A three-stage Data Envelopment Analysis model with application to banking industry
Introduction
Data Envelopment Analysis (DEA) is a non-parametric performance evaluation method that was originally developed by Charnes et al. [19] and later extended by Banker et al. [8] to include variable returns to scale. DEA generalizes the Farrell’s [36] single-input single-output technical efficiency measure to the multiple-input multiple-output case to evaluate the relative efficiency of peer units with respect to multiple performance measures [18], [26]. The units under evaluation in DEA are called Decision Making Units (DMUs). A DMU is considered efficient when no other DMU can produce more outputs using an equal or lesser amount of inputs. The DEA generalizes the usual efficiency measurement from a single-input single-output ratio to a multiple-input multiple-output ratio by using a ratio of the weighted sum of outputs to the weighted sum of inputs [27]. Unlike parametric methods which require detailed knowledge of the process, DEA does not require an explicit functional form relating inputs and outputs (see Cooper et al. [27] and Cook and Seiford [25] for an appraisal of the theoretical foundations and developments in DEA).
Although DEA can evaluate the relative efficiency of a set of DMUs, it cannot identify the sources of inefficiency in the DMUs because conventional DEA models view DMUs as black boxes that consume a set of inputs to produce a set of outputs [4]. In such cases, using single-stage DEA may result in inaccurate efficiency evaluation [82]. In contrast, a two-stage DEA model allows one to further investigate the structure and processes inside the DMU, to identify the misallocation of inputs among sub-DMUs and generate insights about the sources of inefficiency within the DMU [31], [65].
The existing multi-stage DEA models in the literature can be classified into two categories: closed-system and open-system models. In the closed-system DEA models, the intermediate outputs remain unchanged from one stage to another. In contrast, in the open-system DEA models, the intermediate outputs in one stage are partial inputs in a subsequent stage.
In this type of systems, unlike the first stage, the second stage has inputs that are the intermediate variables since the outputs of the first stage are the inputs of the second stage. Fig. 1 presents a graphical representation of a closed two-stage DEA system.
Seiford and Zhu [88] used a two stage network model to measure the profitability and marketability of American commercial banks. In the first stage, they use labor and assets as inputs to produce profitability as output of the first-stage. In the second stage, they use profitability from the first stage and marketability as inputs in the second stage to produce market value and earnings per share as outputs of the second-stage. Zhu [99] also used this two-stage network for Fortune Global 500 companies. Chilingerian and Sherman [23] used a two-stage procedure to measure the physician care. This two-stage procedure has also been used to evaluate the performance of mental health care programs [86], the education sector [69], information technology [21], [22], and purchasing and supply management [84].
These methods produce three separate efficiency measures for the first stage, second stage, and the DMU as a whole with no consideration of the interactions between these components. Kao and Hwang [59] showed that the performance of the DMU is a combination of the performance of two stages with a chain relation between them. The efficiencies estimated from this two-stage DEA approach was more meaningful than those estimated from the independent two-staged DEA approaches. Kao and Hwang [59] used this method in a Taiwanese insurance company and compared their results with the results from the independent stage performance measurement models. Chen et al. [20] also proposed a DEA model similar to Kao and Hwang’s [59] two-stage model, but in additive format.
In this type of system, unlike for the first stage, the second stage has other inputs in addition to the intermediate variables and the outputs of the first stage are not necessarily inputs of the second stage. Fig. 2 presents a graphical representation of an open two-stage DEA system.
There are many cases in the real-world systems in which some outputs of one stage (e.g., parts in an automobile manufacturing plant) might be delivered to customers and the rest of the output will proceed to the next stages in the manufacturing process. Most of these models are in the form of DEA network systems. In the open-system models, each stage operates as an open system and gets the inputs from outside just as it may get some from the previous stages. Golany et al. [45] designed a performance measurement system that comprised of two linked sub-systems. Each sub-system uses separate resources and produces outputs. These resources could be labor or capital. Their network DEA could calculate the performance in each sub-system as well the overall performance in the entire system.
There are a number of examples for these chained processes where each sub-process uses other resources than the outputs of the previous stage. For example, consider the production and delivery sub-systems in a manufacturing system. Labor and raw materials are the inputs in the production sub-process and the finished goods are the outputs of this sub-process. The finished goods are also considered as inputs of the delivery sub-process. Other inputs of the delivery sub-process could be drivers and trucks and the final delivered product could be the output of the delivery sub-process. Liang et al. [66] applied this network concept to the performance measurement in a supply chain using Stackelberg game strategy (or leader–follower). In their two-stage model, the second stage receives inputs other than outputs of the first stage. Another network DEA was created for systems with more than two processes based on this assumption. Castelli et al. [14] studied two-stage and two-layer DMUs. Other examples of open-system multi-stage DEA models include Färe Fare and Whirraker [35], Färe and Grosskopf [30] and Tone and Tsutsui [94], [95].
Modeling and consideration of undesirable outputs in productivity and performance measurement date back to 1983 when Pittman [79] enhanced the multilateral productivity index of Caves et al. [16], [17]. Following Pittman, Färe et al. [34] integrated this concept into Farrell’s [36] technical efficiency measurement framework by introducing the new concept of “weak” versus “strong” disposability of undesirable outputs. DEA assumes that either making more output with the same input or making the same output with less input is a criterion for efficiency. In the presence of undesirable outputs, DMUs with more good (desirable) outputs and less bad (undesirable) outputs (relative to less input resources) should be categorized as efficient units. Färe et al. [34] modeled the weak disposability of undesirable outputs by changing the inequality constraint of the undesirable outputs to an equality constraint in the envelopment model. They expanded desirable outputs (by a linear constraint) and contracted undesirable outputs (by a non-linear constraint).
Färe et al.’s [34] work was the turning point in undesirable output modeling in DEA and other researchers joined him in studying undesirable outputs a decade later [50], [32], [49], [64]. Recent DEA studies on performance measurement in the presence of undesirable outputs include Fukuyama and Weber [41], Avkiran [5], Barros et al. [9], and Hwang et al. [55] among others. Liu et al. [68] has given a systematic classification of DEA models which consider undesirable factors, including radial models, non-radial models, and slacks-based models.
Scheel [85] divided modeling of undesirable outputs into two groups: direct and indirect approaches. The direct approaches use the original values and perform the transformation by: (i) multiplying its value by −1 [62]; (ii) using the inverse of the undesirable outputs [46]; and (iii) multiplying the undesirable output by −1 and adding a positive scalar that is large enough to make the transformed value greater than zero [89]. In contrast, the indirect approaches transform the values of undesirable outputs by a monotone decreasing function to desirable outputs. As noted by Gomes and Lins [47] and Mahdiloo et al. [73], Seiford and Zhu’s [89] approach is only applicable to models with the translation invariance property. Färe and Grosskopf [32] argued that undesirable outputs should be modeled under the assumption of weak disposability to be consistent with physical laws. Färe and Grosskopf [33] modeled the undesirable outputs with the directional distance function approach since Seiford and Zhu’s [89] approach did not consider the disposability assumption for the undesirable outputs.
On the other hand, two of the most famous direct approaches deal with the undesirable outputs as inputs [52], [80] and nonlinear abatement of undesirable outputs [34]. The idea of considering undesirable outputs as inputs adopted in the model proposed in this study is also discussed and justified in detail by Hailu and Veeman [50] and Korhonen and Luptacik [63].
We investigate the efficiency decomposition in a three-stage performance measurement system that has two independent parallel stages linking to a third final stage in series. In this three-stage performance measurement system, the outputs of the two parallel stages are the inputs in the third stage. We consider both desirable and undesirable variables and use the Chen and Zhu’s [22] two-stage model to develop the three-stage DEA model proposed in this study.
The remainder of this paper is organized as follows. In Section 2, we review the DEA literature in banking. In Section 3, we show the structure of the proposed DEA model. In Section 4, we use a real-world case study in the banking industry to demonstrate the applicability of the proposed models and exhibit the efficacy of the procedures and algorithms. In Section 5, we conclude with our conclusions and future research direction.
Section snippets
DEA banking literature review
The field of DEA has grown immensely since the pioneering papers of Farrell [36] and Charnes et al. [19]. Numerous applications in recent years have been accompanied by new extensions and developments in expanding the concept and methodology of DEA (see Seiford [87] and Emrouznejad et al. [29] for an extensive bibliography of DEA). Evaluating the overall performance and monitoring the financial condition of commercial banks has been the focus of numerous research studies since the early works
The proposed DEA model
As we argued in the previous section, in order to properly evaluate the performance of multi-stage DMUs, we need to change the conventional approach and consider the intermediate measures in addition to the initial inputs and the final outputs. In this section we first present the Chen and Zhu’s [22] two-stage model and we then describe the conventional closed system for a two-stage performance evaluation system with two parallel sub-DMUs and a third sub-DMU in series. Finally, we describe an
Peoples Bank1 case study
The primary role of a bank is to efficiently transform savings into investments. Successful investments build up the capital in the economy and foster future growth. Although banks are not the only financial institutions, they play a dominant role in the local and regional economy.
Conclusion and future research directions
The conventional DEA models view DMUs as black boxes that consume a set of inputs to produce a set of outputs and do not take into consideration the intermediate measures within a DMU. As a result, some intermediate measures are lost in the process of changing the inputs to outputs. In this study, we investigated the efficiency decomposition in a three-stage performance measurement system with two independent parallel stages linking to a third final stage in series. Herewith, we have extended
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions.
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