Cone ratio models with shared resources and nontransparent allocation parameters in network DEA

Abstract

Many studies have examined the performance of production systems with shared resources through the application of data envelopment analysis (DEA). The present models are based on the multiplier-type frameworks and resource allocation variables (RAVs) in simple yet edifying network settings, such as multi-component and two-stage structures. Two issues associated with RAVs, however, are relevant to both the theory and practice. First, the existing models with RAVs are nonlinear in general. Second, a potential conflict of interest between a central evaluator (CE) and the managers of decision-making units (DMUs), due to the unknown allocation parameters, has not been addressed. The current study contributes to the resolution of these issues by presenting conflict free models (CFMs). Two striking features of the proposed models include their convenient transformation into linear programs and that they are conflict free. Thus, the manager of a DMU has no basis to argue against the evaluation results by simply claiming that faulty information regarding the split of shared resources has been used, as no such information relating to the CE’s preference or pertaining to RAVs is included in the model formulation. Furthermore, we investigate the relations between CFMs and the existing models, strengthening the interpretations of the existing models. Finally, the proposed models incorporate partial ordering preferences expressed as the cone ratio constraints, which are suitable for a wide range of real life applications. A dataset extracted from literature is used to illustrate the main concept that drives this research.

Appendix

Theorem 1

$\theta_{CRM}^{*} = \mathop {\hbox{max} }\limits_{{\beta_{i} \in [LB_{i} ,UB_{i} ]}} \quad \theta_{CRD}^{*} ({\varvec{\upbeta}})$, where $\theta_{CRM}^{*}$ is the optimal value of model ( 3 ), and $\theta_{CRD}^{*} ({\varvec{\upbeta}})$ is the optimal value of model ( 6 ) with given ${\varvec{\upbeta}}$.

Proof

Note that θ ^* _CRM can be secured by the decomposition approach, i.e.,$\theta_{CRM}^{*} = \mathop {\hbox{max} }\limits_{{\beta_{i} \in [LB_{i} ,UB_{i} ]}} \, \theta_{CMP}^{*} ({\varvec{\upbeta}})$, where $\theta_{CMP}^{*} ({\varvec{\upbeta}})$ is the optimal value of the inner problem of model (5). This theorem holds since the primal–dual gap between $\theta_{CMP}^{*} ({\varvec{\upbeta}})$ and $\theta_{CMD}^{*} ({\varvec{\upbeta}})$ is zero. Q.E.D.

Lemma 1

$f_{i} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} )(i = 1, \ldots ,t)$ are quasi-convex with respect to ${\varvec{\upbeta}}$.

Proof

Suppose ${\varvec{\upbeta}}_{1}$,${\varvec{\upbeta}}_{2}$ in [LU ₁, UB ₁] × [LU ₂, UB ₂] × ··· × [LU _l , UB _l ]. For any λ ∊ [0, 1], we have

$$\begin{array}{l} f_{i} (\lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ,{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} ) = {{{\mathbf{a}}^{iT} \left( \begin{array}{l} \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} X^{s} {\varvec{\uplambda}}_{1} \hfill \\ ({\mathbf{E}} - \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ){\mathbf{X}}^{s} {\varvec{\uplambda}}_{2} \hfill \\ {\mathbf{X}}^{1} {\varvec{\uplambda}}_{1} \, \hfill \\ \, {\mathbf{X}}^{2} {\varvec{\uplambda}}_{2} \hfill \\ \end{array} \right)} \mathord{\left/ {\vphantom {{{\mathbf{a}}^{iT} \left( \begin{array}{l} \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} X^{s} {\varvec{\uplambda}}_{1} \hfill \\ ({\mathbf{E}} - \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ){\mathbf{X}}^{s} {\varvec{\uplambda}}_{2} \hfill \\ {\mathbf{X}}^{1} {\varvec{\uplambda}}_{1} \, \hfill \\ \, {\mathbf{X}}^{2} {\varvec{\uplambda}}_{2} \hfill \\ \end{array} \right)} {{\mathbf{a}}^{iT} \left( \begin{array}{l} \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda )\beta_{2} X_{o}^{s} \hfill \\ ({\mathbf{E}} - \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}}} \right. \kern-0pt} {{\mathbf{a}}^{iT} \left( \begin{array}{l} \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda )\beta_{2} X_{o}^{s} \hfill \\ ({\mathbf{E}} - \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}} \hfill \\ \quad = \rho {{{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{1} {\mathbf{X}}^{s} {\varvec{\uplambda}}_{1} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{1} ){\mathbf{X}}^{s} {\varvec{\uplambda}}_{2} \hfill \\ {\mathbf{X}}^{1} {\varvec{\uplambda}}_{1} \, \hfill \\ \, {\mathbf{X}}^{2} {\varvec{\uplambda}}_{2} \hfill \\ \end{array} \right)} \mathord{\left/ {\vphantom {{{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{1} {\mathbf{X}}^{s} {\varvec{\uplambda}}_{1} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{1} ){\mathbf{X}}^{s} {\varvec{\uplambda}}_{2} \hfill \\ {\mathbf{X}}^{1} {\varvec{\uplambda}}_{1} \, \hfill \\ \, {\mathbf{X}}^{2} {\varvec{\uplambda}}_{2} \hfill \\ \end{array} \right)} {{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{1} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{1} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}}} \right. \kern-0pt} {{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{1} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{1} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}} + (1 - \rho ){{{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{2} {\mathbf{X}}^{s} {\varvec{\uplambda}}_{1} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{2} ){\mathbf{X}}^{s} {\varvec{\uplambda}}_{2} \hfill \\ {\mathbf{X}}^{1} {\varvec{\uplambda}}_{1} \, \hfill \\ \, {\mathbf{X}}^{2} {\varvec{\uplambda}}_{2} \hfill \\ \end{array} \right)} \mathord{\left/ {\vphantom {{{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{2} {\mathbf{X}}^{s} {\varvec{\uplambda}}_{1} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{2} ){\mathbf{X}}^{s} {\varvec{\uplambda}}_{2} \hfill \\ {\mathbf{X}}^{1} {\varvec{\uplambda}}_{1} \, \hfill \\ \, {\mathbf{X}}^{2} {\varvec{\uplambda}}_{2} \hfill \\ \end{array} \right)} {{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{2} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{2} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}}} \right. \kern-0pt} {{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{2} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{2} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}} \hfill \\ \end{array}$$

where

$$\rho = {{{\mathbf{a}}^{iT} \left( \begin{array}{l} {\varvec{\upbeta}}_{1} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{1} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)} \mathord{\left/ {\vphantom {{{\mathbf{a}}^{iT} \left( \begin{aligned} {\varvec{\upbeta}}_{1} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - {\varvec{\upbeta}}_{1} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{aligned} \right)} {{\mathbf{a}}^{iT} \left( \begin{array}{l} \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}}} \right. \kern-0pt} {{\mathbf{a}}^{iT} \left( \begin{array}{l} \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} {\mathbf{X}}_{o}^{s} \hfill \\ ({\mathbf{E}} - \lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ){\mathbf{X}}_{o}^{s} \hfill \\ {\mathbf{X}}_{o}^{1} \hfill \\ {\mathbf{X}}_{o}^{2} \hfill \\ \end{array} \right)}}.$$

Therefore, it follows that $f_{i} (\lambda {\varvec{\upbeta}}_{1} + (1 - \lambda ){\varvec{\upbeta}}_{2} ,{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} ) \le \hbox{max} \{ f_{i} ({\varvec{\upbeta}}_{1} ,{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} ),f_{i} ({\varvec{\upbeta}}_{2} ,{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} )\}$. Q.E.D.

Theorem 2

The CFM is equivalent to the Model (14).

Proof

The constraints $\theta \ge f_{i} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} ) ,\quad i = 1, \ldots ,t,\beta_{i} \in [LB_{i} ,UB_{i} ]$ in model (13) are equivalent:

$$\begin{gathered} \theta \ge \mathop {\hbox{max} }\limits_{{\beta_{i} \in [LB_{i} ,UB_{i} ]}} \{ f_{1} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} ), \ldots ,f_{t} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} )\} \hfill \\ \quad = \hbox{max} \left\{ {\mathop {\hbox{max} }\limits_{{\beta_{i} \in [LB_{i} ,UB_{i} ]}} f_{1} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} ), \ldots ,\mathop {\hbox{max} }\limits_{{\beta_{i} \in [LB_{i} ,UB_{i} ]}} f_{t} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} )} \right\} \hfill \\ \end{gathered}$$

Note, $f_{i} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} )$ (i = 1, …, t) are quasi-convex with respect to ${\varvec{\upbeta}}$ over the feasible region [LU ₁, UB ₁] × [LU ₂, UB ₂] × ··· × [LU _l , UB _l ] according to Lemma 1. The maximal value of $\mathop {\hbox{max} }\limits_{{\beta_{i} \in [LB_{i} ,UB_{i} ]}} f_{i} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} )$ is attained at some extreme point in EXT. Therefore, the constraints can be alternatively written as $\theta \ge f_{i} ({\varvec{\upbeta}},{\varvec{\uplambda}}_{1} ,{\varvec{\uplambda}}_{2} ),\quad {\varvec{\upbeta}} \in \varvec{EXT},i = 1, \ldots ,t.$ Q.E.D.

Proposition 1

The efficiency of a DMU measured by CFM is not less than that measured by CRM.

Proof

Suppose the optimal RAV of CRM is β ^* _i . Then the optimal value of CRM equals $\theta_{CRD}^{*} (\beta_{i}^{*} )$, where $\theta_{CRD}^{*} (\beta_{i}^{*} )$ is the optimal value of model (6) with β ^* _i . Note that any feasible solution of model (6), given β ^* _i , is a feasible solution of model (14). This is because the set of constraints of model (6), given β ^* _i , are a subset of the set of constraints of model (14). Therefore, the optimal value of model (14), i.e., CFM is not less than $\theta_{CRD}^{*} (\beta_{i}^{*} )$. Hence, the proposition follows.  Q.E.D.

Proposition 2

If U and V are nonnegative orthants, CRM is equivalent to (20), and the optimal RAVs can be any nonzero value in [LB ₁, UB ₁] × ··· × [LB _l , UB _l ].

Proof

It suffices to show that the optimal value of model (20) equals $\mathop {\hbox{max} }\limits_{{\beta_{i} \in [LU_{i} ,UB_{i} ]}} \theta_{CRD}^{*} ({\varvec{\upbeta}})$ (Theorem 1). For any nonzero value in [LB ₁, UB ₁] × ··· × [LB _l , UB _l ], it is sufficient to show that the feasible region of model (20) is the smallest possible, or the feasible region of model (20) is contained in the feasible region of model (6) with arbitrary feasible ${\varvec{\upbeta}}$. Note, the constraints related to ${\varvec{\upbeta}}$ of model (6) are as follows:

$$\left. \begin{array}{l} \beta_{i} \left( {\theta x_{io}^{s} - \sum\limits_{j = 1}^{n} {\lambda_{j}^{1} x_{ij}^{s} } } \right) \ge 0 \hfill \\ (1 - \beta_{i} )\left( {\theta x_{io}^{s} - \sum\limits_{j = 1}^{n} {\lambda_{j}^{2} x_{ij}^{s} } } \right) \ge 0 \hfill \\ \end{array} \right\}\quad i = 1, \ldots ,l$$

Further note that if β _i is set to zero, the consequence is to delete the associated constraints for model (6). In other words, the closer β _i is to zero, the smaller the number of effective constraints remain in model (6), i.e., the feasible region becomes the smallest when β _i is nonzero for i = 1, …, l. Therefore, the optimal value of the model (20) is $\mathop {\hbox{max} }\limits_{{\beta_{i} \in [LU_{i} ,UB_{i} ]}} \, \theta_{CRD}^{*} ({\varvec{\upbeta}})$. Q.E.D.

Proposition 3

If U and V are nonnegative orthants, the reduced CRM is equal to ( 21 ). Therefore, the optimal values are bound-dependent.

Proof

To see this, we first note that, if U and V are nonnegative orthants, the reduced CRM can be transformed into the linear equivalent model through substitution of variables. For the sake of space, only the dual of the linear equivalent program is presented here:

$$\begin{array}{*{20}l} {\hbox{min} } \hfill & \theta \hfill \\ {s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j}^{1} y_{rj}^{1} } \ge y_{ro}^{1} } \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j}^{2} y_{rj}^{2} } \ge y_{ro}^{2} } \hfill \\ {} \hfill & { \, \sum\limits_{j = 1}^{n} {\lambda_{j}^{1} x_{ij}^{1} } \le \theta x_{io}^{1} } \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j}^{2} x_{ij}^{2} } \le \theta x_{io}^{2} } \hfill \\ {} \hfill & {b_{i} - a_{i} + \sum\limits_{j = 1}^{n} {\lambda_{j}^{2} (x_{ij}^{s} )} = \sum\limits_{j = 1}^{n} {\lambda_{j}^{1} (x_{ij}^{s} )} } \hfill \\ {} \hfill & {b_{i} UB_{i} - a_{i} LB_{i} + \sum\limits_{j = 1}^{n} {\lambda_{j}^{2} (x_{ij}^{s} )} \le \theta x_{io}^{s} } \hfill \\ {} \hfill & {\lambda_{j}^{1} ,\lambda_{j}^{2} ,a_{i} ,b_{i} \ge 0.} \hfill \\ \end{array}$$

(23)

For any i (i = 1, …, l), the optimal β ^* _i is obtained as: β ^* _i  = v ^s* _i /v ^* _i , where v ^s* _i and v ^* _i are the shadow prices associated with constraint $b_{i} - a_{i} + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2} \left( {x_{ij}^{s} } \right)} = \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1} \left( {x_{ij}^{s} } \right)}$ and $b_{i} UB_{i} - a_{i} LB_{i} + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2} \left( {x_{ij}^{s} } \right) \le \theta x_{io}^{s} }$, respectively, if v ^* _i  ≠ 0, or β ^* _i can take any value in[LB _i , UB _i ].

By Proposition 1, the optimal value of model (21), denote θ ^* _CFM , is greater or equal to the optimal value of the reduced CRM, denoteθ ^* _RCRM , orθ ^* _CFM  ≥ θ ^* _RCRM  = θ ^* _A1 , where θ ^* _A1 is the optimal value of mode (23). It remains to be shown that θ ^* _CFM  ≤ θ ^* _RCRM  = θ ^* _A1 . We first present the following fact summarized in Lemma A1.

Lemma (A1)

There exists an optimal solution (λ ^1* _j , λ ^2* _j , a ^* _i , b ^* _i , θ ^* _A1 ) to model ( 23 ) with a ^* _i b ^* _i  = 0.

Proof

To see this, suppose a ^* _i b ^* _i  ≠ 0. Let a ^* _i  − b ^* _i  = c _i , where c _i can be positive, negative or zero. Assume, without loss of generality, that c _i  < 0 for illustrative purpose. Let $\bar{b} = b_{i}^{*} - a_{i}^{*} = c_{i} ,\bar{a} = 0$. Then,

$$\left\{ \begin{gathered} \bar{b} - \bar{a} + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} (x_{ij}^{s} )} = \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} \hfill \\ (\bar{b} - \bar{a})UB_{i} + \bar{a}(UB_{i} - LB_{i} ) + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2} \left( {x_{ij}^{s} } \right)} \le \hfill \\ (b_{i}^{*} - a_{i}^{*} )UB_{i} + a_{i}^{*} (UB_{i} - LB_{i} ) + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2} \left( {x_{ij}^{s} } \right)} \le \theta^{*} x_{io}^{s} \hfill \\ \end{gathered} \right.$$

Note, $(\lambda_{j}^{1*} ,\lambda_{j}^{2*} ,\bar{a},\bar{b},\theta_{A1}^{*} )$ is a feasible solution to model (23), and so is an optimal solution. Thus, Lemma A1 holds. Q.E.D.

Based on Lemma A1, we have $\left. \begin{array}{l} b_{i}^{*} + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} = \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} \\ b_{i}^{*} UB_{i} + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} \le \theta^{*} x_{io}^{s} \\ \end{array} \right\}$ if a ^* _i  = 0; otherwise, $\left. \begin{gathered} - a_{i}^{*} + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} = \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} \hfill \\ - a_{i}^{*} LB_{i} + \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} \le \theta^{*} x_{io}^{s} \hfill \\ \end{gathered} \right\}$ if b ^* _i  = 0; which means:

$$\begin{gathered} \sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \left( {\sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } } \right)UB_{i} } \le \theta^{*} x_{io}^{s} \quad\quad {\text{if}}\;\sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } \le 0 \hfill \\ \sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \left( {\sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} - \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } \right)LB_{i} } \le \theta^{*} x_{io}^{s} \quad\quad {\text{if}}\;\sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } \ge 0 \hfill \\ \end{gathered}$$

Therefore, (λ ^1* _j , λ ^2* _j , θ ^* _A1 ) is a feasible solution to model (21). To see this, note that

(i) If $\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } \le 0$, then

$\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \left( {\sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } } \right)LB_{i} } \le \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \left( {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} - \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } \right)UB_{i} } \le \theta^{*} x_{io}^{s}$, (ii) If $\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right) \ge 0} }$, then $\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \left( {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} - \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } \right)UB_{i} } \le \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right) - \left( {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{2*} \left( {x_{ij}^{s} } \right)} - \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{1*} \left( {x_{ij}^{s} } \right)} } \right)LB_{i} } \le \theta^{*} x_{io}^{s} .$

It is then clear that the constraints related to the shared inputs can be fully satisfied.

Thus, θ ^* _CFM  ≤ θ ^* _A1 . Q.E.D.

Proposition 4

If β ₁ = β ₂ = ··· = β _l  ∊ [LB, UB], or l = 1, $\mathop {\hbox{max} }\limits_{{\beta_{i} \in [LB,UB]}} \, \theta_{CRD}^{*} ({\varvec{\upbeta}})$ is attained at some end point LB or UB.

Proof

Suppose (λ ^11* _j , λ ^21* _j , θ ^* _CRD (β ₁)) and (λ ^12* _j , λ ^22* _j , θ ^* _CRD (β ₂)) are optimal solutions to model (6), respectively, corresponding to β = β ₁ and β = β ₂. Now let us construct λ ^1* _j and λ ^2* _j as follows:

$$\left\{ \begin{array}{l} \lambda_{j}^{1*} = \frac{{\lambda \beta_{1} \lambda_{1}^{11*} + (1 - \lambda )\beta_{2} \lambda_{1}^{12*} }}{{\lambda \beta_{1} + (1 - \lambda )\beta_{2} }} \hfill \\ \lambda_{j}^{2*} = \frac{{\lambda (1 - \beta_{1} )\lambda_{1}^{21*} + (1 - \lambda )(1 - \beta_{2} )\lambda_{1}^{22*} }}{{1 - \lambda \beta_{1} - (1 - \lambda )\beta_{2} }} \hfill \\ \end{array} \right..$$

where λ is an arbitrary value in [0,1]. Obviously, λ ^1* _j and λ ^2* _j are nonnegative. Now substitute them into the model (6), we have

$$\begin{gathered} \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} y_{rj}^{1} } \ge y_{ro}^{1} \quad r = 1, \ldots ,s_{1} \hfill \\ \sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} y_{rj}^{2} } \ge y_{ro}^{2} \quad r = 1, \ldots ,s_{2} \hfill \\ \left[ {\beta^{*} \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} X_{j}^{s} } { + (1} - \beta^{*} )\sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} X_{j}^{s} } } \right] \le \left( {\lambda \theta_{CRD}^{*} (\beta_{1} ) + (1 - \lambda )\theta_{CRD}^{*} (\beta_{2} )} \right)X_{o}^{s} \quad i = 1, \ldots ,l \, \hfill \\ \sum\limits_{j = 1}^{n} {\lambda_{j}^{1*} x_{ij}^{1} } \le \left( {\frac{{\lambda \beta_{1} }}{{\lambda \beta_{1} + (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{1} ) + \frac{{(1 - \lambda )\beta_{2} }}{{\lambda \beta_{1} + (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{2} )} \right)x_{io}^{1} \quad i = 1, \ldots ,m_{1} \hfill \\ \sum\limits_{j = 1}^{n} {\lambda_{j}^{2*} x_{ij}^{2} } \le \left( {\frac{{\lambda (1 - \beta_{1} )}}{{1 - \lambda \beta_{1} - (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{1} ) + \frac{{(1 - \lambda )(1 - \beta_{2} )}}{{1 - \lambda \beta_{1} - (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{2} )} \right)x_{io}^{2} \quad i = 1, \ldots ,m_{2} \hfill \\ \end{gathered},$$

where β ^* = λβ ₁ + (1 − λ)β ₂.

Let,

$$\begin{gathered} \theta^{*} = \hbox{max} \left\{ {\lambda \theta_{CRD}^{*} (\beta_{1} ) + (1 - \lambda )\theta_{CRD}^{*} (\beta_{2} ),\frac{{\lambda \beta_{1} }}{{\lambda \beta_{1} + (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{1} ) + \frac{{(1 - \lambda )\beta_{2} }}{{\lambda \beta_{1} + (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{2} ),} \right. \hfill \\ \quad \left. {\frac{{\lambda (1 - \beta_{1} )}}{{1 - \lambda \beta_{1} - (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{1} ) + \frac{{(1 - \lambda )(1 - \beta_{2} )}}{{1 - \lambda \beta_{1} - (1 - \lambda )\beta_{2} }}\theta_{CRD}^{*} (\beta_{2} )\} } \right\} \hfill \\ \end{gathered}$$

It then follows that (λ ^1* _j , λ ^2* _j , θ ^*) is a feasible solution to model (6) with β = β ^*. Therefore, the optimal solution with respect to β ^*, θ ^* _CRD (β ^*) should be less than or equal to θ ^*.

Given that θ ^* is a convex combination of θ ^* _CRD (β ₁) and θ ^* _CRD (β ₂), we have θ ^* _CRD (β ^*) ≤ max {θ ^* _CRD (β ₁), θ ^* _CRD (β ₂)}, where β ^* = λβ ₁ + (1 − λ)β ₂.

Since λ is an arbitrarily chosen convex multiplier, we conclude that $\mathop {\hbox{max} }\limits_{{\beta_{{}} \in [LB,UB]}} \, \theta_{CRD}^{*} (\beta )$ is attained at LB or UB. Q.E.D.

Proposition 5

If LB _i  = 0, UB _i  = 1, CRM, the reduced CRM, and the associated CFMs (20) and (21) are all equivalent.

Proof

The equivalence between CFMs (20) and (21) is straightforward, if we substitute LB _i  = 0, UB _i  = 1in (21). Following Proposition 2 and 3, the equivalence across CRM, reduced CRM, and CFMs (20) and (21) is established. Q.E.D.