O.R. Applications
Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan

https://doi.org/10.1016/j.ejor.2006.11.041Get rights and content

Abstract

The efficiency of decision processes which can be divided into two stages has been measured for the whole process as well as for each stage independently by using the conventional data envelopment analysis (DEA) methodology in order to identify the causes of inefficiency. This paper modifies the conventional DEA model by taking into account the series relationship of the two sub-processes within the whole process. Under this framework, the efficiency of the whole process can be decomposed into the product of the efficiencies of the two sub-processes. In addition to this sound mathematical property, the case of Taiwanese non-life insurance companies shows that some unusual results which have appeared in the independent model do not exist in the relational model. In other words, the relational model developed in this paper is more reliable in measuring the efficiencies and consequently is capable of identifying the causes of inefficiency more accurately. Based on the structure of the model, the idea of efficiency decomposition can be extended to systems composed of multiple stages connected in series.

Introduction

Since Charnes et al. (1978), data envelopment analysis (DEA) has been widely applied to measure the relative efficiency of a set of decision making units (DMUs) which apply the same inputs to produce the same outputs. The results indicate how efficient each DMU has performed as compared to other DMUs in converting inputs to outputs. An issue which is of greater concern to the inefficient DMUs is what factors that cause the inefficiency, although it is obvious that either reducing inputs or increasing outputs will improve their performance.

To answer this question, much effort has been devoted to breaking down the overall efficiency into components so that the sources of inefficiency can be identified. One type of decomposition focuses on the structure of the DEA model. Banker et al. (1984) break the overall efficiency of a DMU into the product of scale efficiency and technical efficiency. Byrnes et al. (1984) further separate the congestion effect from the technical efficiency. Kao (1995) decomposes the overall efficiency into a weighted arithmetic mean of the efficiencies of individual outputs. A similar decomposition from the input side is also derived. Another type of decomposition emphasizes the stages of the production process. The complicated production process is divided into sub-processes, in that some intermediate products are the outputs of a sub-process on the one hand and the inputs of another sub-process on the other hand. The works of Färe and Grosskopf, 1996, Färe and Grosskopf, 2000, Seiford and Zhu, 1999 are some examples of this approach. In the former type of decomposition, there exists some mathematical relationship between the overall efficiency and the component efficiencies, while for the latter type there is no specific relationship between those two parts. The reason is because the sub-processes in the latter type are considered as independent processes in calculating their efficiencies. The model for calculating the efficiencies of the sub-processes does not reflect any relationship between the components and the whole system.

The simplest case of a complicated production process is a tandem system, in which the whole production process is composed of two sub-processes connected in series. Seiford and Zhu (1999) divide a commercial bank’s production process into the stages of profitability and marketability. The inputs of the bank production process are employees, assets, and shareholders’ equity, which are also the inputs of the first stage. The outputs of the bank production process are market value, total return on investments, and earnings per share, which are also the outputs of the second stage. In addition to the inputs and outputs of the system, there are two intermediate products, revenues and profits, which are the outputs of the first stage as well as the inputs of the second stage. The efficiencies of the first stage, second stage, and the whole production process are calculated via three independent DEA models for 55 US commercial banks. Decomposition of the production process helps identify the source of inefficiency. Zhu (2000) follows the same idea to analyze the financial efficiency of the best 500 companies as ranked by Fortune. This two-stage concept has also been applied to measure the performance of mental health care programs (Schinnar et al., 1990), education sector (Lovell et al., 1994), American Major League Baseball teams (Sexton and Lewis, 2003), information technology (Chen and Zhu, 2004, Chen et al., 2006), etc.

The objective of this paper is to investigate efficiency decomposition in a two-stage production process where the outputs of the first stage are the inputs of the second stage. Different from previous studies, which treat the whole production process and the two sub-processes as independent, this paper takes the series relationship of the two sub-processes into account in measuring the efficiencies. We will show that the overall efficiency is the product of the efficiencies of the two sub-processes. This mathematically sound property reflects the physical linkage of the two sub-processes with the whole process in a tighter way. The efficiencies calculated from this relational two-stage DEA approach are more meaningful than those calculated from the independent two-stage DEA approach. The non-life insurance companies in Taiwan, whose production process resembles the two-stage process, are utilized to illustrate the whole idea. The results are compared to those calculated from the independent two-stage approach to make suitable discussions.

The structure of this paper is organized as follows. In the next section, we develop the relational two-stage DEA model for measuring the efficiencies of the whole process as well as the two sub-processes. In Section 3, we use the non-life insurance companies in Taiwan as an example to calculate the efficiencies by using the relational model. Finally, the results are compared to those calculated from the independent model to draw conclusions.

Section snippets

Relational two-stage DEA model

Denote Xij, i = 1,  , m and Yrj, r = 1,  , s as the ith input and rth output, respectively, of DMU j, j = 1,  , n. The conventional DEA model for measuring the efficiency of DMU k under the assumption of constant returns-to-scale is the CCR model (Charnes et al., 1978):Ek=maxr=1surYrki=1mviXiks.t.r=1surYrji=1mviXij1,j=1,,n,ur,viε,r=1,,s;i=1,,m,where ε is a small non-Archimedean number (Charnes et al., 1979, Charnes and Cooper, 1984). Each DMU applies m inputs to produce s outputs, and Ek is the

Non-life insurance companies in Taiwan

Similar to other service industries, the non-life insurance industry provides services to their clients to generate profit. There are several studies which use the DEA technique to measure the managerial performance of this industry (Fecher et al., 1993, Noulas et al., 2001). Notably, the profit is not earned from insurance services alone. Non-life insurance companies use the insurance premiums acquired through the systems of agencies, brokers, solicitors, etc. as capital for investment. Hence,

Relational model vs independent model

For the same data set contained in Table 2, Hwang and Kao (2006) have calculated the overall efficiencies, stage 1 efficiencies, and stage 2 efficiencies independently by using the conventional CCR model. The results are reported in the right half of Table 3 under the heading “Independent two-stage model”. There are four companies, Chung Kuo (No. 2), Fubon (No. 5), Union (No. 12), and Asia (No. 22), which perform efficiently in the whole production process. Interestingly, none of them performs

Conclusion

The objective of efficiency measurement is to detect the weak areas so that appropriate effort can be devoted to improve performance. When a production system can be separated into two sub-processes, several studies find that in addition to calculating the efficiency of the whole system by using the conventional DEA model, the efficiencies of the two sub-processes can also be calculated to identify the source that causes the inefficiency of the whole system. A deficiency of these studies is

Acknowledgement

This research is supported by the National Science Council, Republic of China, under contract number: NSC95-2416-H-006-026-MY3. The authors are grateful to Dr. Wen-Shiang Wu for his help in calculating the efficiencies.

References (21)

There are more references available in the full text version of this article.

Cited by (0)

View full text