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A new multi-component DEA approach using common set of weights methodology and imprecise data: an application to public sector banks in India with undesirable and shared resources

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Abstract

Owing to the importance of internal structure of decision making units (DMUs) and data uncertainties in real situations, the present paper focuses on multi-component data envelopment analysis (MC-DEA) approach with imprecise data. The undesirable outputs and shared resources are also incorporated in the production process of multi-component DMUs to validate real problems. The interval efficiencies of DMUs and their components in MC-DEA are often challenging with imprecise data. In many practical situations, different set of weights may be resulted into valid efficiency intervals for DMUs but invalid interval efficiencies for their components. Therefore, the present study proposes a new common set of weights methodology, based on interval arithmetic and unified production frontier, to determine unique weights for measuring these interval efficiencies. It is a two-level mathematical programming approach that preserves linearity of DEA and exhibits stronger discrimination power among the DMUs as compared to some existing approaches. Theoretically, the aggregate efficiency interval of each DMU lies between the components’ interval efficiencies. Further, the proposed approach is also applied to banks in India for proving its acceptability in practical applications. The performance of each bank is investigated in terms of two components: general business and bancassurance business for the years 2011–2013. The present study emphasized expanding pattern of bancassurance business in current market scenario with more percentage increase as contrasted to general business.

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Acknowledgements

The authors are thankful to the editor and anonymous reviewers for their constructive comments and suggestions that helped us in improving the paper significantly.

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Correspondence to Jolly Puri.

Appendices

Appendix A

See Table 10.

Table 10 The list of PuSBs during the periods 2011–2012 and 2012–2013

Appendix B: Proofs of theorems

1.1 Proof of Theorem 1

Proof

The aggregate efficiency \({E}_{k}^{ (a)} \) is given by

$$\begin{aligned} {E}_{k}^{ (a)}= & {} {\left( {\sum \limits _{i=1}^d {{U}_{k}^{g(i)} {Y}_{k}^{g(i)} } -\sum \limits _{i=1}^d {{U}_{k}^{b(i)} {Y}_{k}^{b(i)} } +\sum \limits _{i=1}^d {{W}_{k}^{S(i)} \left( \beta _{ik} {Y}_{k}^S \right) } } \right) }\Big /\\&{\left( {\sum \limits _{i=1}^d {{V}_{k}^{(i)} {X}_{k}^{(i)} } +\sum \limits _{i=1}^d {{V}_{k}^{S(i)} \left( \alpha _{ik} {X}_{k}^S \right) } } \right) }\\ \hbox {Let } H= & {} \sum \limits _{i=1}^d {{V}_{k}^{(i)} {X}_{k}^{(i)} } +\sum \limits _{i=1}^d {{V}_{k}^{S(i)} \left( \alpha _{ik} {X}_{k}^S \right) } .\hbox { Then}\\ {E}_{k}^{ (a)}= & {} \frac{{U}_{k}^{g(1)} {Y}_{k}^{g(1)} -{U}_{k}^{b(1)} {Y}_{k}^{b(1)} +{W}_{k}^{S(1)} \left( \beta _{1k} {Y}_{k}^S \right) }{H}\\&+\frac{{U}_{k}^{g(2)} {Y}_{k}^{g(2)} -{U}_{k}^{b(2)} {Y}_{k}^{b(2)} +{W}_{k}^{S(2)} \left( \beta _{2k} {Y}_{k}^S \right) }{H}+\cdots + \\&\frac{{U}_{k}^{g(d)} {Y}_{k}^{g(d)} -{U}_{k}^{b(d)} {Y}_{k}^{b(d)} +{W}_{k}^{S(d)} \left( \beta _{dk} {Y}_{k}^S \right) }{H} \\= & {} \frac{{U}_{k}^{g(1)} {Y}_{k}^{g(1)} -{U}_{k}^{b(1)} {Y}_{k}^{b(1)} +{W}_{k}^{S(1)} \left( \beta _{1k} {Y}_{k}^S \right) }{{V}_{k}^{(1)} {X}_{k}^{(1)} +{V}_{k}^{S(1)} \left( \alpha _{1k} {X}_{k}^S \right) }\times \frac{{V}_{k}^{(1)} {X}_{k}^{(1)} +{V}_{k}^{S(1)} \left( \alpha _{1k} {X}_{k}^S \right) }{H}\\&+\frac{{U}_{k}^{g(2)} {Y}_{k}^{g(2)} -{U}_{k}^{b(2)} {Y}_{k}^{b(2)} +{W}_{k}^{S(2)} \left( \beta _{2k} {Y}_{k}^S \right) }{{V}_{k}^{(2)} {X}_{k}^{(2)} +{V}_{k}^{S(2)} \left( \alpha _{2k} {X}_{k}^S \right) }\times \\&\frac{{V}_{k}^{(2)} {X}_{k}^{(2)} +{V}_{k}^{S(2)} \left( \alpha _{2k} {X}_{k}^S \right) }{H}+ \cdots + \frac{{U}_{k}^{g(d)} {Y}_{k}^{g(d)} -{U}_{k}^{b(d)} {Y}_{k}^{b(d)} +{W}_{k}^{S(d)} \left( \beta _{dk} {Y}_{k}^S \right) }{{V}_{k}^{(d)} {X}_{k}^{(d)} +{V}_{k}^{S(d)} \left( \alpha _{dk} {X}_{k}^S \right) }\times \\&\frac{{V}_{k}^{(d)} {X}_{k}^{(d)} +{V}_{k}^{S(d)} \left( \alpha _{dk} {X}_{k}^S \right) }{H} \\= & {} {E}_{k}^{ (1)} \times \frac{{V}_{k}^{(1)} {X}_{k}^{(1)} +{V}_{k}^{S(1)} \left( \alpha _{1k} {X}_{k}^S \right) }{H}+ {E}_{k}^{ (2)} \times \frac{{V}_{k}^{(2)} {X}_{k}^{(2)} +{V}_{k}^{S(2)} \left( \alpha _{2k} {X}_{k}^S \right) }{H}\\&+\cdots +{E}_{k}^{ (d)} \times \frac{{V}_{k}^{(d)} {X}_{k}^{(d)} +{V}_{k}^{S(d)} \left( \alpha _{dk} {X}_{k}^S \right) }{H} \end{aligned}$$

Let \(\lambda _i ={\left( {{V}_{k}^{(i)} {X}_{k}^{(i)} +{V}_{k}^{S(i)} \left( \alpha _{ik} {X}_{k}^S \right) } \right) }/H, i=1,2,3,\ldots ,d.\) Also \(\lambda _i \ge 0 \,\forall i\).\(\square \)

Hence, \({E}_{k}^{ (a)} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)} } \) for each \(\lambda _i \ge 0\) and \(\sum \limits _{i=1}^d {\lambda _i } =1\). This completes the proof.

1.2 Proof of Theorem 3

Proof

Let \(H=\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)} x_{lk}^{{(i)} {U}}} } +\sum \limits _{i=1}^d {\sum \limits _{t=1}^{I^{S}} {v_{tk}^{S(i)} (\alpha _{ik}^t x_{tk}^{S {U}})} } \). Then,

$$\begin{aligned} {E}_{k}^{ (a)L}= & {} {\left( {\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i}^{g} } {u_{rk}^{g(i)} y_{rk}^{{g(i)} {L}}} } -\sum \limits _{i=1}^d {\sum \limits _{p=1}^{{K}_{i}^{b} } {u_{pk}^{b(i)} y_{pk}^{{b(i)}{U}}} } +\sum \limits _{i=1}^d {\sum \limits _{q=1}^{K^{S}} {w_{qk}^{S(ni)} \left( \beta _{ik}^q y_{qk}^{S {L}}\right) } } } \right) }\Big / \\&{\left( {\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)} x_{lk}^{{(i)} {U}}} } +\sum \limits _{i=1}^d {\sum \limits _{t=1}^{I^{S}} {v_{tk}^{S(i)} \left( \alpha _{ik}^t x_{tk}^{S {U}}\right) } } } \right) }\\= & {} \frac{\sum \nolimits _{r=1}^{{K}_{1}^{g} } {u_{rk}^{g(1)} y_{rk}^{{g(1)} {L}}} -\sum \nolimits _{p=1}^{K_{1}^{b} } {u_{pk}^{b(1)} y_{pk}^{{b(1)} {U}}} +\sum \nolimits _{q=1}^{K^{S}} {w_{qk}^{S(1)} \left( \beta _{1k}^q y_{qk}^{S {L}}\right) } }{H}\\&+\frac{\sum \nolimits _{r=1}^{{K}_{2}^{g} } {u_{rk}^{g(2)} y_{rk}^{{g(2)} {L}}} -\sum \nolimits _{p=1}^{K_{2}^{b} } {u_{pk}^{b(2)} y_{pk}^{{b(2)} {U}}} +\sum \nolimits _{q=1}^{K^{S}} {w_{qk}^{S(2)} \left( \beta _{2k}^q y_{qk}^{S {L}}\right) } }{H} \\&+\cdots +\frac{\sum \nolimits _{r=1}^{{K}_{d}^{g} } {u_{rk}^{g(d)} y_{rk}^{{g(d)} {L}}} -\sum \nolimits _{p=1}^{{K}_{d}^{b} } {u_{pk}^{b(d)} y_{pk}^{{b(d)} {U}}} +\sum \nolimits _{q=1}^{K^{S}} {w_{qk}^{S(d)} \left( \beta _{dk}^q y_{qk}^{S {L}}\right) } }{H}\\= & {} \frac{\sum \nolimits _{r=1}^{{K}_{1}^{g} } {u_{rk}^{g(1)} y_{rk}^{{g(1)} {L}}} -\sum \nolimits _{p=1}^{K_{1}^{b} } {u_{pk}^{b(1)} y_{pk}^{{b(1)} {U}}} +\sum \nolimits _{q=1}^{K^{S}} {w_{qk}^{S(1)} \left( \beta _{1k}^q y_{qk}^{S {L}}\right) } }{\sum \nolimits _{l=1}^{{I}_{1} } {v_{lk}^{(1)} x_{lk}^{{(1)} {U}}} +\sum \nolimits _{t=1}^{I^{S}} {v_{tk}^{S(1)} \left( \alpha _{1k}^t x_{tk}^{S {U}}\right) } } \\&\times \frac{\sum \nolimits _{l=1}^{{I}_{1} } {v_{lk}^{(1)} x_{lk}^{{(1)}^{U}}} +\sum \nolimits _{t=1}^{I^{S}} {v_{tk}^{S(1)} \left( \alpha _{1k}^t x_{tk}^{S {U}}\right) } }{H}\\&+\frac{\sum \nolimits _{r=1}^{{K}_{2}^{g} } {u_{rk}^{g(2)} y_{rk}^{{g(2)} {L}}} -\sum \nolimits _{p=1}^{K_{2}^{b} } {u_{pk}^{b(2)} y_{pk}^{{b(2)} {U}}} +\sum \nolimits _{q=1}^{K^{S}} {w_{qk}^{S(2)} \left( \beta _{2k}^q y_{qk}^{S {L}}\right) } }{\sum \nolimits _{l=1}^{I_2 } {v_{lk}^{(2)} x_{lk}^{{(2)} {U}}} +\sum \nolimits _{t=1}^{I^{S}} {v_{tk}^{S(2)} \left( \alpha _{2k}^t x_{tk}^{S {U}}\right) } }\\&\times \frac{\sum \nolimits _{l=1}^{{I}_{2} } {v_{lk}^{(2)} x_{lk}^{{(2)} {U}}} +\sum \nolimits _{t=1}^{I^{S}} {v_{tk}^{S(2)} \left( \alpha _{2k}^t x_{tk}^{S {U}}\right) } }{H} \\&+ \cdots +\frac{\sum \nolimits _{r=1}^{{K}_{d}^{g} } {u_{rk}^{g(d)} y_{rk}^{{g(d)} {L}}} -\sum \nolimits _{p=1}^{{K}_{d}^{b} } {u_{pk}^{b(d)} y_{pk}^{{b(d)} {U}}} +\sum \nolimits _{q=1}^{K^{S}} {w_{qk}^{S(d)} \left( \beta _{dk}^q y_{qk}^{S \,{L}}\right) } }{\sum \nolimits _{l=1}^{I_d } {v_{lk}^{(d)} x_{lk}^{{(d)} {U}}} +\sum \nolimits _{t=1}^{I^{S}} {v_{tk}^{S(d)} \left( \alpha _{dk}^t x_{tk}^{S {U}}\right) } } \\&\times \frac{\sum \nolimits _{l=1}^{I_d } {v_{lk}^{(d)} x_{lk}^{{(d)} {U}}} +\sum \nolimits _{t=1}^{I^{S}} {v_{tk}^{S(d)} \left( \alpha _{dk}^t x_{tk}^{S {U}}\right) } }{H}\\= & {} {E}_{k}^{ (1)L} \times \lambda _1 + {E}_{k}^{ (2)L} \times \lambda _2 +\cdots +{E}_{k}^{ (d)L} \times \lambda _d \end{aligned}$$

where \(\lambda _i ={\left( {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)} x_{lk}^{{(i)} {U}}} +\sum \limits _{t=1}^{I^{S}} {v_{tk}^{S(i)} (\alpha _{ik}^t x_{tk}^{S {U}})} } \right) }\Big /H, i=1,2,3,\ldots ,d.\) Also \(\lambda _i \ge 0 \,\forall i\).\(\square \)

Hence, \({E}_{k}^{ (a)L} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)L} } \) for each \(\lambda _i \ge 0\) and \(\sum \limits _{i=1}^d {\lambda _i } =1.\) This completes the proof.

1.3 Proof of Theorem 6

Proof

Assume that \({E}_{k}^{ (i)U^*} = 1, \forall i=1,2,\ldots ,d.\) By Theorem 4, \({E}_{k}^{ (a)U^*} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)U^*} } \) and since \(\sum \limits _{i=1}^d {\lambda _i } =1\) for \(\lambda _i \in [0,1] \,\forall i\), it follows that each \({E}_{k}^{ (a)U^*} =1\).\(\square \)

Now assume that \({E}_{k}^{ (a)U^*} =1.\) If any \({E}_{k}^{ (i)U^*} <1 \,\forall i\in \{1,2,\ldots ,d\}\) then, \({E}_{k}^{ (a)U^*} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)U^*} } <1\) which is a contradiction. Hence, \({E}_{k}^{ (a)U^*} = 1\Leftrightarrow {E}_{k}^{ (i)U^*} =1, \forall i=1,2,\ldots ,d\).

1.4 Proof of Theorem 7

Proof

Let \((u_{1k}^{(i)^*} ,u_{2k}^{(i)^*} ,\ldots ,u_{rk}^{(i)^*} ;v_{1k}^{(i)^*} ,v_{2k}^{(i)^*} ,\ldots ,v_{lk}^{(i)^*} ), \forall r,l\) be the optimal solution to Model-7 when the production process of multi-component \(\hbox {DMU}_{k}\) is of the form of Fig. 2. Then, the following equalities and inequalities hold:

$$\begin{aligned}&\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {L}}} } =1,\\&\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {U}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)}{L}}} } \le 0, \forall j,\nonumber \\&\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {U}}} -\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)}{L}}} \le 0, \forall i, j.\nonumber \end{aligned}$$
(9)

For \(\hbox {DMU}_{k}\), we have

$$\begin{aligned}&\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)} {U}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {L}}} } \le 0, \end{aligned}$$
(10)
$$\begin{aligned}&\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)} {U}}} -\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {L}}} \le 0, \forall i. \end{aligned}$$
(11)

For DMU\(_{j} \quad (j=1,2,\ldots ,n; j\ne k),\) we get

$$\begin{aligned}&\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {L}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)} {U}}} } \le \sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {U}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)}{L}}} } \le 0,\nonumber \\ \end{aligned}$$
(12)
$$\begin{aligned}&\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {L}}} -\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)} {U}}} \le \sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {U}}} -\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)}{L}}} \le 0. \end{aligned}$$
(13)

Equations (9)–(13) implies that \((u_{1k}^{(i)^*} ,u_{2k}^{(i)^*} ,\ldots ,u_{rk}^{(i)^*} ;v_{1k}^{(i)^*} ,v_{2k}^{(i)^*} ,\ldots ,v_{lk}^{(i)^*} ), \forall r,l\) is a feasible solution to Model-4b. Thus, we have the following inequality relation:

$$\begin{aligned} {E}_{k}^{ (a)U^*} =\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)} {U}}} } ={H}_{k}^{ (a)U} \le {H}_{k}^{ (a)U^*} . \end{aligned}$$

Similarly, if \((u_{1k}^{(i)^*} ,u_{2k}^{(i)^*} ,\ldots ,u_{rk}^{(i)^*} ;v_{1k}^{(i)^*} ,v_{2k}^{(i)^*} ,\ldots ,v_{lk}^{(i)^*} ), \forall r,l\) be the optimal solution to Model-8 when the production process of multi-component \(\hbox {DMU}_{k}\) is of the form of Fig. 2.

Therefore, for \(\hbox {DMU}_{k}\), we have

$$\begin{aligned}&\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {U}}} } =1, \end{aligned}$$
(14)
$$\begin{aligned}&\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)} {U}}} } -\left( {\left( {{E}_{k}^{ (a)U^*} } \right) *\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {L}}} } } \right) =0 \hbox { and } {E}_{k}^{ (a)U^*} \le 1 \hbox { implies that}\nonumber \\&\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)} {U}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {L}}} } \le 0, \nonumber \\&\hbox {So}, \sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)}{L}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {U}}} } \le \sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)} {U}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {L}}} } \le 0, \end{aligned}$$
(15)
$$\begin{aligned}&\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)}{L}}} -\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {U}}} \le \sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)} {U}}} -\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lk}^{{(i)} {L}}} \le 0, \forall i. \end{aligned}$$
(16)

For DMU\(_{j} \quad (j=1,2,\ldots ,n; j\ne k),\) we have

$$\begin{aligned}&\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {U}}} } -\sum \limits _{i=1}^d {\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)}{L}}} } \le 0, \end{aligned}$$
(17)
$$\begin{aligned}&\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rj}^{{(i)} {U}}} -\sum \limits _{l=1}^{{I}_{i} } {v_{lk}^{(i)^*} x_{lj}^{{(i)}{L}}} \le 0. \end{aligned}$$
(18)

Equations (14)–(18) implies that \((u_{1k}^{(i)^*} ,u_{2k}^{(i)^*} ,\ldots ,u_{rk}^{(i)^*} ;v_{1k}^{(i)^*} ,v_{2k}^{(i)^*} ,\ldots ,v_{lk}^{(i)^*} ), \forall r,l\) is a feasible solution to Model-4a. Thus, we have the following inequality relation:

$$\begin{aligned} {E}_{k}^{ (a)L^*} =\sum \limits _{i=1}^d {\sum \limits _{r=1}^{{K}_{i} } {u_{rk}^{(i)^*} y_{rk}^{{(i)}{L}}} } ={H}_{k}^{ (a)L} \le {H}_{k}^{ (a)L^*} . \end{aligned}$$

This completes the proof.\(\square \)

Appendix C: Proofs of propositions

1.1 Proof of Proposition 1

Proof

By Theorem 3, we have \({E}_{k}^{ (a)L^*} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)L^*} } \) for each \(\lambda _i \ge 0\) and \(\sum \limits _{i=1}^d {\lambda _i } =1.\)

$$\begin{aligned}&{E}_{k}^{ (a)L^*} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)L^*} } = \lambda _1 {E}_{k}^{ (1)L^*} +\lambda _2 {E}_{k}^{ (2)L^*} +\cdots +\lambda _d {E}_{k}^{ (d)L^*} \nonumber \\&\ge \lambda _1 {E}_{k}^{ (e)L^*} +\lambda _2 {E}_{k}^{ (e)L^*} +\cdots +\lambda _d {E}_{k}^{ (e)L^*} \nonumber \\&= \left( {\sum \limits _{i=1}^d {\lambda _i } } \right) {E}_{k}^{ (e)L^*} = {E}_{k}^{ (e)L^*} \nonumber \nonumber \\&\hbox {Thus},\,\, {E}_{k}^{ (a)L^*} \ge {E}_{k}^{ (e)L^*} ={M}_{k} . \end{aligned}$$
(19)
$$\begin{aligned}&\hbox {Now},\,\, {E}_{k}^{ (a)L^*} = \sum \limits _{i=1}^d {\lambda _i } {E}_{k}^{ (i)L^*} \le \lambda _1 {E}_{k}^{ (f)L^*} +\lambda _2 {E}_{k}^{ (f)L^*} +\cdots +\lambda _d {E}_{k}^{ (f)L^*} \nonumber \\&= \left( {\sum \limits _{i=1}^d {\lambda _i } } \right) {E}_{k}^{ (f)L^*} = {E}_{k}^{ (f)L^*} \nonumber \\&\hbox {Thus, } {E}_{k}^{ (a)L^*} \le {E}_{k}^{ (f)L^*} ={N}_{k} . \end{aligned}$$
(20)

Equations (19) and (20) implies that \({M}_{k} \le {E}_{k}^{(a)L^*} \le {N}_{k}\).\(\square \)

1.2 Proof of Proposition 2

Proof

By Theorem 4, we have \({E}_{k}^{ (a)U^*} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)U^*} } \) for each \(\lambda _i \ge 0\) and \(\sum \limits _{i=1}^d {\lambda _i } =1\).

$$\begin{aligned}&{E}_{k}^{ (a)U^*} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)U^*} } \ge \lambda _1 {E}_{k}^{ (e)U^*} +\lambda _2 {E}_{k}^{ (e)U^*} +\cdots +\lambda _d {E}_{k}^{ (e)U^*} \nonumber \\&\quad = \left( {\sum \limits _{i=1}^d {\lambda _i } } \right) {E}_{k}^{ (e)U^*} = {E}_{k}^{ (e)U^*}\nonumber \\&\hbox {Thus, }{E}_{k}^{ (a)U^*} \ge {E}_{k}^{ (e)U^*} ={P}_{k} . \end{aligned}$$
(21)
$$\begin{aligned}&\hbox {Now, }{E}_{k}^{ (a)U^*} =\sum \limits _{i=1}^d {\lambda _i {E}_{k}^{ (i)U^*} } \le \lambda _1 {E}_{k}^{ (f)U^*} +\lambda _2 {E}_{k}^{ (f)U^*} +\cdots +\lambda _d {E}_{k}^{ (f)U^*} \nonumber \\&\quad = \left( {\sum \limits _{i=1}^d {\lambda _i } } \right) {E}_{k}^{ (f)U^*} = {E}_{k}^{ (f)U^*}\nonumber \\&\hbox {Thus, }{E}_{k}^{ (a)U^*} \le {E}_{k}^{ (f)U^*} ={Q}_{k} . \end{aligned}$$
(22)

Equations (21) and (22) implies that \({P}_{k} \le {E}_{k}^{(a)U^*} \le {Q}_{k} .\) \(\square \)

Appendix D: Minimax regret-based approach (MRA) for ranking interval efficiencies in MC-DEA

In interval efficiency assessment problem, the final efficiency score of every DMU is characterized by an interval. Wang et al. (2005) proposed an interesting approach, named as minimax regret-based approach (MRA), to compare and rank the DMUs that possess interval efficiencies. The main advantage of this approach is that it can be used to compare and rank the intervals even if they are equi-centered but different in widths. To rank efficiency intervals, Wang et al. (2005) defined maximum regret as follows:

Let \({A}_{k} =[{a}_{k}^{L} ,{a}_{k}^{U} ] = \left\langle {m({A}_{k} ), w({A}_{k} )} \right\rangle (k=1,2,\ldots ,n)\) be the efficiency interval of the kth DMU where \(m({A}_{k} )={({a}_{k}^{U} +{a}_{k}^{L} )}/2\) is the midpoint of \({A}_{k} \) and \(w({A}_{k} )={({a}_{k}^{U} -{a}_{k}^{L} )}/2\) is the width of \({A}_{k} \). The maximum loss of efficiency (or maximum regret) of each \({A}_{k}\, (k=1,2,\ldots ,n)\) is defined as

$$\begin{aligned} R({A}_{k} )=\max \left[ {\mathop {\max }\limits _{j\ne k} \{a_j^U \}-{a}_{k}^{L} ,0} \right] =\max \left[ {\mathop {\max }\limits _{j\ne k} \{m({A}_{j} )+w({A}_{j} )\}-(m({A}_{k} )-w({A}_{k} )),0} \right] . \end{aligned}$$

It is obvious that the efficiency interval with the smallest maximum loss of efficiency is the best (most desirable) efficiency interval. In order to generate a ranking for a set of efficiency intervals using the above discussed maximum loss of efficiency, Wang et al. (2005) suggested minimax regret-based approach which is an eliminating process. In this ranking approach, firstly, evaluate the maximum loss of efficiency (regret) of all n efficiency intervals and eliminate a most desirable efficiency interval (say \(A_{{k}_{1} } , 1\le {k}_{1} \le n)\) that has the smallest maximum loss of regret. Further, recalculate the maximum loss of regret of each efficiency interval and eliminate a most desirable efficiency interval (say (say \(A_{{k}_{2} } , 1\le {k}_{2} \le n, {k}_{2} \ne {k}_{1} )\) from the remaining \((n-1)\) efficiency intervals. Repeat the above eliminating process until only one efficiency interval \(A_{k_n } \)is left. Then, the final ranking is given by \(A_{{k}_{1} } \succ A_{{k}_{2} } \succ \cdots \succ A_{k_n }\), where the symbol ‘\(\succ \)’ means ‘is superior to’. This ranking approach is referred to as MRA.

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Puri, J., Yadav, S.P. & Garg, H. A new multi-component DEA approach using common set of weights methodology and imprecise data: an application to public sector banks in India with undesirable and shared resources. Ann Oper Res 259, 351–388 (2017). https://doi.org/10.1007/s10479-017-2540-1

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