Elsevier

Expert Systems with Applications

Volume 99, 1 June 2018, Pages 213-230
Expert Systems with Applications

A novel two-stage DEA production model with freely distributed initial inputs and shared intermediate outputs

https://doi.org/10.1016/j.eswa.2017.11.005Get rights and content

Highlights

  • Standard data envelopment analysis (DEA) models ignore intermediate stages making hard to detect sources of inefficiency.

  • We define a two-stage DEA model with initial inputs freely allocated in both stages.

  • Intermediate outputs in Stage 1 are split into inputs used in Stage 2 and final outputs.

  • Overall efficiency is modelled in a cooperative setting via upper and lower bounds obtained in a non-cooperative one.

  • The proposed approach is applied to evaluate 15 bank branches.

Abstract

Conventional data envelopment analysis (DEA) models treat the decision-making units (DMUs) as black-boxes: inputs enter the system and outputs exit the system, with no consideration for the intermediate steps characterizing the DMUs. As a result, intermediate measures are lost in the process of changing the inputs to outputs and it becomes difficult, if not impossible, to provide individual DMU managers with specific information on what part of a DMU is responsible for the overall inefficiency. This study defines a two-stage DEA model, where each DMU is composed of two sub-DMUs in series, the intermediate products by the sub-DMU in Stage 1 are partly consumed by the sub-DMU in Stage 2, and the initial inputs of the DMU can be freely allocated in both stages. Also, there are additional inputs directly consumed in Stage 2 while part of the outputs of Stage 1 are final outputs. We develop four new linear models to determine the upper and lower bounds of the efficiencies of the two sub-DMUs in a non-cooperative setting and a linear model to calculate the overall efficiency of DMU in a cooperative setting. That is, the overall efficiency of a DMU is modelled in a cooperative setting via upper and lower bounds obtained in the non-cooperative one. The proposed two-stage DEA method allows for important applications to several management areas. A case study in the banking industry is presented to demonstrate the applicability and exhibit the efficacy of the proposed models.

Introduction

Data envelopment analysis (DEA) is an effective non-parametric evaluation method for measuring the relative efficiency of a set of decision making units (DMUs) each of which uses multiple inputs to produce multiple outputs. In the traditional DEA methods, DMUs are treated as black-boxes, that is, the internal structure of the DMUs is often ignored. As a result, the focus of the investigation is on the single operational processes with a set of initial inputs and final outputs that is unable of pinpointing the sources of inefficiency within the DMUs (Lewis et al., 2013, Wang et al., 2014). On the other hand, in the two-stage DEA models, DMUs are modelled as systems composed of two sub-DMUs in series where the outputs of the sub-DMU in Stage 1, known as intermediate measures/products/outputs, are considered as inputs of the sub-DMU in Stage 2. As a result, a two-stage DEA model allows one to further investigate the structure of a DMU and its processes and, hence, to identify the misallocation of the inputs among the sub-DMUs (Du et al., 2011, Ebrahimnejad et al., 2014, Yu et al., 2013).

Despite appearing as the simplest multi-stage approach to efficiency evaluation, two-stage DEA models are the building blocks for the study of series systems whose DMUs consist of multiple sub-DMUs operating through procedures of different complexity. The work of Charnes et al. (1986) on army recruitment using a two-stage approach was the first study to discuss the loss of information intrinsic to single-stage models. In recent years, many researchers have studied several applications of two-stage DEA models to diverse decision making situations such as healthcare management programs (Schinnar, Kamis-Gould, Delucia, & Rothbard, 1990), education programs (Lovell, Walters, & Wood, 1994), and sport teams (Sexton & Lewis, 2003), among others. Among the many possible applications, those to the banking industry have been attracting a considerable amount of attention (Seiford and Zhu, 1999, Ebrahimnejad et al., 2014). More in general, Castelli, Pesenti, and Ukovich (2010) provide a comprehensive categorized overview of the models and methods that have been developed for different multi-stage production architectures.

The problem of evaluating the efficiency of DMUs can also be considered from the game theoretical perspective. In this sense, there are two main approaches to the problem: one approach models two-stage processes as non-cooperative games, the other as centralized systems aiming at maximizing their overall efficiency (Li, 2017). One of the first studies to employ the concepts of cooperative and non-cooperative games in an efficiency evaluation context referring to two-stage DMUs is due to Liang, Yang, Cook, and Zhu (2006). In the non-cooperative setting, one of the two stages is considered more important than the other: the former is identified with the leader while the latter with the follower. As for the efficiency scores, one starts by measuring and optimizing the leader's efficiency without considering the follower; hence, the follower's efficiency is measured under the constraint that the leader's efficiency remains constant. In the cooperative setting, both stages have the same importance and their efficiency scores must be measured and optimized at the same time (Hosseinzadeh Lotfi et al., 2012, Mahdiloo et al., 2016). For example, interpreting the offices of a regional R&D program as two-stage DMUs whose first stage consists of Premium Acquisition and the second stage is Profit Generation, we could assume that the Stage 1 is the leader. Thus, the efficiency of Stage 2 would be computed subject to the constraint that the efficiency of Stage 1 remains fixed. Alternatively, we could assume that Stage 2 is the leader and, hence, measure the efficiency of Stage 1 keeping the efficiency of Stage 2 constant (Li et al., 2012, Yu and Shi, 2014).

Kao and Hwang (2008) presented a two-stage DEA model where the efficiency of the entire process can be decomposed into the product of the efficiencies of its two sub-processes. However, their approach has a few limitations (Guo, Abbasi Shureshjani, Foroughi, & Zhu, 2017). To start, it can be applied only when the assumption of constant returns to scale is satisfied. Moreover, it allows to model only closed multi-stage processes, that is, processes where the outputs of a stage are entirely used as inputs in the following stage with no extra input added. If, for instance, additional inputs are introduced from one stage to the following one, the resulting model becomes non-linear in the multiplicative approach. This is one of the main drawbacks associated with two-stage structures. Zha and Liang (2010) slightly improved the situation by presenting a method for studying a two-stage production process in series where the initial inputs can be freely allocated in the two stages. However, their method yields a parametric model that has other pitfalls. In particular, it does not allow for additional direct inputs to be used in Stage 2 or the existence of shared intermediate products. In order to improve both the method of Zha and Liang (2010) and the structure of two-stage systems, Yu and Shi (2014) proposed a two-stage DEA model where additional inputs are allocated in Stage 2 and part of the outputs of Stage 1 are used as inputs in Stage 2. However, they formulated a parametric model to solve their two-stage structure that does not allow for shared inputs or final outputs to be produced directly in Stage 1. The method proposed in this study deals with the aforementioned problems providing a viable solution.

In this paper, we consider a novel two-stage DEA model, where each DMU is composed of two sub-DMUs in series, the intermediate products of the sub-DMU in Stage 1 are partly consumed by the sub-DMU in Stage 2, and the initial inputs of the DMU can be freely allocated to either one of the two sub-DMUs. Also, there are additional inputs directly consumed in Stage 2 while the sub-DMU in Stage 1 is allowed to produce final outputs.

Building on the concept of Stackelberg leader–follower game and the work of Yu and Shi (2014), we formulate suitable fractional programming problems that allow to determine the upper and lower bounds of the efficiencies of the two sub-DMUs in the non-cooperative setting. Afterwards, we consider the cooperative setting and define a fractional programming model to calculate the overall efficiency of DMUs. All the models are linearized by applying a Charnes–Cooper transformation (Charnes & Cooper, 1962).

To show the advantages of using the proposed method in place of the existing ones, we provide some numerical comparisons with the methods defined by Kao and Hwang, 2008, Zha and Liang, 2010 and Yu and Shi (2014). These comparisons show that our method is much more general than the existing models in terms of closeness and consistency of the solutions.

According to Zha and Liang, 2010, Matthews, 2013, and Akther, Fukuyama, and Weber (2013), a banking system can be viewed as a two-stage process. Moreover, following Seiford and Zhu, 1999, Zhu, 2000, Luo, 2003, and Liu (2011b), the first stage can be used to measure the profitability of a financial company (its ability to generate revenue and profit using its labor, assets, and capital resources) while the second stage measures its marketability (the ability of the company to generate revenue and profit through its stock market performance). We will modify the two-stage evaluation framework proposed by Seiford and Zhu (1999) and Liu (2011b) and adapt it to analyze 15 branches of a commercial bank in the Philadelphia metropolitan area. The results obtained by using our intermediate-like approach are compared with those delivered by a classical black-box approach showing that the latter approach does not suffice to adequately evaluate the aggregate performances of the DMUs composing a system.

The paper unfolds as follows. Section 2 presents a literature review while Section 3 provides a detailed review of three specific two-stage DEA models that will be used as reference points to define our approach. Section 4 extends Yu and Shi (2014)’s method to the proposed two-stage DEA production model with freely distributed inputs and shared intermediate outputs. Section 5 provides some numerical comparisons among the proposed method and those defined by Kao and Hwang, 2008, Zha and Liang, 2010 and Yu and Shi (2014) showing the advantages of our model over the existing ones. Section 6 describes a case study in the US banking industry and discusses the corresponding numerical results. Finally, Section 7 concludes and suggests some future research directions.

Section snippets

Literature review

In this section, we review some the vast literature related to the present study.

Methodological background

Consider n decision making units (DMUs) each of which uses m inputs to produce s outputs. A two-stage DMU is a DMU consisting of two sub-DMUs working in series. The outputs of the first stage (intermediate products) are usually used as inputs of the second stage to produce the final outputs. Finally, all inputs and outputs are assigned a weight. The following notations are used to review three keynote models in the literature that will be extended by our approach.

DMUjj = 1,  2,  ...,  njth DMU

Proposed methodology

We consider a two-stage DEA model where each DMU consists of two sub-DMUs in series with the following characteristics:

  • intermediate outputs of the sub-DMU in Stage 1 are only partially consumed by the sub-DMU in Stage 2;

  • part of the inputs of the DMU can be freely allocated to each one of the two sub-DMUs;

  • additional inputs are directly consumed in Stage 2;

  • part of the outputs of Stage 1 are considered as final outputs.

Fig. 4 depicts a two-stage DMU such as the one just described.

In summary,

Proposed method versus existing methods

There are many novel performance measurement studies in the two-stage DEA literature. Table 1 presents a comparative overview of the most widely used two-stage DEA models outlining the advantages provided by the method introduced in this study when compared to them.

Our method is more general than the other existing models. It allows for many more possibilities when assigning direct inputs and final outputs and, consequently, for a wider range of applications. In order to support this statement

Case study

Defining performance evaluation procedures for commercial banks constitutes one of the most important and promising application areas of DEA. In this section, we apply the proposed method to evaluate 15 branches of the Philadelphia National Bank (PNB)1 in the Philadelphia metropolitan area.

Seiford and Zhu (1999) investigated the efficiencies of a set of listed U.S. commercial banks by interpreting them as

Conclusion and future research directions

Conventional DEA models interpret DMUs as black-boxes that consume a set of inputs to produce a set of outputs without taking into consideration the intermediate performance measures that characterize a DMU. That is, traditional DEA studies view a system as a sole block and ignore the performance of the internal processes in calculating the relative efficiency within a production system. As a result, intermediate measures are lost in the process of transforming the inputs into outputs and it

Acknowledgement

The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions.

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