#### 3.1. Dynamic Network DEA

Data Envelopment Analysis is based on a Pareto optimal solution concept and uses linear programming techniques to evaluate the relative efficiencies of Decision Making Units (DMU). The first use of this type of research method was 1957 in Farrell’s [

23] study on the “Measurement of Productive Efficiency”. However, Farrell’s efficiency assessment model was only suitable for a single input and a single output. Because such research generally involves multiple inputs and multiple outputs, Charnes, Cooper and Rhodes [

24] proposed the CCR model in 1978, which extended Farrell’s model to allow for multiple inputs and multiple outputs, with the optimal solution being solved using linear programming, which is now known as Data Envelopment Analysis. In 1984, Banker, Charnes and Cooper [

25] proposed the BCC model that included a variable return to scale (VRS) assumption to replace the constant return to scale (CRS) assumption. Tone [

26] then proposed a Slacks-Based Measure (SBM) to allow for any output and input slacks. Using non-radial estimation and a single scalar to present the SBM efficiency, the efficiency values are typically between 0 and 1; when the efficiency of a decision making unit is 1, there is no input or output slack on the production frontier.

Färe et al. [

27] proposed Network Data Envelopment Analysis (NDEA) in 2007, in which the production processes were broken up into secondary production technologies called Sub-DMUs, and traditional CCR or BCC models employed to determine the optimal solutions. In 2013, Tone and Tsutsui [

22] proposed the SBM (weighted slack-based measures) Dynamic Network DEA, which was based on the links between the decision-making units, included each department as a Sub-DMU, and had carry-over link activities.

Dynamic Network DEA Model and Solution:

It is assumed that the number of DMUs is n (j = 1, ..., n), with each DMU being divided into a number of k, (k = 1, …, K), and time periods t, (t = 1, …, T). Each DMU has an input and output in period t through a carry over (link) to the next period t + 1.

Set m_{k} and r_{k} as the input and output for each division K, in which (k,h)i indicates division k to h, and L_{hk} denotes the set of k and h.

Inputs and Outputs

${X}_{ijk}^{t}\in {R}_{+}^{}$$(i=1,\dots ,{m}_{k}^{};\text{}j=1,\dots ,n;K=\mathrm{1...},K;t=1,\dots ,T)$: indicates input $i$ in period $t$ for division $k$ in $DM{U}_{j}$.

${y}_{rjk}^{t}\in {R}_{+}^{}$$(r=1,\dots ,{r}_{k}^{};\text{}j=1,\dots ,n;K=\mathrm{1...},K;t=1,\dots ,T)$: indicates output $r$ in period $t$ for division $k$ in $DM{U}_{j}$.

If part of the output is not good, it is considered an input to division $k$.

Links

${Z}_{j(kh)t}^{t}\in {R}_{+}^{}$$(j=1;\dots ;n;l=1;\mathrm{..};{L}_{hk}^{};t=1;\dots ;T)0$: denotes the link between division $k$ and division $h$ in $DM{U}_{j}$ in period $t$, where ${L}_{hk}^{}$ is the number of links between $k$ and $h$.

Z^{t}_{j}_{(kh)t} ∊ R_{+} (j = 1; …; n; l = 1; …; L_{kh}; t = 1; …; T).

Carry-overs

${Z}_{jkl}^{(t,t+1)}\in {R}_{+}^{}$$(j=1,\dots ,n;l=1,\mathrm{..},{L}_{k}^{};k=1,\dots k,t=1,\dots ,T-1)$: denotes the carry-overs from division $k$ to $h$ in $DM{U}_{j}$ from period $t$ to $t+1$ where ${L}_{k}^{}$ is the number of carry-overs from division $k$.

#### Objective Function

Constraints:

${x}_{ok}^{t}={X}_{k}^{t}{{\displaystyle \lambda}}_{k}^{t}+{s}_{ko}^{t-}$$(\forall k,\forall t)$

${y}_{ok}^{t}={Y}_{k}^{t}{{\displaystyle \lambda}}_{k}^{t}-{s}_{ko}^{t+}$$(\forall k,\forall t)$

$e{\lambda}_{k}^{t}=1$$(\forall k,\forall t)$

${\lambda}_{k}^{t}\ge 0,$${s}_{ko}^{t-1}\ge 0,$${s}_{ko}^{t+}\ge 0,$$(\forall k,\forall t)$

${Z}_{(kh)free}^{t}{\lambda}_{h}^{t}={Z}_{(kh)free}^{t}{{\displaystyle \lambda}}_{k}^{t}$$(\forall (k,h)free,\forall t)$

${Z}_{(kh)free}^{t}=({Z}_{1(kh)free}^{t},\dots ,{Z}_{n(kh)free}^{t})\in {R}^{{L}_{(h)free}\times n}$

${Z}_{o(kh)fix}^{t}={Z}_{(kh)fix}^{t}{{\displaystyle \lambda}}_{h}^{t}$$(\forall (k,h)fix,\forall t)$

${Z}_{o(kh)fix}^{t}={Z}_{(kh)fix}^{t}{{\displaystyle \lambda}}_{k}^{t}$$(\forall (k,h)fix,\forall t)$

${Z}_{o(kh)in}^{t}={Z}_{(kh)in}^{t}{\lambda}_{k}^{t}+{S}_{o(kh)in}^{t}$$((kh)in=1,\dots ,linki{n}_{k})$

${Z}_{o(kh)out}^{t}={Z}_{(kh)out}^{t}{\lambda}_{k}^{t}-{S}_{o(kh)out}^{t}$$((kh)out=1,\dots ,linkou{t}_{k})$

${\sum}_{j=1}^{n}{{\displaystyle z}}_{j{k}_{1}\alpha}^{(t,(t+1))}}{\lambda}_{jk}^{t}={\displaystyle {\sum}_{j=1}^{n}{z}_{j{k}_{1}\alpha}^{(t,(t+1))}}{\lambda}_{jk}^{t+1$$(\forall k;\forall {k}_{l};t=1,\dots ,T-1)$

${Z}_{o{k}_{l}good}^{(t,(t+1))}={{\displaystyle {\sum}_{j=1}^{n}z}}_{j{k}_{l}good}^{(t,(t+1))}{\lambda}_{jk}^{t}-{s}_{o{k}_{l}good}^{(t,(t+1))}$${k}_{l}=1,\dots ,ngoo{d}_{k};\forall k;\forall t)$

${Z}_{o{k}_{l}bad}^{(t,(t+1))}={{\displaystyle {\sum}_{j=1}^{n}z}}_{j{k}_{l}bad}^{(t,(t+1))}{\lambda}_{jk}^{t}-{s}_{o{k}_{l}bad}^{(t,(t+1))}$${k}_{l}=1,\dots ,nba{d}_{k};\forall k;\forall t)$

${Z}_{o{k}_{l}free}^{(t,(t+1))}={{\displaystyle {\sum}_{j=1}^{n}z}}_{j{k}_{l}free}^{(t,(t+1))}{\lambda}_{jk}^{t}-{s}_{o{k}_{l}free}^{(t,(t+1))}$${k}_{l}=1,\dots ,nfre{e}_{k};\forall k;\forall t)$

${Z}_{o{k}_{l}fix}^{(t,(t+1))}={{\displaystyle {\sum}_{j=1}^{n}z}}_{j{k}_{l}fix}^{(t,(t+1))}{\lambda}_{jk}^{t}-{s}_{o{k}_{l}fix}^{(t,(t+1))}$${k}_{l}=1,\dots ,nfi{x}_{k};\forall k;\forall t)$

Period and Division Efficiencies

Division Period efficiency: From the above results, the overall efficiency, period efficiency, division efficiency and division period efficiency can be determined.

#### 3.2. Fixed Assets, Labor, Energy Consumption, GDP, Health Expenditure, Birth Rate, Respiratory Disease, and Death Rate Efficiencies

Hu and Wang’s [

28] total-factor energy efficiency index is employed here to overcome any possible biases in the traditional energy efficiency indicators. There are nine key features used in this present study: fixed assets, labor, energy consumption, GDP, health expenditure, birth rate, respiratory diseases and death rate. In our study, “I” represents area and “t” represents time.

#### 3.2.1. Fixed Asset Efficiency

Fixed asset efficiency is the ratio of target fixed asset input to actual fixed asset input, the model for which is:

If the target fixed asset input is equal to the actual input level, then the fixed asset efficiency equals 1, indicating efficiency; however, if the target fixed asset input is less than the actual input level, then the fixed asset efficiency is less than 1, indicating inefficiency.

#### 3.2.2. Labor Efficiency

Labor efficiency is the ratio of target labor input to actual labor input, the model for which is:

If the target labor input is equal to the actual input level, then labor efficiency equals 1, indicating efficiency; however, if the target labor input is less than the actual input, then the labor efficiency is less than 1, indicating inefficiency.

#### 3.2.3. Energy Consumption Efficiency

Energy consumption efficiency is the ratio of target energy input to actual energy input, the model for which is:

If the target energy input is equal to the actual input level, then energy consumption efficiency equals 1, indicating efficiency; however, if the target energy input is less than the actual input level, then the energy consumption efficiency is less than 1, indicating inefficiency.

#### 3.2.4. GDP Efficiency

GDP efficiency is the ratio of actual desirable GDP output to target desirable GDP output, the model for which is:

If the target desirable GDP output is equal to the actual desirable GDP output level, then the GDP efficiency equals 1, indicating efficiency. If the actual desirable GDP output is less than the target desirable GDP output level, then the GDP efficiency is less than 1, indicating inefficiency.

#### 3.2.5. Health Expenditure Efficiency

Health expenditure efficiency is the ratio of target health expenditure input to actual health expenditure input, the model for which is:

If the target health expenditure input is equal to the actual health expenditure input level, then the health expenditure efficiency equals 1, indicating efficiency; however, if the target health expenditure input is less than the actual health expenditure input level, then the health expenditure efficiency is less than 1, indicating inefficiency.

#### 3.2.6. Birth Rate Efficiency

Birth rate efficiency is the ratio of actual desirable birth rate output to target desirable birth rate output, the model for which is:

If the target desirable birth rate output is equal to the actual desirable birth rate output level, then the birth rate efficiency equals 1, indicating efficiency; however, if the actual desirable birth rate output is less than the target desirable birth rate output level, then the birth rate efficiency is less than 1, indicating inefficiency.

#### 3.2.7. Respiratory Disease Efficiency

Respiratory disease efficiency is the ratio of target undesirable respiratory disease output to actual undesirable respiratory disease output, the model for which is;

Respiratory

If the target undesirable respiratory disease output is equal to the actual undesirable respiratory disease output, then the respiratory diseases efficiency equals 1, indicating efficiency; however, if the target undesirable respiratory disease output is less than the actual undesirable respiratory disease output, then the respiratory disease efficiency is less than 1, indicating inefficiency.

#### 3.2.8. Death Rate Efficiency

The death rate efficiency is the ratio of target undesirable death rate output to actual undesirable death rate output, the model for which is;

If the target undesirable death rate output is equal to the actual undesirable death rate output, then the death rate efficiency equals 1, indicating efficiency; however, if the undesirable target death rate output is less than the actual undesirable death rate output, then the death rate efficiency is less than 1, indicating inefficiency.