A note on “A New Approach for the Selection of Advanced Manufacturing Technologies: Data Envelopment Analysis with Double Frontiers”

Recently, using the data envelopment analysis (DEA) with double frontiers approach, Wang and Chin (2009) proposed a new approach for the selection of advanced manufacturing technologies: DEA with double frontiers and a new measure for the selection of the best advanced manufacturing technologies (AMTs). In this note, we show that their proposed overall performance measure for the selection of the best AMT has an additional computational burden. Moreover, we propose a new measure for developing a complete ranking of AMTs. Numerical examples are examined using the proposed measure to show its simplicity and usefulness in the AMT selection and justification.


Introduction
Selection of advanced manufacturing technologies (AMTs) is an important decisionmaking process for the explanation and implementation of AMTs. This requires careful consideration of various performance criteria (Wang & Chin, 2009). As an excellent method for performance evaluation based on data when a set of decision-making units (DMUs) has multiple inputs and outputs, data envelopment analysis (DEA) has proven its value. Therefore, the DEA has been widely used for AMT selection and justification.
For best use of the DEA, Wang and Chin (2009) introduced a new DEA method called "DEA with double frontiers" for AMTs selection and justification. The DEA with double frontiers considers two different efficiencies, i.e. optimistic and pessimistic efficiencies for decision-making. In this note, we show that the overall performance measure proposed by Wang and Chin (2009) for selecting the best AMT has an additional computational burden and may affect the ranking results. Finally, we propose a new measure to develop a complete ranking of AMTs.
The remainder of the paper is organized as follows: Section 2 starts with an overview on the measure proposed by Wang and Chin (2009  Optimistic and pessimistic efficiencies are measured from different perspectives, and often lead to two different rankings for AMTs. Therefore, an overall performance measure is needed to obtain a single overall ranking of AMTs. To this end, Wang and Chin (2009) proposed the following overall performance measure for ranking AMTs: It is clear that the overall performance measure defined in (3) is the sum of elements for the normalized vectors of the two vectors derived from optimistic and pessimistic efficiencies. Since the normalization of efficiency vectors has no effect on the ranking of AMTs, the following measure can also be used for ranking AMTs: Measure (5) may provide more correct results compared with measure (3), because measure (3) includes a rounding error.

New overall performance measure
In Wang et al. (2007), the geometric average of two efficiencies was proposed as the overall performance measure. The geometric average efficiency integrates both optimistic and pessimistic efficiency measures for each DMU, so it is more comprehensive than either of these two measures. In Wang and Chin (2009), in a sense, the arithmetic average of both optimistic and pessimistic efficiencies was proposed as an overall performance measure. Since measure (3) is twice the arithmetic average of the normalized efficiencies and their ranking is exactly the same, three different means (i.e., geometric average, arithmetic average, and quadratic mean) can be used for ranking DMUs as follows: The relationship between these means is as follows: Generally, when optimistic and pessimistic efficiencies are larger, the DMU is evaluated better. Thus, according to equation (9), one can use the quadratic mean as the overall performance measure for ranking DMUs. Since the value 2 / 1 does not affect the ranking of DMUs, we consider the following measure as the new overall performance measure for each DMU:

Numerical Examples
In For input and output data related to all the tables presented in Wang and Chin (2009), we run DEA models (1) and (2) for each AMT to obtain optimistic and pessimistic efficiencies. The results are shown in Tables 1-4. Additionally, the overall performance of each AMT is measured by measures (3) and (10) and their ranking is shown in Tables 1-4.  Tables 3 and 4 are not identical. In Table 3 ), their ranking is switched. One of its reasons is the high computational burden of measure (3), and a rounding error. It is clear that measure (10) is more efficient, and can save a lot of calculations compared with measure (3). A similar problem exists in Table 4. The ranking based on measures (3) and (10) has changed the results of 26 AMTs. That is, more than 55% of AMTs are ranked wrongly. We have shown them in bold font. This is the biggest advantage of measure (10) over measure (3) for AMT selection and justification.