Efficiency evaluation of an interactive system by data envelopment analysis approach
Introduction
With fierce global competition, production/service system structures have become more complex to reduce costs and improve competitiveness. Typically, the structure of a system is not linear, but rather a network wherein the relationship between two members is bidirectional. The education and research sectors in a university are two such examples. Better education quality can lead to better research results, while better research capabilities can improve teaching quality. Thus, their interactive relationship cannot be overlooked. These systems must improve performance to adapt to competition; measuring the system efficiency is critical toward that goal.
Data envelopment analysis, originally proposed by Charnes, Cooper, and Rhodes (1978), is a non-parametric programming method for efficiency evaluation of a group of homogenous decision making units (DMUs) in which multiple inputs are used to produce multiple outputs. It has been widely used to benchmark and evaluate the efficiency of schools, bank branches, hospitals, firms, and so on (Charnes et al., 2013, Khalili-Damghani and Shahmir, 2015, Wu et al., 2016). In traditional DEA models (such as CCR model and BCC model), a DMU’s internal structure is regarded as a “black box” (An et al., 2016, Andersen and Petersen, 1993, Banker et al., 1984, Charnes et al., 1978, Tone, 2001). To further analyze the efficiencies of the DMUs considering their different internal structures (such as the serial structure, parallel structure, and mixed structure), many scholars propose using network DEA models. Among these network DEA models, two-stage structure networks are simple, popular, and well-studied. Cook, Liang, and Zhu (2010) reviewed the related DEA works for the two-stage system where the outputs from the first stage were intermediate measures and taken as the inputs for the second stage. Based on this review and recent developments on two-stage DEA models, Halkos, Tzeremes, and Kourtzidis (2014) reviewed the DEA works for extensive two-stage systems that allow “exogenous” inputs at the intermediate measure. Based on these two reviews, we distinguished the research of two-stage DEA into four categories. The first category is the independent two-stage DEA approach in which two separate DEA runs are applied to the two stages to calculate their respective efficiencies (Seiford and Zhu, 1999, Wang et al., 1997). The second category is the connected two-stage DEA approach. In this type, the interactions between the two stages are considered when evaluating the overall efficiency of the DMUs, and the DMUs are regarded as DEA efficient if and only if they are fully efficient in both of the two stages. This research can be seen in Chen and Zhu, 2004, Liu and Lu, 2012, and so on. The third category is the relational two-stage DEA approach. In this research, mathematical relationships are assumed between the overall efficiency and individual efficiencies of the two stages. The multiplicative model of Kao and Hwang (2008), and the additive models of Chen, Cook, Li, and Zhu (2009) and Wang and Chin (2010) are representative studies of this type of research. The last category is the Game-theoretic two-stage DEA approach in which the two DMU stages are regarded as players in a game under the assumption of cooperative or non-cooperative states. This approach was firstly proposed by Liang, Cook, and Zhu (2008).
Besides two-stage DEA models, there are more complicated network models. It should be noted that network models in the following text do not refer to the aforementioned network DEA approach. Here, we define all DEA models for addressing the internal structure of a system as network models; thus, it must contain all two-stage DEA models. These models are generalized network models. We can strictly classify these network structures into three categories: serial structure, parallel structure, and mixed structure. In serial structure models, two or more internal stages are linked by intermediate measures. The simplest is the two-stage structure mentioned in the last paragraph, in which all outputs from the first stage are taken as inputs for the second stage. A general structure in this category is given in Cook, Zhu, Bi, and Yang (2010), where each stage has its own inputs (and/or outputs) in addition to the intermediate measures. The evident differences between the simple and the general form are the number of internal procedures (in the general form there are more than two stages), exogenous inputs may enter in any stage, final outputs may be produced in any stage, and intermediate measures may not be consumed entirely by one stage. Tone and Tsutsui (2009) proposed a serial structure DEA model to evaluate the electric power company efficiency in the United States. Monfared and Safi (2013) applied a serial slacks-based measure DEA structure to assess the academic performance of universities in Alzahra. They measured sub-functional efficiencies, such as teaching quality, research productivity, and overall efficiency of universities. An, Chen, Wu, and Liang (2015) built some slacks-based measure DEA models to evaluate the overall and divisional efficiency of Chinese commercial banks. In the parallel structure models, the individual stages operated parallel to each other. An important work is Kao (2009), which evaluated the efficiency of parallel production systems composed of independent production units. An extension of this type of model is the shared flows system, where the inputs were shared among the individual stages (Kao and Hwang, 2010, Wu et al., 2015). Amirteimoori (2013) proposed a more general parallel structure that some inputs are shared among the individual stages and some inputs are individually used by one stage. In the mixed structure, both serial processes and parallel processes exist. A general mixed network structure can be seen in Cook, Zhu, et al. (2010). There are few real applications that only have a serial or parallel structure; most practical applications have mixed structures. Lewis and Sexton (2004) applied a mixed structure model to analyze major league baseball. Two parallel Sub-DMUs were in “Font Office” (the first stage). Two parallel Sub-DMUs and a serial Sub-DMU were in “On-Field” (the second stage). “Font Office” and “On-Field” were linked with a serial structure. Overall efficiency and each Sub-DMU’s efficiency were measured. Avkiran, 2009, Wang et al., 2014 used mixed network models to measure commercial banking system efficiency. The author measured the overall and divisional bank efficiency of three individual bank profit centers; they were linked with serial and parallel structures. An, Yan, Wu, and Liang (2016) built a two-stage DEA model for a mixed system to measure its efficiency and investigate the relationship between internal resource waste and centralization degree.
There are many works on the efficiency measurement of a network system, including serial system and parallel system, especially the two-stage system where the outputs from the first stage are used as inputs for the second stage to produce the final outputs. These works help us understand the internal structure of a system. However, in previous works, the relationship among divisions (Sub-DMUs) in a system is assumed to be unidirectional or independent, see Kao and Hwang, 2008, Chen et al., 2009, Huang et al., 2014 and Li, Lei, Dai, and Liang (2015). So far, to our best knowledge, few works consider the interactive (bidirectional) relationship among the members of a production system where two or more sub-processes provide resources to each other. It should be noted that the relationship we focus on in this paper refers to the relationship between sub-DMUs of a DMU, which is different from the relationship studied in some “black box” DEA models, such as zero sum gains DEA model (Lins, Gomes, de Mello, & de Mello, 2003) and the Beasley model (Beasley, 2003), where the relationship among all DMUs is investigated. For example, zero sum gains DEA model assumes the sum of output of all DMUs are fixed, that is, losses (or gains) of one DMU must be gained (or lost) by others.
Since the interactive relationship commonly exists in sub-DMUs of a system, such as two production departments in a factory, it is therefore necessary to build DEA models to analyze the performance of this system. In this paper, we focus on a simple but representative parallel production system with two interactive production units. In this production system, two Sub-DMUs provide some of their outputs (intermediate products) to each other. A novel parallel DEA method is developed to measure the system’s overall and individual efficiencies in the centralized and non-centralized mode. In the centralized mode, we assume two sub-processes cooperate to achieve the maximum system efficiency. In the non-centralized mode, we assume two sub-processes do not cooperate but work with a leader-follower relationship to achieve maximum efficiencies. Based on the results, the decision makers can find a system’s weakness so that steps can be taken to improve system performance.
The rest of this paper is organized as follows: Section 2 shows our game approach in evaluating interactive parallel system efficiencies; Section 3 provides an example; Section 4 discusses the conclusion and future possibilities.
Section snippets
Interactive parallel system with two Sub-DMUs
We consider a new structure where the parallel sub-processes have an interaction that one invests some of its outputs to another sub-process, and also consumes some outputs from another sub-process. Similar to the studies of Wang et al., 2014, Huang et al., 2014, Li et al., 2015, and Liu, Zhou, Ma, Liu, and Shen (2015), we assume that the sub-processes do not have shared inputs and the time-lag effects of the interactive parts between the sub-DMUs can be ignored. Fig. 1 shows a simple structure
Application to Chinese “985 Project” Universities
In this section, we apply the proposed approach for efficiency evaluation of the 38 Chinese “985 Project” universities in 2012. “985 Project” universities are derived from “985 Project,” which is a constructive project for founding world-class universities in the 21st century, conducted by the People’s Republic of China. Nine universities were included in the project’s initial phase. By the end of the second phase, 39 universities were sponsored. It was announced in 2011 that no more schools
Conclusions
In this paper, we investigate an interactive parallel system by a data envelopment analysis game approach. In this system, one sub-process invests some of its outputs to another stage and also consumes some outputs of the other sub-process. We first built a centralized model for measuring the efficiency of a DMU in a centralized mode. This non-linear program can be considered as a parametric program and solved via searching the best value during an interval. Based on this, we give an algorithm
Acknowledgments
This research is supported by the National Natural Science Foundation of China under Grants (No. 71501189, 71271196, 71601067), and the Fundamental Research Fund for the Central Universities (No. JZ2014HGBZ0339), the State Key Program of National Natural Science of China (No. 71431006, 71631008), the Projects of Major International Cooperation NSFC (No. 71210003); It is also supported by Research Fund for Innovation-driven Plan of Central South University (2015CX010), and the ANR (French
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