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Relative partial efficiency: network and black box SBM DEA interpretations in multiplier form

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Abstract

In traditional black-box DEA when the ratio-based multiplier DEA model is estimated to obtain a technical efficiency score, the estimated multipliers (shadow prices) serve as the weights that maximize the ratio of the aggregation of weighted sum of outputs (virtual output) to that of inputs (virtual input) of the assessed DMU in comparison with the other decision making units (DMUs). With respect to the ratio-based multiplier model of non-radial slack-based measure (SBM), however, there does not exist such a nice efficiency interpretation. For the purpose of providing a reasonable efficiency interpretation for both black-box and network SBM models, this paper introduces a concept called relative partial efficiency (RPE). In the black box structure, RPEs are defined for each input–output pair and a multi objective programming is formed in order to maximize RPEs. Then, it is proved that its equivalent single objective programming problem is the same SBM multiplier DEA model. The obtained explicit efficiency interpretation coming from this novel concept is then generalized for the multiplier network SBM DEA model represented by Boloori (Comput Ind Eng 95:83–96, 2016).

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Notes

  1. The axioms are: (A1) Positivity of network flows and (A2) Divisional feasible production. These axioms indicate that the vector of contributing factors should be feasible in each division. For more details refer to the related paper.

  2. Note that only \( (X,Y) \) are used in Lozano (2011, 2015).

  3. If we write model (5) in terms of slacks by \( x_{q}^{k} = x_{qo}^{k} - s_{q}^{k} \), \( y_{q}^{k} = y_{qo}^{k} + t_{q}^{k} \) and \( z_{q}^{(k,h)} = z_{qo}^{(k,h)} + \alpha_{q}^{(k,h)} \), then despite other SBM models \( s_{q}^{k} \ge 0 \), \( t_{q}^{k} \ge 0 \) are not required. Instead, the total slacks of a main input/output in all divisions must be nonnegative (i.e. \( \sum\nolimits_{{k:q \in N_{k} }} {s_{q}^{k} } \ge 0 \) and \( \sum\nolimits_{{k:q \in M_{k} }} {t_{q}^{k} } \ge 0\)).

  4. Refer to the descriptions before statement (5) in preliminaries section for the input/output entities.

  5. Here divisions’ indicator is denoted by g instead of k. This is because we wanted to use k and k′′ as the indicator of corresponding divisions of q and q′.

References

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Acknowledgements

The financial support of the University of Tabriz in the context of the research fund (No. 31/27) devoted to the first author is gratefully acknowledged.

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Correspondence to Fatemeh Boloori.

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Appendix

Appendix

Proof of Theorem 1

Part I We must prove that model (15) is the equivalent dual form for the linearized form of model (2). In model (2), consider that \( \sum\nolimits_{q \in M} {\frac{{y_{q}^{{}} }}{{y_{qo}^{{}} }}} = \frac{1}{\tau } \), in which variable \( \tau \) is strictly positive. By multiplication of all constraints of model (2) by \( \tau \) and setting \( \varLambda_{j} = \tau \lambda_{j} \), \( X_{q} = \tau x_{q} \) and \( Y_{q} = \tau y_{q} \), model (2) is linearized as in (24). Note that writing \( \tau \ge 0 \) instead of \( \tau > 0 \) has no problem here. This is because, according to (24-a), all \( Y_{q} \)’s cannot be zero at the same time and consequently in constraint (24-c), all \( \varLambda_{j}^{{}} \)’s cannot be zero simultaneously. Therefore (24-f) ensures that \( \tau \) can never be zero.

$$ \begin{aligned} & \theta_{o}^{*} = \hbox{min} \quad \frac{|M|}{|N|} \times \sum\limits_{q \in N} {\frac{{X_{q} }}{{x_{qo}^{{}} }}} \\ & \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{q \in M} {\frac{{Y_{q}^{{}} }}{{y_{qo}^{{}} }}} = 1} \hfill & {} \hfill & {} \hfill & {(a)} \hfill \\ {} \hfill & { - \sum\limits_{j} {\varLambda_{j}^{{}} x_{qj}^{{}} + X_{q}^{{}} \ge 0} } \hfill & {\forall q \in N} \hfill & {(q \in N)} \hfill & {(b)} \hfill \\ {} \hfill & {\sum\limits_{j} {\varLambda_{j}^{{}} y_{qj}^{{}} - Y_{q}^{{}} \ge 0} } \hfill & {\forall q \in M} \hfill & {(q \in M)} \hfill & {(c)} \hfill \\ {} \hfill & { - X_{q}^{{}} + \tau .x_{qo}^{{}} \ge 0} \hfill & {\forall q \in N} \hfill & {} \hfill & {(d)} \hfill \\ {} \hfill & {Y_{q}^{{}} - \tau .y_{qo}^{{}} \ge 0} \hfill & {\forall q \in M} \hfill & {} \hfill & {(e)} \hfill \\ {} \hfill & {\left[ {\sum\nolimits_{j} {\varLambda_{j}^{{}} } - \tau = 0} \right]} \hfill & {} \hfill & {} \hfill & {(f)} \hfill \\ {} \hfill & {\varLambda_{j}^{{}} ,\quad Y_{q}^{{}} ,\quad X_{q}^{{}} \ge 0,\quad \tau \ge 0} \hfill & {} \hfill & {} \hfill & {(g)} \hfill \\ \end{array} \\ \end{aligned} $$
(24)

On the other hand, models (24) and (15) are equivalent based on the duality theory. To see this consider that \( \bar{\tau },\bar{\varLambda }_{j}^{{}} ,\bar{X}_{q} ,\bar{Y}_{q} \) are corresponding dual variables to constraints (15-a)-(15-d). The dual form would be as in (25). Again, we can substitute \( \bar{\tau } \ge 0 \) instead of \( \bar{\tau }\;free \) here. Because, some \( \bar{Y}_{q} \) must be strictly positive due to (25-a) and consequently in constraint (25-c), some \( \varLambda_{j}^{{}} \)’s must be positive too. Hence, (25-c) it implies that \( \bar{\tau } \ge 0 \). Now in (25), suppose that \( \bar{\tau } \ge 0 \) and perform a variable change such that \( \bar{Y}_{q} = \frac{|M|}{|N|}Y_{q} ,\;\bar{X}_{q} = \frac{|M|}{|N|}X_{q} ,\;\bar{\varLambda }_{j}^{{}} = \frac{|M|}{|N|}\varLambda_{j}^{{}} ,\;\bar{\tau } = \frac{|M|}{|N|}\tau \). The resultant model would be the same model (24) then.

$$ \begin{aligned} & \theta_{o}^{*} = \hbox{min} \quad \sum\limits_{q \in N} {\frac{{\bar{X}_{q} }}{{x_{qo}^{{}} }}} \\ & \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{q \in M} {\frac{{\bar{Y}_{q}^{{}} }}{{y_{qo}^{{}} }}} = \frac{|M|}{|N|}} \hfill & {} \hfill & {(\eta_{o} )} \hfill & {(a)} \hfill \\ {} \hfill & { - \sum\limits_{j} {\bar{\varLambda }_{j}^{{}} x_{qj}^{{}} + \bar{X}_{q}^{{}} \ge 0} } \hfill & {\forall q \in N\,} \hfill & {(w_{q} (q \in N))} \hfill & {(b)} \hfill \\ {} \hfill & {\sum\limits_{j} {\bar{\varLambda }_{j}^{{}} y_{qj}^{{}} - \bar{Y}_{q}^{{}} \ge 0} } \hfill & {\forall q \in M\,} \hfill & {(w_{q} (q \in M))} \hfill & {(c)} \hfill \\ {} \hfill & { - \bar{X}_{q}^{{}} + \bar{\tau }.x_{qo}^{{}} \ge 0} \hfill & {\forall q \in N} \hfill & {(v_{q} )} \hfill & {(d)} \hfill \\ {} \hfill & {\bar{Y}_{q}^{{}} - \bar{\tau }.y_{qo}^{{}} \ge 0} \hfill & {\forall q \in M} \hfill & {(u_{q} )} \hfill & {(e)} \hfill \\ {} \hfill & {\left[ {\sum\nolimits_{j} {\bar{\varLambda }_{j}^{{}} } - \bar{\tau } = 0} \right]} \hfill & {} \hfill & {(\mu )} \hfill & {(f)} \hfill \\ {} \hfill & {\bar{\varLambda }_{j}^{{}} ,\quad \bar{Y}_{q}^{{}} ,\quad \bar{X}_{q}^{{}} \ge 0,\quad \bar{\tau }free} \hfill & {} \hfill & {} \hfill & {(g)} \hfill \\ \end{array} \\ \end{aligned} $$
(25)

Now we can conclude that models (15) and (24) and also model (2) are equivalent.


Part II Now we prove that model (14) is equivalent to model (15), i.e., model (14) is equivalent to SBM model (2). In (14), let \( \eta_{o} = \mathop {\hbox{min} }\nolimits_{{(q,q^{{\prime }} ) \in M \times N}} \left( {{{y_{qo}^{{}} (w_{q}^{{}} - u_{q}^{{}} )} \mathord{\left/ {\vphantom {{y_{qo}^{{}} (w_{q}^{{}} - u_{q}^{{}} )} {x_{{q^{{\prime }} o}}^{{}} (w_{{q^{{\prime }} }}^{{}} - v_{{q^{{\prime }} }}^{{}} )}}} \right. \kern-0pt} {x_{{q^{{\prime }} o}}^{{}} (w_{{q^{{\prime }} }}^{{}} - v_{{q^{{\prime }} }}^{{}} )}}} \right) \) and therefore \( \eta_{o} = {{\mathop {\hbox{min} }\nolimits_{q \in M} \left( {y_{qo}^{{}} (w_{q}^{{}} - u_{q}^{{}} )} \right)} \mathord{\left/ {\vphantom {{\mathop {\hbox{min} }\nolimits_{q \in M} \left( {y_{qo}^{{}} (w_{q}^{{}} - u_{q}^{{}} )} \right)} {\mathop {\hbox{max} }\nolimits_{q \in N} \left( {x_{qo}^{{}} (w_{q}^{{}} - v_{q}^{{}} )} \right)}}} \right. \kern-0pt} {\mathop {\hbox{max} }\nolimits_{q \in N} \left( {x_{qo}^{{}} (w_{q}^{{}} - v_{q}^{{}} )} \right)}} \). In order to make the model linear, we use the Charnes–Cooper transformation and let \( \mathop {\hbox{max} }\nolimits_{q \in N} \left( {x_{qo}^{{}} (w_{q}^{{}} - u_{q}^{{}} )} \right) = 1 \) and \( \eta_{o} = \mathop {\hbox{min} }\nolimits_{q \in M} \left( {y_{qo}^{{}} (w_{q}^{{}} - v_{q}^{{}} )} \right) \). The first equality implies that \( x_{qo}^{{}} (w_{q}^{{}} - v_{q}^{{}} ) \le 1 \) for all \( q \in N \) and the second one implies that \( y_{qo}^{{}} (w_{q}^{{}} - u_{q}^{{}} ) \ge \eta_{o} \) for all \( q \in M \). These are the same inequalities as stated in (15-c) and (15-d). Now it is enough to prove that the constraints in (15) imply that \( v_{q}^{{}} < w_{q}^{{}} \) and \( u_{q}^{{}} < w_{q}^{{}} \) which in turn implies that these redundant constraints do not appear in model (15). To see this, consider the complementary-slackness condition for (15-c) \( w_{q}^{{}} X_{q}^{{}} = v_{q}^{{}} X_{q}^{{}} + {{X_{q}^{{}} } \mathord{\left/ {\vphantom {{X_{q}^{{}} } {x_{qo}^{{}} }}} \right. \kern-0pt} {x_{qo}^{{}} }} \). On the other hand, \( \tau > 0 \) accompanied with (24-f) implies that there are some strictly positive \( \varLambda_{j} \)’s. Hence, according to (24-b) \( X_{q}^{{}} \) is also strictly positive. Thus the \( {{X_{q}^{{}} } \mathord{\left/ {\vphantom {{X_{q}^{{}} } {x_{qo}^{{}} }}} \right. \kern-0pt} {x_{qo}^{{}} }} \) term is strictly positive and \( w_{q}^{{}} X_{q}^{{}} > v_{q}^{{}} X_{q}^{{}} \). This indicates that \( v_{q}^{{}} < w_{q}^{{}} \). Also, according to the complementary-slackness condition for (15-d), we have \( w_{q}^{{}} Y_{q}^{{}} = u_{q}^{{}} Y_{q}^{{}} + {{\eta_{0} Y_{q}^{{}} } \mathord{\left/ {\vphantom {{\eta_{0} Y_{q}^{{}} } {y_{qo}^{{}} }}} \right. \kern-0pt} {y_{qo}^{{}} }} \). On the one hand, \( Y_{q}^{{}} > 0 \) (because (24-e) implies \( {{Y_{q}^{{}} } \mathord{\left/ {\vphantom {{Y_{q}^{{}} } {y_{qo}^{{}} }}} \right. \kern-0pt} {y_{qo}^{{}} }} \ge \tau > 0 \)). On the other hand, since \( \sum\limits_{q \in N} {\frac{{X_{q} }}{{x_{qo}^{{}} }}} > 0 \)(because all \( X_{q}^{{}} > 0 \) for all inputs) and the optimal values in objective function of (15), (24) are equal, we have \( \eta_{o}^{{}} > 0 \). Hence we have \( {{\eta_{0} Y_{q}^{{}} } \mathord{\left/ {\vphantom {{\eta_{0} Y_{q}^{{}} } {y_{qo}^{{}} }}} \right. \kern-0pt} {y_{qo}^{{}} }} > 0 \), which indicates \( w_{q}^{{}} Y_{q}^{{}} > u_{q}^{{}} Y_{q}^{{}} \). This statement accompanied with \( Y_{q}^{{}} > 0 \) implies that \( u_{q}^{{}} < w_{q}^{{}} \).

Part III It is concluded based on part I and II. □

Proof of Lemma 1

It is obvious that \( 0 \le NRPE_{{q,q^{{\prime }} }} \).We just prove that \( \frac{|M|}{|N|}NRPE_{{q,q^{{\prime }} }} \le 1 \).

First, note that for each intermediate factor q where \( k\mathop \to \limits_{q} h \) we have \( w_{q}^{k} - w_{q}^{h} \ge 0 \) based on (20-c). Hence we would have the following fact:

$$ \sum\limits_{{q \in \tilde{N} \cup \tilde{M}}} {\sum\limits_{{(k,h):k\mathop \to \limits_{q} h}} {\left( {w_{q}^{h} - w_{q}^{k} } \right)z_{qj}^{(k,h)} } = } \sum\limits_{h} {\sum\limits_{{q \in \tilde{N}_{h} }} {w_{q}^{h} a_{qj}^{h} e_{{\tilde{N}_{h} }}^{q} } - } \sum\limits_{k} {\sum\limits_{{q \in \tilde{M}_{k} }} {w_{q}^{k} b_{qj}^{k} e_{{\tilde{M}_{k} }}^{q} } } \le 0 $$

Secondly, let us add up constraints (20-b) over different divisionsFootnote 5 and rearrange the terms. The obtained statement is:

$$ \sum\limits_{g} {\sum\limits_{{q \in M_{k} }} {w_{q}^{g} y_{qj}^{g} } } - \sum\limits_{g} {\sum\limits_{{q \in N_{k} }} {w_{q}^{g} x_{qj}^{g} } } \left[ { + \sum\limits_{g} {\mu_{g} } } \right] \le \sum\limits_{g} {\sum\limits_{{q \in \tilde{N}_{k} }} {w_{q}^{g} a_{qj}^{g} } } - \sum\limits_{g} {\sum\limits_{{q \in \tilde{M}_{k} }} {w_{q}^{g} b_{qj}^{g} } } $$

Based on the previous statement, the right hand side of this inequality is lower than or equal to zero. Hence we would have:

$$ \sum\limits_{g} {\sum\limits_{{q \in M_{g} }} {w_{q}^{g} y_{qj}^{g} } } - \sum\limits_{g} {\sum\limits_{{q \in N_{g} }} {w_{q}^{g} x_{qj}^{g} } } \left[ { + \sum\limits_{g} {\mu_{g} } } \right] \le 0,\quad \forall j $$

Now let us add up \( - \sum\nolimits_{g} {\sum\nolimits_{{q \in M_{g} }} {\Delta u_{q}^{g} y_{qo}^{g} } + } \sum\nolimits_{g} {\sum\nolimits_{{q \in N_{g} }} {\Delta v_{q}^{g} x_{qo}^{g} } } \) to both sides of this inequality where \( u_{q} = w_{q}^{g} - \Delta u_{q}^{g} \) and \( v_{q} = w_{q}^{g} - \Delta v_{q}^{g} \). In fact the recent inequality in compare with (20-a), has a non-negative gap as follows:

$$ 0 \le - \sum\limits_{g} {\sum\limits_{{q \in M_{g} }} {\Delta u_{q}^{g} y_{qo}^{g} } } + \sum\limits_{g} {\sum\limits_{{q \in N_{g} }} {\Delta v_{q}^{g} x_{qo}^{g} } } $$

This non-negative gap shows the loss value between the current status of DMUo (with current prices of u,v) and its target status (with target prices of w). This gap of loss value can be decomposed to the non-negative gap of losses for different input–output pairs (q′, q) as follows:

$$ \sum\limits_{g} {\sum\limits_{{q \in N_{g} }} {\Delta v_{q}^{g} x_{qo}^{g} } } - \sum\limits_{g} {\sum\limits_{{q \in M_{g} }} {\Delta u_{q}^{g} y_{qo}^{g} } } = \sum\limits_{{(q^{{\prime }} ,q) \in N \times M}} {\left( {\frac{1}{|M|}\sum\limits_{{k^{{\prime }} :q^{{\prime }} \in N_{{k^{{\prime }} }} }} {\Delta v_{{q^{{\prime }} }}^{{k^{{\prime }} }} x_{{q^{{\prime }} o}}^{{k^{{\prime }} }} } - \frac{1}{|N|}\sum\limits_{{k:q \in M_{k} }} {\Delta u_{q}^{k} y_{qo}^{k} } } \right)} \ge 0 $$

Now consider the corresponding gap of loss for input–output pair \( (q^{{\prime }} ,q) \):

$$ \frac{1}{|M|}\sum\limits_{{k^{{\prime }} :q^{{\prime }} \in N_{{k^{{\prime }} }} }} {\Delta v_{{q^{{\prime }} }}^{{k^{{\prime }} }} x_{{q^{{\prime }} o}}^{{k^{{\prime }} }} } - \frac{1}{|N|}\sum\limits_{{k:q \in M_{k} }} {\Delta u_{q}^{k} y_{qo}^{k} } \ge 0. $$

Since \( \mathop {\hbox{min} }\nolimits_{{k:q \in M_{k} }} \Delta u_{q}^{k} \le \Delta u_{q}^{k} \) for each \( (k,q) \) which \( q \in M_{k} \) and also \( - \mathop {\hbox{max} }\nolimits_{{k^{{\prime }} :q \in M_{{k^{{\prime }} }} }} \Delta v_{{q^{{\prime }} }}^{{k^{{\prime }} }} \le - \Delta v_{{q^{{\prime }} }}^{{k^{{\prime }} }} \) for each \( (k^{{\prime }} ,q^{{\prime }} ) \) where \( q^{{\prime }} \in N_{{k^{{\prime }} }} \), then we can write:

$$ \frac{1}{|M|}\sum\limits_{{k^{{\prime }} :q^{{\prime }} \in N_{{k^{{\prime }} }} }} {\left( {\mathop {\hbox{max} }\limits_{{k^{{\prime }} :q^{{\prime }} \in N_{{k^{{\prime }} }} }} \Delta v_{{q^{{\prime }} }}^{{k^{{\prime }} }} } \right)x_{{q^{{\prime }} o}}^{{k^{{\prime }} }} } - \frac{1}{|N|}\sum\limits_{{k:q \in M_{k} }} {\left( {\mathop {\hbox{min} }\limits_{{k^{{\prime }} :q \in M_{{k^{{\prime }} }} }} \Delta u_{q}^{{k^{{\prime }} }} } \right)y_{qo}^{k} } \ge \frac{1}{|M|}\sum\limits_{{k^{{\prime }} :q^{{\prime }} \in N_{{k^{{\prime }} }} }} {\Delta v_{{q^{{\prime }} }}^{{k^{{\prime }} }} x_{{q^{{\prime }} o}}^{{k^{{\prime }} }} } - \frac{1}{|N|}\sum\limits_{{k:q \in M_{k} }} {\Delta u_{q}^{k} y_{qo}^{k} } \ge 0 $$

or:

$$ \frac{1}{|M|}\left( {\mathop {\hbox{max} }\limits_{{k^{{\prime }} :q \in N_{{k^{{\prime }} }} }} \Delta v_{q}^{{k^{{\prime }} }} } \right)x_{qo}^{total} - \frac{1}{|N|}\left( {\mathop {\hbox{min} }\limits_{{k:q \in M_{k} }} \Delta u_{q}^{k} } \right)y_{qo}^{total} \ge 0 $$

This implies that \( \frac{|M|}{|N|}NRPE_{{q,q^{{\prime }} }} = \frac{{|M|\mathop {\hbox{min} }\nolimits_{{k:q \in M_{k} }} y_{qo}^{total} (w_{q}^{k} - u_{q}^{{}} )}}{{|N|\mathop {\hbox{max} }\nolimits_{{k^{{\prime }} :q^{{\prime }} \in N_{{k^{{\prime }} }} }} x_{{q^{{\prime }} o}}^{total} (w_{{q^{{\prime }} }}^{{k^{{\prime }} }} - v_{{q^{{\prime }} }}^{{}} )}} \le 1 \) as desired. □

Proof of Theorem 2

Part I First, we prove that model (23) is the equivalent dual form for the linearized form of model (5). In model (5), consider that \( \sum\nolimits_{q \in M} {\sum\nolimits_{{k:q \in M_{k} }} {\frac{{y_{q}^{k} }}{{y_{qo}^{total} }}} } = \frac{1}{\tau } \), in which variable \( \tau \) is strictly positive. By multiplying all constraints of model (5) by \( \tau \) and considering \( \varLambda_{j}^{k} = \tau \lambda_{j}^{k} \), \( X_{q}^{k} = \tau x_{q}^{k} \) and \( Y_{q}^{k} = \tau y_{q}^{k} \), model (5) would be linear as shown in (26). On the other hand, models (26) and (23) are equivalent based on the duality theory. To see this, with a subjective calculation, temporarily remove multiplier |M|/|N| from the both objective functions in (26) and (23) and write the dual and finally bring back the multiplier form to both objective functions. Corresponding dual variable for each constraint is written at the end of each constraint in model (26).

$$ \begin{aligned} & \frac{|M|}{|N|}\hbox{min} \quad \sum\limits_{q\in N} {\sum\limits_{{k:q \in N_{k} }} {\frac{{X_{q}^{k}}}{{x_{qo}^{total} }}} } \\ & \begin{array}{*{20}l}\begin{aligned}&s.t. \hfill \\ \hfill \\ \hfill \\ \hfill \\\end{aligned}\hfill & \begin{aligned}&\sum\limits_{q \in M}{\sum\limits_{{k:q \in M_{k} }} {\frac{{Y_{q}^{k}}}{{y_{qo}^{total} }} = 1} } \hfill \\&- \sum\nolimits_{j}{\varLambda_{j}^{k} \left( {x_{qj}^{k} e_{{N_{k} }}^{q} + a_{qj}^{k}e_{{\tilde{N}_{k} }}^{q} } \right)} \hfill \\&\quad \quad + \left({X_{q}^{k} e_{{N_{k} }}^{q} + \left( {\sum\limits_{{h:h\mathop \to\limits_{q} k}} {Z_{q}^{(h,k)} } } \right)\;e_{{\tilde{N}_{k} }}^{q}} \right) \ge 0 \hfill \\ \end{aligned} \hfill & \begin{aligned}\hfill \\ \hfill \\ \forall k,\quad \forall q \in (\tilde{N}_{k}\cup N_{k} )\, \hfill \\ \end{aligned} \hfill & \begin{aligned}(\eta_{o} )\quad \quad (a) \hfill \\ \hfill \\ (w_{q}^{k} )\quad\quad (b) \hfill \\ \end{aligned} \hfill \\ {} \hfill &\begin{aligned}&\sum\nolimits_{j} {\varLambda_{j}^{k} \left({y_{qj}^{k} e_{{M_{k} }}^{q} + b_{qj}^{k} e_{{\tilde{M}_{k} }}^{q} }\right)} \hfill \\&\quad \quad - \left( {Y_{q}^{k} e_{{M_{k} }}^{q}+ \left( {\sum\limits_{{h:k\mathop \to \limits_{q} h}}{Z_{q}^{(k,h)} } } \right)e_{{\tilde{M}_{k} }}^{q} } \right) \ge 0\hfill \\ \end{aligned} \hfill & {\forall k,\quad \forall q \in(\tilde{M}_{k} \cup M_{k} )} \hfill & {(w_{q}^{k} )\quad \quad(c)} \hfill \\ {} \hfill & { - \sum\nolimits_{{k:q \in N_{k} }}{X_{q}^{k} } + \tau \sum\nolimits_{{k:q \in N_{k} }} {x_{qo}^{k} }\ge 0} \hfill & {\forall q \in N\left( { = \bigcup\limits_{k}{N_{k} } } \right)} \hfill & {(v_{q}^{{}} )\quad \quad (d)}\hfill \\ {} \hfill & {\sum\nolimits_{{k:q \in M_{k} }}{Y_{q}^{k} } - \tau \sum\nolimits_{{k:q \in M_{k} }} {y_{qo}^{k} }\ge 0} \hfill & {\forall q \in M\left( { = \bigcup\limits_{k}{M_{k} } } \right)} \hfill & {(u_{q}^{{}} )\quad \quad (e)}\hfill \\ {} \hfill & {\sum\nolimits_{j} {\varLambda_{j}^{k} } -\tau = 0} \hfill & {\forall k} \hfill & {(\mu_{k} )\quad\;\;\;(f)} \hfill \\ {} \hfill & {\varLambda_{j}^{k} ,\quad Z_{q}^{(k,h)} ,\quad X_{q}^{k} ,\quad Y_{q}^{k} \ge 0,\quad \tau> 0} \hfill & {} \hfill & {\quad \quad \quad \;\;(g)}\hfill \\ \end{array} \\ \end{aligned} $$
(26)

Since N and M are distinct sets, considering \( w_{q}^{k} \) can be considered as dual variables both for (26-b) and (26-c). Moreover, \( \tau > 0 \) can be considered to be \( \tau \ge 0 \). This is because according to (26-a), all \( Y_{q} \)’s cannot be zero at the same time and consequently in constraint (26-b), all \( \varLambda_{j}^{{}} \)’s cannot be zero simultaneously. Therefore (26-f) ensures that \( \tau \) can never be zero. Hence, models (5), (23) and (26) are equivalent.

Part II Now we prove that if we linearize model (22), it would be equivalent to model (26) and therefore equivalent to NSBM model (5). In (22), let \( \eta_{o} = {{\mathop {\hbox{min} }\nolimits_{{(q,k):q \in M_{k} }} \left( {y_{qo}^{total} (u_{q}^{{}} - w_{q}^{k} )} \right)} \mathord{\left/ {\vphantom {{\mathop {\hbox{min} }\nolimits_{{(q,k):q \in M_{k} }} \left( {y_{qo}^{total} (u_{q}^{{}} - w_{q}^{k} )} \right)} {\mathop {\hbox{max} }\nolimits_{{(q^{{\prime }} ,k^{{\prime }} ):q^{{\prime }} \in N_{{k^{{\prime }} }} }} \left( {x_{{q^{{\prime }} o}}^{total} (v_{{q^{{\prime }} }}^{{}} - w_{{q^{{\prime }} }}^{{k^{{\prime }} }} )} \right)}}} \right. \kern-0pt} {\mathop {\hbox{max} }\nolimits_{{(q^{{\prime }} ,k^{{\prime }} ):q^{{\prime }} \in N_{{k^{{\prime }} }} }} \left( {x_{{q^{{\prime }} o}}^{total} (v_{{q^{{\prime }} }}^{{}} - w_{{q^{{\prime }} }}^{{k^{{\prime }} }} )} \right)}} \). In order to make the model linear, we use the Charnes-Cooper transformation and let \( \mathop {\hbox{max} }\nolimits_{{(q,k):q \in N_{k} }} \left( {x_{qo}^{total} (w_{q}^{k} - v_{q}^{{}} )} \right) = 1 \) and \( \eta_{o} = \mathop {\hbox{min} }\nolimits_{{(q,k):q \in M_{k} }} \left( {y_{qo}^{total} (w_{q}^{k} - u_{q}^{{}} )} \right) \). The obtained equalities are the same inequalities as stated in (23-d) and (23-e). Constraints (a-c) are the same in both (23) and (22). Moreover, note that \( 0 < w_{q}^{k} - v_{q}^{{}} \) from model (22) is not brought into model (23). We previously noted that there exists a feasible solution for (22) in which all the fractions in its objective function are positive [as stated in (21)]. Therefore, since \( 0 < w_{q}^{k} - u_{q}^{{}} \) holds, the model will not automatically searches for the solutions which \( w_{q}^{k} - v_{q}^{{}} \le 0 \) for some pairs (k,q) (because it causes to have some negative fractions in the objective function and maximization model will not choose them automatically).

$$ \begin{aligned} & \frac{|M|}{|N|}\mathop {\hbox{max}}\limits_{{}} \eta_{o} \\ & \begin{array}{*{20}l} {s.t.} \hfill& {\sum\limits_{k} {\sum\limits_{{q:q \in M_{k} }} {u_{q}^{{}}y_{qo}^{k} } } + \left[ {\sum\limits_{k} {\mu_{k} } } \right] =\sum\limits_{k} {\sum\limits_{{q:q \in N_{k} }} {v_{q}^{{}}x_{qo}^{k} } } ,} \hfill & {} \hfill & {(\tau )\;\;\quad\;\quad (a)} \hfill \\ {} \hfill & \begin{aligned}\sum\limits_{{q \in (\tilde{M}_{k} \cup M_{k} )}} {w_{q}^{k}\left( {y_{qj}^{k} e_{{M_{k} }}^{q} + b_{qj}^{k} e_{{\tilde{M}_{k}}}^{q} } \right)} + \left[ {\mu_{k} } \right] \hfill \\ \quad \quad\quad \le \sum\limits_{{q \in (\tilde{N}_{k} \cup N_{k} )}}{w_{q}^{k} \left( {x_{qj}^{k} e_{{N_{k} }}^{q} + a_{qj}^{k}e_{{\tilde{N}_{k} }}^{q} } \right)} , \hfill \\ \end{aligned} \hfill& \begin{aligned} \hfill \\ \forall j,\quad \forall k \hfill \\ \end{aligned} \hfill & \begin{aligned} \hfill \\(\varLambda_{j}^{k} )\quad \;\;\;\;\,(b) \hfill \\ \end{aligned}\hfill \\ {} \hfill & {w_{q}^{h} - w_{q}^{k} \le 0,} \hfill& {\forall (k,h,q):k\mathop \to \limits_{q} h} \hfill &{(Z_{q}^{(k,h)} )\quad \;(c)} \hfill \\ {} \hfill &\begin{aligned}&- w_{q}^{k} + u_{q}^{{}} + \eta_{o} /y_{qo}^{total}\le 0, \hfill \\&w_{q}^{k} - v_{q}^{{}} \le 1/x_{qo}^{total} \hfill\\&w_{q}^{k} - u_{q}^{{}} > 0 \hfill \\&w_{q}^{k} ,\quad u_{q}^{{}} ,\quad v_{q}^{{}} ,\quad \eta_{o} \ge 0,\quad \left[{\mu_{k} free} \right]. \hfill \\ \end{aligned} \hfill &\begin{aligned} \forall q \in M_{{}}^{{}} ,\quad \forall k:q \in M_{k} \hfill \\ \forall q \in N_{{}}^{{}} ,\quad \forall k:q \in N_{k} \hfill \\ \forall q \in M_{{}}^{{}} ,\quad \forall k:q \in M_{k} \hfill \\ \end{aligned} \hfill & \begin{aligned}(Y_{q}^{k} )\quad \;\;\;\;(d) \hfill \\ (X_{q}^{k} )\quad \;\;\;(e)\hfill \\ \quad \quad \;\;\;\;\;\;(f) \hfill \\ \quad \quad\;\;\;\;\;\;(g) \hfill \\ \end{aligned} \hfill \\ \end{array} \\ \end{aligned} $$
(27)

The only thing remaining is to prove that constraint (f) in model (27) as the linearized form of (22) is redundant, and hence if we omit this constraint in model (23) then model (22) is equivalent to (23). To see this, note that according to (27-d) and (27-f), we have \( {{0 \le \eta_{0} } \mathord{\left/ {\vphantom {{0 \le \eta_{0} } {y_{qo}^{total} }}} \right. \kern-0pt} {y_{qo}^{total} }} \le w_{q}^{k} - u_{q}^{k} \). So we have \( w_{q}^{k} \ge u_{q}^{k} \). Moreover, note that \( \sum\nolimits_{q \in N} {\sum\nolimits_{{k:q \in N_{k} }} {\frac{{X_{q}^{k} }}{{x_{qo}^{total} }}} } > 0 \) (because at least one main input q exists where it plays only the main input role for a division like k (i.e. \( e_{{N_{k} }}^{q} = 1 \) and \( e_{{\tilde{N}_{k} }}^{q} = 0 \)). So, according to (27-f) at least some positive \( \varLambda_{j}^{k} \) exist which make the left hand side of (27-b) to be positive. This implies that at least one \( X_{q}^{k} \) is positive). Therefore, since the optimal objective function values of (27) and (26) are equal, we must have \( \eta_{o}^{{}} > 0 \). Consequently, we have \( 0 < w_{q}^{k} - u_{q}^{k} \) automatically.

Part III It is simply obtained from parts I and II. □

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Boloori, F., Khanjani-Shiraz, R. & Fukuyama, H. Relative partial efficiency: network and black box SBM DEA interpretations in multiplier form. Oper Res Int J 21, 2689–2718 (2021). https://doi.org/10.1007/s12351-019-00532-x

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