Fuzzy efficiency ranking in fuzzy two-stage data envelopment analysis

Data envelopment analysis (DEA) is a useful tool for efficiency measurement of firms and organizations. Many production systems in the real world are composed of two processes connected in series. Measuring the system efficiency without taking the operation of each process into consideration will obtain misleading results. Two-stage DEA models show the performance of individual processes, thus is more informative than the conventional one-stage models for making decisions. When input and output data are fuzzy numbers, the derived efficiencies become fuzzy as well. This paper proposes a method to rank the fuzzy efficiencies when the exact membership functions of the overall efficiencies derived from fuzzy two-stage model are unknown. By incorporating the fuzzy two-stage model with the fuzzy number ranking method, a pair of nonlinear program is formulated to rank the fuzzy overall efficiency scores of DMUs. Solving the pair of nonlinear programs determines the efficiency rankings. An example of the ranking of the 24 non-life assurance companies in Taiwan is illustrated to explain how the proposed method is applied.     

Decomposition of technical and scale efficiencies in two-stage production systems

Conventional data envelopment analysis (DEA) models are used to measure the technical and scale efficiencies of a system when it is considered as a whole unit. This paper extends the efficiency measurement to two-stage systems where each stage has one process and all the outputs from the first process become the inputs of the second. An input-oriented DEA model for the first process is developed to separate the process efficiency into the input technical and scale efficiencies, and an output-oriented model is developed for the second process to separate the process efficiency into the output technical and scale efficiencies. Combining the two models, the system efficiency is expressed as the product of the overall technical and scale efficiencies, where the overall technical and scale efficiencies are the products of the corresponding efficiencies of the two processes, respectively. The detailed decomposition allows the sources of inefficiency to be identified.     

Reliability and cost analysis of series system models using fuzzy parametric geometric programming

In this paper, we have discussed series system models with system reliability and cost. We have considered two types of the model; the former focuses on a problem of optimal reliability for series system with cost constraint and the latter is a center system cost model with reliability goal. It is necessary to improve the reliability of the system under limited available cost of system and also to minimize the systems cost subject to target goal of the reliability. Practically, cost of components has always been imprecise with vague in nature. So they are taken as fuzzy in nature and the reliability models are formulated as a fuzzy parametric geometric programming problem. Numerical examples are given to illustrate the model through fuzzy parametric geometric programming technique.     

Multi-period efficiency and Malmquist productivity index in two-stage production systems

Conventional two-stage data envelopment analysis (DEA) models measure the overall performance of a production system composed of two stages (processes) in a specified period of time, where variations in different periods are ignored. This paper takes the operations of individual periods into account to develop a multi-period two-stage DEA model, which is able to measure the overall and period efficiencies at the same time, with the former expressed as a weighted average of the latter. Since the efficiency of a two-stage system in a period is the product of the two process efficiencies, the overall efficiency of a decision making unit (DMU) in the specified period of time can be decomposed into the process efficiency of each period. Based on this decomposition, the sources of inefficiency in a DMU can be identified. The efficiencies measured from the model can also be used to calculate a common-weight global Malmquist productivity index (MPI) between two periods, in that the overall MPI is the product of the two process MPIs. The non-life insurance industry in Taiwan is used to verify the proposed model, and to explain why some companies performed unsatisfactorily in the specified period of time.     

Improving weak efficiency frontiers in the fuzzy data envelopment analysis models

Evaluating the performance of activities or organization by common data envelopment analysis models requires crisp input/output data. However, the precise inputs and outputs of production processes cannot be always measured. Thus, the data envelopment analysis measurement containing fuzzy data, called “fuzzy data envelopment analysis”, has played an important role in the evaluation of efficiencies of real applications. This paper focuses on the fuzzy CCR model and proposes a new method for determining the lower bounds of fuzzy inputs and outputs. This improves the weak efficiency frontiers of the corresponding production possibility set. Also a numerical example illustrates the capability of the proposed method.     

Efficiency decomposition for general multi-stage systems in data envelopment analysis

Conventional data envelopment analysis (DEA) models only consider the inputs supplied to the system and the outputs produced from the system in measuring efficiency, ignoring the operations of the internal processes. The results thus obtained sometimes are misleading. This paper discusses the efficiency measurement and decomposition of general multi-stage systems, where each stage consumes exogenous inputs and intermediate products (produced from the preceding stage) to produce exogenous outputs and intermediate products (for the succeeding stage to use). A relational model is developed to measure the system and stage efficiencies at the same time. By transforming the system into a series of parallel structures, the system efficiency is decomposed into the product of a modification of the stage efficiencies. Efficiency decomposition enables decision makers to identify the stages that cause the inefficiency of the system, and to effectively improve the performance of the system. An example of an electricity service system is used to explain the idea of efficiency decomposition.     

The dual simplex method and sensitivity analysis for fuzzy linear programming with symmetric trapezoidal numbers

In this paper, we first extend the dual simplex method to a type of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. The results obtained lead to a solution for fuzzy linear programming problems that does not require their conversion into crisp linear programming problems. We then study the ranges of values we can achieve so that when changes to the data of the problem are introduced, the fuzzy optimal solution remains invariant. Finally, we obtain the optimal value function with fuzzy coefficients in each case, and the results are described by means of numerical examples.     

Supply chain performance evaluation using fuzzy network data envelopment analysis: a case study in automotive industry

Supply chain performance evaluation problems are evaluated using data envelopment analysis. This paper proposes a fuzzy network epsilon-based data envelopment analysis for supply chain performance evaluation. In the common data envelopment analysis models which are used for evaluation of decision-maker units efficiency, there are several inputs and outputs. One of the bugs of such models is that the intermediate products and linking activities are overlooked. Considering these intermediate activities and products, the current study evaluates the performance of decision-maker units in an automotive supply chain. There are ten decision-maker units in the supply chain in which there are three suppliers, two manufacturers, two distributors, and four customers. Moreover, the overall efficiency of input-oriented (input-based) model and input-oriented divisional efficiency are calculated. In order to improve the efficiencies, the projections onto the frontiers are obtained by using the outputs of the solved model and Lingo software. In order to show the applicability of the proposed model, it is applied on automotive industry, as a case study, to evaluate supply chain performance. Then, the overall efficiencies of DMUs and each sections (divisions) of DMUs were calculated separately. Therefore, every organization can apply this evaluation method for improving the performance of alternative factors.

     

Solving generalized fuzzy data envelopment analysis model: a parametric approach

Data envelopment analysis (DEA) is a non-parametric technique to assess the performance of a set of homogeneous decision making units (DMUs) with common crisp inputs and outputs. Regarding the problems that are modelled out of the real world, the data cannot constantly be precise and sometimes they are vague or fluctuating. So in the modelling of such data, one of the best approaches is using the fuzzy numbers. Substituting the fuzzy numbers for the crisp numbers in DEA, the traditional DEA problem transforms into a fuzzy data envelopment analysis (FDEA) problem. Different methods have been suggested to compute the efficiency of DMUs in FDEA models so far but the most of them have limitations such as complexity in calculation, non-contribution of decision maker in decision making process, utilizable for a specific model of FDEA and using specific group of fuzzy numbers. In the present paper, to overcome the mentioned limitations, a new approach is proposed. In this approach, the generalized FDEA problem is transformed into a parametric programming, in which, parameter selection depends on the decision maker’s ideas. Two numerical examples are used to illustrate the approach and to compare it with some other approaches.     

Dominance stochastic models in data envelopment analysis

In this paper stochastic models in data envelopment analysis (DEA) are developed by taking into account the possibility of random variations in input-output data, and dominance structures on the DEA envelopment side are used to incorporate the modelbuilder's preferences and to discriminate efficiencies among decision making units (DMUs). The efficiency measure for a DMU is defined via joint dominantly probabilistic comparisons of inputs and outputs with other DMUs and can be characterized by solving a chance constrained programming problem. Deterministic equivalents are obtained for multivariate symmetric random errors and for a single random factor in the production relationships. The goal programming technique is utilized in deriving linear deterministic equivalents and their dual forms. The relationship between the general stochastic DEA models and the conventional DEA models is also discussed.     

Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis

This paper replaces ordinary DEA formulations with stochastic counterparts in the form of a series of chance constrained programming models. Emphasis is on technical efficiencies and inefficiencies which do not require costs or prices, but which are nevertheless basic in that the achievement of technical efficiency is necessary for the attainment of ‘allocative’, ‘cost’ and other types of efficiencies.     

含直觉模糊弹性约束的广义模糊变量线性规划

本文基于模糊结构元方法建立并讨论了一类含有直觉模糊弹性约束的广义模糊变量线性 规划问题。首先,简单介绍了结构元方法并对结构元加权排序中权函数表征决策者风险态度进行了深入分析。然后,通过选取风险中立型决策态度来定义序关系并拓展Verdegay模糊线性规划方法,将新型模糊变量线性规划问题转化为两个含一般模糊弹性约束的模糊变量线性规划模型,给出了此类规划最优直觉模糊解的求法。最后,通过数值算例进一步说明该方法的有效性。     

The Advantages of Fuzzy Optimization Models in Practical Use

Classical mathematical programming models require well-defined coefficients and right hand sides. In order to avoid a non satisfying modeling usually a broad information gathering and processing is necessary. In case of real problems some model parameters can be only roughly estimated. While in case of classical models the vague data is replaced by "average data", fuzzy models offer the opportunity to model subjective imaginations of the decision maker as precisely as a decision maker will be able to describe it. Thus the risk of applying a wrong model of the reality and selecting solutions which do not reflect the real problem can be clearly reduced. The modeling of real problems by means of deterministic and stochastic models requires extensive information processing. On the other hand we know that an optimum solution is finally defined only by few restrictions. Especially in case of larger systems we notice afterwards that most of the information is useless. The dilemma of data processing is due to the fact that first we have to calculate the solution in order to define, whether the information must be well-defined or whether vague data may be sufficient. Based on multicriteria programming problems it should be demonstrated that the dilemma of data processing in case of real programming problems can be handled adequately by modeling them as fuzzy system combined with an interactive problem-solving. Describing the real problem by means of a fuzzy system first of all only the available information or such information which can be achieved easily will be considered. Then we try to develop an optimum solution. With reference to the cost-benefit relation further information can be gathered in order to describe the solution more precisely. Furthermore it should be pointed out that some interactive fuzzy solution algorithms, e.g. FULPAL provide the opportunity to solve mixed integer multicriteria programming models as well.     

Three-way analysis of imprecise data

Data are often affected by uncertainty. Uncertainty is usually referred to as randomness. Nonetheless, other sources of uncertainty may occur. In particular, the empirical information may also be affected by imprecision. Also in these cases it can be fruitful to analyze the underlying structure of the data. In this paper we address the problem of summarizing a sample of three-way imprecise data. In order to manage the different sources of uncertainty a twofold strategy is adopted. On the one hand, imprecise data are transformed into fuzzy sets by means of the so-called fuzzification process. The so-obtained fuzzy data are then analyzed by suitable generalizations of the Tucker3 and CANDECOMP/PARAFAC models, which are the two most popular three-way extensions of Principal Component Analysis. On the other hand, the statistical validity of the obtained underlying structure is evaluated by (nonparametric) bootstrapping. A simulation experiment is performed for assessing whether the use of fuzzy data is helpful in order to summarize three-way uncertain data. Finally, to show how our models work in practice, an application to real data is discussed.     

Methods of critical value reduction for type-2 fuzzy variables and their applications

A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a type-2 fuzzy variable. Secondly, this paper applies the reduction methods to data envelopment analysis (DEA) models with type-2 fuzzy inputs and outputs, and develops a new class of generalized credibility DEA models. According to the properties of generalized credibility, when the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we can turn the proposed fuzzy DEA model into its equivalent parametric programming problem, in which the parameters can be used to characterize the degree of uncertainty about type-2 fuzziness. For any given parameters, the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers. Finally, one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed DEA model.     

Performance evaluation: An integrated method using data envelopment analysis and fuzzy preference relations

In a multi-attribute decision-making (MADM) context, the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives. This paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations. This proposed approach includes three stages. First, pairwise efficiency scores are computed using two DEA models: the CCR model and the proposed cross-evaluation DEA model. Second, the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation. Third, by use of the row wise summation technique, we yield a priority vector, which is used for ranking decision-making units (DMUs). For the case of a single output and a single input, the preference relation can be directly obtained from the original sample data. The proposed approach is validated by two numerical examples.     

Resource allocation based on cost efficiency

In this paper, we consider a resource allocation (RA) problem and develop an approach based on cost (overall) efficiency. The aim is to allocate some inputs among decision making units (DMUs) in such way that their cost efficiencies improve or stay unchanged after RA. We formulate a multi-objective linear programming problem using two different strategies. First, we propose an RA model which keeps the cost efficiencies of units unchanged. This is done assuming fixed technical and allocative efficiencies. The approach is based on the assumption that the decision maker (DM) may not have big changes in the structure of DMUs within a short term. The second strategy does not impose any restrictions on technical and allocative efficiencies. It guarantees that none of the cost efficiencies of DMUs get worse after RA, and the improvement for units is possible if it is feasible and beneficial. Two numerical examples and an empirical illustration are also provided.     

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.     

Fuzzy data envelopment analysis (DEA): Model and ranking method

Data Envelopment Analysis (DEA) is a very effective method to evaluate the relative efficiency of decision-making units (DMUs). Since the data of production processes cannot be precisely measured in some cases, the uncertain theory has played an important role in DEA. This paper attempts to extend the traditional DEA models to a fuzzy framework, thus producing a fuzzy DEA model based on credibility measure. Following is a method of ranking all the DMUs. In order to solve the fuzzy model, we have designed the hybrid algorithm combined with fuzzy simulation and genetic algorithm. When the inputs and outputs are all trapezoidal or triangular fuzzy variables, the model can be transformed to linear programming. Finally, a numerical example is presented to illustrate the fuzzy DEA model and the method of ranking all the DMUs.     

包含股指期货的投资组合之风险研究——Copula方法在金融风险管理中的应用

本文运用Copula方法研究了含股指期货的投资组合的风险度量问题.由于股指期货和股票现货之间存在很大的相关性,因此在度量组合的风险时,各资产间的相关结构起到了关键作用,但这一相关结构很难用线性的相关系数去刻画,本文采用Copula模型来描述相关结构。而后,我们构建了基于Copula理论的风险度量指标PVaR,并验证了不同Copula模型的拟合效果.我们利用沪深300指数的数据来研究股指期货和现货的相关结构,并使用了多种Copula函数结合不同的边际分布假设进行了模拟,说明了Copula方法在风险度量尤其是包含了股指期货的投资组合的风险度量上具有较高的精确性.