Cross-efficiency aggregation by ordered visibility graph averaging: method, and application in portfolio selection

Abstract

1. Introduction

Portfolio selection refers to an investor’s decision to diversify their financial assets to reduce investment risk while maintaining a specific level of projected return. Portfolio selection offers guidance to an investor for allocating resources among various investment opportunities. A major breakthrough in the field came in 1952 when Harry Markowitz published his modern portfolio theory commonly known as mean-variance portfolio theory. Markowitz (1952) mean-variance portfolio selection model is based on the trade-off between risk and return under various economic conditions. To alleviate the limitations of the mean-variance framework, various alternative risk measures have been proposed in the literature. Although variance is widely accepted as a risk measure, it has limitations. Semi-variance is another risk measure, originally introduced by Markowitz (1959), and is used in the mean-semi-variance portfolio selection model in Guo & Ching (2021).

Data envelopment analysis (DEA) is a non-parametric empirical approach for evaluating the relative efficiency of decision-making units (DMUs) in the presence of multiple inputs and multiple outputs. DEA model was first introduced by Farrell (1957), and then further developed by Charnes et al. (1978) (called CCR model) and Banker et al. (1984) (called BCC model). Due to the lack of translation invariance, the CCR and BCC models cannot handle the negative data values in inputs and outputs unless the data get transformed into positive values. The directional distance function based DEA models have been described in the literature to handle the non-positive data values without converting them into positive ones.

One of the applications of DEA in finance is portfolio selection. Few articles in the literature have studied the problem of portfolio selection using DEA. Chen (2008) presented an application of DEA in portfolio selection. Dia (2009) introduced a four-step methodology based on DEA for portfolio selection. Chen et al. (2018) constituted fuzzy portfolio assessment models based on DEA using various risk indicators to address the uncertainty involved in the financial market. Zhou et al. (2018) offered a DEA frontier improvement technique as a rebalancing strategy for investors. Gupta et al. (2020) introduced credibilitic fuzzy portfolio selection models based on DEA. Zhu & Zhu (2023) pioneered the concept of eco-efficiency of industrial investment in terms of energy consumption, economic benefits and environmental impact using a slack-based DEA model. Omid et al. (2023) introduced a two-stage multiobjective optimization problem incorporating network DEA. Soltanifar et al. (2023) developed DEA-R models for merger analysis that can deal with negative data and applied it to the Iranian banking sector. Ghiyasi & Zhu (2020) proposed an inverse semi-oriented DEA model to assess the efficiency of Chinese commercial banks following the global financial crisis, when negative outputs were present. Amirteimoori et al. (2022) developed the chance-constrained multistage DEA model for the relative performances of supply chains. Emrouznejad et al. (2023) recently presented a review of the eco-efficiency evaluation using DEA. Recently, Lim et al. (2014) introduced a novel way of portfolio selection using the DEA cross-efficiency. They considered the Markowitz’s mean-variance optimization model, which identifies a portfolio consisting of DMUs with minimum variation in their cross-efficiency scores and having a desirable expected mean efficiency.

Conventional DEA approaches permit each DMU to assess its efficiency using the most favourable weights, which might lead to exaggerated weight schemes. One of the most prominent ways of dealing with limited discriminative power is the cross-efficiency assessment, which presented citesexton1986data as an expansion of DEA and further improved by Doyle & Green (1994). These authors initiated aggressive and benevolent strategies as secondary objectives to resolve the problem of non-unique optimum weights in DEA. Davtalab-Olyaie & Asgharian (2021) discussed the problem of a non-Pareto optimum solution in CE evaluation. They proposed a multiobjective program to obtain a set of Pareto-optimal CE scores. Chen et al. (2020a) highlighted the problem of non-uniqueness in CE assessments and developed a meta-frontier analysis framework to develop a novel CE assessment method. The approach presented in the paper Oukil & Amin (2015) derives the importance of each DMU from a common consensus among all DMUs, which is then reflected via a preference voting matrix. Because self and peer evaluation scores represent DMU’s mutual appreciation for one another, the CE matrix is viewed as an election framework in which each DMU is both a candidate and a voter. Oukil (2020) introduced a new method based on collective weight profile for each DMU using the preference voting system embedded within the weight matrix, which views the assessing DMUs as voters and the input/output factors as candidates. For a more detailed study of cross-efficiency innovations, limitations and extensions, see Wu et al. (2021).

DEA cross-efficiency is helpful for portfolio selection problems since it removes the disadvantage of shifting weights in traditional DEA models. To achieve the maximum efficiency value, the DMU can give some inputs and outputs exceptionally high weights. As a result, portfolio selection that uses the traditional DEA technique is not recommended. To resolve this issue, Lim et al. (2014) proposed an MV-DEA portfolio selection method that selects a portfolio of DMUs with the least volatility in their CE scores that meets a specified anticipated return. Mashayekhi & Omrani (2016) applied the suggested MV-DEA cross-efficiency portfolio selection assuming fuzzy returns. Recently, Amin & Hajjami (2021) examined the effect of the alternative optimal solution in portfolio selection and indicated that significant improvement in portfolio selection is obtained when alternate optimum solutions are used to obtain a cross-efficiency matrix. Chen et al. (2021) describe a portfolio selection method using a fuzzy DEA cross-efficiency assessment, which considers undesirable ambiguous inputs and outputs. Deng & Fang (2019) developed a novel mean-variance-maverick DEA cross-efficiency approach based on the prospect of selecting a fuzzy portfolio. Gong et al. (2021) presented a method based on regret theory, in which they applied a fuzzy multiobjective portfolio optimization model with four objectives, namely, mean, variance, skewness and efficiency, bounded by several realistic constraints. Essid et al. (2018) offered an innovative framework for portfolio optimization that combines the Maverick index with a DEA game cross-efficiency model.

Mehlawat et al. (2018) looked at the portfolio selection issue by adding additional facts about the non-normality of asset returns using skewness and kurtosis. Efficiency evaluation is also a critical factor to consider in the portfolio selection problem.

One of the drawbacks of the DEA cross-efficiency approach in portfolio selection is that all DMUs/stocks in the optimal portfolio are given identical weights. Amin & Hajjami (2021) suggested the development of a novel DEA cross-efficiency portfolio selection model to obtain more reasonable weights for DMUs/stocks. Although many scholars have realized the importance of aggregation methods for cross-efficiency assessment, there is still room for improvement in the existing aggregation methods and their applications in portfolio optimization problems. This paper uses the ordered visibility graph averaging (OVGA) operator to develop a cross-efficiency aggregation technique to analyse the units more thoroughly. The weights in the OVGA operator are determined by the relevance of the data in the obtained visibility graph (VG) (Jiang et al., 2016). The OVGA operator is vital in aggregating CE scores since it allows values to be given varied weights depending on the importance of CE scores of different DMUs.

1.1 Literature review

The cross-efficiency technique, an extended form of DEA, considers both self and peer appraisal, resulting in a unique ranking of DMUs and removing the problem of artificial weight selections. Most cross-efficiency studies emphasize the cross-evaluation technique, and the arithmetic average is frequently used as a CE aggregation method in the existing research. The arithmetic average technique has certain drawbacks. First, self-evaluated efficiency does not play a sufficient role in the efficiency aggregation procedure, as each DMU has only one self-evaluated efficiency value with multiple peer-evaluated efficiency values. Second, there is no other methodology to consider the decision-maker’s subjective preferences, as the arithmetic average approach provides equal weight to each efficiency score (Wang & Chin (2011); Oukil (2019)). To remedy these issues, Song et al. (2017) and Li et al. (2018) devised aggregation methods that are motivated by the Shannon entropy weight and balanced adjustment weights, respectively, which provide both a solution for the inadequacies of the average arithmetic technique and an acceptable theoretical justification for CE aggregation. Wang & Chin (2011) introduced the ordered weighted averaging (OWA) operator weight to cross-efficiency aggregation. Using weights based on the OWA operator for CE aggregation permits the decision-maker’s optimism level towards the best relative efficiencies. Liu & Chen (2022) pointed out that different efficiency scores affect integration outcomes differently. The relative importance of each efficiency value is not considered in an average aggregation of cross-efficiency. Liu & Chen (2022) first applied regret theory to the CE aggregation process to understand the regret aversion behaviour of DMs, which plays a crucial role in the decision-making process. To determine the regret-based aggressive and benign cross-efficiency, weight matrices are built into their study. Wu et al. (2012) suggested a common weight technique to aggregate the cross-efficiency by considering DMUs as players in a cooperative game sense and calculating their Shapley values to address these limitations. Chen et al. (2020b) developed prospect theory to represent DMs’ subjective preferences in their innovative technique to aggregate CE scores based on the prospect-consensus procedure. Li et al. (2022) outlined a novel strategy for generating aggregation weights from subjective and objective viewpoints. To reflect DM preferences from a subjective standpoint, the authors established prospect theory. Their method offers reference points as intervals that can be chosen based on the DMs’ decision aims and preferences. Chen et al. (2020b) introduced a novel method based on prospect consensus (APC) theory for CE aggregation. The APC method keeps as much of the cross-efficiency matrix’s decision-making information as feasible and then aggregates the cross-efficiency that suits decision-making demands. Combining cumulative prospect theory, Fang & Yang (2019) proposed an ordering method for interval CE aggregation.

All the studies mentioned above have different methodologies, indicators and characteristics, requiring deep knowledge to understand the various procedures. Additionally, the above-discussed methods require many parameters in the calculation process that can be very tedious in handling large-scale data.

1.2 Research motivation and novelty of the proposed approach

To the best of our knowledge, DEA cross-efficiency aggregation has been the subject of numerous research studies using a variety of frameworks; however, a new aggregation technique that needs the information from the cross-efficiency matrix remains possible. Also, we find fewer studies in the literature where DEA cross-efficiency has been applied to portfolio optimization problems. The proposed study makes the following contributions:

1. Compared with the aggregated method existing in the literature (Wang & Chin (2011); León et al. (2014); Oukil (2019)), our proposed OVGA method aggregates the cross-efficiencies with unique weights and provides weightage according to the importance of the data value, while the weights depend upon different characteristics such as DMs’ optimism level and orness degree value, which is uncertain; furthermore, different orness degrees provide different sets of weights, as seen in Wang & Chin (2011) and León et al. (2014).

2. Compared with different aggregating operators, the OVGA operator fully considers the information of orders and the correlation between the values. The main advantage of OVGA is that it is completely data-driven, and the aggregated results can be produced from a series of original values.

3. We develop a new application of OVGA compared with that seen in Jiang et al. (2016, 2017) and Wang et al. (2022) in regard to DEA cross-efficiency aggregation.

4. Furthermore, compared with Lim et al. (2014); Deng & Fang (2019) and Essid et al. (2018), we integrate the MV-DEA cross-efficiency model with OVGA cross-efficiency scores and deal with the existence of an alternate optimal solution in CE for portfolio optimization problems.

1.3 Organization of the paper

The remainder of this paper is structured as follows. Section 2 is separated into two subsections; its subsequent subsections review the fundamental concepts of the DEA models and OVGA operator. Section 3 first presents the terminology of the OVGA operator for cross-efficiency aggregation, and then introduces the portfolio performance evaluation model. In Section 4, a numerical data set from the literature is used to demonstrate the applicability of the proposed model; further, a case study of the Indian banking sector is presented and the managerial implications are also discussed there. Section 5 contains the concluding remarks.

2. Preliminaries

This section presents the DEA models existing in the literature, used to investigate the efficiency and cross-efficiency of a DMU. Further, the ordered visibility graph weighted averaging (OVGWA) operator, developed by Wang et al. (2015), is also presented here. For basic definitions and notations regarding DEA models and OVGA operator, we may refer to Charnes et al. (1978) and Wang et al. (2015), respectively.

2.1 DEA models for efficiency evaluation

Suppose we have n |$DMUs$| to be evaluated with m inputs and s outputs. Let |$j^{th}$| DMU, |$DMU_{j}$|⁠, |$j=1, 2, \ldots , n$| use |$x_{ij}, \; i=1,2,\ldots ,m$| inputs and produce |$y_{rj}, \; r=1,2,\ldots , s $| outputs. Let |$DMU_{k}$|⁠, |$k=1,2, \ldots , n$| be the DMU that is under evaluation. Let |$\omega _{ik} \;i=1,2, \ldots , m $| and |$\mu _{rk}\; r=1,2, \ldots , s$| be the weight given by the |$DMU_{k}, \; k=1,2, \ldots , n $| to the |$i^{th}$| input and |$r^{th}$| outputs, respectively. The best relative self-efficiency |$e_{kk}^{\star }$|⁠, under the assumption of constant return to scale, can be determined using the following input-oriented CCR model (Charnes et al. (1978)).

Let |$\mu ^{\star }_{rk}\; r=1,2, \ldots , s$| and |$\omega ^{\star }_{ik} \;i=1,2, \ldots , m $| be the optimal solution of Model I, then |$e_{kk}^{\star } = \sum _{r=1}^{s} \mu ^{\star }_{rk}y_{rk} $| is the optimal efficiency that |$DMU_{k}$| can attain, which is also referred to the CCR-efficiency of |$DMU_{k}$| under self-evaluation. If |$e_{kk}^{\star } =1$| and all the optimal weights |$\mu ^{\star }_{rk}\; r=1,2, \ldots , s$| and |$\omega ^{\star }_{ik} \;i=1,2, \ldots , m $| are positive, then |$DMU_{k}$| is CCR-efficient, otherwise, it is CCR-inefficient.

The DEA cross-efficiency assessment approach uses peer review in place of sole self-evaluation to assess efficiency. Thus, the peer evaluation of |$DMU_{j}, \; j=1,2,\ldots , n$| by the |$DMU_{k}$| is defined as follows:

It should be noted that the optimal weights derived from model I are rarely unique. As a result, the cross-efficiency score defined in (2.1) will not produce a unique set of cross-efficiency values. To deal with this issue Oral et al. (2015) introduced the maximum resonated appreciative model, which was further discussed in Amin & Hajjami (2021) and Amin & Oukil (2019). In Oral et al. (2015), the authors have shown that the cross-efficiency score of any |$DMU_{j}$| with respect to any |$DMU_{k}, \; k=1,2,\ldots ,n$| can take any value between the interval. So the minimum and the maximum cross-efficiency score of |$DMU_{j}$|⁠, evaluated by |$DMU_{k}, \; k=1,2,\ldots , n $| can be computed by those optimal weights for which the self-efficiency score value will remain unchanged. They presented the following two formulations, Models II and Model III, to determine the cross-efficiency interval of |$DMU_{j}$| evaluated by |$DMU_{k},\; k=1,2,\ldots , n $|⁠.

Let |${\mu }_{rk}^{\star }\; r=1,2, \ldots , s$| and |${\omega }_{ik}^{\star } \;i=1,2, \ldots , m $| be the optimal solutions of Model II, then |$\hat{e}_{kj}^{L} = \sum _{r=1}^{s} {\mu }_{rk}^{\star } y_{rj} $| is the optimal efficiency that |$DMU_{j}$| can attain when evaluated by |$DMU_{k}$|⁠. We can find out the Min cross-efficiency matrix for n |$DMUs$| corresponding to Model II. The n efficiency values constitute a cross-efficiency matrix, as shown in Table 1.

Let |${\mu }_{rk}^{\star }, \; r=1,2, \ldots , s$| and |${\omega }_{ik}^{\star }, \;i=1,2, \ldots , m $| be the optimal solutions of Model III, then |$\hat{e}_{kj}^{U} = \sum _{r=1}^{s} {\mu }_{rk}^{\star } y_{rj} $| is the optimal efficiency that |$DMU_{j}$| can attain when evaluated by |$DMU_{k}$|⁠.

Across all possible optimal solutions of CCR, Model II aims to determine the worst cross-efficiency score for |$DMU_{k}$|⁠, and Model III aims to determine the best cross-efficiency score for |$DMU_{k}$|⁠.

Similarly, we can find out the Max cross-efficiency matrix for n |$DMUs$| corresponding to Model III.

The most popular method is averaging each row or column’s efficiency in the cross-efficiency matrix with equal weights to determine a DMU’s overall performance measurement. Assigning identical weights for cross-efficiency aggregation is not the best option because the self-evaluated efficiency score contributes less than peer-evaluated efficiencies do. In the next section, to follow we have the notion of the OVGA operator for aggregation of efficiency of |$DMU^{\prime}s$|⁠.

2.2 OVGWA operator

A VG is an undirected graph of n nodes. The VG method converts a time series into a graph. The associated graph derived from a time series has the following properties:

1. Connected: a node can see its nearest neighbours.

2. Undirected: the associated graph extracted from a time series.

3. Invariant under affine transformations of the series data: Although both the horizontal and vertical axes are rescaled, the visibility criterion remains unchanged. This means that rescaling does not affect the ability to clearly distinguish or identify the elements on the graph.

Further, Wang et al. (2015) introduced an algorithm for ordered visible graphs (OVGs). The VG method is to convert a time series into a graph, while the OVG maps a set of ordered data into a network.

These argument values are connected in a network of connections by the visibility criteria. These connections are made depending on both the argument values themselves and the order in which they are presented.

The visibility criteria of OVG are similar to VG and are defined as follows:

The Python program that calculates the adjacency matrix, degree and weight of an ordered set, on the lines of Definitions 3, 4 and 5, is described as Algorithm 1 in ‘Appendix’. To give more clarity, we consider an example considered in Wang et al. (2015).

Fig. 1.

OVG of ordered set |$O.$|

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It should be noted that if a vertex is connected to another vertex by an edge, it is visible from that vertex since the two vertices have a direct line of sight. On the other hand, if a vertex is not connected to another vertex by an edge, it means that there is an obstruction in the way and the vertex cannot be seen from that specific vertex. In a visible network, the vertex with the highest degree is the one with the greatest number of connections to other vertices. It resembles a graph’s centre that is visible from several other spots. This vertex is crucial to the network as a whole because it is connected to a larger number of other vertices.

3. Portfolio optimization model based on the cross-efficiency matrix: proposed method

Lim et al. (2014) studied Markowitz (1952) portfolio selection model based on the efficiencies of their assets. Lim et al. (2014) considered return and risk aspects of Markowitz’s model in terms of mean efficiencies and the co-variance between the cross-efficiency scores of all |$DMU^{\prime}s$|⁠, respectively. Before introducing the suggested model, it is necessary to comprehend the terminology from the previous part in terms of DMU efficiency.

Portfolio selection model corresponding to min-cross-efficiency 

Let us consider a portfolio |$\varOmega $| constituted with |$n$|  |$DMUs$| say |$DMU_{i},\; i=1,2,\ldots , n$|⁠. Following the approach of Lim et al. (2014), this paper describes the return and risk characteristics of Markowitz’s model (Markowitz 1952). Specifically, the mean cross-efficiency scores represent the model’s return, while the variance of the cross-efficiencies represents the model’s risk.

Let |$[\hat{e}_{ij}^{L}]_{n\times n}$| be the min-cross-efficiency matrix on n DMUs taken under consideration for portfolio selection. Let |$x_{i}$| be the proportion of the total efficiency invested on |$DMU_{i}$|⁠. The portfolio |$\varOmega $| of |$n$|  |$DMUs$| by the OVGA operator is obtained by solving the following cross-efficiency portfolio optimization model:

where |$\tilde{e}_{i}^{L}$| is the mean cross-efficiency score of |$DMU_{i}$| by OVGA operator, defined in Definition 7 and |$\tilde{\sigma }_{ij}^{L} = \frac{1}{n}\sum _{k=1}^{n} (\hat{e}_{ki}^{L} -\tilde{e}_{i}^{L})(\hat{e}^{L}_{kj} -\tilde{e}_{j}^{L})$| is the covariance between the cross-efficiency scores of |$DMU_{i}$| and |$DMU_{j}$|⁠. Here, |$\gamma $| is the trade-off parameter between mean efficiency and co-variance of their efficiency. Further, for a better selection of a DMU, the budget constraint |$\sum _{i=1}^{n} x_{i} =1 $|⁠, cardinality constraint |$\sum _{i=1}^{n} z_{i}=d, \, z_{i} \in \{0,1\}, \quad i= 1,2,\ldots ,n,$|⁠, buy-in threshold |$\epsilon _{i} z_{i} \leq x_{i} \leq \delta _{i} z_{i}, \quad i= 1,2,\ldots ,n, $| and no short-selling criteria |$ x_{i} \geq 0,\; i= 1,2,\ldots ,n,$| are also considered here. Here, |$E_{\varOmega }^{b}$| is the maximum OVGA efficiency achievable by the portfolio under the budget constraint, cardinality constraint, buy-in threshold and no short-selling criteria.

Let |$\tilde{E}_{\varOmega ^{\star }}$| and |$\tilde{V}_{\varOmega ^{\star }}$| be the optimal mean efficiency and optimal risk corresponding to the optimal portfolio |$\varOmega ^{\star }$| so obtained.

Similarly, the mean-variance cross-efficiency model for simple average efficiency as defined in Definition 8 is as follows:

where |$\overline{e}_{i}^{L}$| is the mean cross-efficiency score of |$DMU_{i}$| defined in Definition 8 and |$\overline{\sigma }_{ij}^{L} = \frac{1}{n}\sum _{k=1}^{n} (e_{ki}^{L} -\overline{e}_{i}^{L})(e_{kj}^{L} -\overline{e}_{j}^{L}) $| is the covariance between the cross-efficiency scores of |$DMU_{i}$| and |$DMU_{j}$|⁠. |$E_{\varOmega }^{b}$| is the maximum mean efficiency, achievable by the portfolio |$E_{\varOmega }$|⁠. Further, |$\gamma $| is the trade-off parameter between mean efficiency and co-variance of their efficiency. Further, other constraints have the same interpretation as explained in Model IV. Let |$\tilde{E}_{\varOmega ^{\star }}$| and |$\tilde{V}_{\varOmega ^{\star }}$| be the optimal mean efficiency and optimal risk corresponding to the optimal portfolio |$\varOmega ^{\star }$| so obtained.

4. Numerical illustration

In this section, we apply our proposed method to a numerical example consisting of 22 DMUs to demonstrate the empirical effectiveness of the method. The numerical dataset is taken from Lim et al. (2014), which was further analysed by Amin & Hajjami (2021). The input and output data for the DMUs are presented in Table 3.

On solving Model I, for |$DMU_{1}$|⁠, we get |$\mu ^{\star }_{11}= 0.85, \; \omega ^{\star }_{11}= 0.0769, \; \omega ^{\star }_{21}=0.1538 $|⁠, with the optimal value of the objective function as |$e^{\star }_{11}=0.85$|⁠. Similarly on solving Model I, for other |$DMUs$|⁠, the self-efficiency of respective |$DMUs$| are presented in Table 4.

For cross-efficiency interval evaluation of |$DMU_{j}$|⁠, we have to solve Model II and Model III, respectively, for different values of k for a fixed value of j. For example if we take |$k=2$| and |$j=1$|⁠, we can find |$\tilde{e}^{L}_{21}$| and |$\overline{e}^{U}_{21}$|⁠. The minimum cross-efficiency matrix is shown in Table 6. As in the cross-efficiency matrix, |$DMU_{j}$| is evaluated by |$DMU_{k}$|⁠, so to determine the weights given by |$DMU_{j}$| to |$DMU_{k}$| by the OVGA operator, we need a re-ordered minimum cross-efficiency matrix.

To get more clarity, let us calculate the weights given by |$DMU_{1}$| to |$DMU_{k}, \; k=1,2,\ldots , n$| for the OVGWA operator to get its aggregated efficiency. In a similar fashion, we can find out the weights given by other |$DMUs$| to |$DMU_{k}, \; k=1,2,\ldots , n$|⁠. The set of ordered cross-efficiencies of |$DMU_{1}$|⁠, |$\hat{e}_{k1}^{L}, \; k= 1,2,\ldots ,22$| (as per Table 7) for |$DMU_{1}$| is

To calculate the weights given by |$DMU_{1}$| to |$DMU_{k}, \; k=1,2,\ldots ,22$|⁠, for evaluating its OVGWA efficiency, Fig. 2 depicts the ordered VG of all |$DMUs$| with respect to ordered cross-efficiency set |$\hat{E}_{1}^{L}$| and Table 5 presents the weights corresponding to the first DMU.

Fig. 2.

Ordered VG of ordered efficiency set |$\hat{E}_{1}^{L}$|⁠.

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Now on lines of Definition 6, from Algorithm 1, the corresponding adjacency matrix is

So the aggregated cross-efficiency by OVGWA operator and by simple average is |$ \tilde{e}_{1}^{L} = \sum _{p=1}^{22} w_{p}^{1} \hat{e}_{p1}^{L}=0.76 $| and |$\overline{e}_{1}^{L} = \sum _{p=1}^{22} (1/n)\hat{e}_{p1}^{L} =0.75, $| respectively. These values are the same as given in Table 9, and in a similar manner, we can calculate the aggregated cross-efficiency scores of other DMUs. For this, we first need to solve Model II; for |$k,j=1,2,\ldots ,n$| the minimum cross-efficiency matrix is depicted in Table 9. Further, Table 7 presents the re-ordered cross-efficiency matrix. As per the requirement of the calculation Table 8 illustrates the weights given by |$DMU_{j}$| to |$DMU_{p}$| for |$j,p=1,2,\ldots ,22$|⁠.