Uncertain DEA-Malmquist productivity index model and its application

Abstract

Uncertain data envelopment analysis (DEA) model make an estimate of the efficiency of decision making unit (DMU) under data uncertainty. The current research on uncertain DEA model is only based on sectional data to calculate DMU’s static efficiency for the DMU’s set in the same period. From this article, we attempt to combine Malmquist productivity index and uncertain DEA model (the uncertain DEA-Malmquist productivity index model) to calculate the dynamic change of DMU’s efficiency over time. Additionally, the impact of technical factors and scale factors on DMU’s efficiency can be further explored and the Malmquist productivity index will be decomposed into pure technical efficiency change, scale efficiency change and technical change. Finally, the article uses the model to analyze the provincial environmental efficiency from 2014 to 2016 in China.

Keywords

Uncertainty theory uncertain DEA model malmquist productivity index decision making unit

1 Introduction

Data envelopment analysis (DEA) is an effective method to calculate DMU’s relative efficiency. The DMU is an entity, such as a province or a department. Charnes et al. [1] proposed the first DEA model referred to CCR model in 1978. The CCR model can evaluate DMU’s comprehensive technical efficiency. Later, Banker et al. [2] presented another model (BCC model) to evaluate DMU’s scale efficiency. Here, each DMU is “self-evaluated" based on the DMU set, which is used to evaluate the situation of DMU in the same period. When evaluating DMU data set is panel data, the traditional DEA method is no longer applicable to its premise and needs to be improved. In addition, DEA has many applications. For example, Ganji et al. [3, 4] applied DEA to road safety evaluation.

Malmquist Productivity Index (MPI) is a significantly mainstream way of measuring productivity over time and it was first proposed by Malmquist [5] in 1953. Then, Caves et al. [6] first applied the Malmquist index to the measurement of productivity changes under the DEA in 1982. Later, Ray and Desli [7] put forward the decomposition mode of MPI, which is referred to as RD decomposition mode. The mode can decompose the index value to the technical change (TC) and technical efficiency change (TEC), among which the latter was further decomposed into pure technical efficiency change (PTEC) and scale efficiency change (SC). Since then, the combination of DEA and Malmquist productivity index (the DEA-Malmquist productivity index model) has been widely applied to reflect the dynamic efficiency changes of DMU. Such as Ganji and Rassafi [8] use the DEA-Malmquist productivity index to assess Iranian road safety in 2019, Singh et al. [9] applied DEA-Malmquist productivity index on Health Care System Efficiency in 2021, Bansal et al. [10] applied the network DEA and Malmquist productivity index to banking data in 2022, and so on.

However, the above mentioned studies on traditional MPI require precise input-output data. However, some input or output data are imprecise in reality, such as carbon dioxide emissions. At this time, the traditional MPI is not suitable to measure the dynamic efficiency change of DMU when the evaluation data is imprecise. For this situation, researchers have done a lot of research on how to measure MPI of DMU with imprecise data. In recent ten years, such as Hatami-Marbini et al. [9] applied the MPI to health care in 2012, Oruc [12] proposed a new grey MPI model to measure DMU’s productivity changes with interval data in 2015, Kordrostami and Noveiri [13] proposed a new fuzzy DEA with the expected value method to the multi-period systems, where MPI is used to calculate the efficiency change in two periods through two examples in 2017, Huang et al. [14] combined fuzzy BWM-DEA-AR model with the MPI to analysis China’s energy security from 2008 to 2017 in 2021, Peykani and Seyed Esmaeili [15] extended the possibilistic programming and chance-constrained programming to deal with fuzzy data in 2021, Huang et al. [16] applied the interval MPI to the new patterns in China’s regional green development in 2021, Yang et al. [17] applied the DEA-MPI model to measure the efficiency of scientific and technological innovation in regional industrial enterprises in 2022, and so on.

In this paper, we will start from uncertainty theory [18] and propose a new uncertain DEA-Malmquist productivity index model with imprecise data to measure DMU’s dynamic efficiency. Uncertainty theory is an understandable mathematical structure for dealing with the doubt in data. Wen et al. [19] established the first uncertain DEA model by the uncertain chance constraint method in 2014. Subsequently, Lio and Liu [20] improved the uncertain DEA model studied by Wen et al. [19] to describe DMU’s technical efficiency through the expected value method in 2018. Then, Jiang et al. [21] separated the pure technical efficiency and scale efficiency of the uncertain DEA model based on the research results by Lio and Liu [20]. So far, these uncertain DEA models are focused on sectional data to analysis DMU’s static efficiency value relative to DMU’s set over the same period. In order to explore the dynamic changes of DMU efficiency over different periods and the influence of technical factors and scale factors on DMU’s dynamic efficiency, the paper proposed the new uncertain DEA-Malmquist productivity index model with imprecise data. In the measurement of production efficiency in financial, industrial, medical, environmental, and other sectors, it plays a very important role in measuring productivity changes over time in the case of imprecise panel data. Therefore, the purpose of this study is to establish the uncertain DEA-Malmquist productivity index model to reflect the change of DMU’s efficiency value in different periods when the data is imprecise. In addition, researchers will answer the following research questions: (a) How to deal with imprecise data? (b) What is the specific DEA model involved in the uncertain DEA-Malmquist productivity index model? (c) How to model imprecise data to calculate MPI? (d) How to demonstrate the application of the uncertain DEA-Malmquist productivity index model in practice through examples? To sum up, the main contributions of this paper are as follows: In order to study the change of DMU productivity over time under imprecise data, we use uncertain variables to describe the imprecise data, and then construct the uncertain DEA-Malmquist productivity index model. The specific construction process of this model is shown in Section 3, in which DEA models (1) and (2) are involved. Then, a numerical example is used to demonstrate the practical application of this model.

The rest of this paper is structured as shown below. In Section 2, some basic knowledge of uncertainty theory and two basic uncertain DEA models are introduced. In Section 3, the uncertain DEA-Malmquist productivity index model is proposed to describe the total factor productivity of DMU. Additionally, the Malmquist productivity index model is decomposed. In Section 4, a numerical example is given to empirically analyze each China’s province environmental efficiency from 2014 to 2016. In Section 5, some conclusions are given.

2 Preliminaries

2.1 Uncertainty theory

In this section, we will introduce some knowledge about uncertainty theory that is involved in this paper. Uncertainty theory uses uncertain measure $M$ to model the degree of people’s belief [18]. The uncertain measure $M$ is a set function on a σ-algebra $L$ over a nonempty set Γ. It satisfies the normality, duality, and subadditivity axioms. The triplet ( $Γ, L, M$ ) is the uncertainty space.

Definition 1. (Liu [18]) Uncertain variable ξ is a measurable function from the uncertainty space $(Γ, L, M)$ to the set of real numbers, i.e., ${ξ \in B} = {γ \in Γ ∣ ξ (γ) \in B}$ is an event for any Borel set B of real numbers.

Moveover, the uncertainty distribution of ξ is $Φ (x) = M {ξ \leq x}, \forall x \in R .$ For example, if ξ is a linear uncertain variable, then $Φ (x) = {\begin{matrix} 0, & if x \leq a \\ (x - a) / (b - a), & if a < x \leq b \\ 1, & if x > b \end{matrix}$ is its linear uncertainty distribution. It is denoted as $ξ \sim L (a, b)$ .

In addition, if the uncertainty distribution Φ (x) is a continuous function and strictly increasing with respect to x at which 0 < Φ (x) <1, and $lim_{x \to - \infty} Φ (x) = 0, lim_{x \to \infty} Φ (x) = 1,$ so Φ (x) is called regular [18].

Definition 2. (Liu [22]) Let Φ (x) is the uncertainty distribution of the uncertain variable ξ, the inverse function Φ^-1 (α) is the inverse uncertainty distribution of ξ.

Example 1. If the uncertain variable $ξ \sim L (a, b)$ , the inverse uncertainty distribution of ξ is $Φ^{- 1} (α) = (1 - α) a + α b .$

Definition 3. (Liu [23]) Uncertain variable ξ₁, ξ₂, …, ξ_n are called independent if $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, …, B_n of real numbers.

Theorem 1. (Liu [22]) Let ξ₁, ξ₂, …, ξ_n be indepent uncertain variables with regular uncertain distributions Φ₁, Φ₂, …, Φ_n, respectively. If f is strictly increasing with respect to ξ₁, ξ₂, …, ξ_k and strictly decreasing with respect to ξ_k+1, ξ_k+2, …, ξ_n, then the inverse uncertainty distribution of ξ = f (ξ₁, ξ₂, …, ξ_n) is $Φ^{- 1} (α) = f (Φ_{1}^{- 1} (α), \dots, Φ_{k}^{- 1} (α), Φ_{k + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) .$

Definition 4. (Liu [18]) Let ξ is an uncertain variable, the expected value of ξ is defined as $E [ξ] = \int_{0}^{+ \infty} M (ξ \geq x) d x - \int_{- \infty}^{0} M (ξ \leq x) d x$ provided that at least one of the two integrals is finite.

Theorem 2. (Liu and Ha [24]) Let ξ₁, ξ₂, …, ξ_n be independent uncertain variables with regular uncertain distributions Φ₁, Φ₂, …, Φ_n, respectively. If f is strictly increasing with respect to ξ₁, ξ₂, …, ξ_k and strictly decreasing with respect to ξ_k+1, ξ_k+2, …, ξ_n, then the expected value of ξ = f (ξ₁, ξ₂, …, ξ_n) is $E [ξ] = \int_{0}^{1} f (Φ_{1}^{- 1} (α), \dots, Φ_{k}^{- 1} (α), Φ_{k + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) d α .$

2.2 Uncertain DEA models

With the application and development of the uncertainty theory, DEA has a broader development prospect on account of imprecise data. In 2018, Lio and Liu [20] proposed the uncertain DEA model to measure whether the DMU is technically efficient. The basic symbols of the model are as follows:

DMU_i: the ith DMU, i = 1, 2, …, n.

DMU_o: the target DMU.

${\tilde{x}}_{i} = ({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{ir})$ : the uncertain inputs vector of DMU_i, i = 1, 2, …, n.

${\tilde{x}}_{o} = ({\tilde{x}}_{o 1}, {\tilde{x}}_{o 2}, \dots, {\tilde{x}}_{or})$ : the uncertain inputs vector of DMU_o.

${\tilde{y}}_{i} = ({\tilde{y}}_{i 1}, {\tilde{y}}_{i 2}, \dots, {\tilde{y}}_{is})$ : the uncertain outputs vector of DMU_i, i = 1, 2, …, n.

${\tilde{y}}_{o} = ({\tilde{y}}_{o 1}, {\tilde{y}}_{o 2}, \dots, {\tilde{y}}_{os})$ : the uncertain outputs vector of DMU_o.

u = (u₁, u₂, …, u_r): the vector of input weights.

v = (v₁, v₂, …, v_s): the vector of ouput weights.

There are n DMUs, r inputs, and s outputs. DMU’s relative efficiency evaluation is relative to the overall DMUs. Uncertain DEA model to measure technical efficiency was presented by Lio and Liu [20] under the condition of constant returns to scale is as follows: ${\begin{matrix} max_{u, v} & θ = E [\frac{v^{T} {\tilde{y}}_{o}}{u^{T} {\tilde{x}}_{o}}] \\ s . t . & E [\frac{v^{T} {\tilde{y}}_{i}}{u^{T} {\tilde{x}}_{i}}] \leq 1, i = 1, 2, \dots, n \\ u \geq 0 \\ v \geq 0 . \end{matrix}$ (1)

Definition 5. (Lio and Liu [20]) In model (1), the DMU is efficient only when the $\bar{θ}$ value achieve 1. When $\bar{θ} < 1$ , it shows that the DMU is inefficient. In addition, DMU’s efficiency is not only affected by technology, but may also be affected by scale in real life. In certain circumstances, we only need pure technical efficiency or scale efficiency. Jiang et al. [21] proposed a specific model to solve this problem. It can calculate the pure technical efficiency and scale efficiency of DMU under the condition of variable returns to scale. To measure the DMU’s scale efficiency, the uncertain DEA model is defined as ${\begin{matrix} max_{u, v, ω} & φ = E [\frac{v^{T} {\tilde{y}}_{o} - ω}{u^{T} {\tilde{x}}_{o}}] \\ s . t . & E [\frac{v^{T} {\tilde{y}}_{i} - ω}{u^{T} {\tilde{x}}_{i}}] \leq 1, i = 1, 2, \dots, n \\ u \geq 0 \\ v \geq 0 . \end{matrix}$ (2)

Definition 6. (Jiang et al. [21]) In model (2), the DMU is efficient only when the $\bar{φ}$ value achieve 1. When $\bar{φ} < 1$ , it shows that the DMU is inefficient.

3 Uncertain DEA-Malmquist productivity index model

In 2018, Lio and Liu [20] upgraded an uncertain DEA model based on the expected method to measure DMU’s technical efficiency. After that, Jiang et al. [21] proposed an uncertain DEA model to measure DMU’s scale efficiency and pure technical efficiency based on Lio and Liu [20]. The existing uncertain DEA models measure static relative efficiency values of DMU in the same period. However, the evaluation data of DMU includes not only sectional data but also panel data. When it is panel data, model (1) and (2) are no longer applicable. In order to further reflect the change of DMU’s efficiency value in different periods, the uncertain DEA-Malmquist productivity index model is presented in this article. The calculation of MPI is based on distance function of period t and period t+1, where the distance function happens to be the inverse of the calculation results of the uncertain DEA model. Taking the construction of the distance function at period t as an example. Suppose there are n DMUs. Each DMU at period t has r inputs and s outputs. The basic symbols are as follows:

${\tilde{x}}_{i}^{t} = ({\tilde{x}}_{i 1}^{t}, {\tilde{x}}_{i 2}^{t}, \dots, {\tilde{x}}_{ir}^{t})$ : the uncertain inputs vector of DMU_i at period t, i = 1, 2, …, n.

${\tilde{x}}_{o}^{t} = ({\tilde{x}}_{o 1}^{t}, {\tilde{x}}_{o 2}^{t}, \dots, {\tilde{x}}_{or}^{t})$ : the uncertain inputs vector of DMU_o at period t.

${\tilde{y}}_{i}^{t} = ({\tilde{y}}_{i 1}^{t}, {\tilde{y}}_{i 2}^{t}, \dots, {\tilde{y}}_{is}^{t})$ : the uncertain outputs vector of DMU_i at period t, i = 1, 2, …, n.

${\tilde{y}}_{o}^{t} = ({\tilde{y}}_{o 1}^{t}, {\tilde{y}}_{o 2}^{t}, \dots, {\tilde{y}}_{os}^{t})$ : the uncertain outputs vector of DMU_o at period t.

Thus, based on constant returns to scale, let $D_{T}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ be the distance function of $({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ at period t, let $D_{T}^{t + 1} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ be the distance function of $({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ at period t+1, let $D_{T}^{t} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ be the distance function of $({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ at period t, and let $D_{T}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ be the distance function of $({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ at period t+1. To simplify the representation, let’s denote the period by γ, so that γ=t or t+1. According to model (1), the corresponding distance function model can be obtained as follows:

The distance function $D_{T}^{γ} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ of $({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ is defined as ${\begin{matrix} max_{u, v} & (D_{T}^{γ} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t}))^{- 1} = E [\frac{v^{T} {\tilde{y}}_{o}^{t}}{u^{T} {\tilde{x}}_{o}^{t}}] \\ s . t . & E [\frac{v^{T} {\tilde{y}}_{i}^{γ}}{u^{T} {\tilde{x}}_{i}^{γ}}] \leq 1, i = 1, 2, \dots, n \\ u \geq 0 \\ v \geq 0, \end{matrix}$ (3) where $D_{T}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ can be calculated when γ=t and $D_{T}^{t + 1} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ can be calculated when γ=t+1. In addition, the distance function $D_{T}^{γ} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ of $({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ is defined as ${\begin{matrix} max_{u, v} & (D_{T}^{γ} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1}))^{- 1} = E [\frac{v^{T} {\tilde{y}}_{o}^{t + 1}}{u^{T} {\tilde{x}}_{o}^{t + 1}}] \\ s . t . & E [\frac{v^{T} {\tilde{y}}_{i}^{γ}}{u^{T} {\tilde{x}}_{i}^{γ}}] \leq 1, i = 1, 2, \dots, n \\ u \geq 0 \\ v \geq 0, \end{matrix}$ (4) where $D_{T}^{t} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ can be calculated when γ=t and $D_{T}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ can be calculated when γ=t+1. The model (3) and model (4) are uncertain DEA models, which are too complex to be calculated. Therefore, through Theorem 2, we give their equivalent crisp models to simplify the calculation. Let both inputs ${\tilde{x}}_{i 1}^{t}, {\tilde{x}}_{i 2}^{t}, \dots, {\tilde{x}}_{ir}^{t}$ and outputs ${\tilde{y}}_{i 1}^{t}, {\tilde{y}}_{i 2}^{t}, \dots, {\tilde{y}}_{is}^{t}$ be independent uncertain variables with regular uncertainty distributions $Φ_{i 1}^{t}, Φ_{i 2}^{t}, \dots, Φ_{ir}^{t}$ and $Ψ_{i 1}^{t}, Ψ_{i 2}^{t}, \dots, Ψ_{is}^{t}$ , respectively. The equivalent form of model (4) is drived as (hereafter model II):

${\begin{matrix} max_{u, v} & (D_{T}^{γ} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t}))^{- 1} = \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{oq}^{t})^{- 1} (α)}{\sum_{p = 1}^{r} u_{p} (Φ_{op}^{t})^{- 1} (1 - α)} d α \\ s . t . & \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{iq}^{γ})^{- 1} (α)}{\sum_{p = 1}^{r} u_{p} (Φ_{ip}^{γ})^{- 1} (1 - α)} d α \leq 1, i = 1, 2, \dots, n \\ u = (u_{1}, u_{2}, \dots, u_{r}) \geq 0 \\ v = (v_{1}, v_{2}, \dots, v_{s}) \geq 0 . \end{matrix}$

The equivalent form of model (6) is drived as (hereafter model IV):

${\begin{matrix} max_{u, v} & (D_{T}^{γ} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1}))^{- 1} = \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{oq}^{t + 1})^{- 1} (α)}{\sum_{p = 1}^{r} u_{p} (Φ_{op}^{t + 1})^{- 1} (1 - α)} d α \\ s . t . & \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{iq}^{γ})^{- 1} (α)}{\sum_{p = 1}^{r} u_{p} (Φ_{ip}^{γ})^{- 1} (1 - α)} d α \leq 1, i = 1, 2, \dots, n \\ u = (u_{1}, u_{2}, \dots, u_{r}) \geq 0 \\ v = (v_{1}, v_{2}, \dots, v_{s}) \geq 0 . \end{matrix}$

Similarly, based on variable returns to scale, let $D_{S}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ be the distance function of $({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ at period t and $D_{S}^{t + 1} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ be the distance function of $({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ at period t+1, let $D_{S}^{t} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ be the distance function of $({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ at period t and $D_{S}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ be the distance function of $({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ at period t+1. According to model (2), the corresponding distance function model can be obtained as follows:

Distance function $D_{S}^{γ} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ of $({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ is defined as ${\begin{matrix} max_{u, v, ω} & (D_{S}^{γ} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t}))^{- 1} = E [\frac{v^{T} {\tilde{y}}_{o}^{t} - ω}{u^{T} {\tilde{x}}_{o}^{t}}] \\ s . t . & E [\frac{v^{T} {\tilde{y}}_{i}^{γ} - ω}{u^{T} {\tilde{x}}_{i}^{γ}}] \leq 1, i = 1, 2, \dots, n \\ u \geq 0 \\ v \geq 0, \end{matrix}$ (5) where $D_{S}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ can be calculated when γ=t and $D_{S}^{t + 1} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})$ can be calculated when γ=t+1. In addition, the distance function $D_{S}^{γ} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ of $({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ is defined as ${\begin{matrix} max_{u, v, ω} & (D_{S}^{γ} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1}))^{- 1} = E [\frac{v^{T} {\tilde{y}}_{o}^{t + 1} - ω}{u^{T} {\tilde{x}}_{o}^{t + 1}}] \\ s . t . & E [\frac{v^{T} {\tilde{y}}_{i}^{γ} - ω}{u^{T} {\tilde{x}}_{i}^{γ}}] \leq 1, i = 1, 2, \dots, n \\ u \geq 0 \\ v \geq 0, \end{matrix}$ (6) where $D_{S}^{t} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ can be calculated when γ=t and $D_{S}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})$ can be calculated when γ=t+1. The model (5) and model (6) are uncertain DEA models, which are too complex to be calculated. Therefore, through Theorem 2, we give their equivalent crisp models to simplify the calculation. Let both inputs ${\tilde{x}}_{i 1}^{t}, {\tilde{x}}_{i 2}^{t}, \dots, {\tilde{x}}_{ir}^{t}$ and outputs ${\tilde{y}}_{i 1}^{t}, {\tilde{y}}_{i 2}^{t}, \dots, {\tilde{y}}_{is}^{t}$ be independent uncertain variables with regular uncertainty distributions $Φ_{i 1}^{t}, Φ_{i 2}^{t}, \dots, Φ_{ir}^{t}$ and $Ψ_{i 1}^{t}, Ψ_{i 2}^{t}, \dots, Ψ_{is}^{t}$ , respectively. The equivalent form of model (5) is drived as follows (hereafter model III):

${\begin{matrix} max_{u, v, ω} & (D_{S}^{γ} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t}))^{- 1} = \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{oq}^{t})^{- 1} (α) - ω}{\sum_{p = 1}^{r} u_{p} (Φ_{op}^{t})^{- 1} (1 - α)} d α \\ s . t . & \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{iq}^{γ})^{- 1} (α) - ω}{\sum_{p = 1}^{r} u_{p} (Φ_{ip}^{γ})^{- 1} (1 - α)} d α \leq 1, i = 1, 2, \dots, n \\ u = (u_{1}, u_{2}, \dots, u_{r}) \geq 0 \\ v = (v_{1}, v_{2}, \dots, v_{s}) \geq 0 . \end{matrix}$

The equivalent form of model (4) is drived as (hereafter model II):

${\begin{matrix} max_{u, v, ω} & (D_{S}^{γ} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1}))^{- 1} = \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{oq}^{t + 1})^{- 1} (α) - ω}{\sum_{p = 1}^{r} u_{p} (Φ_{op}^{t + 1})^{- 1} (1 - α)} d α \\ s . t . & \int_{0}^{1} \frac{\sum_{q = 1}^{s} v_{q} (Ψ_{iq}^{γ})^{- 1} (α) - ω}{\sum_{p = 1}^{r} u_{p} (Φ_{ip}^{γ})^{- 1} (1 - α)} d α \leq 1, i = 1, 2, \dots, n \\ u = (u_{1}, u_{2}, \dots, u_{r}) \geq 0 \\ v = (v_{1}, v_{2}, \dots, v_{s}) \geq 0 . \end{matrix}$

Remark 1. In model I and model II, $Φ_{o 1}^{t}, Φ_{o 2}^{t}, \dots, Φ_{or}^{t}$ and $Ψ_{o 1}^{t}, Ψ_{o 2}^{t}, \dots, Ψ_{os}^{t}$ are the regular uncertain distributions of ${\tilde{x}}_{o 1}^{t}, {\tilde{x}}_{o 2}^{t}, \dots, {\tilde{x}}_{or}^{t}$ and ${\tilde{y}}_{o 1}^{t}, {\tilde{y}}_{o 2}^{t}, \dots, {\tilde{y}}_{os}^{t}$ , respectively. In model III and model IV, $Φ_{o 1}^{t + 1}, Φ_{o 2}^{t + 1}, \dots,$ $Φ_{or}^{t + 1}$ and $Ψ_{o 1}^{t + 1}, Ψ_{o 2}^{t + 1}, \dots, Ψ_{os}^{t + 1}$ are the regular uncertain distributions of ${\tilde{x}}_{o 1}^{t + 1}, {\tilde{x}}_{o 2}^{t + 1}, \dots, {\tilde{x}}_{or}^{t + 1}$ and ${\tilde{y}}_{o 1}^{t + 1}, {\tilde{y}}_{o 2}^{t + 1},$ $\dots, {\tilde{y}}_{os}^{t + 1}$ , respectively.

Then, MPI reflects the DMU’s productivity change. The DMU’s productivity change on account of the technical level at period γ is defined as $M^{γ} = \frac{D_{T}^{γ} ({\tilde{x}}^{t + 1}, {\tilde{y}}^{t + 1})}{D_{T}^{γ} ({\tilde{x}}^{t}, {\tilde{y}}^{t})} .$ (7) The MPI is defined by the geometric mean of M^t +1 and M^t as follows: $M = {[\frac{D_{T}^{t} ({\tilde{x}}^{t + 1}, {\tilde{y}}^{t + 1})}{D_{T}^{t} ({\tilde{x}}^{t}, {\tilde{y}}^{t})} \times \frac{D_{T}^{t + 1} ({\tilde{x}}^{t + 1}, {\tilde{y}}^{t + 1})}{D_{T}^{t + 1} ({\tilde{x}}^{t}, {\tilde{y}}^{t})}]}^{\frac{1}{2}} .$ (8) From period t to period t+1, The value of M express the DMU’s total productivity change. If M > 1, it reflects the DMU’s total factor productivity increased. If M < 1, it means the DMU’s total factor peoductivity decreased. If M = 1, it indicated the DMU’s total productivity unchanged.

According to Ray and Desli [7], the MPI can be divided into technical efficiency change (UTEC) and technology change (UTC). Moreover, UTEC can be further divided into pure technical efficiency change (UPTEC) and scale efficiency change (USEC). Besides, the results of uncertain DEA model include technical efficiency (UTE), pure technical efficiency (UPTE), and scale efficiency (USE). UTE reflects the resourse alllocation efficiency for the DMU, UPTE shows the DMU’s input utilization level determined by the specific production technology level, and USE reflects the DMU’s input utilization level determined by the specific scale efficiency. The model (1) can calculate the value of UTE. The model (2) model can calculate the value of UPTE. In addition, USE is defined as $USE = \frac{UTE}{UPTE} .$ (9) UTEC is defined as $UTEC = \frac{D_{T}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})}{D_{T}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})} .$ (10) UPTEC is defined as $UPTEC = \frac{D_{S}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})}{D_{S}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})} .$ (11) From period t to period t+1, UTEC reflects technical efficiency change based on constant returns to scale, i.e., the catch-up degree of DMU relative to production frontier. When UTEC > 1, it reflects the production of DMU is closer to production frontier in period t+1 exceed period t, indicating UPTEC of DMU has improved. When UTEC < 1, it indicates the production of DMU is farther away from the production frontier in period t+1 exceed period t, indicating UPTEC of DMU has decreased. When UTEC = 1, it means the DMU’s production remains the same production frontier from period t to period t+1, indicating DMU’s UPTEC is invariable. In addition, UTC is defined as $UTC = {[\frac{D_{S}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})}{D_{S}^{t + 1} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})} \times \frac{D_{S}^{t} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})}{D_{S}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})}]}^{\frac{1}{2}} .$ UTC reflects the extent of the production frontier movement of DMU from period t to period t+1. In other words, the indicator expresses the degree of technological progress or innovation. When UTC > 1, it indicates that the production frontier moves outward, i.e., the overall technology tends to increase. When UTC < 1, it indicates that the production frontier moves to origin, i.e., the overall technology tends to decline. When UTC = 1, it signs the production frontier is unchanged, meaning no change in technical level. USEC is defined as $\begin{matrix} USEC = & [\frac{D_{T}^{t} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1}) / D_{S}^{t} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})}{D_{T}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t}) / D_{S}^{t} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})} \times \\ {\frac{D_{T}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1}) / D_{S}^{t + 1} ({\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1})}{D_{T}^{t + 1} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t}) / D_{S}^{t + 1} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t})}]}^{\frac{1}{2}} . \end{matrix}$ USEC represents the scale’s contribution to productivity change. When USEC > 1, it reflects scale change promotes productivity change. When USEC < 1, it reflects scale change reduces productivity change. Combined with the above definitions, the RD decomposition mode of Malmquist productivity index model is as follows: $\begin{matrix} M_{RD} ({\tilde{x}}_{o}^{t}, {\tilde{y}}_{o}^{t}, {\tilde{x}}_{o}^{t + 1}, {\tilde{y}}_{o}^{t + 1}) \\ = UTC \times UTEC \\ = UPTEC \times UTC \times USEC . \end{matrix}$ (12) From period t to period t+1, Malmquist productivity index reflects the dynamic change of DMU’s total factor productivity. When M_RD > 1, DMU’s total factor productivity shows a rising trend. When M_RD < 1, DMU’s total factor productivity shows a declining trend. When M_RD = 1, it reflects DMU’s total factor productivity remains unchanged.

4 Numerical Example

With the continuous progress of society and the improvement of people’s living standards, environmental pollution has gradually entered people’s vision. How to control environmental pollution and improve the quality of people’s living environment has become a major problem in today’s society. Therefore, it’s practical significance to study province’s environmental efficiency. In this section, we apply the uncertain DEA-Malmquist productivity index model to assess the dynamics of environmental efficiency of China’s provinces from 2014 to 2016.

The paper selects specific statistics from 30 provinces in China from 2014 to 2016. Each province acts as a DMU. Moreover, The statistics of Tibet, Hong Kong, Macau, and Taiwan are incomplete, so this article does not consider them for the time being. Then, we need to fully consider and choose input indicators and output indicators. The existing research’s input indicators with respect to the environmental efficiency are usually selected from three perspectives: capital, labor, and energy (e.g., Du, Chen, and Huang [25], Zhu et al. [26], Choi, Oh, and Zhang [27]). In this paper, we select provincial employed population (EP) as the labor input variable, provincial energy consumption (EC) as energy input variable, and capital stock (CS) of each province as the capital input index according to the practical situation. Among them, CS represents all existing capital resources of a province, which can usually reflect the current province’s production technological level and scale. However, there are no accurate statistics on hand of the CS. Therefore, we can regard CS as uncertain variable by using uncertainty theory [18], and obtain its uncertainty distribution by using expert scoring method. In the selection of output variables, the output variables may be divided into desirable output and undesirable output. Among them, the desirable output is mainly economic variable, and undesirable output is mainly the emission index of related environmental pollutants. In this paper, we select four output variables, gross domestic product (GDP), total CO₂ emissions (TCE), total SO₂ emissions (TSE) and total wastewater emissions (TWE) according to the practical situation. Table 1 indicates that the selection of input indicators and output indicators. Furthermore, one of the traditional DEA model purposes is to obtain the maximum output. TCE, TSE and TWE are undesirable outputs because they belong to environmental pollutants. Thus, one of the solutions is that we regard these undesirable output variables as input variables (Zhou et al. [28]). Furthermore, under the current technical conditions, we may not get accurate data about the emissions of environmental pollutants, so we may use the uncertainty theory [18] to treat TCE, TSE, and TWE as uncertain variables, and then use the expert scoring method to obtain their uncertainty distribution. Except CS, TCE, TSE, and TWE, the precise statistics of other indicators come from China Energy Statistical Yearbook [29] and the employed population statistics comes from the provincial statistical yearbooks 1 in China.

Table 1
The selection of indicators

Type Variable Units

Inputs Capital stock(CS) 100 Million RMB

Employed population(EP) 10 Thousand persons

Energy consumption(EC) 10 Thousand tons

Desirable output GDP 100 Million RMB

Undesirable outputs Total CO₂ emissions(TCE) 10,000 tons

Total SO₂ emissions(TSE) 10,000 tons

Total wastewater emissions(TWE) 10,000 tons

Type	Variable	Units
Inputs	Capital stock(CS)	100 Million RMB
	Employed population(EP)	10 Thousand persons
	Energy consumption(EC)	10 Thousand tons
Desirable output	GDP	100 Million RMB
Undesirable outputs	Total CO₂ emissions(TCE)	10,000 tons
	Total SO₂ emissions(TSE)	10,000 tons
	Total wastewater emissions(TWE)	10,000 tons

Since we can’t obtain specific values of CS, TCE, TSE, and TWE, we have to invite some environmental experts and economic experts to assess people’s level of belief degrees of environmental pollutants and CS, respectively. Take Beijing’s total CO₂ emissions in 2014 as an example, the specific consultation procedure was as shown below.

(Q) What’s your opinion about the minimum Beijing’s total CO₂ emissions in 2014?

(A) 85 million tons. (We got a set of the expert’s experimental data (85,0), which represents that the experts believe that the minimum Beijing’s CO₂ emissions in 2014 is 85 million tons and the belief degree is recorded as 0.)

(Q) What’s your opinion about the maximum Beijing’s total CO₂ emissions in 2014?

(A) 94 million tons. (We got a set of the expert’s experimental data (94,1), which represents that the experts believe that the maximum Beijing’s CO₂ emissions in 2014 is 94 million tons and the belief degree is recorded as 1.)

Two experimental values from the expert’s evaluation as (85,0) and (94,1) about the value of Beining’s total CO₂ emissions in 2014 was obtained. Therefore, the uncertain variable is $L$ (85,94) and its uncertainty distribution is $Φ (x) = {\begin{matrix} 0, & if x \leq 85 \\ (x - 85) / 9, & if 85 < x \leq 94 . \\ 1, & if x > 94 \end{matrix}$ (13)

Similiary, use the same approach to get Beijing’s CS, TSE and TWE in 2014 as $L$ (32660,32664), $L$ (6,10), and $L$ (150710,150716), respectively. Further, we can also get their linear uncertainty distributions similar to (13). The same method was used in 2015 and 2016. Appendix Tables 6 and 7 presents the specific input statistics and output statistics from 2014 to 2016. Next, to comprehensively analyze changes in environmental efficiency across China’s provinces, we firstly calculate the environmental efficiency in China’s 30 provinces from 2014 to 2016. In model I, we let γ = t, t take 2014, 2015, or 2016. Then, we substituted the data in Tables 6 and 7 into model I to calculate the specific environmental efficiency value as shown in Table 2.

Table 2

The 30 provinces’ environental efficiency of China

Province	2014	2015	2016	Average
Beijing	1.000	1.000	1.000	1.000
Tianjin	0.739	0.654	0.846	0.746
Hebei	0.576	0.531	0.659	0.589
Shanxi	0.596	0.534	0.567	0.566
Inner mongolia	0.600	0.573	0.886	0.686
Liaoning	0.492	0.456	0.492	0.480
Jilin	0.507	0.443	0.713	0.554
Heilongjiang	0.568	0.482	0.578	0.543
Shanghai	1.000	1.000	1.000	1.000
Jiangsu	0.919	0.936	0.928	0.928
Zhejiang	0.870	0.875	0.853	0.866
Anhui	0.956	0.896	0.890	0.914
Fujian	0.784	0.754	0.830	0.789
Jiangxi	0.920	0.905	0.845	0.890
Shandong	0.657	0.647	0.779	0.695
Henan	0.588	0.543	0.660	0.597
Hubei	0.872	0.824	0.814	0.836
Hunan	0.858	0.857	0.846	0.854
Guangdong	1.000	0.991	0.968	0.986
Guangxi	0.497	0.443	0.556	0.499
Hainan	0.630	0.687	0.673	0.664
Chongqing	0.855	0.835	0.839	0.843
Sichuan	0.899	0.968	0.855	0.907
Guizhou	0.767	0.763	0.773	0.767
Yunnan	0.655	0.594	0.608	0.619
Shaanxi	0.775	0.672	0.751	0.732
Gansu	0.735	0.606	0.683	0.675
Qinghai	0.495	0.423	0.453	0.457
Ningxia	0.438	0.442	0.515	0.465
Xinjiang	0.629	0.559	0.684	0.624

In Table 3, Beijing and Shanghai performed eximiously from 2014 to 2016. Their environmental efficiencies were achieved 1. Thus, they were efficient. It is worth noting that Guangdong province’s environmental efficiency value in 2014 was 1, which was effective. However, the efficiency values were 0.991 and 0.968 in 2015 and 2016, respectively. It indicates the Guangdong’s environmental efficiency has declined year by year in the past three years. In addition, Ningxia performed the worst since the environmental efficiencies from 2014 to 2016 were 0.438, 0.442 and 0.515. Moreover, its average efficiency over the 3 years was only 0.465. Then, there are 4 provinces with an average environmental efficiency below 0.5, including Liaoning with 0.480, Guangxi with 0.499, Qinghai with 0.457 and Ningxia with 0.465, which is lower than half of the optimal environmental efficiency value. Obviously, the developed provinces are generally more environmentally efficient than less developed provinces. The above results show that the environmental development of China’s provinces is unbalanced.

Table 3

The specific division of China’s regions

Region	Provinces
the eastern region	Beijing, Tianjin, Hebei, Liaoning, Shandong, Shanghai, Jiangsu, Zhejiang, Fujian, Guangdong, Hainan
the central region	Shaanxi, Inner Mongolia, Jilin, Heilongjiang, Anhui, Jiangxi, Henan, Hubei, Hunan, Guangxi
the western region	Chongqing, Sichuan, Guizhou, Yunnan, Shanxi, Gansu, Qinghai, Ningxia, Xinjiang

For the purpose of further analyzing the impact of regional differences on each province’s environmental efficiency, we divided the 30 provinces into three categories: the eastern region, the central region, and the western region. Table 3 shows the specific division.

Then, we calculated the average environmental efficiency of the provinces included in each region and plotted it as a line chart as shown in Fig. 1. Observing Fig. 1, we can easily find that the average environmental conditions in the eastern region are the best, followed by the central region and the worst in the western region. Moreover, the average environmental efficiency of the three regions showed a trend of decreasing first and then increasing from 2014 to 2016. Additionally, it is worth noting that the average environmental efficiency in the central region was slightly higher than the whole country environmental level in 2016, while it was significantly lower than the whole country environmental level in 2014 and 2015. It shows that the growth of the central region’s environmental efficiency is faster than other two regions from 2015 to 2016.

Fig. 1

Changes in China’s regional environmental efficiency.

To further analyze the dynamic changes of environmental efficiency over time in each province from 2014 to 2016, we calculated the MPI for China’s 30 provinces. First, according to the data in Appendix Tables 6 and 7, we use the method in Example 1 to obtain the inverse uncertainty distribution of each uncertain variable. Then, according to models I, II, III, and IV, we calculate the distance functions (i.e., $D_{T}^{2014} ({\tilde{x}}_{o}^{2014}, {\tilde{y}}_{o}^{2014}), D_{T}^{2014} ({\tilde{x}}_{o}^{2015}, {\tilde{y}}_{o}^{2015}), D_{T}^{2015}$ $({\tilde{x}}_{o}^{2014}, {\tilde{y}}_{o}^{2014}), D_{T}^{2015} ({\tilde{x}}_{o}^{2015}, {\tilde{y}}_{o}^{2015}), D_{T}^{2015} ({\tilde{x}}_{o}^{2016}, {\tilde{y}}_{o}^{2016}),$ $D_{T}^{2016} ({\tilde{x}}_{o}^{2015}, {\tilde{y}}_{o}^{2015})$ , $D_{T}^{2016} ({\tilde{x}}_{o}^{2016}, {\tilde{y}}_{o}^{2016})$ , $D_{S}^{2014} ({\tilde{x}}_{o}^{2014},$ ${\tilde{y}}_{o}^{2014})$ , $D_{S}^{2014} ({\tilde{x}}_{o}^{2015}, {\tilde{y}}_{o}^{2015})$ , $D_{S}^{2015} ({\tilde{x}}_{o}^{2014}, {\tilde{y}}_{o}^{2014})$ , $D_{S}^{2015}$ $({\tilde{x}}_{o}^{2015}, {\tilde{y}}_{o}^{2015}), D_{S}^{2015} ({\tilde{x}}_{o}^{2016}, {\tilde{y}}_{o}^{2016})$ , $D_{S}^{2016} ({\tilde{x}}_{o}^{2015}, {\tilde{y}}_{o}^{2015})$ , and $D_{S}^{2016} ({\tilde{x}}_{o}^{2016}, {\tilde{y}}_{o}^{2016})$ ) by using the specific calculation method of integral values in Theorem 2. Finally, according to MPI calculation methods (7)-(12), the average growth of MPI and its decomposition value of China’s 30 province are shown in Table 4 during the 3 years.

Table 4

China’s 30 provinces’ average MPIs and its decomposition

Year	M_RD	UTEC	UTC	USEC	UPTEC
2014-2015	1.010	1.020	0.993	0.994	1.029
2015-2016	0.994	0.980	1.045	1.028	0.950
Average	1.002	1.000	1.019	1.011	0.989

The results show that China’s total factor productivity decreased by 0.016, but the average value was 1.002, indicating that the overall province’s environmental efficiency was rising. From the decomposition of the total factor productivity index, UTEC decreases by 0.04, and UTC increases by 0.052. In contrast, UTC plays a major role in the change the total factor productivity index, indicating that the improvement of the environment from 2014 to 2016 mainly comes from technical progress and technical innovation. In addition, USEC increased by 0.034 and UPTEC decreased by 0.079, indicating that the UPTEC has a greater impact on the UTEC. Then, we calculated the average MPI of China’s each province. Table 5 displays the results.

Table 5

The decomposition of MPI in China’s different provinces

Province	M_RD	UTEC	UTC	USEC	UPTEC
Beijing	0.991	0.970	1.021	0.970	1.000
Tianjin	0.998	1.105	0.903	1.187	0.931
Hebei	0.938	0.922	1.018	0.979	0.941
Shanxi	1.048	1.035	1.013	0.990	1.046
Inner mongolia	1.021	0.856	1.192	0.993	0.863
Liaoning	1.070	1.347	0.794	1.242	1.084
Jilin	0.923	0.882	1.047	1.007	0.876
Heilongjiang	1.052	1.010	1.041	0.995	1.015
Shanghai	0.998	0.998	1.000	0.998	1.000
Jiangsu	0.992	0.978	1.013	0.983	0.996
Zhejiang	0.970	1.037	0.936	1.028	1.009
Anhui	1.036	1.034	1.001	0.991	1.044
Fujian	1.002	0.969	1.035	0.996	0.973
Jiangxi	1.040	1.005	1.034	0.971	1.035
Shandong	0.969	0.877	1.106	1.019	0.861
Henan	0.995	0.963	1.033	0.994	0.969
Hubei	1.056	1.057	0.999	0.996	1.062
Hunan	1.026	1.022	1.004	1.011	1.011
Guangdong	1.016	1.014	1.002	0.999	1.015
Guangxi	0.929	0.893	1.040	0.940	0.950
Hainan	0.959	0.952	1.007	0.937	1.017
Chongqing	1.018	1.009	1.009	1.006	1.003
Sichuan	1.050	1.049	1.001	1.012	1.036
Guizhou	1.026	1.015	1.011	1.021	0.994
Yunnan	1.066	1.053	1.013	1.020	1.032
Shaanxi	1.051	1.026	1.024	1.001	1.025
Gansu	1.057	1.021	1.035	0.985	1.036
Qinghai	1.122	1.028	1.091	1.060	0.970
Ningxia	1.021	0.939	1.087	1.036	0.907
Xinjiang	1.000	0.943	1.060	0.969	0.974
Average	1.015	1.000	1.019	1.011	0.989

In Table 5, from 2014 to 2016, some provinces’ average MPIs are less than 1, indicating that these provinces’ environmental efficiency exhibited a downward trend. However, the 30 provinces’ average MPI are growing with an average growth rate of 1.5 percent, indicating that China’s overall environmental efficiency is improving. Then, from the perspective of technological progress, except for Tianjin, Hubei, Liaoning and Zhejiang, other provinces’ average UTC values are all exceed 1. It shows that environmental technology innovation is increasingly competitive with China’s rapid economic development, and technological progress has become the key factor to promote environmental efficiency in each province. From the perspective of pure technical efficiency changes, UPTEC values of all provinces fluctuated around 1, and nearly half of 30 provinces had UPTEC values less than 1. In addition, the average UPTEC value of the 30 provinces is 0.989, which is less than 1. These indications suggest that pure technical efficiency is an unimportant factor in promoting provincial environmental efficiency. From the perspective of scale changes, the overall average USEC value of the 30 provinces is 1.011, which is greater than 1. It shows that China’s environmental returns to scale are generally on the rise, and scale is an important factor for affecting the environmental efficiency of Chinese provinces. From these analysis, it can be seen that the growth momentum of environmental total factor productivity in China’s provinces mainly from technological progress and the promotion of returns to scale. How to improve pure technical efficiency is an issue that needs attention in China’s environment.

For the sake of reflecting the environmental total factor productivity’s change of China’s provinces well, we have drawn a distribution map as shown in Fig. 2 according to the size of the data and the geographic location of each province. Intuitively, Hebei, Yunnan and Jilin have the lowest degree of coloration, except for provinces such as Tibet that are inconsiderable. Among them, Hebei and Jinlin are the northern provinces of China, and its economic development and technological level are relatively weak. Yunnan, as one of the China’s southernmost provinces, due to the impact of geographical location and other factors, its economic development and technological level still are relatively weak.

Fig. 2

Distribution map of MPI in different provinces of China.

5 Conclusions

The current research on uncertain DEA models is only based on sectional data to calculate DMU’s static efficiency value. This arcticle introduced the uncertain DEA-Malmquist productivity index model to describe the dynamic change of DMU efficiency. Moveover, the Malmquist productivity index was decomposed to technology change, pure technical efficiency change, and scale efficiency change to explore the impact of technology and scale. Then, we applied this model to measure the environmental efficiency changes in China’s provinces from 2014 to 2016.

The main conclusions were as follows: (1) From 2014 to 2016, Beijing and Shanghai achieved the optimal environmental efficiency and they could serve as benchmarks for inefficient provinces to improve their environmental efficiency. (2) From the regional division, the eastern region of China has the best environmental efficiency, followed by the central region, and the western region has the lowest environmental efficiency. It showed the imbalance of regional environmental development in China. In addition, from 2015 to 2016, the growth of the environmental efficiency in central China was significantly faster than other regions. (3) From the Malmquist productivity index and its decomposition, China’s average environmental total factor productivity has been on the rise during the 3 years. The improvement of China’s environmental total factor productivity primarily from technological progress and the improvement of returns to scale. How to improve pure technical efficiency was an issue that needs more attention.

In this paper, the selection of input and output indicators is based on previous research results and the actual situation. Human subjective factors are relatively large. In future research, Tobit regression model may be introduced into the selection of indicators of the model to make more accurate selection of indicators. In addition, the model proposed in this paper may be extended based on the double-frontier DEA model for empirical analysis.

Footnotes

Acknowledgments

This work was funded by the National Natural Science Foundation of China (Grant Nos. 12061072 and 62162059) and the Xinjiang Key Laboratory of Applied Mathematics (Grant No. XJDX1401).

Appendix

See Tables 6 and 7.

The special data of the provincial statistical yearbooks in China are from .

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