COORDINATION EFFICIENCY FOR GENERAL TWO-STAGE NETWORK SYSTEM

. Two-stage network data envelopment analysis (DEA) is widely used to evaluate efficiency of different organizations with multiple operations processes or hierarchical structures. Although existing two-stage network DEA assumes two-stage systems resolve the inherent conflicts between two stages, the coordination effect between the two stages is usually ignored. Recently, the relation of two-stage network DEA to traditional “black box” DEA has been studied from the perspective of system coordination. A coordination efficiency was defined and measured by a DEA-based approach based on simple two-stage network systems. In this paper, we propose an extended DEA-based approach for measuring the coordination efficiency for general two-stage network systems. The paper shows that the coordination efficiency based on the multiplier DEA and envelopment DEA approaches is equivalent to each other under both constant returns to scale (CRS) and variable returns to scale (VRS) assumptions. The proposed approach is verified via two numerical examples finally.


Introduction
Data envelopment analysis (DEA) is a mathematical programming approach for measuring the relative efficiency of entities using multiple inputs to produce multiple outputs [3].The entities are usually called decision making units (DMUs).After more than 40 years of development, DEA has been significantly extended and widely applied to various areas [8,10,24,25].In recent years, DEA has been used as a data-driven tool for balanced benchmarking [27].Most recently, it was advocated as data enabled analytics [31].
Traditional DEA models regard each DMU as a black box and the composite efficiency index is evaluated according to the inputs and outputs of the black box.This consideration makes us estimate the efficiencies of DMUs conveniently.However, the intermediate links and the coordination effects among the sub-processes are ignored.Network DEA [11,13] opens the black boxes and estimate the overall efficiencies of the black boxes themselves and the divisional efficiencies of the sub processes embedded in them.The estimation of divisional efficiencies can help decision maker find out where the inefficiencies come from more clearly.
Two-stage network DEA [17,22], which is a special case of network DEA, is based on the basic series network structure.Kao and Hwang [17] deemed that either constraint of the two sub-stages should also constrain the other sub-stage and the whole production system.Following this idea, they proposed a rational efficiency estimation and decomposition model that regarded the overall efficiency as the product of divisional efficiencies.Note that the weights in terms of the intermediate products were assumed the same [17,22].
Inspired by Kao and Hwang [17] and Liang et al. [22], a lot of interesting studies have enriched the methodologies for two-stage network DEA.Some studies proposed to estimate the overall efficiency and divisional efficiencies based on radial measures [5,9,20].Kao and Liu [18] further extended their discussions to cross efficiency measurement and decomposition.Based on non-radial measures, especially slacks-based measure (SBM) [28], some other studies developed their approaches for basic two-stage network system [7], Although the SBMbased models in Kao [15,16] are proposed for closed series production systems and general network system, they are also suitable for two-stage network systems.Besides the efficiency measurement, a few studies investigated frontier projection and duality [7,23].See reviews and references by Kao [13,14] for more comprehensive discussions on two-stage network DEA.
Chen et al. [6] claimed that traditional DEA and two-stage network DEA were two separate and different methods.Although two-stage network DEA solves the conflicts between two sub-stages, the relationship between traditional DEA and two-stage network DEA is usually neglected.As discussed in the existing studies [2,26], the coordination effect widely exists in two-stage systems [12] such as supply chain.However, the coordination effect in two-stage network DEA is seldom considered although Li [19] mentioned the coordination in two-stage network DEA.Zhao et al. [30] firstly defined the coordination efficiency and proposed a DEA-based approach for measuring the coordination effect of a special simple two-stage DEA system, i.e., two-stage supply chain system [4,21,29].
In this paper, we further extend the work of Zhao et al. [30] to general two-stage network system.Particularly, we propose both multiplier and envelopment DEA models to calculate the coordination efficiency.It shows that the multiplier DEA approach is equivalent to the envelopment DEA one for general two-stage network system.In addition, the conclusion is tenable under both constant returns to scale (CRS) and variable returns to scale (VRS) [1].The proposed approach is illustrated by two numerical examples.
The reminder of the paper is presented as follows.Section 2 roughly introduces the proposed approach in Zhao et al. [30].Section 3 proposes multiplier DEA models for estimating the coordination efficiency while Section 4 proposes envelopment DEA models for doing so.The equivalence between the multiplier and envelopment DEA approaches is discussed in Section 5. Section 6 concludes the paper.
2. The approach in Zhao et al. [30] In this section, we roughly introduces the approach in Zhao et al. [30] to measure the coordination efficiency under basic two-stage DEA system.Assume there are  DMUs and each DMU is a simple two-stage network system as shown in Figure 1.Each DMU  uses inputs X  ( 1 , . . .,   ) to produce intermediate products Z  ( 1 , . . .,   ) in the first stage, then the intermediate products Z  ( 1 , . . .,   ) is used to produce outputs Y  ( 1 , . . .,   ) in the second stage.
In Zhao et al. [30], coordination efficiency is defined as the ratio of the system efficiency with coordination to the system efficiency without coordination.Under the VRS assumption, let  VRS  be the system efficiency with coordination for DMU  .It can be obtained as follows: where  2 VRS  is stage 2's VRS efficiency calculated as follows: If   and   is deleted from the above models, we can obtain the coordination efficiency under CRS.For example, if there are no variables   and   in model (1), then it measures the system efficiency with coordination under CRS, which is exactly the overall efficiency in Kao and Hwang [17].See Zhao et al. [30] for more details.

CRS case
In the following, we use the same definition of coordination efficiency in Zhao et al. [30], i.e., the ratio of the system efficiency with coordination to the one without coordination.Similar to the approach for the simple two-stage network system in Section 2, the multiplier DEA model for estimating the system efficiency with coordination ẼCRS  of DMU  under CRS is as follows: where   ,   ,   ,   , and  ℎ are weight variables (i.e., multipliers) in terms of  2  ,   ,  1  ,  1  , and  2 ℎ , respectively.It should be noted that model ( 5) is exactly model (11.13) of Kao [14] if we neglect the non-Archimedean number constrained to the weight variables.
For the general two-stage network system, due to the existence of the outputs ( 1  ,  = 1, 2, . . ., ) in the first stage and the inputs ( 2 ℎ , ℎ = 1, 2, . . ., ) in the second stage, we cannot obtain the system efficiency without coordination (i.e., πCRS  ) directly by following the same approach in the previous section.Hence, the coordination efficiency ρCRS The difference between models ( 5) and ( 6) is the inclusion of Ẽ1

VRS case
The VRS DEA model for estimating the system efficiency with coordination of DMU  is as follows: where Ẽ2 VRS  is the VRS efficiency of the second stage.Ẽ1 VRS

𝑜
is the minimum reduction ratio of input  1  , which could be estimated by the following model:

Coordination efficiency for general two-stage network system based on envelopment DEA models
In this section, we further propose the envelopment DEA model for measuring the coordination efficiency in general two-stage network system.Likewise, both CRS and VRS cases are considered.

Equivalence between multiplier and envelopment DEA models
In this section, we show the equivalence between multiplier and envelopment DEA models proposed in Sections 3 and 4. Both CRS and VRS cases are considered.

CRS case
Models ( 5)-( 7) can be converted to their linear ones via Charnes-Cooper transformation as follows: It should be noted that Ẽ2 CRS  and Ẽ2 CRS env  are CCR efficiency of the second stage based on multiplier and envelopment models, respectively.Obviously, they are equal, i.e., Ẽ2 CRS  = Ẽ2 CRS env  .It can be found that models ( 19)-( 21) are equivalent dual models of model (11), model (14), and model (12), respectively.Hence, the system efficiency with coordination ẼCRS env .It indicates that under the CRS assumption the multiplier DEA approach is equivalent to the envelopment DEA approach for general two-stage network system.

VRS case
Similarly, models ( 8)-( 10) can be converted to their linear ones via Charnes-Cooper transformation as follows: .Based on the above findings, it shows that under the VRS assumption the multiplier DEA approach is equivalent to the envelopment DEA approach for general two-stage network system.

Numerical example
To verify the proposed approach in the previous section, we use a numerical example to illustrate in this section.Considering that the VRS case is similar to the CRS case, we only show the results under the CRS.
Particularly, we use two data sets in Liang et al. [21] and Chen et al. [4], respectively.The results are shown in Tables 1 and 2. As is shown in the tables, for general two-stage network system, the efficiency estimates (system efficiency and coordination efficiency) based on multiplier and envelopment DEA approach are totally the same.

Conclusions
Although two-stage network DEA and traditional DEA are widely studied, the relation between them is seldom discussed.Zhao et al. [30] estimated the coordination effect for the simple two-stage network system.Their proposed coordination efficiency builds a link between two-stage network DEA and traditional DEA, which is usually neglected by the existing studies, especially supply chain DEA.In this paper, we further extend the research of Zhao et al. [30] on coordination efficiency to general two-stage network system.
Particularly, we develop both multiplier and envelopment DEA approaches for estimating coordination efficiency based on both CRS and VRS assumptions.As is shown in Chen et al. [6], for network DEA the multiplier and envelopment DEA models were two different approaches.However, we still show that under the general two-stage network system the multiplier DEA approach for measuring the coordination efficiency is equivalent to the envelopment DEA approach under both CRS and VRS cases.Two numerical examples are used to verify the proposed approach in this paper.
As pointed out by Zhao et al. [30], the traditional "black box" efficiency is unobtainable for a two-stage network system such as a supply chain with two members.Some endeavors or sacrifices must be made in order to coordinate the two-stage system, which inevitably gives rise to an efficiency loss [30].As a result, the coordination efficiency defined in Zhao et al. [30] and this paper could be used to trace any potential efficiency losses and quantify contributions or responsibilities of each division in the two-stage network system.
For future research, it may use the proposed approach to real applications with good insights.In this paper, we define the coordination efficiency under general two-stage network system, which is still a series network structure.Future research may apply the proposed approach to general network system.Moreover, whether the finding in this paper holds for efficiency decomposition of the coordination efficiency deserves for further research.In short, we hope that our study could attract the community's more attention on the interesting characteristics of network DEA, one of the most active and exciting areas currently.

𝑜
and coordination efficiency ρCRS env  based on envelopment DEA models are equal to ẼCRS  and ρCRS  based on multiplier DEA models, i.e., ẼCRS env , and   are weight variables (i.e., multipliers) in terms of   ,   , and   , respectively.  and   are free variables for modelling VRS.

Table 1 .
Result comparison based on data set in Liang et al. [21].

Table 2 .
Result comparison based on data set in Chen et al. [4].