Efficiency measurement and decomposition in hybrid two-stage DEA with additional inputs

Highlights

• Parallel and series internal structure of system are modeled as hybrid network DEA.

• Additive and multiplicative efficiency evaluation methods are jointly used.

• A heuristic search procedure is introduced to achieve a global optimal solution.

• The efficiency of the hybrid system can be evaluated and decomposed simultaneously.

• The model provides more detailed inefficiency sources hidden in system “black-box”.

Abstract

The simulation and analysis of structure within decision making unit (DMU) is the basis on which network data envelopment analysis (DEA) opens the “black box” and evaluates the efficiency of system with complex internal structure. The efficiency measurements of system with sub-DMU in series or with sub-DMUs in parallel are two common cases in the theory development and applications of two-stage DEA. However, the research on parallel-series hybrid system is not rich enough. The paper develops a set of DEA models to treat a two-stage system comprised of three sub-DMUs in hybrid form with additional inputs to the second stage. The proposed models simulate precisely the system's parallel-series internal structure, employ synthetically additive and multiplicative DEA approaches to estimate and decompose the efficiencies of system, and adopt a heuristic method to convert non linear program due to the additional inputs into a linear program. This approach gives more information about the sources of inefficiency by penetrating into the depth of system and modeling the efficiency formation mechanism. A model application is provided.

Introduction

Data envelopment analysis (DEA) is a non-parametric technique for assessing the relative efficiency of a set of decision making units (DMUs), which are homogenous and responsible for converting multiple inputs into multiple outputs. Since the initiation of DEA by Charnes, Cooper, and Rhodes in 1978, this approach has been developed by a large number of researchers and extensively used for identifying sources of inefficiency, ranking the DMUs, evaluating management, evaluating the effectiveness of program or policies, create a quantitative basis for reallocating resources, etc (Golany & Roll, 1989). Over the last decade, DEA has gained considerable development both in the methodology and application, and become an important managerial tool for evaluating the performance of system.

The conventional DEA models neglect the internal operations or structure of DMUs and treat each DMU as a “black box” with single-process transforming the initial inputs to the final outputs. These approaches tend to over-evaluate the efficiency of system with complex internal structure (Kao, 2009a), so that sometimes there will be many efficient DMUs with the same ranking but with different performances in actual practice. In other words, the conventional DEA models lack discrimination power to distinguish between DEA efficient units (Kritikos, 2017). On the other hand, the negligence of internal structure of DMUs leads the conventional DEA models unable to provide the specific information regarding the sources of inefficiency within DMUs (Lewis & Sexton, 2004). In fact, the shortcoming of the conventional DEA models, especially the insufficient information about inefficiency, incites many earlier scholars to decompose the efficiency of DMUs into different components. For example, Banker, Charnes, and Cooper (1984) break down the efficiency of a DMU into scale efficiency and technical efficiency; Byren, Färe, and Grosskopf (1984) then identify the congestion effect from the technical efficiency; and Kao (1995) decomposes the overall efficiency of a DMU into a weighted arithmetic mean of the efficiencies of individual outputs. Even if these previous works get more details about the efficiency of DMUs, they still focus on the structure of DEA models without giving an explicit representation of operation processes inside a DMU to open the “black box” in a real sense.

Inspired by the research about multi-stage production process (Färe & Whittaker, 1995), Färe and Grosskopf, 1996, Färe and Grosskopf, 2000) develop a general multi-stage model with intermediate inputs-outputs, commonly called network DEA. This approach applies to the DMUs that consist of a network of sub-DMUs, some of which consume resources produced by others and some of which produce resources consumed by others (Lewis & Sexton, 2004). The advantages of network-DEA in the efficiency evaluation of the DMUs with complex internal structure are mainly threefold: (1) the models are able to simulate the internal structure of system, and the efficiencies of sub-DMUs and the overall efficiency of system can be evaluated simultaneously; (2) the network-DEA proposes and proves the divisibility of the overall efficiency of system with complex internal structure, and the calculation can reduce the number of efficiency DMU and improve the accuracy of results; (3) the method provides the possibility to study the resources allocation within the system and to explore the relationship among the sub-DMUs. The simulation and analysis of DMUs internal structure is therefore one of the key contents of network-DEA, and it makes this approach a real evaluation method which is able to open the “black-box” of DMUs indeed, even from a dynamic perspective (Kou et al., 2016, Lee et al., 2016).

The sub-DMUs in series and sub-DMUs in parallel are two very common cases in the description of DMU's internal structure by network-DEA approach. Taking two-stage system as an example, the serial structure assumes that each system is comprised of two sub-stages connected in series, wherein the first stage consumes the external inputs to produce intermediate measures which are used by the second stage to produce the final outputs. Kao and Hwang (2008) define the overall efficiency of such a system as the product of the efficiency ratios of stages 1 and 2. This multiplicative method is used by many researchers (see example, Chen, Liang, & Zhu, 2009; Wanke & Barros, 2014) to evaluate and decompose the efficiency of the system within which there are two sub-DMUs connected in series. Nevertheless, the parallel structure considers that the internal structure of a two-stage system is comprised of two independent sub-stages in parallel, and the sum of the inputs or outputs of each stage is the inputs or outputs of the whole system. Chen, Cook, Li, and Zhu (2009) model the overall efficiency of two-stage process as a weighted sum of the efficiencies of the two individual stages. Kao, 2009a, Kao, 2009b) also adopt the thought of additive method to carry out an efficiency measurement for parallel production systems, and the author recently apply additive approach to evaluate and decompose the Malmquist productivity index for parallel production systems (Kao, 2017).

Although not all EDA efficiency evaluations of the system with serial/parallel structure rely respectively on the multiplicative/additive decomposition approach, it can be thought that the multiplicative efficiency decomposition method underlines the ‘link’ role of the intermediate measures between two stages, while the additive method emphasizes the relative contribution of the efficiency of each stage to the performance of whole system. In fact, the overall efficiency definition of the multiplicative approach permits for the linear formulation for overall efficiency evaluation model since the intermediate measures exist in the denominator of the efficiency ratio of the first stage and in the numerator of the efficiency ratio of the second stage (Kao & Hwang, 2008); Nevertheless, the weight associated with the efficiency of each stage defined by the additive method is the portion of total resources devoted to concerned stage, which can reasonably represent its relative ‘size’ in the whole system and determine the importance of individual stage efficiency in the overall efficiency of system (Chen, Cook et al., 2009). In this perspective, the additive approach seems more appropriate when each DMU is comprised of two independent sub-DMUs in parallel, because the intermediate measures do not exist and the relative importance of individual stage is more meaningful.

This thought is embodied, to a certain extent, in the research of Kao, 2009a, Kao, 2009b on the efficiency evaluation of an integrated system within which each stage in the series is of a parallel structure composed of a set of processes. That is to say, the efficiency measurement of parallel system and of serial system follows the additive approach and the multiplicative approach logic, respectively. However, the serial system proposed by Kao (2009b) is a “closed system”, that is where the outputs from one stage become the inputs to the next stage, and where no other inputs enter the sub-DMUs at any intermediate stage (Cook, Zhu, Bi, & Yang, 2010). Based on these studies, Kao (2014) proposed a DEA approach to evaluate and decompose the efficiency of general multi-stage systems with consideration of exogenous inputs and intermediate products for each stage. The general multi-stage structure DEA transforms the system into a series of parallel structures and decomposes the system efficiency into the product of a modification of the stage efficiencies. This transformation is the key to find solution to the efficiency decomposition, but the complicated notation used with the general multi-stage system make this transformation more complex. Without system's internal structure transformation assumption, we attempt to propose an intuitive method by exploiting a combination of multiplicative and additive efficiency decomposition methods in DEA with a heuristic process.

Based on the above problems and consideration, we put emphasis on a two-stage system comprised of three sub-DMUs where the two first sub-DMUs form a parallel structure in the first stage, and the latter is connected to the second stage in series. The objective of this paper is to investigate efficiency measurement and decomposition of such a system while considering the additional inputs to the second stage. In doing so, we will develop a hybrid two-stage DEA model to illustrate the features of efficiency formation in a system with complex internal structure by adopting the multiplicative and additive methods. A heuristic technique will be put into use to cope with the linear program transformation in presence of the additional inputs. The proposed model is able to simulate the internal structure of system and the structure of exogenous inputs in detail, and apply different methods to describe the efficiency formation mechanism by adopting a heuristic method to deal with the linear program transformation problems.

The remainder of this paper is organized as follows. Section 1 introduces the mathematical details of the proposed models. Section 2 presents the solution of the models and the method of efficiency decomposition. Section 3 is devoted to a case study to exhibit the efficacy of the procedures and to demonstrate the applicability of the proposed method. Section 4 outlines conclusions and future research directions.