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A new uncertain DEA model and application to scientific research personnel

  • Meilin Wen
  • Xue Yu
  • Fei WangEmail author
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Abstract

Data envelopment analysis (DEA), which has been widely used since it was introduced by Charnes et al. (J Econom 30:91–107, 1985), is an effective method for evaluating the relative efficiency of decision-making units. DEA models require accurate input data and output data; however, if no sample is available to estimate accurate data, then uncertain DEA is introduced. This paper reports on several new studies on uncertain DEA using the Hurwicz criterion, which attempts to find the intermediate area between extremes. Some uncertain DEA models, as well as their crisp equivalent models, are presented. Then, the Hurwicz ranking method is proposed based on these models, which can give an evaluation to all the decision-making units. By varying the parameter \(\beta \) in the Hurwicz criterion, which reflects the optimism of the decision maker, the new ranking method can exhibit various forms. Finally, an application to scientific personnel is provided to prove the advantage of the proposed method.

Keywords

Data envelopment analysis Hurwicz criterion Ranking method Uncertainty theory Uncertain measure 

1 Introduction

Data envelopment analysis (DEA), which was introduced by Charnes et al. (1985), is a methodology that was first used in the area of operational research to compare the productive efficiency of different independent units. Since then, many extensions of DEA have been developed, such as those proposed by Banker et al. (1984), Cooper et al. (1999), and Petersen (1990). DEA has greatly enriched the microeconomic theory (Liu 2009) and its application in production function technology. Simultaneously, this method exhibits superiority that cannot be underestimated in terms of avoiding competent factors, simplifying algorithms, reducing errors, and so on.

Decision-making units (DMUs) (Yang et al. 2011) are grouping into two sets, namely efficient and inefficient DMUs. In some cases, ranking is required for DMUs; hence, studies on ranking have become popular. Andersen and Petersen (1993) developed the super-efficiency approach, which may exhibit better performance in establishing a ranking value than methods that evaluate DMU exclusion from linear constraints. If a DMU is chosen as an effective target for other DMUs in the benchmarking ranking method (Wen and Li 2009), then the DMU is highly ranked.

The original DEA models assume that inputs and outputs are measured by exact values. In many situations, however, inputs and outputs are volatile and complex and thus are difficult to measure accurately. Consequently, some researchers have applied probability theory to establish stochastic DEA models, including Sengupta (1982), Banker (1993), Olesen and Petersen (1995), and Cooper et al. (1996). In addition, numerous documents have referred to fuzzy DEA when some inputs or outputs are fuzzy numbers. Kao and Liu (2000), Liu (2008) developed a method for determining the membership functions of fuzzy efficiency scores. Other researchers, including Entani et al. (2002), Guo and Tanaka (2001), Rashidi and Cullinane (2019), and Lertworasirikul et al. (2003) further explored possible measures.

Liu (2007) presented an uncertainty theory to deal mathematically with human belief degree, which he refined in Liu (2010). To obtain probability distribution, numerous samples are required. Sometimes, however, no sample is available because of economical or technological reasons. In such case, we have to invite domain experts to evaluate the belief degree that each possible event will happen. Given that humans tend to overweigh unlikely events (Kahneman and Tversky 1979), the belief degree has a considerably larger variance than the frequency and thus cannot be regarded as a probability distribution of a random variable. In such case, we can consider the belief degree as an uncertainty distribution of several uncertain variables and manage it using uncertainty theory. Some researchers focused on uncertainty theory and propose several different uncertainty DEA models. Nejad and Ghaffari-Hadigheh (2018) proposed a new model that could acquire the highest belief degree that made the evaluated DMU efficient. However, the feasibility of this model cannot be proved, and the model is infeasible for some instances at higher belief degree rates. Jiang et al. (2018) introduced an uncertain DEA model for scale efficiency evaluations using imprecise inputs and outputs. Additionally, they provided a sensitivity and stability analysis of the uncertain DEA model for scale efficiency. In another paper, Jiang et al. (2018) used uncertain variables and uncertain data envelopment analysis model to measure seaports sustainability efficiency of China in 2018. Moreover, they captured the quantity to be improved of each output. Lio and Liu (2018) introduced a new uncertain DEA model; in their paper, the inputs and outputs were considered as uncertain variables. However, this model was very sensitive to data changes, which was not conducive to the practical application of the model. In this study, DMUs will be considered as uncertain variables; thus, a new ranking method based on uncertainty theory is developed.

In previous works, the optimistic criterion and pessimistic criterion, which are both extreme cases, are widely used in uncertain environments. To balance between these two extremes, the Hurwicz criterion was proposed by Hurwicz (1951a, b). Since then, this criterion has been applied to many problems, such as the facility location–allocation problem. Inspired by this work, the current study presents a new ranking method for uncertain DEA using the Hurwicz criterion. In this way, the results are neither too optimistic nor too pessimistic.

This paper is organized as follows: In Sect. 2, we introduce the basic concepts and some results of uncertainty theory. In Sect. 3, we present a new uncertain ranking method using the Hurwicz criterion. Finally, a numerical example is provided in Sect. 4 to illustrate the proposed uncertain ranking method.

2 Preliminaries

Uncertainty theory was founded by Liu (2007) and refined by Liu (2010). As extensions of uncertainty theory, uncertain process and uncertain differential equations (Liu and Ha 2010) and uncertain calculus were proposed (Liu 2009). Besides, uncertain programming was first proposed by Liu (2009), which wants to deal with the optimal problems involving uncertain variable (Liu 2009). This work was followed by an uncertain multiobjective programming and an uncertain goal programming (Liu 2008). Since that, uncertainty theory was used to solve variety of real optimal problems, including risk analysis, reliability analysis (Liu 2010; Zeng et al. 2013; Zhang et al. 2018; Wen and Kang 2016), and decision making (Yao and Ji 2014). 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 \(\Gamma \) be a nonempty set, and \(\L \) a \(\sigma \)-algebra over \(\Gamma \). Each element \(\Lambda \in \L \) is assigned a number \(M\{ \Lambda \}\in [0,1]\). In order to ensure that the number \(M\{ \Lambda \}\) has certain mathematical properties, Liu presented the four axioms (Liu 2007):
Axiom 1.

\(M\{\Gamma \}=1\) for the universal set \(\Gamma \).

Axiom 2.

\(M\{\Lambda \}+M\{\Lambda ^c\}=1\) for any event \(\Lambda \).

Axiom 3.
For every countable sequence of events \(\Lambda _1,\Lambda _2,\ldots ,\) , we have
$$\begin{aligned} M\left\{ \bigcup _{i=1}^{\infty }\Lambda _i \right\} \le \sum _{i=1}^{\infty } M\{\Lambda _i\}. \end{aligned}$$
Axiom 4.
Let \((\Gamma _k,\L _k,M_k)\) be uncertainty spaces for \(k=1,2,\ldots \). Then the product uncertain measure \(M\) is an uncertain measure satisfying
$$\begin{aligned} M\left\{ \prod _{k=1}^{\infty }\Lambda _k\right\} =\bigwedge _{k=1}^{\infty }M_k\{\Lambda _k\} \end{aligned}$$
where \(\Lambda _k\) are arbitrarily chosen events from \(\L _k\) for \(k=1,2,\ldots \), respectively.

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

Definition 1

(Liu (2007)) Let \(\Gamma \) be a nonempty set, \(\L \) a \(\sigma \)-algebra over \(\Gamma \), and \(M\) an uncertain measure. Then the triplet \((\Gamma ,\L ,M)\) is called an uncertainty space.

Definition 2

(Liu (2007)) An uncertain variable \(\xi \) is a measurable function from an uncertainty space \((\Gamma ,\L ,M)\) to the set of real numbers, i.e., for any Borel set B of real numbers, the set
$$\begin{aligned} \{\xi \in B\}=\{\gamma \in \Gamma \,|\,\xi (\gamma )\in B\} \end{aligned}$$
is an event.

Definition 3

(Liu (2007)) The uncertainty distribution \(\Phi \) of an uncertain variable \(\xi \) is defined by
$$\begin{aligned} \Phi (x)=M\{\xi \le x\} \end{aligned}$$
for any real number x.

Example 1

The linear uncertain variable \(\xi \sim {{\mathcal {L}}}(a,b)\) has an uncertainty distribution
$$\begin{aligned} \Phi (x)=\left\{ \begin{array}{cc} 0, &{}\quad \text{ if }\ x\le a \\ (x-a)/(b-a), &{}\quad \text{ if }\ a\le x\le b \\ 1, &{}\quad \text{ if } \ x\ge b. \end{array}\right. \end{aligned}$$
(1)

Example 2

An uncertain variable \(\xi \) is called zigzag if it has a zigzag uncertainty distribution
$$\begin{aligned} \Phi (x)=\left\{ \begin{array}{cl} 0,&{}\quad \text{ if } x\le a\\ (x-a)/2(b-a),&{}\quad \text{ if } a\le x\le b\\ (x+c-2b)/2(c-b),&{}\quad \text{ if } b\le x\le c\\ 1,&{}\quad \text{ if } x\ge c \end{array}\right. \end{aligned}$$
(2)
denoted by \({{\mathcal {Z}}}(a,b,c)\) where abc are real numbers with \(a<b<c\).

Definition 4

(Liu (2009)) The uncertain variables \(\xi _1,\xi _2,\ldots ,\xi _n\) are said to be independent if
$$\begin{aligned} M\left\{ \bigcap _{i=1}^{n}(\xi _i\in B_i)\right\} =\bigwedge _{i=1}^nM\left\{ \xi _i\in B_i\right\} \end{aligned}$$
for any Borel sets \(B_1,B_2,\ldots ,B_n\).

Definition 5

(Liu (2010)) An uncertainty distribution \(\Phi \) of an uncertain variable \(\xi \) is said to be regular if its inverse function \(\Phi ^{-1}(\alpha )\) exists and is unique for each \(\alpha \in (0,1)\). In this case, the inverse function \(\Phi ^{-1}(\alpha )\) is called the inverse uncertainty distribution of \(\xi .\)

Example 3

The inverse uncertainty distribution of a zigzag uncertain variable \({{\mathcal {Z}}}(a,b,c)\) is
$$\begin{aligned} \Phi ^{-1}(\alpha )=\left\{ \begin{array}{cl} (1-2\alpha )a+2\alpha b,&{}\quad \text{ if } \alpha \le 0.5\\ (2-2\alpha )b+(2\alpha -1)c,&{}\quad \text{ if } \alpha >0.5. \end{array}\right. \end{aligned}$$

Theorem 1

(Liu (2010)) Let \(\xi _1,\xi _2,\ldots ,\xi _n\) be independent uncertain variables with regular uncertainty distributions \(\Phi _1,\Phi _2, \ldots ,\Phi _n\), respectively. If f is a strictly increasing function, then
$$\begin{aligned} \xi =f(\xi _1,\xi _2,\ldots ,\xi _n) \end{aligned}$$
(3)
is an uncertain variable with inverse uncertainty distribution
$$\begin{aligned} \Psi ^{-1}(\alpha )=f(\Phi _1^{-1}(\alpha ),\Phi _2^{-1}(\alpha ),\ldots ,\Phi _n^{-1}(\alpha )). \end{aligned}$$
(4)

Example 4

Let \(\xi \) be an uncertain variable with regular uncertainty distribution \(\Phi \). Since \(f(x)=ax+b\) is a strictly increasing function for any constants \(a>0\) and b, the inverse uncertainty distribution of \(a\xi +b\) is
$$\begin{aligned} \Psi ^{-1}(\alpha )=a\Phi _1^{-1}(\alpha )+b. \end{aligned}$$
(5)

Example 5

Let \(\xi _1,\xi _2,\ldots ,\xi _n\) be independent uncertain variables with regular uncertainty distributions \(\Phi _1,\Phi _2, \ldots ,\Phi _n\), respectively. Since
$$\begin{aligned} f(x_1,x_2,\ldots ,x_n)=x_1+x_2+\cdots +x_n \end{aligned}$$
(6)
is a strictly increasing function, the sum
$$\begin{aligned} \xi =\xi _1+\xi _2+\cdots +\xi _n \end{aligned}$$
(7)
is an uncertain variable with inverse uncertainty distribution
$$\begin{aligned} \Psi ^{-1}(\alpha )=\Phi _1^{-1}(\alpha )+\Phi _2^{-1}(\alpha )+\cdots +\Phi _n^{-1}(\alpha ). \end{aligned}$$
(8)

Theorem 2

(Liu (2010)) Assume the constraint function \(g(\varvec{x},\xi _1,\xi _2,\ldots ,\xi _n)\) is strictly increasing with respect to \(\xi _1,\xi _2,\)\(\ldots ,\xi _k\) and strictly decreasing with respect to \(\xi _{k+1},\xi _{k+2},\ldots ,\xi _n\). If \(\xi _1,\xi _2,\ldots ,\xi _n\) are independent uncertain variables with uncertainty distributions \(\Phi _1,\Phi _2,\ldots ,\Phi _n\), respectively, then the chance constraint
$$\begin{aligned} M\left\{ g(\varvec{x},\xi _1,\xi _2,\ldots ,\xi _n)\le 0\right\} \ge \alpha \end{aligned}$$
(9)
holds if and only if
$$\begin{aligned} g(x,\Phi _1^{-1}(\alpha ),\ldots ,\Phi _k^{-1}(\alpha ), \Phi _{k+1}^{-1}(1-\alpha ),\ldots ,\Phi _n^{-1}(1-\alpha ))\le 0. \end{aligned}$$
(10)

3 Uncertain ranking method

The original DEA models assume that inputs and outputs are measured by exact values. In many cases, however, inputs and outputs cannot be given exactly and thus can be considered as uncertain. Wen (2014) provided an uncertain DEA model, where in the symbols and notations are given as follows:

\(\hbox {DMU}_i\): the \(i\hbox {th}\) DMU, \(i=1,2,\ldots ,n\); \(\hbox {DMU}_0\): the target DMU;

\({\widetilde{\varvec{x}}}_k=({\widetilde{x}}_{k1},{\widetilde{x}}_{k2},\ldots , {\widetilde{x}}_{kp})\): the uncertain inputs vector of \(\hbox {DMU}_k\), \(k=1,2,\ldots ,n\);

\(\Phi _{ki}(x)\): the uncertainty distribution of \({\widetilde{x}}_{ki}\), \(k=1,2,\ldots ,n\), \(i=1,2,\ldots ,p\);

\({\varvec{\Phi }}_k(x)=(\Phi _{k1}(x),\Phi _{k2}(x),\ldots , \Phi _{kp}(x))\): the uncertainty distribution vector of \({\widetilde{\varvec{x}}}_k=({\widetilde{x}}_{k1},{\widetilde{x}}_{k2},\ldots , {\widetilde{x}}_{kp})\), \(k=1,2,\ldots ,n\);

\(\varvec{x}_0=(x_{01},x_{02},\ldots ,x_{0p})\): the uncertain inputs vector of the target \(\hbox {DMU}_0\);

\(\Phi _{0i}(x)\): the uncertainty distribution of \({\widetilde{x}}_{0i}\), \(i=1,2,\ldots ,p\);

\({\widetilde{\varvec{y}}}_k=({\widetilde{y}}_{k1},{\widetilde{y}}_{k2},\ldots , {\widetilde{y}}_{kq})\): the uncertain outputs vector of \(\hbox {DMU}_k\), \(k=1,2,\ldots ,n\);

\(\Psi _{kj}(x)\): the uncertainty distribution of \({\widetilde{x}}_{kj}\), \(k=1,2,\ldots ,n\), \(j=1,2,\ldots ,q\);

\({\varvec{\Psi }}_k(x)=(\Psi _{k1}(x),\Psi _{k2}(x),\ldots , \Psi _{kq}(x))\): the uncertainty distribution vector of \({\widetilde{\varvec{y}}}_k=({\widetilde{y}}_{k1},{\widetilde{y}}_{k2},\ldots , {\widetilde{y}}_{kq})\), \(k=1,2,\ldots ,n\);

\(\varvec{y}_0=(y_{01},y_{02},\ldots ,y_{0q})\): the outputs vector of the target \(\hbox {DMU}_0\);

\(\Psi _{0j}(x)\): the uncertainty distribution of \({\widetilde{x}}_{0j}\), \(j=1,2,\ldots ,q\);

The model is given by Wen (2014) as follows:
$$\begin{aligned} \left\{ \begin{array}{l} \theta _1=\max \ \sum \limits _{i=1}^{p}{s_{i}^{-}}+\sum \limits _{j=1}^{q}{s_{j}^{+}}, \\ \text{ which } \text{ is } \text{ subject } \text{ to } \text{: }\\ \quad \quad \quad \quad M\left\{ \sum \limits _{k=1}^{n}{{\widetilde{x}}_{ki}\lambda _{k}}\le {\widetilde{x}}_{0i}-s_{i}^{-}\right\} \ge \alpha _1,\quad i=1,2,\ldots , p;\\ \quad \quad \quad \quad M\left\{ \sum \limits _{k=1}^{n}{{\widetilde{y}}_{kj}\lambda _{k}} \ge {\widetilde{y}}_{0j}+s_{j}^{+}\right\} \ge \alpha _1, \quad j=1,2 \ldots ,q;\\ \quad \quad \quad \quad \sum \limits _{j=1}^{n}{\lambda _{j}}=1; \\ \quad \quad \quad \quad \lambda _j\ge 0, \quad j=1,2,\ldots ,n;\\ \quad \quad \quad \quad s_i^{-}\ge 0,\quad i=1,2,\ldots , p;\\ \quad \quad \quad \quad s_j^{+}\ge 0, \quad j=1,2 \ldots ,q, \end{array}\right. \end{aligned}$$
(11)
which considers the total distances to an efficient frontier.

Definition 6

(Wen (2014)) \(\hbox {DMU}_0\) is \(\alpha \)-efficient if \(s_i^{-*}\) and \(s_j^{+*}\) are zero for \(i=1,2,\ldots ,p\) and \(j=1,2 \ldots ,q\), where \(s_i^{-*}\) and \(s_j^{+*}\) are optimal solutions of (11).

$$\begin{aligned} \left\{ \begin{array}{l} \theta _2=\max \ \sum \limits _{i=1}^{p}{s_{i}^{-}}+\sum \limits _{j=1}^{q}{s_{j}^{+}}, \\ \text{ which } \text{ is } \text{ subject } \text{ to } \text{: }\\ \quad \quad \quad \quad M\left\{ \sum \limits _{k=1}^{n}{{\widetilde{x}}_{ki}\lambda _{k}}\ge {\widetilde{x}}_{0i}+s_{i}^{-}\right\} \ge \alpha _2,\quad i=1,2,\ldots , p;\\ \quad \quad \quad \quad M\left\{ \sum \limits _{k=1}^{n}{{\widetilde{y}}_{kj}\lambda _{k}} \le {\widetilde{y}}_{0j}-s_{j}^{+}\right\} \ge \alpha _2, \quad j=1,2 \ldots ,q;\\ \quad \quad \quad \quad \sum \limits _{j=1}^{n}{\lambda _{j}}=1; \\ \quad \quad \quad \quad \lambda _j\ge 0, \quad j=1,2,\ldots ,n;\\ \quad \quad \quad \quad s_i^{-}\ge 0,\quad i=1,2,\ldots , p;\\ \quad \quad \quad \quad s_j^{+}\ge 0, \quad j=1,2 \ldots ,q. \end{array}\right. \end{aligned}$$
(12)
This model clearly considers the total distance to an efficient frontier. The higher the optimal objective, the less efficient the \(\hbox {DMU}_0\) ranking. The DMUs are compared with the best performances; hence, the method can be regarded as optimistic. By contrast, Jahanshahloo and Afzalinejad (2006) presented a new model that compared DMUs with the worst performances; hence, this method can be regarded as pessimistic. Similarly, the pessimistic model in uncertain environments, which considers the total distance to an inefficient frontier, can be given as follows:

Definition 7

(\(\alpha \)-inefficiency)\(\hbox {DMU}_0\) is \(\alpha \)-inefficient if \(s_i^{-*}\) and \(s_j^{+*}\) are zero for \(i=1,2,\ldots ,p\) and \(j=1,2 \ldots ,q\), where \(s_i^{-*}\) and \(s_j^{+*}\) are optimal solutions of (12).

These definitions are aligned more closely with deterministic optimistic and pessimistic models. However, they also differ because a credibility measure is involved. For example, as determined by the choice of \(\alpha \), a risk that \(\hbox {DMU}_0\) will not be efficient or inefficient exists even when the conditions of Definition 3 or 4 are satisfied.

Given that \(j=0\) is one of the \(\hbox {DMU}_j\), we can always get a solution with \(\lambda _0=1, \lambda _j=0 \ (j\ne 0)\) and all slacks zero. Thus, uncertain DEA models (11) and (12) have feasible solution and the optimal values \(s_i^{-*}=s_j^{+*}=0\) for all \(i, \ j\).

The aforementioned two models are both extreme cases: the first is too optimistic and the second is too pessimistic. Thus, we employ the Hurwicz criterion, which was proposed by Hurwicz (1951a). This criterion incorporates a measure for both by assigning a certain percentage weight \(\beta \) to \(\theta _1^*\) and \(1-\beta \) to \(-\theta _2^*\), with \(0\le \beta \le 1\):
$$\begin{aligned} \theta ^*=\beta \theta _1^*+(1-\beta )(-\theta _2^*) \end{aligned}$$
(13)
which can be rewritten as follows:
$$\begin{aligned} \theta ^*=\beta \theta _1^*-(1-\beta )\theta _2^*. \end{aligned}$$
(14)
Ranking criterion The greater the value \(\theta ^*\), the less efficient the \(\hbox {DMU}_0\) ranking.

In the Hurwicz criterion, parameter \(\beta \in [0,1]\), which reflects the optimism degree of the decision maker, must be determined by the decision maker. In general, determining the appropriate \(\beta \) for decision makers is difficult because this value varies from person to person. By varying parameter \(\beta \), the Hurwicz criterion is transformed into various models. For example, when \(\beta =1\), the criterion is the traditional DEA model (11); meanwhile, when \(\beta =0\), the criterion is transformed into model (12). This phenomenon indicates that the Hurwicz criterion is fairly flexible.

In some cases, \(\theta _1^*=0\) and \(\theta _2^*=0\), and thus, the ranking value \(\theta ^*=0\). which indicates that \(\hbox {DMU}_0\) is both efficient and inefficient. This phenomenon occurs when \(\hbox {DMU}_0\) is the best in some inputs or outputs and the worst in some inputs or outputs. For example, if \(\hbox {DMU}_0\) is the only DMU with the largest value for input 1 and least amount for input 2, then \(\hbox {DMU}_0\) is both efficient and inefficient.

In Wen (2014), the uncertain DEA model (11) can be converted into the crisp model, as follows:
$$\begin{aligned} \left\{ \begin{array}{l} \theta _1^*=\max \quad \sum \limits _{i=1}^{p}{s_{i}^{-}}+\sum \limits _{j=1}^{q}{s_{j}^{+}}, \\ \hbox {which is subject to :}\\ \quad \quad \quad \quad \sum \limits _{k=1,k\ne 0}^{n}\lambda _{k}\Phi _{ki}^{-1}(\alpha _1)+\lambda _{0}\Phi _{0i}^{-1}(1-\alpha _1)\\ \quad \quad \quad \quad \quad \le \Phi _{0i}^{-1}(1-\alpha )-s_{i}^{-}, \qquad i=1,2,\ldots , p; \\ \quad \quad \quad \quad \sum \limits _{k=1,k\ne 0}^{n}\lambda _{k}\Psi _{kj}^{-1}(1-\alpha _1)+\lambda _{0}\Psi _{0j}^{-1}(\alpha _1)\\ \quad \quad \quad \quad \quad \ge \Psi _{0j}^{-1}(\alpha _1)+s_{j}^{+}, \ \quad \qquad j=1,2,\ldots , q; \\ \quad \quad \quad \quad \sum \limits _{k=1}^{n}{\lambda _{k}}=1, \\ \quad \quad \quad \quad \lambda _k\ge 0, \quad k=1,2,\ldots ,n;\\ \quad \quad \quad \quad s_i^{-}\ge 0, \quad i=1,2 \ldots , p;\\ \quad \quad \quad \quad s_j^{+}\ge 0,\quad j=1,2,\ldots ,q, \end{array}\right. \end{aligned}$$
(15)
which is a linear programming model. Thus, this model can be easily solved by many traditional methods.
Table 1

Evaluation criteria of investment in human resources \(X_1\)

\(X_1\)

The number of students

Quality score

Level 1

More than 10

10

Level 2

5–9

5

Level 3

1–4

2

Similarly, the pessimistic model (12) can be transformed into the following linear programming model:
$$\begin{aligned} \left\{ \begin{array}{l} \theta _2^*=\max \quad \sum \limits _{i=1}^{p}{s_{i}^{-}}+\sum \limits _{j=1}^{q}{s_{j}^{+}}, \\ \hbox {which is subject to :}\\ \quad \quad \quad \quad \sum \limits _{k=1,k\ne 0}^{n}\lambda _{k}\Phi _{ki}^{-1}(1-\alpha 2)+\lambda _{0}\Phi _{0i}^{-1}(\alpha 2)\\ \quad \quad \quad \quad \quad \ge \Phi _{0i}^{-1}(\alpha 2)+s_{i}^{+}, \qquad \qquad i=1,2,\ldots , p; \\ \quad \quad \quad \quad \sum \limits _{k=1,k\ne 0}^{n}\lambda _{k}\Psi _{kj}^{-1}(\alpha _2)+\lambda _{0}\Psi _{0j}^{-1}(1-\alpha _2)\\ \quad \quad \quad \quad \quad \le \Psi _{0j}^{-1}(1-\alpha _2)-s_{j}^{-}, \qquad j=1,2,\ldots , q; \\ \quad \quad \quad \quad \sum \limits _{k=1}^{n}{\lambda _{k}}=1; \\ \quad \quad \quad \quad \lambda _k\ge 0, \quad k=1,2,\ldots ,n; \\ \quad \quad \quad \quad s_i^{-}\ge 0, \quad i=1,2 \ldots , p;\\ \quad \quad \quad \quad s_j^{+}\ge 0,\quad j=1,2,\ldots ,q. \end{array}\right. \end{aligned}$$
(16)
The preceding two models are both crisp models. Thus, they can be easily solved using many traditional methods.

4 Application to scientific research personnel

In college, the quality of education greatly depends on the quality of scientific research personnel. Also, their professional ability determines the quality of their own work. Hence, their performance evaluation is of great significance in colleges and universities.

In this section, we present scientific research personnel as an example to demonstrate the applicability and validity of the developed model. For simplicity, we only consider six research stuff as DMUs with two inputs \(X_1\), \(X_2\) and four outputs \(Y_1\), \(Y_2\), \(Y_3\), \(Y_4\).
\(X_1\):

investment in human resources, which can be quantified accurately;

\(X_2\):

investment of financial resources, which is an uncertain variable;

\(Y_1\):

papers and publications, which can be quantified accurately;

\(Y_2\):

awards for teachers, which can be quantified accurately;

\(Y_3\):

professional influence, which is an uncertain variable;

\(Y_4\):

degree of recognition, which is an uncertain variable;

\(X_1\), \(Y_1\), \(Y_2\)

can be easily quantified; the evaluation methods are shown in Tables 12, and 3.

Table 2

Evaluation criteria of professional influence

\(Y_1\)

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

Less than 5 papers appear in the JCR Q1 or Q2

2

Table 3

Evaluation criteria of degree of recognition

\(Y_2\)

Standard for evaluation

Quality score

Level 1

National awards

10

Level 2

Provincial and ministerial awards

5

Level 3

Municipal rewards

2

Table 4

Scientific researchers with two inputs and four outputs

Number of research stuff

1

2

3

4

5

6

\(X_1\)

2

10

10

5

2

5

\(X_2\)

\({{\mathcal {Z}}}\)(2,5,8)

\({{\mathcal {Z}}}\)(5,6,15)

\({{\mathcal {Z}}}\)(6,8,15)

\({{\mathcal {Z}}}\)(5,6,13)

\({{\mathcal {Z}}}\)(6,7,9)

\({{\mathcal {Z}}}\)(6,8,12)

\(Y_1\)

10

10

5

5

2

10

\(Y_2\)

5

2

2

2

5

2

\(Y_3\)

\({{\mathcal {Z}}}\)(8,9,15)

\({{\mathcal {Z}}}\)(6,8,12)

\({{\mathcal {Z}}}\)(3,5,9)

\({{\mathcal {Z}}}\)(4,4.3,7.25)

\({{\mathcal {Z}}}\)(3,4,9)

\({{\mathcal {Z}}}\)(3,5,8)

\(Y_4\)

\({{\mathcal {Z}}}\)(6,8,12)

\({{\mathcal {Z}}}\)(3,6,12)

\({{\mathcal {Z}}}\)(3.5,4,6)

\({{\mathcal {Z}}}\)(3,5,9)

\({{\mathcal {Z}}}\)(2,4,8)

\({{\mathcal {Z}}}\)(3,5,9)

Table 5

Results of evaluating the scientific research

Number of research stuff

1

2

3

4

5

6

\(\theta ^*\)

\(-\) 9.68

0

12.96

\(-\) 1.1

7.23

5.88

As for \(X_2\), \(Y_3\), and \(Y_4\), the evaluation of these four points involves a variety of factors, which cannot be evaluated simply by quantification, so, we obtained the distribution of scientific researchers’ performance levels by issuing questionnaires to relevant personnel. We thought that they are all zigzag uncertain variables denoted by \({{\mathcal {Z}}}(a,b,c)\). Table 4 gives the information of the scientific researchers.

Table 5 shows the final evaluation results; then, we can easily determine the ranking of the values. The smaller the evaluation value, the better the scientific research stuff ranking. We set \(\lambda =0.6\); then, the results are ranked as follows: No. 1, No. 4, No. 2, No. 5, No. 6, No. 3. From this, we can judge that the scientific researcher represented by No. 1 whose comprehensive ability is the best among the six and the scientific researcher represented by No. 3 has the lowest professional ability.

5 Conclusion

DEA has become a popular area of research because of its wide practical applications and extensive background. Decision makers are always interested in complete ranking, which leads to studies on complete ranking methods in DEA. Thus, a new ranking method that employs the Hurwicz criterion in uncertain environments is proposed in this study. The uncertain ranking method can be transformed into various forms by varying parameter \(\beta \) in the Hurwicz criterion. A new DEA model, as well as its equivalent deterministic model, is presented. Finally, an application to scientific personnel is provided to illustrate the uncertain ranking method. In this case, we also found the limitations of this method. Inputs and outputs need to be determined before the evaluation. However, in actual situation, it is difficult to express all the characteristics of the evaluation object with the model inputs and outputs. Therefore, the evaluation method also has room for improvement.

Notes

Acknowledgements

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

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Science and Technology on Reliability and Environmental Engineering LaboratoryBeijingChina
  2. 2.School of Reliability and Systems EngineeringBeihang UniversityBeijingChina
  3. 3.Research Institute of Frontier ScienceBeihang UniversityBeijingChina

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