Inefficiency Identification in IDEA (Imprecise Data Envelopment Analysis) via Additive Model

Abstract

DEA(Data Envelopment Analysis)는 복수의 투입-산출요소들을 갖는 DMU(Decision Making Unit)들의 효율성을 분석하기 위한 수리계획 접근법이다. 기존 DEA에서는 투입-산출요소들에 관한 정확한 데이터가 주어 진다고 가정하고 있는 반면, IDEA (Imprecise DEA)는 불완전 데이터를 DEA에서 취급하기 위하여 개발되었다. 그러나, IDEA는 각각의 DMU에 대한 종합적 비효율성 만을 측정/제공한다. 따라서, 기존의 DEA 평가에서 이루어졌던 것처럼, DMU들에 관한 세부적인 비효율성(예를 들어 투입-잔여분, 산출-부족 등)을 측정할 수 있는 방법론 개발이 필요하다. 이러한 세부적인 (또는 구체적인) 비효율성을 밝혀내는 것이 본 논문의 목적이다. 한가지 중요한 점은, 세부적인 비효율성을 결정하기 위한 수리계획 모형이 비선형 문제가 된다는 것이다. 따라서 본 논문에서는 비선형 문제를 동일한 해를 갖는 선형문제로 전환한 후, 세부적인 비효율성을 결정하기 위한 two-stage 방법을 개발한다. 첫번째 단계에서는 전환된 선형문제로부터 종합적 비효율성을 먼저 얻는다. 이때 선형문제에 관한 변수들의 최적해를 도출한다. 도출된 최적해는 정확한 데이터를 포함하며, 이 정확한 데이터는 두번째 단계에서 이용되어 기존의 DEA 모델을 형성하게 된다. 따라서 본 논문의 목적인 세부적인 비효율성을 결정할 수 있다.

Data Envelopment Analysis (DEA) is a mathematical programming approach to evaluating the relative efficiency of Decision Making Units (DMUs) that use multiple inputs to produce multiple outputs. While assuming exact data in ordinary DEA, development of Imprecise Data Envelopment Analysis (IDEA) makes possible to deal with imprecise data in DEA. However, IDEA only provides an aggregated measure of inefficiency. for each DMU. It is thus needed to develop methods from which we can obtain specific inefficiencies such as slacks, as well as peer groups and scale sizes, as have been done in ordinary DEA evaluations. The purpose of this paper is hence on the identification of specific inefficiencies in IDEA. This is done via employing an additive model which we refer to as additive IDEA model. A point to be noted is that the original formulation becomes a nonlinear programming problem. We thus transform it into a linear programming equivalent and then present a two-stage method to identify specific inefficiencies. In the first stage, we obtain an aggregated measure of inefficiency from solving the linear version of additive IDEA model. We then retrieve exact data based upon the optimal solutions obtained in the first stage. These exact data retrieved are used in the next stage which implies that an ordinary additive DEA model is constructed. We can thus obtain the specific inefficiencies in terms of slacks as well as peer groups and scale sizes for each DMU to be considered in IDEA problem.

Data Envelopment Analysis (DEA) is a mathematical programming approach to evaluating the relative efficiency of Decision Making Units (DMUs) that use multiple inputs to produce multiple outputs. While assuming exact data in ordinary DEA, development of Imprecise Data Envelopment Analysis (IDEA) makes possible to deal with imprecise data in DEA. However, IDEA only provides an aggregated measure of inefficiency. for each DMU. It is thus needed to develop methods from which we can obtain specific inefficiencies such as slacks, as well as peer groups and scale sizes, as have been done in ordinary DEA evaluations. The purpose of this paper is hence on the identification of specific inefficiencies in IDEA. This is done via employing an additive model which we refer to as additive IDEA model. A point to be noted is that the original formulation becomes a nonlinear programming problem. We thus transform it into a linear programming equivalent and then present a two-stage method to identify specific inefficiencies. In the first stage, we obtain an aggregated measure of inefficiency from solving the linear version of additive IDEA model. We then retrieve exact data based upon the optimal solutions obtained in the first stage. These exact data retrieved are used in the next stage which implies that an ordinary additive DEA model is constructed. We can thus obtain the specific inefficiencies in terms of slacks as well as peer groups and scale sizes for each DMU to be considered in IDEA problem.