# Hybrid Dynamic Network Data Envelopment Analysis

Authors: Ling Li; Fengshan Wang
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

Conventional DEA models make 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 divisions with assignable inputs in the network structure and, horizontally, we extend network structure by means of a dynamic pattern with unrelated activities between two succeeding periods. The hybrid dynamic network DEA model 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.


---

## Body

## 1. 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., [2–5]). 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 t are inputs in the next period, t+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 Section2, 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.


## 2. Network DEA Model

### 2.1. 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 aren DMUs to be evaluated where each DMUj, j=1,2,…,n, consumes m inputs, xij (i = 1,2,…,m), producing s outputs, yrj(r=1,2,…,s). The structure of conventional DEA can be viewed as a “black box” described in Figure 1.Figure 1
The structure of conventional DEA.The CCR (Charnes, Cooper, and Rhodes) model for evaluating the technical input-efficiency ofj0th DMU (DMU0) under constant returns to scale (CRS) is represented as(1)PCCR:max⁡h0=∑r=1sμryr0∑i=1mωixi0=1∑i=1mωixij-∑r=1sμryrj≥0;j=1,2,…,nωi,μr≥0,∀i,r.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.Figure 2
The structure of network DEA with chain links and nonassignable inputs.


### 2.2. Network 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 aren homogeneous DMUs which involve s sub-DMUs Pl (l=1,2,…,s). The input and output of DMUj are Xj0 and Yje, respectively. Inputs of sub-DMUs received from input node are Xj0l and Xj0r, which are assignable, and outputs of sub-DMUs exported to output node are Yjle and Yjre. There are input and output values between sub-DMUs which we call intermediate products; X~jrl is defined as input from Pr to Pl; Y~jlr is defined as output from Pl to Pr, where r, l∈{1,2,…,s}, r≠l. We can deduce that X~jrl=Y~jrl and X~jlr=Y~jlr. Figure 3 illustrates the structure of network DEA with intermediate products and assignable inputs.Figure 3
The structure of network DEA with intermediate products and assignable inputs.When we take intermediate products into consideration in DEA model, we can formulate network DEA as follows:(2)PNetwork:max⁡μeTY0e=VPω0lTXj0l+∑r=1,r≠lsω~rlTX~jrl-μleTYjle-∑r=1,r≠lsμ~lrTY~jlr≥0,j=1,2,…,n;l=1,2,…,s;ω0TX00=1,ω0≥ω0lμ~rl≥ω~rl,ω0l≥0,ω~rl≥0,μe≥0,μle≥0,μ~rl≥0.The dual model of network DEA above can be transformed as (3), where θ is the network DEA efficiency of DMU0, which reflects the overall efficiency of network DEA with intermediate products. θ is proportional to the performance of the evaluated DMU:(3)DNetwork:min⁡θ=VD∑l=1sX00l=θX00,∑j=1nλjlXj0l+S0l-=X00l,∑j=1nλjlX~jrl+S~rl-=X~0rl∑j=1nλjlYjle-Sle+=Y0le,∑j=1nλjlY~jlr-S~lr+=Y~0lr,l,r∈1,2,…,s,l≠rλj≥0,j=1,2,…,n;X00l≥0,S0l-≥0,S~rl-≥0,Sle+≥0,S~lr+≥0.


## 2.1. 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 aren DMUs to be evaluated where each DMUj, j=1,2,…,n, consumes m inputs, xij (i = 1,2,…,m), producing s outputs, yrj(r=1,2,…,s). The structure of conventional DEA can be viewed as a “black box” described in Figure 1.Figure 1
The structure of conventional DEA.The CCR (Charnes, Cooper, and Rhodes) model for evaluating the technical input-efficiency ofj0th DMU (DMU0) under constant returns to scale (CRS) is represented as(1)PCCR:max⁡h0=∑r=1sμryr0∑i=1mωixi0=1∑i=1mωixij-∑r=1sμryrj≥0;j=1,2,…,nωi,μr≥0,∀i,r.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.Figure 2
The structure of network DEA with chain links and nonassignable inputs.


## 2.2. Network 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 aren homogeneous DMUs which involve s sub-DMUs Pl (l=1,2,…,s). The input and output of DMUj are Xj0 and Yje, respectively. Inputs of sub-DMUs received from input node are Xj0l and Xj0r, which are assignable, and outputs of sub-DMUs exported to output node are Yjle and Yjre. There are input and output values between sub-DMUs which we call intermediate products; X~jrl is defined as input from Pr to Pl; Y~jlr is defined as output from Pl to Pr, where r, l∈{1,2,…,s}, r≠l. We can deduce that X~jrl=Y~jrl and X~jlr=Y~jlr. Figure 3 illustrates the structure of network DEA with intermediate products and assignable inputs.Figure 3
The structure of network DEA with intermediate products and assignable inputs.When we take intermediate products into consideration in DEA model, we can formulate network DEA as follows:(2)PNetwork:max⁡μeTY0e=VPω0lTXj0l+∑r=1,r≠lsω~rlTX~jrl-μleTYjle-∑r=1,r≠lsμ~lrTY~jlr≥0,j=1,2,…,n;l=1,2,…,s;ω0TX00=1,ω0≥ω0lμ~rl≥ω~rl,ω0l≥0,ω~rl≥0,μe≥0,μle≥0,μ~rl≥0.The dual model of network DEA above can be transformed as (3), where θ is the network DEA efficiency of DMU0, which reflects the overall efficiency of network DEA with intermediate products. θ is proportional to the performance of the evaluated DMU:(3)DNetwork:min⁡θ=VD∑l=1sX00l=θX00,∑j=1nλjlXj0l+S0l-=X00l,∑j=1nλjlX~jrl+S~rl-=X~0rl∑j=1nλjlYjle-Sle+=Y0le,∑j=1nλjlY~jlr-S~lr+=Y~0lr,l,r∈1,2,…,s,l≠rλj≥0,j=1,2,…,n;X00l≥0,S0l-≥0,S~rl-≥0,Sle+≥0,S~lr+≥0.


## 3. Hybrid Dynamic Network DEA

### 3.1. Dynamic Network DEA Model

Network DEA has the structure with intermediate products between input node and output node, which shows a network form. However, it is a static modality essentially. On the basis of decision making space of network DEA, we introduce the time parametert to formulate three-dimensional decision making issue, called dynamic network DEA (DN-DEA). During the tth time span, the input of DMUj is Xjt0 and the output of DMUj is Yjte, which can be formulated as (4)Xjt0=x1t0,x2t0,…,xnt0=x11t0⋯x1jt0⋯x1nt0⋯⋯⋯⋯⋯xi1t0⋯xijt0⋯xint0⋯⋯⋯⋯⋯xm1t0⋯xmjt0⋯xmnt0m×n,Yjte=y1te,y2te,…,ynte=y11te⋯y1jte⋯y1nte⋯⋯⋯⋯⋯yk1te⋯ykjte⋯yknte⋯⋯⋯⋯⋯ys1te⋯ysjte⋯ysntes×n.Inputs of sub-DMUs received from input node are Xjt0l and Xjt0r, and outputs of sub-DMUs exported to output node are Yjtle and Yjtre. There are input and output values between sub-DMUs in thetth time span; X~jtrl is defined as input from Pr to Pl; Y~jtlr is defined as output from Pl to Pr, where r, l∈{1,2,…,s}, r≠l, t=1,2,…,T. We can also deduce that X~jtrl=Y~jtrl and X~jtlr=Y~jtlr. We can describe the structure of DN-DEA in thetth time span as in Figure 4.Figure 4
The structure of DN-DEA in one time span.The DN-DEA model can be formed as follows:(5)PDN:max⁡μteTY0te=VPtωt0lTXjt0l+∑r=1,r≠lsω~trlTX~jtrl-μtleTYjtle-∑r=1,r≠lsμ~tlrTY~jtlr≥0,j=1,2,…,n;l=1,2,…,s;t=1,2,⁡…,T,ωt0TX0t0=1,ωt0≥ωt0lμ~trl≥ω~trl,ωt0l≥0,ω~trl≥0,μte≥0,μtle≥0,μ~trl≥0.The dual model of the above formulation can be transformed as (6), where θ(t) is the network DEA efficiency of DMU0 in thetth time span (t=1,2,…,T), showing dynamic efficiency of DN-DEA:(6)DDN:min⁡θt=VDt∑l=1sX0t0l=θtX0t0∑j=1nλjtlXjt0l+St0l-=X0t0l,∑j=1nλjtlX~jtrl+S~trl-=X~0trl∑j=1nλjtlYjtle-Stle+=Y0tle,∑j=1nλjtlY~jtlr-S~tlr+=Y~0lr,l,r∈1,2,…,s,l≠r;λjt≥0,j=1,2,…,n;t=1,2,…,T,X0t0l≥0,St0l-≥0,S~trl-≥0,Stle+≥0,S~tlr+≥0.


### 3.2. Dynamic Evaluation of DN-DEA Efficiencies

θ
j
(
t
) is the DN-DEA efficiency of DMUj in thetth time span. We define θj(t) 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:(7)θjt=θ1t,θ2t,…,θnt=θ11⋯θj1⋯θn1⋯⋯⋯⋯⋯θ1t⋯θjt⋯θnt⋯⋯⋯⋯⋯θ1T⋯θjT⋯θnTT×n.The column vector of θj(t) 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, θj(t) represents the network DEA efficiency of DMUj in thetth 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 θj(t) is proportional to the performance of DMUj in period t; thus we can treat it as output of the new DEA. Moreover, input can be formed as 1×n dimensional vector. In order to operate easily, we treat it as 1×n dimensional unit vector. The structure diagram of the new DEA is displayed as shown in Figure 5.Figure 5
The constructed DEA for measuring dynamic efficiency of DN-DEA.From the new constructed DEA as shown in Figure5, we can build the dynamic evaluation model of DN-DEA efficiency, which can be formulated as follows:(8)DIε:min⁡σ-εe^TS-+eTS+=ξ∑j=1n1×λj+S-=σ×1∑j=1nθjtλj-S+=θ0tλj≥0,j=1,2,…,n;S-≥0,S+≥0,where e^=1,1,…,1T∈E1 and e = 1,1,…,1T∈ET are unit vectors, ε (ε>0) is a non-Archimedean infinitesimal, and S- and S+ are slacks of DMUs. ξ is the dynamic efficiency of DMU0 all over T period in DN-DEA model. We can further transform the model to (9)DIε:min⁡σ-εe^TS-+eTS+=ξ∑j=1nλj=σ-S-,∑j=1nθjtλj=θ0t+S+λj≥0⁡,j=1,2,…,n,S-≥0,S+≥0.


## 3.1. Dynamic Network DEA Model

Network DEA has the structure with intermediate products between input node and output node, which shows a network form. However, it is a static modality essentially. On the basis of decision making space of network DEA, we introduce the time parametert to formulate three-dimensional decision making issue, called dynamic network DEA (DN-DEA). During the tth time span, the input of DMUj is Xjt0 and the output of DMUj is Yjte, which can be formulated as (4)Xjt0=x1t0,x2t0,…,xnt0=x11t0⋯x1jt0⋯x1nt0⋯⋯⋯⋯⋯xi1t0⋯xijt0⋯xint0⋯⋯⋯⋯⋯xm1t0⋯xmjt0⋯xmnt0m×n,Yjte=y1te,y2te,…,ynte=y11te⋯y1jte⋯y1nte⋯⋯⋯⋯⋯yk1te⋯ykjte⋯yknte⋯⋯⋯⋯⋯ys1te⋯ysjte⋯ysntes×n.Inputs of sub-DMUs received from input node are Xjt0l and Xjt0r, and outputs of sub-DMUs exported to output node are Yjtle and Yjtre. There are input and output values between sub-DMUs in thetth time span; X~jtrl is defined as input from Pr to Pl; Y~jtlr is defined as output from Pl to Pr, where r, l∈{1,2,…,s}, r≠l, t=1,2,…,T. We can also deduce that X~jtrl=Y~jtrl and X~jtlr=Y~jtlr. We can describe the structure of DN-DEA in thetth time span as in Figure 4.Figure 4
The structure of DN-DEA in one time span.The DN-DEA model can be formed as follows:(5)PDN:max⁡μteTY0te=VPtωt0lTXjt0l+∑r=1,r≠lsω~trlTX~jtrl-μtleTYjtle-∑r=1,r≠lsμ~tlrTY~jtlr≥0,j=1,2,…,n;l=1,2,…,s;t=1,2,⁡…,T,ωt0TX0t0=1,ωt0≥ωt0lμ~trl≥ω~trl,ωt0l≥0,ω~trl≥0,μte≥0,μtle≥0,μ~trl≥0.The dual model of the above formulation can be transformed as (6), where θ(t) is the network DEA efficiency of DMU0 in thetth time span (t=1,2,…,T), showing dynamic efficiency of DN-DEA:(6)DDN:min⁡θt=VDt∑l=1sX0t0l=θtX0t0∑j=1nλjtlXjt0l+St0l-=X0t0l,∑j=1nλjtlX~jtrl+S~trl-=X~0trl∑j=1nλjtlYjtle-Stle+=Y0tle,∑j=1nλjtlY~jtlr-S~tlr+=Y~0lr,l,r∈1,2,…,s,l≠r;λjt≥0,j=1,2,…,n;t=1,2,…,T,X0t0l≥0,St0l-≥0,S~trl-≥0,Stle+≥0,S~tlr+≥0.


## 3.2. Dynamic Evaluation of DN-DEA Efficiencies

θ
j
(
t
) is the DN-DEA efficiency of DMUj in thetth time span. We define θj(t) 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:(7)θjt=θ1t,θ2t,…,θnt=θ11⋯θj1⋯θn1⋯⋯⋯⋯⋯θ1t⋯θjt⋯θnt⋯⋯⋯⋯⋯θ1T⋯θjT⋯θnTT×n.The column vector of θj(t) 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, θj(t) represents the network DEA efficiency of DMUj in thetth 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 θj(t) is proportional to the performance of DMUj in period t; thus we can treat it as output of the new DEA. Moreover, input can be formed as 1×n dimensional vector. In order to operate easily, we treat it as 1×n dimensional unit vector. The structure diagram of the new DEA is displayed as shown in Figure 5.Figure 5
The constructed DEA for measuring dynamic efficiency of DN-DEA.From the new constructed DEA as shown in Figure5, we can build the dynamic evaluation model of DN-DEA efficiency, which can be formulated as follows:(8)DIε:min⁡σ-εe^TS-+eTS+=ξ∑j=1n1×λj+S-=σ×1∑j=1nθjtλj-S+=θ0tλj≥0,j=1,2,…,n;S-≥0,S+≥0,where e^=1,1,…,1T∈E1 and e = 1,1,…,1T∈ET are unit vectors, ε (ε>0) is a non-Archimedean infinitesimal, and S- and S+ are slacks of DMUs. ξ is the dynamic efficiency of DMU0 all over T period in DN-DEA model. We can further transform the model to (9)DIε:min⁡σ-εe^TS-+eTS+=ξ∑j=1nλj=σ-S-,∑j=1nθjtλj=θ0t+S+λj≥0⁡,j=1,2,…,n,S-≥0,S+≥0.


## 4. 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 Figure6. It is worth mentioning that there are intermediate products between sub-DMU1 and sub-DMU2 and between sub-DMU2 and sub-DMU3. 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.Table 1
Evaluation data of DN-DEA (t=1).
DM
U
j
Efficiency
Input
Sub-DMU1
Sub-DMU2
Sub-DMU3
θ
j
1
X
j
0
X
j
01
X
~
j
12
Y
j
1
e
X
j
02
X
~
j
23
Y
j
2
e
X
j
03
Y
j
3
e
1
1
4170
2412
4310
195
1066
722
198
692
910
2
0.76
6349
4662
7710
191
963
414
189
724
492
3
0.79
8046
6395
7939
183
936
510
183
715
510
4
0.83
4679
2522
5533
188
1252
568
177
905
599
5
0.92
5938
4211
5194
177
1012
621
199
715
814
6
0.94
4442
3148
4426
195
891
204
193
403
355
7
0.79
7036
5338
6623
185
956
493
187
742
493
8
1
5122
3606
6490
194
906
681
196
610
858
9
0.90
5740
4253
5285
185
775
289
192
712
355
10
0.95
4680
3087
3839
189
780
580
193
813
630Table 2
Evaluation data of DN-DEA (t=2).
DMU
j
Efficiency
Input
Sub-DMU1
Sub-DMU2
Sub-DMU3
θ
j
1
X
j
0
X
j
01
X
~
j
12
Y
j
1
e
X
j
02
X
~
j
23
Y
j
2
e
X
j
03
Y
j
3
e
1
1
3820
2308
4209
207
932
677
213
580
996
2
0.79
6082
4564
7608
211
840
408
198
678
512
3
0.76
8199
6432
7966
168
997
518
143
770
448
4
0.89
4457
2443
5411
190
1130
550
187
884
625
5
0.87
6144
4321
5233
141
1136
664
178
687
905
6
0.96
4290
3028
4297
212
870
189
197
392
392
7
0.70
8307
6408
6667
176
1102
535
168
797
445
8
1
4929
3503
6370
209
886
656
208
540
967
9
0.93
5583
4105
5123
194
770
248
197
708
377
10
0.91
4867
3134
3899
170
884
598
168
849
628Table 3
Evaluation data of DN-DEA (t=3).
DMU
j
Efficiency
Input
Sub-DMU1
Sub-DMU2
Sub-DMU3
θ
j
1
X
j
0
X
j
01
X
~
j
12
Y
j
1
e
X
j
02
X
~
j
23
Y
j
2
e
X
j
03
Y
j
3
e
1
1
3500
2150
4100
230
884
654
222
466
1033
2
0.80
5762
4406
7499
234
792
385
207
564
549
3
0.77
7879
6274
7857
191
949
495
152
656
485
4
0.90
4137
2285
5302
213
1082
527
196
770
662
5
0.89
5824
4163
5124
164
1088
641
187
573
942
6
0.97
3970
2870
4188
235
822
166
206
278
429
7
0.74
7987
6250
6558
199
1054
512
177
683
482
8
1
4609
3345
6261
232
838
633
217
426
1004
9
0.95
5263
3947
5014
217
722
225
206
594
414
10
0.92
4547
2976
3790
193
836
575
177
735
665Table 4
Evaluation data of DN-DEA (t=4).
DMU
j
Efficiency
Input
Sub-DMU1
Sub-DMU2
Sub-DMU3
θ
j
1
X
j
0
X
j
01
X
~
j
12
Y
j
1
e
X
j
02
X
~
j
23
Y
j
2
e
X
j
03
Y
j
3
e
1
1
4509
2531
4523
178
1202
809
183
776
947
2
0.71
6688
4781
7923
174
1099
501
174
808
529
3
0.74
8385
6514
8152
166
1072
597
168
799
547
4
0.86
5018
2641
5746
171
1388
655
162
989
636
5
0.82
6277
4330
5407
160
1148
708
184
799
851
6
0.90
4781
3267
4639
178
1027
291
178
487
392
7
0.79
7375
5457
6836
168
1092
580
172
826
530
8
0.83
5461
3725
6703
177
1042
768
181
694
895
9
0.80
6079
4372
5498
168
911
376
177
796
392
10
0.85
5019
3206
4052
172
916
667
178
897
667Figure 6
DN-DEA with 3 sub-DMUs in the one time span.By running (6), we can obtain the network DEA efficiency of ten DMUs in thetth time span (t = 1, 2, 3, 4), also summarized in Tables 1–4. Thus, we can gain DN-DEA efficiency matrix as shown in (10)θjt=10.760.790.830.920.940.7910.900.9510.790.760.890.870.960.7010.930.9110.800.770.900.890.970.7410.950.9210.710.740.860.820.900.790.830.800.85.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 information between inefficient DMUs and optimum values by means of projection. We can know that DMU1 performs best and DMU7 displays worst. The ranking sequence shows that DMU1 > DMU8 > DMU6 > DMU9 > DMU10 > DMU5 > DMU4 > DMU2 > DMU3 > DMU7.Table 5
Evaluation results of DN-DEA.
DMU
j
ξ
j
S
1
-
S
1
+
S
2
+
S
3
+
S
4
+
Ranking
1
1
0
0
0
0
0
1
2
0.80
0
0.04
0.01
0
0.09
8
3
0.79
0
0
0.03
0.02
0.05
9
4
0.90
0
0.07
0.01
0
0.04
7
5
0.92
0
0
0.05
0.03
0.10
6
6
0.97
0
0.03
0.01
0
0.66
3
7
0.78
0
0
0.09
0.05
0
10
8
0.99
0
0
0
0
0.17
2
9
0.95
0
0.05
0.02
0
0.15
4
10
0.94
0
0
0.04
0.03
0.10
5Figure 7
Dynamic fluctuation of network DEA efficiency.Figure 8
Dynamic network DEA efficiency of 10 DMUs.


## 5. Conclusions

In this paper, we put forward a hybrid dynamic network DEA model, taking dynamic assessment and network structure in DEA into consideration simultaneously. The hybrid system measures the relative efficiencies of a set of DMUs with network structure of intermediate products between sub-DMUs with assignable inputs over observed time periods. The dynamic efficiency reflects overall network efficiency during the same discrete time spans, showing more practical meaning. What is more, improved information between inefficient DMUs and optimum values can be provided to decision makers by means of non-Archimedean infinitesimal and slacks of DMUs. Notably, different dynamic structure of network 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, super-efficiency 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.



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*Source: 989581-2015-06-04.xml*