Elsevier

Economic Analysis and Policy

Volume 69, March 2021, Pages 613-628
Economic Analysis and Policy

Modelling economic policy issues
Undesirable factors in stochastic DEA cross-efficiency evaluation: An application to thermal power plant energy efficiency

https://doi.org/10.1016/j.eap.2021.01.013Get rights and content

Abstract

In this study using an input-oriented data envelopment analysis (DEA) model with undesirable outputs a new stochastic model called Expected Ranking Criterion is proposed. The proposed model employs statistical techniques to evaluate the efficiency of decision making units (DMUs) with stochastic data. Based on the proposed model, a stochastic DEA (SDEA) cross-efficiency model is suggested for ranking and discrimination of DMUs. Then, given the non-uniqueness of resulting optimal solution, a stochastic model is introduced for rating priorities by which cross-efficiency evaluation is performed using aggressive approach. Finally, the proposed models are implemented for evaluating 32 thermal power plants. The results show the applicability of the proposed models.

Introduction

Charnes, Cooper and Rhodes (CCR) proposed DEA as a nonparametric method for evaluating the efficiency of DMUs and estimating efficiency frontier in 1978. Based on which BCC (Banker, Cooper, Charnes) or Variable Returns to Scale (VRS) model was introduced in 1984. Since then evaluating the efficiency of DMUs with multiple inputs/outputs attains widespread applications. DMUs are categorized as efficient and inefficient based on the DEA efficiency scores. One of the interesting topics in DEA is the introducing of methods that rank and discriminate DMUs. The following 6 methods were mentioned according to a report by Adler et al. (2002) based on extensive studies: (a) Cross Efficiency Ranking and Modeling: Doyle and Green (1994), and Yang et al. (2012). (b) Super Efficiency Ranking technique: Andersen and Petersen (1993), Sueyoshi (1999), Noura et al. (2011), Hadi-Vencheh and Esmaeilzadeh (2013), Liu and Wang (2018) and Esmaeilzadeh and Hadi-Vencheh, 2013, Esmaeilzadeh and Hadi-Vencheh, 2015). (c) Benchmarking Ranking technique: Torgersen et al. (1996). (d) Ranking with multivariate statistics in DEA: Wang et al. (2011) and Sinuany-Stern et al. (1996). (e) Ranking inefficient decision making units: Bardhan et al. (1996). (f) Multi-criteria decision-making models: Halme et al. (1999), Thanassoulis and Dyson (1992), Yu and Lee (2013).

The models proposed in DEA are all applicable for definite and deterministic inputs and output. However, in calculating the efficiency of units, the values can be non-deterministic or random. Huang and Li (1996) conducted extensive studies in the case of stochastic data. For uncertain situations, Kao and Liu (2019) and Lertworasirikul et al. (2003) utilized fuzzy theory, Despotis and Smirlis (2002) examined confidence interval, and Olesen (2006) investigated stochastic constraint programming. Bruni et al. (2009) proposed probabilistic joint constraints. Simar and Zelenyuk (2011) introduced spatial maximum likelihood estimate and Wong (2009) and Kuah et al. (2012) proposed a method for simulating Monte Carlo.

In the presence of random variables Land et al. (1993) used Chance-constrained programming (CCP) model proposed by Charnes and Cooper (1959) to calculate the efficiency. They considered joint normal distribution for output distribution and inputs of DMUs and developed random constraints for the model. Olesen and Petersen (1995) developed random constraints model in DEA in multiplier form. Cooper et al. (1996) introduced joint random constraints for DEA model. For more reference one can read: Cooper et al., 1998, Cooper et al., 2002, Cooper et al., 2004, Morita and Seiford (1999), Huang and Li (1996), Bruni et al. (2009), Wanke et al. (2016), Olesen and Petersen (2016), Dotoli et al. (2016), Branda and Kopa (2016), Simar et al. (2017), Zhou et al. (2017), Chen et al. (2017), Charles and Cornillier (2017), Liu et al., 2017a, Liu et al., 2017b, Jradi and Ruggiero (2019), Park et al. (2018) and Kao and Liu (2019).

The general attitude in the performance evaluation of units is to minimize the inputs and maximizing the outputs, as done in conventional CCR and BCC models. But it should be noted that organizations are not always looking to maximize output and minimize input because outputs and inputs can be desirable or undesirable. For example, the number of defective goods, or the amount of pollution and waste, or the release of CO2 in the production process are all undesirable which should be reduced. Accordingly, models with undesirable inputs/outputs should be considered. For instance, to assess energy efficiency of cement industry in India, Mandal (2010) showed that if undesirable output is overlooked, biased results are observed in efficiency calculations. Furthermore, Shi et al. (2010) utilized undesirable outputs for assessing energy efficiency in China manufacturing industry and Yeh et al. (2010) compared total factors of energy efficiency in China. Sueyoshi and Goto (2010) proposed an alternative perspective in DEA to measure the efficiency of electric fossil fuels taking into account the CO2 produced by production units. In addition, Liu et al. (2013) used DEA to study efficiency and environmental efficiency of national energy production. Jin et al. (2014) compared APEC member countries in terms of efficiency in Gross Domestic Product (GDP) considering undesirable stochastic input and output (i.e. CO2 production) with given risk. Wu et al. (2013) compared several provinces in China in terms of Gross Domestic Product (GDP) considering wasting water, emission of toxic gases, and the production of useless solid material as undesirable stochastic outputs with given error. Chen et al. (2017) used a stochastic network DEA model for Chinese airline efficiency under CO2 emissions and flight delays. Izadikhah and Saen (2018) proposed a chance-constrained two-stage DEA model in the presence of undesirable factors to evaluate the sustainability of supply chains. Liu et al. (2020) proposed a multi-attribute decision making based on stochastic DEA cross-efficiency with ordinal variable and its application to evaluation of banks’ sustainable development by considering undesirable outputs with weak disposability. Ren et al. (2020) proposed a measuring the energy and carbon emission efficiency of regional transportation systems in China by chance-constrained DEA models.

As mentioned above the DEA cross-efficiency model plays an important role in efficiency analysis and ranking priority. However, the researchers published numerous papers related to cross-efficiency model, but to best of our knowledge so far no one considered the stochastic cross-efficiency model in the presence of undesirable outputs for ranking priority and secondary goal setting in DEA. The contribution of this article is defining a new model for Stochastic Data Envelopment Analysis (SDEA) with undesirable outputs. Due to existence of alternative optimal solution in DEA-models, this study proposes a secondary goal as well.

This paper is organized as follow: In the following section, we first review input oriented-BCC (IBCC) model with undesirable outputs. Then, in the third section, we present the random form of IBCC model with risk α and a new model under the Expected Ranking Criterion is presented as well. In section four, given the non-uniqueness of the cross-efficiency optimal solution, using aggressive approach, we present a stochastic cross-efficiency model with respect to the defined models. In section five, to show applicability of the proposed models, we apply the models on a real case. The case is related to 32 thermal power plants located in Angola. Finally, conclusions are given in section six.

Section snippets

IBCC model with undesirable output

In this section, we briefly describe IBCC model with undesirable outputs in definitive mode. Then, a stochastic model with risk α is proposed for evaluating stochastic cross-efficiency with undesirable outputs. Assume there are n DMUs for evaluation, each with m inputs and s outputs. Inputs and outputs of DMUj (j=1,,n) are represented by xij (i=1,,m) and yrj (r=1,,s), respectively. Suppose DMUd is under evaluation, d1,,n, then the IBCC model is defined as follows: Edd=maxr=1sμrdyrd+ud

Stochastic form of proposed models

The proposed model (4) is unable to measure the efficiency of DMUs when input and output are randomly changed. Assume the stochastic variables x̃ij, ỹrj and z̃pj are inputs, desirable outputs and undesirable stochastic outputs, respectively, each with normal distribution. That is, x̃ijNx¯ij,σijx2i,jỹrjNy¯rj,σrjy2r,jz̃pjNz¯pj,σpjz2p,j j=VAR(x̃1j)COV(x̃1j,z̃kj)COV(z̃kj,x̃1j)VAR(z̃kj)

Besides, assume each DMU has the variance–covariance matrix , with major diagonal being the

The stochastic ranking priority model

It may that optimal solution of the model (8) not be unique and there exists alternative optimal solution. Hence the stochastic cross-efficiency scores will be somewhat arbitrary. To solve this problem, a new model is introduced to rank DMUs. This model not only maintains the stochastic efficiency values, but also increases the stochastic cross-efficiency. Rpd=minj=1nzj

s.t. i=1mωidx¯id=1 r=1sμrdy¯rd+p=1kφpdz¯pd+ud=Edd i=1mωidx¯ijr=1sμrdy¯rjp=1kφpdz¯pjud+AjΦ1α0j=1,,n

Aj2=i=1mk

Application

In this section, we apply the proposed models for 32 thermal power plants located in Angola. Angola thermal power generation plants are public plants located near cities that burn fuel to produce electricity for local city consumption. Each thermal power plant as a DMU has two input variables and three output variables, two of which are undesirable. The first input variable (x̃1) is the capacity of thermal power generation in terms of MW and the second input variable (x̃2) is the number of

Conclusion and further research directions

In this study we proposed a BCC-SDEA approach to efficiency evaluation in the presence of undesirable outputs. Given the non-uniqueness of optimal solutions, some aggressive-based models were proposed. Finally, implementing the proposed models for 32 thermal power plants showed the applicability of proposed approach. This research can be extended by considering at least the following three aspects. Firstly, the stochastic models proposed in this paper are based on radial models. They can be

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank two anonymous referees and the Editor-in-Chief Prof. Clevo Wilson for their helpful comments and suggestions which improved the first draft of this paper.

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