Shared and unsplittable performance links in network DEA

Abstract

Data envelopment analysis (DEA) is a broadly used non-parametric technique for performance evaluation and data analytics. While conventional single-stage DEA models overlook the internal interactions of decision making units (DMUs), network DEA opens this black box to investigate the internal structure of DMUs. Practically, many network DEA models involve shared performance measures that are not easily divisible among individual components of a network. Based upon a two-stage network DEA model, the current study treats such performance measures as inseparable links, implying that no proportions are optimized and allocated to the two stages of the network. The shared and unsplittable links in the proposed two-stage DEA model manifest integrality while both ends of the link are maximized or minimized simultaneously, and this setting has not been modeled in any existing DEA studies. The shared and unsplittable links in our model can be considered intermediate measures, but they are different from the two existing types of dual-role intermediate measures, which are traditional intermediate measures and feedback measures. Our performance link is a new type of intermediate measure that is minimized or maximized in both stages of the network. The resulting network DEA model is highly non-linear. To address the non-linearity, a parametric linear model is adopted. The proposed approach is construed in four variants, and then illustrated using a set of 100 banks in the United States.

Appendices

Appendix

The aforementioned measures are based on the network DEA framework with intermediate links z. Here we elaborate on a complementary case of the two-stage process without the intermediate links z but with the unsplittable minimizing links. This is a special type of network model without the sequential relationship between two stages, and the shared and unsplittable links have become the only links between the two stages. The conceptual figure is depicted as follows.

Shared and unsplittable minimizing measure only

We first construct the basic model with the shared and unsplittable minimizing links as model (A1). Notably, without intermediate measures Z, the first stage would be modeled as a measure with only inputs. To express this feature in network DEA, we adopt the method of Seiford and Zhu (1998) and Lovell and Pastor (1999) and use a weight factor c for the output of constant 1 in the first stage.

$$ \begin{aligned} \tilde{\varTheta }_{0}^{*} = & \hbox{max} \left( {\alpha \cdot \frac{{c^{'} }}{{\sum\nolimits_{i = 1}^{m} {v^{'}_{i} x_{i0} } + \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{l0} } }} + (1 - \alpha ) \cdot \frac{{{\kern 1pt} \sum\nolimits_{r = 1}^{s} {\mu^{'}_{r} y_{r0} } }}{{\sum\nolimits_{l = 1}^{L} {\rho^{'}_{l} u_{l0} } }}} \right) \\ & {\text{s}} . {\text{t}} . { }\frac{{c^{'} }}{{\sum\nolimits_{i = 1}^{m} {v^{'}_{i} x_{ij} } + \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{lj} } }} \le 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \ldots ,n{\kern 1pt} , \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\kern 1pt} \sum\nolimits_{r = 1}^{s} {\mu^{'}_{r} y_{rj} } {\kern 1pt} }}{{\sum\nolimits_{l = 1}^{L} {\rho^{'}_{l} u_{lj} } }} \le 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \ldots ,n, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu_{r}^{'} ,\nu_{i}^{'} ,{\kern 1pt} {\kern 1pt} \rho_{l}^{'} ,c^{'} \ge \varepsilon . \\ \end{aligned} $$

(A1)

Let $h\mu_{r}^{'} = \mu_{r} ,h\nu_{i}^{'} = \nu_{i} ,{\kern 1pt} {\kern 1pt} h\rho_{l}^{'} = \rho_{l} ,{\kern 1pt} {\kern 1pt} hc^{'} = c$. Then we obtain model (A2).

$$ \begin{aligned} \tilde{\varTheta }_{0}^{*} = & \hbox{max} \left( {\alpha \cdot {\kern 1pt} \frac{c}{{\sum\limits_{i = 1}^{m} {v_{i} x_{i0} } + \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{l0} } }} + (1 - \alpha ) \cdot \sum\nolimits_{r = 1}^{s} {\mu_{r} y_{r0} } } \right) \\ & {\text{s}} . {\text{t}} . { }{\kern 1pt} \frac{c}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} } + \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{lj} } }} \le 1,j = 1,2, \ldots ,n{\kern 1pt} , \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\kern 1pt} {\kern 1pt} \sum\nolimits_{r = 1}^{s} {\mu_{r} y_{rj} } }}{{\sum\nolimits_{l = 1}^{L} {\rho_{l} u_{lj} } }}{\kern 1pt} \le 1,j = 1,2, \ldots ,n{\kern 1pt} , \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{l0} } = 1, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu_{r} ,\nu_{i} ,{\kern 1pt} {\kern 1pt} \rho_{l} ,c \ge \varepsilon .{\kern 1pt} \\ \end{aligned} $$

(A2)

Also, by introducing parameter k, we have model (A3), and let $kc = \tilde{c}$.

$$ \begin{aligned} \tilde{\varTheta }_{0}^{*} = & \hbox{max} \left( {\alpha \cdot {\kern 1pt} \tilde{c} + (1 - \alpha ) \cdot \sum\nolimits_{r = 1}^{s} {\mu_{r} y_{r0} } } \right) \\ & {\text{s}} . {\text{t}} . { }\frac{{\tilde{c}}}{{k(\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} } + \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{lj} } )}} \le 1,j = 1,2, \ldots ,n, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\sum\nolimits_{r = 1}^{s} {\mu_{r} y_{rj} } }}{{\sum\nolimits_{l = 1}^{L} {\rho_{l} u_{lj} } }}{\kern 1pt} \le 1,j = 1,2, \ldots ,n, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{l0} } = 1, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k\left( {\sum\nolimits_{i = 1}^{m} {v_{i} x_{i0} } + \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{l0} } } \right) = 1, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu_{r} ,\nu_{i} ,{\kern 1pt} {\kern 1pt} \rho_{l} ,\tilde{c} \ge \varepsilon . \\ \end{aligned} $$

(A3)

To solve model (A3), the feasible region of k should be identified. Since $\sum\nolimits_{l = 1}^{L} {\rho_{l} u_{l0} } = 1$ and $\sum\nolimits_{i = 1}^{m} {v_{i} x_{i0} } \ge 0,$ we have $0 \le {\kern 1pt} k = \frac{1}{{\left( {\sum\nolimits_{i = 1}^{m} {v_{i} x_{i0} } + \sum\nolimits_{l = 1}^{L} {\rho_{l} u_{l0} } } \right)}} \le 1{\kern 1pt}$. This is the feasible region of parameter k, and the lower bound and upper bound of k is 0 and 1 respectively. We can solve model (A3) by varying k with fixed step-lengths and solving each resulting linear program, then we compare the solutions yielded from different ks and find the best one (Fig. 3).

Fig. 3

Network with only shared unsplittable links in-between two stages

Practically, this model can be applied to a two-stage structure with only undesirable outputs from the first stage, and such undesirable outputs should be minimized to improve substage efficiency. One way of dealing with these undesirable outputs is modelling them as inputs (Hailu and Veeman 2001) to the first stage. A materialization of this problem, for example, is the balance between industrialization and forestation in a city. There is no sequential relationship between the industrialization stage and the forestation stage. One may be interested in studying the severity of environmental impact of factories and households, and how efficiently forestation could mollify the effect of carbon-dioxide (CO2). Thus, the focus is on evaluating the environmental factor of the factories and households, rather than the production or profit, and the only output from the industrialization stage is an undesirable output: CO2. Since CO2 is used as an input to the second stage, forestation stage, to release oxygen (O2) through photosynthesis, and it is also an undesirable output treated as an input to the first stage, CO2 becomes a shared and unsplittable minimizing link. Figure (4) provides a visualization of this basic two-stage network.

Fig. 4

Basic Industrialization vs. Forestation Problem

If we allow for more output(s) from the first stage and inputs to the second stage besides the shared and unsplittable link, a more complete and comprehensive design of this problem can be achieved, for instance, like the one formulated in Fig. (5). Although this illustration incorporates another possible output from the first stage and one more input into the second stage, it does not violate our setting of the shared and unsplittable performance link model without conventional intermediate measures, as the two new measures are not intermediate measures.

Fig. 5

Extended Industrialization vs. Forestation problem

Shared and unsplittable maximizing measure only

We further propose a shared and unsplittable maximizing measure without the intermediate links as model (B1). $c^{'}$ stands for the weight of the output of constant 1 in the second stage.

$$ \begin{aligned} \tilde{\varPsi }_{0}^{*} = & \hbox{min} \left( {\alpha \cdot \frac{{\sum\nolimits_{i = 1}^{m} {v^{'}_{i} x_{i0} } }}{{\sum\nolimits_{q = 1}^{Q} {\rho^{'}_{q} \varOmega_{q0} } }} + (1 - \alpha ) \cdot \frac{{c^{'} }}{{\sum\nolimits_{r = 1}^{s} {\mu^{'}_{r} y_{r0} } + \sum\nolimits_{q = 1}^{Q} {\rho^{'}_{q} \varOmega_{q0} } }}} \right) \\ & {\text{s}} . {\text{t}} . { }\frac{{\sum\nolimits_{i = 1}^{m} {v^{'}_{i} x_{ij} } }}{{\sum\nolimits_{q = 1}^{Q} {\rho^{'}_{q} \varOmega_{qj} } }} \ge 1,j = 1,2, \ldots ,n, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{c^{'} }}{{\sum\nolimits_{r = 1}^{s} {\mu^{'}_{r} y_{rj} } + \sum\nolimits_{q = 1}^{Q} {\rho^{'}_{q} \varOmega_{qj} } }} \ge 1,j = 1,2, \ldots ,n, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu_{r}^{'} ,\nu_{i}^{'} ,{\kern 1pt} {\kern 1pt} \rho_{q}^{'} ,c^{'} \ge \varepsilon . \\ \end{aligned} $$

(B1)

By introducing two parameters h and k, we have model (B2) to conduct the Charnes and Cooper’s transformation twice, and $\tilde{c} = kc = ktc^{'}$ in model (B2).

$$ \begin{aligned} \tilde{\varPsi }_{0}^{*} = & \hbox{min} \left( {\alpha \cdot {\kern 1pt} \sum\nolimits_{i = 1}^{m} {v_{i} x_{i0} } + (1 - \alpha ) \cdot \tilde{c}} \right) \\ & {\text{s}} . {\text{t}} . { }\frac{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} } }}{{\sum\nolimits_{q = 1}^{Q} {\rho_{q} \varOmega_{qj} } }} \ge 1,j = 1,2, \ldots ,n{\kern 1pt} , \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\tilde{c}}}{{k(\sum\nolimits_{r = 1}^{s} {\mu_{r} y_{rj} } + \sum\nolimits_{q = 1}^{Q} {\rho_{q} \varOmega_{qj} } )}} \ge 1,j = 1,2, \ldots ,n, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\nolimits_{q = 1}^{Q} {\rho_{q} \varOmega_{q0} } = 1, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k(\sum\nolimits_{r = 1}^{s} {\mu_{r} y_{r0} } + \sum\nolimits_{q = 1}^{Q} {\rho_{q} \varOmega_{q0} } ) = 1, \\ & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu_{r} ,\nu_{i} ,{\kern 1pt} {\kern 1pt} \rho_{q} ,\tilde{c} \ge \varepsilon .{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ \end{aligned} $$

(B2)

Similarly, we should calculate the feasible region of k. Since it is set as $\sum\nolimits_{q = 1}^{Q} {\rho_{q} \varOmega_{q0} } = 1$ and $\sum\nolimits_{r = 1}^{s} {\mu_{r} y_{r0} } \ge 0$ in model (B2), it is inferred that $0 \le {\kern 1pt} k = \frac{1}{{\left( {\sum\nolimits_{i = 1}^{m} {v_{i} x_{i0} } + \sum\nolimits_{q = 1}^{Q} {\rho_{q} \varOmega_{q0} } } \right)}} \le 1{\kern 1pt}$. Then we can obtain the lower bound and upper bound of k as 0 and 1 respectively. This means the feasible region of parameter k is [0, 1]. We can solve model (B2) through linear programming by varying k with fixed step-lengths, then compare and find the best solution.

An embodiment of this shared and unsplittable measure is rainfall, particularly in regions with drought. The severe lack of precipitation could increase the likelihood of wildfires, jeopardize crops and livestock and lead to famine, and impair the region’s economy. Generally speaking, precipitation is an event that occurs naturally, but in countries and regions that are prone to drought, artificial rainmaking provides an alternative solution. One commonly used method of rainmaking is asphalt coating (Black and Tarmy, 1963), which initiates rising convection currents through surface heating. We present an example that aims to appraise the rainfall irrigation efficiencies of the nature and artificial rainmaking, using the proposed structure with the shared and unsplittable maximizing link but without intermediate measures, as demonstrated in Fig. (6). In this case, the shared and unsplittable measure, rainfall, should be maximized for both stages.

Fig. 6

Artificial Rainmaking vs. Natural Rainfall Problem