj}})+{\mu }_{n+1}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}})\leq {\theta }_{R}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s,\\ \hspace{-1cm} \displaystyle\sum_{j=1}^{n}{\mu }_{j}+{\mu }_{n+1}=1,\hspace{0.2cm}{\mu }_{n+1}\geq 0,\hspace{0.2cm} {\mu }_{j}\geq 0,\hspace{0.2cm} j=1,\cdots ,n. \end{array} \end{equation} \noindent \textbf{Theorem 4.} \textit{Suppose that ${DMU}_o=\left(x_o,y_o\right)$ perturbs its output from $y_o$ to ${\eta }^o=y_o+\Delta y_o$, $\Delta y_o\ge -y_o$, $\Delta y_o\in R$. Then $\left({\gamma }^o,\mu \right)$ is a weak efficient solution of MOLP Model $(9)$ if and only if ${\theta }^+_R\mathrm{=}{\theta }^I_R$.}\\ \noindent \textbf{Proof}: First assume that $\left({\gamma }^o,\mu \right)$ is a weak efficient solution of MOLP Model $(9)$ . We show that the efficiency score of ${DMU}_o=\left(x_o,y_o\right)$ and ${DMU}^{new}_o=\left(x_o+\Delta x_o,y_o+\Delta y_o\right)=\left({\gamma }^o,{\eta }^o\right)$ are equal, i.e.$\ {\theta }^+_R\mathrm{=}{\theta }^I_R$. Put $\mu^{'}=(\mu,0)^{T}$, it is easily seen that $(\theta^{I}_{R},\mu^{'})$ is a feasible solution for Model $(10)$, so we will have ${\theta }^+_R\mathrm{\le }{\theta }^I_R$. Now suppose ${\theta }^+_R<{\theta }^I_R$, let that $\left({\theta }^+_R,{\mu }^+\right)\ $is an optimal solution of Model $(10)$ , so according to the constraints of Model $(10)$ , we will have \begin{equation}\begin{array}{cc} \hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\mu }_{j}^{+}(\frac{x_{ij}}{y_{rj}})+{\mu }_{n+1}^{+}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}})\leq {\theta }_{R}^{+}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s,\\ \hspace{-1.5cm} \displaystyle\sum_{j=1}^{n}{\mu }_{j}^{+}+{\mu }_{n+1}^{+}=1,\hspace{0.2cm}{\mu }_{n+1}^{+}\geq 0,\hspace{0.2cm} {\mu }_{j}^{+}\geq 0,\hspace{0.2cm} j=1,\cdots ,n. \end{array} \end{equation} Given that $\theta^{I}_{R}\leq 1$ and $\left({\gamma }^o,\mu \right)\ $is a weak efficient solution of MOLP Model $(9)$, so we will have \begin{equation}\begin{array}{cc} \hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\mu }_{j}(\frac{x_{ij}}{y_{rj}})\leq {\theta }_{R}^{I}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}})\leq (\frac{\gamma_{i}^{o}}{\eta_{r}^{o}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s,\\ \hspace{-4cm} \displaystyle\sum_{j=1}^{n}{\mu }_{j}=1,\hspace{0.2cm} {\mu }_{j}\geq 0,\hspace{0.2cm} j=1,\cdots ,n. \end{array} \end{equation} By comparing relations $(11)$ and $(12)$ we will have \begin{equation}\begin{array}{cc} \hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\mu }_{j}^{+}(\frac{x_{ij}}{y_{rj}})+{\mu }_{n+1}^{+}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}})\geq \displaystyle\sum_{j=1}^{n} {\mu }_{j}^{+}(\frac{x_{ij}}{y_{rj}})+{\mu }_{n+1}^{+}(\displaystyle\sum_{j=1}^{n} {\mu }_{j}(\frac{x_{ij}}{y_{rj}}))=\\ \displaystyle\sum_{j=1}^{n}({\mu }_{j}^{+}+{\mu }_{n+1}^{+}(\displaystyle\sum_{j=1}^{n} {\mu }_{j}))(\frac{x_{ij}}{y_{rj}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s. \end{array} \end{equation} Now, we put \[{\widetilde{\mu }}_j={\mu }^+_j+{\mu }^+_{n+1}\left(\sum^n_{j=1}{{\mu }_j}\right), j=1,\dots ,n. \] And so according to relation Model $(13)$, we will have \begin{equation}\begin{array}{cc} \hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\widetilde{\mu} }_{j}(\frac{x_{ij}}{y_{rj}})\leq {\theta }_{R}^{+}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s. \end{array} \end{equation} Given that $\sum^n_{j=1}{{\mu }_j}=1$ and $\sum^n_{j=1}{{\mu }^+_j}+{\mu }^+_{n+1}=1,$ ${\mu }^+_{n+1}\ge 0,$ ${\mu }^+_j\ge 0,$ $j=1,\dots ,n$. Then $\sum^n_{j=1}{{\widetilde{\mu }}_j}=1,\ \ {\widetilde{\mu }}_j\ge 0,\ \ j=1,\dots ,n$. \begin{equation}\begin{array}{cc} \hspace{0.2cm}\displaystyle\sum_{j=1}^{n} {\widetilde{\mu} }_{j}(\frac{x_{ij}}{y_{rj}})\leq {\theta }_{R}^{+}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}})< {\theta }_{R}^{I}(\frac{\gamma_{i}^{o}}{\eta_{r}^{o}}),\hspace{0.2cm} i=1,\cdots ,m,\hspace{0.2cm} r=1,\cdots ,s,\\ \hspace{-4.4cm} \displaystyle\sum_{j=1}^{n}{\widetilde{\mu} }_{j}=1,\hspace{0.2cm} {\widetilde{\mu}}_{j}\geq 0,\hspace{0.2cm} j=1,\cdots ,n. \end{array} \end{equation} Therefore $\left({\gamma }^o,\widetilde{\mu }\right)\ $that $\widetilde{\mu }\mathrm{=}\left({\widetilde{\mu }}_1{,\dots ,\widetilde{\mu }}_n\right)$ will be a feasible solution for model $(9)$. According to relation $(15)$ there exists a $0