Cross-efficiency Matrix Research Articles

Cross-efficiency aggregation based on interval conditional entropy: An application to forest carbon sink efficiency

Cross-efficiency evaluation is an extension of data envelopment analysis (DEA), which can effectively distinguish between decision-making units (DMUs) through self- and peer-evaluation. The cross-efficiency of each DMU in a set of DMUs is measured in terms of intervals when the input–output data are represented by the number of intervals. Based on the interval cross-efficiency matrix, the interval entropy is defined in terms of the likelihood. Then, considering the influence of peer evaluation, the interval conditional cross-efficiency entropy is proposed and an aggregation model of the interval conditional cross-efficiency entropy is presented to create a ranking index for DMUs. Finally, a simple example is provided to illustrate the effectiveness of the proposed method, which is applied to the evaluation of forest carbon sink efficiency in China. The results indicate that the final cross-efficiencies of all 30 provinces range from 0 to 0.6. Among these provinces, those with a relatively high efficiency include Guangdong, Guizhou, Hainan, Shandong, and Qinghai.

On value efficiency analysis and targeted benevolence cross efficiency

In this article, we examine the relation between cross efficiency and Value Efficiency Analysis, two Data Envelopment Analysis related models for peer appraisal assessment. Particularly, we show that the Targeted Benevolence cross efficiency formulation is equivalent to the Value Efficiency Analysis model if either (i) the “reference” unit, i.e. the one whose vector of optimal multipliers is used to evaluate all others in the Targeted Benevolence formulation, or (ii), its radial projection if it is inefficient, is chosen as the Most Preferred Solution in Value Efficiency Analysis. An immediate implication of this result is an alternative and most frequently shorter way to compute the Targeted Benevolence cross-efficiency matrix, as we can do so by solving a series of Value Efficiency Analysis problems, each of which uses a different unit or its radial projection if it is not efficient as the Most Preferred Solution. For this purpose, we provide a four-step algorithm, from which it is clear that in most of the cases it is not necessary to solve all the relevant problems. This depends on the number of inefficient units that are (radially) projected in the strongly efficient frontier and the number of different sets of peers for these units.

Cross-Efficiency aggregation based on the denominator rule

In this paper we propose an alternative cross-efficiency aggregation scheme that is based on the theory underlying the aggregation of self-appraisal efficiency scores. For this purpose, we apply the denominator rule to aggregate the cross-efficiency scores obtained from the multiplier form of the DEA model. This results in a share-weighted average cross-efficiency that takes explicitly into account the relative importance of each element of the cross efficiency matrix by means of either the opportunity cost of resources used or the value of outputs produced, depending upon the orientation chosen to measure efficiency. It turns out that the proposed share-weighting average cross-efficiency uses the simple arithmetic average of the estimated input and output weights from the self-appraisal form of the DEA model to compute the ultimate cross-efficiency. As this is a set of common (across decision-making units) input and output weights, it allows for complete ranking and comparison of all (efficient and inefficient) decision-making units.

Fuzzy cross-efficiency evaluation based on prospect theory and regret theory

Cross-efficiency in data envelopment analysis is widely used in production as an evaluation method that includes input and output indicators and allows for self-evaluation and mutual evaluation of decision making units (DMUs). However, as the application scenarios continue to expand, the traditional methods gradually fail to meet the needs. Many researchers have proposed improved methods and made great progress in weight determination, but the existing studies still have shortcomings in considering the psychological behavior of decision makers (DMs) and there is still relatively little research on cross-efficiency in fuzzy environments. In this paper, we proposed a method to apply CRITIC to determine weights and introduce both prospect theory and regret theory into the evaluation method of cross-efficiency to obtain the prospect cross-efficiency matrix and regret cross-efficiency matrix respectively, and then applied the Pythagorean hesitant fuzzy operator to aggregate them to achieve the ranking of DMUs through the fraction function. This largely takes into account the subjective preference and regret avoidance psychology of DMs. The applicability of this paper’s method is also verified through an example of shopping for a new energy vehicle. Finally, the effectiveness of this paper’s method is verified by comparing three traditional methods with this paper’s method, which provides an effective method for considering risk preferences in the decision-making process.

An ICE aggregation method based on optimal rally points in fuzzy environment

In order to avoid the hesitation of choosing between aggressive and benevolent strategies, we propose two cross-efficiency models to get interval cross-efficiency (ICE) from the relatively neutral angle in fuzzy environment, and then propose a novel aggregation method for ICE to solve the full ranking of Decision-Making Units (DMUs). Firstly, regard the expected value of fuzzy data as the input and output of Data Envelopment Analysis (DEA) method based on fuzzy set theory. Secondly, construct the cross-efficiency models based on the fuzzy expected values from the relatively neutral angle, and generate the lower and upper bounds of ICE for all DMUs, which determines the interval cross-efficiency matrix (ICEM). Thirdly, project all ICE onto the plane as points, then seek the optimal rally point for each DMU based on ICEM as the comprehensive ICE. Fourthly, rank the comprehensive ICE to obtain the complete ranking of DMUs by using the optimal number sorting method. Finally, the proposed model is applied to the evaluation of manufacturing enterprises, and the results are compared with different models to prove its effectiveness.

A ranking framework based on interval self and cross-efficiencies in a two-stage DEA system

The evaluation of the performance of a decision-making unit (DMU) can be measured by its own optimistic and pessimistic multipliers, leading to an interval self-efficiency score. While this concept has been thoroughly studied with regard to single-stage systems, there is still a gap when it is extended to two-stage tandem structures, which better correspond to a real-world scenario. In this paper, we argue that in this context, a meaningful ranking of the DMUs is obtained; this outcome simultaneously considers the optimistic and pessimistic viewpoints within the self-appraisal context, and the most favourable and unfavourable weight sets of each of the other DMUs in a peer-appraisal setting. We initially extend the optimistic-pessimistic Data Envelopment Analysis (DEA) models to the specifications of such a two-stage structure. The two opposing self-efficiency measures are merged to a combined self-efficiency measureviathe geometric average. Under this framework, the DMUs are further evaluated in a peer settingviathe interval cross-efficiency (CE). This methodological tool is applied to evaluate the target DMU in relation to the most favourable and unfavourable weight profiles of each of the other DMUs, while maintaining the combined self-efficiency measure. We, thus, determine an interval individual CE score for each DMU and flow. By treating the interval CE matrix as a multi-criteria decision making problem and by utilising several well-established approaches from the literature, we delineate its remaining elements; we show how these lead us to a meaningful ultimate ranking of the DMUs. A numerical example about the efficiency evaluation of ten bank branches in China illustrates the applicability of our modelling approaches.

Energy-aware job scheduling in a multi-objective production environment – An integrated DEA-OWA model

Manufacturing is a major source of energy consumption and, therefore, a significant contributor to emissions and greenhouse gases. This paper is concerned with evaluating different scheduling policies in a job shop system where energy-efficient scheduling is incorporated with multiple other scheduling criteria. In the production systems being investigated, the electrical energy is offered on a time-of-use (TOU) pricing regime. The objective of minimizing TOU energy costs conflicts sharply with most other traditional objectives in production scheduling. The aim is to identify best performing scheduling rules for different scenarios based on different shop congestion levels, and devise new rules to enable an improved integration of energy cost with other scheduling criteria.A ranking approach based on data envelopment analysis (DEA) and Ordered Weighting Average (OWA) concepts is presented. The proposed methodology exploits the preference voting system embedded under the cross-efficiency (CE) matrix to derive a collective importance scale for the aggregation process. The approach is applied to 28 dispatching rules (DRs) for scheduling jobs that arrive continuously at random points in time during the production horizon. Computational results highlight the effect of energy costs on the overall ranking of the DRs, and unveil the superiority of certain rules under multi-objective performance criteria.

Consensus reaching for prospect cross-efficiency in data envelopment analysis with minimum adjustments

The expected utility theory and the arithmetic average method have been widely adopted for the cross-efficiency strategy formulation and cross-efficiencies aggregation. However, traditional cross-efficiency methods often fail to consider the decision maker’s (DM’s) risk attitude and neglect the consensus among decision-making units (DMUs) in the cross-evaluation process. Thereby, this paper utilizes the prospect cross-efficiency (PCE) model to describe the DM’s subjective preference in the evaluation process and a consensus framework to manage the consensus among DMUs in the aggregation process. First, the PCE model is utilized to derive the cross-efficiency matrix, and it can reflect DMs’ psychological factors in the strategy formulation of cross-efficiency. Then, a two-stage consensus optimization framework (TSCOF) with minimum adjustments is presented to improve the consensus degree (CD) among the DMUs in the cross-evaluation process. The TSCOF generates a sequential optimization strategy that minimizes the adjustment distance and the number of modifications sequentially. Next, we develop an interactive consensus reaching process (CRP) for PCE problems based on the consensus rule. The CRP does well in helping improve the CD among DMUs. Finally, an illustrative example related to the performance of the S&T activities of 10 universities is offered to elaborate on the feasibility and effectiveness of our approach.

Evaluation of Road Safety Performance Based on Self-Reported Behaviour Data Set

Self-reported behavioural data, being often linguistic variables that represent a qualitative measure of respondents’ opinions/attitudes, are vague, uncertain, and fuzzy in nature. A road safety performance index, based on these fuzzy data, should consider this uncertainty. In this study, fuzzy numbers were used to describe self-reported behaviour on Montenegrin roads, which was further integrated into the data envelopment analysis (DEA), a technique for measuring the relative performance of decision-making units (DMUs). The vagueness of the performance scores obtained in this way was treated with grey relational analysis (GRA). GRA was applied to the cross-efficiency (CE) matrix constructed by the DEA to distinguish Montenegrin municipalities’ performance, with the main goal of describing road safety in the observed territories in the environment of uncertain/grey data. It is concluded that the proposed DEA–GRA model, based on fuzzy data, provides a more reasonable and encompassing measure of performance, and with which the overall ranking position of municipalities can be obtained.

Cross-efficiency intervals integrated ranking approach based on the generalized Fermat-Torricelli point

As a significant extension of data envelopment analysis studies, cross-efficiency has been commenly adopted to rank the performances of decision-making units (DMUs). Interval cross-efficiency techniques can solve the nonuniqueness problem by considering all possible weight sets in weight space. Most existing cross-efficiency approaches employ the average cross-efficiency to aggregate the cross-efficiency matrix (CEM), but the consensus preferences among DMUs acquire little consideration. In this paper, we develop a new integrated ranking technique for cross-efficiency intervals. Cross-efficiency methods with crisp and interval input–output data are used to construct generalized interval CEMs. The cross-efficiency intervals are projected into two-dimensional coordinates, and the optimal rally point is generated using the plant growth simulation algorithm to solve the generalized Fermat-Torricelli problem. A possibility distribution function is applied to transform the aggregated interval CEMs, and then we obtain the ultimate cross-efficiency rankings of all DMUs. Two illustrations are provided to demonstrate the validity of the proposed approach.

A new cross-efficiency ranking method in group decision making environment

As an extended model of data envelopment analysis (DEA), cross-efficiency is utilized in various fields. It applies the relative evaluation values of homogeneous decision-making units (DMUs) in ranking completely. Due to the non-uniqueness of DEA optimal weights, and the cross-efficiency methods focus on the aggregation of cross-efficiency matrix values. There is a low correction between the final efficiency evaluation value and the value assigned to the criteria weight or the value of the criteria. This paper uses the modified aggressive cross-efficiency model to calculate the weights of the decision-making units (DMUs). Then according to the consistency of peer-evaluation criteria, the entropy weight method is used to aggregate the index weights to generate a set of common weights, and a new Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) method is used to calculate the utility score of DMUs that can rank them completely. Finally, a numerical case of supplier selection is offered to illustrate the feasibility and effectiveness of the proposed method.

Variable returns to scale DEA\u2014Taguchi approach for ternary additives optimization in expansive soil subgrade enhancement

In this study, variable returns to scale (VRS) data envelopment analysis was integrated into the Taguchi approach to optimize ternary additives for expansive soil enhancement. The ternary additives selected were sawdust ash (SDA), quarry dust (QD) and ordinary Portland cement (OPC). The additives were set as the input variables while multiple responses obtained from the experiments performed with the Taguchi orthogonal array were set as the output variables. Each row in the orthogonal array were defined as a decision making unit (DMU) in the optimization process and output-oriented VRS model was used to obtain the efficiency score for each DMU. Next, benevolent formulation was utilized to obtain the multipliers for the inputs and outputs which were subsequently used to determine the cross efficiency scores for each DMU. The cross-efficiency scores were used to construct the cross-efficiency matrix. Thereafter, the mean cross-efficiency score (MCES) was determined for each DMU. Parameter level that maximizes the MCES was chosen as the optimal level for that parameter. Optimum combination of additives was found at A6 B2 C3. Lastly, confirmatory experiments performed by blending the soil with the optimum combination of additives showed the effectiveness of this method in the enhancement of expansive soil properties.

Ranking DMUs using a novel combination method for integrating the results of relative closeness benevolent and relative closeness aggressive models

In this paper, a novel combination method is offered to integrate the results of two new relative closeness models, called relative closeness benevolent (RCB) and relative closeness aggressive (RCA) models, for ranking all DMUs. To prove the applicability of the proposed method, it is examined in three numerical examples, performance assessment problem, six nursing homes and fourteen international passenger airlines. Firstly, RCB and RCA models were formulated in order to generate the cross-efficiency intervals matrix (CEIM). After obtaining CEIM, the RC index was utilized to generate a combined cross-efficiency matrix (combined CEM). In combined CEM, target DMUs were viewed as criteria and DMUs were viewed as alternatives. After that, the weights of each criterion were generated using a new weighting method based on standard deviation technique (MSDT). Finally, all DMUs were evaluated and ranked. Comparison with existing cross-efficiency models indicates the more reliable results through the use of the proposed method.

Improving DEA cross-efficiency optimization in portfolio selection

This paper deals with the role of alternative optimal solutions existing in the data envelopment analysis (DEA) models for cross-efficiency evaluation in portfolio selection. The paper shows that incorporating alternative optimal solutions for constructing cross-efficiency matrix improves the result of the mean-variance portfolio selection method. This improvement means that building portfolios with lower risk and higher expected return is possible when alternative optimal solutions are considered. The proposed method in this paper is applied to stock portfolio selection in the Tehran stock market.

An Alternative Approach to Rank Efficient DMUs in DEA via Cross-Efficiency Evaluation, Gini Coefficient, and Bonferroni Mean

In this paper, we focus on a critical problem in data envelopment analysis (DEA) and propose a simple resolution for it. The major problem of the DEA is the existence of several efficient decision-making units (DMUs). To deal with this issue, we introduce a method that involves cross-efficiency evaluation, Gini coefficient, and Bonferroni mean. First, a cross-efficiency matrix is developed. Then, mixing the Gini coefficient and Bonferroni mean, a Gini–Bonferroni (GB) index is proposed for ranking efficient DMUs, where the DMUs with bigger GB are ranked higher. The proposed method broke the tie between efficient DMUs. Finally, a numerical example and real application of this method are presented in the ranking of research and development (R&D) investment companies in the pharmaceutical and biotechnology industries.

DEA Cross-Efficiency Ranking Method Based on Grey Correlation Degree and Relative Entropy

The ranking of decision-making units (DMUs) is one of the most significant issues in efficiency evaluation. However, the calculation results from the traditional Data envelopment analysis(DEA), method sometimes include multiple efficient DMUs or multiple DMUs with the same efficiency value, in which case the approach is weak in distinguishing among these DMUs. Therefore, this study proposes a DEA cross-efficiency ranking method based on the relative entropy evaluation method and the grey relational analysis method. First, the approach uses the cross-efficiency matrix as the decision matrix of multiple criteria decision-making (MCDM), and the relationship between DMU and the ideal solution is analyzed by the grey relational analysis method and the relative entropy evaluation method. Then, the degree of the criteria is determined by Shannon entropy, and the weighted grey correlation degree and the weighted relative entropy are obtained. Finally, with the comprehensive relative closeness degree between the DMU and the ideal solution, we can sort all the DMUs accordingly. In a comparative analysis, it shows that this method analyzes the similarity between DMUs and the ideal solution from the information distance and the similarity of the data sequence curve, and has certain advantages for analyzing the ranking of DMUs.

Two-Stage Fuzzy Cross-Efficiency Aggregation Model Using a Fuzzy Information Retrieval Method

In an uncertain environment, the process of fuzzy cross-evaluation in data envelopment analysis (DEA) models is explained by different types of fuzzy data. In this paper, the fuzzy cross-efficiency matrix with different types of fuzzy data will be changed into a cross-evaluation similarity matrix by similarity theory of generalized trapezoidal fuzzy numbers. Based on the fuzzy information retrieval (FIR) method, a new ordered-geometric-mean averaging operator is defined to yield aggregation weights by the relevance between the self-evaluation and cross-evaluation of each decision-making unit (DMU). Furthermore, the common neighbor DMU selection algorithm is constructed to classify the DMUs. Then, the fuzzy logic rules are proposed to modify the aggregation results by combining the clustering results. Finally, this paper demonstrates the applicability of fuzzy cross-efficiency aggregation using the FIR method by taking an example of 12 suppliers from the semiconductor industry.

A novel DEACM integrating affinity propagation for performance evaluation and energy optimization modeling: Application to complex petrochemical industries

Data Envelopment Analysis (DEA) has been widely used in performance and energy efficiency evaluation. However, in the traditional DEA, the effective of each decision making unit (DMU) is evaluated through its own optimized perspective and regardless of other DMUs influence, which may result in too many effective DMUs. And the DEA cross-model (DEACM) can distinguish the effective DMUs better by constructing a cross-efficiency matrix, but the optimal weight of the DMU may not be unique, so the cross efficiency of the DEACM may be different. Therefore, this paper proposes a novel DEACM based on the affinity propagation (AP) clustering algorithm (AP-DEACM). Through the AP clustering algorithm, the high impact data affecting the performance capacity and energy saving are obtained. Then the better effective DMU is identified through the high discrimination of the AP–DEACM. Finally, the proposed AP-DEACM is used for performance evaluation and energy optimization modeling of the Pure Terephthalic Acid (PTA) production process and the ethylene industrial process in complex petrochemical industries. The experimental results show that the energy saving potential of PTA production plants and ethylene production plants are 2.78% and 1.26%, respectively, and the average value of carbon emission savings potential is 3.62% in ethylene production plants.

A method to rank the efficient units based on cross efficiency matrix without involving the zero weights

One of the basic objections of the previous models of cross efficiency (CE) is the possibility for the weights to equal zero. This case takes place for the inputs and the outputs in the efficient responses in CE models. Therefore, the input and the output weights which equal zero do not play a role in computing the score of the CE. In this paper, to overcome this problem, an idea to prevent the optimal weights to equal zero in the CE method is offered. This new method can be expanded to all CE models. Based on the offered method, a zero-one mixed linear programming problem is proposed to obtain a set of non-zero weights among the optimal solutions of the preliminary CE model. Following, the zero-one mixed linear programming problem is changed into an equivalent linear program. Then, according to a consistent CE matrix the efficient units are ranked. In order to explain the model and indicate its advantage, an example is given.

A method to rank the efficient units based on cross efficiency matrix without involving the zero weights

A method to rank the efficient units based on cross efficiency matrix without involving the zero weights