Network DEA: efficiency analysis of organizations with complex internal structure

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Abstract

DEA models treat the DMU as a “black box.” Inputs enter and outputs exit, with no consideration of the intervening steps. Consequently, it is difficult, if not impossible, to provide individual DMU managers with specific information regarding the sources of inefficiency within their DMUs. We show how to use DEA to look inside the DMU, allowing greater insight as to the sources of organizational inefficiency. Our model applies to DMUs that consist of a network of Sub-DMUs, some of which consume resources produced by other Sub-DMUs and some of which produce resources consumed by other Sub-DMUs. Our Network DEA Model allows for either an input orientation or an output orientation, any of the four standard assumptions regarding returns to scale in any Sub-DMU, and adjustments for site characteristics in each Sub-DMU. We demonstrate how to incorporate reverse quantities as inputs, intermediate products, or outputs. Thus, we can apply the Network DEA Model presented here in many managerial contexts. We also prove some theoretical properties of the Network DEA Model.

By applying the Network DEA Model to Major League Baseball, we demonstrate the advantages of the Network DEA Model over the standard DEA Model. Specifically, the Network DEA Model can detect inefficiencies that the standard DEA Model misses. Perhaps of greatest significance, the Network DEA Model allows individual DMU managers to focus efficiency-enhancing strategies on the individual stages of the production process.

Introduction

Data envelopment analysis (DEA) is a linear programming-based methodology for evaluating the relative efficiency of each member of a set of organizational units. The units, called decision-making units (DMUs), consume various levels of each specified input and produce various levels of each specified output. DEA evaluates the efficiency of a DMU relative to an empirical production possibility frontier determined by all DMUs under appropriate assumptions regarding returns to scale and orientation.

DEA makes no assumptions concerning the internal operations of a DMU. Rather, DEA treats each DMU as a “black box” by considering only the inputs consumed and outputs produced by each DMU. This perspective is often appropriate and sufficient. For example, if the purpose of the analysis is to identify inefficient DMUs and evaluate the extent of their inefficiency, then a “black box” approach is adequate. However, such an approach provides no insight regarding the sources of inefficiency and cannot provide process-specific guidance to DMU managers to help them improve the DMU's efficiency.

Castelli et al. [1] described a DEA-like model that evaluates the efficiencies of each of a number of interdependent sub-units within a larger DMU. Their analysis assesses sub-unit efficiency relative to other sub-units within the same DMU. This model is attractive because it allows a DMU manager to assess efficiencies within the DMU without access to data from other DMUs. However, their approach may result in very small reference sets, thereby limiting the number of inputs and outputs that can be included and possibly leading to inflated efficiency scores.

In Sexton and Lewis [2], we considered an extension of the DEA model in which each DMU is comprised of two Sub-DMUs connected in series. In that model, the Stage 1 Sub-DMU consumes inputs to produce intermediate products. These, in turn, are the inputs to the Stage 2 Sub-DMU, which uses them to produce the DMU's outputs. The objective of the two-stage DEA Model is to evaluate the relative efficiencies of each DMU and each of its Sub-DMUs.

While we believe that the two-stage DEA Model describes many important settings, we recognize that we may require more complexity in other cases. In this article, we develop a Network DEA Model in which the internal design of the production process is more intricate. Specifically, we assume that each DMU is comprised of a set of Sub-DMUs. Each input to a Sub-DMU is either exogenous to the DMU or is the output of another Sub-DMU. Similarly, each output from a Sub-DMU either leaves the DMU or is an input to another Sub-DMU. See Fig. 1. Consider the directed graph in which the nodes correspond to Sub-DMUs and the arcs correspond to flows from one Sub-DMU to another. We assume that this graph is acyclic.

We are not the first to consider the network DEA Model. Färe and Whittaker [3] and Färe and Grosskopf [4] have presented similar models. Those models define the reference set for the DMU as the set of all convex linear combinations of DMUs, as is the common practice in DEA. In our model, we define the reference set for the DMU based on the hypothetical Sub-DMUs identified for each Sub-DMU.

We apply the Network DEA Model to major league baseball (MLB). Two Sub-DMUs correspond to the team's front office operation, which uses money in the form of player salaries to acquire talent. Three Sub-DMUs correspond to the team's on-field operation, which utilize the talent to win games. This analysis is a generalization of that presented in Sexton and Lewis [2].

We argue that it is crucial for the team's ownership to identify the extent of inefficiency within each Sub-DMU. Inefficiency in a front office Sub-DMU suggests that the team evaluates offensive or defensive talent poorly and pays players more than they are worth. Inefficiency in an on-field Sub-DMU, however, indicates that the team is unable to produce or prevent as many runs as warranted by their talent level or produce as many victories as warranted by the runs they score and prevent. (DEA is a deterministic method that ignores random variation such as players having exceptionally good or bad years. At the team level, DEA assumes that such random variations cancel one another sufficiently to allow a deterministic analysis. Other methods, such as stochastic frontier analysis (see Kumbhakar and Knox Lovell [5]), explicitly allow for such random variation). Knowing the sources of inefficiency within an organization allows management to develop focused interventions designed to improve organizational performance.

Section snippets

Network DEA Model formulation

We first present the Network DEA Model using an output orientation, indicating the modifications necessary for an input orientation. A DEA Model is output oriented if it seeks to increase outputs without increasing inputs. Similarly, a DEA model is input oriented if it seeks to decrease inputs without decreasing outputs. Our approach to the Network DEA Model is an extension of that used in the two-stage DEA Model. In brief, we begin by solving a DEA problem for each Sub-DMU to obtain its

Network DEA theory

Unlike standard DEA, our Network DEA Model does not guarantee the existence of an organizationally efficient DMU. It is possible that all DMUs under consideration are organizationally inefficient. Consider the output-oriented, constant returns to scale example shown in Fig. 2 and Table 1, Table 2. The Network DEA Model computes the organizational inverse efficiency for each DMU to be 1.33. DMU A's inefficiency comes from the fact that it should have been able to produce 10 rather than 5 units

Application to major league baseball

The following discussion of MLB follows that in Lewis and Sexton [11]. MLB is comprised of 30 teams divided into two leagues. The teams are the DMUs in our analysis. There are 16 teams in the National League, and 14 in the American League. Each league is comprised of three divisions, each containing between 4 and 6 teams. The leagues play under essentially identical rules with one major exception: the American League allows the use of a designated hitter who bats in place of the pitcher. This

Computational results

We applied the model described above to evaluate the performance of MLB teams during the 1999 season. We obtained data (see Table 3) from a variety of sources. See Appendix A for complete team names. We extracted player salary data from the USA Today web site [12]. We gathered games won and the team performance data from the Baseball Archive Database [13]. We obtained population data for the markets in which teams play from the US Census Bureau web site [14].

Table 4 shows the organizational

Discussion and conclusion

DEA models treat the DMU as a “black box.” Inputs enter and outputs exit, with no consideration of the intervening steps. Consequently, it is difficult, if not impossible, to provide individual DMU managers with specific information regarding the sources of inefficiency within their DMUs.

We have shown how to use DEA to look inside the DMU, allowing greater insight as to the sources of organizational inefficiency. Our model applies to DMUs that consist of two or more Sub-DMUs, some of which

Herbert F. Lewis is a Lecturer at the W. Averell Harriman School for Management and Policy at the State University of New York at Stony Brook. He received his Ph.D. in 1996 in operations research from the Department of Applied Mathematics and Statistics at the State University of New York at Stony Brook. His research interests include productivity and efficiency analysis using data envelopment analysis, and solution methods for combinatorial problems in scheduling, vehicle routing, and facility

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Herbert F. Lewis is a Lecturer at the W. Averell Harriman School for Management and Policy at the State University of New York at Stony Brook. He received his Ph.D. in 1996 in operations research from the Department of Applied Mathematics and Statistics at the State University of New York at Stony Brook. His research interests include productivity and efficiency analysis using data envelopment analysis, and solution methods for combinatorial problems in scheduling, vehicle routing, and facility location.

Thomas R. Sexton is the Director of the W. Averell Harriman School for Management and Policy at the State University of New York at Stony Brook. He earned his Ph.D. in Applied Mathematics and Statistics from the State University of New York at Stony Brook in 1979. He has published extensively on topics related to productivity, health care management, and vehicle routing and scheduling. His recent research is in the areas of productivity and efficiency measurement methodologies, student perceptions of Internet learning, and the effects of mandatory auditor rotation and retention on audit market share.

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