Congestion measurement and elimination under the framework of data envelopment analysis

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Abstract

Congestion is a wasteful stage of the production process where outputs are reduced due to excessive amount of inputs. This paper investigates three types of models prevailing in the data envelopment analysis literature, i.e., the Färe-family model, the Cooper-family model, and the Wei–Tone model, for measuring the congestion effect. The Färe-family model is able to decompose technical efficiency into the product of pure technical efficiency and congestion effect, yet fails to detect congestion in some situations. The Cooper-family model successfully detects congestion in all situations, yet is unable to differentiate pure technical efficiency with the congestion effect. The Wei–Tone model does not have the aforementioned deficiencies, yet is unable to identify the excessive amounts of each input. To illustrate how to eliminate congestion in reality, the Wei–Tone model is utilized to measure the congestion effect of Taiwan forests. There are three forest districts found to have different extents of congestion. The theoretical method of deleting excessive inputs to improve efficiency is actually impractical in this case. By means of reorganization, this paper finds that the congestion effects of two districts are eliminated and the other is alleviated.

Introduction

Economies of scale concern the optimal size of a production unit when increasing the amount of input results in larger amount of output. There are a lot of papers addressing this issue (see, for example, the recent article by Hsu and Li, 2009). Congestion is a special phenomenon in the production process where excessive amounts of the input cause a reduction of the output. Mining is a typical example. When too many workers are crowded in a narrow underground mining pit, the amount of minerals excavated will be reduced.

The first paper that discusses congestion with a solid theoretical basis is probably the one by Färe and Svensson (1980). In that paper, three forms of congestion were defined and characterized for a production function of single output. Färe and Grosskopf (1983) and Färe et al. (1985) developed a data envelopment analysis (DEA) model to calculate the congestion effect. Their model is of the radial approach, in that the congestion effect is measured as a ratio of the observed amounts to the expected amounts. Based on that model, Byrnes et al., 1984, Byrnes et al., 1988 used examples of coal mines to illustrate the decomposition of production efficiency, where the congestion effect is a component. Another approach initially discussed in Cooper et al. (1996) is a slack-based approach, where the congestion effect is measured as the difference between the observed amounts and the expected amounts. This approach has been applied to identify congesting inputs in Chinese industries by Brockett et al. (1998) and Cooper et al. (2001).

These two approaches are based on different assumptions regarding congestion. Consequently, one approach may produce superficially better solutions than the other under some conditions. There are several articles discussing the differences between these two approaches (Cherchye et al., 2001; Cooper et al., 2000, Cooper et al., 2001a, Cooper et al., 2001b, Cooper et al., 2001c, Cooper et al., 2002; Färe and Grosskopf, 2000a, Färe and Grosskopf, 2001). According to Färe and Svensson (1980), congestion means that “if a proper subset of production factors (inputs) is kept fixed, increases in the others may obstruct output” The definition of Cooper et al. (1996) is: “increases in one or more inputs can be associated with decreases in one or more outputs or, proceeding in reverse, reductions in one or more inputs can be associated with increases in one or more outputs”. Verbally, these two definitions are the same. They differ only in the way of measurement.

In addition to the studies conducted by the research teams of Färe and Cooper, respectively, there is another type of study conducted independently by Wei and Yan (2004) and Tone and Sahoo (2004). The previous two families of studies express the congestion effect in terms of excessive inputs. According to the definition, congestion occurs when increases in some inputs results in decreases in some outputs. In this sense, congestion can also be measured as shortfalls in outputs. Diagrammatically, expressing the congestion effect in terms of output is clearer. The model of Wei and Yan (2004) and Tone and Sahoo (2004) is developed from the output point of view. Sueyoshi and Sekitani (2009) proposed a modified approach which is able to measure the degree of congestion under the occurrence of multiple solutions. Flegg and Allen (2007) applied these three approaches to examine whether the rapid growth in the number of students in British universities has led to congestion. This paper will investigate these three types of studies from the input–output space, the input space, and the output space to find their pros and cons.

One merit of the DEA technique is it provides the decision making units (DMUs) being evaluated with a target to become efficient. Usually, either the excessive input must be reduced or the insufficient output must be increased. For a congesting DMU, the efficiency can be improved only via a reduction in inputs because, technically, the output cannot be increased. However, in the real world, there are situations that a reduction in inputs is hardly possible, especially in laying off employees in state-owned organizations. In this paper, we propose the idea of reorganization, either splitting or merging DMUs to eliminate congestion. A case of Taiwan forests is used for illustration.

In the sections that follow, firstly, we review the three representative congestion measures in the DEA literature. Three examples are used to discuss the differences among these models with the aid of diagrams. Next, the congestion effects of the Taiwan forests are calculated. How to eliminate the congestion effect via reorganization is also illustrated. Finally, some conclusions are drawn and directions for future research are suggested.

Section snippets

Three congestion measures

Suppose there are n DMUs, each produces s different outputs Yrj, r=1,…,s using m different inputs Xij, i=1,…,m; j=1,…,n. The BCC model (Banker et al., 1984) for measuring the efficiency has two variations, input- and output-orientation. Since the input-oriented model may produce erroneous results in measuring congestion (Cooper et al., 2000), this paper will concentrate on the output-orientation model. The efficiency of a specific DMU 0 is calculated via the following mathematical program:θ^=min

Geometric interpretation

A single-input single-output example will clarify the idea of these three models. Consider the example of Fig. 1 taken from Cooper et al. (2000). There are five DMUs labeled as A, B, C, D, and G. The line segments connecting A, B, and C, and extending horizontally to the right is the efficiency frontier constructed by Model (1), where A and B are efficient, C is weakly efficient (Cooper et al., 2001a) with θ*=1 and θ^<1, and D and G are inefficient. From C to D a phenomenon of congestion has

Taiwan forests

Taiwan is a small island of 36,000 square kilometers. Of which approximately one-half is forested and managed by the Taiwan Forestry Bureau. Before 1989, the national forests were divided into 13 districts. Kao and Yang, 1991, Kao and Yang, 1992 applied the DEA methodology to evaluate the performance of those 13 districts. There were four inputs and three outputs being considered:

Land (X1): area in hectares.
Labor (X2): number of employees.
Expenditures (X3): expenses per year in US dollars.

Conclusion

Several methods for measuring the effect of congestion have been developed under the framework of data envelopment analysis. The first part of this paper investigates three of them which are representative in the literature, viz., the FGL model of Färe et al. (1985), the CTT model of Cooper et al. (1996), and the WY–TS model of Wei and Yan (2004) and Tone and Sahoo (2004).

The efficiency calculated from the FGL model is a radial measure. The radial feature enables the decomposition of the

Acknowledgment

This research is supported by the National Science Council of the Republic of China under contract NSC95-2416-H-006-026-MY3.

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