An Integrated Artificial Neural Network Fuzzy C-Means-Normalization Algorithm for performance assessment of decision-making units: The cases of auto industry and power plant

Efficiency frontier analysis has been an important approach of evaluating firms’ performance in private and public sectors. There have been many efficiency frontier analysis methods reported in the literature. However, the assumptions made for each of these methods are restrictive. Each of these methodologies has its strength as well as major limitations. This study proposes two non-parametric efficiency frontier analysis sub-algorithms based on (1) Artificial Neural Network (ANN) technique and (2) ANN and Fuzzy C-Means for measuring efficiency as a complementary tool for the common techniques of the efficiency studies in the previous studies. Normal probability plot is used to find the outliers and select from these two methods. The proposed computational algorithms are able to find a stochastic frontier based on a set of input–output observational data and do not require explicit assumptions about the functional structure of the stochastic frontier. In these algorithms, for calculating the efficiency scores, a similar approach to econometric methods has been used. Moreover, the effect of the return to scale of decision-making unit (DMU) on its efficiency is included and the unit used for the correction is selected by notice of its scale (under constant return to scale assumption). Also in the second algorithm, for increasing DMUs’ homogeneousness, Fuzzy C-Means method is used to cluster DMUs. Two examples using real data are presented for illustrative purposes. First example which deals with power generation sector shows the superiority of Algorithm 2 while the second example dealing auto industries of various developed countries shows the superiority of Algorithm 1. Overall, we find that the proposed integrated algorithm based on ANN, Fuzzy C-Means and Normalization approach provides more robust results and identifies more efficient units than the conventional methods since better performance patterns are explored.     

An integrated artificial neural network-genetic algorithm clustering ensemble for performance assessment of decision making units

This study proposes a non-parametric efficiency frontier analysis method based on artificial neural network (ANN) and genetic algorithm clustering ensemble (GACE) for measuring efficiency as a complementary tool for the common techniques of the efficiency studies in the previous studies. The proposed ANN GA algorithm is able to find a stochastic frontier based on a set of input–output observational data and do not require explicit assumptions about the functional structure of the stochastic frontier. Furthermore, it uses a similar approach to econometric methods for calculating the efficiency scores. Moreover, the effect of the return to scale of decision making unit (DMU) on its efficiency is included and the unit used for the correction is selected based on its scale (under constant return to scale assumption). Also, in this algorithm, GA is used to cluster DMUs to increase DMUs’ homogeneousness. It should be noted that data envelopment analysis (DEA) is sensitive to the presence of the outliers and statistical noise. It is also not capable of performing prediction and forecasting. This is shown by two examples related to outlier situations. However, the proposed algorithm is capable of handling outliers and noise and DEA is used as a benchmark to show advantages of the proposed algorithm. Also, the proposed algorithm and conventional algorithm are compared in viewpoint of DEA through statistical t-test. The proposed approach is applied to a set of actual conventional power plants to show its applicability and superiority.     

An ideal-seeking fuzzy data envelopment analysis framework

Data envelopment analysis (DEA) is a widely used mathematical programming approach for evaluating the relative efficiency of decision making units (DMUs) in organizations. Crisp input and output data are fundamentally indispensable in traditional DEA evaluation process. However, the input and output data in real-world problems are often imprecise or ambiguous. In this study, we present a four-phase fuzzy DEA framework based on the theory of displaced ideal. Two hypothetical DMUs called the ideal and nadir DMUs are constructed and used as reference points to evaluate a set of information technology (IT) investment strategies based on their Euclidean distance from these reference points. The best relative efficiency of the fuzzy ideal DMU and the worst relative efficiency of the fuzzy nadir DMU are determined and combined to rank the DMUs. A numerical example is presented to demonstrate the applicability of the proposed framework and exhibit the efficacy of the procedures and algorithms.     

DEA efficiency analysis: A DEA approach with double frontiers

Data envelopment analysis (DEA) is a method for measuring efficiency of peer decision-making units (DMUs). Conventional DEA evaluates the performance of each DMU using a set of most favourable weights. As a result, traditional DEA models can be considered methods for the analysis of the best relative efficiency or analysis of the optimistic efficiency. DEA efficient DMUs obtained from conventional DEA models create an efficient production frontier. Traditional DEA can be used to identify units with good performance in the most desirable scenarios. There is a similar approach that evaluates the performance indicators of each DMU using a set of most unfavourable weights. Accordingly, such models can be considered models for analysing the worst relative efficiency or pessimistic efficiency. This approach uses the inefficient production frontier for determining the worst relative efficiency that can be assigned to each DMU. DMUs lying on the inefficient production frontier are referred to as DEA inefficient while those neither on the efficient frontier nor on the inefficient frontier are declared DEA inefficient. It can be argued that both relative efficiencies should be considered simultaneously and any approach with only one of them would be biased. This paper proposed the integration of both efficiencies as an interval so that the overall performance score would belong to this interval. It was shown that efficiency interval provided more information than either of the two efficiencies, which was illustrated using two numerical examples.     

基于SBM区间模型的决策单元相似度

在数据包络分析方法的研究中,当决策单元的输入输出值为区间数时,区间效率的测算取决于生产参照集的选择.对此,针对非径向非角度的松弛测度模型(SBM),通过比较计算认为,以最佳生产状态为统一参照来测算决策单元的区间效率最有利于排序.同时,利用区间数相似度理论建立模糊相似矩阵,进而提出基于参照单元区间效率的相似度排序方法,对决策单元聚类和排序.最后通过算例表明了所提出方法的可行性和有效性.     

Positive and normative use of fuzzy DEA‐BCC models: A critical view on NATO enlargement

Data envelopment analysis (DEA) is a widely used mathematical programming approach for comparing the input and output of a set of comparable decision‐making units (DMUs) by evaluating their relative efficiency. The traditional DEA methods require accurate measurement of both the inputs and outputs. However, the real evaluation of the DMUs is often characterized by imprecision and uncertainty in data definitions and measurements. The development of fuzzy DEA (FDEA) with imprecise and ambiguous data has extended the scope of application for efficiency measurement. The purpose of this paper is to develop a fuzzy DEA framework with a BCC model for measuring crisp and interval efficiencies in fuzzy environments. We use an α‐level approach to convert the fuzzy Banker, Charnes, and Cooper (BCC) (variable returns to scale) model into an interval programming model. Instead of comparing the equality (or inequality) of the two intervals, we define a variable in the interval to satisfy our constraints and maximize the efficiency value. We present a numerical example to show the similarities and differences between our solution and the solutions obtained from four fuzzy DEA methods in the literature. In addition, a case study for NATO enlargement is presented to illustrate the applicability of the proposed method.     

Expansion of the cross-efficiency evaluation by using the equations forming the efficient frontier in DEA

In this paper, we propose an algorithm to calculate cross-efficiency scores which used the equations forming the efficient frontier in data envelopment analysis (DEA). In many standard DEA models, each decision-making unit (DMU) is evaluated by using the advantageous weight for itself. Then, many DMUs are evaluated as efficient, and those efficient DMUs are not ranked by the models. The cross-efficiency evaluation is a method to rank DMUs by using the advantageous weights for all DMUs. Previously, the cross-efficiency scores based on different ideas are calculated by solving multiple linear or nonlinear programming problems. However, it is often hard to solve such a nonlinear programming problem. Therefore, by analysing the efficient frontier, we construct an algorithm to calculate alternative cross-efficiency scores.     

A method of stepwise benchmarking for inefficient DMUs based on the proximity-based target selection

DEA is a useful nonparametric method of measuring the relative efficiency of a DMU and yielding a reference target for an inefficient DMU. However, it is very difficult for inefficient DMUs to be efficient by benchmarking a target DMU which has different input use. Identifying appropriate benchmarks based on the similarity of input endowment makes it easier for an inefficient DMU to imitate its target DMUs. But it is rare to find out a target DMU, which is both the most efficient and similar in input endowments, in real situation. Therefore, it is necessary to provide an optimal path to the most efficient DMU on the frontier through several times of a proximity-based target selection process. We propose a dynamic method of stepwise benchmarking for inefficient DMUs to improve their efficiency gradually.The empirical study is conducted to compare the performance between the proposed method and the prior methods with a dataset collected from Canadian Bank branches. The comparison result shows that the proposed method is very practical to obtain a gradual improvement for inefficient DMUs while it assures to reach frontier eventually.     

A modified slacks‐based ranking method handling negative data in data envelopment analysis

Performance ranking for a set of comparable decision‐making units (DMUs) with multiple inputs and outputs is an important and often‐discussed topic in data envelopment analysis (DEA). Conventional DEA models distinguish efficient units from inefficient ones but cannot further discriminate the efficient units, which all have a 100% efficiency score. Another weakness of these models is that they cannot handle negative inputs and/or outputs. In this paper, a new modified slacks‐based measure is proposed that works in the presence of negative data and provides quantitative data that helps decision makers obtain a full ranking of DMUs in situations where other methods fail. In addition, the new method has the properties of unit invariance and translation invariance, and it can give targets for inefficient DMUs to guide them to achieve full efficiency. Two numerical examples are analysed to demonstrate the usefulness of the new method.     

A concept of fuzzy input mix-efficiency in fuzzy DEA and its application in banking sector

Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiency of homogeneous decision making units (DMUs) on the basis of multiple inputs and multiple outputs. There exist radial and non-radial models in DEA. Radial models only deal with proportional changes of inputs/outputs and neglect the input/output slacks. On the other hand, non-radial models directly deal with the input/output slacks. The slack-based measure (SBM) model is a non-radial model in which the SBM efficiency can be decomposed into radial, scale and mix-efficiency. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure requires crisp data which may not always be available in real world applications. In real world problems, data may be imprecise or fuzzy. In this paper, we propose (i) a concept of fuzzy input mix-efficiency and evaluate the fuzzy input mix-efficiency using α – cut approach, (ii) a fuzzy correlation coefficient method using expected value approach which calculates the expected intervals and expected values of fuzzy correlation coefficients between fuzzy inputs and fuzzy outputs, and (iii) a new method for ranking the DMUs on the basis of fuzzy input mix-efficiency. The proposed approaches are then applied to the State Bank of Patiala in the Punjab state of India with districts as the DMUs.     

Robust clustering by pruning outliers

In many applications of C-means clustering, the given data set often contains noisy points. These noisy points will affect the resulting clusters, especially if they are far away from the data points. In this paper, we develop a pruning approach for robust C-means clustering. This approach identifies and prunes the outliers based on the sizes and shapes of the clusters so that the resulting clusters are least affected by the outliers. The pruning approach is general, and it can improve the robustness of many existing C-means clustering methods. In particular, we apply the pruning approach to improve the robustness of hard C-means clustering, fuzzy C-means clustering, and deterministic-annealing C-means clustering. As a result, we obtain three clustering algorithms that are the robust versions of the existing ones. In addition, we integrate the pruning approach with the fuzzy approach and the possibilistic approach to design two new algorithms for robust C-means clustering. The numerical results demonstrate that the pruning approach can achieve good robustness.     

A new ranking method to fuzzy data envelopment analysis

Due to its wide practical use, data envelopment analysis (DEA) has been adapted to many fields to deal with problems that have occurred in practice. One adaptation has been in the field of ranking decision-making units (DMUs). Most methods of ranking DMUs assume that all input and output data are exactly known, but in real life the data cannot be precisely measured. Thus this paper will carry out some researches to DEA under fuzzy environment. A fuzzy comparison of fuzzy variables is defined and the CCR model is extended to be a fuzzy DEA model based on credibility measure. In order to rank all the DMUs, a full ranking method will be given. Since the ranking method involves a fuzzy function, a fuzzy simulation is designed and embedded into the genetic algorithm to establish a hybrid intelligent algorithm. However, it is shown to be possible to avoid some of the need for dealing with these nonlinear problems by identifying conditions under which they can be replaced by linear problems. Finally we will provide a numerical example to illustrate the fuzzy DEA model and the ranking method.     

基于自组织神经网络的非线性系统建模

针对基于T-S模糊模型的非线性系统建模问题,提出了一种基于自组织神经网络的新方法.在T-S模糊模型的建模中,目前常用的模糊C均值聚类算法存在迭代次数多,计算耗时的缺点.首先,利用竞争学习算法对输入空间进行聚类,基于此结果,借助于模糊C均值聚类算法进一步优化聚类结果,提取T-S模糊模型的规则前件隶属函数参数.然后,采用最小二乘法求得T-S模糊模型的规则后件参数,从而建立起非线性系统的T-S模糊模型.最后,仿真结果表明,该方法可以为模糊建模提供好的模型结构,并且有较高的计算效率和精度.     

A learning performance evaluation with benchmarking concept for English writing courses

This paper adopts data envelopment analysis (DEA), a robust and reliable evaluation method widely applied in various fields to explore the key indicators contributing to the learning performance of English freshmen writing courses in a university of Taiwan from the academic year 2004 to 2006. The results of DEA model applied in learning performance change our original viewpoint and reveal that some decision-making units (DMUs) with higher actual values of inputs and outputs have lower efficiency because the relative efficiency of each DMU is measured by their distance to the efficiency frontier. DMUs may refer to different facet reference sets according to their actual values located in lower or higher ranges. In the managerial strategy of educational field, the paper can encourage inefficient DMUs to always compare themselves with efficient DMUs in their range and make improvement little by little. The results of DEA model can also give clear indicators and the percentage of which input and output items to improve. The paper also demonstrates that the benchmarking characteristics of the DEA model can automatically segment all the DMUs into different levels based on the indicators fed into the performance evaluation mechanism. The efficient DMUs on the frontier curve can be considered as the boundaries of the classification which are systematically defined by the DEA model according to the statistic distribution.     

基于MapReduce的并行模糊C均值算法

模糊C均值是一种重要的软聚类算法,针对模糊C均值的随着数据量的增加,时间复杂度过高的缺点,提出了一种基于MapReduce的并行模糊C均值算法。算法重新设计模糊C均值,使其符合MapReduce的基于key/value的编程模型,并行计算数据集到中心点的隶属度,并重新计算出新的聚类中心,提高了模糊C均值处理大容量数据的计算效率。实验结果表明,基于MapReduce的并行模糊C均值算法具有较高的加速比和扩展性。     

Ranking efficient DMUs using cooperative game theory

The problem of ranking Decision Making Units (DMUs) in Data Envelopment Analysis (DEA) has been widely studied in the literature. Some of the proposed approaches use cooperative game theory as a tool to perform the ranking. In this paper, we use the Shapley value of two different cooperative games in which the players are the efficient DMUs and the characteristic function represents the increase in the discriminant power of DEA contributed by each efficient DMU. The idea is that if the efficient DMUs are not included in the modified reference sample then the efficiency score of some inefficient DMUs would be higher. The characteristic function represents, therefore, the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient units is dropped from the sample. Alternatively, the characteristic function of the cooperative game can be defined as the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient DMUs are the only efficient DMUs that are included in the sample. Since the two cooperative games proposed are dual games, their corresponding Shapley value coincide and thus lead to the same ranking. The more an efficient DMU impacts the shape of the efficient frontier, the higher the increase in the efficiency scores of the inefficient DMUs its removal brings about and, hence, the higher its contribution to the overall discriminant power of the method. The proposed approach is illustrated on a number of datasets from the literature and compared with existing methods.     

Sensitivity analysis on modified variable returns to scale model in Data Envelopment Analysis using facet analysis

One of the concerns in Data Envelopment Analysis (DEA) is the sensitivity and stability analysis of specific Decision Making Unit (DMU), which is under evaluation. In economical point of view, the stability region in input–output space for maintaining the efficiency score of efficient DMU is important. In this paper, a new sensitivity analysis approach based on Banker, Charnes and Cooper (BCC) model which is modified by facet analysis, is developed. An extended stability region is determined especially for DMUs that are placed on intersection of efficient and weak efficient frontier. The results are shown by numerical examples.     

A cross-dependence based ranking system for efficient and inefficient units in DEA

Decision making units (DMUs) can be ranked using data envelopment analysis (DEA) technologies. This paper develops a new ranking system under the condition of variable returns to scale (VRS) based on a measure of cross-dependence efficiency (MCDE), where the evaluation for an efficient DMU is dependent of the efficiency changes of all inefficient units due to its absence in the reference set, while the appraisal of inefficient DMUs depends on the influence of the exclusion of each efficient unit from the reference set. The infeasibility problem that arises from the conventional super-efficiency models is eliminated. A new approach to ranking inefficient units is embedded in the proposed ranking system. Through an example, advantages of our proposal are demonstrated. A real application illustrates the effectiveness of the cross-dependence based ranking system.     

Value of the stochastic efficiency in data envelopment analysis

This article examines the potential benefits of solving a stochastic DEA model over solving a deterministic DEA model. It demonstrates that wrong decisions could be made whenever a possible stochastic DEA problem is solved when the stochastic information is either unobserved or limited to a measure of central tendency. We propose two linear models: a semi-stochastic model where the inputs of the DMU of interest are treated as random while the inputs of the other DMUs are frozen at their expected values, and a stochastic model where the inputs of all of the DMUs are treated as random. These two models can be used with any empirical distribution in a Monte Carlo sampling approach. We also define the value of the stochastic efficiency (or semi-stochastic efficiency) and the expected value of the efficiency.     

基于改进PSO和FCM的模糊辨识

为了提高T-S模糊模型的辨识精度和效率,本文提出了一种改进的粒子群算法和模糊C均值聚类算法相结合的模糊辨识新方法。在该方法中,针对粒子群算法在处理高维复杂函数时容易陷入局部极值的问题,提出了一种粒子群局部搜索和全局搜索动态调整的全新优化算法。模糊C均值聚类算法是模糊辨识最常用的方法之一,该算法简单,计算效率高,但是对初始化特别敏感,容易陷入局部最优。为了解决这一问题,利用改进粒子群算法的全局搜索能力优化聚类中心,显著地提高了算法的辨识精度和效率。最后,针对非线性系统进行建模仿真,仿真结果表明了本文方法的有效性和优越性。