Abstract
This paper aims to contribute to the contemporary and imperative research on the performance and productivity growth of the oil industry. Among cutting edge methods, frontier analysis is a successful approach that has been widely used to assess the efficiency and productivity of entities with multiple resources and multiple outputs. This study first develops a unique framework based upon data envelopment analysis (DEA) to measure efficiency and productivity in the way that it tackles the uncertainty in data and undesirable outputs and, in turn, provides useful information to decision-makers. An adaptive robust optimisation is utilised to combat uncertain data whose distributions are unknown and consider the nexus between the level of conservatism and decision-makers’ risk preference. The key advantage of the proposed robust DEA approach is that the results remain relatively unchanged when uncertain conditions exist in the problem. An empirical study on the oil refinery is presented in situations of data uncertainty along with considering CO2 emissions as the undesirable output to conduct environmental efficiency and productivity analysis of the 25 countries over the period 2000–2018. The empirical results obtained from the proposed approach give some imperative implications. First, results show that the price of robustness does not affect identically for varying technologies when assessing productivity in a global oil market, and the USA oil industry is observed as the highest productivity growth in all cases confirming its efforts for the rapid rise in oil extraction and production at low costs. There may be practical lessons for other nations to learn from the USA oil industry to improve productivity. Findings also support a considerable regress during the 2008 Global Financial Crisis in the oil industry compared to the rest of the periods in question, and due to monetary and fiscal stimulus, there is a sharp productivity growth from 2009 to 2011. The other implication that can be drawn is that the GDP growth rate and technology innovation can more effectively improve the productivity of the oil industry across the globe.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Efficiency in global petroleum production has become an overriding issue of concern around the world due to extreme competition and the effects of global uncertainties such as the COVID-19 pandemic, sanctions on Iranian oil industry, the Russia–Ukraine war, and the US–China trade war. Needless to say, the oil industry is among the essential industries in the world today. It is the primary source of global energy supply and plays a significant role in long-run economic growth. According to the international labour organisation (ILO), roughly 6 million people are employed directly, and over 50 million are indirectly employed in the oil industry. The industry invariably faces multiple challenges and uncertainties associated with crude oil production and refining petroleum product at the lowest possible cost to retain market competitiveness. These uncertainties range from global demand, price volatility, increasing rise in a global pandemic, international cooperation, technological advantage, and increasing stagnant environmental controls. With this in mind, it is imperative to continuously measure the efficiency and productivity growth of petroleum production activities. Performance measurement brings the strength and vulnerabilities of a production system to the fore to ensure a wider and smarter system, encourage customer satisfaction, enhance internal and external relations, and direct the system objectively.
1.1 Performance and productivity assessment
Data envelopment analysis (DEA) is a non-parametric technique aiming to measure the relative efficiency of a group of production units with multiple inputs and multiple outputs. In comparison with parametric techniques such as statistical regression approaches, DEA does not require prescribing the functional forms of the relationship between inputs and outputs. CCR and BCC are two basic DEA models which are commonly used in the literature. CCR was initially introduced by Charnes et al. (1978) under the assumption of constant returns to scale (CRS), and afterwards, BCC was developed by Banker et al. (1984) based upon the assumption of variable returns to scale (VRS). Banker et al. (1984) also defined the scale efficiency (SE) as the ratio of the CCR efficiency to the BCC efficiency. DEA models are structurally classified into radial and non-radial models (see Cooper et al. (2007) for a detailed description).
Most traditional DEA models have been developed in static directions and the findings are set in stone, while many of the real-world problems require a dynamic setting. Therefore, the development of non-parametric DEA models with the dynamic (time-dependent) focus has jumped out at scholars and researchers. There are two streams of time-dependent DEA models in the existing literature to measure efficiency change over time. The first stream is called “window analysis” which was put forth by Klopp (1985) who was a chief statistician for the US Army Recruiting Command. The second stream is the Malmquist productivity index (MPI) originated by Malmquist (1953), and then, the MPI was extended to the non-parametric DEA framework by a plethora of researchers such as Caves et al. (1982) (hereafter CCD), Färe and Grosskopf (1992), Färe et al. (1994a), Färe et al. (1998) and Thrall (2000). The MPI represents the total factor productivity (TFP) growth of a given unit in a way that shows, simultaneously, the progress and regress in the efficiency and frontier technology between two periods of time where the production system includes the multiple inputs and multiple outputs. While there are a significant number of MPI approaches in the related literature, we here intend to provide an overview of the key attempts only.
Färe et al. (1994b) utilised the input and output-oriented radial DEA model to compute the MPI. Despite the usefulness and popularity of the radial approach in many applications, it has been criticised for neglecting the non-radial input/output slacks (Charnes et al. 1985). To overcome this problem, the non-radial approach was introduced to consider the slacks and allow inputs and outputs to change in different scales (see e.g. Tone 2001, 2002). In this respect, Tone (2004) developed an input-oriented SBM (slacks-based measure) model in view of Tone (2001, 2002)’s models and then computed the MPI and its components to attend to the foregoing trouble in measuring productivity growth. Since the change in efficiency and productivity may be affected by scale economies, many researchers have made attempts to enhance the MPI by looking into the effects of scale efficiency change (see e.g. Färe et al. 1994a; Grifell-Tatjé and Lovell 1999; Ray and Desli 1997; and Balk 2001).
A new presentation of the MPI was suggested by Chen and Ali (2004) to aid in revealing more practical information for managerial implications where each of the individual components was studied on its own. To be more specific, the Malmquist approach of Chen and Ali (2004) has the capability to identify the strategic shift of each firm in a particular time period and help the assessor to examine if the strategic shift is desirable or a course of action is of the essence to achieve the organisational objectives. The MPI is based upon the geometric mean of adjacent period measures leading to three typical problems; (i) the lack of circularity, (ii) two different measures of productivity change for the adjacent periods, and (iii) linear programming (LP) infeasibility (Pastor and Lovell 2005). A new global Malmquist productivity index (GMPI) developed by Pastor and Lovell (2005) aims to attend to the foregoing problems. In other words, the GMPI and its components are circular and provide single measures of productivity change and its components, as well as not being suffered from LP infeasibility. The aggregation of MPI measures developed by Zelenyuk (2006) is an interesting topic from the practical aspect aiming to aggregate MPIs over individual units into a group MPI. This area of research has been extended by Mayer and Zelenyuk (2014) to aggregate MPIs, which allow inputs to be reallocated within the group when the output orientation is considered.
Pastor et al. (2011) introduced a biennial MPI (BMPI) that takes advantage of three important properties; (i) being immune from LP infeasibility under the VRS technology, (ii) allowing for technical regress, and (iii) being immune against a substantial change in productivity when a new time period is included in the data set. With these advantages in mind, some researchers have started using the BMPI to measure the efficiency and productivity growth in real-world applications (see e.g. Kao 2017; Zhu et al. 2020). There is a snag in the Malmquist–Luenberger index decomposition, which may lead to inconsistent and questionable productivity measures in a given technology, in particular, when undesirable outputs are available. To circumvent this problem, Aparicio et al. (2013) considered a new postulate into the technology associated with the production of undesirable outputs. Some studies have been conducted to measure the MPI in the area of allocative efficiency with the aim of showing how DEA-MPI can be used to analyse productivity change when information on prices and costs are known precisely (see e.g. Maniadakis and Thanassoulis 2004; Tohidi et al. 2012; Thanassoulis et al. 2015 and Walheer 2018).
Karagiannis and Lovell (2016) proposed the MPI based on radial DEA models with a single constant input and compared it with the Hicks–Moorsteen productivity index. It was concluded that the Malmquist and Hicks–Moorsteen productivity indices coincide regardless of the structure of technology and the orientation of the MPI analyses. Of late, Pastor et al. (2020) used a proportional directional distance function to define the CCD Malmquist index as a productivity index. Besides, a new decomposition of the CCD Malmquist index was developed showing productivity change as the ratio of “productivity change due to output change” to “productivity change due to input change”.
1.2 Environmental efficiency
Many existing studies have been used DEA as a powerful tool for assessing the operational, productivity, and environmental performance of private and public energy organisations at the regional, national, and international levels. In this regard, there are some comprehensive surveys on energy productivity and efficiency in the literature that review related studies, give further insight, and raise major challenges. Sueyoshi et al. (2017) reviewed around 700 DEA papers from the 1980s to 2010s on energy and environment issues and classified them based on applications. Mardani et al. (2017) reviewed and summarised 144 published studies over 10 years from 2006 to 2015 to show the capabilities of DEA models in energy efficiency evaluation. Recently, Yu and He (2020) provided a bibliometric study on energy efficiency based on DEA to provide a comprehensive perspective of this area and its applicability.
Here we intend to emphasise the literature concerning DEA applications in the oil and gas industries. In doing so, we outline the key characteristics of pertinent studies in Electronic Supplementary Materials. The scholars applied various DEA models to assess the oil and gas performance over different periods of years. Amongst them, original CCR and BCC models have been widely used and recognised to be helpful and insightful for decision-makers and policy-makers (see e.g. Thompson et al. 1992, 1994; Bevilacqua and Braglia 2002; Mekaroonreung and Johnson 2010; Vikas and Bansal 2019; Atris 2020; Dalei and Joshi 2020). It is worth mentioning that several non-radial DEA models such as slack based measure (SBM), range-adjusted measure (RAM), and inverse DEA were employed to assess the oil and gas industries’ performance (see e.g. Sueyoshi and Goto 2012a, 2012b, 2015; Sueyoshi and Wang 2014, 2018; Hosseini and Stefaniec 2019; Wegener and Amin 2019). Despite the efficiency measurement, many energy-based studies attempted to assess productivity growth over time (see e.g. Managi et al. 2004; Barros and Managi 2009; Barros and Antunes 2014; Ike and Lee 2014; Sueyoshi and Goto 2015; Sheng et al. 2015; Wang et al. 2019; Tavana et al. 2020; Tachega et al. 2021). Of late, the structural complexity of the oil and gas industry has received some attention, and hence a few papers used network DEA models to combat this type of complexity (see e.g. Tavana et al. 2019, 2020; Atris and Goto 2019; Nemati et al. 2020).
Having looked at the above literature review, several studies on the oil refinery industry used DEA and MPI to assess environmental efficiency and productivity by considering greenhouse gas emissions. Bevilacqua and Braglia (2002) developed a DEA model in the presence of six different types of gas emissions to evaluate the environmental efficiency of the seven oil refinery companies in Italy. Mekaroonreung and Johnson (2010) measured the efficiency of 113 oil refineries in the USA by taking into account toxic releases as undesirable output in the production process. Sueyoshi and Goto (2012a, 2012b, 2015) evaluated the environmental assessment of several local and international oil companies from 2005 to 2009 in which CO2 emissions were defined as an undesirable output in the model. Sueyoshi and Wang (2014, 2018) leveraged DEA assessment methods to help managers of petroleum firms in the USA as to how they can invest in the production process to decrease the impact of undesirable outputs such as CO2 emissions and improve their relative efficiency simultaneously. Hosseini and Stefaniec (2019) used mazut as an undesirable output in a dynamic network DEA model to assess the efficiency of the petroleum refinery industry in Iran. Atris and Goto (2019) examined operational and environmental efficiency measures to evaluate 34 USA oil and gas companies. Wegener and Amin (2019) first developed an inverse DEA model to minimise GHG emissions and then applied their model to oil and gas companies in the USA and Canada. Recently, Tachega et al. (2021) assessed the energy efficiency and productivity of 14 oil-producing countries in Africa using an SBM-DEA model and MPI, respectively, and their model includes both economical (GDP) and environmental (CO2 emission) factors.
1.3 Uncertainty
The related literature in the previous subsections considers situations where input and output data are deterministic while uncertainty and incomplete data are an unavoidable property of numerous real-life problems. For instance, regulatory agencies and corporations in the oil industry mostly rely on accurate performance measurement to help increase oil revenue and mitigate business risks and reduce greenhouse gas emissions. However, the inherent uncertainty in view of data fluctuations is often observable when assessing the performance of oil firms and, consequently, finding the best practice frontier (i.e. efficient firms) might be confusing and untrue. Needless to say, data uncertainty which is the inevitable feature of most business environments affects improvement decisions made by the top management team and it is of importance to deal with uncertainty in real-world applications; particularly in this study the aim is to attend to the problem of uncertainty in data when measuring the performance and productivity growth of the oil industry.
The pertinent literature has received scant attention to the question of how to deal with uncertain situations in measuring MPI. In the context of DEA, there are various approaches to overcome data uncertainty which can be classified into imprecise DEA (see e.g. Cooper et al. 1999; Hatami-Marbini et al. 2014, 2018; Toloo et al. 2021), stochastic DEA (see e.g. Olesen and Petersen 1995, 2016; Tavana et al. 2014; Chen and Zhu 2019), bootstrap DEA (see e.g. Simar and Wilson 1998; 1999; Bădin et al. 2019), fuzzy DEA (see e.g. Hatami-Marbini et al. 2011, 2017; Hatami-Marbini and Saati 2018), robust DEA approaches (see e.g. Shokouhi et al. 2010; Hatami-Marbini and Arabmaldar 2021).
Imprecise DEA received a flurry of interest from its introduction by Cooper et al. (1999) where the uncertain data are characterised by bounded intervals. The main drawback of imprecise DEA methods is that the upper and lower efficiency bounds are merely determined and no information within the efficiency interval is provided (Shokouhi et al. 2010). A stochastic DEA approach is based upon stochastic optimisation to deal with stochastic uncertainty. This approach necessitates specifying a probability distribution function for the noisy data although in real-world problems this assumption may not duly work since there is not ample empirical evidence to choose the specified distribution function. In addition, it is arduous to apply stochastic optimisation in DEA when the sample size is small (Sengupta 1992). Bootstrap DEA originated by Simar and Wilson (1998) examines the statistical propertiesFootnote 1 of efficiency measures estimated by DEA models using bootstrap algorithms. In this approach, there are two main difficulties: (i) finding a suitable value of the smoothing parameter and (ii) the number of iterations required in this algorithm (Daraio and Simar 2007). Fuzzy DEA handles vague and ambiguous data such as linguistic variables, and the existing methods can be grouped into (i) tolerance approach, (ii) \(\alpha\)-cut-based approach, (iii) fuzzy ranking approach, and (iv) possibility approach (Hatami-Marbini et al. 2011). However, the applicability of fuzzy DEA is often questionable since there is no clear way for defining the membership functions of fuzzy inputs and outputs as well as some extant theoretical and computational complexity in fuzzy mathematical programming problems (Hatami-Marbini et al. 2011). The robust DEA approach is used in this paper to deal with some drawbacks of the aforesaid methods. A concise literature review for the robust DEA models is provided in Section 3.
Let us ultimately pay special attention to the trend of develo** MPI under uncertainty and review the related papers in DEA used in oil industries in the presence of uncertainty (see Table 1). Emrouznejad et al. (2011) focused on situations where imprecise input and output data are characterised by fuzzy numbers or vary within intervals. They re-formulated the conventional profit MPI problem in the presence of imprecise data and proposed two novel methods for measuring the overall profit MPI. Furthermore, Hatami-Marbini et al. (2012) applied a new productivity measurement approach with uncertain data to measure the MPI of hospitals for a State Office of Inspector General in the USA. Kevork et al. (2017) explored the probabilistic version of directional distance functions introduced by Daraio and Simar (2014) to first measure the productivity index and its components and then compare it with traditional technical, pure and scale efficiency. To the best of our knowledge, the MPI studies under uncertainty are hedged in the DEA literature and received insufficient attention from both theoretical and practical viewpoints. There are also a few studies in which uncertain inputs and outputs data are assumed to be involved in the applications associated with the oil industry (Sueyoshi 2000; Eller et al. 2011; Tavana et al. 2019; Al-Mana et al. 2020). Sueyoshi (2000) presented a stochastic DEA model to plan the restructuring strategy of a Japanese petroleum company. Eller et al. (2011) applied DEA and stochastic frontier analysis (SFA) to a panel of 78 national oil companies (NOCs) and private international oil companies (IOCs) to explore the inefficiency sources based upon empirical evidence on the revenue efficiency. Tavana et al. (2019) introduced a novel fuzzy multi-objective multi-period network DEA model that was aimed to assess the dynamic efficiency of oil refineries when undesirable outputs are available. To compare the efficiency of NOCs and IOCs, Al-Mana et al. (2020) employed an integrated framework containing four methods: SFA, DEA, financial, and operational analysis to ensure the validity of the obtained results as well as providing managerial implications. Table 1 shows a summary of those papers in the oil industry using DEA and MPI to measure economic efficiency and productivity in the presence of uncertainty.
1.4 Contributions
The above literature review shows that measuring productivity growth in the oil industry under uncertainty has been massively neglected due to its complexity. This paper aims to focus on the productivity growth in oil refineries when uncertainty is a matter of concern and, in turn, a suitable remedy requires. We use the RO approach to model data uncertainty and undesirable outputs in DEA models when measuring the BMPI. The key contributions of the paper are sevenfold: (i) presenting a new method in the presence of undesirable outputs for measuring the productivity growth under uncertain environments in which the proposed models are feasible with high probability, (ii) decision-makers can choose their risk preference by controlling the level of conservatism, (iii) introducing the concept of price of robustness and finding out the refineries are more robust against different levels of perturbations, (iv) the proposed robust BMPI circumvents LP infeasibilities under the VRS technology, and it does not need to make re-calculation when a new time period is added to the data set, (v) identifying and evaluating the main factors in order to measure the productivity growth of the oil refineries when dealing with uncertainty and undesirable outputs are inevitable, (vi) selecting the more appropriate inputs and outputs to present the production function considering both economical and environmental factors, and (vii) applying the proposed method to investigate efficiencies and productivities of the oil refining industry in terms of different levels of perturbations in the inputs, desirable and undesirable outputs data over the subsequent 10 years.
1.5 Organisation
The rest of the paper is organised as follows. Section 2 first reviews the basic concepts of DEA models, the MPI and BMPI approaches and then presents an extension of BMPI in the presence of undesirable outputs. The effect of data uncertainty on the BMPI is studied and, in turn, the mathematical details of the proposed robust BMPI are presented in Section 3. Section 4 examines the applicability of the proposed robust BMPI method using a real case study of the 25 oil refineries. Conclusions and future research directions are discussed in Section 5.
2 Background
This section first provides an overview of the DEA-based models for measuring efficiency and productivity and then presents the BMPI with undesirable outputs.
2.1 Measuring efficiency and productivity
Amongst non-parametric frontier techniques, DEA is a well-established tool to measure the efficiency and productivity of firms. DEA has been widely used to measure the technical efficiency and productivity of a set of firms over time. Based upon the ideas of Farrell (1957) and Caves et al. (1982), the original study for a Malmquist index of productivity change was carried out by Färe and Grosskopf (1992) and Färe et al. (1994a) with the aim of measuring the productivity and efficiency of a firm.
Suppose that \({\varvec{x}}^{k}\) and \({\varvec{y}}^{k}\) denote the input and output vectors, respectively, in which the \(j\)th firm \(\left( {j = 1, \ldots ,n} \right)\) transforms \(m\) inputs denoted by a vector \({\varvec{x}}_{j}^{k} = \left( {x_{1j}^{k} ,x_{2j}^{k} , \ldots ,x_{mj}^{k} } \right)^{T} \in R^{m}\) into \(s\) outputs denoted by a vector \({\varvec{y}}_{j}^{k} = \left( {y_{1j}^{k} ,y_{2j}^{k} , \ldots ,y_{sj}^{k} } \right)^{T} \in R^{s}\) over period \(k \left( {k = t,\;t + 1} \right)\). The reference set of n observations is represented in the matrices \({\varvec{X}}^{k} = \left[ {{\varvec{x}}_{1}^{k} ,{\varvec{x}}_{2}^{k} \ldots ,{\varvec{x}}_{n}^{k} \left] { \in {\mathbb{R}}^{m \times n} , {\varvec{Y}}^{k} = } \right[{\varvec{y}}_{1}^{k} ,{\varvec{y}}_{2}^{k} \ldots ,{\varvec{y}}_{n}^{k} } \right] \in {\mathbb{R}}^{s \times n}\) where \({\varvec{x}}_{j}^{k} > 0\) and \({\varvec{y}}_{j}^{k} > 0\) for all firms. The production possibility, \(\psi^{k}\), can be defined as follows:
.
The Malmquist index includes a focus on evaluating the productivity change of a firm between two subsequent time periods. Referring to Caves et al. (1982), the MPI for the \(o\)th firm, can be expressed as follows:
where \(E^{t} \left( {{\varvec{x}}_{o}^{t} , {\varvec{y}}_{o}^{t} } \right)\) and \(E^{t + 1} \left( {{\varvec{x}}_{o}^{t + 1} , {\varvec{y}}_{o}^{t + 1} } \right)\) represent the (Farrell) technical efficiency at t and t + 1, respectively, and \(E^{t} \left( {{\varvec{x}}_{o}^{t + 1} , {\varvec{y}}_{o}^{t + 1} } \right)\) and \(E^{t + 1} \left( {{\varvec{x}}_{o}^{t} , {\varvec{y}}_{o}^{t} } \right)\) are the maximum (intertemporal) proportional change to make \(\left( {{\varvec{x}}_{o}^{t + 1} , {\varvec{y}}_{o}^{t + 1} } \right)\) and \(\left( {{\varvec{x}}_{o}^{t} , {\varvec{y}}_{o}^{t} } \right)\) feasible at t and t + 1, respectively. As an alternative way, the MPI can be decomposed into the product of Catch-up (CU) and Frontier shift (FS) (Färe et al. 1994a):
\({\text{MPI}}_{o}^{F} = \left[ {\overbrace {{\left[ {\frac{{E^{t + 1} \left( {{\varvec{x}}_{o}^{t + 1} , {\varvec{y}}_{o}^{t + 1} } \right)}}{{E^{t} \left( {{\varvec{x}}_{o}^{t} , {\varvec{y}}_{o}^{t} } \right)}}} \right]}}^{{\left( {\text{Catch - up}} \right)}} \times \overbrace {{\left[ {\frac{{E^{t} \left( {{\varvec{x}}_{o}^{t} , {\varvec{y}}_{o}^{t} } \right)}}{{E^{t + 1} \left( {{\varvec{x}}_{o}^{t} , {\varvec{y}}_{o}^{t} } \right)}} \times \frac{{E^{t} \left( {{\varvec{x}}_{o}^{t + 1} , {\varvec{y}}_{o}^{t + 1} } \right)}}{{E^{t + 1} \left( {{\varvec{x}}_{o}^{t + 1} , {\varvec{y}}_{o}^{t + 1} } \right)}}} \right]^{0.5} }}^{{\left( {\text{Frontier - shift}} \right)}}} \right]\).
The extent to which a firm improves, stagnates, or deteriorates its technical efficiency describes the CU component and the extent to which the efficient frontier of a firm shift indicates the FS component. The MPI value of a firm helps describe the productivity status. There is progress, no change or regress in the total factor productivity from period \(t\) to \(t + 1\) provided that the MPI value is greater than, equal to, or less than the unity, respectively.
Despite returns to scale, Färe et al. (1994a)’s approach exploits the radial DEA models to capture the MPIs. Several studies such as Ray and Desli (1997), Grifell-Tatjé and Lovell (1999) and Balk (2001) strived to refine the MPI by taking the scale efficiency change into account. One influential effort is concerned with Ray and Desli (1997)’s approach to deal with a problem of internal consistency when using CRS and VRS within the same decomposition. The decomposition of Ray and Desli (1997) uses a VRS frontier and includes one additional component, namely scale efficiency (SE) in comparison with Färe et al. (1994a)’s approach as presented below:
where \({\text{SE}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} } \right) = \frac{{E_{{{\text{crs}}}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} } \right)}}{{E_{{{\text{vrs}}}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} } \right)}}, k = t, t + 1\) and \({\text{SE}}^{k} \left( {{\varvec{x}}_{o}^{l} , {\varvec{y}}_{o}^{l} } \right) = \frac{{E_{{{\text{crs}}}}^{k} \left( {{\varvec{x}}_{o}^{l} , {\varvec{y}}_{o}^{l} } \right)}}{{E_{{{\text{vrs}}}}^{k} \left( {{\varvec{x}}_{o}^{l} , {\varvec{y}}_{o}^{l} } \right)}}, k,l = t, t + 1, k \ne l\). With a reference technology exhibiting CRS, the [radial] technical efficiency \(E_{{{\text{crs}}}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} } \right)\) in adjacent periods \(k = t, \;t + 1\) for firm \(o = 1,2, \ldots ,n\) is calculated as:
\({\varvec{v}} = \left( {v_{1} , \ldots ,v_{m} } \right)\) and \({\varvec{u}} = \left( {u_{1} , \ldots ,u_{s} } \right)\) denote the input weights and output weights, respectively. It also needs to estimate the [radial] technical efficiency, \(E_{{{\text{crs}}}}^{k} \left( {X_{o}^{l} , Y_{o}^{l} } \right)\), for the cross-time periods, \(k,l = t,\; t + 1, k \ne l\) as expressed below:
If a free variable \(u_{0}\) is added to the objective function and common constraints of models (2) and (3), \(E_{{{\text{vrs}}}}^{k} \left( {{\varvec{x}}_{o}^{k} ,{ }{\varvec{y}}_{o}^{k} } \right)\), \({ }k = t, t + 1\) and \(E_{{{\text{vrs}}}}^{k} \left( {{\varvec{x}}_{o}^{l} , {\varvec{y}}_{o}^{l} } \right),k,l = t, t + 1, k \ne l{ }\) can be, respectively, computed with respect to the assumption of VRS.
The above decomposition (1) plausibly suffers from the infeasibility problem when calculating mixed-period efficiencies (see e.g. Ray and Desli 1997; Balk 2001; Balk and Zofio 2018). The literature includes some efforts to avoid this infeasibility problem such as the sequential Malmquist index (Shestalova 2003), the GMPI (Pastor and Lovell 2005) and the BMPI (Pastor et al. 2011). The main pitfall of the sequential Malmquist index is concerned with the lack of capabilities to identify technical regress of productivity. Noticeably, the GMPI needs to be recalculated when adding a new time period to the data set, and this method is computationally uneconomic in practice. The BMPI presently comes to the fore in the literature to circumvent the above-mentioned drawbacks and this paper aims to focus on this well-developed approach. Let us here give a brief review of the BMPI.
Given two technologies \(\psi^{k} \left( {k = t, t + 1} \right)\), the biennial technology \(\psi_{B}^{k}\) is defined as the convex hull of the period \(t\) and \(t + 1\) technologies, i.e. \(\psi_{B}^{k} = {\text{conv}}\left\{ {\psi^{k} \left( {k = t, t + 1} \right)} \right\}\), and consequently, the decomposition of the BMPI is thus given as follows:
where \(E_{{{\text{vrs}}}}^{B} \left( {{\varvec{x}}_{o}^{t} , {\varvec{y}}_{o}^{t} } \right)\) measures the technical efficiency of firms in period t using the biennial technology under the VRS technology.
2.2 Measuring productivity with undesirable outputs
In many production settings, undesirable (environmental) outputs such as pollutants, noise or wastes in addition to desirable outputs are generated from inputs, and, in such situations, undesirable outputs need to be reduced to improve the production unit’s efficiency.
There are three major DEA approaches to deal with undesirable outputs. The first approach is to assume weak disposability for desirable and undesirable outputs in a production process implying that desirable and undesirable outputs can be decreased simultaneously (Färe et al. 1989; Kuosmanen 2005). The second approach is to consider the undesirable outputs as inputs while this change does not preserve the original production process (Korhonen and Luptacik 2004). The third approach is to obtain the new adjusted values for undesirable outputs by applying a monotone decreasing transformation (Ramanathan 2006). The current study leverages the linear monotone decreasing transformation proposed by Seiford and Zhu (2002) to handle undesirable outputs since this type of transformation preserves the convexity relations (strong disposability).
Suppose that each production unit \(\left( {j = 1, \ldots ,n} \right)\) uses \(m\) inputs \({\varvec{x}}_{j}^{k} = \left( {x_{1j}^{k} ,x_{2j}^{k} , \ldots ,x_{mj}^{k} } \right)^{T} \in R^{m}\) to produce \(s\) desirable (good) outputs \({\varvec{y}}_{j}^{k} = \left( {y_{1j}^{k} ,y_{2j}^{k} , \ldots ,y_{sj}^{k} } \right)^{T} \in R^{s}\) and \(d\) undesirable (bad) outputs \({\varvec{b}}_{j}^{k} = \left( {b_{1j}^{k} ,b_{2j}^{k} , \ldots ,b_{dj}^{k} } \right)^{T} \in R^{d}\). Following Seiford and Zhu (2002)’s approach, we multiply each undesirable output by \(\left( { - 1} \right)\) and then by finding a proper translation vector \({\varvec{c}}\) the negative value of the undesirable output is transformed into the non-negative one, i.e. \(\overline{\varvec{b}}_{j}^{k} = - {\varvec{b}}_{j}^{k} + {\varvec{c}} > 0\) where \({\varvec{c}} > \mathop {\max }\limits_{1 \le j \le n} \left( {{\varvec{b}}_{j}^{k} } \right)\). Resultantly, models (2) and (3) are transformed into the following models:
and
where \({\varvec{v}} = \left( {v_{1} , \ldots ,v_{m} } \right)\), \({\varvec{u}} = \left( {u_{1} , \ldots ,u_{s} } \right)\) and \({\varvec{w}} = \left( {w_{1} , \ldots ,w_{d} } \right)\) are the weights assigned to inputs, desirable and undesirable outputs, respectively. Therefore, in situations of undesirable outputs the BMPI decomposition (4) can be extended as follows:
3 Uncertain data in the Malmquist index
The inherent presence of uncertainty in data might affect the efficiency and productivity performance of production units. As minutely discussed by Ehrgott et al. (2018), it is essential to assess the performance analysis without closing the eyes to uncertainty in input and output data. Two main reasons inspire us to investigate original DEA models under uncertainty: (i) sensitivity of the efficient frontier leading to the dislocation of the enveloped frontier as a result of the uncertain data, and (ii) small data perturbation often gives rise to changes of the unit’s efficiencies since the idea of DEA models is to measure efficiency relative to the benchmark frontier. Therefore, it is of importance to deem data uncertainty when analysing productivity growth using BMPI. Put differently, the decision-maker would be confident about results under uncertainty to make decisions about the performance improvement across time and take appropriate actions.
The RO approach has been initially put forth to treat uncertainties in the input data by way of classical engineering problems. The seminal work dates back to 1973 when Soyster (1973) developed the mathematical models to immunise column-wise uncertainty in LP problems where uncertain parameters are restricted to a convex set. Soyster’s robust model is linear as well as being too conservatism because the worst-case value of each uncertain parameter is considered. An ellipsoidal set that solves the robust counterparts as a conic quadratic program deals with the over-conservatism issue of robust solutions and a user-defined adjustment parameter is devised to control the trade-off between conservativeness and performance of the subsequent robust counterpart (Ben-Tal and Nemirovski 1998, 2000). Alternatively, uncertainty can be regarded as a polyhedral set through the RO approach which is a vital way in the literature to immunise real-world problems against uncertainty (Bertsimas and Sim 2004). Together with the widespread use of the RO approach in diverse fields, several studies have been carried out to tackle the problems of uncertainty in DEA (see e.g. Shokouhi et al. 2010, 2014; Sadjadi et al. 2011; Omrani 2013; Lu 2015; Arabmaldar et al. 2017; Tavana et al. 2021; Arabmaldar et al. 2021; Hatami-Marbini and Arabmaldar 2021;). This strand of DEA models is generally based on the three RO approaches proposed by Mulvey et al. (1995), Ben-Tal and Nemirovski (2000) and Bertsimas and Sim (2004). The current paper draws on Bertsimas and Sim (2004) to measure efficiency and productivity in ways that attend to the uncertainty and provide useful information to decision-makers. Some features of RO approaches come to the fore in comparison with existing uncertainty approaches: (i) there is no need to make predefined assumptions on probability distributions of uncertain inputs and outputs data, (ii) it is an established and fast-growing approach to identify the solutions that remain feasible with high probability under data perturbations, (iii) it enables managers to make a trade-off between the level of conservatism and the price of achieving robustness, (iv) it provides additional insights to complement sensitivity analysis and stochastic programming, (v) Bertsimas and Sim (2004)’s robust approach, in particular, preserves the class of models under analysis and the robust counterpart models are thus tractable that is beneficial to large scale problems. These merits motivate us to focus on the RO approach and explore the BMPI under uncertainty. In this section, we first provide a simple example for two adjacent periods to delineate our motivation and some key ingredients of this study in situations of uncertainty and then present a robust BMPI with our emphasis on dealing with inherent uncertainties.
3.1 An illustrative example
Let us give a small example to observe the effect of uncertain data when measuring the BMPI. Suppose that there are three units labelled as A, B and C which use one input to produce one output for two adjacent periods (\(t = 1,{ }2\)). It is also assumed that some data are changed somewhat in light of uncertainty. Table 2 shows the original and amended input and output data.
The input–output production diagram in the adjacent periods is depicted in Fig. 1 under the CRS and VRS technologies. In the case of exact data, the CRS and VRS frontiers for periods 1 and 2 are indicated by the solid blue lines (\({\text{CRS}}_{1} ,{\text{ VRS}}_{1}\)) and the solid black lines (\({\text{CRS}}_{2} ,{\text{ VRS}}_{2}\)), respectively. In the situation of uncertain data, the CRS and VRS frontiers are presented by the dashed blue lines (\({\text{CRS}}_{1}^{\prime } , {\text{VRS}}_{1}^{\prime }\)) and the dashed black lines (\({\text{CRS}}_{2}^{\prime } , {\text{VRS}}_{2}^{\prime }\)), respectively. It should be noted that the biennial CRS and VRS frontiers coincide with those of period 2 in both exact and uncertain conditions. Figure 1 shows that the CRS and VRS frontiers are changed when input and output data are subject to uncertainty. As shown in Fig. 1, A, for example, remains efficient in period 2 for both CRS and VRS technologies even if the uncertainty matters. However, A which is not efficient in period 1 under the CRS technology is moved onto the efficient frontier in the uncertain situation. Therefore, the identification of the inefficient units in comparison with the best practice frontier and the right amount for improving the inefficient units turns out to be a complicated process.
Frontiers of exact and uncertain data under the CRS and VRS technologies in periods 1 and 2
Table 3 summarises the productivity growth and its related components when the input and output data are certain and uncertain (in parentheses). Since all three components of BMPI are constructed based upon comparing the units with regard to different frontiers in periods 1 and 2, it is envisaged that a small perturbation in input and/or output data can affect the values of BMPI’s components and, resultantly, managers might face with quite difficult conditions to make a correct decision. To attend to this problem, the next section aims to focus on the RO approach to immunise data against a prescribed uncertainty set, and the productivity analysis based on the BMPI is robust and reliable from a practical viewpoint.
3.2 The proposed robust BMPI with undesirable outputs
Assume that the true values of the uncertain input, desirable and undesirable outputs data (\(\tilde{\varvec{x}}_{j}^{k} ,\tilde{\varvec{y}}_{j}^{k} ,\widetilde{{\overline{\varvec{b}}}}_{j}^{k}\)) for two subsequent time periods, \(k = t,\; t + 1\), are expressed as \(\tilde{x}_{ij}^{k} = x_{ij}^{k} + \xi_{ij}^{{x^{k} }} \hat{x}_{j}^{k}\), \(\tilde{y}_{rj}^{k} = y_{rj}^{k} + \xi_{rj}^{{y^{k} }} \hat{y}_{rj}^{k}\) and \(\widetilde{{\overline{b}}}_{hj}^{k} = \overline{b}_{hj}^{k} + \xi_{hj}^{{\overline{b}^{k} }} \widehat{{\overline{b}}}_{hj}^{k}\) where \(\hat{x}_{ij}^{k} = ex_{ij}^{k}\), \(\hat{y}_{rj}^{k} = ey_{rj}^{k}\), and \(\widehat{{\overline{b}}}_{hj}^{k} = e\overline{b}_{hj}^{k}\) represent the maximum deviations. Note that \(e\) is the perturbation percentage showing the amount of deviation between the uncertain data and their true values. The true values are modelled as variables \(\tilde{x}_{ij}^{k}\), \(\tilde{y}_{rj}^{k}\), and \(\widehat{{\overline{b}}}_{hj}^{k}\) taking values in the symmetric intervals \(\left[ {x_{ij}^{k} - \hat{x}_{ij}^{k} , x_{ij}^{k} + \hat{x}_{ij}^{k} } \right]\), \(\left[ {y_{rj}^{k} - \hat{y}_{rj}^{k} , y_{rj}^{k} + \hat{y}_{rj}^{k} } \right]\), and \(\left[ {\overline{b}_{hj}^{k} - \widehat{{\overline{b}}}_{hj}^{k} , \overline{b}_{hj}^{k} + \widehat{{\overline{b}}}_{hj}^{k} } \right]\), respectively. In the view of uncertain input, desirable, and undesirable output data \(\tilde{x}_{ij}^{k}\), \(\tilde{y}_{rj}^{k}\), and \(\widetilde{{\overline{b}}}_{hj}^{k}\), we define the random variables \(\xi_{ij}^{{x^{k} }} = \left( {\tilde{x}_{ij}^{k} - x_{ij}^{k} } \right)/\hat{x}_{ij}^{k}\), \(\xi_{rj}^{{y^{k} }} = \left( {\tilde{y}_{rj}^{k} - y_{rj}^{k} } \right)/\hat{y}_{rj}^{k}\), and \(\xi_{hj}^{{b^{k} }} = \left( {\widetilde{{\overline{b}}}_{hj}^{k} - {{\overline{b}}}_{hj}^{k} } \right)/\widehat{{\overline{b}}}_{rj}^{k}\) which are unknowingly distributed but symmetrically in the interval [− 1, 1], respectively (Bertsimas and Sim 2004). This study uses notations \(I_{j}\), \(R_{j}\) and \(B_{j}\) to present, respectively, the set of inputs, desirable, and undesirable outputs of the jth firm that are subject to uncertainty, i.e. \(I_{j} = \left\{ {i{|}\hat{x}_{ij}^{k} > 0} \right\}, \forall ,\) \(R_{j} = \left\{ {r{|}\hat{y}_{rj}^{k} > 0} \right\}, \forall j\), and \(B_{j} = \left\{ {h{|}\widehat{{\overline{b}}}_{hj}^{k} > 0} \right\}, \forall j\). Thereby, a firm \(\left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} ,\overline{\varvec{b}}_{o}^{k} } \right)\) is uncertain if there exists \(i \in I_{o}\) or \(r \in R_{o}\) or \(h \in B_{o}\). A degree of control is allowed to prevent over-conservative solutions. This control is given rise to by imposing a so-called budget of uncertainty as \(\Gamma_{j}^{{x^{k} }} \in \left[ {0, \left| {I_{j} } \right|} \right]\), \(\Gamma_{j}^{{y^{k} }} \in \left[ {0, \left| {R_{j} } \right|} \right]\), and \(\Gamma_{j}^{{\overline{b}^{k} }} \in \left[ {0, \left| {B_{j} } \right|} \right]\) for input, desirable, and undesirable output parameters, respectively, ascertaining that the deviation from the true values is only achieved by some uncertain parameters onlyFootnote 2. To adjust the level of robustness, let \(\mathop \sum \limits_{{i \in I_{j} }} \left| {\xi_{ij}^{{x^{k} }} } \right| \le \Gamma_{j}^{{x^{k} }} \le m\), \(\mathop \sum \limits_{{r \in R_{j} }} \left| {\xi_{rj}^{{y^{k} }} } \right| \le \Gamma_{j}^{{y^{k} }} \le s\), \(\mathop \sum \limits_{{h \in B_{j} }} \left| {\xi_{hj}^{{\overline{b}^{k} }} } \right| \le \Gamma_{j}^{{\overline{b}^{k} }} \le d\). Accordingly, the budget of uncertainty setsFootnote 3 in relation to inputs, desirable and undesirable outputs is, respectively, defined as follows:
The above sets have attracted numerous researchers because of their feature for remaining the linearity of the problems.
Let us now translate the robust approach into models (5) and (6). Given \(\mathop \sum \limits_{r = 1}^{s} u_{r} y_{rj}^{k} + \mathop \sum \limits_{h = 1}^{d} w_{d} \overline{b}_{hj}^{k} - \mathop \sum \limits_{i = 1}^{m} v_{i} x_{ij}^{k} \le 0, \forall j\), the uncertainty set \(\mathcal{U}_{{\Gamma_{j}^{{x^{k} }} + \Gamma_{j}^{{y^{k} }} + \Gamma_{j}^{{\overline{b}^{k} }} = \Gamma_{j} }}\) is defined as \(\mathop \sum \limits_{{i \in I_{j} }} \left| {\xi_{ij}^{{x^{k} }} } \right| + \mathop \sum \limits_{{r \in R_{j} }} \left| {\xi_{rj}^{{y^{k} }} } \right| + \mathop \sum \limits_{{h \in B_{j} }} \left| {\xi_{hj}^{{\overline{b}^{k} }} } \right| \le \Gamma_{j}^{{x^{k} }} + \Gamma_{j}^{{y^{k} }} + \Gamma_{j}^{{\overline{b}^{k} }} = \Gamma_{j}\) for all firms. The level of the budget of uncertainty \(\Gamma_{j} \in \left[ {0, \left| {I_{j} } \right| + \left| {R_{j} } \right| + \left| {B_{j} } \right|} \right]\) represents the allowable level of robustness (or conservatism) for any perturbation in the constraints. For simplicity but without loss of generality, an equal level of the budget of uncertainty is set for each constraint, i.e. \(\Gamma_{j} = \Gamma\). The constraints \(\Gamma_{j}^{{x^{k} }} + \Gamma_{j}^{{y^{k} }} + \Gamma_{j}^{{\overline{b}^{k} }} = \Gamma\), \(\Gamma_{o}^{{x^{k} }} \le m\), \(\Gamma_{o}^{{y^{k} }} \le s\), and \(\Gamma_{j}^{{\overline{b}^{k} }} \le d\) are imposed to the robust counterpart of the models (5) and (6) to control the level of conservatism robust solution against uncertainty. In this respect, \(\Gamma_{j}^{{x^{k} }} + \Gamma_{j}^{{y^{k} }} + \Gamma_{j}^{{\overline{b}^{k} }} = \Gamma\) enables the decision-makers to appreciate the effect of input, desirable and undesirable outputs uncertainty jointly on the technical efficiency measures where \(\Gamma\)s fluctuate within \(\left[ {0, m + s + d} \right]\) and \(\Gamma_{j}^{{x^{k} }}\), \(\Gamma_{j}^{{y^{k} }}\), and \(\Gamma_{j}^{{\overline{b}^{k} }}\) are non-negative variables. Under the CRS technology, the robust technical efficiency in two adjacent periods \(k = t, t + 1\) and cross-time periods, \(k,l = t, t + 1, k \ne l\) for firm \(o = 1,2, \ldots ,n\) can be expressed as followsFootnote 4:
where \(z_{j} ,p_{ij} ,q_{rj} ,\) and \(\gamma_{hj}\) are non-negative auxiliary decision variables of the set of uncertain inputs and outputs, and they are defined when the classical dualisation technique is applied to the deterministic models (5) and (6). Furthermore, \(z_{j} ,p_{ij} ,q_{rj}\), and \(\gamma_{hj}\) determine the robustness of models (9) and (10) when the level of conservativeness Γ is varied by an infinitesimally small amount. Table 4 illustrates the notations of the proposed models (9) and (10).
It is a straightforward exercise to develop the above models under the assumption of VRS by the inclusion of a free variable \(u_{0}\) into the objective function and common constraints. Let us emphasise that the developed models \({\text{RE}}_{{{\text{crs}},\left( \Gamma \right)}}^{k} \left( {{\varvec{x}}_{o}^{k} ,{\varvec{y}}_{o}^{k} ,\overline{\varvec{b}}_{o}^{k} } \right)\), \({\text{RE}}_{{{\text{crs}},\left( \Gamma \right)}}^{k} \left( {{\varvec{x}}_{o}^{l} ,{\varvec{y}}_{o}^{l} ,\overline{\varvec{b}}_{o}^{l} } \right)\), \({\text{RE}}_{{{\text{vrs,}}\left( \Gamma \right)}}^{k} \left( {{\varvec{x}}_{o}^{k} ,{\varvec{y}}_{o}^{k} ,\overline{\varvec{b}}_{o}^{k} } \right)\), and \({\text{RE}}_{{{\text{vrs}},\left( \Gamma \right)}}^{k} \left( {{\varvec{x}}_{o}^{k} ,{\varvec{y}}_{o}^{l} ,\overline{\varvec{b}}_{o}^{l} } \right)\) are all linear parametric programming problems due to the parameters \(\Gamma_{o}^{{x^{k} }} , \Gamma_{o}^{{x^{l} }} \in \left[ {0, m} \right]\), \(\Gamma_{o}^{{y^{k} }} , \Gamma_{o}^{{y^{l} }} \in \left[ {0, s} \right]\), and \(\Gamma_{o}^{{\overline{b}^{k} }} , \Gamma_{o}^{{\overline{b}^{l} }} \in \left[ {0, d} \right]\) associated with \(\Gamma \in \left[ {0, m + s + d} \right]\). Put differently, for any fixed \(\Gamma_{o}^{{x^{k} }} ,{ }\Gamma_{o}^{{x^{l} }} ,\Gamma_{o}^{{y^{k} }}\), \(\Gamma_{o}^{{y^{l} }}\), \(\Gamma_{o}^{{\overline{b}^{k} }}\) and \(\Gamma_{o}^{{\overline{b}^{l} }}\), a parametric programming problem is a linear programming (convex programming) problem. Due to the complexity and computational aspects of the analysis of parametric programming problems, we assume that \(\Gamma_{o}^{{x^{k} }} ,{ }\Gamma_{o}^{{x^{l} }} ,\Gamma_{o}^{{y^{k} }}\), \(\Gamma_{o}^{{y^{l} }}\), \(\Gamma_{o}^{{\overline{b}^{k} }}\) and \(\Gamma_{o}^{{\overline{b}^{l} }}\) are decision variables in this study resulting in non-linear programming problems. BARON as one of the fastest and most robust solvers is used in this study to solve these non-linear programming models via general algebraic modelling systems (GAMS) to obtain a global optimal or near-global optimal solution (Hatami-Marbini and Arabmaldar 2021). Further, in large-scale problems, it is rationale to transform the developed parametric models to the linear optimisation forms and carry out a sensitivity analysis by giving different values to parameter \(\Gamma_{j}^{{x^{k} }}\), \(\Gamma_{j}^{{y^{k} }}\), and \(\Gamma_{o}^{{\overline{b}^{k} }}\) separately. The efficiency values obtained from models (9) and (10) are called robust efficiency values indicating the worst possible performance of a firm. That is, robust efficiency is a conservative efficiency that is concerned with an assured level of performance a firm can realise across all feasible input and output data. Accordingly, the BMPI can be accommodated to yield the respective decomposition based on the budget of uncertainty and the level of conservatism Γ as follows:
where \({\text{RCU}}_{\left( \Gamma \right)} = \left( {\frac{{{\text{RE}}_{{{\text{vrs}},\left( \Gamma \right)}}^{t + 1} \left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)}}{{{\text{RE}}_{{{\text{vrs}},\left( \Gamma \right)}}^{t} \left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)}}} \right)\), \({\text{RFS}}_{\left( \Gamma \right)} = \left( {\frac{{{\text{RE}}_{{{\text{vrs}},\left( \Gamma \right)}}^{t} \left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)}}{{{\text{RE}}_{{{\text{vrs}},\left( \Gamma \right)}}^{B} \left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)}} \times \frac{{{\text{RE}}_{{{\text{vrs}}, \left( \Gamma \right)}}^{B} \left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)}}{{{\text{RE}}_{{{\text{vrs}},\left( \Gamma \right)}}^{t + 1} \left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)}}} \right)\) and \({\text{RSE}}_{\left( \Gamma \right)} = \left( {\frac{{{\text{RE}}_{{{\text{CRS}},\left( \Gamma \right)}}^{B} \left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)}}{{{\text{RE}}_{{{\text{vrs}},\left( \Gamma \right)}}^{B} \left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)}} \times \frac{{E_{{{\text{vrs}},\left( \Gamma \right)}}^{B} \left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)}}{{E_{{{\text{crs}},\left( \Gamma \right)}}^{B} \left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)}}} \right)\) are called the biennial robust catch-up, the biennial robust frontier-shift and the biennial robust scale efficiency change, respectively. Given a level of conservatism, the robust BMPI including the three aforesaid components provides useful information about the productivity change over time.
Let us expound the rationale behind the employment of the notion of robustness on the three above mentioned components. The \({\text{RCU}}_{\left( \Gamma \right)}\) calculates the robust efficiency of the firm \(\left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)\) with respect to period \(t + 1\) frontier relative to the robust efficiency of the firm \(\left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)\) with respect to period \(t\) frontier for each level of conservatism Γ. Vividly, changing \(\Gamma\) perturbs the input and output data and resultantly the efficiency frontier is shifted at time periods \(t\) and \(t + 1\). The \({\text{RFS}}_{\left( \Gamma \right)}\) takes account of the movement of the frontier between periods \(t\) and \(t + 1\), that is, the \({\text{RFS}}_{\left( \Gamma \right)}\) effect can be viewed by the product of (i) robust efficiency of \(\left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)\) with respect to the frontier at period \(t\) relative to the robust efficiency of \(\left( {{\varvec{x}}_{o}^{t} ,{\varvec{y}}_{o}^{t} ,\overline{\varvec{b}}_{o}^{t} } \right)\) with respect to the biennial VRS frontier and (ii) robust efficiency of \(\left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)\) with respect to the biennial VRS frontier relative to robust efficiency of \(\left( {{\varvec{x}}_{o}^{t + 1} ,{\varvec{y}}_{o}^{t + 1} ,\overline{\varvec{b}}_{o}^{t + 1} } \right)\) with respect to the frontier at period \(t + 1\). Likewise, \(\Gamma\) helps the decision-maker adjust the efficient frontier at time periods \(t\) and \(t + 1\) to investigate the impression of data fluctuations. Finally, the \({\text{RSE}}_{\left( \Gamma \right)}\) analyses the sensitivity of the scale efficiency changes of \(\left( {{\varvec{x}}_{o} , {\varvec{y}}_{o} ,\overline{\varvec{b}}_{o} } \right)\) evaluated by the ratio of the scale effect of the time periods \(t\) and \(t + 1\) with respect to the biennial CRS and VRS frontiers by adjusting the level of conservatism Γ.
The overall standing of productivity of a firm from period \(t\) to \(t + 1\) can be obtained from \({\text{BMPI}}_{ \left( \Gamma \right)}^{R}\) for a given \(\Gamma \in \left[ {0, m + s + d} \right]\). The progress, regress, or no change in the total factor can be reached if \({\text{BMPI}}_{\left( \Gamma \right)}^{R} > 1\), \({\text{BMPI}}_{\left( \Gamma \right)}^{R} < 1\) or \({\text{BMPI}}_{\left( \Gamma \right)}^{R} = 1\), respectively. Note that \(\Gamma = m + s + d\) leads to the worst-case formulation while \(\Gamma = 0\) results in the nominal BMPI.
Theorem 1
Models (9) and (10) are always feasible.
Proof
Without loss of generality, let us consider model (9). To prove the feasibility of model (9), it needs to find a feasible solution. Let us assume that \(x_{go}^{k} = \max \left\{ {x_{io}^{k} {|}1 \le i \le m} \right\} > 0\) and consider \(\left( {u_{1} , \ldots ,u_{s} ,w_{1} , \ldots ,w_{d} ,v_{1} , \ldots ,v_{m} } \right) = \left( {0, \ldots ,0,0, \ldots 0,0, \ldots ,1/x_{go}^{k} , \ldots ,0} \right)\). In addition, let \(z_{j} ,p_{ij} , q_{rj} ,\gamma_{hj} , \Gamma_{o}^{{x^{k} }} ,\Gamma_{o}^{{y^{k} }} , \Gamma_{o}^{{\overline{b}^{k} }} = 0\, \left( {\forall i,r,h,j} \right)\). It is clear that this feasible solution completes the proof. The proof for model (10) is identical which is omitted. □
Theorem 1 is mathematically important and shows that the proposed robust BMPI models are invariably feasible in each time period in the presence of uncertainty. Besides, Theorem 2 states that the presence of uncertainty leads to a decrease in efficiency measures which would be a key fact for practitioners and managers who should understand that ignorance of uncertainty may result in biased decisions.
Theorem 2
The optimal objective function value of the robust model (9) (or (10)) is smaller than or equal to the objective function value of the deterministic model (5) (or (6)).
Proof
Let us consider two cases in respect of the relationship between models (9) and (5).
-
(i).
If there is no uncertainty, viz., \(z_{j} ,p_{ij} , q_{rj} , \gamma_{hj} , \Gamma_{o}^{{x^{k} }} ,\Gamma_{o}^{{y^{k} }} ,\Gamma_{o}^{{\overline{b}^{k} }} = 0\), then model (9) is transformed into model (5).
-
(ii).
If \(\Gamma_{o}^{{x^{k} }} + \Gamma_{o}^{{y^{k} }} + \Gamma_{o}^{{\overline{b}^{k} }} = \Gamma > 0\) and since all the decision variables \(z_{j} ,p_{ij} , q_{rj} , \gamma_{hj} , \Gamma_{o}^{{x^{k} }} ,\Gamma_{o}^{{y^{k} }}\) and \(\Gamma_{o}^{{\overline{b}^{k} }}\) are non-negative, then comparing the objective function of model (5), i.e. \(\mathop \sum \limits_{r = 1}^{s} u_{r} y_{ro}^{k} + \mathop \sum \limits_{h = 1}^{d} w_{d} \overline{b}_{ho}^{k}\) with model (9), i.e. \(\mathop \sum \limits_{r = 1}^{s} u_{r} y_{ro}^{k} + \mathop \sum \limits_{h = 1}^{d} w_{d} \overline{b}_{ho}^{k} - z_{o} \Gamma_{o}^{{y^{k} }} - \mathop \sum \limits_{r = 1}^{s} q_{ro} - z_{o} \Gamma_{o}^{{y^{k} }} - \mathop \sum \limits_{h = 1}^{d} \gamma_{ho}\), it can be easily seen that \({\text{RE}}_{{{\text{crs}},\left( \Gamma \right)}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} ,\overline{\varvec{b}}_{o}^{k} } \right) < E_{{{\text{crs}}}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} ,\overline{\varvec{b}}_{o}^{k} } \right)\)Footnote 5.
4 A case study of the oil refinery industry
The oil industry is highly technologically driven and requires a continuous performance and productivity assessment for business continuity. It is the industry with a huge capital investment with high risk all through its value chain. Given that it supplies a large percentage of the world’s energy and its key role in global economic development, it is important to frequently understand the state of the art in terms of performance and productivity growth.
The data used for this research are gathered from secondary sources including the US Energy Information AdministrationFootnote 6 (EIA), B.P. statistical bulletin, and other published annual reports of selected countries in question. The criteria for selecting input and output variables are based on the raw materials (resources) injected into the production system in order to produce the expected outcomes. The current study focuses on the oil refineries of the 25 countries (production units) to calculate the productivity growth and its components. Notably, the 25 chosen countries are among the highest refinery capacity, percentage utilisation, and refinery throughput in the world. In other words, this sample of countries produces about 77% of the total refined petroleum product around the globe. It is initially targeted to analyse productivity growth in the oil refinery of the 25 countries over the period 2000–2018. However, the missing data for some countries at the beginning and end of the targeted period leads us to focus on the 2005–2014 period.
The input data for the current research are selected to cover raw materials and intermediate materials processed at the oil refinery to produce finished petroleum products. Similarly, the output data are selected to cover all secondary products produced from the refinery. Therefore, the specification of the estimated production function in respect of oil refineries includes two inputs, 6 desirable outputs, and one undesirable output for all 25 countries for 10 periods. Although the selected countries may have different levels of technological advancement, regulatory framework, closeness to raw materials, and political and environmental stability, all oil refineries are assumed to be almost in the same line of business and are operating under very similar market conditions and global uncertainty. Notwithstanding, we account for the GDP and CO2 emissions as desirable and undesirable output factors, respectively. The GDP shows the impact of this key industry on the economic growth of countries while the CO2 emission factor indicates the environmental impact of the oil refinery industry in the sampled countries. CO2 emissions are one of the greenhouse gas emissions with the highest level of emission causing global warming and climate change that could result in extreme weather conditions. The refinery analysis model is depicted in Fig. 2, and the input and output variables included are defined as follows:
A refinery analysis model
Input Variables:
-
Crude oil throughput (I1) This is the total amount of crude oil that goes into the refinery to produce refined petroleum products over a given period and is commonly expressed in barrels per day. Refinery throughput is used to determine the capacity utilisation in a ratio of the refinery throughput to the refinery capacity.
-
Energy Consumption (I2) This is the total amount of energy used in the refinery production process over a given period.
Output Variables:
-
Motor Gasoline (O1) This is a light hydrocarbon with a density of 0.78 g/cm3 produced from the refinery processing of crude oil and other petroleum mixtures. Motor gasoline is used as a fuel, mostly for land-based combustion engines.
-
Jet fuel (O2) This is produced from crude oil and other petroleum mixtures. It is composed of either unleaded kerosene or naphtha kerosene. Jet fuel is used in turbine and compression ignition engines.
-
Distillate fuel oil (O3) This is a product from the conventional distillation process of a petroleum refinery. It is classified into diesel fuel and fuel oil, mainly used on public roads and cross-country diesel engines such as farm machinery and railway locomotive engines.
-
Residual fuel oil (O4) This is a heavy oil that stays put after lighter hydrocarbon, and distillate fuel oil are distilled out of the refinery production process. This product is used mainly for industrial purposes such as in steam power vessels and inshore plants.
-
Other petroleum products (O5) These are all other finished products obtained from the petroleum refinery operation apart from the listed above.
-
Gross domestic product (O6) This is the most important indicator of the value of an economy showing the total market value of all goods and services produced in the country over a given period.
-
CO2 emissions (O7) CO2 makes up the vast majority of greenhouse gas emissions and is known as the main driver of global climate change that could cause extreme weather events such as wildfires, severe droughts, heat waves, and tropical storms.
The descriptive statistics of inputs and outputs are presented in Table 5, showing the maximum, minimum, mean, and standard deviation values of the input and output variables of the petroleum refinery country data of selected countries for 10 years.Footnote 7 For the input variables, the USA (Iraq) recorded the maximum (minimum) value of refinery throughput with a value of 15.844 Mb/d (0.342 Mb/d) in 2014. China (Iraq) recorded the maximum (minimum) value of energy consumption with a value of 136.9 quad (0.985quad) in 2014 (2007). Referring to the output variables, the USA produced the maximum value of 9571 Mb/d of motor gasoline in 2014, 9058 Mb/d of jet fuel in 2011, 4916 Mb/d of distillate fuel oil in 2014, and 4916 Mb/d of residual fuel oil in 2014 while China produced the maximum value of other products in 2013. Furthermore, the United Arab Emirates documented the minimum value of motor gasoline, distillate fuel oil, and residual fuel in 2005, 2014, and 2008, with values given as 44 Mb/d, 79 Mb/d and 21 Mb/d, respectively. Iraq produced the minimum value of jet fuel and other products in 2012 and 2014, with the value of 3 Mb/d and 73.3 Mb/d, respectively. France (Kuwait) recorded the maximum (minimum) value of GDP with a value of 26,651.1 ppp (227.1 ppp), and finally, China (Singapore) recorded the maximum (minimum) value of CO2 emissions with a value of 10,827,895 (107.9425015) MM tonnes in 2013 (2000), respectively, indicating a continuous increase in CO2 emissions overs the study period.
Likewise, the standard deviation values in most variables show heterogeneity in the data set, although the refinery throughput with a low standard deviation implies that the data are closer to the mean value in refinery throughput. However, most countries indicated maximum values in 2014 because of the rebound in the global economic activities following the global crisis. The overestimated economic growth after 2009 led to the rise in oil production that helped trigger a sharp drop in oil prices from $110 to $35 between 2015 and 2016. Iraq recorded the minimum value of input resources in 2007 just before the 2008 economic crisis because of the political and civil unrest caused by the 2007 Iraq war.
Figure 3 interestingly shows the average energy consumption, CO2 emission and GDP per capita of countries for ten years at once. It can be viewed that some countries such as Brazil, France, Germany, India, Japan, and the UK with high values of GDP consume lower energy and in turn emit lower CO2 emissions. Meanwhile, countries with a larger population and GDP, in particular China and USA, emit much more CO2 emissions than other countries. It should be pointed out that the USA with a higher GDP than China emits less CO2 emissions on average from 2005 to 2014.
A comparison between countries
Refineries often operate in complex environments in the world where there may include uncertainty and fluctuations. For instance, the prices of crude oil products, and CO2 emissions usually vary randomly as a result of market demand and other political and economical decisions associated with the oil refinery worldwide. In consequence, it is vital to apply advanced analytical tools to cope with uncertainty in performance and productivity assessment. This study, in situations of uncertainty, considers three different values of perturbations (\(e = 1\% , \;5\% ,\; 10\%\)) in all the inputs and outputs data aiming to assess the refinery’s performance of the 25 countries via the robust DEA models (9) and (10) and calculate the productivity growth and its components using the proposed method. We point out that each country \(\left( {j = 1,2, \ldots ,25} \right)\) is known as an uncertain observation if any of the inputs or outputs are uncertain. Bertsimas and Sim (2004)’s robust approach put forth a given method to select the appropriate level of conservatism Γ. To do so, it suffices to select Γ approximately at \(\Gamma = 1 + \varphi^{ - 1} \left( {1 - e} \right)\sqrt n\), where the parameter \(\varphi\) represents the cumulative distribution of the standard Gaussian variable, and \(n\) is the number of parameters that are subject to the uncertainty associated with each constraint. It should be noted that selecting an appropriate level of conservatism Γ can be guaranteed that the optimal solutions obtained from robust DEA models are feasible with high probability. This application includes two uncertain inputs and seven uncertain outputs that can be resulted that the level of conservatism Γ can be varied within [0, 9]. Considering three different values of perturbations 1%, 5%, and 10%, the appropriate values for Γ, respectively, are approximately 7.96, 5.92, and 4.84. These three conservatism values imply that they adequately protect the problem against 88%, 66%, and 54% of the uncertain parameters to arrive at robust solutions.
The robust DEA model (9) is applied under the CRS and VRS versions of technology to measure the technical efficiency of each country over different values of perturbation. Table 6 reports the average efficiency measures of all countries and the ranking order of the countries (in parenthesis) over 2005–2014. Notice that the deterministic DEA model (5) occurs when \(\Gamma = 0\). A country is said to be efficient and constructs the efficient frontier if the efficiency measure equals unity. Given the deterministic case, Iraq, Indonesia, Italy, Kuwait, Singapore, and UAE are efficient for the CRS technology, and Brazil, China, France, Iraq, Indonesia, India, Italy, Kuwait, Russia, Singapore, the UAE, and the USA are efficient for the VRS technology (see Table 6). In the case of the developed robust models, efficiency measures of the countries in both the CRS and VRS cases significantly decrease as the perturbation (uncertainty) level increases from 1% to 10% (See Theorem 2). In detail, the average performance of robust DEA models for CRS (VRS) technology at 0%Footnote 8, 1%, 5%, and 10% is 0.948 (0.990), 0.900 (0.953), 0.787 (0.831), and 0.666 (0.709), respectively. Resultantly, although the number of efficient countries reduces and only a few countries are close to efficient, the results are highly robust against uncertainty and decision-makers in a specific country are more confident of results to make decisions on how the refinery industry can learn from the best practices across the world.
Furthermore, the Spearman’s rank correlation coefficient is here used to verify the relationship between different levels of perturbations for both the CRS and VRS technologies. Table 7 reports results showing that as perturbations increase from 0% to 10% the correlation for CRS and VRS increases from 0.446 to 0.872 (see the main diagonal). It can be concluded that the correlations for the CRS technology (above diagonal) considering different levels of perturbations are higher than those for the VRS technology (below diagonal).
Let us here borrow the concept of the price of robustness originated by Bertsimas and Sim (2004) and widely used in the literature to help the decision-maker comprehend the detrimental effect of uncertainty on the objective function value (see e.g. Zhou-Kangas and Miettinen 2019; Kasperski and Zieliński 2021). The price of robustness is therefore calculated for the developed robust DEA models under CRS and VRS technologies. It implies that the decision-maker needs to provide a trade-off between countries’ performance and the uncertainty degree. In this paper, the price paid for catching up with a high level of robustness, i.e. protection against uncertainty, can be calculated as follows:
where \(\overline{{{\text{RE}}}}_{{{\text{crs}},\left( 0 \right)}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} ,\overline{\varvec{b}}_{o}^{k} } \right)\) is the average of efficiency measures of production units (DMUs) in the deterministic situation as presented in the 2nd and 6th columns of Table 6, and \(\overline{{{\text{RE}}}}_{{{\text{crs}},\left( \Gamma \right)}}^{k} \left( {{\varvec{x}}_{o}^{k} , {\varvec{y}}_{o}^{k} ,\overline{\varvec{b}}_{o}^{k} } \right)\) is the average of robust efficiency measure of each country for varying levels of Γ. Straightforwardly, the identical formulation can be defined to obtain the price of robustness for VRS technology. Figure 4 shows the average price of robustness for the three different perturbation levels over 10 years. Appealingly, the price of robustness increases as the perturbation level increases from 1% to 10%. Besides, it can be shown that the VRS technology is more robust than the CRS technology.
Price of robustness in the CRS and VRS technologies
Figure 5 is depicted to report the average price of robustness for all the perturbations at the country level over 10 years. This proxy enables us to provide a more accurate picture of the uncertainty effect on efficiency. In other words, a higher amount of the price of robustness obtained from the inherent uncertainty implies that more attention must be drawn by policymakers because it leads to a decrease in the efficiency of the oil refinery industry. Under the VRS technology, Iraq has the smallest price of robustness (6.41) in comparison with other countries; that is, this country is more robust against the data fluctuations. On the other hand, Iran has the biggest price of robustness against all perturbations. Given the CRS technology, Kuwait and Iran are the countries with the smallest and biggest price of robustness, respectively. It can be inferred that as a result of embedded uncertainty Iran has the most negative impact on its performance in the oil refinery while Kuwait is the most robust country with the least negative impact against the data fluctuations.
Price of robustness of the CRS and VRS technologies for the 25 countries
We now intend to analyse the productivity growth of the refinery of each country using the proposed robust BMPI and its components including CU, FS, and SE when the input and output data are subject to uncertainty. Figure 6a–d presents the productivity growth of the refinery industry during the years 2005–2014 for the three various levels of perturbation in observations. As shown in Fig. 6, BMPI and its components have almost constant values except around 2008. It is inferred that the financial crisis in 2008 did hit the oil industries’ construction and caused a steep decline in the price of crude oil from $150 to $35 within a short period, resulting in unemployment, lower spending, and less demand for oil. Although after the 2008 financial crisis, there was a sharp growth from 2009 to 2011 due to monetary and fiscal stimulus, and then, the oil industries’ level came back to its previous level before 2008 approximately.
Average of BMPI and its components over 10 years
Figure 7 reveals the change in GDP for each country along with the average BMPI during the 10 years. This also shows a sharp decline between 2008 and 2009 and an upsurge between 2009 and 2010, while soft fluctuation is observed in the remaining adjacent time periods. The sharp decline in GDP between 2008 and 2009 resulted from low capital accumulation and a reduced growth rate of potential GDP due to the fall in investment during the 2008 economic contraction. Moreover, the improvement between 2009 and 2010 occurs because of the rise in economic activities just after the economic crisis. In addition, Fig. 7 implies that the BMPI is correlated with GDP.
GDP change with average BMPI
Table 8 presents the productivity growth results including the average BMPI, CU, FS, and SE over the entire 2005–2014 period. It should be noted that if the BMPI is less (greater) than 1, it means a regress (progress) in Malmquist productivity. USA (28%), Iraq (48%), Iraq (29%), and France (19%) take the highest BMPI values per year on average when 0% (deterministic), 1%, 5%, and 10% perturbation are, respectively, applied in observations. A glance at Table 8 shows that the SE of the USA is the highest in all cases implying that the scale efficiency effect on the BMPI is stronger during the study period. There is evidence that a rapid influx of domestically sourced tight light crude oil was used as feedstock to the USA refineries during the study period. Refinery and production efficiency values are highly sensitive to variables such as crude quality (light crude oil), seasonal and regional factors, and refinery configuration and complexity (Forman et al. 2014), implying that the quality of crude used as feedstock to the USA refineries could be the primary reason for the overall efficiency of the refineries. Similarly, the USA has considered advanced technological innovations as long-term plans towards emission reductions; such innovations include fuel switching; electrification; and carbon capture, utilisation, and storage as part of activities to decarbonise the petroleum refineries. Table 8 also shows that China and Russia recorded the lowest performance over time with the BMPI value at 0.8999 and 0.9171, respectively, mainly caused by the FS and SE. These findings are corresponding to the fact that China’s economic output has become less oil intensive; for example, in 1995, China required 200,000 bbl of oil to produce one billion RMB worth of GDP; however, in 2015, it required 64,500 bbl of oil to produce one billion RMB worth GDP. These are because of the rising concern about pollution and unsustainable financial burden from underutilised infrastructure, leading China’s oil intensity to a continuous gradual downward trend as the service and light industry takes over a greater portion of the economic activities (Collins 2016). Equally, Russia exhibited low performance because of the numerous issues facing the downstream sector including low capital investment and the refining technologies available. Other reasons for the decline may include that most of Russia’s oil refineries were under construction during the study period with a proposed completion in 2015. These caused a decline in primary production, forcing the shutdown of most minor and least sophisticated oil refineries (Kapustin and Grushevenko 2018).
Similarly, Table 8 also shows that China and Russia recorded a low FS over time, resulting in a regressed BMPI over time. This suggests that an increase (decrease) in FS could increase (decrease) the BMPI.
Figure 8 exhibits the average BMPI for each country at 0%, 1%, 5%, and 10% perturbation levels over the period 2005–2014 as well as their means represented by a solid black line. Results from Fig. 8 reveal that the USA recorded the highest productivity growth in the presence of perturbation and the deterministic case with an average value of 1.1981. However, China and Russia had the lowest productivity growth with an average value of 0.9404 and 0.9853, respectively, due to technological and environmental issues. Therefore, the USA has the highest BMPI, and it inspires us to analyse this country in more detail at the end of this section.
Average BMPI for 25 countries 2005–2014
Table 9 aims to present the descriptive statistics of results to explore dynamic changes in the oil industry over 10 periods. The results show that the average rate of change (ARC) for the BMPI is constantly positive with above 0.89% growth in all cases and FS as the main contributing factor. It can be concluded that productivity growth has been progressive on average in the oil industry during the 10 years of study, even if uncertainty matters for managers and decision-makers. Given the presented ranking orders in parentheses next to the BMPIs, it confirms that the financial crisis in 2008 causes a considerable regress in the oil industry compared to the rest of the periods in question. Interestingly, in both deterministic and uncertain cases, the inferior and superior growth rates occur over the 2009–2010 and 2008–2009 periods, respectively.
Given both country by country and year by year cases, Table 10 summarises Spearman’s correlation coefficients for BMPI rankings obtained from different levels of perturbations presented in Tables 8 and 9, respectively. The entries above and below the diagonal of Table 10 indicate Spearman’s rank coefficients of year-wise and country-wise results, respectively. The results show that (i) all coefficients are positive indicating a direction of the relationship between each pair of rankings and (ii) there is a strong enough correlation between the rankings considering different levels of perturbations. In other words, if a country arrives at a higher-ranking order than another country at a given perturbation’s level, then one can envisage the same order at another perturbation’s level.
In the end, let us limit our focus to the USA with the highest BMPI to gain some valuable insight. Figure 9 shows the relative efficiency measure of the USA over 2005–2014 under the CRS and VRS technologies in situations of deterministic and data uncertainty. As it is envisaged from Theorem 2, the efficiency measures of the USA decrease as the perturbation level increases. In other words, it is inescapable to pay the price for gaining robustness when the uncertainty level increases (Bertsimas and Sim 2004).
Relative efficiency of the USA in the CRS and VRS
Table 11 hence reports the price of robustness for the USA from 2005 to 2014 where the data perturbation is assumed to be 1%, 5%, and 10%. It is deserved to mention that the increase in the perturbation level leads to the rise in the price of robustness amount. Furthermore, the highest price of robustness occurs in 2008 and 2009 for CRS and VRS technologies which are 18.94% and 17.41% on average, respectively.
Finally, let us analyse the CU, FS, SE, and BMPI for the USA over 2005–2014. Table 12 demonstrates the BMPI and its relevant components for different levels of perturbation and conservatism on an annual basis. It is obvious from Table 12 that FS plays a key role in productivity growth over time.
To emphasise the analysis of the case study, it is notable that performance measurement and its application to determine productivity growth are of importance to companies in the oil and natural gas supply chain, enabling managers and policymakers to assess the performance of the system against a set of objectives. With this in mind, the current study considers multiple inputs against multiple outputs to evaluate productivity growth in a situation of uncertainty. The findings demonstrate that there is a growing need for managers and policymakers in the oil industry to consider alternative production (refinery) channels in times of periodic maintenance or total overhauling to keep the security of supply which would impact the GDP positively. In addition, industries need to continually attempt to diversify energy sources that could help stimulate economic growth and reduce carbon emissions by develo** renewable resources. It also might be of importance to indicate the superiority of the proposed method in this study compared to other existing methods. The proposed framework allows the decision-maker to adjust the level of conservatism based on his/her risk preference as well as making the decision-maker confident in observing the most robust results in situations of uncertainty.
5 Conclusion and policy implications
Uncertainty is an imperative feature of real-world problems, and if uncertainty is neglected in productivity assessment, this may lead decision-makers to wrong decisions and invest in the wrong places for improvement. The world oil industry encompasses both endogenous and exogenous uncertainties such as uncertain oil prices, non-deterministic demand, variable consumer behaviour, unstable environmental and emission regulations, supply chain disruption, and unprecedented events (e.g. the COVID-19 pandemic, wars, sanctions, etc.). This paper presents a robust DEA-based model to measure efficiency and productivity when uncertainty and undesirable outputs exist. Drawing upon the robust optimisation approach, we combat uncertain data with unknown distributions as well as consider the link between the level of conservatism and decision-makers’ risk preference. We hence generalise the extension of Ray and Desli (1997)’s approach for measuring the MPI, i.e. BMPI proposed by Pastor et al. (2011) to cases where undesirable outputs are available. As such, a combination of the generalised BMPI and Bertsimas and Sim (2004)’s approach is used to develop our theoretical foundations. The proposed approach introduces the concept of the price of robustness in DEA using the average of obtained efficiencies under CRS and VRS technologies over varying periods and, resultantly, the average values of efficiencies, BMPIs, and their related components are utilised to simplify the interpretation and illustration of findings.
We analyse the productivity growth and efficiency measurement of the oil refinery of the 25 different countries from 2005 to 2014 using the proposed robust approach to help deal with data uncertainty and undesirable outputs under different levels of perturbations. Under different level perturbations in the input and output data of the oil refinery at the country level, it can be concluded that productivity measures have almost constant values except for a steep decline around 2008 because of the financial crisis and a sharp growth from 2009 to 2011 owing to monetary and financial stimulus. The obtained results show that VRS technology is more robust than CRS technology. Having a positive correlation between the productivity change and the growth rate of GDP is another insight that can be of interest to managers and policymakers. Appealingly, among the 25 biggest oil refining countries, the USA’s productivity growth is the highest, while China and Russia have the lowest productivity growth.
Energy policy and reform in the petroleum sector are under tremendous pressure. Policy agent has a direct impact on the efficiency and productivity growth of the oil industry. Most significantly, the refinery sector has a direct link to domestic consumers and the increased environmental and socio-political regulations. Therefore, it is imperative to consider all groups of consumers (commercial, industrial, and domestic) and business continuity during the development of energy policies. Oil industries need to consider alternative means of production in routine maintenance and construction to avoid a drop in the production level as identified in the result (case of Russia). On the national government side, relative to the downstream sector, policies should be focused on higher standards on quality, safety, environmental protection, and energy efficiency. In this respect, it is advised that environment and safety policies are revamped throughout the whole supervision process and safety system to achieve effective risk management. It is also important to note that technology plays a significant role in refinery operations in energy consumption and processing time. Advance technology should be employed for greater efficiency and productivity growth.
This study is not free of limitations. First, data uncertainty in this research is limited to polyhedral uncertainty sets even though other types of uncertainty sets such as ellipsoidal, correlated polyhedral, and variable polyhedral sets are neglected because we only aim to shed light on how important uncertainty is in analysing efficiency and productivity. Second, data are not available for every year but only for ten years over the time span from 2000 to 2018 and expanding this period would provide meaningful insights which would help us further interpret the results. Third, because of data availability, we only include CO2 emissions as the undesirable output in the analysis. Fourth, setting levels of perturbation in observations should be fairly justified based on decision-makers’ preferences and objectives when applying the proposed approach in practice but access to such information is extremely difficult. Hence, interactive approaches are needed to be developed for arranging true level perturbations. Fifth, this study considers the oil refinery structure as a black-box system, and in future research, it is of interest to model network structures of oil refineries in order to identify the sources of inefficiencies (see e.g. Färe and Grosskopf 2000). Finally, exploring the desirable properties of unit invariance and translation invariance in robust DEA models remains for further research as well.
Notes
It should be noted that there are some studies in which uncertain data attend to statistical properties in the literature (e.g., Cazals et al. 2002; Daraio and Simar 2007; Landete et al. 2017; Daouia et al. 2020) which are dubbed as “robust non-parametric estimator” that is completely different with the concept of robust optimisation used in this paper.
The sign |.| denotes the absolute value of a variable or parameter.
The similar proof can be followed for models (6) and (10).
The full data set can be found in Electronic Supplementary Materials.
Note that it is a deterministic case.
References
Al-Mana AA, Nawaz W, Kamal A, Koҫ M (2020) Financial and operational efficiencies of national and international oil companies: an empirical investigation. Resour Policy 68:101701
Aparicio J, Pastor JT, Zofio JL (2013) On the inconsistency of the Malmquist-Luenberger index. Eur J Oper Res 229(3):738–742
Arabmaldar A, Jablonsky J, Saljooghi FH (2017) A new robust DEA model and super-efficiency measure. Optimization 66(5):723–736
Arabmaldar A, Kwasi Mensah E, Toloo M (2021) Robust worst-practice interval DEA with non-discretionary factors. Expert Syst Appl 182:115256
Atris AM, Goto M (2019) Vertical structure and efficiency assessment of the US oil and gas companies. Resour Policy 63:101437
Atris AM (2020) Assessment of oil refinery performance: application of data envelopment analysis-discriminant analysis. Resour Policy 65:101543
Bădin L, Daraio C, Simar L (2019) A bootstrap approach for bandwidth selection in estimating conditional efficiency measures. Eur J Oper Res 277(2):784–797
Balk BM (2001) Scale efficiency and productivity change. J Prod Anal 15:159–183
Balk BM, Zofio J (2018) The many decompositions of total factor productivity change. ERIM Rep Ser Res Manag. https://doi.org/10.2139/ssrn.3167686
Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manage Sci 30(9):1078–1092
Barros C, Antunes OS (2014) Productivity change in the oil blocks of Angola. Energy Sour Part B 9(4):413–424
Barros CP, Managi S (2009) Productivity assessment of Angola’s oil blocks. Energy 34(11):2009–2015
Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805
Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88(3):411–424
Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press
Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52(1):35–53
Bevilacqua M, Braglia M (2002) Environmental efficiency analysis for ENI oil refineries. J Clean Prod 10(1):85–92
Caves DW, Christensen LR, Diewert WE (1982) The economic theory of index numbers and the measurement of input, output, and productivity. Econometrica 50(6):1393
Cazals C, Florens JP, Simar L (2002) Nonparametric frontier estimation: a robust approach. J Econom 106(1):1–25
Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429–444
Charnes A, Cooper WW, Golany B, Seiford L, Stutz J (1985) Foundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functions. J Econom 30(1–2):91–107
Chen Y, Ali AI (2004) DEA malmquist productivity measure: new insights with an application to computer industry. Eur J Oper Res 159(1):239–249
Chen K, Zhu J (2019) Computational tractability of chance constrained data envelopment analysis. Eur J Oper Res 274(3):1037–1046
Collins G (2016) China’s evolving oil demand: slowing overall growth, gasoline replacing diesel as demand driver, refined product exports rising substantially. Energy Environ Regul Aff
Cooper WW, Park KS, Yu G (1999) IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Manag Sci 45(4):597–607
Cooper WW, Seiford LM, Tone K (2007) Data envelopment analysis a comprehensive text with models, applications, references and dea-solver software, 2nd edn. Springer
Dalei NN, Joshi JM (2020) Estimating technical efficiency of petroleum refineries using DEA and tobit model: an India perspective. Comput Chem Eng 142:107047
Daouia A, Florens JP, Simar L (2020) Robust frontier estimation from noisy data: a Tikhonov regularization approach. Econom Stat 14:1–23
Daraio C, Simar L (2007) Advanced robust and nonparametric methods in efficiency analysis. methodology and applications. Springer
Daraio C, Simar L (2014) Directional distances and their robust versions: computational and testing issues. Eur J Oper Res 237(1):358–369
Ehrgott M, Holder A, Nohadani O (2018) Uncertain data envelopment analysis. Eur J Oper Res 268:231–242
Eller SL, Hartley PR, Medlock KB (2011) Empirical evidence on the operational efficiency of national oil companies. Empir Econom 40(3):623–643
Emrouznejad A, Rostamy-Malkhalifeh M, Hatami-Marbini A, Tavana M, Aghayi N (2011) An overall profit Malmquist productivity index with fuzzy and interval data. Math Comput Model 54(11–12):2827–2838
Färe R, Grosskopf Sh (2000) Network DEA. Socioecon Plann Sci 34(1):35–49
Färe R, Grosskopf Sh, Lovell CAK, Pasurka C (1989) Multilateral productivity comparisons when some outputs are undesirable: a nonparametric approach. Rev Econ Stat 71(1):90–98
Färe R, Grosskopf SH, Lindgren B, Roos P (1994a) Productivity developments in swedish hospitals: a malmquist output index approach. Data Envel Anal Theory Methodol Appl. Springer, Dordrecht, pp. 253–272. https://doi.org/10.1007/978-94-011-0637-5_13
Färe R, Grosskopf SH, Norris M, Zhang Z (1994b) Productivity growth, technical progress, and efficiency change in industrialized countries. Am Econom Rev 84:66–83
Färe R Grosskopf SH, Russell RR (1998) Index numbers: essays in honour of sten malmquist. In: Färe R, Grosskopf S, Russell RR (ed), 1st edn. Springer, Netherlands
Fare R, Grosskopf Sh (1992) Malmquist productivity indexes and fisher ideal indexes. Econ J 102(410):158
Farrell MJ (1957) The measurement of productive efficiency. J. R Stat Soc. Series A (General) 120(3):253–290
Forman GS, Divita VB, Han J, Cai H, Elgowainy A, Wang M (2014) US refinery efficiency: impacts analysis and implications for fuel carbon policy implementation. Environ Sci Technol 48(13):7625–7633
Grifell-Tatjé E, Lovell CAK (1999) A generalized malmquist productivity index. TOP 7(1):81–101
Hatami-Marbini A, Arabmaldar A (2021) Robustness of farrell cost efficiency measurement under data perturbations: Evidence from a US manufacturing application. Eur J Oper Res 295(2):604–620
Hatami-Marbini A, Saati S (2018) Efficiency evaluation in two-stage data envelopment analysis under a fuzzy environment: a common-weights approach. Appl Soft Comput 72:156–165
Hatami-Marbini A, Emrouznejad A, Tavana M (2011) A taxonomy and review of the fuzzy data envelopment analysis literature: two decades in the making. Eur J Oper Res 214(3):457–472
Hatami-Marbini A, Tavana M, Emrouznejad A (2012) Productivity growth and efficiency measurements in fuzzy environments with an application to health care. Int J Fuzzy Syst Appl 2(2):1–35
Hatami-Marbini A, Emrouznejad A, Agrell PJ (2014) Interval data without sign restrictions in DEA. Appl Math Model 38(7–8):2028–2036
Hatami-Marbini A, Agrell PJ, Fukuyama H, Gholami K, Khoshnevis P (2017) The role of multiplier bounds in fuzzy data envelopment analysis. Ann Oper Res 250(1):249–276
Hatami-Marbini A, Ghelej Beigi Z, Hougaard JL, Gholami K (2018) Measurement of returns-to-scale using interval data envelopment analysis models. Comput Ind Eng 117:94–107
Hosseini K, Stefaniec A (2019) Efficiency assessment of Iran’s petroleum refining industry in the presence of unprofitable output: a dynamic two-stage slacks-based measure. Energy 189:116112
Ike CB, Lee H (2014) Measurement of the efficiency and productivity of national oil companies and its determinants. Geosyst Eng 17(1):1–10
Kao C (2017) Measurement and decomposition of the malmquist productivity index for parallel production systems. Omega 67:54–59
Kapustin NO, Grushevenko DA (2018) Exploring the implications of Russian energy strategy project for oil refining sector. Energy Policy 117:198–207
Karagiannis G, Knox Lovell CA (2016) Productivity measurement in radial DEA models with a single constant input. Eur J Oper Res 251(1):323–328
Kasperski A, Zieliński P (2021) Soft robust solutions to possibilistic optimization problems. Fuzzy Sets Syst 422:130–148
Kevork IS, Pange J, Tzeremes P, Tzeremes NG (2017) Estimating malmquist productivity indexes using probabilistic directional distances: an application to the European banking sector. Eur J Oper Res 261(3):1125–1140
Klopp GA (1985) The analysis of the efficiency of production system with multiple inputs and outputs. PhD dissertation, University of Illinois, Industrial and System Engineering College, Chicago
Korhonen PJ, Luptacik M (2004) Eco-efficiency analysis of power plants: an extension of data envelopment analysis. Eur J Oper Res 154(2):437–446
Kuosmanen T (2005) Weak disposability in nonparametric production analysis with undesirable outputs. Am J Agric Econ 87(4):1077–1082
Landete M, Monge JF, Ruiz JL (2017) Robust DEA efficiency scores: a probabilistic/combinatorial approach. Expert Syst Appl 86:145–154
Lu C (2015) Robust data envelopment analysis approaches for evaluating algorithmic performance. Comput Ind Eng 81:78–89
Malmquist S (1953) Index numbers and indifference surfaces. Trabajos De Estadistica 4(2):209–242
Managi S, Opaluch JJ, ** D, Grigalunas TA (2004) Technological change and depletion in offshore oil and gas. J Environ Econ Manag 47(2):388–409
Maniadakis N, Thanassoulis E (2004) A cost malmquist productivity index. Eur J Oper Res 154(2):396–409
Mardani A, Zavadskas EK, Streimikiene D, Jusoh A, Khoshnoudi M (2017) A comprehensive review of data envelopment analysis (DEA) approach in energy efficiency. Renew Sustain Energy Rev 70:1298–1322
Mayer A, Zelenyuk V (2014) Aggregation of malmquist productivity indexes allowing for reallocation of resources. Eur J Oper Res 238(3):774–785
Mekaroonreung M, Johnson AL (2010) Estimating the efficiency of American petroleum refineries under varying assumptions of the disposability of bad outputs. Int J Energy Sect Manag 4(3):356–398
Mulvey JM, Vanderbei RJ, Zenios SA (1995) Robust optimization of large-scale systems. Oper Res 43(2):264–281
Nemati M, Kazemi Matin R, Toloo M (2020) A two-stage DEA model with partial impacts between inputs and outputs: application in refinery industries. Ann Oper Res 295(1):285–312
Olesen OB, Petersen NC (1995) Chance constrained efficiency evaluation. Manag Sci 41(3):442–457
Olesen OB, Petersen NC (2016) Stochastic data envelopment analysis - a review. Eur J Oper Res 251(1):2–21
Omrani H (2013) Common weights data envelopment analysis with uncertain data: a robust optimization approach. Comput Ind Eng 66(4):1163–1170
Pastor JT, Lovell CAK (2005) A global malmquist productivity index. Econ Lett 88(2):266–271
Pastor JT, Asmild M, Lovell CAK (2011) The biennial malmquist productivity change index. Socioecon Plann Sci 45(1):10–15
Pastor JT, Lovell CAK, Aparicio J (2020) Defining a new graph inefficiency measure for the proportional directional distance function and introducing a new Malmquist productivity index. Eur J Oper Res 281(1):222–230
Ramanathan R (2006) A multi-factor efficiency perspective to the relationships among world GDP, energy consumption and carbon dioxide emissions. Technol Forecast Soc Change 73(5):483–494
Ray SC, Desli E (1997) Productivity growth, technical progress, and efficiency change in industrialized countries: comment. Am Econ Rev 87:1033–1039
Sadjadi SJ, Omrani H (2008) Data envelopment analysis with uncertain data: an application for Iranian electricity distribution companies. Energy Policy 36(11):4247–4254
Sadjadi SJ, Omrani H, Abdollahzadeh S, Alinaghian M, Mohammadi H (2011) A robust super-efficiency data envelopment analysis model for ranking of provincial gas companies in Iran. Expert Syst Appl 38(9):10875–10881
Seiford LM, Zhu J (2002) Modeling undesirable factors in efficiency evaluation. Eur J Oper Res 142:16–20
Sengupta JK (1992) A fuzzy systems approach in data envelopment analysis. Comput Math Appl 24(8–9):259–266
Sheng Y, Wu Y, Shi X, Zhang D (2015) Energy trade efficiency and its determinants: a malmquist index approach. Energy Econom 50:306–314
Shestalova V (2003) Sequential malmquist indices of productivity growth: an application to OECD industrial activities. J Prod Anal 19(2–3):211–226
Shokouhi AH, Hatami-Marbini A, Tavana M, Saati S (2010) A robust optimization approach for imprecise data envelopment analysis. Comput Ind Eng 59(3):387–397
Shokouhi AH, Shahriari H, Agrell PJ, Hatami-Marbini A (2014) Consistent and robust ranking in imprecise data envelopment analysis under perturbations of random subsets of data. OR Spectrum 36(1):133–160
Simar L, Wilson PW (1998) Sensitivity analysis of efficiency scores: how to bootstrap in nonparametric frontier models. Manag Sci 44(1):49–61
Simar L, Wilson PW (1999) Estimating and bootstrap** malmquist indices. Eur J Oper Res 115(3):459–471
Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper Res 21(5):1154–1157
Sueyoshi T (2000) Stochastic DEA for restructure strategy: an application to a japanese petroleum company. Omega 28(4):385–398
Sueyoshi T, Goto M (2012a) Data envelopment analysis for environmental assessment: comparison between public and private ownership in petroleum industry. Eur J Oper Res 216(3):668–678
Sueyoshi T, Goto M (2012b) Returns to scale and damages to scale under natural and managerial disposability: Strategy, efficiency and competitiveness of petroleum firms. Energy Econom 34(3):645–662
Sueyoshi T, Goto M (2015) DEA environmental assessment in time horizon: radial approach for malmquist index measurement on petroleum companies. Energy Econom 51:329–345
Sueyoshi T, Wang D (2014) Sustainability development for supply chain management in U.S. petroleum industry by DEA environmental assessment. Energy Econom 46:360–374
Sueyoshi T, Wang D (2018) DEA environmental assessment on US petroleum industry: non-radial approach with translation invariance in time horizon. Energy Econom 72:276–289
Sueyoshi T, Yuan Y, Goto M (2017) A literature study for DEA applied to energy and environment. Energy Econom 62:104–124
Tachega MA, Yao X, Liu Y, Ahmed D, Li H, Mintah C (2021) Energy efficiency evaluation of oil producing economies in Africa: DEA, malmquist and multiple regression approaches. Clean Environ Syst 2:100025
Tavana M, Khanjani Shiraz R, Hatami-Marbini A (2014) A new chance-constrained DEA model with birandom input and output data. J Oper Res Soc 65(12):1824–1839
Tavana M, Khalili-Damghani K, Santos Arteaga FJ, Hosseini A (2019) A fuzzy multi-objective multi-period network DEA model for efficiency measurement in oil refineries. Comput Ind Eng 135:143–155
Tavana M, Khalili-Damghani K, Santos Arteaga FJ, Hashemi A (2020) A Malmquist productivity index for network production systems in the energy sector. Ann Oper Res 284(1):415–445
Tavana M, Toloo M, Aghayi N, Arabmaldar A (2021) A robust cross-efficiency data envelopment analysis model with undesirable outputs. Expert Syst Appl 167:114117
Thanassoulis E, Shiraz RK, Maniadakis N (2015) A cost malmquist productivity index capturing group performance. Eur J Oper Res 241(3):796–805
Thompson RG, Dharmapala PS, Rothenberg LJ, Thrall RM (1994) DEA ARs and CRs applied to worldwide major oil companies. J Prod Anal 5(2):181–203
Thompson RG, Lee E, Thrall R (1992) Dea/AR-efficiency of US independent oil/gas producers over time. Comput Oper Res 19(5):377–391
Thrall RM (2000) Measures in DEA with an application to the malmquist index. J Prod Anal 13(2):125–137
Tohidi G, Razavyan S, Tohidnia S (2012) A global cost malmquist productivity index using data envelopment analysis. J Oper Res Soc 63(1):72–78
Toloo M, Keshavarz E, Hatami-Marbini A (2021) An interval efficiency analysis with dual-role factors. Or Spectrum 43:255–287
Tone K (2001) A slacks-based measure of efficiency in data envelopment analysis. Eur J Oper Res 130(3):498–509
Tone K (2002) A slacks-based measure of super-efficiency in data envelopment analysis. Eur J Oper Res 143(1):32–41
Tone K (2004) Malmquist productivity index. In: Cooper WW, Seiford LM, Zhu J (eds) Handbook on data envelopment analysis. Springer, Boston
Vikas V, Bansal R (2019) Efficiency evaluation of Indian oil and gas sector: data envelopment analysis. Int J Emerg Mark 14(2):362–378
Walheer B (2018) Disaggregation of the cost malmquist productivity index with joint and output-specific inputs. Omega 75:1–12
Wang Y, Zhu Z, Liu Z (2019) Evaluation of technological innovation efficiency of petroleum companies based on BCC–malmquist index model. J Pet Explor Prod Technol 9:2405–2416
Wegener M, Amin GR (2019) Minimizing greenhouse gas emissions using inverse DEA with an application in oil and gas. Expert Syst Appl 122:369–375
Yu D, He X (2020) A bibliometric study for DEA applied to energy efficiency: trends and future challenges. Appl Energy 268:115048
Zelenyuk V (2006) Aggregation of malmquist productivity indexes. Eur J Oper Res 174(2):1076–1086
Zhou-Kangas Y, Miettinen K (2019) Decision making in multiobjective optimization problems under uncertainty: balancing between robustness and quality. Or Spectrum 41:391–413
Zhu W, Zhu Y, Yu Y (2020) China’s regional environmental efficiency evaluation: a dynamic analysis with biennial malmquist productivity index based on common weights. Environ Sci Pollut Res 27(32):39726–39741
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of 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.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Hatami-Marbini, A., Arabmaldar, A. & Asu, J.O. Robust productivity growth and efficiency measurement with undesirable outputs: evidence from the oil industry. OR Spectrum 44, 1213–1254 (2022). https://doi.org/10.1007/s00291-022-00683-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00291-022-00683-y