Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1619798, 13 pages

https://doi.org/10.1155/2017/1619798

## A Revised Model of the Neutral DEA Model and Its Extension

School of Management, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Li-Fang Wang

Received 11 February 2017; Accepted 30 March 2017; Published 13 June 2017

Academic Editor: Vladimir Turetsky

Copyright © 2017 Peng Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The neutral data envelopment analysis (DEA) model is an alternative way to determine the weights in DEA cross-efficiency evaluation, while avoiding the difficulty in making a choice between the aggressive and benevolent formulations. However, the weights determined by the neutral model merely make the efficiency of part output bigger than other sets of weights. The neutral model is not able to make the efficiency of each output of the DMU biggest among the favorable weights. This neutral model is not purely “neutral” and not most favorable to the DMU. We proposed a revised model for the neutral model. Based on the idea that the DMU should choose a set of weights to maximize its own efficiency, this paper proposes a new cross-efficiency model. The weights determined by the two models are neutral, neither aggressive nor benevolent.

#### 1. Introduction

Data envelopment analysis proposed by Charnes et al. [1] is a nonparametric method to identify the production frontier and to measure the efficiency of homogenous decision-making units (DMUs) with multiple inputs and multiple outputs. DEA traditional model (CCR and BCC) measures the relative efficiency of DMU relative to its own subpoint on the production frontier. As long as the DMU is on the production frontier, it is the DEA efficient point and the CCR value is equal to 1. Since there is often more than one DMU on the production frontier, more than one DMU is often evaluated as DEA efficient and cannot be discriminated any further. Lacking of the discrimination power becomes the major drawback of the traditional DEA model.

To solve this problem, some scholars have extended the traditional DEA model and proposed some new approaches to improve the discrimination power of traditional DEA model. One such method is DEA super efficiency model [2] that will lead to the efficiency value of efficient DMUs to be larger than 1, which will further distinguish the efficient DMUs. Another method is DEA common-weight evaluation [3]. Unlike the traditional DEA model in which each DMU uses its own favorable weights to measure the efficiency, the DEA common-weight evaluation adopts the common set of weights to measure the efficiency of all DMUs. There are still other methods, such as the multivariate statistics ranking technique [4], the benchmark ranking method [5], the DEA multicriteria decision-making method [6], and the context-dependent DEA technique [7].

DEA cross-efficiency evaluation as an extension to DEA is a commonly used method to improve the discriminative power of the traditional DEA model. The cross-efficiency evaluation utilizes self-evaluation and peer-evaluation to evaluate the efficiency of each DMU. The concept of cross-efficiency evaluation was first proposed by Sexton et al. [8] and was later developed by Doyle and Green [9, 10]. In the cross-efficiency evaluation, each DMU determines a set of weights among all the favorable weights based on some ways. That will lead to sets of weights. sets of weights will lead to efficiency values for each DMU. The final average efficiency value for each DMU will be computed by assembling the efficiency values. The rank of DMUs is determined by the final average efficiency value of DMU. It is believed that the overall efficiency value calculated by cross-efficiency evaluation can guarantee a unique ordering for the DMUs and can be used with few DMUs (e.g., four or five) to produce a unique ranking ordering [10].

Because of the powerful discrimination among DMUs, the cross-efficiency evaluation has been widely applied in efficiency evaluation of nursing homes [8], project ranking and preference voting [11], efficiency evaluation of countries in the Olympic Games (Wu, Liang, and Chen 2009), fixed cost and resource allocation [12], portfolio selection in the Korean stock market [13], supplier selection under uncertainty [14], and so on. However, the nonunique solution to the traditional DEA model (CCR) limits the application of DEA ross-efficiency. How to solve this problem is the focus of the research of DEA cross-efficiency. Using the secondary goal models is the main method to solve this question. Up to now, there are many secondary goal models proposed by many scholars such as aggressive and benevolent models proposed by Doyle and Green [9], the neutral model by Wang (2010), the weight-balanced model by Wu et al. [15], the maximin model by Lim [16], and the satisfaction degree model by Wu et al. [17]. The aggressive and benevolent models proposed by Doyle and Green [9] are two classical ways among the many models. The aggressive (benevolent) model minimizes (maximizes) the average efficiency of other DMUs. The neutral DEA cross-efficiency model suggested by Ying-Ming Wang (2010) offers the new idea to solve this question. Wang considered that the particular DMU should choose a set of weights which is most favorable to itself not considering whether it is favorable or not to other DMUs. The neutral model maximinimizes the efficiency of each output.

To avoid the difficulty in making a choice between the aggressive and benevolent formulations in DEA cross-efficiency evaluation, Wang and Chin [18] put forward a neutral DEA model for the cross-efficiency evaluation, which maximinimizes the relative efficiency of each output. The neutral DEA model was proposed on the idea that “when a DMU is given an opportunity to decide a set of input and output weights, what the DMU is concerned most about is whether the weights can be as favorable as possible to itself.” However, the weights determined by the neutral DEA model can just make the efficiency of part outputs higher. That leads to the neutral DEA model proposed by Wang and Chin [18] being not very consistent with its modeling idea. To address this issue, we propose a revised model. We further propose a new model based on the idea that according to self-interest principle the DMU will select a set of input and output weights among the favorable weights to maximize its own efficiency.

The rest of the paper is organized as follows. Section 2 briefly introduces the cross-efficiency evaluation and its aggressive and benevolent formulations. Section 3 reviews the neutral DEA mode suggested by Wang and Chin [18] and proposes a revised model. Section 4 proposes a new model based on the self-interest principle. Section 5 makes a comparison among these above models through the numerical examples. Section 6 offers the conclusions.

#### 2. The Cross-Efficiency Evaluation

Suppose there are DMUs to be evaluated with inputs and outputs. Denote by and the input and output values of , based on the input and output ratio, whose efficiencies are defined as where and are, respectively, the input and output weights.

Consider a DMU, say, , , whose DEA CCR-efficiency value can be computed by the following CCR model [1]:When the value is equal to 1, it means that is DEA efficient. The input and output weights determined by the model are favorable to . Because the weights determined by CCR model are favorable to the , the CCR value has been viewed as the self-evaluation.

By using Charnes and Cooper transformation (Charnes and Cooper, 1962), model (2) can be equivalently transformed into the linear program (LP) below for solution:Let and be the optimal solution to the above model. Then, is referred to as the CCR-efficiency value of , which is the self-evaluated efficiency of As such, is referred to as a cross-efficiency value of according to the favorable weights of and reflects the peer-evaluation of to .

Model (3) is solved times, each time for each DMU. As a result, there will be sets of input and output weights available for and each DMU will have cross-efficiency values that are peer-evaluation and one CCR-efficiency value that is self-evaluation, which form a cross-efficiency matrix, as shown in Table 1, where are the CCR-efficiencies of the ; that is, . The average efficiency of DMU is measured by the one CCR-efficiency value and cross-efficiency values; namely,Based on their average cross-efficiencies, the DMUs can usually be fully ranked.