Regret cross-efficiency evaluation using attitudinal entropy approach

Abstract

Data envelopment analysis (DEA) is a popular nonparametric tool for measuring the performance of decision making unit (DMU). Cross-efficiency evaluation is an efficient means to rank DMUs, which compensates for efficiency overestimation and incomplete ranking of conventional DEA models. With respect to incomplete rational cross-efficiency evaluation, the common methods construct models with two extreme efficiencies as reference, such as efficiency values of the aggressive and benevolent models. In addition, they only explore the irrational preferences of decision maker (DM) at a certain stage in the evaluation process. They fail to investigate the cross-efficiency evaluation synthetically under incomplete rationality and characterize the complexity of the individuals’ decision making. To fill this gap, a new method must be proposed. This paper proposes a new cross-efficiency termed regret cross-efficiency model using attitudinal entropy approach (RACE), which constructs a secondary goal model from a comprehensive perspective under the framework of regret theory, and introduces attitudinal entropy to aggregate the cross-efficiencies. Besides, this paper presents some empirical examples to illustrate the validity and robustness of the RACE method.

Introduction

Data envelopment analysis (DEA) originally proposed by Charnes et al. (1978), is a famous nonparametric method to evaluate the relative efficiency value for a cluster of decision making units (DMUs) with multiple inputs and outputs. It has a wide range of applications and it is used for performance evaluation in various fields. For instance, evaluation of banking related business efficiency (Parkan and Wu, 1999; Soteriou and Zenios, 1999; Saha and Ravisankar, 2000), related work efficiency evaluation in education (Colbert et al., 2000; Johnes and Li, 2008; Li, 2011), evaluation of medical efficiency (Birman et al., 2003; O’Neill et al., 2008; Kazley and Ozcan, 2009), and measures of environmental performance (Dyckhoff and Allen, 2001; Zaim, 2004; Hua et al., 2007).

Although the DEA method has been widely applied, it still has several limitations. For instance, the weights derived from DEA are often highly favorable to the evaluated DMU, potentially leading to extremely unreasonable weight distributions and resulting in the overestimation of the DMU’s efficiency (Dyson and Thanassoulis, 1988; Wong and Beasley, 1990). Moreover, since efficient DMUs are assigned the same efficiency score, DEA is unable to effectively differentiate and rank these efficient DMUs (Sun et al., 2013). Hence, a series of DEA extension methods have been proposed. The cross-efficiency evaluation method, first introduced by Sexton et al. (1986), is a typical representative among them. The primary distinction between cross-efficiency evaluation and traditional DEA lies in the fact that cross-efficiency evaluation combines self-evaluation with peer-evaluation (Anderson et al., 2002). The cross-efficiency evaluation overcomes the shortcoming of the DEA evaluation and implements complete sorting of all DMUs (Boussofiane et al., 1991). Benefiting from the advantages of cross-efficiency evaluation, this method has been applied in numerous domains, such as airline environmental efficiency (Cui and Li, 2020), energy efficiency (Khodadadipour et al., 2021; Wang et al., 2021), and road safety efficiency (Zhu et al., 2021). However, there are also some problems in the practical application of cross efficiency. The self-evaluation efficiency of cross-efficiency is solved by traditional DEA model, but the optimal weight solution may be non-unique, it leads to the problem that cross-efficiency solution values are not unique. Alternative optimal weight solutions seriously reduce the usefulness of cross-efficiency evaluation (Jahanshahloo et al., 2011; Alcaraz et al., 2013; Khodabakhshi and Aryavash, 2017; Liu, 2018).

In regard to non-unique efficiency value caused by the multiple optimal weight solutions, a secondary goal model was developed to determine unique optimal weights by Doyle and Green (1994). As a precursory work in developing the secondary objective model, they constructed the aggressive and benevolent cross-efficiency models. The aggressive model aims to maximize the efficiency value of the evaluated DMU, while minimizing the gross cross-efficiency value of other DMUs. The benevolent model not only maximizes the efficiency value of the evaluated DMU but also maximizes the total cross-efficiency value of other DMUs as much as possible. Afterwards, Liang et al. (2008) extended the model of Doyle and Green (1994) by introducing three alternative secondary goal model represented disparate efficiency evaluation criterion, the relationship between the stability of the cross-efficiency and the multiple solutions of the DEA model is discussed to certain extent. Lim (2012) further extended the secondary objective model by introducing minimax and maximin functions. This allows for increasing the efficiency value of the worst-performing DMU or reducing the efficiency value of the best-performing DMU when selecting weights, thereby balancing the performance across all DMUs. Nevertheless, most secondary objective model approaches focus solely on the efficiency values of individual DMUs while overlooking the more practical issue of ranking. Given this situation, Wu et al. (2009b) put forward a consideration ranking DMUs optimally and determined the final cross-efficiency values on account of the secondary goal model. Two conventional strategies of the aggressive and the benevolent models often lead to diverse cross-efficiency values and sorting results, confuse decision makers (DMs) when making decision. Wang et al. (2010) developed a neutral DEA cross-efficiency model that synthesizes the two classic strategies mentioned above. They suggested that DMUs should focus on whether the weight selection is most conducive to the DMU being evaluated, rather than concentrating on maximizing or minimizing the efficiency values of other DMUs. The traditional DEA model may produce extreme weight solutions. These highly asymmetric weights can significantly impact the credibility of the resulting cross-efficiency values, a concern that has garnered considerable attention from scholars. Weight balancing models are continually being developed to prevent the occurrence of extreme weights. (Lam, 2010; Jahanshahloo et al., 2011; Ramón et al., 2011; Wang et al., 2012; Wu et al., 2012a).

Another important issue in cross-efficiency is weight aggregation. In assembling the cross-efficiency matrix to determine the final cross-efficiency value for each DMU, the traditional approach involves calculating the simple arithmetic mean of all cross-efficiency values. Utilizing the average cross-efficiency value to evaluate all DMUs has several shortcomings. It weakens the connection between cross-efficiency values and their corresponding weights, and fails to provide DMs with the necessary weight information to improve their efficiencies. At the same time, the results obtained by this method may not be acceptable to all DMs. In response to the aforementioned issues, scholars have proposed various methods to address these challenges. Wu et al. (2009a) applied cooperative game theory, treating the DMUs as players in a cooperative game. They derived the weights for each DMU during the cross-efficiency assembly by calculating the Shapley value for each player. Angiz et al. (2013) proposed a new aggregation scheme that first transformed the cross-efficiency matrix into a corresponding ranking order matrix, and then introduced an order priority model to obtain the cross-efficiency aggregation weights and determine the final cross-efficiency values. Wu et al. (2022) exploited the Manhattan distance to measure the consensus degree of each DMU and designed an algorithm to converge the consensus degree for weight aggregation. Information entropy was initially proposed by Shannon (1948), then it was widely used in the field of information science. Many scholars have also applied it to weight aggregation. Wu et al. (2011) employed the entropy method to obtain cross-efficiency values based on both the aggressive and benevolent models. Shuai and Wu (2011) adopted gray entropy method to appraise hotel performance. Subsequently, the entropy method was continuously expanded and applied (Wu et al., 2012b; Song and Liu, 2018; Song et al., 2017).

The above-mentioned methods of cross-efficiency evaluation are restricted to a research scope of complete rationality. In reality, DMs often make decisions under bounded rationality rather than complete rationality. Regret theory is a well-known behavioral decision theory introduced by Bell (1982) and Loomes and Sugden (1982). It posits that individuals not only care about the outcomes they achieve but also compare them with the results of alternative options. If they find that choosing another option would have led to a better outcome, they experience regret; otherwise, they feel joy. Regret theory plays a crucial role in behavioral decision-making and has led to numerous significant research contributions (Connolly and Zeelenberg, 2002; Humphrey, 2004; Bourgeois-Gironde, 2010; Mengash and Ayadi, 2022; Wang et al., 2023). Gong et al. (2021) constructed a cross-efficiency evaluation model based on regret theory to deal with fuzzy portfolio selection problems. Liu and Chen (2022) proposed a regret-based cross-efficiency aggregation method, incorporating both aggressive and benevolent cross-efficiency models, and designed a regret cross-efficiency adjustment algorithm to aggregate the evaluation results. Jin et al. (2022) incorporated the characteristics of the DM through language distribution and regret aversion psychology, establishing a language distribution DEA model to assess self-efficiency, while constructing a regret-rejoice super-efficiency model to evaluate cross-efficiency values. In addition to regret theory, prospect theory is also widely applied in decision-making. Kahneman and Tversky (2013) proposed prospect theory to capture the DM’s irrational psychological behavior under conditions of risk. From the perspective of prospect theory, many scholars have applied it to cross-efficiency evaluation to measure the risk preferences of DMs. Liu et al. (2019) developed a secondary goal model which considered two reference points including the optimal point and the worst ideal point based on prospect theory, and appraised the cross-efficiency value of each DMU. Chen et al. (2020) addressed the issue of cross-efficiency evaluation from a different perspective and proposed a cross-efficiency aggregation method based on prospect theory. The primary distinction between prospect theory and regret theory lies in their focus: prospect theory employs an S-shaped value function to represent human perceptions of gains and losses, whereas regret theory focuses on emotions and regret, aiming to minimize potential future regrets. Besides, the issue of subjective preference is also considered in weight aggregation (Wang and Chin, 2011; Yang et al., 2012; Yang et al., 2013; Fang and Yang, 2019).

Based on the above discussion, it is evident that while regret theory has been applied in cross-efficiency evaluation, researchers often focus on ideal and non-ideal reference points or use aggressive and benevolent models as benchmarks, disregarding the comprehensiveness of the evaluation. Moreover, despite the fact that a small number of scholars have considered subjective preference factors in the weight aggregation stage, they are always subject to the partial stage of cross-efficiency evaluation. Consequently, we attempt to comprehensively examine the endogenous and exogenous preferences of the DMs under the framework of regret theory, and introduce the attitudinal entropy to assemble weight and develop a regret cross-efficiency model using attitudinal entropy (RACE) method to portray the psychological factors in the whole cross-efficiency evaluation process. We analyze the performance of R&D Activities in High-tech Industry by region in China, compare the RACE method with some classical models and further conduct comparative analysis between our model and some existing relevant models.

The rest of the paper is organized as follow: Section “Preliminaries” reviews the relevant concepts and knowledge of cross-efficiency evaluation, introduces the regret theory and attitudinal entropy in brief. Section “Regret cross-efficiency evaluation using attitudinal entropy approach (RACE method)” proposes a new method evaluating the cross-efficiency value. Section “Empirical analyses” demonstrates the application of the new model through some examples. Section “Conclusions” presents the conclusions and discussions.

Preliminaries

DEA cross-efficiency method

Suppose there are n DMUs to be evaluated, indexed by $j=1,2,\ldots n$. Each $DM{U}_{j}$ consumes m inputs to produce s outputs, variables ${x}_{ij}(i=1,2,\ldots m)$ and ${y}_{rj}(r=1,2,\ldots s)$ are denoted as the input and output values, respectively. For a particular $DM{U}_{k}(k=1,2,\ldots n)$, its efficiency relative to other DMUs can be measured by the CCR model (Charnes et al., 1978):

$$\begin{array}{ll}{\theta }_{kk}=\,\max \mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{y}_{rk}\\ s.t.\left\{\begin{array}{ll}\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{ik}=1,\\ \mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{y}_{rj}-\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{ij}\le 0,j=1,2,\ldots n,\\ {\mu }_{rk}\ge 0,r=1,2,\ldots s,\\ {\omega }_{ik}\ge 0,i=1,2,\ldots m.\end{array}\right.\end{array}$$

(1)

Where ${\mu }_{rk}$ and ${\omega }_{ik}$ are the non-negative weights assigned to outputs and inputs, respectively. ${\theta }_{kk}$ is the self-evaluation efficiency of $DM{U}_{k}$. Assume ${\mu }_{rk}^{\ast }$, ${\omega }_{ik}^{\ast }$ are the optimal weight solutions of model (1), then ${\theta }_{kk}^{\ast }=\mathop{\sum }\nolimits_{r=1}^{s}{\mu }_{rk}^{\ast }{y}_{rk}$ is denoted as the optimal self-evaluation efficiency of $DM{U}_{k}$. The peer evaluation efficiency defined by Sexton et al. (1986) utilizes the optimal weights of $DM{U}_{k}$ to evaluate the other DMUs. The peer evaluation efficiency of $DM{U}_{d}$ evaluated by $DM{U}_{k}$ is defined as follow:

$${\theta }_{dk}=\frac{\mathop{\sum }\nolimits_{r=1}^{s}{\mu }_{rk}^{\ast }{y}_{rd}}{\mathop{\sum }\nolimits_{i=1}^{m}{\omega }_{ik}^{\ast }{x}_{id}},d,k=1,2,\ldots n,d\,\ne\, k$$

(2)

In cross-efficiency evaluation, model (1) would be calculated n times and the optimal weight solutions are obtained to calculate the cross-efficiency scores for all DMUs. Consequently, every DUM has one self-evaluation efficiency and n-1 peer evaluation efficiencies. Table 1 shows the $n\times n$ cross-efficiency matrix where the efficiency on the diagonal is self-evaluation efficiency values. The average cross-efficiency score is defined as follow:

$$\overline{{\theta }_{d}}=\frac{1}{n}\mathop{\sum }\limits_{k=1}^{n}{\theta }_{dk}$$

(3)

Table 1 Cross-efficiency matrix.

Note that the optimal solution of model (1) may be non-unique, which would lead to non-unique optimal cross-efficiency values for DMUs. Sexton et al. (1986) introduced a secondary goal model to tackle the issue. Doyle and Green (1994) developed the benevolent and aggressive strategies, which are the most commonly used in the secondary goal models. These strategies are presented as follows:

$$\begin{array}{ll}{\rm{max}}\mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}\left(\mathop{\sum}\limits_{j=1,j\ne k}^{n}{y}_{rj}\right)\\ s.t.\left\{\begin{array}{ll}\mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{y}_{rk}-{\theta }_{kk}^{\ast }\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{ik}=0,\\ \mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}\left(\mathop{\sum }\limits_{j=1,j\ne k}^{n}{x}_{ij}\right)=1,\\ \mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{y}_{rj}-\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{ij}\le 0,j=1,2,\ldots n,j\ne k\\ {\mu }_{rk}\ge 0,r=1,2,\ldots s,\\ {\omega }_{ik}\ge 0,i=1,2,\ldots m.\end{array}\right.\end{array}$$

(4)

and

$$\begin{array}{ll}{\rm{min}}\mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}\left(\mathop{\sum }\limits_{j=1,j\ne k}^{n}{y}_{rj}\right)\\ s.t.\left\{\begin{array}{ll}\mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{y}_{rk}-{\theta }_{kk}^{\ast }\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{ik}=0,\\ \mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}\left(\mathop{\sum }\limits_{j=1,j\ne k}^{n}{x}_{ij}\right)=1,\\ \mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{y}_{rj}-\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{ij}\le 0,j=1,2,\ldots n,j\,\ne\, k\\ {\mu }_{rk}\ge 0,r=1,2,\ldots s,\\ {\omega }_{ik}\ge 0,i=1,2,\ldots m.\end{array}\right.\end{array}$$

(5)

In benevolent strategy model (4) and aggressive strategy model (5), ${\theta }_{kk}^{\ast }$ is the optimal efficiency of CCR model (1). The core idea of model (4) is to treat other evaluated DMUs as cooperators and endeavor to maximize the gross efficiency of other DUMs, while model (5) regards other DUMs as competitors and strives to minimize the total efficiency of other DUMs. Compared with the benevolent and aggressive strategies in polarization, Wang and Chin (2010) proposed a neutral DEA model for cross-efficiency evaluation. In addition to these methods, many scholars have also proposed a variety of methods and applications for appraising cross-efficiency (Liang et al., 2008; Wu et al., 2016; Kao and Liu, 2020).

Previous cross-efficiency evaluation models always follow the assumption that DMs are completely rational, but ignore the bounded rationality. Therefore, irrational factors have been taken into the cross-efficiency evaluation models for consideration. Liu et al. (2019) constructed a cross-efficiency model under the framework of prospect theory. Fang and Yang (2019) extended the cross-efficiency evaluation model from prospect theory to cumulative prospect theory. Wu et al. (2022) have also considered consensus reaching for prospect cross-efficiency. The subjective preference of individuals has received much attention from scholars. In order to capture the irrational factors of individuals, we introduce regret theory, which will be briefly introduced in the following sections.

Regret theory

Regret theory, first proposed by Bell (1982) and Loomes and Sugden (1982), incorporates emotions and motivations into the traditional expectation framework. According to expected regret theory, individuals assess their anticipated emotional reactions to future outcomes, and these expected emotions influence the utility function, leading DMs to minimize regret in their decision-making. When presented with a choice between a familiar option and an unfamiliar one, DMs often prefer the familiar option, as it tends to evoke less regret compared to unfamiliar alternatives. Zeelenberg et al. (1996) discovered that individuals tend to prefer choices that minimize regret rather than risk, which helps explain why people sometimes choose safer options and at other times opt for riskier ones, exhibiting seemingly contradictory decision-making behaviors. Owing to its effectiveness in modeling irrational behavior, regret theory has been widely applied in behavioral decision-making research. Specifically, it captures individuals’ perceptions of regret or rejoicing under risk and uncertainty (Fujii et al., 2021; Chi and Zhuang, 2022). DMs experience regret when they choose a worse alternative over a better one and feel rejoicing when they make the opposite choice. To examine the role of irrational behavior in cross-efficiency evaluation, we define and elucidate the relevant theoretical concepts and functions of regret-rejoice utility and perceived utility. The overall construction process is illustrated in Fig. 1.

Fig. 1: The construction process of perceived utility function.

The figure shows the idea of constructing the perceived utility function.

Definition 1: Suppose t denote variable of input attribute value, the utility function $\phi (t)$ of input attribute is as follows:

$$\phi (t)={a}^{-1}(1-{e}^{at}),0 \,<\, a \,<\, 1$$

(6)

Where a is risk aversion parameter of the DM, the larger a is, the higher level of risk aversion of the DM. The input attribute represents the cost of the DM in reality, The graph of Fig. 2 demonstrates the effect of a on $\phi (t)$ for cost attribute. As can be seen the utility function $\phi (t)$ is a monotonically decreasing function of input attribute under constant absolute risk aversion, satisfies $\phi (t){\text{'}} \,<\, 0$ and $\phi (t){\text{''}} \,<\, 0$. It is in line with DMs’ expectation that the less the cost, the better.

Fig. 2: Utility function $\phi (t)$ for input attribute.

The figure shows that the utility function of input attribute monotonically decreases with the increase of input under different aversion parameters.

Definition 2: Suppose $t$ denote variable of output attribute value, the utility function $\phi (t)$ of output attribute is as follows:

$$\phi (t)={b}^{-1}(1-{e}^{-bt}),0 \,<\, b \,<\, 1$$

(7)

Where b is risk aversion parameter of the DM, the larger b is, the higher level of risk aversion of the DM. The output attribute represents the revenue of the DM in reality, the graph of Fig. 3 demonstrates the effect of b on $\phi (t)$ for revenue attribute. As can be seen the utility function $\phi (t)$ is a monotonically increasing concave function of output attribute under constant absolute risk aversion, satisfies $\phi (t)\text{'} > 0$ and $\phi (t)\text{'}\text{'} < 0$, it is in line with DMs’ expectation that the more the revenue, the better.

Fig. 3: Utility function $\phi (x)$ for output attribute.

The figure shows that the utility function of output attribute monotonically increases with the increase of revenue under different aversion parameters.

Definition 3: The regret-rejoice function is defined as follow:

$$R(\Delta \phi )=1-{e}^{-\delta \Delta \phi }$$

(8)

Where $\delta$ is the regret aversion parameter and $\delta \in [0,+\infty )$. It reflects the degree of regret aversion of the DM. The larger the $\delta$ is, the higher level of regret aversion of the DM (Peng and Yang, 2017). Assume ${t}_{1}$ and ${t}_{2}$ are two different variables, then $\Delta \phi =\phi ({t}_{1})-\phi ({t}_{2})$, here $\Delta \phi$ is the utility difference between the DM’s chosen point and reference point. The regret-rejoice function $R(\Delta \phi )$ is a monotonically increasing function, $R(\Delta \phi )$ with diverse $\delta$ is shown in Fig. 4. When $R(\varDelta \phi ) \,<\, 0$, the DMs feel regret on account that the reference point is better than the chosen point. When $R(\varDelta \phi ) \,>\, 0$, the DMs feel rejoice because the reference point is worse than the chosen point. It is easily known in Fig. 4 that the curve is steeper when $\Delta \phi \,<\, 0$, which implies that the DM is more sensitive to regret than rejoice. The DM is more incline to regret perception than rejoice perception, it is consistent with the valuation of gain and loss in prospect theory (Kahneman and Tversky, 2013).

Fig. 4: The regret-rejoice function $R(\varDelta \phi )$.

The figure reflects that individuals perceive regret more than joy under different aversive coefficients.

On the basis of regret theory, regret-rejoice function seeks to capture individuals’ feelings of regret and rejoice. In order to mirror the DMs’ feelings availably, we use the perceived utility function which consists of a regret–rejoice function and a utility function representing current result. Let ${t}_{A}$ and ${t}_{B}$ denote the DM’s perceived evaluation values of alternatives A and B. The perceived utility value of A is shown as follow:

$$U({t}_{A},{t}_{B})=\phi ({t}_{A})+R(\phi ({t}_{A})-\phi ({t}_{B}))$$

(9)

Information entropy

Information entropy, also known as Shannon entropy, was proposed by Shannon (1948). According to information theory, information represents the state of motion and the manner in which things change, both of which involve some degree of uncertainty. Shannon discovered that these changes follow specific statistical laws, leading him to construct a mathematical concept to measure uncertainty. In relation to the conventional aggregation method of cross-efficiency weights, Wu et al. (2011) utilized Shannon entropy to determine the weights for ultimate cross-efficiency scores. Song et al. (2017) further improved Shannon entropy weights for cross-efficiency. Additionally, some scholars have extended other theories into information entropy weight aggregation. For instance, Wen et al. (1998) incorporated gray theory entropy weighting. With regard to information entropy, we delineate the pertinent definitions associated with efficiency evaluation.

Definition 4: For each $DM{U}_{k}(k=1,2,\cdots n)$, the Shannon entropy value is defined as:

$${H}_{k}^{Sh}=\mathop{\sum }\limits_{d=1}^{n}{H}_{dk}^{Sh}=-\mathop{\sum }\limits_{d=1}^{n}{E}_{dk}\,\mathrm{ln}\,{E}_{dk}$$

(10)

Where ${H}_{k}^{Sh}$ is the Shannon entropy of $DM{U}_{k}$, here ${H}_{dk}^{Sh}=-{E}_{dk}\,\mathrm{ln}\,{E}_{dk}$ and ${E}_{dk}={\theta }_{dk}/\mathop{\sum }\nolimits_{d=1}^{m}{\theta }_{dk}$. We map the cross-efficiency values ${\theta }_{dk}$ to approximate probability value ${E}_{dk}$. ${H}_{dk}^{Sh}$ manifests the chaos among cross-efficiency information and mirrors the lability of information. To a certain extent, entropy can reflect the inconsistency of mutual evaluation. Therefore, for cross-efficiency the lower the entropy value, the higher the acceptability of the evaluation results.

Definition 5: For each $DM{U}_{k}(k=1,2,\ldots n)$, the attitudinal entropy value is defined as:

$${H}_{k}^{ASh}={\log }_{\alpha }\left(\mathop{\sum }\limits_{d=1}^{n}{E}_{dk}{\alpha }^{{g}_{d}}\right)={\log }_{\alpha }\left(\mathop{\sum }\limits_{i=1}^{n}{E}_{dk}{\alpha }^{{\log }_{\beta }(\frac{1}{{E}_{dk}})}\right)$$

(11)

Aggarwal (2019) proposed a concept of subjective utility, considered information entropy with subjective factor. Base on this idea, we construct attitudinal entropy of cross-efficiency. ${E}_{dk}$ is the approximate probability value of cross-efficiency score ${\theta }_{dk}$. ${g}_{d}={\log }_{\beta }(1/{E}_{dk})$ denotes information gain function (Aggarwal, 2021a, 2021b), it is a perceived performance of information uncertainty in regard to ${E}_{dk}$. $\beta$ is the base of logarithmic gain function, which is used to control information receiving. In formula (11), $\alpha \in (0,+\infty )$ represents the DM’s preference of information uncertainty, a large parameter $\alpha$ indicates that the DM is more inclined to believe in low-probability events, reflects the individual’s emphasis on relevant low probability among different $\alpha$. The impact of different $\alpha$ on the entropy value with a specific $\beta =2$ under a two-state system is shown in Fig. 5. It is more intuitive to find that attitudinal entropy and individual’s information preference change in the same direction under the 0–1 distribution. When probability is 0.5, attitudinal entropy with different $\alpha$ is equal, it indicates uniform distribution that ${E}_{dk}=1/n$. And as it can be seen attitudinal entropy value is consistent with the original entropy curve when $\alpha$ approaches to 1.

Fig. 5: Attitudinal entropy under different α values.

The figure shows the distribution shape of the entropy function under different information uncertainty preferences.

Regret cross-efficiency evaluation using attitudinal entropy approach (RACE method)

In this section, we are committed to addressing the issues raised above. Section “Regret cross-efficiency model” proposed a secondary goal model which takes both endogenous and exogenous references into account. Section “Aggregation of regret cross-efficiency” considers weight aggregation using attitudinal entropy on the basis of regret cross-efficiency. The overall framework of the method we proposed is shown in the Fig. 6.

Fig. 6: The overall framework of the RACE method.

The figure shows the multiple ideological factors considered in the evaluation process of the regret cross-efficiency model.

Regret cross-efficiency model

Each DMU uses its own optimal weights to evaluate other DUMs in the process of cross-efficiency evaluation. In the meantime, it frequently encounters a problem of non-unique solutions, which results in uncertainty in peer-evaluation. Many scholars have solved this problem by constructing a secondary goal model based on different orientations. Bounded rational decision-making issues have attracted much attention, especially in the field of cross-efficiency evaluation. Such as the extension and application of prospect theory in cross-efficiency evaluation (Liu et al., 2019; Shi et al., 2021). As one of the important theories of bounded rational behavior theory, regret theory has also been applied to cross-efficiency evaluation (Chen et al., 2023; Jin et al., 2023). The DMs’ perception of regret-rejoice comes from the comparison of the results or expectations that can be obtained from the alternatives in the form of reference points. In practical applications, some specific reference points can be selected as following way: (1) the maximum value, (2) the minimum value, (3) the median point, (4) the mean value. Utilizing the reference points above, we construct the secondary goal model. The secondary goal model of bounded rationality previously considered problems from the perspective of exogeneity, while in real life, individuals usually consider problems from both the endogenous and exogenous perspectives of the system when making decisions. To balance between endogeneity and exogeneity, we set both endogenous and exogenous as reference points. Ideal and anti-ideal reference points are set as exogenous reference points (Wang et al., 2011), they are virtual values and can be derived from the sample or specific set by the DMs. Since the mean value is correlated with all relevant individuals’ values in the sample and keeps up with individuals’ concerns about group average level in reality, it is chosen as the endogenous reference point. Some essential definitions of different attributes and the multi-attribute secondary goal model are introduced as follows:

Definition 6: If the reference point is the ideal point, then the DMs would gain regret feeling. The ideal point is regarded as regret attribute point. For $DM{U}_{k}$ the composite perceived functions of input and output are defined as follows:

$${U}^{-}({x}_{ik})=\phi ({x}_{ik})+{R}_{I}^{-}(\varDelta \phi )=\phi ({x}_{ik})+1-\exp (-\delta (\phi ({x}_{ik})-\phi (min\{{x}_{i\cdot }\})))$$

(12)

$${U}^{-}({y}_{rk})=\phi ({y}_{rk})+{R}_{O}^{-}(\varDelta \phi )=\varphi ({y}_{rk})+1-\exp (-\delta (\phi ({y}_{rk})-\phi (\max \{{y}_{r\cdot }\})))$$

(13)

Where ${x}_{i\cdot }={x}_{ij}(j=1,2,\ldots n)$, ${y}_{r\cdot }={y}_{rj}(j=1,2,\ldots n)$. ${R}_{I}^{-}$ and ${R}_{O}^{-}$ are the regret functions of input and output respectively when the ideal point is chosen as reference point. It is obviously known that ${R}_{I}^{-}(\varDelta \phi ) \,<\, 0$ and ${R}_{O}^{-}(\varDelta \phi ) \,<\, 0$. To maximize perceived utility, the absolute values of $|{R}_{I}^{-}(\varDelta \phi )|$ and $|{R}_{O}^{-}(\varDelta \phi )|$ should be minimized.

Definition 7: If the reference point is the anti-ideal point, then the DMs would gain rejoice feelings. The anti-ideal point is regarded as rejoice attribute point. For $DM{U}_{k}$ the composite perceived functions of input and output are defined as follows:

$${U}^{+}({x}_{ik})=\phi ({x}_{ik})+{R}_{I}^{+}(\varDelta \phi )=\phi ({x}_{ik})+1-\exp (-\delta (\phi ({x}_{ik})-\phi (\max \{{x}_{i\cdot }\})))$$

(14)

$${U}^{+}({y}_{rk})=\phi ({y}_{rk})+{R}_{O}^{+}(\varDelta \phi )=\phi ({y}_{rk})+1-\exp (-\delta (\phi ({y}_{rk})-\phi (min\{{y}_{r\cdot }\})))$$

(15)

It is obviously known that ${R}_{I}^{+}(\varDelta \phi ) \,>\, 0$ and ${R}_{O}^{+}(\varDelta \phi ) > 0$. To maximize perceived utility, the values of ${R}_{I}^{+}(\varDelta \phi )$ and ${R}_{O}^{+}(\varDelta \phi )$ should be maximized.

Definition 8: If the reference point is the mean point, then the DMs would gain heterogeneous feelings. The mean reference point is regarded as heterogeneous attribute point. For $DM{U}_{k}$, the composite perceived functions of input and output are defined as follows:

$${U}_{h}({x}_{ik})=\phi ({x}_{ik})+{R}_{I}^{h}(\varDelta \phi )=\phi ({x}_{ik})+1-\exp (-\delta (\phi ({x}_{ik})-\phi (\mathop{{x}_{i\cdot }^{ge}}\limits^{\_})))$$

(16)

$${U}_{h}({y}_{rk})=\phi ({x}_{ik})+{R}_{O}^{h}(\varDelta \phi )=\phi ({y}_{rk})+1-\exp (-\delta (\phi ({y}_{rk})-\phi (\mathop{{y}_{r\cdot }^{ge}}\limits^{\_})))$$

(17)

Where ${x}_{i\cdot }^{\bar{ge}}$ is the geometric mean of ${x}_{ij}(j=1,2,\ldots n)$, $\mathop{{y}_{r\cdot }^{ge}}\limits^{\_}$ is the geometric mean of ${y}_{r\cdot }={y}_{rj}(j=1,2,\ldots n)$, namely ${\rm{x}}_{{\rm{i}}\cdot }^{\bar{\rm{g}}{\rm{e}}}=\root{\rm{n}}\of{{{\rm{x}}}_{{\rm{i}}1}{\rm{x}}_{{\rm{i}}2}\ldots {{\rm{x}}_{\rm{in}}}}$, ${\rm{y}}_{{\rm{r}}\cdot}^{\bar{\rm{g}}{\rm{e}}}=\root{\rm{n}}\of{{\rm{y}}_{{\rm{r}}1}{\rm{y}}_{{\rm{r}}2}\ldots {\rm{y}}_{\rm{rn}}}$. Considering that the arithmetic mean is easily affected by extreme values, the geometric mean is less affected by extreme values and can better reflect the group’s average level. Therefore, the geometric mean is utilized as the endogenous reference point.

Combine formula (12) to (17) and above analysis, a new regret cross-efficiency secondary goal model considering endogeneity and exogeneity can be constructed as follow:

$$\begin{array}{ll}\max \lambda I+(1-\lambda )(\varsigma G+(1-\varsigma )L)\\ s.t.\left\{\begin{array}{ll}I=\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{U}_{h}({x}_{ik})+\mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{U}_{h}({y}_{rk}),\\ G=\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{U}^{+}({x}_{ik})+\mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{U}^{+}({y}_{rk}),\\ L=\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{U}^{-}({x}_{ik})+\mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{U}^{-}({y}_{rk}),\\ \mathop{\sum }\limits_{r=1}^{s}{\mu }_{rk}{y}_{rd}-\mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{id}\le 0,d=1,2,\ldots n,d\,\ne\, k,\\ \mathop{\sum }\limits_{i=1}^{m}{\omega }_{ik}{x}_{ik}=1,\\ {\theta }_{kk}^{\ast }=\mathop{\sum }\limits_{r=1}^{s}{\mu }_{jk}{y}_{jk},\\ {\mu }_{rk}\ge 0,r=1,2,\ldots s,\\ {\omega }_{ik}\ge 0,i=1,2,\ldots m,\\ \lambda ,\varsigma \in [0,1].\end{array}\right.\end{array}$$

(18)

In model (18), the objective function is the DM’s combined perceived utility including endogeneity and exogeneity. Where $\lambda$ denotes the degree of emphasis on endogenous perception, which reflects the DM’s concern on endogenous perception. $\varsigma$ denotes the degree of emphasis on rejoice perception, which reflects the DM’s concern on rejoice perception. Different $\lambda$ and $\varsigma$ can be used to reveal the DM’s laterality. For instance, if $\lambda =1$, the DM will fully focus on the perception of endogenous reference. If $\lambda =0$, the DM will totally concentrate on the perception of exogenous reference, this scenario is similar to some models (Liu et al., 2019; Gong et al., 2021). If $0 \,<\, \lambda \,<\, 1$ and $0 \,<\, \varsigma \,<\, 1$, the DM will decide on the appropriate focus according to individual’s own priority.

Aggregation of regret cross-efficiency

This section interprets how to aggregate weights and acquire the cross-efficiency values of all DMUs. Attitudinal entropy method is used in the above models to calculate the final cross-efficiency, which is the RACE method. The pattern of weight aggregation is similar to those (Song et al., 2017; Wu et al., 2011). Nevertheless, we incorporate the DM’s information preference into weight aggregation for the first time to our best knowledge.

The main steps of the RACE method for cross-efficiency evaluation are as follow:

Setp1: Construct the cross-efficiency matrix based on regret theory.

Calculate the self-evaluated efficiency of each DMU according to the traditional model (see model 1), and evaluate other DMUs’ efficiencies with the optimal weight solutions of the self-evaluated efficiency. Construct objective function to deal with multiple optimal solutions in self-evaluation from different perspectives of the DM. Endogenous and exogenous reference points are selected to build the secondary goal model, obtain cross-efficiency matrix:

$$\begin{array}{*{20}{l}} {} & {\mathrm{DMU}}_1 & {\mathrm{DMU}}_2 & \ldots & {\mathrm{DMU}}_{\rm{n}} \\\begin{array}{l}{\mathrm{DMU}}_1 \\{\mathrm{DMU}}_2 \\\vdots \\{\mathrm{DMU}}_{\rm{n}}\end{array}& \left[\begin{array}{l} \theta_{11} \\\theta_{21} \\\vdots \\\theta_{n1}\end{array}\right.& \begin{array}{l} \theta_{12} \\\theta_{22} \\\vdots \\\theta_{n2}\end{array}& \begin{array}{l} \ldots \\\ldots \\\ddots \\\ldots\end{array}& \left.\begin{array}{l} \theta_{1n} \\\theta_{2n} \\\vdots \\\theta_{nn}\end{array}\right]\end{array}$$

Step 2: Calculate attitudinal entropy value of each DMU.

Standardize the values of the cross-efficiency matrix by column. ${E}_{dk}={\theta}_{dk}/\mathop{\sum }\nolimits_{d=1}^{n}{\theta }_{dk},d=1,2,\cdots n,k=1,2,\ldots n$. Obtain the attitudinal entropy ${H}_{k}^{ASh}$ from formula (11).

Step 3: Acquire the weight coefficient of each DM.

Suppose ${H}_{k}=1-{H}_{k}^{ASh}$, which represents information utility. Since entropy reflects the degree of information uncertainty, ${H}_{k}$ mirrors the degree of information certainty or importance for each DMU. We can obtain weight coefficient ${\upsilon }_{k}={H}_{k}/\mathop{\sum }\nolimits_{k=1}^{n}{H}_{k}$.

Step 4: Synthesize comprehensive cross-efficiency values.

Utilize the cross-efficiency matrix values and comprehensive weight values obtained by above calculation, we can acquire RACE evaluation values: ${E}_{d}^{RACE}=\mathop{\sum }\nolimits_{k=1}^{n}{\upsilon }_{k}{\theta }_{dk}$.

Empirical analyses

In this section, we explore the effectiveness of the RACE method through some practical examples. Firstly, the RACE method is used to evaluate the efficiency of scientific and technological activities in provincial high-tech industries of China. Then we compare the RACE method to some traditional models through this example. Secondly, sensitivity analysis is operated to measure the influence of each parameter on all DMUs. Finally, the comparison between the RACE method and other relevant models is conducted by using an example of Chinese universities’ scientific and technological activities.

An illustration

High-tech industries are profoundly changing the global economic structure and industrial layout, and the technological innovation efficiency of high-tech industries has become a key determinant of regional economic growth and core competitiveness of industrial. At the same time, China is facing many problems such as lack of independent intellectual property rights, excessive reliance on imports, low R&D investment, and unbalanced development between industries and regions in the development of high-tech industries. These are not conducive to the high-quality development of the country’s high-tech industries. Therefore, how to effectively improve the technological innovation efficiency of the country’s high-tech industries and narrow the gap in the technological innovation efficiency of high-tech industries among regions has become a hot topic of concern for scholars. We select 12 provinces of different regions of China to measure the efficiencies of their scientific and technological activities in high-tech industries. The sample data is shown in Table 2, which is from “China Statistical Yearbook on Science and Technology 2021”. The data in Table 2 was published in 2021, while the actual statistical time of the data was 2020. The 12 sample provinces are from the eastern region, the middle region, the western region and the northeast region of China. As with most studies, we chose human resource (Full-time equivalent of R&D personnel) and funding (Expenditure on new products development) as input indicators. In addition, we choose number of inventions in force as one of the output indicators to highlight the high-tech industry’s ownership of innovation. Secondly, sales revenue of new products is the quantification of the conversion income of technological innovation achievements, so we choose it as another output indicator.

Table 2 Input and output of 12 provinces of China in 2020.

After collecting the data, we can utilize the corresponding method to appraise the efficiency of scientific and technological activities in regional high-tech industries. First of all, the self-evaluation efficiency of each DMU was obtained by CCR model (see model 1), which is shown as Table 4. As can be seen from the results of self-evaluation efficiencies, multiple efficiency values of DMUs reach to 1. CCR model makes it impossible to effectively distinguish the efficiency values of 12 DMUs. At the same time, the model (4) and model (5) are used to calculate the cross-efficiency values except of the aggressive method and the benevolent method. The calculation methods above are the traditional cross-efficiency models.

Then, we set the relevant parameters of the model for the RACE method. When the individual faces an event, his/her risk attitude is usually consistent. However, he/she may exhibit different risk attitudes at different stages of processing the same event. For the individual’s consistent risk attitude attribute, we suppose $a=b=0.02$ and $\delta =0.3$ (c.f., Peng and Yang, 2017). Without loss of generality, it is assumed that the DM attaches equal importance to endogeneity and exogeneity, and have the same preference for regret-rejoice perception in exogeneity, so let $\lambda =\varsigma =0.5$, and assume $\alpha =1.5$, $\beta =2$ (c.f., Aggarwal, 2021a, 2021b). Likewise, appraise the date in Table 2 by using the model (18), we can obtain the cross-efficiency matrix in Table 3. Aggregate the weight for each DMU based on the attitudinal entropy method to acquire the cross-efficiency values in the penultimate column of Table 3.

Table 3 The RACE cross-efficiency of 12 regions in 2020.

Besides, through previous calculation we get the corresponding results of the conventional models and the RACE method, which is shown in Table 4. Figure 7 exhibits the radar comparison of 12 DMUs under different methods. It can be seen intuitively in Fig. 7 that the efficiency values obtained by the RACE method are generally intersperse between the benevolent and aggressive values. For one thing, the benevolent model maximizes the total efficiency values of other DMUs, while the aggressive model minimizes the total efficiency values of other DMUs. For another, the traditional models are absolutely rational decision-making method, while the RACE method is a bounded rational evaluation model based on regret theory, which considers different degrees of comprehensive preferences of the DMs. The RACE method can fully sort all DMUs, that is better than the CCR model. As can be noticed, the efficiency value of $DM{U}_{3}$ using the RACE method is more than that of the aggressive method or benevolent method, but the efficiency value of $DM{U}_{11}$ is the opposite. Meanwhile, the rankings of $DU{M}_{4}$ and $DM{U}_{7}$ are different from the traditional models’. In terms of setting the model parameters, the DMs’ preference for reference is balanced, the DMs are risk-seeking with low regret perception and prefer uncertainty. $DM{U}_{3}$ has a better return but $DM{U}_{11}$ gains less, which leads to the efficiency value result of the RACE method. The input of $DM{U}_{7}$ is nearly three times as that of $DM{U}_{4}$, but the output of $DM{U}_{7}$ is less than three times as that of $DM{U}_{4}$. From the perspective of the RACE method, $DM{U}_{4}$ should be assigned a higher efficiency value than $DM{U}_{7}$.

Table 4 Comparison results of traditional models.

Fig. 7: Comparison results of traditional models.

The figure shows the contrast among the proposed RACE model and the traditional CCR, aggressive and benevolent models in the form of a radar map.

Sensitivity analysis

In this subsection, we conduct a series of sensitivity analysis to investigate how different aversion parameters ($a$, $b$, $\delta$) for the DMs to affect the results of assessment. In the meantime, the preference parameters ($\lambda$, $\varsigma$, $\alpha$) for the DMs are presented and analyzed.

Sensitivity analysis of single coefficient

We investigate the sensitivity of risk aversion coefficient as well as regret aversion coefficient, and still follow the assumption of consistent risk attitude for decision-making individuals, namely a = b. First, we keep the other parameters constant, and let the parameter under study vary within a certain interval. The corresponding evaluation cross-efficiency results are shown in Fig. 8. From Fig. 8a, it can be found that the overall efficiency values change relatively smoothly. However, the efficiency values of some DMUs fluctuate with the increase of the risk aversion coefficient. The main reason for this distribution phenomenon is that the influence of a single risk aversion coefficient on individual perceived utility (formula 9) is not direct. In addition, it can be seen that the efficiency of $DM{U}_{10}$ is lower than that of $DM{U}_{4}$ in (0, 0.14), and the efficiency value is greater than that of $DM{U}_{4}$ in (0.14, 0.6). Similarly, the efficiency value of $DM{U}_{9}$ in (0, 0.8) is less than that of $DM{U}_{8}$, and the efficiency value in (0.8, 1) is greater than that of $DM{U}_{8}$. From Fig. 8b, with the increase of the regret aversion coefficient, the overall efficiency of most DMUs show a slight and steady downward trend. The regret-aversion coefficient $\delta$ represents the individual’s regret attitude, it indicates the degree of regret aversion. A great $\delta$ implies large steepness of the regret curve.

Fig. 8: The impact of different parameter variations on efficiency scores.

a represents the sensitivity analysis of the parameter a/b, (b) represents the changes in efficiency values for each DMU as the parameter δ changes, (c) represents the sensitivity analysis for the parameter λ, (d) represents the changes for parameter ς, and (e) shows the efficiency changes with respect to parameter α.

Subsequently, we analyze the influence of different preference coefficients on efficiency values. Figure 8c–e shows the DM’s preferences of endogeneity, rejoice and information uncertainty. Remain the rejoice preference coefficient unchanged, the efficiency value of each DMU change with the endogenous preference coefficient in Fig. 8c. For example, when $\lambda \,<\, 0.64$, $DM{U}_{8}$’s efficiency overtake $DM{U}_{6}$’s efficiency, when $\lambda \,>\, 0.64$, $DM{U}_{6}$’s efficiency exceeds $DM{U}_{8}$’s efficiency. Remain the endogenous preference coefficient unchanged, the efficiency value of each DMU change with the rejoice preference coefficient in Fig. 8d. For instant, $DM{U}_{8}$’s efficiency value is greater than $DM{U}_{6}$’s in (0, 0.07) and (0.35, 0.95), but $DM{U}_{6}$’s is larger than $DM{U}_{8}$’s in (0.07, 0.35) and (0.95, 1). In order to facilitate the weight aggregation of cross-efficiency, we set the value range of the information preference parameter $\alpha$ in (0,2). In Fig. 6e, it can be found that $DM{U}_{1}$ reaches its relative maximum at $\alpha =0.8$, while other $DMUs$ reach their relative minimum at the same point.

Sensitivity analysis of double parameters

In the previous section on the sensitivity analysis of single parameter, we found that obvious changes are mainly concentrated in the intermediate efficiency level with the alteration of the parameter, and it is difficult to look into the variation of the most efficient DMU. In order to explore the influence of different parameter combinations on the efficiency value more synthetically, we analyzed the influence of different parameter combinations on the efficiency value of each DMU. Figure 9 shows the sensitivity analysis of the evaluation double parameters, (a) $\lambda$ and $\varsigma$, (b) a/b and $\alpha$, (c) $\alpha$ and $\delta$, binary parameter values and the corresponding efficiency value are utilized to fit the 3D curves of each DUM. It is can be seen the influence of the endogenous preference and exogenous preference of each DMU on efficiency value in Fig. 9. When $\lambda \,<\, 0.19$ and $\varsigma > 0.8$, contrary to the understanding in single parameter analysis, the most efficient DMU in this region is not $DM{U}_{1}$ but $DM{U}_{5}$ in Fig. 9a. This illustrates that the DM’s diverse comprehensive emphasis on endogenous and exogenous factors will have an effect on the cross-efficiency value. Specially, When the DM has higher perceived preference ($\lambda$, $\varsigma$) for rejoice and exogeneity, the top evaluation ranking will change. And if there is no obvious bias between $\lambda$ and $\varsigma$, then the evaluation ranking is relatively stable. From Fig. 9b, we detect that when $0.67 \,<\, \alpha \,<\, 0.95$ and $0.75 \,<\, a/b \,<\, 0.85$, $DM{U}_{9}$’s efficiency overtakes $DM{U}_{8}$’s efficiency. Be similar to Fig. 9b, when $0.76 \,<\, \alpha \,<\, 1$ and $0 \,<\, \delta \,<\, 0.3$, $DM{U}_{3}$’s efficiency exceeds $DM{U}_{11}$’s efficiency in Fig. 9c. The combination variation of the above parameters is the manifestation of individual’s behavior psychology in the decision-making process. Such as $\alpha \in (1,2)$ and $a/b\in (0,0.5)$, it seems to be divergent between two intervals, but in reality they portray the ambivalence of decision-making individuals.

Fig. 9: Three-dimensional sensitivity analysis of the evaluation results with respect to double parameters.

a represents the sensitivity analysis of the double parameters ς and λ, (b) shows the sensitivity analysis of the double parameters a/b and α, and (c) presents the sensitivity analysis for the double parameters α and δ. Each surface represents the efficiency of different DMUs as the two parameters vary. The different colors represent different DMUs, and the efficiency is plotted on the z-axis.

Discussion on parameters

In reality, many evaluation problems can be addressed using the cross-efficiency model under bounded rationality. The regional efficiency evaluation of high-tech industries mentioned above is an example. Our proposed model fully considers the overall process of self-evaluation and peer evaluation. Based on the empirical results of the model, we derive the following implications for the main parameters within the model.

• (1) The risk aversion parameter primarily reflects an individual’s tolerance for risk, influencing their choice of conservative or aggressive decision-making strategies. Regret aversion parameter indicates the sensitivity of an individual to potential future regret, affecting their choice between safe and risky options. The DMs with high regret preferences may tend toward stable and conservative policies, encouraging continuous improvement and steady development. In contrast, the DMs with high rejoice preferences may be more inclined to take risks and innovate, providing incentive mechanisms to encourage high-return projects. Furthermore, the DMs with high endogenous preferences may formulate long-term development strategies focused on internal improvements and performance enhancement. Those with high exogenous preferences may develop competition-oriented strategies, emphasizing the improvement of relative rankings and competitiveness under external standards.

• (2) The attitudinal parameter $\alpha$ characterizes the DM’s preference for uncertainty, with adjustments to the parameter reflecting the DM’s personality. For instance, the parameter can be used to flexibly respond to uncertainty; a larger $\alpha$ can help manage potential technological breakthroughs and market changes in high-tech industries by emphasizing low-probability, high-impact events.

• (3) In fact, the bounded rationality is not confined to a particular segment but rather permeates the entire process of events. Observations of multiple parameter changes reveal differences from the effects of single parameter changes, notably in the shifts of the top-ranked DMUs, which differ from the typical influence on lower-ranked units seen with single parameter changes. Additionally, the RACE model is somewhat suitable for explaining the contradictory mindsets of individuals during the evaluation process. Strategically setting risk aversion parameter, regret aversion parameter, endogenous preference parameter, rejoice preference parameter and attitudinal parameter can help the DMs formulate more scientific and optimized decisions. For instance, rational allocation of multiple parameters can optimize resource allocation, improve decision quality, and balance risk and reward, thereby promoting the sustained healthy development and innovative breakthroughs in regional high-tech industries.

Comparison with some current cross-efficiency evaluation models

In Section 4.1, we compared the RACE method with some traditional cross-efficiency models. For purpose of further detecting the robustness and validity of the model we developed, we present comparison with some frequently used cross-efficiency models, including the conventional CCR model (Charnes et al., 1978), the PCE model (Liu et al., 2019), Wu’s model (Wu et al., 2012b) and the RCEC method (Liu and Chen, 2022). The example data we adopted is derived from Liu and Chen (2022), which is shown in Table 5. The input indexes are R&D fund, full-time R&D staff. The output indexes are technology transfer revenue, number of papers published in domestic academic journals and number of papers published in foreign academic journals, respectively. We set the model parameters as before. The cross-efficiency matrix in Table 6 can be acquired through the proposed RACE method. The efficiency values and ranking results are displayed in Table 7. From Table 7, we can learn that compared with the models of the PCE model, the RECE model and Wu’s model, the efficiency values of evaluated DMUs calculated by our proposed method are generally higher than those of other three models, which is due to the RACE method considering the preferences of the DMs in a more holistic way. One can find that only the efficiency score of the $DM{U}_{11}$ is less than that of the RCEC method, which suggests that the $DM{U}_{11}$ has a relatively large loss compared with endogenous reference point of the input attribute.

Table 5 Input and output of 13 universities in 2017.

Table 6 The RACE cross-efficiency of 13 universities in 2017.

Table 7 Comparison results of some improved models.

The rankings of the five models are depicted in Fig. 10. It is shown that the ranking orders of our proposed model are roughly the same except $DM{U}_{7}$. The main reason is the efficiency value of the $DM{U}_{7}$ is similar to those of the PCE model, Wu’s model and the RCEC model, nevertheless, the efficiency scores of other DMUs appraised by the RACE method have been improved, but $DM{U}_{7}$ has not altered significantly. Wu’s model does not think over the factor of the DM’s preference, it is an improved model of Shannon entropy method, while our model can reflect the influence of the DM’s preference on the cross-efficiency value through the preference coefficient. The PCE model based on prospect theory considers individual’s risk attitude. From Fig. 10, it can be seen that the ranking of $DM{U}_{2}$ compared to our model differs widely. The reason is that $DM{U}_{2}$’s input is 608330 and 743, its input is less than that of the endogenous reference point, so higher efficiency values are obtained in the RACE method. In addition, in terms of efficiency distinction, we can find that the difference in efficiency values between rank 1 ($DM{U}_{9}$) and rank 2 ($DM{U}_{5}$) using the RCEC method is no more than 0.01, It is somewhat difficult to distinguish between the first and the second, especially when only a few decimal places are retained. However, our model can distinguish these DMUs more efficiently than the RCEC method. It can be seen that the above-mentioned efficiency values among DMUs lack effective discrimination to a certain extent. The efficiency scores refined by our method are stable and reasonable. Definitely, for the sake of the valid and robust results, the DM should analyze each DMUs using these models according to the realistic needs.

Fig. 10: The ranking orders of different models.

The figure shows the distribution of the DMUs’ ranking under different methods.

A Spearman rank correlation test (as shown in Fig. 11) revealed a high correlation with RCEC (0.93) and Wu (0.86), indicating that the RACE model aligns well with these advanced methods. Additionally, we analyzed the mean efficiency score, interquartile range (IQR), and standard deviation (SD) (as presented in Table 7). The mean score of RACE (0.7642) is higher than models like PCE (0.7274), indicating a tendency toward higher efficiency evaluations. The IQR of RACE (0.7125), larger than most other models, demonstrates its strong discriminatory ability among middle-performing DMUs. However, its SD (0.1025), smaller than RCEC’s (0.1200), suggests that RACE provides more consistent evaluations for extreme DMUs, avoiding excessive fluctuations.

Fig. 11: Spearman rank correlations of different models.

The figure shows the degree of correlation among different methods.

While the RACE model demonstrates strong discriminatory power, its slightly higher mean score suggests a potential for overestimating efficiency in certain DMUs. Additionally, the larger IQR, though beneficial for distinguishing between DMUs, may lead to wider efficiency gaps, which might not be ideal in scenarios requiring more uniform assessments.

Based on the previous discussion, the characteristic and advantage of the RACE method can be summarized as follows:

• (1) In traditional cross-efficiency evaluation models, DMs are assumed to assess efficiency from a fully rational perspective. However, the RACE model incorporates regret theory, which allows for the inclusion of bounded rationality by considering both regret and rejoice feelings in decision-making. This approach enables a more realistic reflection of decision makers’ attitudes towards risk and uncertainty, making the evaluation process more aligned with real-world decision-making scenarios.

• (2) Conventional cross-efficiency models typically rely on exogenous reference points, such as the maximum, minimum, ideal, or anti-ideal points, in constructing secondary goal models. The RACE model, however, addresses the limitations of this approach by considering both endogenous reference points (e.g., geometric mean) and exogenous reference points. This dual consideration provides a more comprehensive and balanced framework for cross-efficiency evaluation, ensuring that the evaluation accounts for both internal system characteristics and external benchmarks, leading to more robust results.

• (3) The issue of weight aggregation in cross-efficiency models is addressed in the RACE model through the application of attitudinal entropy, which differs from the traditional equal-weight averaging and Shannon entropy methods. Attitudinal entropy takes into account the DM’s preference for information uncertainty, which can influence the magnitude of Shannon entropy and, subsequently, the aggregated weight used in cross-efficiency evaluation. This method allows for a more nuanced reflection of individual preferences, risk attitudes, and regret feelings, thereby offering a more realistic and practical approach to efficiency evaluation in real-world contexts.

• (4) In terms of validity and robustness, comparative analysis shows that the RACE model performs favorably when evaluated against both traditional and contemporary cross-efficiency models. The model demonstrates effective performance in ranking DMUs and provides stable and applicable results, particularly in environments where decision-making is influenced by irrational factors such as preference uncertainty and ambivalence.

Conclusions

Cross-efficiency evaluation has long been a central topic in data envelopment analysis. It is widely recognized as an effective method for evaluating the relative efficiency of the DMUs. Despite improvements in cross-efficiency models, traditional approaches often assume complete rationality, which overlooks the bounded rationality of the DMs. Additionally, conventional models relying on average strategies often ignore individual differences in self-evaluation and peer evaluation preferences.

This study introduces the RACE model, which expands the cross-efficiency framework by incorporating regret theory. This allows the model to consider DMs’ risk attitudes and regret aversion, offering a more realistic view of decision-making. The model also introduces a secondary goal that balances both endogenous (e.g., geometric mean) and exogenous reference points. This provides a more balanced evaluation by considering both internal and external performance comparisons. The use of preference coefficients further allows the model to reflect individual differences in how DMs’ value internal versus external performance indicators. Furthermore, the attitudinal entropy method used in the RACE model addresses the limitations of traditional weight aggregation. By considering DMs’ preferences for information uncertainty, this method provides a flexible approach to both self-evaluation and peer evaluation. As a result, the model produces cross-efficiency results that better reflect the psychological and informational preferences of DMs, often overlooked in traditional models.

The RACE model has broader implications, as it better reflects the complexity of human decision-making across different fields. Its flexibility allows it to be applied in various areas: in financial investment, it can optimize portfolios by balancing risk and return according to investor preferences; in business management, it can harmonize conservative and aggressive strategies, promoting both stability and innovation; in public project evaluation, it can enhance resource allocation and stakeholder satisfaction, improving operational efficiency.

Despite its advantages, the RACE model has certain limitations. The current use of fixed reference points could be enhanced by incorporating interval numbers, providing greater flexibility. Additionally, the model is currently based on a static analysis framework, which may not fully capture the dynamic nature of real-world decision-making processes. Future research could explore extending the model to incorporate multi-stage or dynamic evaluations, allowing for a more comprehensive analysis of complex, evolving scenarios.