Additive efficiency decomposition in two-stage DEA

Kao and Hwang (2008) [Kao, C., Hwang, S.-N., 2008. Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan. European Journal of Operational Research 185 (1), 418–429] develop a data envelopment analysis (DEA) approach for measuring efficiency of decision processes which can be divided into two stages. The first stage uses inputs to generate outputs which become the inputs to the second stage. The first stage outputs are referred to as intermediate measures. The second stage then uses these intermediate measures to produce outputs. Kao and Huang represent the efficiency of the overall process as the product of the efficiencies of the two stages. A major limitation of this model is its applicability to only constant returns to scale (CRS) situations. The current paper develops an additive efficiency decomposition approach wherein the overall efficiency is expressed as a (weighted) sum of the efficiencies of the individual stages. This approach can be applied under both CRS and variable returns to scale (VRS) assumptions. The case of Taiwanese non-life insurance companies is revisited using this newly developed approach.     

Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan

The efficiency of decision processes which can be divided into two stages has been measured for the whole process as well as for each stage independently by using the conventional data envelopment analysis (DEA) methodology in order to identify the causes of inefficiency. This paper modifies the conventional DEA model by taking into account the series relationship of the two sub-processes within the whole process. Under this framework, the efficiency of the whole process can be decomposed into the product of the efficiencies of the two sub-processes. In addition to this sound mathematical property, the case of Taiwanese non-life insurance companies shows that some unusual results which have appeared in the independent model do not exist in the relational model. In other words, the relational model developed in this paper is more reliable in measuring the efficiencies and consequently is capable of identifying the causes of inefficiency more accurately. Based on the structure of the model, the idea of efficiency decomposition can be extended to systems composed of multiple stages connected in series.     

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.     

Decomposing technical efficiency and scale elasticity in two-stage network DEA

The constant returns to scale assumption maintained by neoclassical theorists for justifying the black-box structure of production technology in long run does not necessarily allow one to infer that there are no scale benefits available in its sub-technologies. Most of real-life production technologies are multi-stage in nature, and the sources of increasing returns lie in the sub-technologies. It is, therefore, imperative to estimate the scale economies of a firm not only for the network technology but also for the sub-technologies. To accomplish this, two approaches are suggested in this contribution, based on the premise concerning whether a network technology construct considers allocative inefficiency. The first approach, which is ours, makes use of a single network technology for two interdependent sub-technologies. The second approach, which is due to Kao and Hwang (2011), however, assumes complete allocative efficiency by considering two independent sub-technology frontiers, one for each sub-technology. The distinction between these two approaches is important from a policy point of view since the network efficiencies revealed from these two approaches have distinctive causative factors that do not permit them to be used interchangeably.     

Determining best practice production in an aluminium smelter involving sub-processes based substantially on tacit knowledge: an application of Communities of Practice

This paper considers the difficulties associated with a production process that contains a sub-process that is not fully understood and for which data for many parameters are only able to be approximately obtained. The aluminium smelting industry epitomizes such a situation. Here, the critical sub-process that exemplifies these difficulties is the actual heart of the smelter, the electrolytic processing of alumina. This sub-process of aluminium production is at best ‘fuzzy’ and relies on the smelter operators to use their experience and tacit knowledge on a day-to-day basis, that is, the sub-process involves ‘alchemy’. In this paper, this is referred to as the tacit knowledge problem. The impact of such sub-processes on production is significant and the development of a methodology that will lead to a reduced reliance on uncertain alchemy associated with them, highly beneficial. The role of Communities of Practice in finding a solution to the tacit knowledge problem is discussed together with its integration into a mixed-mode model for the determination of best practice production for the smelter.     

Efficiency analysis in two-stage structures using fuzzy data envelopment analysis

Two-stage data envelopment analysis (TsDEA) models evaluate the performance of a set of production systems in which each system includes two operational stages. Taking into account the internal structures is commonly found in many situations such as seller-buyer supply chain, health care provision and environmental management. Contrary to conventional DEA models as a black-box structure, TsDEA provides further insight into sources of inefficiencies and a more informative basis for performance evaluation. In addition, ignoring the qualitative and imprecise data leads to distorted evaluations, both for the subunits and the system efficiency. We present the fuzzy input and output-oriented TsDEA models to calculate the global and pure technical efficiencies of a system and sub-processes when some data are fuzzy. To this end, we propose a possibilistic programming problem and then convert it into a deterministic interval programming problem using the α-level based method. The proposed method preserves the link between two stages in the sense that the total efficiency of the system is equal to the product of the efficiencies derived from two stages. In addition to the study of technical efficiency, this research includes two further contributions to the ancillary literature; firstly, we minutely discuss the efficiency decompositions to indicate the sources of inefficiency and secondly, we present a method for ranking the efficient units in a fuzzy environment. An empirical illustration is also utilised to show the applicability of the proposed technique.     

外源投入型两阶段制造系统网络DEA效率分析研究

制造过程评价是改善制造系统效率的重要一环,传统的评价方法将每个制造系统决策单元视为黑箱来研究整体效率,忽略了中间产品转化信息及投入要素在各子过程中的配置信息。针对两阶段(第二阶段有外源性新投入)制造系统的效率评估问题,分别在固定规模报酬和可变规模报酬假设下,充分利用制造系统中间产品的转化及外源投入要素的配置信息,建立了制造系统网络DEA效率测度及分解模型,建模方法遵循客观评价原则,无需事先主观确定子效率和系统效率之间的组合关系。并将其应用于钢铁制造系统效率测度与分解,研究结果表明该方法能够挖掘决策单元内部子单元的效率情况,帮助决策者发现复杂制造过程非有效的根源,为复杂制造过程的整体效率测度及分解提供了有效的分析方法。     

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.     

Optimal Consecutive-k-out-of-(2k+1): G Cycle

We present a complete proof for the invariant optimal assignment for consecutive-k-out-of-(2k+1): G cycle, which was proposed by Zuo and Kao in 1990 with an incomplete proof, pointed out recently by Jalali, Hawkes, Cui and Hwang.     

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.     

Utility-based two-stage models with fairness concern

This paper studies the impact of social comparison and fairness concerns on the efficiency of a two-stage process. The decision maker of each stage cares about not only the absolute score of his efficiency, but also the relative status when comparing with the other. By incorporating the utility theory and the concept of fairness, an efficiency-based Neumann–Morgenstern cardinal utility is defined to compose of basic utility from his self-efficiency and additional utility from the fairness concern of the other’s efficiency. Utility-based two-stage models are proposed to optimize the utilities of the stages rather than only the efficiencies instead. We characterize the concern of fairness as advantageous and disadvantageous inequity based on equitable outcome comparison. By investigating the non-cooperating relationship between two stages, we show that the stage dominating the process has the incentive to optimize his efficiency without ignoring that of the other, which is contrary to the conventional situation. In addition, the coefficients of equitable outcome and inequity significantly affect the efficiencies of both stages. We further investigate the cooperation between two stages and find that the efficiencies of the stages vary with the coefficients of unfairness perceptions. Numerical analysis verifies the validity of the proposed models and identifies the impacts of the coefficients of equitable outcome and inequity on the stages’ and overall efficiencies.     

Analysis of nonpreemptive priority queues with multiple servers and two priority classes

In this paper, we model a priority multiserver queueing system with two priority classes. A high priority customer has nonpreemptive priority over low priority customers. The approaches for solving the problem are the state-reduction based variant of Kao, the modified boundary algorithm of Latouche, the logarithmic reduction algorithm of Latouche and Ramaswami, and the power-series method of Blanc. The objectives of this paper are to present a power-series implementation for the priority queue and to evaluate the relative efficiencies of alternative procedures to compute various performance characteristics. In the paper, we find that at times the logarithmic reduction algorithm may not perform as well as expected and the power-series approach can occasionally pose numerical difficulties.     

Some remarks on the two-level DEA model

The two-level DEA model was introduced to increase the discriminational power of Data Envelopment Analysis (DEA) models. This nonlinear model was presented by Meng et al. (2008) [3], and then converted into a linear model by Kao (2008) [4].In this paper two subjects will be discussed: first, we show that the two-level DEA model is a special case of DEA models where weight restrictions are applied. Then, we express that the nonlinear model is equivalent to the conventional DEA model.     

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.     

The interval efficiency based on the optimistic and pessimistic points of view

Data envelopment analysis (DEA) is a data-oriented approach for evaluating the performances of a set of peer entities called decision-making units (DMUs), whose performance is determined based on multiple measures. The traditional DEA, which is based on the concept of efficiency frontier (output frontier), determines the best efficiency score that can be assigned to each DMU. Based on these scores, DMUs are classified into DEA-efficient (optimistic efficient) or DEA-non-efficient (optimistic non-efficient) units, and the DEA-efficient DMUs determine the efficiency frontier. There is a comparable approach which uses the concept of inefficiency frontier (input frontier) for determining the worst relative efficiency score that can be assigned to each DMU. DMUs on the inefficiency frontier are specified as DEA-inefficient or pessimistic inefficient, and those that do not lie on the inefficient frontier, are declared to be DEA-non-inefficient or pessimistic non-inefficient. In this paper, we argue that both relative efficiencies should be considered simultaneously, and any approach that considers only one of them will be biased. For measuring the overall performance of the DMUs, we propose to integrate both efficiencies in the form of an interval, and we call the proposed DEA models for efficiency measurement the bounded DEA models. In this way, the efficiency interval provides the decision maker with all the possible values of efficiency, which reflect various perspectives. A numerical example is presented to illustrate the application of the proposed DEA models.     

A dynamic transfer scheme deriving from the dual similar associated game

Dynamic process is an approach to cooperative games, and it can be defined as that which leads the players to a solution for cooperative games. Hwang et al. (2005) adopted Hamiache’s associated game (2001) to provide a dynamic process leading to the Shapley value. In this paper, we propose a dynamic transfer scheme on the basis of the dual similar associated game, to lead to any solution satisfying both the inessential game property and continuity, starting from an arbitrary efficient payoff vector.     

Network DEA: A slacks-based measure approach

Traditional DEA models deal with measurements of relative efficiency of DMUs regarding multiple-inputs vs. multiple-outputs. One of the drawbacks of these models is the neglect of intermediate products or linking activities. After pointing out needs for inclusion of them to DEA models, we propose a slacks-based network DEA model, called Network SBM, that can deal with intermediate products formally. Using this model we can evaluate divisional efficiencies along with the overall efficiency of decision making units (DMUs).     

基于效用比的多工序并联制造过程能力分析

多工序制造过程通常包含串联和并联两种结构,具有串并联混合结构的多工序制造过程是实践中最为常见的形式,而不同模式的并联结构其上游工序质量波动对下游工序及总过程能力的影响不同。针对多工序制造过程并联结构特点,本文从波动减少的角度重点对多工序并联制造过程中并行、分散和收敛三种基本模式进行过程能力分析,研究多工序制造过程中各子过程波动对整体过程能力的影响,并根据各子过程质量波动减少的“困难度”和“效用比”评价质量改进的效果,给出多工序并联过程能力改进策略选择依据。通过实例表明,本方法能较好地识别各工序质量波动减少对本工序过程能力和总过程能力的不同影响,确定质量改进的优先顺序,实现多工序制造过程的经济性质量改进和优化。     

Measuring the performances of decision-making units using interval efficiencies

Efficiency is a relative measure because it can be measured within different ranges. The traditional data envelopment analysis (DEA) measures the efficiencies of decision-making units (DMUs) within the range of less than or equal to one. The corresponding efficiencies are referred to as the best relative efficiencies, which measure the best performances of DMUs and determine an efficiency frontier. If the efficiencies are measured within the range of greater than or equal to one, then the worst relative efficiencies can be used to measure the worst performances of DMUs and determine an inefficiency frontier. In this paper, the efficiencies of DMUs are measured within the range of an interval, whose upper bound is set to one and the lower bound is determined through introducing a virtual anti-ideal DMU, whose performance is definitely inferior to any DMUs. The efficiencies turn out to be all intervals and are thus referred to as interval efficiencies, which combine the best and the worst relative efficiencies in a reasonable manner to give an overall measurement and assessment of the performances of DMUs. The new DEA model with the upper and lower bounds on efficiencies is referred to as bounded DEA model, which can incorporate decision maker (DM) or assessor's preference information on input and output weights. A Hurwicz criterion approach is introduced and utilized to compare and rank the interval efficiencies of DMUs and a numerical example is examined using the proposed bounded DEA model to show its potential application and validity.     

Parameters identification of nonlinear system with impulse effect and its application on conversion from glycerol to 1,3-propanediol

Based on the continuous culture model we begin with introducing the nonlinear impulsive system to describe the fed-batch culture and some properties of its solution. And then the whole fermentation process is divided into two sub-processes, growth period and steady period according to the rule of the microorganisms fermentation. Instead of identifying all the uncertain parameters we only select several sensitive ones by analyzing their sensitivity to determine two pairs of parameters that are suitable for the two periods and to construct the control variable. In the end apart from establishing the identification system(PDP) of the sensitive parameters and discussing the existence of its solution we construct a algorithm and use it to obtain the optimal solution of PDP.