The role of multiplier bounds in fuzzy data envelopment analysis

Abstract

The non-Archimedean epsilon $\varepsilon$ is commonly considered as a lower bound for the dual input weights and output weights in multiplier data envelopment analysis (DEA) models. The amount of $\varepsilon$ can be effectively used to differentiate between strongly and weakly efficient decision making units (DMUs). The problem of weak dominance particularly occurs when the reference set is fully or partially defined in terms of fuzzy numbers. In this paper, we propose a new four-step fuzzy DEA method to re-shape weakly efficient frontiers along with revisiting the efficiency score of DMUs in terms of perturbing the weakly efficient frontier. This approach eliminates the non-zero slacks in fuzzy DEA while keeping the strongly efficient frontiers unaltered. In comparing our proposed algorithm to an existing method in the recent literature we show three important flaws in their approach that our method addresses. Finally, we present a numerical example in banking with a combination of crisp and fuzzy data to illustrate the efficacy and advantages of the proposed approach.

Appendices

Appendix 1

Proof of Proposition 1

(i) Feasibility of model (6):

Let us consider the following envelopment model that is evaluated the upper bound of $\hbox {DMU}_{{o}}$:

$$\begin{aligned} \min&\left( {\theta _o^U } \right) _{\alpha }\nonumber \\ s.t.\,\,&\sum \limits _{j=1}^{n}{\lambda _j}(x_{ij})_{\alpha }^{L} \le (\theta _{o}^{U})_{\alpha }(x_{io})_{\alpha }^{L}, \quad i=1,\ldots ,\hbox {m},\nonumber \\&\sum \limits _{j=1}^n {\lambda _j \left( {y_{rj} } \right) _{\alpha }^U}\ge \left( {y_{ro}}\right) _{\alpha }^U , \quad r=1,\ldots ,s, \nonumber \\&\lambda _j \ge 0,j=1,\ldots ,n. \end{aligned}$$

(11)

where $\left( {\theta _o^U}\right) _{\alpha }$ and $\lambda _j, j=1,\ldots ,n,$ are the decision variables at ${\alpha }$. The above model has at least one feasible solution as follows:

$$\begin{aligned} \left( {\theta _o^U}\right) _{\alpha }= & {} 1, \\ \lambda _o= & {} 1, \\ \lambda _j= & {} 0,\quad j=1,\ldots ,n,j\ne o. \\ \end{aligned}$$

Hence, model (11) always has a feasible solution. If $\left( {\left( {\theta _{o}^{U}}\right) _{{\alpha }}^{*} ,\lambda _{j}^{*}}\right) ,j=1,\ldots ,n$, is an optimal solution of (11) for $\hbox {DMU}_{{o}}$, the inequality constraints of model (11) can be transformed into the following equality form:

$$\begin{aligned} s_i^+= & {} \left( {\theta _o^U } \right) _{{\alpha }}^{*} \left( {x_{io}}\right) _{\alpha }^L -\sum \limits _{j=1}^n {\lambda _j ^{*}}\left( {x_{ij} } \right) _{\alpha }^L ,\quad i=1,\ldots ,\hbox {m}, \\ s_{r}^{-}= & {} \sum \limits _{j=1}^n {\lambda _j ^{*}\left( {y_{rj} } \right) _{\alpha }^U } -\left( {y_{ro} } \right) _{\alpha }^U ,\qquad {r}=1,\ldots ,{s}, \end{aligned}$$

Obviously, the above constraints are the same as the constraints of model (6) when $\lambda _j=\lambda _{j}^{*} , j=1,\ldots ,n$ and it leads to the following feasible solution to model (6):

$$\begin{aligned}&\lambda _{j}^{*},\quad j=1,\ldots ,n, \\&s_i^+,\quad i=1,\ldots ,m; \\&s_{r}^{-},\quad r=1,\ldots ,s. \\ \end{aligned}$$

(ii) Non-negativity of the optimal objective function value of model (6):

The proof is trivial.

(iii) Boundedness of the optimal objective function value of model (6):

Let $\left( {\lambda _{j}^{*} ,j=1,\ldots ,n; s_i^{+ ^{*}},i=1,\ldots ,m; s_{r}^{{-}^{*}},r=1,\ldots ,s}\right)$ be the optimal solution of (6). The proof is by contradiction.

Assume that the optimal objective function value of model (6) is unbounded. This implies that there exists at least one unbounded value in $\left( {s_i^{+^{*}},i=1,\ldots ,m,s_{r}^{{-}^{*}},r=1,\ldots ,s} \right)$. Without loss of generality, assume that $s_t^{{-}^{*}}=\infty$. Thus, the following relation follows from the second set of constraints of model (6)

$$\begin{aligned} s_t^{{-}^{*}}=\sum \limits _{j=1}^n{\lambda _j ^{*}\left( {y_{tj} } \right) _{\alpha }^U}-\left( {y_{to}}\right) _{\alpha }^U =\infty . \end{aligned}$$

Since $\left( {y_{tj}}\right) _{\alpha }^U ,j=1,\ldots ,n,$ is bounded, at least one $\lambda _{j}^{*}, j=1,\ldots ,n$ is unbounded. Let $\lambda _h ^{*}=\infty$ and its substitution into the first constraint of model (6) is

$$\begin{aligned} \underbrace{\underbrace{\lambda _h \left( {x_{ih} } \right) _\alpha ^L }_\infty +\underbrace{{\mathop {\mathop {\sum }\limits _{j=1}}\limits _{j\ne h}}^n{\lambda _j ^{*}} \left( {x_{ij} } \right) _\alpha ^L +s_i^+}_{\ge 0}}_\infty =\left( {\theta _o^U } \right) _\alpha ^*\left( {x_{io}}\right) _\alpha ^L , i=1,\ldots ,\hbox {m}, \end{aligned}$$

The left hand side of the above equation is infinity (unbounded) while its right hand side is bounded which is a contradiction and completes the proof. $\square$

Appendix 2

Proof of Proposition 2

(i) Feasibility of model (7):

Model (7) is solved for those DMUs whose upper efficiency scores are unity (100%), i.e., belonging to set E. Model (5-2) calculates the upper efficiency score and the solution set is non-empty as proven by the following constructive proof. Consider the following point:

$$\begin{aligned} \left( {x_{ko}}\right) _{\alpha }^L= & {} \mathop {\max }\limits _{1\le i\le m} \left\{ {\left( {x_{io} } \right) _{\alpha }^L } \right\} \\ (u_1 ,\ldots ,u_s)= & {} (0,\ldots ,0), (v_1 ,\ldots ,v_k ,\ldots ,v_m )=\left( 0,\ldots ,\frac{1}{\left( {x_{ko}}\right) _{\alpha }^L },\ldots ,0\right) \end{aligned}$$

Let $\left( {\left( {\theta _o^U}\right) _{{\alpha }}^{*} ; u_{r}^*, r=1,\ldots ,s; v_i^*, i=1,\ldots ,m} \right)$ be the optimal solution of (5-2) for $\hbox {DMU}_{\mathrm{o}}$, thus, we have:

$$\begin{aligned}&\sum \limits _{i=1}^m {v_i ^{*}\left( {x_{io}}\right) _{\alpha }^L}=1,\\&\sum \limits _{r=1}^s {u_r ^{*}\left( {y_{rj}}\right) _{\alpha }^U}-\sum \limits _{i=1}^m {v_i ^{*}\left( {x_{ij}}\right) _{\alpha }^L}\le 0, \quad j=1,\ldots ,n. \\ \end{aligned}$$

Since $\hbox {DMU}_{\mathrm{o}}$ belongs to set $E, \left( {\theta _o^U}\right) _{\alpha }^{*}=\sum \limits _{r=1}^s {u_r ^{*}\left( {y_{ro}}\right) _{\alpha }^U}=1$. It implies that the optimal solution of (5-2) is a feasible solution for (7).

(ii) Non-negativity of the optimal objective function value of model (7):

The proof is trivial.

(iii) Boundedness of the optimal objective function value of model (7):

Let $\left( v_i^*, i=1,\ldots ,m; u_{r}^*, r=1,\ldots ,s\right)$ be the optimal solution of (7). The proof is by contradiction.

Let the optimal objective function value of model (7) is unbounded. This implies that at least one optimal variable is infinity. Without loss of generality, assume that $u_t^{*}=\infty$. Thus, we have

$$\begin{aligned} \sum \limits _{r=1}^s {u_r ^{*}\left( {y_{ro}}\right) _{\alpha }^U } =\underbrace{\underbrace{u_t ^{*}\left( {y_{to}} \right) _{\alpha }^U}_\infty +\underbrace{\sum \limits _{\begin{array}{l} r=1 \\ r\ne t \\ \end{array}}^s {u_r^{*}\left( {y_{ro}}\right) _{\alpha }^U} }_{\ge 0}}_\infty \end{aligned}$$

The above constraint is contradicted with the first constraint, $\sum \nolimits _{r=1}^s {u_r \left( y_{ro}\right) _{\alpha }^U} =1$, and it completes the proof. $\square$

Appendix 3

Proof

A DMU in class $E^{+}$ is such that $\left( {x_{io} } \right) _{\alpha }^L <\sum \nolimits _{j=1}^n {\lambda _j } \left( {x_{ij} } \right) _{\alpha }^U$for at least one i, or $\left( {y_{ro} } \right) _{\alpha }^U >\sum \nolimits _{j=1}^n {\lambda _j } \left( {y_{rj} } \right) _{\alpha }^L$ for at least one r. Thus, constraint 1 and/or constraint 2 of (8) are violated and the problem has no feasible solution. $\square$

Appendix 4

Proof

Let us first introduce the dual of model (4-2) to prove this proposition as:

$$\begin{aligned} \min&\;\theta _o^U \\ s.t.\,\,&\sum \limits _{{\begin{array}{l} j=1 \\ j\ne o \\ \end{array}}}^n {\lambda _j (x_{ij})_{\alpha }^U } +\lambda _o \left( {x_{io} } \right) _{\alpha }^L \le \theta _o^U \left( {x_{io}}\right) _{\alpha }^L , i=1,\ldots ,m, \\&\sum \limits _{{\begin{array}{l} j=1 \\ j\ne o \\ \end{array}}}^n {\lambda _j \left( {y_{rj}}\right) _{\alpha }^L } +\lambda _o \left( {y_{ro} } \right) _{\alpha }^U \ge \left( {y_{ro} } \right) _{\alpha }^U,\quad r=1,\ldots ,s, \\&\lambda _j \ge 0, j=1,\ldots ,n. \\ \end{aligned}$$

Suppose that a $\hbox {DMU}_{\mathrm{o}}$ is in class $E^{-}$. Due to the inefficiency of $\hbox {DMU}_{\mathrm{o}}$, $\lambda _{o}^{*} =0$ and $0<\theta _o^{U*} <1$ in the optimal solution of model (10). Therefore, the constraints of (10) can be converted to

$$\begin{aligned} \left\{ {\begin{array}{l} \sum \limits _{{\begin{array}{l} j=1 \\ j\ne o \\ \end{array}}}^n{\lambda _{j}^{*} \left( {x_{ij} } \right) _{\alpha }^U } <\left( {x_{io} } \right) _{\alpha }^L , i=1,\ldots ,m, \\ \sum \limits _{{\begin{array}{l} j=1 \\ j\ne o \\ \end{array}}}^n{\lambda _{j}^{*} \left( {y_{rj} } \right) _{\alpha }^L } \ge \left( {y_{ro} } \right) _{\alpha }^U , r=1,\ldots ,s, \\ \end{array}}\right. \end{aligned}$$

which is equivalent to the following slack-variable formulation:

$$\begin{aligned} \left\{ {\begin{array}{l} \sum \limits _{j=1}^n {\lambda _{j}^{*} \left( {x_{ij}}\right) _{\alpha }^U } +s_i^+ =\left( {x_{io} } \right) _{\alpha }^L , i=1,\ldots ,m, \\ \sum \limits _{j=1}^n {\lambda _{j}^{*} \left( {y_{rj}}\right) _{\alpha }^L } -s_{r}^{-} =\left( {y_{ro} } \right) _{\alpha }^U , r=1,\ldots ,s, \\ \end{array}} \right. \end{aligned}$$

Consequently, $(\lambda _{j}^{*} ,s_i^+ ,s_{r}^{-})$ is a feasible solution for model (8). $\square$

Appendix 5

Proof

The use of $\varepsilon _{i}^{v}$ and $\varepsilon _{r}^{u}$ as the lower bounds for the weights in model (4-2) cannot guarantee feasibility. According to Step 3 of the Khoshfetrat and Daneshvar (2011) method, we need to individually solve $\left| S\right| \times (m+s)$ ¹¹ models (9) for calculating the lower bounds of the input and output weights, $\varepsilon _{i}^{v}$ and $\varepsilon _{r}^{u}$ that are incorporated into model (4-2). Due to solving $\left| S\right| \times (m+s)$ models without interacting with each other, it is possible that dual weights $v_i\ge \varepsilon _{i}^{v} \left( u_{r}\ge \varepsilon _{r}^{u}\right)$ jeopardize the feasibility. We prove this infeasibility problem by existence the numerical example in Khoshfetrat and Daneshvar (2011, p. 343). C, F and H are the weakly efficient units and that $\varepsilon$ for $v_{1}, u_{1}$ and $u_{2}$ are set as 0.1, 0.2 and 0.1, respectively.¹² The corresponding upper efficiency bound of model (4-2) is infeasible for all DMUs with the given $\varepsilon$ for the weights. $\square$

Appendix 6

The GAMS code associated with this article can be found in the online version.