Hybrid Dynamic Network Data Envelopment Analysis

Conventional DEAmodelsmake no hypothesis concerning the internal operations in a static situation. To open the “black box” and work with dynamic assessment issues synchronously, we put forward a hybrid model for evaluating the relative efficiencies of a set of DMUs over an observed time period with a composite of network DEA and dynamic DEA.We vertically deal with intermediate products between divisionswith assignable inputs in the network structure and, horizontally, we extend network structure bymeans of a dynamic pattern with unrelated activities between two succeeding periods.The hybrid dynamic network DEAmodel proposed in this paper enables us to (i) pry into the internal operations of DEA by another network structure, (ii) obtain dynamic change of period efficiency, and (iii) gain the overall dynamic efficiency of DMUs over the entire observed periods. We finally illustrate the calculation procedure of the proposed approach by a numerical example.


Introduction
Data envelopment analysis (DEA), originated by Charnes, Cooper, and Rhodes, is a nonparametric approach for measuring the relative efficiency of homogeneous decision making units (DMUs), which use similar inputs to produce similar outputs [1].DEA evaluates the efficiency of each DMU relative to an estimated production possibility frontier constructed by all DMUs.Traditional DEA approach is based on the thinking of "black box, " which requires no assumption on the appearance of the frontier surface and it makes no hypothesis concerning the internal operations of a DMU.However, the actual transformation process is generally not modeled explicitly.
Network DEA allows us to look into these boxes and to evaluate organizational performance and its component performance (see, e.g., [2][3][4][5]).The development of network DEA model has come through four phases as follows: Firstly, Färe et al. [6] studied allocation of farmland to various crops, which allows for allocation of a (fixed) factor or input among alternative uses.This general structure could also be used to introduce allocation of a budget or allocation of resources across units or branches.This is the rudiment of network DEA model.Secondly, Löthgren and Tambour [7] proposed a kind of network model that allows inclusion of customer satisfaction in efficiency and productivity measures.The network consists of a production node and a consumption node and offers flexibility in modelling the production and consumption process where a firm-specific allocation of input resources to production and customer oriented activities is allowed.In this phase, intermediate products were emphasized.Thirdly, Färe and Grosskopf [8] suggested dynamic DEA model which considered time factor.The dynamic DEA model introduced here is used to study the dynamic efficiency of APEC (Asian-Pacific Economic Community) countries.The network formulation is a dynamic DEA model in which some outputs at period  are inputs in the next period,  + 1.However, the third phase ignores the structure of network itself which is discussed in the second phase.Thus, it is a simple network model with time parameter, but not real dynamic network DEA model.Fourthly, Tone and Tsutsui [9] dealt with intermediate products formally to evaluate divisional efficiencies along with the overall efficiency of DMUs by a slack-based measure.
They handled any network structure in which nodes were connected and intermediates were directed.
Measurement of intertemporal efficiency change has long been an interesting subject of concern in DEA (called dynamic DEA).The dynamic DEA was explored to assess DMUs performance from a long-term perspective using carry-over variables.The dynamic DEA originated by Färe et al. [6] is the first innovative algorithm for dealing with interconnecting activities.It was further developed by several authors from different perspectives.In recent years, Chen and van Dalen [10] developed a dynamic DEA that incorporates dynamic factors via a linear parametric formulation.Soleimani-damaneh [11] provided a recursive enumeration algorithm for solving the nonconvex dynamic DEA models.Soleimani-damaneh [12] also utilized the envelopment models for estimating returns to scale in dynamic DEA.To cope with long time point of view, Tone and Tsutsui [13] put forward a dynamic DEA model that adds carry-over activities into the model to measure period specific efficiency based on the long time optimization during the whole period.Then, Tone and Tsutsui extended the model by a combination of network DEA and dynamic DEA [14].However, sub-DMUs of their hybrid dynamic network DEA were connected by links by means of chain network structure.The intermediate products between sub-DMUs were neglected and inputs cannot be allocated to them.
Hence, this paper is a hybrid composite of dynamic and network DEA.However, our dynamic network DEA model deals with network structure with intermediate products between any sub-DMUs and copes with assignable inputs over observed dynamic periods.The hybrid model proposed in this paper enables us to open the "black box" to reveal the internal connections of DMUs as well as realizing dynamic evaluation of network DEA over the entire observed periods.The rest of this paper unfolds as follows.In Section 2, we formulate network DEA model with intermediates and assignable inputs.In Section 3, we propose a hybrid model called dynamic network DEA (DNDEA), combining dynamic DEA with network DEA.We also construct a new DEA unit to obtain the dynamic efficiency of network DEA.In Section 4, we exhibit an illustrative example to explain the operational process of the hybrid model.Some conclusions are presented in Section 5.

Introduction of DEA.
DEA is a widely used operational programming approach for comparing the multiple inputs and outputs of a set of homogenous DMUs by measuring their relative efficiency.Suppose that there are  DMUs to be evaluated where each DMU  ,  = 1, 2, . . ., , consumes  inputs,   ( = 1, 2, . . ., ), producing  outputs,   ( = 1, 2, . . ., ).The structure of conventional DEA can be viewed as a "black box" described in Figure 1.
The CCR (Charnes, Cooper, and Rhodes) model for evaluating the technical input-efficiency of  0 th DMU (DMU 0 ) under constant returns to scale (CRS) is represented as As an extension of the "black box" model above, Färe and Grosskopf [8] introduced the network DEA model with chain links and nonassignable inputs illustrated in Figure 2. Tone and Tsutsui [13] developed the network structure by applying it in the dynamic network DEA using a slack-based measure.When we take intermediate products into consideration in DEA model, we can formulate network DEA as follows:

Network
The dual model of network DEA above can be transformed as (3), where  is the network DEA efficiency of DMU 0 , which reflects the overall efficiency of network DEA with intermediate products. is proportional to the performance of the evaluated DMU: (3)

Hybrid Dynamic Network DEA
) ) .We can describe the structure of DN-DEA in the tth time span as in Figure 4.
The DN-DEA model can be formed as follows: (5)

Input node
Output node The dual model of the above formulation can be transformed as (6), where  () is the network DEA efficiency of DMU 0 in the tth time span ( = 1, 2, . . ., ), showing dynamic efficiency of DN-DEA:

Dynamic Evaluation of DN-DEA Efficiencies. 𝜃 (𝑡)
is the DN-DEA efficiency of DMU  in the tth time span.We define  ()   as the DN-DEA efficiency matrix, which means efficiencies of DN-DEA model in discrete time spans.The DN-DEA efficiency matrix can be formulated as follows: The column vector of  ()  reflects the dynamic efficiency of DN-DEA over the whole period of each DMU.Hence, measuring the dynamic network efficiency of DN-DEA is a typical multiple attributes decision making issue essentially.In addition to DEA model, many scholars have put forward some mathematical approaches to solve the problem, for example, analytic hierarchy process, grey correlation analysis, and fuzzy comprehensive evaluation, which have their own advantages and disadvantages.In this work, we apply DEA approach for the "second round" on the basis of hybrid DN-DEA model formulated above.In view of the structure of DN-DEA efficiency matrix to be evaluated,  ()   represents the network DEA efficiency of DMU  in the tth time span.However, the matrix itself has no network structure inside.Thus, we can build a new "black box" DEA to assess overall dynamic efficiency for DN-DEA.For DEA model, input and output index have their normal requirements: input is inversely proportional to the DEA efficiency; output is proportional to the DEA efficiency.From the analysis above we know that  ()   is proportional to the performance of DMU  in period ; thus we can treat it as output of the new DEA.Moreover, input can be formed as 1 ×  dimensional vector.In order to operate easily, we treat it as 1 ×  dimensional unit vector.The structure diagram of the new DEA is displayed as shown in Figure 5.
From the new constructed DEA as shown in Figure 5, we can build the dynamic evaluation model of DN-DEA efficiency, which can be formulated as follows:  where ê = (1, 1, . . ., 1)  ∈  1 and  = (1, 1, . . ., 1)  ∈   are unit vectors,  ( > 0) is a non-Archimedean infinitesimal, and  − and  + are slacks of DMUs. is the dynamic efficiency of DMU 0 all over  period in DN-DEA model.We can further transform the model to

Numerical Example
In this section, a numerical example is presented to illustrate DN-DEA model.To this end, ten DMUs with one input node, one output node, and three sub-DMUs are considered over four time periods.We can describe the DN-DEA structure of this example in Figure 6.It is worth mentioning that there are intermediate products between sub-DMU 1 and sub-DMU 2 and between sub-DMU 2 and sub-DMU 3 .The data of inputs, outputs, and intermediate products for these ten DMUs over the observed four time periods have been listed in Tables 1-4.By running (6), we can obtain the network DEA efficiency of ten DMUs in the tth time span ( = 1, 2, 3, 4), also  (10) Figure 7 reflects the dynamic fluctuation of network DEA efficiency for each DMU over the four time periods, from which we can conclude that the network DEA efficiency of the observed time periods vibrates irregularly.We apply model (9) for the sake of evaluating the overall dynamic network efficiency of DN-DEA on the basis of the DEA structure proposed in Figure 5.The overall dynamic network efficiency and ranking sequence of DN-DEA can be finally provided in Table 5, which can be compared in Figure 8. Table 5 also provides slacks from which we can get improved

Conclusions
In

Figure 3 :
Figure 3: The structure of network DEA with intermediate products and assignable inputs.

Figure 4 :
Figure 4: The structure of DN-DEA in one time span.

Figure 6 :
Figure 6: DN-DEA with 3 sub-DMUs in the one time span.
The input and output of DMU  are  0  and    , respectively.Inputs of sub-DMUs received from input node are  0  and  0  , which are assignable, and outputs of sub-DMUs exported to output node are    and    .There are input and output values between sub-DMUs which we call intermediate products; X DEA with Intermediate Products.In this paper, we propose a network structure with intermediate products between any two sub-DMUs.And the inputs can be assigned to sub-DMUs as well.We suppose that there are  homogeneous DMUs which involve  sub-DMUs   ( = 1, 2, . . ., ). is defined as input from   to   ; Ỹ  is defined as output from   to   , where ,  ∈ {1, 2, . . ., },  ̸ = .We can deduce that X  = Ỹ  and X  = Ỹ  .Figure 3 illustrates the structure of network DEA with intermediate products and assignable inputs.

Table 5 :
Evaluation results of DN-DEA.DEA such as carry-over dynamic type or continuous one may need diverse models for measuring corresponding dynamic efficiency of DN-DEA.The dynamic evaluation results of DN-DEA will depend on the constructed dynamic model related to the defined dynamic structure.The "black box" DEA model set up above adapts to the structure of DN-DEA efficiency matrix.However, other DEA models give more importance to more recent years.For instance, superefficiency DEA will help to solve unsatisfactory differentiation of overall dynamic network efficiencies.Further work based on the framework proposed in this paper could extend the hybrid dynamic network model to include other dynamic patterns as well as different network structures. network