Elsevier

Omega

Volume 38, Issue 6, December 2010, Pages 423-430
Omega

Review
Measuring performance of two-stage network structures by DEA: A review and future perspective

https://doi.org/10.1016/j.omega.2009.12.001Get rights and content

Abstract

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). An important area of development in recent years has been devoted to applications wherein DMUs represent two-stage or network processes. One particular subset of such processes is those in which all the outputs from the first stage are the only inputs to the second stage. The current paper reviews these models and establishes relations among various approaches. We show that all the existing approaches can be categorized as using either Stackelberg (leader-follower), or cooperative game concepts. Future perspectives and challenges are discussed.

Introduction

Data envelopment analysis (DEA), introduced by Charnes et al [1], is an approach for identifying best practices of peer decision making units (DMUs) in the presence of multiple inputs and outputs. In many cases DMUs may consist of two-stage network structures with intermediate measures. In other words, DMUs under evaluation share a common feature found in many two-stage network structures, namely that outputs from the first stage become the inputs to the second stage. Outputs from the first stage are referred to as intermediate measures. For example, Seiford and Zhu [2] use a two-stage network structure to measure the profitability and marketability of US commercial banks. In their study, profitability is measured relative to labor and assets as inputs, and the outputs are profits and revenues. In the second stage, for marketability, the profits and revenue are then used as inputs, while market value, returns and earnings per share constitute the outputs. Zhu [3] applies the same two-stage network structure to the Fortune Global 500 companies.

Seiford and Zhu [2] use the standard DEA approach which does not address potential conflicts between the two stages arising from the intermediate measures. For example, the second stage may have to reduce its inputs (intermediate measures) in order to achieve an ‘efficient’ status. Such an action would, however, imply a reduction in the first stage outputs, thereby reducing the efficiency of that stage.

In the current study, we review the various existing DEA models for measuring efficiency in the aforementioned two-stage network structures or processes. We classify these DEA models into four categories: standard DEA approach; efficiency decomposition approach; network-DEA approach; and game-theoretic approach. Except for the standard DEA approach, all other approaches attempt to correct for the above-referenced conflict issue.

Throughout the paper, we use the Kao and Hwang [4] data set involving non-life insurance companies in Taiwan. The rest of the paper is organized as follows. Section 2 presents the generic two-stage process and a general literature review and classification of papers dealing with DMUs with two stage processes.

In 3 Centralized model, 4 Stackelberg game, we discuss the efficiency decomposition methodology and game-theoretic approaches. We begin with the work by Liang, et al. [5] where DEA models are developed to measure the performance of supply chains with two members. In their study, because some of the inputs to the second stage are not from the first stage, one of the DEA models is non-linear. However, if we apply their approach to our two-stage processes, and use the overall efficiency definition from Kao and Hwang [4], we can obtain linear DEA models as in [6]. This establishes the relationships among the works of Liang et al. [5], [7], Kao and Hwang [4] and [6]. These approaches are then re-categorized as (1) the centralized models of Kao and Hwang [4] and [6], and (2) the non-cooperative (or leader-follower) model. It is shown how to test for uniqueness of the efficiency decomposition.

We then proceed to the network DEA approach in Section 5. We show that the Kao and Hwang [4] model and the centralized model of [6] are equivalent to the network DEA approach of Färe and Grosskopf [8]. Note the fact that, as demonstrated in Chen et al. [25], Chen and Zhu's [10] model under the CRS assumption is equivalent to the Kao and Hwang's [4] model. As a result, we establish the equivalence among these models in dealing with two-stage processes. We discuss as well the determination of the efficient frontier of the two-stage process. Since it is possible that no single DMU is efficient, the standard DEA projections can no longer be used to generate the DEA frontier.

Section 6 presents some future challenges and perspectives. For example, Chen et al. [9] consider the sharing of some input resources between the two stages. As a result, their model is non-linear and the solution may not be globally optimal. We discuss possible ways to model the situation in Chen et al. [9] via linear models.

Section snippets

Literature review and classification

Consider a generic two-stage network structure or process as shown in Fig. 1, for each of a set of n DMUs. Using the notation of Chen and Zhu [10] and Kao and Hwang [4], we assume each DMUj (j=1, 2, …, n) has m inputs xij, (i=1, 2, …, m) to the first stage, and D outputs zdj, (d=1, 2, …, D) from that stage. These D outputs then become the inputs to the second stage and will be referred to as intermediate measures. The outputs from the second stage are yrj, (r =1, 2, …, s).

We denote the

Centralized model

Liang et al. [5] show that using the concept of cooperative game theory, or centralized control, the two stage process can be viewed as one where the stages jointly determine a set of optimal weights on the intermediate factors to maximize their efficiency scores. This would be the case in situations where the manufacturer and retailer jointly determine prices, order quantities, etc, to achieve maximum profit ([21]). In other words, the cooperative or centralized approach is characterized by

Stackelberg game

In the previous section we examined the cooperative or centralized game approach to the two-stage problem. In this section we look at the two-stage process from the perspective of the non-cooperative game. The non-cooperative approach is characterized by the leader–follower, or Stackelberg game. For example, consider a case of a supply chain where there is non-cooperative advertising on the part of the manufacture (leader) and the retailer (follower). The manufacturer determines its optimal

Network DEA

If we model the two-stage process shown in Fig. 1 using the network approach of Färe and Grosskopf [8], we haveminΘ,λj,μj,z˜Θsubjectto(stage1)j=1nλjxijΘxijoi=1,,mj=1nλjzdjz˜djod=1,,Dλj0,j=1,,n(stage2)j=1nμjzdjz˜djod=1,,Dj=1nμjyrjyrjor=1,,sμj0,j=1,,nwhere z˜djo are set as decision variables related to the intermediate measures.

Model (11) is obviously equivalent to the following modelminΘ,λj,μj,z˜Θsubjecttoj=1nλjxijΘxijoi=1,,mj=1n(λjμj)zdj0d=1,,Dj=1nμjyrjyrjor=1,,sλj,μj0

Perspectives and challenges

This article has attempted to shed light on the more pertinent literature related to the evaluation of two-stage processes. Such processes are important in many real-world settings, with one being that involving supply chains. Capturing supply chain efficiency and identifying the extent to which individual players in the chain contribute to the overall efficiency has become an issue of critical importance in the Operations Management field. For this reason, this review is believed to be

Acknowledgements

The authors are grateful to the comments and suggestions made by three anonymous reviewers. Professor Liang thanks the support by the National Natural Science Foundation of China (Grant No. 70821001).

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