基于DEA理论的固定产出DMU评价与满意度分摊方法研究
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摘要
在一些组织中,为了节约成本,往往会搭建一个公共平台以实现资源共享,此时,各决策单元(Decision Making Unit,简称DMU)在从公共平台获取利益的同时,也要承担相应的公共费用支出,如银行总行为各分行建立统一的交易系统等。因此,设计一个公平合理的成本分摊机制就显得尤为重要。数据包络分析(Data Envelopment Analysis,简称DEA)已逐渐成为解决这一分摊问题的一个重要工具。然而,传统基于DEA方法的固定成本分摊机制仅从决策单元效率的角度出发,没有考虑被分摊决策单元对分摊结果是否满意,为了使得分摊结果更容易被DMU接受,本文首次从决策单元对分摊结果满意程度的角度提出了一种基于满意度的DEA固定成本分摊方法。
在现实生活中,我们也经常遇到一些决策单元,它们的某个或某些产出的和是固定的情况,举例来说,一个行业的市场份额的总和是一个固定值(为1)。然而在产出和固定的情况下,某个决策单元产出的增加就会需要其它决策单元通过减少自身的产出来弥补这个增量。而传统DEA在评价决策单元的效率评价时却很少考虑到这一点,目前关于这方面的研究也相对较少,所以本文还将致力于固定产出决策单元的DEA评价方法研究。
全文共有六章,主要内容介绍如下：
在第一章绪论中,我们首先介绍了数据包络分析(DEA)的基本理论知识,主要包括DEA理论的一些基本概念、基本模型、DEA效率评价的基本原理以及DEA方法的一些相关应用领域。然后,我们将上述DEA基本理论作为本文研究的背景知识,并由此引入到本文将要研究的两个要点,它们分别是固定成本分摊和固定产出DMU效率评价。最后简要介绍了这两个研究要点的研究现状和研究意义。
第二章提出了一种基于DEA理论和满意度的固定成本分摊方法,首次将决策单元对分摊结果的满意程度考虑其中。本章首先根据固定成本分摊方案集求出分摊成本的最大值和最小值,并由此构成成本分摊区间,再由这个分摊区间定义满意度的概念,该满意度的定义满足了以下两个条件,即当被分摊的成本为区间最小值时该DMU最满意,此时满意度达到最大值为1,反之,当被分摊的成本为区间最大值时该DMU最不满意,此时满意度达到最小值为O。接下来我们根据满意度的定义构建了一个Maxmin满意度模型,根据这个模型我们可以得出一组唯一的分摊方案,并且能够使得所有DMUs都能够接受。
第三章提出一种叫做均衡有效前沿面DEA(简称EEFDEA)的方法来评价含有固定产出的DMUs.本章首先回顾了以往关于DEA方法评价固定产出DMUs的相关文献,指出了之前文献在处理该问题上的不足,如评价时所基于的有效前沿面不一致等,为了弥补这一不足,本章首先基于最小调整策略提出一种均衡有效前沿面构建模型,运用这个模型,可以使得所有DMUs能够按照一定顺序依次达到有效前沿面上,最终形成一个均衡有效前沿面。最后所有DMUs都是基于这样一个公共的均衡前沿面上来进行评价。
考虑到前一章提出的EEFDEA方法在构建前沿面上需要给定一个顺序通过多步才能实现,为了简化构建前沿面步骤,第四章提出了一种更为一般的均衡有效前沿面DEA(简称GEEFDEA)方法用以评价具有固定产出的DMUs.运用GEEFDEA方法,均衡有效前沿面只要一步就可实现,而且不需要事先给定调整顺序,这样大大减轻了运算负担。最后,我们将运用GEEFDEA方法来评价2012伦敦奥运会,考虑到奥运会的三个固定产出(金、银、铜牌)重要性是不同的,我们又在模型中加入了保证域的约束,以确保评价结果更符合实际情况。
第五章是将第四章的固定产出拓展到非期望产出的范畴,本章首先回顾了DEA领域中关于非期望产出研究的相关文献,从文献中我们发现尽管关于DEA非期望产出的研究已经有很多,但是却很少有考虑非期望产出的和是固定的情况,为此本章提出了考虑非期望固定产出的GEEFDEA模型,该模型不仅保持了第四章GEEFDEA模型的所有优势,如在公共的均衡有效前沿面上评价等,而且还弥补了之前非期望固定产出DMU评价方法可能会出现无可行解的缺陷。最后运用本章提出的模型来评价我国31个省市自治区的工业环境效率。
第六章是总结和展望。本章首先对前面几章的主要内容分别做个总结,同时指出了本文研究的不足之处。并针对这些不足,提出了几个要点作为未来的研究展望。
本文的最主要创新之处体现在如下几个方面：1)本文在运用DEA方法进行固定成本分摊时,首次将决策单元对分摊成本的满意程度考虑其中,并最终得出一组唯一的可接受的分摊方案。2)本文提出的均衡有效前沿面DEA方法在评价固定产出决策单元时,能同时满足以下四个条件：a)基于一个公共的有效前沿面进行评价；b)它考虑了固定产出的情况；c)它能将固定产出从一维拓展到多维；d)它能够对DMUs进行完全效率排序。3)本文首次提出的一般形式的均衡有效前沿面DEA方法,除了保持了EEFDEA方法的所有优势外,还将均衡有效前沿面的构建过程由多步简化至一步,大大降低了计算的复杂度。4)本文首次提出了非期望固定产出的GEEFDEA模型,它避免了以往非期望固定产出DEA模型可能出现无可行解的缺陷。
In some organizations, in order to save costs, they tend to build a common platform for resources sharing. At this time, each decision-making unit (DMU for short) must bear the corresponding cost of common expenses when obtaining benefits from the common platform. For example, bank headquarter will build a united trading systems for its branches. Therefore, to design a fair and reasonable cost-allocation mechanism is particularly necessary. Data envelopment analysis (DEA for short) has been proved to be a more and more important tool to solve this allocation problem. However, traditional DEA-based approaches for fixed cost allocation are only from the perspective of efficiency, which do not consider whether the DMU satisfies with the results of allocation. In order to make the results to be more reasonable, this paper firstly proposes a fixed cost allocation method based on DEA and satisfaction degree from the perspective of DMUs'satisfaction relative to the results of allocation.
In real life, we often encounter with the situation that some DMUs whose outputs' sum are fixed. For example, the sum of an industry's market share is a constant (1). In the fixed outputs environment, one DMU expanding its outputs may reduce others' outputs to make up for the increase. However, traditional DEA models do not take this constraint into account during the evaluation, and the existing researches relative to this field are rare, therefore, this paper also will research on evaluating DMUs with fixed outputs.
This paper contains six chapters, and the main contents of each chapter are displayed as follows.
In the first chapter, we introduce the basic knowledge about data envelopment analysis (DEA), including some basic concepts, some basic models, the basic principle of evaluation as well as some relative applications of DEA. Then, we view the above knowledge as the background of our research and clarify two key points of this study:fixed cost allocation and DMUs with fixed outputs evaluation. Lastly, we respectively introduce the meaning and the status of these two researches briefly.
Chapter2proposes a fixed cost allocation approach based on DEA and satisfaction degree, which firstly consider the DMUs'satisfaction degree to the results of allocation. In the beginning of this chapter, we calculate the maximum and minimum fixed cost of each DMU according to allocation scheme equations, which are used to constitute the cost interval. Then we define the concept of satisfaction degree by the cost interval. This concept meets the following two conditions. One is that the DMU most satisfies with its cost allocation when its satisfaction degree reaches its maximum value1. The other is that, on the contrary, the DMU most dissatisfies with its cost allocation when its satisfaction degree reaches its minimum value0. Then we construct a Maxmin model based on this satisfaction degree. In light of this model, we get a unique allocation, which can be accepted by all DMUs。
Chapter3proposes a method named equilibrium efficiency frontier DEA (EEFDEA) to evaluate DMUs with fixed outputs. First of all, we review some of prior DEA literatures relative to evaluation on DMUs with fixed outputs and point out the shortages of them such as they evaluated DMUs based on different efficient frontiers and so on. To fill those gapes, this chapter first proposes an equilibrium efficiency frontiers constructed model based on minimum adjustment strategy. By this model, all DMUs can achieve efficient frontier one by one with a given order and, finally, an equilibrium efficient frontier is formed. Then all DMUs are evaluated based on such a common equilibrium efficient frontier.
Consider the proposed EEFDEA method in Chapter3needs multiple steps and a given order for constructing a common equilibrium frontier, in order to simplify steps of equilibrium frontier achievement, Chapter4proposes a general equilibrium efficient frontier DEA (GEEFDEA for short) approach to evaluate DMUs with fixed outputs. According to GEEFDEA method, the equilibrium efficient frontier can be achieved by only one step and without any given adjusted order, which reduce the computational complexity greatly. Finally, we applied GEEFDEA method to2012London Olympic Games. Consider the importance of three fixed outputs of Olympics (gold, silver and bronze medal) is different, we add the constraint of assurance region in the model to ensure the results of evaluation more reasonable.
Chapter5extends the fixed outputs in Chapter4to undesirable outputs. In the beginning of this chapter, we review some DEA literatures relative to undesirable outputs. From literatures we find that although there have been so many the researches about undesirable outputs, few researches consider the situation regards to undesirable fixed outputs. Therefore, this chapter proposes a GEEFDEA model which takes undesirable fixed outputs into account. The proposed model not only maintains all advantages of GEEFDEA model, but also overcomes the shortage in prior models that those models may appear infeasibility solution when evaluating DMUs with undesirable fixed outputs. In the end, we utilize the proposed model to evaluate the environmental efficiency of industry of31provinces and autonomous regions.
Chapter6is conclusion and future research. We first conclude the main contents in all previous chapters, respectively. Meanwhile, we point out the shortcomings of this study. In order to overcome these shortcomings, we put forward some key points for future research.
The main innovations of this paper are listed as follows:1) this paper firstly takes DMUs'satisfaction degree to the results of allocation into account when allocating fixed cost via DEA method. We get a unique and acceptable allocation.2) The proposed equilibrium efficiency frontier DEA approach is unique one which can satisfy the following four conditions simultaneously when evaluating competitive DMUs with fixed outputs:(i) evaluating based on a common efficient frontier,(ii) taking account of fixed outputs,(iii) extending fixed outputs from one dimension to multi-dimensions, and (iv) providing a full rank order of DMUs.3) This paper firstly proposes a general equilibrium efficiency frontier DEA approach. It not only maintains all advantages of EEFDEA model, but also reduces the computational complexity greatly since the equilibrium efficient frontier can be achieved by only one step and without any given adjusted order.4) This paper firstly proposes undesirable fixed outputs GEEFDEA model which overcomes the shortage in prior models that those models may appear infeasibility solution when evaluating DMUs with undesirable fixed outputs.
在现实生活中,我们也经常遇到一些决策单元,它们的某个或某些产出的和是固定的情况,举例来说,一个行业的市场份额的总和是一个固定值(为1)。然而在产出和固定的情况下,某个决策单元产出的增加就会需要其它决策单元通过减少自身的产出来弥补这个增量。而传统DEA在评价决策单元的效率评价时却很少考虑到这一点,目前关于这方面的研究也相对较少,所以本文还将致力于固定产出决策单元的DEA评价方法研究。
全文共有六章,主要内容介绍如下：
在第一章绪论中,我们首先介绍了数据包络分析(DEA)的基本理论知识,主要包括DEA理论的一些基本概念、基本模型、DEA效率评价的基本原理以及DEA方法的一些相关应用领域。然后,我们将上述DEA基本理论作为本文研究的背景知识,并由此引入到本文将要研究的两个要点,它们分别是固定成本分摊和固定产出DMU效率评价。最后简要介绍了这两个研究要点的研究现状和研究意义。
第二章提出了一种基于DEA理论和满意度的固定成本分摊方法,首次将决策单元对分摊结果的满意程度考虑其中。本章首先根据固定成本分摊方案集求出分摊成本的最大值和最小值,并由此构成成本分摊区间,再由这个分摊区间定义满意度的概念,该满意度的定义满足了以下两个条件,即当被分摊的成本为区间最小值时该DMU最满意,此时满意度达到最大值为1,反之,当被分摊的成本为区间最大值时该DMU最不满意,此时满意度达到最小值为O。接下来我们根据满意度的定义构建了一个Maxmin满意度模型,根据这个模型我们可以得出一组唯一的分摊方案,并且能够使得所有DMUs都能够接受。
第三章提出一种叫做均衡有效前沿面DEA(简称EEFDEA)的方法来评价含有固定产出的DMUs.本章首先回顾了以往关于DEA方法评价固定产出DMUs的相关文献,指出了之前文献在处理该问题上的不足,如评价时所基于的有效前沿面不一致等,为了弥补这一不足,本章首先基于最小调整策略提出一种均衡有效前沿面构建模型,运用这个模型,可以使得所有DMUs能够按照一定顺序依次达到有效前沿面上,最终形成一个均衡有效前沿面。最后所有DMUs都是基于这样一个公共的均衡前沿面上来进行评价。
考虑到前一章提出的EEFDEA方法在构建前沿面上需要给定一个顺序通过多步才能实现,为了简化构建前沿面步骤,第四章提出了一种更为一般的均衡有效前沿面DEA(简称GEEFDEA)方法用以评价具有固定产出的DMUs.运用GEEFDEA方法,均衡有效前沿面只要一步就可实现,而且不需要事先给定调整顺序,这样大大减轻了运算负担。最后,我们将运用GEEFDEA方法来评价2012伦敦奥运会,考虑到奥运会的三个固定产出(金、银、铜牌)重要性是不同的,我们又在模型中加入了保证域的约束,以确保评价结果更符合实际情况。
第五章是将第四章的固定产出拓展到非期望产出的范畴,本章首先回顾了DEA领域中关于非期望产出研究的相关文献,从文献中我们发现尽管关于DEA非期望产出的研究已经有很多,但是却很少有考虑非期望产出的和是固定的情况,为此本章提出了考虑非期望固定产出的GEEFDEA模型,该模型不仅保持了第四章GEEFDEA模型的所有优势,如在公共的均衡有效前沿面上评价等,而且还弥补了之前非期望固定产出DMU评价方法可能会出现无可行解的缺陷。最后运用本章提出的模型来评价我国31个省市自治区的工业环境效率。
第六章是总结和展望。本章首先对前面几章的主要内容分别做个总结,同时指出了本文研究的不足之处。并针对这些不足,提出了几个要点作为未来的研究展望。
本文的最主要创新之处体现在如下几个方面：1)本文在运用DEA方法进行固定成本分摊时,首次将决策单元对分摊成本的满意程度考虑其中,并最终得出一组唯一的可接受的分摊方案。2)本文提出的均衡有效前沿面DEA方法在评价固定产出决策单元时,能同时满足以下四个条件：a)基于一个公共的有效前沿面进行评价；b)它考虑了固定产出的情况；c)它能将固定产出从一维拓展到多维；d)它能够对DMUs进行完全效率排序。3)本文首次提出的一般形式的均衡有效前沿面DEA方法,除了保持了EEFDEA方法的所有优势外,还将均衡有效前沿面的构建过程由多步简化至一步,大大降低了计算的复杂度。4)本文首次提出了非期望固定产出的GEEFDEA模型,它避免了以往非期望固定产出DEA模型可能出现无可行解的缺陷。
In some organizations, in order to save costs, they tend to build a common platform for resources sharing. At this time, each decision-making unit (DMU for short) must bear the corresponding cost of common expenses when obtaining benefits from the common platform. For example, bank headquarter will build a united trading systems for its branches. Therefore, to design a fair and reasonable cost-allocation mechanism is particularly necessary. Data envelopment analysis (DEA for short) has been proved to be a more and more important tool to solve this allocation problem. However, traditional DEA-based approaches for fixed cost allocation are only from the perspective of efficiency, which do not consider whether the DMU satisfies with the results of allocation. In order to make the results to be more reasonable, this paper firstly proposes a fixed cost allocation method based on DEA and satisfaction degree from the perspective of DMUs'satisfaction relative to the results of allocation.
In real life, we often encounter with the situation that some DMUs whose outputs' sum are fixed. For example, the sum of an industry's market share is a constant (1). In the fixed outputs environment, one DMU expanding its outputs may reduce others' outputs to make up for the increase. However, traditional DEA models do not take this constraint into account during the evaluation, and the existing researches relative to this field are rare, therefore, this paper also will research on evaluating DMUs with fixed outputs.
This paper contains six chapters, and the main contents of each chapter are displayed as follows.
In the first chapter, we introduce the basic knowledge about data envelopment analysis (DEA), including some basic concepts, some basic models, the basic principle of evaluation as well as some relative applications of DEA. Then, we view the above knowledge as the background of our research and clarify two key points of this study:fixed cost allocation and DMUs with fixed outputs evaluation. Lastly, we respectively introduce the meaning and the status of these two researches briefly.
Chapter2proposes a fixed cost allocation approach based on DEA and satisfaction degree, which firstly consider the DMUs'satisfaction degree to the results of allocation. In the beginning of this chapter, we calculate the maximum and minimum fixed cost of each DMU according to allocation scheme equations, which are used to constitute the cost interval. Then we define the concept of satisfaction degree by the cost interval. This concept meets the following two conditions. One is that the DMU most satisfies with its cost allocation when its satisfaction degree reaches its maximum value1. The other is that, on the contrary, the DMU most dissatisfies with its cost allocation when its satisfaction degree reaches its minimum value0. Then we construct a Maxmin model based on this satisfaction degree. In light of this model, we get a unique allocation, which can be accepted by all DMUs。
Chapter3proposes a method named equilibrium efficiency frontier DEA (EEFDEA) to evaluate DMUs with fixed outputs. First of all, we review some of prior DEA literatures relative to evaluation on DMUs with fixed outputs and point out the shortages of them such as they evaluated DMUs based on different efficient frontiers and so on. To fill those gapes, this chapter first proposes an equilibrium efficiency frontiers constructed model based on minimum adjustment strategy. By this model, all DMUs can achieve efficient frontier one by one with a given order and, finally, an equilibrium efficient frontier is formed. Then all DMUs are evaluated based on such a common equilibrium efficient frontier.
Consider the proposed EEFDEA method in Chapter3needs multiple steps and a given order for constructing a common equilibrium frontier, in order to simplify steps of equilibrium frontier achievement, Chapter4proposes a general equilibrium efficient frontier DEA (GEEFDEA for short) approach to evaluate DMUs with fixed outputs. According to GEEFDEA method, the equilibrium efficient frontier can be achieved by only one step and without any given adjusted order, which reduce the computational complexity greatly. Finally, we applied GEEFDEA method to2012London Olympic Games. Consider the importance of three fixed outputs of Olympics (gold, silver and bronze medal) is different, we add the constraint of assurance region in the model to ensure the results of evaluation more reasonable.
Chapter5extends the fixed outputs in Chapter4to undesirable outputs. In the beginning of this chapter, we review some DEA literatures relative to undesirable outputs. From literatures we find that although there have been so many the researches about undesirable outputs, few researches consider the situation regards to undesirable fixed outputs. Therefore, this chapter proposes a GEEFDEA model which takes undesirable fixed outputs into account. The proposed model not only maintains all advantages of GEEFDEA model, but also overcomes the shortage in prior models that those models may appear infeasibility solution when evaluating DMUs with undesirable fixed outputs. In the end, we utilize the proposed model to evaluate the environmental efficiency of industry of31provinces and autonomous regions.
Chapter6is conclusion and future research. We first conclude the main contents in all previous chapters, respectively. Meanwhile, we point out the shortcomings of this study. In order to overcome these shortcomings, we put forward some key points for future research.
The main innovations of this paper are listed as follows:1) this paper firstly takes DMUs'satisfaction degree to the results of allocation into account when allocating fixed cost via DEA method. We get a unique and acceptable allocation.2) The proposed equilibrium efficiency frontier DEA approach is unique one which can satisfy the following four conditions simultaneously when evaluating competitive DMUs with fixed outputs:(i) evaluating based on a common efficient frontier,(ii) taking account of fixed outputs,(iii) extending fixed outputs from one dimension to multi-dimensions, and (iv) providing a full rank order of DMUs.3) This paper firstly proposes a general equilibrium efficiency frontier DEA approach. It not only maintains all advantages of EEFDEA model, but also reduces the computational complexity greatly since the equilibrium efficient frontier can be achieved by only one step and without any given adjusted order.4) This paper firstly proposes undesirable fixed outputs GEEFDEA model which overcomes the shortage in prior models that those models may appear infeasibility solution when evaluating DMUs with undesirable fixed outputs.
引文
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