\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]{S. Asdadi Rahmati, R.Fallahnejad} \fancyhead[CO]{Evaluating Groups of Decision Making Units in the Data Envelopment Analysis based on Cooperative Games} %\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 \\ Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\ ISSN: 1735-8299\\ URL: http://www.ijmex.com\\ \vspace*{9mm} \begin{center} {\Large \bf Evaluating Groups of Decision Making Units in the Data Envelopment Analysis based on Cooperative Games\\} \let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)} {\bf Sanaz Asadi Rahmati}\vspace*{-2mm}\\ \vspace{2mm} {\small Department of Mathematics, Khorramabad branch, Islamic Azad University, Khorramabad, Iran} \vspace{2mm} {\bf Reza Fallahnejad$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\ \vspace{2mm} {\small Department of Mathematics, Khorramabad branch, Islamic Azad University, Khorramabad, Iran} \vspace{2mm} \end{center} \vspace{4mm} {\footnotesize \begin{quotation} {\noindent \bf Abstract.} Data Envelopment Analysis(DEA) is a non-parametric method for evaluating the efficiency of those Decision Making Units (DMUs) that have the same functionality and use multiple inputs to generate multiple outputs. DMUs may sometimes be divided into several groups according to a series of criteria, and it is intended to evaluate a group of similar DMUs. In this paper, each group was considered as a player in a cooperative game and a subset of groups was considered as a coalition. Assuming the Production Possibility Set is made up of the members of the coalition, a characteristic function was defined in terms of the sum of the efficiency of all units to determine the marginal effect of each group in different coalitions. The groups were then evaluated using the Shapley Value as a solution of the cooperative game. Some Examples were provided to describe and apply the method. \end{quotation} \begin{quotation} \noindent{\bf AMS Subject Classification:}91A12 ;90C08. \noindent{\bf Keywords and Phrases:} Data envelopment analysis (DEA), Group evaluation, Game theory, Cooperative game, Shapley Value. \end{quotation}} \section{Introduction} \label{intro} % It is advised to give each section and subsection a unique label. Data Envelopment Analysis (DEA) is used as a nonparametric instrument to evaluate the relative efficiency of homogeneous DMUs with multiple inputs so that they produce multiple outputs. In contrast to the singular selection of DMUs which provides for the evaluation of DMU's singularly, the grouping of DMUs enhances the dimensions of the evaluation process and, thus, provides for a more intelligent decision making. A brief review of the literature indicates that there is a great need for further research in this field. The following is a brief summary of previous and current research. Charnes et al. \cite{Charnes1} first introduced the CCR model for evaluating the efficiency of DMUs and then in another paper \cite{Charnes2} introduced the group analysis. Banker and Morey \cite{Banker2}, argued that a group of DMUs in nesting mode could be defined as the categorical variable. Further, they compared the efficiency of each DMU with those of the category to which it belonged and the categories below it. In the absence of the possibility of classifying DMU's into homogeneous groups, Cook et al. al. \cite{Cook1} classified the DMUs hierarchically in different levels. In this method, the efficiencies found at one level were counted in the efficiency of the higher levels. They also devised a model for keeping track of various ratings received by a DMU in different possible groupings. Camanho and Dyson \cite{Camanho} concentrated on technical efficiency. Their goal was to evaluate the groups and identify each internally inefficient DMU to be compared and contrasted with inefficiencies of its group. They used Malmquist Productivity Index. Maniadakis and Thanassoulis \cite{Maniadakis} revised the model introduced by Camanho and Dyson in order to show the cost when the input prices are available. In another model, Cook and Zhu \cite{Cook2} used the common weight to minimize the maximum differences between the efficiency of an independent DMU and that of the same DMU as a member of the group to reach a point as close to the optimal case as possible. O'Donnell et al. \cite{ODonnell} used the concept of meta-frontier to compare the technical efficiency of companies that could be divided into various groups. They divided the efficiency measured by the meta-frontier into two parts: the distance of a DMU from its own frontier and the distance of its group's frontier from the meta-frontier. Bagherzade Valami \cite{Valami} used the production technology to devise a model for evaluating the performance of a group in which the efficiency of each DMU is defined as the distance of that DMU from the group's frontier. The model defines the efficiency of a group as the geometric mean of the efficiencies of all the DMUs based on that group's PPS. The higher the geometric mean of the efficiencies of all the DMUs, the higher the group efficiency. This paper evaluates the groups in a common platform, each group was considered as a player in a cooperative game and a subset of groups was considered as a coalition. Assuming PPS is made up of the members of the coalition S, a characteristic function was defined in terms of the sumof the efficiency of all units to determine the marginal effect of each group in different coalitions. Then, the groups were evaluated using the Shapley value as a cooperative game solution. The present study, however, took a different approached compared to that by Li et al. \cite{Li}. On the other hand, in the study of Bagherzadeh Valami \cite{Valami}, the efficiency of the group was calculated based only on the efficiency of all the DMUs with the frontier of that group, indicating that the DMUs have been measured only by a single-member coalition of the groups, meaning that, in fact, the marginal effect of each group has been considered a null coalition. In the present paper, all DMUs were evaluated through different coalitions of groups and thus with more frontiers. Therefore, the marginal effects of a group on all possible coalitions were used to evaluate the efficiency of the respective group, making the evaluation of groups more accurate. The different sections of the present study are as follows: In Section \ref{Sec 2}, the efficiency of a DMU, the efficiency of a DMU in a group and the efficiency of a group are defined and an introduction to game theory is presented. In Section \ref{Sec 3}, a new method is used to measure the efficiency of groups based on the cooperative games. In Section \ref{Sec 4}, the proposed method is first illustrated by two simple examples, and then, a real world example is provided to indicate the applicability of the method. Section \ref{Sec 5} contains the results of the present paper. \section{Definitions of key terms} \label{Sec 2} \subsection{Efficiency of a DMU} Consider $n$ Decision Making Units (DMUs), so that each $DMU _j$ $(j \in \{1,\ldots n\})$ uses $m$ inputs ($ x_{ij} $, $i \in \{1,\ldots m\}$) to generate $s$ outputs ($ y_{rj} $, $r \in \{1,\ldots s\}$). Production Possibility Set (PPS) with constant return to scale (CRS) is a set of $(X, Y) $ pairs where Input $X$ can generate Output $Y$. \[ T_{c} = \{ (X, Y)\big|~X \geq \sum_{j = 1}^n \lambda_jX_j, \quad Y \leq \sum_{j = 1}^n \lambda_jY_j,\quad \lambda_j \geq 0,~j = 1, \ldots ,n \}. \] By adding the constraint $ \sum_{j = 1}^{n} \lambda_j=1 $, a PPS with a variable return to scale (VRS) is obtained. Different methods, including CCR \cite{Charnes1}, BBC \cite{Banker1}, SBM \cite{Tone}, and enhanced Russell measure (ERM) \cite{Bowlin, Fare} have been provided to calculate the efficiency of DMUs in DEA. The ERM model is as follow: \begin{equation}\label{eq1} \begin{aligned} R_o^*=\min\frac{\frac{1}{m}\sum_{i = 1}^{m}\theta_i}{\frac{1}{s}\sum_{r= 1}^s\varphi_r}&\\ \text{s.\,t.}\quad \sum_{j= 1}^{n} \lambda_jx_{ij} \leq \theta_i x_{io}, \quad &i=1, \ldots m\\ \sum_{j= 1}^{n} \lambda_jy_{rj} \geq \varphi_r y_{ro}, \quad &r=1, \ldots s\\ \theta_i \leq 1, \quad &i=1, \ldots m\\ \varphi_r \geq 1, \quad &r=1, \ldots s\\ \lambda_j \geq 0, \quad &j=1, \ldots n. \end{aligned} \end{equation} As indicated in the study of Izadikhah et al.\cite{Izadikhah}, when a DMU leaves the reference set and is located outside of the PPS, the ERM model does not get the correct super-efficiency value, because the direction of movement should actually be towards the frontier. However, this model moves away from the frontier instead of moving from the eliminated DMU located beyond the frontier towards the frontier. Therefore, Izadikhah et al. \cite{Izadikhah} introduced the following super efficiency method based on modified ERM model, which allows both movement directions simultaneously and obtains the right super-efficiency value for the DMUs both inside and outside the frontiers: \begin{equation}\label{eq2} \begin{aligned} &R_o^*=\min\frac{\frac{1}{m}\sum_{i = 1}^{m}\theta_i}{\frac{1}{s}\sum_{r= 1}^s\varphi_r}\\ \text{s.\,t.}\quad & \sum_{\scriptstyle j = 1 \hfill \atop \scriptstyle j \neq 0 \hfill}^n \lambda_j x_{ij} \leq \theta_i x_{io}, \quad i=1, \ldots m\\ &\sum_{\scriptstyle j = 1 \hfill \atop \scriptstyle j \neq 0 \hfill}^n \lambda_j y_{ij} \geq \varphi_r y_{ro}, \quad r=1, \ldots s\\ &\theta_i-1 \leq M \delta , \quad i=1, \ldots m\\ -&\theta_i+1 \leq M (1-\delta), \quad i=1, \ldots m\\ -&\varphi_r+1 \leq M \delta , \quad r=1, \ldots s\\ &\varphi_r-1 \leq M (1-\delta) , \quad r=1, \ldots s\\ &\delta \in \{0,1\} \\ &\theta_i,~\lambda_j \geq 0, \quad \forall ~i,j, \end{aligned} \end{equation} where $ M$ is a number that is large enough and $ \delta $ is a binary variable that establishes only one of the following conditions: \begin{align*} \text{(I)}~\begin{cases}\theta_i \leq 1, \quad &i=1, \ldots m ,\\ \varphi_r \geq 1, \quad &r=1, \ldots s.\end{cases}\\ \text{(II)}~ \begin{cases}\theta_i \geq 1, \quad &i=1, \ldots m,\\ \varphi_r \leq 1, \quad &r=1, \ldots s.\end{cases} \end{align*} $ R_o^* $ is the efficiency of $DMU_o$, when the PPS is made up of all DMUs except $DMU_o$. If $ R_o^* <1 $, then condition $ \text{(I)} $ holds true, that is $DMU_o$ is inside the PPS. If $R_o^*=1$, then $\theta_i=1 $ and $\varphi_r=1$, that is, $DMU_o$ is on the PPS frontier, and if $R_o^*>1$, then condition $\text{(II)} $ holds true, that is $DMU_o$ is outside the PPS. \subsection{Efficiency of a DMU with the frontier of a group} In this section, the definition presented by Izadikhah et al. \cite{Izadikhah} of super efficiency is extended to the case where a subset of the DMU is removed. Consider $n$ number of DMUs where each $DMU_j$ ($j \in \{1, \ldots n\}$) uses $m$ inputs ($ x_{ij} $, $i \in \{1,\ldots m\}$) to generate $s$ outputs ($ y_{rj} $, $r \in \{1,\ldots s\}$) Moreover, suppose these $n$ DMUs are divided into $q$ groups as $A_1$, $A_2$, $...$, and $A_q$. Then, the efficiency of $DMU_o$ with the frontier of the group $A_t$ where $t \in \{1,2, ..., q\}$ is defined as: \begin{equation}\label{eq3} \begin{aligned} &R_o^{A_t}=\min\frac{\frac{1}{m}\sum_{i = 1}^{m}\theta_i}{\frac{1}{s}\sum_{r= 1}^s\varphi_r}\\ \text{s.\,t.}\quad & \sum_{j \in A_t}^n \lambda_j x_{ij} \leq \theta_i x_{io}, \quad i=1, \ldots m\\ &\sum_{j \in A_t}^n \lambda_j y_{rj} \geq \varphi_r y_{ro}, \quad r=1, \ldots s\\ &\theta_i-1 \leq M \delta , \quad i=1, \ldots m\\ -&\theta_i+1 \leq M (1-\delta), \quad i=1, \ldots m\\ -&\varphi_r+1 \leq M \delta , \quad r=1, \ldots s\\ &\varphi_r-1 \leq M \delta , \quad r=1, \ldots s\\ &\delta \in \{0,1\} \\ &\theta_i,~\lambda_j \geq 0, \quad \forall ~i,j, \end{aligned} \end{equation} where $R_o^{A_t }$ is the efficiency of $DMU_o$ assuming that the PPS has been made by DMUs belonging to the group $A_t$. If $R_o^{A_t }=1$, then the $DMU_o$ is located on the frontier of the PPS which is made up of the DMUs belonging to group $A_t$. Therefore, although $DMU_o$ is efficient with respect to this frontier, it may not be efficient with respect to the general frontier belonging to the PPS made up of all the DMUs. This means that the $DMU_o$ may achieve the best efficiency in its group, but it may not perform well compared with the other groups. If $R_o^{A_t }<1$, then the $DMU_o$ is located inside the PPS made by the DMUs belonging to group $A_t$ and, therefore, is inefficient in relation to this frontier. If $R_o^{A_t }>1$, then the $DMU_o$ is located outside the PPS made up of the DMUs belonging to group $A_t$ and is, therefore, super-efficient with respect to this frontier. \subsection{Efficiency of a group } The efficiency of group $A_t$ is defined as the sum of the efficiency of all DMUs relative to the group's frontier, as follows: \begin{equation}\label{eq4} E(A_t)=\sum_{j=1}^n R_j^{A_t}, \end{equation} where $n$ is the number of all DMUs. \subsection{Game theory} The game theory is used in decision-making problems where multiple decision makers have conflicting interests. Consider a competitive condition in which $ N = \{1,2, ..., n\}$ is the set of players in the game. Players can compete in two ways: 1. Non-cooperative game: Players act individually and have personal accomplishments. In this game, it is tried to predict the strategies adopted by each player to obtain the most impressive achievement. 2. Cooperative game: The players are expected to form a coalition to boost their achievement. Cooperative games are usually characterized by the players in the game and a characteristic function ${\mathcal{C}}(S)$ as $\langle N, {\mathcal{C}}(S)\rangle$. Assuming that the coalition $S$ is a subset of the players. The characteristic function ${\mathcal{C}}(S)$, which is the achievement of the players in $S$ from the game, is what the members of the $S$ coalition can be sure to gain together in the coalition. Evidently, what players have gained in the $S$ Coalition must be fairly divided between the players of the coalition $S$. Suppose the prize vector$ X = \{x_1, x_2, ..., x_n\}$ is the prize of $n$ players (e.g. $x_i$ is the prize of the $i^{th}$ player). This prize vector must comply with the two following conditions:\\ 1. $ {\mathcal{C}}(N)=\sum_{i=1}^n x_i \qquad \text{Group rationality}$.\\ 2. $ x_i \geq {\mathcal{C}}(\{i\}) \qquad \text{Individual rationality}$.\\ There are several solutions to obtain the prize vector, including the core, stable set, kernel, nucleolus, and Shapley Value \cite{Driessen}. In the meantime, the Shapley Value is the unique solution that satisfy in three following axioms \cite{Osborne} to fairly allocate overall prize among players:\\ 1. Relabeling of players interchange the player's reward.\\ 2. If ${\mathcal{C}}(S\cup A_k)={\mathcal{C}}(S)$ for all coalitions S, i.e. player$A_k$ adds no value to any coalition, then the reward of player $A_k$ from Shapley Value is zero.\\ 3. If ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ be two characteristic functions for games, with the same payers, for the game with the characteristic function ${\mathcal{C}}_1+{\mathcal{C}}_2$, the reward is equal to the sum of reward for ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$. In the Shapley Value solution, the share of $i^{th}$ player from the prize, i.e. $x_i$ is obtained as follows: \begin{equation}\label{eq5} x_i=\sum_{\scriptstyle S\subseteq N\hfill \atop \scriptstyle i\notin S \hfill} \frac{(s)!(n-s-1)!}{n!}\big({\mathcal{C}}(S)-{\mathcal{C}}(S \cup \{i\})\big). \end{equation} \section{Evaluation of groups based on the cooperative game} \label{Sec 3} Groups are considered as the players of a cooperative game in order to be able to compare them within a common framework. If the coalition $S$ is a subset of $s$ groups, then the characteristic function of the coalition $S$ is defined as the sum of the efficiency of all DMUs, assuming that PPS has been formed by all the DMUs belonging to the $s$ group of the coalition $S$: \begin{equation}\label{eq8} {\mathcal{C}}(S)=\sum_{j=1}^nR_j^S, \end{equation} where $R_j^S$ is calculated using the formula \eqref{eq3}. If $A_k \notin S$, then the characteristic function $ {\mathcal{C}}(S \cup \{k\}) $ is defined as the sum of the efficiency of all DMUs, assuming that the PPS has been generated by all DMUs of the $s$ groups of coalition $S$ and group $A_k$: \begin{equation}\label{eq9} {\mathcal{C}}(S \cup A_k)=\sum_{j=1}^nR_j^{S \cup \{A_k\}}. \end{equation} Given these definitions, the marginal effect of group $A_k$ to the coalition S, which is the alteration in the sum of efficiency of DMUs due to the addition of group $A_k$ to coalition S, is defined as follows: \begin{equation}\label{eq10} ME^S(A_k)={\mathcal{C}}(S) -{\mathcal{C}}(S \cup A_k). \end{equation} The characteristic function ${\mathcal{C}}(S$) is an achievement that the members of coalition $S$ are expected to achieve together. What is gained by the players participating in a coalition should be fairly divided between them.There are different solutions to divide the prize, from which Shapley Value is understandable and easy to interpret. The Shapley Value is used to obtain the solution of this cooperative game, i.e. the share of players participating in the Coalition $S$: \begin{equation}\label{eq11} \begin{aligned} \varphi_{A_k}({\mathcal{C}}) =&\sum_{\scriptstyle S\subseteq \{A_1,A_2, \ldots A_q \}\hfill \atop \scriptstyle \qquad A_k \notin S \hfill} \frac{(s)!(q-s-1)!}{q!}({\mathcal{C}}(S)-{\mathcal{C}}(S \cup \{A_k\}))\\ =&\sum_{\scriptstyle S\subseteq \{A_1,A_2, \ldots A_q \}\hfill \atop \scriptstyle \qquad A_k \notin S \hfill} \frac{(s)!(q-s-1)!}{q!}(ME^S(A_k)), \end{aligned} \end{equation} where $q$ is the total number of groups and $s$ is the number of groups participating in the coalition $S$. $\varphi_{A_k} ({\mathcal{C}})$ is the amount of achievement expected by the $A_k$ player in this cooperative game through participating in the Coalition $S$. The higher the Shapley Value, the better the rank of the group. In the next section, the method is first described using three simple examples, and then a real example is used to demonstrate the applicability of the method. \section{Numerical examples} \label{Sec 4.} \begin{example} \label{example 1} \rm{ A set of 8 DMUs, which includes an input of 1 for all units and two outputs, is divided into three groups $A$, $B$, and $C$ (Table \ref{Table 1}). The frontier of three groups are depicted in Fig \ref{figure 1}. \begin{table}[H] \caption{Data of 8 DMUs} \centering{\begin{tabular}{ccccc} \hline DMU & Group & Input1 & output1 & output2 \\ \hline $DMU_{1}$ & $A$ & 1 & 1 & 1 \\ $DMU_{2}$ & $A$ & 1 & 1 & 2 \\ $DMU_{3}$ & $A$ & 1 & 3 & 1 \\ $DMU_{4}$ & $B$ & 1 & 4 & 2 \\ $DMU_{5}$ & $B$ & 1 & 3 & 4 \\ $DMU_{6}$ & $B$ & 1 & 5 & 3 \\ $DMU_{7}$ & $C$ & 1 & 4 & 6 \\ $DMU_{8}$ & $C$ & 1 & 7 & 3 \\ \hline %\caption{Table 1: data of 8 DMU} \end{tabular}} \label{Table 1}\end{table} %\begin{center} \begin{figure}[H] %\centering \includegraphics[scale=0.5]{figure1} \centering \caption{production posibility sets made by various groups} \label{figure 1}\end{figure} % \end{center} The presented method was used to calculate the marginal effect of these three groups in different coalitions as shown in Table \ref{Table 2}, where their average is the Shapley Value of that group. To illustrate this method, the marginal effect of group $B$ to the coalition $\{A\}$, which appears in the second row of the second column of Table \ref{Table 2}, is as follow:\\ If the PPS is made of the DMUs belonging to the Group $A$ (Figure \ref{figure 1}), it denotes a line that connects the point $n$, $DMU_2$, and $DMU_3$ to the point $p$. With this PPS, the characteristic function of coalition $\{A\}$ which is the sum of efficiency of all DMUs, is equal to 12.49: \[ {\mathcal{C}}(A)=0.50+1+1+1.60+1.60+2.14+2.18+2.47=12.49. \] By adding the DMUs of Group $B$ to the collation$ \{A\}$, PPS made by the DMUs of the Group $A$ and $B$ is indeed the frontier of group $B$, denoted by the line that connects point $m$, $DMU_5$, $DMU_6$ to point $q$. In this PPS, the characteristic function of the Coalition $\{A, B\}$, i.e. sum of efficiency of all DMUs, is equal to 6.12: \[ {\mathcal{C}}(A\cup B)=0.25+0.30+0.40+0.73+1+1+1.26+1.17=6.12. \] Therefore, the marginal effect of adding $B$ to the coalition $\{A\}$ is obtained as follows: \[\ ME^{(A)} (B)={\mathcal{C}}(A)-{\mathcal{C}}(A\cup B)=12.49-6.12=6.57. \] In Table \ref{Table 3}, the obtained Shapley Values are seen in the fourth column, according to which the groups are ranked (column 5 of Table \ref{Table 3}). As expected, Group $C$, with groups $A$ and $B$ as its subgroups, has the most outputs and achieved best rank. Group $B$, with groups $A$ as its subgroups, has more outputs than $A$, and achieved the second rank. Group $A$ with the least outputs has the worst rank. \begin{table}[H] \caption{Marginal effect of groups in various coalitions} \centering{ \begin{tabular}{ccccc} \hline Possible coalitions of groups & $A$& $B$ & $C$ \\ \hline $\{A\}$ & 0 & 6.57 & 7.86 \\ $\{B\}$ & 0 & 0 & 1.49 \\ $\{C\}$ & 0 & 0 & 0 \\ $\{A,B\}$ & 0 & 0 & 1.49 \\ $\{A, C\}$ & 0 & 0 & 0 \\ $\{B,C\}$ & 0 & 0 & 0 \\ \hline \end{tabular} } \label{Table 2}\end{table} \begin{table}[H] \caption{Ranking groups by two proposed method} \centering{ \begin{tabular}{ccc} \hline Group & Shapley Value & Ranking by\\ & &Shapley Value\\ \hline $A$ &0&3 \\ $ B$ &1.09&2 \\ $C$& 1.80&1 \\ \hline \end{tabular} } \label{Table 3}\end{table} }\end{example} \begin{example} \label{example 2} \rm{ A set of 10 DMUs, which includes an input of 1 for all units and two outputs, is divided into four groups $A$, $B$, $C$ and $D$ (Table \ref{Table 4}). The frontier of groups are depicted in Figure \ref{figure 2}. \begin{table}[H] \caption{data of 8 DMUs} \centering{ \begin{tabular}{ccccccc} \hline DMU & Group & Input1 & output1 & output2 \\ \hline $\DMU_{1}$ & $A$& 1 & 1 & 1 \\ $\DMU_{2}$ & $A$& 1 & 4 & 1 \\ $\DMU_{3}$ & $A$& 1 & 2 & 2 \\ $\DMU_{4}$ & $B$ & 1 & 4 & 4 \\ $\DMU_{5}$ & $B$ & 1 & 5 & 3 \\ $\DMU_{6}$ & $C$ & 1 & 2 & 8 \\ $\DMU_{7}$ & $C$ & 1 & 1 & 9 \\ $\DMU_{8}$ & $C$ & 1 & 1 & 8 \\ $\DMU_{9}$ & $D$ & 1 & 5 & 10 \\ $\DMU_{10}$ & $D$ & 1 & 8 & 8 \\ \hline %\caption{Table 1: data of 8 DMU} \end{tabular} } \label{Table 4}\end{table} %%%%%%5 \begin{figure}[H] %\begin{center} \includegraphics[scale=0.4]{figure2} \centering \caption{frontier of groups} %\end{center} \label{figure 2}\end{figure} The presented method was used to calculate the marginal effect of these four groups in different coalitions as shown in Table \ref{Table 5}, where their average is the Shapley Value of that group. As an illustration, the marginal effect of adding group $C$ to the coalition $\{A\}$, which appears in the third row of the fourth column of Table \ref{Table 5}, is as follow:\\ If the PPS is made of the DMUs belonging to the Group $A$ (Figure \ref{figure 2}), it denotes line that connects the point $q$, $DMU_2$, and $DMU_3$ to the point $m$. With this PPS, the characteristic function of coalition $\{A\}$, which is the sum of efficiency of all DMUs, is equal to 16.02. By adding the DMUs of Group $C$ to the collation $\{A\}$, PPS made by the DMUs of the Group $A$ and $C$ is denoted by the line that connects the point $q$, $ DMU_2 $, $DMU_6$, and $DMU_7$ to the point $n$. In this PPS, the characteristic function of the Coalition $\{A, C\}$, i.e. the sum of efficiency of all DMUs, is equal to 9.84. Therefore, the marginal effect of $C$ to the coalition $\{A\}$ is obtained as follows: \[ ME^{\{A\}} (C)={\mathcal{C}}(A)-{\mathcal{C}}(A\cup C)=16.02-9.84=6.18. \] In Table \ref{Table 6}, the obtained Shapley Values are seen in the fourth column, according to which the groups are ranked (column 5 of Table \ref{Table 6}). Although the ranking by two different methods are very close, but in some cases are different. The difference between super efficiency and Shapley Value method is due to multi-platform problem in the super efficiency method. As expected, Group $D$, with groups $B$, $C$ and $D$ as its subgroups, has the most outputs and achieved best rank. Group $A$ that has less outputs achieved worst rank. \begin{table}[H] \caption{Marginal effect of groups in various coalitions} \centering{ \begin{tabular}{cccccc} \hline Possible coalitions of groups & $A$& $B$ & $C$ & $D$ \\ \hline $\{A\}$ & 0 & 5.28 & 6.18 & 11.62 \\ $\{B\}$ & 0 & 0 & 2.06 & 6.34 \\ $\{C\}$ & 0.79& 1.95 & 0 & 6.23 \\ $\{D\}$ & 0 & 0 & 0 & 0 \\ $\{A,B\}$ & 0 & 0 & 2.06 & 6.34 \\ $\{A, C\}$ & 0 & 1.16 & 0 & 5.44 \\ $\{A,D\}$ & 0 & 0 & 0 & 0 \\ $\{B,C\}$ & 0 & 0 & 0 &4.28\\ $\{B,D\}$ & 0 & 0 & 0 & 0\\ $\{C,D\}$ & 0 & 0 & 0 & 0\\ $\{A,B,C\}$ & 0 & 0 & 0 & 4.28 \\ $\{A,B,D\}$ & 0 & 0 & 0 & 0 \\ $\{A,C,D\}$ & 0 & 0 & 0 & 0 \\ $\{B,C,D\}$ & 0 & 0 & 0 & 0 \\ \hline \end{tabular} } \label{Table 5}\end{table} \begin{table}[H] \caption{Ranking groups by two proposed method} \centering{ \begin{tabular}{ccccc} \hline Group &Shapley Value & Ranking by\\ &&Shapley Value\\ \hline $A$& 0.06 &4 \\ $B$ & 0.60 &3 \\ $C$ & 0.74 &2 \\ $D$ & 3.18 &1 \\ \hline \end{tabular} } \label{Table 6}\end{table} }\end{example} In the following, a real word example is presented to identify the applicability of the proposed method. \begin{example} \label{example 3} \rm{ To demonstrate the applicability of this method, a database consisting of 20 branches of an Iranian bank, borrowed from Amirteimoori et al. \cite{Amirteimoori}, were supposed in four groups. Relevant data included in Table \ref{Table 7}: three inputs (number of employees, number of computers, space of the branch); three outputs (amount of deposits, amount of loans and amount of charges). These data, which are divided into four groups, are presented in Table \ref{Table 7}. \begin{table}[H] \caption{The input and output of 20 bank branches divided in 4 groups} \centering{ \begin{tabular}{ccccccccc} \hline Group &DMU&$x_1$ &$x_2$ &$x_3$ &$y_1$ &$y_2$ &$y_3$ & $\theta_{BBC}$ \\ \hline $A_1$ & 1 & 0.950 & 0.700 & 0.155 & 0.190 & 0.521 & 0.293 & 1.00 \\ $A_1$ & 2 & 0.796 & 0.600& 1.000 &0.227 &0.627 & 0.462& 0.90\\ $A_1$& 3 & 0.798 & 0.750& 0.513& 0.228& 0.970&0.261 & 0.99\\ $A_1$&4 &0.865 &0.550 &0.210 &0.193 &0.632 &1.000 &1.00 \\ $A_2$& 5&0.815& 0.850&0.268 &0.233 &0.722 &0.246 &0.90 \\ $A_2$&6 & 0.842&0.650 &0.500 & 0.207&0.603 &0.569 &0.75\\ $A_2$&7 &0.719& 0.600& 0.350&0.182 &0.900 &0.716 &1.00\\ $A_2$&8 & 0.785&0.750 &0.120 &0.125 &0.234 &0.298 &0.80\\ $A_2$&9 &0.476 & 0.600&0.135 &0.080 & 0.364& 0.244&0.79\\ $A_2$& 10& 0.678& 0.550&0.510 &0.082 &0.184 &0.049 &0.29\\ $A_3$& 11&0.711 &1.000 &0.305 &0.212 &0.318 &0.403 &0.60\\ $A_3$&12 &0.811 &0.650 &0.255 &0.123 &0.923 &0.628 &1.00\\ $A_3$&13 &0.659 &0.850 &0.340 &0.176 &0.645 &0.261 &0.82\\ $A_3$&14 & 0.976& 0.800&0.540 &0.144 &0.514 &0.243 &0.47\\ $A_3$&15 & 0.685& 0.950& 0.450& 1.000& 0.262& 0.098&1.00\\ $A_4$&16 &0.613 &0.900 &0.525 &0.115 &0402 &0.464 &0.64\\ $A_4$& 17 & 1.000& 0.600& 0.205& 0.090& 1.000& 0.161&1.00\\ $A_4$&18 &0.634 &0.650 &0.235 &0.059 &0.349 &0.068 &0.47\\ $A_4$& 19& 0.372& 0.700&0.238 &0.039 &0.190 &0.111 &0.41\\ $A_4$& 20&0.583 &0.550 &0.500 &0.110 &0.615 &0.764 &1.00\\ \hline \end{tabular} } \label{Table 7}\end{table} \begin{table}[H] \caption{Ranking by two presented methods} \centering{ \begin{tabular}{ccccc} \hline Group &Shapley Value& Ranking by Shapley Value \\ \hline $A_1$ & 1.04 &2 \\ $A_2$& 0.77 &3 \\ $A_3$ & 1.66&1 \\ $A_4$ & 0.15 &4 \\ \hline \end{tabular} } \label{Table 8}\end{table} In the second column of Table \ref{Table 8}, the Shapley Values obtained using the above-mentioned method are presented and in the third column, the ranking performed by this method is presented. It has been seen that in comparing with a common platform $A_3$ has a better ranking than $A_1$ and $A_2$. }\end{example} \section{Conclusion} \label{Sec 5} In DEA problems and under most conditions, group evaluation of DMUs is much more valuable, resulting in better management decisions. In this paper, to evaluate the groups in a common platform, the groups were evaluated from a cooperative game perspective, in a way that each group was considered as a player, and a subset of the groups was assumed as a coalition. By defining the characteristic function of the coalition $S$ as the sum of the efficiency of all DMUs when the PPS is made up of groups belonging to the coalition $S$, the marginal effect of each group was specified and the Shapley Value of the groups was obtained using that value. The higher the Shapley Value, the better the performance. DMUs were evaluated with respect to different group coalition and thus with more frontiers. Moreover, the marginal effects of a group on all possible coalitions were investigated in order to evaluate the efficiency of that group, which makes the evaluation of groups more precise. In this way, when efficiency with different frontier is desired, we may run into infeasible circumstances where the efficiencies cannot be correctly calculated. To prevent this, the Modified ERM can be used for the evaluation of efficiencies.\\ This research shows that this subject is a dynamic one and requires more research. It also indicates in some circumstances, a large number of groups may cause delay in analysis and the algorithm is time-consuming. In such situations meta-heuristic algorithms (e.g., genetic algorithms) are suggested. \begin{thebibliography}{99} %******************************************************************************************************************************************************** \bibitem{Amirteimoori} A. Amirteimoori, S. Kordrostami, \emph{Efficient Surfaces and an Efficiency Index in Dea: A Constant Returns to Scale}, J. Appl. Math. Comput. 163.2 (2005), pp. 683-691. \bibitem{Banker1} R. D. Banker, A. Charnes, W. W. Cooper,\emph{ Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis}, Management science, 30.9 (1984), pp. 1078-1092. \bibitem{Banker2} R. D. Banker, R. C. Morey, \emph{The Use of Categorical Variables in Data Envelopment Analysis}, Management science, 32.12 (1986), pp. 1613-1627. \bibitem{Bowlin} W. Bowlin, J. Brennan, A. Charnes, W. W. Cooper, T. Sueyoshi, \emph{A Model for Measuring Amounts of Efficiency Dominance}, Unpublished manuscript, (1984). \bibitem{Camanho} A. S. Camanho, R. G. Dyson,\emph{ Data Envelopment Analysis and Malmquist Indices for Measuring Group Performance}, J. Prod. Anal. 26.1 (2006), pp. 35-49. \bibitem{Charnes1} A. Charnes, W. W. Cooper, \emph{Rhodes E Measuring the Efficiency of Decision Making Units}, Eur. J. Oper. Res. 2.6 (1987), pp. 429-444. \bibitem{Charnes2} A. Charnes, W. W. Cooper, R. M. Thrall, \emph{A Structure for Classifying and Characterizing Efficiency and Inefficiency in Data Envelopment Analysis}, J. Prod. Anal. 2.3 (1991), pp. 197-237. \bibitem{Cook1} W. D. Cook, D. Chai, J. Doyle, R. Green \emph{Hierarchies and Groups in Dea}, J. Prod. Anal. 10.2 (1998), pp. 177-198. \bibitem{Cook2} W. D. Cook, J. Zhu, \emph{ Within-Group Common Weights in Dea: An Analysis of Power Plant Efficiency}, Eur. J. Oper. Res. 178.1 (2007), pp. 207-216. \bibitem{Driessen} T. S. Driessen, \emph{Cooperative Games, Solutions and Applications}. Vol. 3: Springer Science $\&$ Business Media, (2013). \bibitem{Fare} R. F$\rm{\ddot{a}}$re, S. Grosskopf, C. K. Lovell, \emph{The Measurement of Efficiency of Production}. Vol. 6: Springer Science $\&$ Business Media, (2013). \bibitem{Izadikhah} M. Izadikhah, R. F. Saen, K. Ahmadi, \emph{How to assess sustainability of suppliers in the presence of dual-role factor and volume discounts? A data envelopment analysis approach}, Asia-Pacific, J.Oper. Res. 34.03 (2017), 1740016. \bibitem{Li} Y. Li, J. Xie, M. Wang, L. Liang, \emph{Super Efficiency Evaluation Using a Common Platform on a Cooperative Game}. Eur. J. Oper. Res. 255.3 (2016), pp. 884-892. \bibitem{Maniadakis} N. Maniadakis, E. Thanassoulis, \emph{A Cost Malmquist Productivity Index}, Eur. J. Oper. Res. 154.2 (2004), pp. 396-409. \bibitem{ODonnell} C. J. O'Donnell, D. P. Rao, G. E, Battese, \emph{Metafrontier Frameworks for the Study of Firm-Level Efficiencies and Technology Ratios}. Empirical economics, 34.2 (2008), pp. 231-255. \bibitem{Osborne} M. J. Osborne, A. Rubinstein, \emph{A Course in Game Theory}, MIT press, (1994). \bibitem{Tone} K. Tone, \emph{A Slacks-Based Measure of Efficiency in Data Envelopment Analysis}, Eur. J. Oper. Res. 130.3 (2001), pp. 498-509. \bibitem{Valami} H. B. Valami, \emph{Group Performance Evaluation an Application of Data Envelopment Analysis}, J. Comput. Appl. 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