A new uncertain DEA model for evaluation of scientific research personnel

Abstract

Data envelopment analysis (DEA) has become an accepted tool for assessing efficiency in a wide range of cases since it was first proposed in 1978. However, traditional DEA models need accurate inputs and outputs, which can’t be obtained or measured in many practical cases. This paper will apply DEA into uncertain environment, and propose a new DEA model with uncertain inputs and outputs based on uncertain theory. Furthermore, the uncertainty theory is utilized to convert the new uncertain DEA model into an equivalent deterministic model for simplification. Finally, this new uncertain DEA model is applied to the evaluation of scientific research personnel to illustrate the effectiveness.

Keywords

Data envelopment analysis uncertainty theory uncertain variable uncertainty distribution efficiency

1. Introduction

Data envelopment analysis (DEA) is a nonparametric approach to evaluate the relative efficiency of decision making units (DMUs) proposed by Charnes et al. in 1978 [1] and Banker et al. in 1984 [2]. Decades passed, DEA has been growing rapidly [3 –5], attracting interest of a large number of researchers from different domains like management science and economics.

Traditional DEA models assume that inputs and outputs of the decision making unit (DMU) are measured by exact values. However, because of uncertainty existing in objective world, exact values cannot be obtained as usual. Researchers proposed many approaches to deal with uncertain problems in DEA. As the most widely utilized theory dealing with uncertainty, probability theory is the earliest theory applied to DEA models. Sengupta proposed the stochastic DEA model by applying the expected value to the stochastic inputs and outputs [6]. Banker combined statistical elements with DEA and developed a statistical method [7]. Wei et al. proposed the stochastic DEA model with a reliability constraint [8]. Chance-constrained programming also has been introduced into DEA model in many papers [9 –13].

As uncertainties in objective world are divided into two kinds, aleatory uncertainty and epistemic uncertainty [14], and probability theory only can handle aleatory uncertainty, some other methods are developed. The Imprecise Data Envelopment Analysis (IDEA) method was founded by Cooper et al. to deal with imprecise information on input and output levels, which results in an efficiency interval rather than a single value [15]. Wei et al. applied the robust efficiency procedure into Imprecise DEA [16]. Moreover, Fuzzy DEA model has been proposed to deal with uncertainty in DMUs. Wen & Li extended the traditional DEA models to a fuzzy framework based on credibility measure [17]. Kao & Liu introduced how to find the membership functions of the fuzzy efficiency scores when some observations are fuzzy numbers [18]. Entani et al. developed a DEA model with an interval efficiency measure by the pessimistic and the optimistic values [19]. Recently, Zhou et al. proposed multi-objective Fuzzy DEA model for evaluating DMUs with type-2 fuzzy input and output variables [20]. Hatami-Marbini et al. investigated the role of pre-determined positive ɛ as decision variables bound to differentiate between strongly and weakly efficient DMUs in Fuzzy DEA approaches [21]. In addition, possibility measure has been introduced into DEA since it is widely used in fuzzy data processing [22 –24].

Consider the situation where historical data lacks or even no historical data exists, we have to invite some domain experts to evaluate their belief degree that each event will occur. The object of belief is an event. A belief degree represents the strength with which you believe the event will happen. Generally, you will assign a number between 0 (complete disbelief) and 1 (complete belief) to the belief degree for each event. The higher the belief degree is, the more strongly you believe the event will happen. The belief degree depends heavily on the personal knowledge and preference concerning the event, and different people hold different belief degrees due to their different knowledge and preference. Generally speaking, belief degrees often are wrong since they deviate far from the frequency. Even though, the effect of those belief degrees for decision making cannot be denied. Belief degree function is a type of distribution function for the imprecise quantity representing the degree with which we believe the quantity falls into the left side of the current point. We should use uncertainty theory to measure belief degree function on account of that deviation between belief degree and frequency may lead to counterintuitive results using probability theory [25].

Uncertainty theory was founded by Liu in 2007 [25] and refined by Liu in 2010 [26]. By introducing the uncertain measure, the uncertain variable, the uncertainty distribution, and other concepts, uncertainty theory is viewed as an appropriate mathematical system to model epistemic uncertainty. Uncertainty theory is thought to be an effective theory to measure epistemic uncertainty concerning belief degree.

Thus, when comes to imprecise data with belief degrees, uncertainty variables are better choices to describe those. For example, evaluation indexes in scientific evaluation of research personnel often rely on personal knowledge and preference, so that belief degree function based on uncertainty theory should be introduced into the evaluation. Wen et al. firstly proposed a DEA model based on uncertainty theory [27], and further studied the stability and sensitivity of their model [28]. Nejad introduced another uncertain DEA model with the objective of acquiring the highest belief degree that the evaluated DMU is efficient [29].

This paper will apply the uncertainty theory into DEA and assume that inputs and outputs of DMUs are uncertain variables, and propose a new uncertain DEA model on the basis of input-oriented BCC model. Moreover, the uncertain DEA model proposed will be applied into evaluation of research personnel to illustrate the effectiveness.

The rest of this paper is organized as follows. Section 2 will introduce some basic concepts and properties about uncertain variables. Traditional DEA model will be introduced in Section 3. Then an uncertain DEA model as well as its equivalent crisp model will be presented in Section 4. Finally, a numerical case about evaluation of research personnel is given in Section 5.

2. Preliminaries

Uncertainty theory was founded by Liu in 2007 and refined by Liu in 2010. As extensions of uncertainty theory, uncertain process and uncertain differential equations [30], uncertain calculus were proposed [31]. Besides, uncertain programming was first proposed by Liu in 2009, which wants to deal with the optimal problems involving uncertain variable [32]. This work was followed by an uncertain multiobjective programming and an uncertain goal programming [33], and an uncertain multilevel programming [34]. Since that, uncertainty theory was used to solve variety of real optimal problems, including finance [35 –37], reliability analysis [38 –41], graph [42, 43], train scheduling [44, 45] and decesion making [46]. In this section, we will state some basic concepts and results on uncertain variables. These results are crucial for the remainder of this paper.

Let Γ be a nonempty set, and Ł a σ-algebra over Γ. Each element Λ∈ Ł is assigned a number M {Λ} ∈ [0, 1]. In order to ensure that the number M {Λ} has certain mathematical properties, Liu presented the four axioms [25]:

Axiom 1. M {Γ} =1 for the universal set Γ.

Axiom 2. M {Λ} + M {Λ^c} =1 for any event Λ.

Axiom 3. For every countable sequence of events Λ₁, Λ₂, ⋯, we have $M {\cup_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} {Λ_{i}} .$

Axiom 4. Let (Γ_k, Ł _k, _Mk) be uncertainty spaces for k = 1, 2, ⋯. Then the product uncertain measure M is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}}$ where Λ_k are arbitrarily chosen events from Ł_k for k = 1, 2, ⋯, respectively.

If the set function M satisfies the first three axioms, it is called an uncertain measure.

Definition 1. (Liu [25]) Let Γ be a nonempty set, Ł a σ-algebra over Γ, and M an uncertain measure. Then the triplet (Γ, Ł , M) is called an uncertaintyspace.

Definition 2. (Liu [25]) An uncertain variable ξ is a measurable function from an uncertainty space (Γ, Ł , M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

Definition 3. (Liu [25]) The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = M {ξ \leq x}$ for any real number x.

Example 1. The linear uncertain variable $ξ \sim L (a, b)$ has an uncertainty distribution $Φ (x) = {\begin{matrix} 0, & if x \leq a \\ (x - a) / (b - a), & if a \leq x \leq b \\ 1, & if x \geq b . \end{matrix}$ (1)

Example 2. An uncertain variable ξ is called zigzag if it has a zigzag uncertainty distribution $Φ (x) = {\begin{matrix} 0, & if x \leq a \\ (x - a) / 2 (b - a), & if a \leq x \leq b \\ (x + c - 2 b) / 2 (c - b), & if b \leq x \leq c \\ 1, & if x \geq c \end{matrix}$ (2) denoted by $Z (a, b, c)$ where a, b, c are real numbers with a < b < c.

Definition 4. (Liu [31]) The uncertain variables ξ₁, ξ₂, ⋯ , ξ_n are said to be 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.

Definition 5. (Liu [26]) An uncertainty distribution Φ of an uncertain variable ξ is said to be regular if its inverse function Φ^-1 (α) exists and is unique for each α ∈ (0, 1). In this case, the inverse function Φ^-1 (α) is called the inverse uncertainty distributionof ξ .

Example 3. The inverse uncertainty distribution of a zigzag uncertain variable $Z (a, b, c)$ is $Φ^{- 1} (α) = {\begin{matrix} (1 - 2 α) a + 2 α b, & if α \leq 0.5 \\ (2 - 2 α) b + (2 α - 1) c, & if α > 0.5 . \end{matrix}$

Theorem 1. (Liu [26]) Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If f is a strictly increasing function, then $ξ = f (ξ_{1}, ξ_{2}, \dots, ξ_{n})$ (3) is an uncertain variable with inverse uncertainty distribution $Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, Φ_{n}^{- 1} (α)) .$ (4)

Example 4. Let ξ be an uncertain variable with regular uncertainty distribution Φ. since f (x) = ax + b is a strictly increasing function for any constants a > 0 and b, the inverse uncertainty distribution of aξ + b is $Ψ^{- 1} (α) = a Φ^{- 1} (α) + b .$ (5)

Example 5. Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. Since $f (x_{1}, x_{2}, \dots, x_{n}) = x_{1} + x_{2} + \dots + x_{n}$ (6) is a strictly increasing function, the sum $ξ = ξ_{1} + ξ_{2} + \dots + ξ_{n}$ (7) is an uncertain variable with inverse uncertainty distribution $Ψ^{- 1} (α) = Φ_{1}^{- 1} (α) + Φ_{2}^{- 1} (α) + \dots + Φ_{n}^{- 1} (α) .$ (8)

Theorem 2. (Liu [26]) Assume the constraint function g (bx, ξ₁, ξ₂, ⋯ , ξ_n) is strictly increasing with respect to ξ₁, ξ₂, ⋯, ξ_k and strictly decreasing with respect to ξ_k+1, ξ_k+2, ⋯ , ξ_n. If ξ₁, ξ₂, ⋯ , ξ_n are independent uncertain variables with uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively, then the chance constraint $M {g (x, ξ_{1}, ξ_{2}, \dots, ξ_{n}) \leq 0} \geq α$ (9) holds if and only if

$\begin{matrix} g (x, Φ_{1}^{- 1} (α), \dots, Φ_{k}^{- 1} (α), \\ Φ_{k + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) \leq 0 . \end{matrix}$ (10)

3. DEA model

This section will start with the introduction of the symbols and notations. Suppose there are n DMUs and the symbols and notations are listed as follows:

DMU_k: the kth DMU, k = 1, 2, ⋯ , n;

DMU₀: the target DMU;

x_k = (x_k1, x_k2, ⋯ , x_kp): the inputs vector of DMU_k, k = 1, 2, ⋯ , n;

x₀ = (x₀₁, x₀₂, ⋯ , x_0p): the inputs vector of the target DMU₀;

y_k = (y_k1, y_k2, ⋯ , y_kq): the outputs vector of DMU_k, k = 1, 2, ⋯ , n;

y₀ = (y₀₁, y₀₂, ⋯ , y_0q): the outputs vector of the target DMU₀;

The input-oriented BCC model proposed by Banker et al. [2] evaluates the efficiency of DMU₀ by solving the following linear program in two steps: ${\begin{matrix} \begin{array}{l} \min θ - ε (\sum_{i = 1}^{p} s_{i}^{-} + \sum_{j = 1}^{q} s_{j}^{+}) \\ subject t o : \\ \sum_{k = 1}^{n} x_{i k} λ_{k} + s_{j}^{-} = θ x_{i 0}, i = 1, 2, …, p \\ \sum_{k = 1}^{n} y_{r k} λ_{k} + s_{j}^{+} = y_{j 0}, j = 1, 2, …, q \\ \sum_{k = 1}^{n} λ_{k} = 1 \\ λ_{k} \geq 0, k = 1, 2, …, n \\ s_{j}^{-} \geq 0, i = 1, 2, …, p \\ s_{j}^{+} \geq 0, j = 1, 2, …, q \end{array} \end{matrix}$ (11) where the $s_{i}^{-}$ and $s_{j}^{+}$ are slack variables. Here, ɛ > 0 is a so-called non-Archimedean element defined to be smaller than any positive real number.

Definition 6. (Efficiency) If an optimal solution (θ^*, s^-*, s^+*) obtained in this two-phase process for model (11) satisfies θ^* = 1 and has no slack s^-* = s^+* = 0, then the DMU₀ is called BCC-efficient, otherwise it is BCC-inefficient.

4. Uncertain DEA model

Since exact values are difficult to obtain as inputs and outputs in DEA model, probability theory is widely applied to DEA models to measure uncertainty. However, probability theory cannot deal with the situation where epistemic uncertainty exists. As is explained in Section 1, uncertainty theory has an advantage in handling imprecise data with epistemic uncertainty and belief degrees. Therefore, this section will discuss uncertain DEA using the theory introduced in Section 2. The new symbols and notations are given as follows:

DMU_k: the kth DMU, k = 1, 2, ⋯ , n;

DMU₀: the target DMU;

${\tilde{x}}_{k} = ({\tilde{x}}_{k 1}, {\tilde{x}}_{k 2}, \dots, {\tilde{x}}_{kp})$ : the uncertain inputs vector of DMU_k, k = 1, 2, ⋯ , n;

Φ_ki (x): the uncertainty distribution of ${\tilde{x}}_{ki}$ , k = 1, 2, ⋯ , n, i = 1, 2, ⋯ , p;

${\tilde{x}}_{0} = ({\tilde{x}}_{01}, {\tilde{x}}_{02}, \dots, {\tilde{x}}_{0 p})$ : the inputs vector of the target DMU₀;

Φ_0i (x): the uncertainty distribution of ${\tilde{x}}_{0 i}$ , i = 1, 2, ⋯ , p;

${\tilde{y}}_{k} = ({\tilde{y}}_{k 1}, {\tilde{y}}_{k 2}, \dots, {\tilde{y}}_{kq})$ : the uncertain outputs vector of DMU_k, k = 1, 2, ⋯ , n;

Ψ_kj (x): the uncertainty distribution of ${\tilde{y}}_{kj}$ , k = 1, 2, ⋯ , n, j = 1, 2, ⋯ , q;

${\tilde{y}}_{0} = ({\tilde{y}}_{01}, {\tilde{y}}_{02}, \dots, {\tilde{y}}_{0 q})$ : the outputs vector of the target DMU₀;

Ψ_0j (x): the uncertainty distribution of ${\tilde{y}}_{0 j}$ , j = 1, 2, ⋯ , q.

Similar to traditional DEA model [2], the objective of the uncertain DEA model is to maximize the total slacks in inputs and outputs subject to the constraints. Then the uncertain DEA model can be given as follows:

$\begin{matrix} {\begin{matrix} min θ - ɛ (\sum_{i = 1}^{p} s_{i}^{-} + \sum_{j = 1}^{q} s_{j}^{+}) \\ subject to : \\ M {\sum_{k = 1}^{n} x_{ik} λ_{k} + s_{i}^{-} \leq θ x_{i 0}} \geq α, i = 1, 2, \dots, p \\ M {\sum_{k = 1}^{n} y_{rk} λ_{k} - s_{j}^{+} \geq y_{j 0}} \geq α, j = 1, 2, \dots, q \\ \sum_{k = 1}^{n} λ_{k} = 1 \\ λ_{k} \geq 0, k = 1, 2, \dots, n \\ s_{i}^{-} \geq 0, i = 1, 2, \dots, p \\ s_{j}^{+} \geq 0, j = 1, 2, \dots, q, \end{matrix} \end{matrix}$ (12)

Definition 7. (α-efficiency) DMU₀ is α-efficient if θ^* = 1 and $s_{i}^{- *} = 0, s_{j}^{+ *} = 0$ for all i = 1, 2, ⋯ , p and j = 1, 2 ⋯ , q, where θ^*, $s_{i}^{- *}$ and $s_{j}^{+ *}$ are optimal solutions of (12).

Since the uncertain measure is involved, this definition is different from traditional efficiency definition. For instance, as determined by the choice of α, there is a risk that DMU₀ will not be efficient even when the condition of Definition 1 is satisfied.

Since j = 0 is one of the DMU_j, we can always get a solution with θ = 1, λ₀ = 1, λ_j = 0 (j ≠ 0) and all slacks zero. Thus this uncertain DEA model has feasible solution.

Model (12) is an uncertain programming model, which is too complex to compute directly. This section will give its equivalent crisp model to simplify the computation process.

Following Theorem 2, the uncertain DEA model can be converted to the crisp model as follows: ${\begin{array}{l} \min θ - ε (\sum_{i = 1}^{p} s_{i}^{-} + \sum_{j = 1}^{q} s_{j}^{+}) \\ subject to : \\ \sum_{k = 1, k \neq 0}^{n} λ_{k} Φ_{k i}^{- 1} (α) + λ_{0} Φ_{0 i}^{- 1} (1 - α) \\ \leq θ Φ_{0 i}^{- 1} (1 - α) - s_{i}^{-}, i = 1, 2, \dots, p \\ \sum_{k = 1, k \neq 0}^{n} λ_{k} Ψ_{k j}^{- 1} (1 - α) + λ_{0} Ψ_{0 j}^{- 1} (α) \\ \geq Ψ_{0 j}^{- 1} (α) + s_{j}^{+}, j = 1, 2, \dots, q \\ \sum_{k = 1}^{n} λ_{k} = 1 \\ λ_{k} \geq 0, k = 1, 2, \dots, n \\ s_{i}^{-} \geq 0, i = 1, 2 \dots, p \\ s_{j}^{+} \geq 0, j = 1, 2, \dots, q \end{array}$ (13) which is a linear programming model. Thus it can be easily solved by many traditional methods.

5. Application to evaluation of scientific research personnel

In this section, we apply the model presented before to the scientific evaluation of research personnel. Scientific researchers play important roles in scientific research institution, since scientific research progress of the institution greatly depends on the quality of scientific researchers. Therefore, the evaluation of their performance is of great significance in scientific research institution. However, since a number of scientific evaluation indexes cannot be measured accurately, subjective assessment is needed to help evaluation, which would bring in belief degrees. Thus, it is necessary to introduce uncertain DEA model proposed to this evaluation problem.

To evaluate the researchers, six research factors are extracted as DMUs for simplicity, which have two inputs X₁, X₂ and four outputs Y₁, Y₂, Y₃, Y₄.

X₁: investment in human resources, which can be quantified accurately;

X₂: investment of financial resources, which is an uncertain variable;

Y₁: papers and publications; which can be quantified accurately;

Y₂: awards for teachers; which can be quantified accurately;

Y₃: professional influence; which is an uncertain variable;

Y₄: degree of recognition, which is an uncertain variable.

X₁, Y₁, Y₂ can be easily quantified, the evaluation methods are show in Tables 1–3.

Table 1
Evaluation criteria of investment in human resources

X₁ The number of students Quality score

Level 1 More than 9 10

Level 2 5–9 5

Level 3 1–4 2

X₁	The number of students	Quality score
Level 1	More than 9	10
Level 2	5–9	5
Level 3	1–4	2

Table 2

Evaluation criteria of professional influence

Y₁	Standard for evaluation	Quality score
Level 1	More than 10 papers appear in the JCR Q1 or Q2;Two or more published works	10
Level 2	More than 5 papers appear in the JCR Q1 or Q2; One published works	5
Level 3	Not exceeding 5 papers appear in the JCR Q1 or Q2	2

Table 3

Evaluation Criteria of degree of recognition

Y₂	Standard for evaluation	Quality score
Level 1	National awards	10
Level 2	Provincial and ministerial awards	5
Level 3	Municipal rewards	2

The evaluation of X₂, Y₃, Y₄ involves a lot of factors, which cannot be quantified directly. Thus, we obtained the uncertainty distribution of those three factors by issuing questionnaires to relevant personnel as belief degree function. Based on the six research factors, two numerical examples are studied as follow.

Case 1

Assuming that those distribution patterns are all zigzag uncertain distribution, Table 4 gives the information of the scientific researchers.

Table 4

Scientific researchers with two inputs and four outputs (case 1)

Number of researcher	1	2	3	4	5	6
X₁	2	10	10	5	2	5
X₂	𝒵(2,5,8)	𝒵(5,6,15)	𝒵(6,8,15)	𝒵(5,6,13)	𝒵(6,7,9)	𝒵(6,8,12)
Y₁	10	10	5	5	2	10
Y₂	5	2	2	2	5	2
Y₃	𝒵(8,9,15)	𝒵(6,8,12)	𝒵(3,5,9)	𝒵(4,4.3,7.25)	𝒵(3,4,9)	𝒵(3,5,8)
Y₄	𝒵(6,8,12)	𝒵(3,6,12)	𝒵(3.5,4,6)	𝒵(3,5,9)	𝒵(2,4,8)	𝒵(3,5,9)

According to the information above, Table 5 shows the result of the DEA model with α=0.8. We can see that the performance of researcher 1,2,4,6 is α-efficiency, indicating that they are better than the other two in terms of scientific research. Researcher 5 is the worst since θ^* of stuff 5 is the largest.

Table 5

Results of evaluating the scientific researcher (case 1)

Number of researcher	1	2	3	4	5	6
$θ^{} - ɛ (\sum_{i = 1}^{p} s_{i}^{- } + \sum_{j = 1}^{q} s_{j}^{+ *})$	1	1	0.2759	1	0.96	1
θ ^*	1	1	1.4118	1	1.5	1

Case 2

Normal uncertain distribution and linear uncertain distribution assumed, Table 6 shows another information of the scientific researchers.

Table 6

Scientific researchers with two inputs and four outputs (case 2)

Number of researcher	1	2	3	4	5	6
X₁	10	2	10	5	10	10
X₂	𝒩(8,0.3)	𝒩(7,0.3)	𝒩(6,0.6)	𝒩(8,0.6)	𝒩(6,0.9)	𝒩(5,0.8)
Y₁	5	2	5	2	10	5
Y₂	5	5	5	2	5	10
Y₃	ℒ(3,8)	ℒ(3,9)	ℒ(4,7.5)	ℒ(3,9)	ℒ(7,11)	ℒ(9,15)
Y₄	𝒩(5,0.6)	𝒩(4,0.6)	𝒩(5,0.4)	𝒩(4,0.25)	𝒩(6,0.9)	𝒩(8,0.7)

According to the information above, Table 7 shows the result of the DEA model with α = 0.8. Obviously, the performance of researcher 2,5,6 is α-efficiency, while the performance of researcher 1 is weak efficiency. Researcher 3 is worse and researcher 4 is the worst.

Table 7

Results of evaluating the scientific researcher (case 2)

Number of researcher	1	2	3	4	5	6
$θ^{} - ɛ (\sum_{i = 1}^{p} s_{i}^{- } + \sum_{j = 1}^{q} s_{j}^{+ *})$	0.3871	1	0.4784	0.6849	1	1
θ ^*	1	1	1.0126	1.36	1	1

6. Conclusion

Considering the epistemic uncertainty existing in the practical cases, this paper proposed a new DEA model with uncertain inputs and outputs. Based on uncertainty theory, the model has been simplified into a linear programming. The application to the evaluation of scientific research personnel shows the effectiveness of the model presented bofore. In the future, the sensitivity and stabilityof the model need to be further studied. Besides, in many conditions, both aleatory uncertainty and epistemic uncertainty exist, which has not been solved in this paper. Uncertain random variables are needed to introduced into DEA model to deal with this situation in future research.

Conflicts of interest

The authors declare no conflict of interest.

Footnotes

Acknowledgments

This work was supported by Science Challenge Project (No. TZ2018007) and National Natural Science Foundation of China (No. 71671009).

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A new uncertain DEA model for evaluation of scientific research personnel

Abstract

Keywords

1. Introduction

2. Preliminaries

Table 1 Evaluation criteria of investment in human resources X1 The number of students Quality score Level 1 More than 9 10 Level 2 5–9 5 Level 3 1–4 2

Conflicts of interest

Footnotes

Acknowledgments

References

Table 1
Evaluation criteria of investment in human resources

X₁ The number of students Quality score

Level 1 More than 9 10

Level 2 5–9 5

Level 3 1–4 2