Measuring sustainable capacity utilization in OECD economies: A new DEA framework with undesirable outputs under chance constraints

Abstract

Examining the sustainable capacity utilization (SCU) of various systems is crucial for effective planning and decision-making. However, many studies face challenges when it comes to assessing capacity utilization due to the presence of random variables and undesirable outputs. In this research, we propose using chance-constrained data envelopment analysis (CCDEA) methods that take into account both managerial and natural disposability to evaluate the SCU of entities with random fixed and variable inputs. Specifically, we introduce non-radial directional distance function (DDF) models that incorporate random measures to provide estimates of input- and output-oriented random SCU metrics in the short term. These innovative techniques are then applied to analyze the SCU of OECD countries, considering both managerial and natural disposability factors. Our findings reveal that uncertainty and risk levels significantly influence the performance outcomes and ratios of the SCU that we calculate. This highlights the importance of accounting for these factors in capacity utilization assessments, ultimately contributing to more informed planning and strategic insights.

Introduction

Assessing capacity utilization (CU) effectively is crucial for the successful functioning of any business. Production capacity refers to the maximum level of output achievable when all inputs are available. However, this potential output may not be realized due to various constraints on resources. The concept of capacity was first introduced by Johansen ¹ in 1968, who defined it as the maximum output that can be achieved within a specific timeframe using available facilities and equipment, without any limitations on resource utilization. Over the years, various definitions of capacity have emerged, leading to extensive research on capacity utilization in different contexts. Furthermore, the productive technology of an enterprise may be altered in compliance with theoretical concepts as well as objectives of practical studies. ²

Singh et al. ³ conducted a thorough literature review on CU and its evaluation methods across various sectors and countries. Shaikh and Moudud ⁴ applied a cointegration approach to measure the CU in OECD areas. Among the techniques explored, data envelopment analysis (DEA) stands out as one of practical methods. ⁵ Färe et al. ⁶ elaborated on DEA in a physical sense to assess CU based on Farrell's efficiency measures, building on Johansen's initial definition. Färe and Grosskopf ⁷ utilized directional distance function (DDF) and DEA models to estimate CU, while Sahoo and Tone ⁸ investigated CU components using both radial and non-radial techniques. An input-oriented measure of plant capacity was introduced by Cesaroni et al. ⁹ Additionally, Yang et al. ¹⁰ applied DEA and DDFs to analyze CU in Chinese manufacturing, focusing on current and changing CU levels over time. Fukuyama et al. ¹¹ developed a tool to evaluate competitive efficiency among Chinese iron and steel companies using DEA, which considers an unconstrained capacity directional output distance function. Shen et al. ² combined weak disposability technology with input- and output-oriented CU measures for both short- and long-term analyses. Karanki and Bilotkach ¹² adapted a revised DEA method from Färe et al. ⁶ to estimate the CU of U.S. airports, allowing for unrestricted variable inputs to determine maximum capacity limits for each output. Yu et al. ¹³ explored capacity utilization and cost differences using an input-oriented slacks-based measures model, while Zhang et al. ¹⁴ assessed CU in the construction industry, accounting for environmental factors and the weak disposability of undesirable outputs. Fukuyama et al. ¹⁵ also developed an alternative DEA approach to address CU while minimizing constant inputs with respect to the average period. Song et al. ¹⁶ investigated CU over a given period. Huang et al. ¹⁷ introduced a multi-period CU estimation approach that incorporates the Technique for Order Preference by Similarity to Ideal Solution and DEA techniques.

Despite the wealth of research on DEA, there remains a gap in studies examining CU in the context of random data and undesirable outputs. As highlighted in Table 1, there are few assessments of CU that consider both natural and managerial disposability. Estimating sustainable capacity utilization (SCU) is particularly vital for OECD countries, as it provides policymakers with essential insights for resource allocation, economic growth, and overall societal well-being. By considering economic, social, and environmental dimensions in SCU estimation, policymakers can promote development that is not only economically viable but also socially inclusive and environmentally sustainable.

Table 1.

Comparative analysis of DEA studies for the assessment of CU.

Study	Weak disposability	Random measures	Sustainability	Managerial and natural disposability	Non-radial DDF	Application
Yang et al. 10	Yes	No	No	No	No	Chinese manufacturing sectors
Sahoo and Tone 8	No	No	No	No	No	Banking sector
Shen et al. 2	Yes	No	No	No	No	Agriculture
Fukuyama et al. 11	No	No	No	No	No	Chinese iron and steel firms
Karanki and Bilotkach 12	No	No	No	No	No	Airports
Fukuyama et al. 15	Yes	No	No	No	No	Chinese steel and iron firms
Zhang et al. 14	Yes	No	No	No	No	Construction industries
Yu et al. 13	No	No	No	No	No	Airlines
Chen et al. 20	No	No	No	No	No	Chinese regions
Arfa et al. 21	No	No	No	No	No	Hospitals
Song et al. 16	No	No	No	No	No	Universities
Huang et al. 17	No	No	No	No	No	Forestry sector
Feng et al. 22	Yes	No	Yes	No	No*	Rapeseed industry
Yang et al. 23	No	Yes	Yes	No	No	Chinese industrial enterprises
This research	No	Yes	Yes	Yes	Yes	OECD countries

*Radial DDF has been applied.

Estimating SCU is essential for fostering long-term economic growth and sustainability. Accurate assessments of capacity in key industries enable policymakers to avoid resource excesses or shortages, which can lead to inefficiencies or economic downturns. Furthermore, SCU estimates help identify investment opportunities and growth areas while highlighting potential risks and vulnerabilities within the economy. The social aspect of SCU is crucial for fostering inclusive growth and development. By evaluating the capacity utilization of industries that significantly impact employment and income generation, policymakers can promote long-term economic growth that benefits all segments of society, including vulnerable populations. This approach allows for the reduction of social inequalities and the enhancement of social cohesion through measures related to labor rights, working conditions, and access to essential services. Finally, considering the environmental context in SCU estimations is vital to ensure that economic activities do not compromise the environment, thus safeguarding the potential for sustainable development for future generations. Evaluating the environmental impacts of major industries and sectors is crucial for policymakers aiming to manage resources sustainably, enhance energy efficiency, and reduce pollution. By estimating SCU, policymakers can establish targets for cutting greenhouse gas emissions, conserving natural resources, and fostering environmentally friendly practices. In summary, understanding SCU across economic, social, and environmental dimensions is vital for countries within the OECD framework. This understanding enables policymakers to make informed decisions that safeguard and advance the three pillars of sustainability. As can be found from Table 1, there is a scarcity of studies to address the sustainable capacity utilization of systems. The renewable energy performance of systems with uncertain data and concerning sustainability dimensions has been studied in some investigations such as.^18,19 However, the sustainable capacity utilization of processes involving random fixed and variable inputs has not been explored in the existing considerations.

Analyzing SCUs can often be quite complex, particularly because many applications involve random variables and undesirable outputs. To address this challenge, this paper introduces chance-constrained data envelopment analysis (CCDEA) methods that take into account both managerial and natural disposability. These methods are specifically designed to assess the SCU of entities that have both fixed and variable inputs that can fluctuate randomly. In our approach, we employ non-radial DDF models that incorporate these random elements, allowing us to calculate both input- and output-oriented random SCU measures over the short term. We then apply these techniques to appraise the SCU of OECD nations, considering the aspects of managerial and natural disposability.

Overall, this study offers several important contributions to the field:

♦ Estimating the SCU of entities when faced with random and undesirable measures,

♦ Proposing CCDEA methods that consider managerial and natural disposability to measure the SCU of decision-making units (DMUs) with random fixed and variable inputs,

♦ Providing input- and output-oriented random SCU measures utilizing the presented non-radial chance-constrained DDF frameworks, and

♦ Assessing the SCU of OECD countries while stochastic and undesirable outputs are presented.

The framework of this research is organized as detailed below. In Section 2, the proposed method is outlined for estimating the SCU for various entities, particularly in contexts where there are random and undesirable outputs. Moving on to Section 3, the SCUs specifically for OECD countries are examined. Section 4 delves into the managerial implications of our findings, addressing the challenges faced and offering recommendations for practitioners. Finally, Section 5 summarizes the conclusions and indicates directions for future research.

Methodology

The main goal of this analysis is to evaluate efficiency and SCU in the presence of random and unwanted variables. To achieve this, the next section introduces output-oriented chance-constrained DEA methods for estimating SCU in processes affected by random factors. Additionally, we expand input-oriented chance-constrained DEA frameworks to approximate random SCUs, incorporating the assumptions of natural and managerial disposability. To clarify, we utilize non-radial DDF models with random measures under these disposability frameworks to assess SCU in various processes.

Chance-constrained DEA models differ from traditional DEA models by factoring in uncertainty within the data. In these models, the DMU faces random variability in its inputs and outputs, which can influence the efficiency ratings derived from the DEA approach. The primary aim of chance-constrained DEA models is to evaluate the efficacy of DMUs while acknowledging data uncertainty. This consideration is crucial in real-world scenarios, where data can be affected by various types of variability, including measurement errors and stochastic processes.

The central concept behind chance-constrained DEA models is to include probabilistic constraints in the DEA formulation. These constraints allow for a specific level of uncertainty in the data while ensuring that the model still identifies the most efficient DMUs.

Overall, chance-constrained DEA models offer a stronger method for assessing efficiency amid uncertainty, helping decision-makers make better-informed choices in situations where data may not be entirely reliable.

Moreover, the natural disposability framework emphasizes that to reduce undesirable outputs, it is essential for an entity to minimize input usage. The DMU aims to enhance positive outputs by implementing strategies focused on reducing both inputs and unwanted outputs. This approach can be seen as a passive way of enforcing environmental regulations since both inputs and emissions are expected to decrease. Additionally, according to managerial disposability theory, a DMU should also work on improving its inputs. It's crucial to adopt measures that will not only cut down on negative outputs but also boost positive ones. Essentially, DMUs seek to leverage changes in environmental legislation by adopting advanced environmental technologies to enhance efficiency.

In summary, the idea of disposability within a business framework can be interpreted through two distinct environmental strategies. The first, referred to as the natural disposability, prioritizes the reduction of inputs to lessen unfavorable outputs while maximizing favorable ones. On the other hand, managerial disposability requires an increase in input levels to leverage regulatory advancements and emerging business opportunities.

In this section, natural and managerial disposability assumptions are applied to assess the SCU of entities with random factors.

Assume there are N DMU, $DM{U}_j(j=1,\dots ,N)$ , with I random natural disposable inputs ${\hat{x}}_{ij}$ , Q random managerial disposable inputs ${\hat{z}}_{qj}$ , T random desirable outputs ${\hat{y}}_{tj}$ and F random undesirable outputs ${\hat{w}}_{fj}$ . Random natural disposable inputs are categorized into fixed ${\hat{x}}_{ij}(i=1,\dots ,I^F)$ and variable inputs ${\hat{x}}_{ij}(i=1,\dots ,I^V)$ that $I=I^F\cup I^V$ . $DM{U}_o$ shows the unit under evaluation. $x_{ij}$ , $z_{qj}$ , $y_{tj}$ and $w_{fj}$ show the expected values of ${\hat{x}}_{ij}$ , ${\hat{z}}_{qj}$ , ${\hat{y}}_{tj}$ , and ${\hat{w}}_{fj}$ . ${\lambda }_j$ indicates the intensity variable of $j\mathrm{th}$ DMU. $\xi$ denotes the predefined risk level that $\xi \in (0,1]$ . $p$ is probability. The superscript $\wedge$ is applied to indicate random factors with normal distribution. We consider that random performance measures are classified into three dimensions, economic, environmental, and social. In the next subsection, the output-oriented SCU is evaluated.

Output-oriented sustainable capacity utilization measure

The following non-radial chance-constrained DDF model under the variable returns to scale assumption can be calculated to adjust desirable and undesirable outputs while all inputs (fixed and variable) are included.

$$\begin{array}{rl}& \begin{array}{c}Max{\alpha }_t+{\gamma }_f\\ s.t.p\left[\sum _{j=1}^N{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}\right]\ge 1-\xi ,i=1,2,\dots ,I,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{z}}_{qj}\ge {\hat{z}}_{qo}\right]\ge 1-\xi ,q=1,2,\dots ,Q,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{y}}_{tj}\ge {\hat{y}}_{to}(1+{\alpha }_t)\right]\ge 1-\xi ,t=1,\dots ,T,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{w}}_{fj}\le {\hat{w}}_{fo}(1-{\gamma }_f)\right]\ge 1-\xi ,f=1,\dots ,F,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}\end{array}$$

(1)

${\alpha }_t$ and ${\gamma }_f$ show the stochastic inefficiency related to desirable and undesirable outputs, respectively. Performance indicators are represented as stochastic variables following a normal distribution and are specific variables used to transform model (1) into a deterministic equivalent. The research conducted by Land et al., ²⁴ Cooper et al., ²⁵ and Zhou et al. ²⁶ is referenced in this approach. By examining the mean value of ${\hat{x}}_{io}$ and the standard deviation of $\sum _{j=1}^n{\lambda }_j{\hat{x}}_{ij}-{\hat{x}}_{io}$ as $x_{io}$ and ${\sigma }_i^o({\lambda }_j)$ , respectively, we can derive constraints related to inputs with natural disposability in the following way:

$$p\left\{\sum _{j=1}^n{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}\right\}=p\left\{{\displaystyle \frac{\left({\sum }_{j=1}^n{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}\right)-\left({\sum }_{j=1}^n{\lambda }_j{x}_{ij}\le x_{io}\right)}{{\sigma }_i^o({\lambda }_j)}\le {\displaystyle \frac{-\left({\sum }_{j=1}^n{\lambda }_j{x}_{ij}\le x_{io}\right)}{{\sigma }_i^o({\lambda }_j)}}}\right\}\ge 1-\xi .$$

(2)

$\sum _{j=1}^n{\lambda }_j{\hat{x}}_{ij}-{\hat{x}}_{io}$ is distributed according to a normal distribution,

$$\sum _{j=1}^n{\lambda }_j{x}_{ij}-x_{io}\le {\mathrm{\Phi }}^{-1}(\xi ){\sigma }_i^o({\lambda }_j).$$

(3)

In which ${\mathrm{\Phi }}^{-1}$ represents the inverse of the standard normal distribution $\mathrm{\Phi }$ . Moreover, it is possible to express the additional constraints of model (1) in a similar manner. Consequently, the shift from model (1) to model (4) occurs.

$$\begin{array}{r}\begin{array}{c}Max{\alpha }_t+{\gamma }_f\\ s.t.\sum _{j=1}^N{\lambda }_j{x}_{ij}-x_{io}\le {\mathrm{\Phi }}^{-1}(\xi ){\sigma }_i^o({\lambda }_j),i=1,2,\dots ,I,\\ z_{qo}-\sum _{j=1}^N{\lambda }_j{z}_{qj}\le {\mathrm{\Phi }}^{-1}(\xi ){\sigma }_q^o({\lambda }_j),q=1,2,\dots ,Q,\\ y_{to}(1+{\alpha }_t)-\sum _{j=1}^N{\lambda }_j{y}_{tj}\le {\mathrm{\Phi }}^{-1}(\xi ){\sigma }_t^o({\lambda }_j,(1+{\alpha }_t)),t=1,\dots ,T,& \\ \sum _{j=1}^N{\lambda }_j{w}_{fj}-w_{fo}(1-{\gamma }_f)\le {\mathrm{\Phi }}^{-1}(\xi ){\sigma }_f^o({\lambda }_j,(1-{\gamma }_f)),f=1,\dots ,F,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}\end{array}$$

(4)

At this moment, it is assumed that each random performance measure is uniquely identified by individual factors, specifically

$${\hat{x}}_{ij}=x_{ij}+e_{ij}\pi ,{\hat{y}}_{tj}=y_{tj}+c_{tj}\omega ,{\hat{z}}_{qj}=z_{qj}+g_{qj}\tau ,{\hat{w}}_{fj}=w_{fj}+h_{fj}\mu .$$

That $e_{ij}$ , $c_{tj}$ , $g_{qj}$ and $h_{fj}$ indicate the standard deviations, and it is assumed that $\pi$ , $\omega$ , $\tau$ and $\mu$ are independent random variables characterized by standard normal distributions. This assumption has been utilized to transform model (4) into a linear problem and has been widely applied in the domain of economics. Readers who are interested may consult Li and Huang ²⁷ as well as Zhou et al. ²⁶ . In accordance with Cooper et al., ²⁸ it is asserted that all factors are non-negative. Therefore, it can be shown that

$$\begin{array}{rl}& {\sigma }_i^o({\lambda }_j)=\left|\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right|,{\sigma }_t^o({\lambda }_j,(1+{\alpha }_t))=\left|\sum _{j=1}^N{\lambda }_j{c}_{tj}-(1+{\alpha }_t)c_{to}\right|,{\sigma }_q^o({\lambda }_j)=\left|\sum _{j=1}^N{\lambda }_j{g}_{qj}-g_{qo}\right|,\\ & {\sigma }_f^o({\lambda }_j,(1-{\gamma }_f))=\left|\sum _{j=1}^N{\lambda }_j{h}_{fj}-(1-{\gamma }_f)h_{fo}\right|.\end{array}$$

Therefore, model (4) is replaced by model (5).

$$\begin{array}{cc}Max{\alpha }_t+{\gamma }_f& \\ s.t.\sum _{j=1}^N{\lambda }_j{x}_{ij}-x_{io}\le {\mathrm{\Phi }}^{-1}(\xi )\left|\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right|,i=1,2,\dots ,I,& \\ z_{qo}-\sum _{j=1}^N{\lambda }_j{z}_{qj}\le {\mathrm{\Phi }}^{-1}(\xi )\left|\sum _{j=1}^N{\lambda }_j{g}_{qj}-g_{qo}\right|,q=1,2,\dots ,Q,& \\ y_{to}(1+{\alpha }_t)-\sum _{j=1}^N{\lambda }_j{y}_{tj}\le {\mathrm{\Phi }}^{-1}(\xi )\left|\sum _{j=1}^N{\lambda }_j{c}_{tj}-(1+{\alpha }_t)c_{to}\right|,t=1,\dots ,T,& \\ \sum _{j=1}^N{\lambda }_j{w}_{fj}-w_{fo}(1-{\gamma }_f)\le {\mathrm{\Phi }}^{-1}(\xi )\left|\sum _{j=1}^N{\lambda }_j{h}_{fj}-(1-{\gamma }_f)h_{fo}\right|,f=1,\dots ,F,& \\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(5)

Model (5) can alternatively be modified to model (6) for the specified value of $\xi \le 0.5$ . To clarify, ${\mathrm{\Phi }}^{-1}(\xi )\le 0$ for $\xi \le 0.5$ and considering the absolute value property that $|x^{\mathrm{\prime }}|\le a^{\mathrm{\prime }}\iff -a^{\mathrm{\prime }}\le x^{\mathrm{\prime }}\le a^{\mathrm{\prime }}$ , each constraint of model (5) can be substituted with two constraints as detailed below:

$$\begin{array}{c}Max{\alpha }_t+{\gamma }_f\\ s.t.-{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\le x_{io}-\sum _{j=1}^N{\lambda }_j{x}_{ij},i=1,2,\dots ,I,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\ge \sum _{j=1}^N{\lambda }_j{x}_{ij}-x_{io},i=1,2,\dots ,I,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{g}_{qj}-g_{qo}\right)\le \sum _{j=1}^N{\lambda }_j{z}_{qj}-z_{qo},q=1,2,\dots ,Q,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{g}_{qj}-g_{qo}\right)\ge z_{qo}-\sum _{j=1}^N{\lambda }_j{z}_{qj},q=1,2,\dots ,Q,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}-(1+{\alpha }_t)c_{to}\right)\le \sum _{j=1}^N{\lambda }_j{y}_{tj}-y_{to}(1+{\alpha }_t),t=1,\dots ,T,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}-(1+{\alpha }_t)c_{to}\right)\ge y_{to}(1+{\alpha }_t)-\sum _{j=1}^N{\lambda }_j{y}_{tj},t=1,\dots ,T,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{h}_{fj}-(1-{\gamma }_f)h_{fo}\right)\le w_{fo}(1-{\gamma }_f)-\sum _{j=1}^N{\lambda }_j{w}_{fj},f=1,\dots ,F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{h}_{fj}-(1-{\gamma }_f)h_{fo}\right)\ge \sum _{j=1}^N{\lambda }_j{w}_{fj}-w_{fo}(1-{\gamma }_f),f=1,\dots ,F,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(6)

The objective function of model (6) shows the output-oriented inefficiency level of the unit under consideration, $DM{U}_o$ , for the risk level $\xi$ . To estimate the stochastic efficiency for each level $\xi$ , the expression (7) can be used while ${\alpha }_t$ and ${\gamma }_f$ are achieved from model (6).

$$E_{\xi }^A=1+{\displaystyle \frac{1}{T+F}\left(\sum _{t=1}^T{\displaystyle \frac{{\alpha }_t{y}_{to}}{y_{to}}+\sum _{f=1}^F{\displaystyle \frac{{\gamma }_f{w}_{fo}}{w_{fo}}}}\right).}$$

(7)

Also, we can define

$$E_{\xi }^{\mathrm{\prime }A}={\displaystyle \frac{1}{1+{\displaystyle \frac{1}{T+F}\left({\sum }_{t=1}^T{\displaystyle \frac{{\alpha }_t{y}_{to}}{y_{to}}+{\sum }_{f=1}^F{\displaystyle \frac{{\gamma }_f{w}_{fo}}{w_{fo}}}}\right)}},}$$

(8)

in order to consider the stochastic efficiency between zero and one, i.e., $(0,1]$ . The DMU under examination is called stochastic efficient if and only if $E_{\xi }^{\mathrm{\prime }A}=1$ . Otherwise, it is stochastic inefficient.

Now, the non-radial chance-constrained DDF model under the variable returns to scale assumption can be calculated to adjust desirable and undesirable outputs while natural disposable variable inputs and managerial disposable inputs are ignored. Thus, we have

$$\begin{array}{c}Max{\alpha }_t+{\gamma }_f\\ s.t.p\left[\sum _{j=1}^N{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}\right]\ge 1-\xi ,i=1,2,\dots ,I^F,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{y}}_{tj}\ge {\hat{y}}_{to}(1+{\alpha }_t)\right]\ge 1-\xi ,t=1,\dots ,T,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{w}}_{fj}\le {\hat{w}}_{fo}(1-{\gamma }_f)\right]\ge 1-\xi ,f=1,\dots ,F,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(9)

Similar to model (1), we can transform model (9) into a linear problem:

$$\begin{array}{c}Max{\alpha }_t+{\gamma }_f\\ s.t.-{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\le x_{io}-\sum _{j=1}^N{\lambda }_j{x}_{ij},i=1,2,\dots ,I^F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\ge \sum _{j=1}^N{\lambda }_j{x}_{ij}-x_{io},i=1,2,\dots ,I^F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}-(1+{\alpha }_t)c_{to}\right)\le \sum _{j=1}^N{\lambda }_j{y}_{tj}-y_{to}(1+{\alpha }_t),t=1,\dots ,T,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}-(1+{\alpha }_t)c_{to}\right)\ge y_{to}(1+{\alpha }_t)-\sum _{j=1}^N{\lambda }_j{y}_{tj},t=1,\dots ,T,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{h}_{fj}-(1-{\gamma }_f)h_{fo}\right)\le w_{fo}(1-{\gamma }_f)-\sum _{j=1}^N{\lambda }_j{w}_{fj},f=1,\dots ,F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{h}_{fj}-(1-{\gamma }_f)h_{fo}\right)\ge \sum _{j=1}^N{\lambda }_j{w}_{fj}-w_{fo}(1-{\gamma }_f),f=1,\dots ,F,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(10)

We can define the stochastic efficiency for each level $\xi$ in the following way:

$$E_{\xi }^F=1+{\displaystyle \frac{1}{T+F}\left(\sum _{t=1}^T{\displaystyle \frac{{\alpha }_t{y}_{to}}{y_{to}}+\sum _{f=1}^F{\displaystyle \frac{{\gamma }_f{w}_{fo}}{w_{fo}}}}\right),}$$

(11)

in which ${\alpha }_t$ and ${\gamma }_f$ are obtained from model (10) and show the inefficiency scores related to desirable and undesirable outputs, respectively. To define the stochastic efficiency between 0 and 1, i.e., $(0,1]$ , we outline the following:

$$E_{\xi }^{\mathrm{\prime }F}={\displaystyle \frac{1}{1+{\displaystyle \frac{1}{T+F}\left({\sum }_{t=1}^T{\displaystyle \frac{{\alpha }_t{y}_{to}}{y_{to}}+{\sum }_{f=1}^F{\displaystyle \frac{{\gamma }_f{w}_{fo}}{w_{fo}}}}\right)}}.}$$

(12)

The entity is considered output-oriented stochastic efficient for risk level $\xi$ if $E_{\xi }^{\mathrm{\prime }F}=1$ ; otherwise, it is considered to be output-oriented stochastic inefficient.

By considering sustainability indicators, the output-oriented short-run stochastic SCU for the level $\xi$ can be estimated as follows:

$$SC{U}_{\xi }^O={\displaystyle \frac{E_{\xi }^A}{E_{\xi }^F}.}$$

(13)

Where the numerator is the output-oriented stochastic efficiency achieved from statement (7) and the denominator is the short-run output-oriented stochastic efficiency obtained from expression (11) (in which natural disposable variable inputs and managerial disposable inputs are ignored). It is clear that the denominator is greater than or equal to the numerator that leads to $0<SCU\le 1$ . It is stated that the short-run capacity is fully utilized when $SCU=1$ . $SCU<1$ demonstrates that the average rate of expansion may be heightened by eliminating limitations on natural disposable variable inputs and managerial disposable inputs.

Assume the optimal value derived from model (10) is referred to as ${\lambda }_j*$ , we consider

$${\overline{x}}_{io}=\sum _{j=1}^N{\lambda }_j*x_{ij},i=1,2,\dots ,I^V,$$

(14)

as the values of the natural disposable variable inputs and define the SCU ratio for each natural disposable variable input i in the following way:

$${\kappa }_{io}*={\displaystyle \frac{{\overline{x}}_{io}}{x_{io}}}$$

(15)

Also, by considering the optimal value ${\lambda }_j*$ obtained from model (10), we have

$${\overline{z}}_{qo}=\sum _{j=1}^N{\lambda }_j{z}_{qj},q=1,\dots ,Q,$$

(16)

as the values of the managerial disposable inputs and describe the SCU ratio for each managerial disposable input q in the next way:

$${\overline{\kappa }}_{qo}*={\displaystyle \frac{{\overline{z}}_{qo}}{z_{qo}}}$$

(17)

Input-oriented sustainable capacity utilization measure

By adjusting random natural and managerial disposable variable inputs for given technology and directions, we can estimate the stochastic inefficiency related to random natural and managerial disposable variable inputs in the following way:

$$\begin{array}{c}Max{\beta }_i+{\eta }_q\\ s.t.p\left[\sum _{j=1}^N{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}(1-{\beta }_i)\right]\ge 1-\xi ,i=1,2,\dots ,I^V,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}\right]\ge 1-\xi ,i=1,2,\dots ,I^F,I=I^F\cup I^V,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{z}}_{qj}\ge {\hat{z}}_{qo}(1+{\eta }_q)\right]\ge 1-\xi ,q=1,2,\dots ,Q,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{y}}_{tj}\ge {\hat{y}}_{to}\right]\ge 1-\xi ,t=1,\dots ,T,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{w}}_{fj}\le {\hat{w}}_{fo}\right]\ge 1-\xi ,f=1,\dots ,F,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(18)

Analogous to model (1), model (18) can be transformed into the subsequent linear problem:

$$\begin{array}{c}Max{\beta }_i+{\eta }_q\\ s.t.-{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-(1-{\beta }_i)e_{io}\right)\le x_{io}(1-{\beta }_i)-\sum _{j=1}^N{\lambda }_j{x}_{ij},i=1,2,\dots ,I^V,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-(1-{\beta }_i)e_{io}\right)\ge \sum _{j=1}^N{\lambda }_j{x}_{ij}-(1-{\beta }_i)x_{io},i=1,2,\dots ,I^V,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\le x_{io}-\sum _{j=1}^N{\lambda }_j{x}_{ij},i=1,2,\dots ,I^F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\ge \sum _{j=1}^N{\lambda }_j{x}_{ij}-x_{io},i=1,2,\dots ,I^F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{g}_{qj}-(1+{\eta }_q)g_{qo}\right)\le \sum _{j=1}^N{\lambda }_j{z}_{qj}-(1+{\eta }_q)z_{qo},q=1,2,\dots ,Q,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{g}_{qj}-(1+{\eta }_q)g_{qo}\right)\ge (1+{\eta }_q)z_{qo}-\sum _{j=1}^N{\lambda }_j{z}_{qj},q=1,2,\dots ,Q,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}-c_{to}\right)\le \sum _{j=1}^N{\lambda }_j{y}_{tj}-y_{to},t=1,\dots ,T,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}-c_{to}\right)\ge y_{to}-\sum _{j=1}^N{\lambda }_j{y}_{tj},t=1,\dots ,T,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{h}_{fj}-h_{fo}\right)\le w_{fo}-\sum _{j=1}^N{\lambda }_j{w}_{fj},f=1,\dots ,F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{h}_{fj}-h_{fo}\right)\ge \sum _{j=1}^N{\lambda }_j{w}_{fj}-w_{fo},f=1,\dots ,F,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(19)

Model (19) shows the stochastic inefficiency value for the risk level $\xi$ in which natural disposable variable inputs and managerial disposable inputs are adjusted.

Statement (20) can be applied to assess the input-oriented stochastic efficiency score for the risk level $\xi$ :

$$E_{\xi }^I=1-{\displaystyle \frac{1}{I^V+Q}\left(\sum _{i=1}^{I^V}{\displaystyle \frac{{\beta }_i{x}_{io}}{x_{io}}+\sum _{q=1}^Q{\displaystyle \frac{{\eta }_q{z}_{qo}}{z_{qo}}}}\right).}$$

(20)

The unit under assessment is called the input-oriented stochastic efficient for the risk level $\xi$ if $E_{\xi }^I=1$ . Otherwise, it is stochastically inefficient.

Next, the following input-oriented chance-constrained DDF model is introduced, which excludes undesirable outputs. In this model, the levels of desirable outputs observed on the right side of the desirable output constraints are adjusted to zero.

$$\begin{array}{c}Max{\beta }_i+{\eta }_q\\ s.t.p\left[\sum _{j=1}^N{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}(1-{\beta }_i)\right]\ge 1-\xi ,i=1,2,\dots ,I^V,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{x}}_{ij}\le {\hat{x}}_{io}\right]\ge 1-\xi ,i=1,2,\dots ,I^F,I=I^F\cup I^V,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{z}}_{qj}\ge {\hat{z}}_{qo}(1+{\eta }_q)\right]\ge 1-\xi ,q=1,2,\dots ,Q,\\ p\left[\sum _{j=1}^N{\lambda }_j{\hat{y}}_{tj}\ge 0\right]\ge 1-\xi ,t=1,\dots ,T,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(21)

Corresponding to the other aforementioned models, model (21) can be reformulated as the following linear issue:

$$\begin{array}{c}Max{\beta }_i+{\eta }_q\\ s.t.-{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-(1-{\beta }_i)e_{io}\right)\le x_{io}(1-{\beta }_i)-\sum _{j=1}^N{\lambda }_j{x}_{ij},i=1,2,\dots ,I^V,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-(1-{\beta }_i)e_{io}\right)\ge \sum _{j=1}^N{\lambda }_j{x}_{ij}-(1-{\beta }_i)x_{io},i=1,2,\dots ,I^V,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\le x_{io}-\sum _{j=1}^N{\lambda }_j{x}_{ij},i=1,2,\dots ,I^F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{e}_{ij}-e_{io}\right)\ge \sum _{j=1}^N{\lambda }_j{x}_{ij}-x_{io},i=1,2,\dots ,I^F,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{g}_{qj}-(1+{\eta }_q)g_{qo}\right)\le \sum _{j=1}^N{\lambda }_j{z}_{qj}-(1+{\eta }_q)z_{qo},q=1,2,\dots ,Q,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{g}_{qj}-(1+{\eta }_q)g_{qo}\right)\ge (1+{\eta }_q)z_{qo}-\sum _{j=1}^N{\lambda }_j{z}_{qj},q=1,2,\dots ,Q,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}\right)\le \sum _{j=1}^N{\lambda }_j{y}_{tj},t=1,\dots ,T,\\ -{\mathrm{\Phi }}^{-1}(\xi )\left(\sum _{j=1}^N{\lambda }_j{c}_{tj}\right)\ge -\sum _{j=1}^N{\lambda }_j{y}_{tj},t=1,\dots ,T,\\ \sum _{j=1}^N{\lambda }_j=1,{\lambda }_j\ge 0.\end{array}$$

(22)

Model (22) indicates the stochastic inefficiency level for the risk level $\xi$ in which natural disposable variable inputs and managerial disposable inputs are adjusted while the desirable output levels on the right side of the desirable output constraints are modified to be zero.

Expression (23) displays the stochastic efficiency of the unit under measurement for the risk level $\xi$ :

$$E_{\xi }^0=1-{\displaystyle \frac{1}{I^V+Q}\left(\sum _{i=1}^{I^V}{\displaystyle \frac{{\beta }_i{x}_{io}}{x_{io}}+\sum _{q=1}^Q{\displaystyle \frac{{\eta }_q{z}_{qo}}{z_{qo}}}}\right),}$$

(23)

that ${\beta }_i$ and ${\eta }_q$ are achieved by model (22).

The input-oriented short-run stochastic SCU for the level $\xi$ can be approximated by examining sustainability indicators in the following manner:

$$SC{U}_{\xi }^I={\displaystyle \frac{E_{\xi }^I}{E_{\xi }^0}.}$$

(24)

It is evident that the numerator is greater than or equal to the denominator, resulting in $SCU\ge 1$ . It should be noted that optimal random short-term sustainable capacity utilization occurs when SCU is equal to 1. This measure indicates the extent to which both random natural disposable variable inputs and random managerial disposable inputs can be adjusted (compared to zero production) to achieve the observed output quantity with the available random fixed natural disposable input resources.

Imagine ${\lambda }_j^{\mathrm{\prime }}$ as the optimal values achieved from model (22); thus, we regard

$$x_{io}^{\mathrm{\prime }}=\sum _{j=1}^N{\lambda }_j^{\mathrm{\prime }}{x}_{ij},i=1,2,\dots ,I^V,$$

(25)

as the values of the natural disposable variable inputs and define the SCU ratio for each natural disposable variable input i in the following way:

$${\kappa }_{io}^{\mathrm{\prime }}={\displaystyle \frac{x_{io}^{\mathrm{\prime }}}{x_{io}}}$$

(26)

By regarding ${\lambda }_j^{\mathrm{\prime }}$ as the best possible result obtained from model (22), we have

$$z_{qo}^{\mathrm{\prime }}=\sum _{j=1}^N{\lambda }_j^{\mathrm{\prime }}{z}_{qj},q=1,\dots ,Q,$$

(27)

as the values of the managerial disposable inputs and establish the SCU ratio for each managerial disposable input q as follows:

$${\overline{\kappa }}_{qo}^{\mathrm{\prime }}={\displaystyle \frac{z_{qo}^{\mathrm{\prime }}}{z_{qo}}}$$

( 28)

The values resulted from statements (15), (17), (26) and (28) show the changes of natural disposable variable inputs and managerial disposable inputs. In fact, factors exhibit surpluses for values below one and deficiencies for values above one. The algorithm proposed in the text outlines the essential steps for identifying the SCU of entities, beginning with the classification of inputs and outputs, as shown in Figure 1. To ensure sustainable development, it is crucial to consider the economic, social, and environmental aspects of the issue. Previous research has provided comprehensive factors for this analysis. Data collected is then considered as random measures before conducting an examination of the SCU using models (13) and (24) for output orientation and input orientation. Non-radial efficiencies are evaluated using expressions (8), (12), (20), and (23). Statements (15) and (26) are used to assess the SCU ratio of natural disposable variable inputs and expressions (17) and (28) are applied to estimate the SCU of managerial disposable inputs. Finally, the results are analyzed to specify the SCU of entities.

Figure 1.

Proposed algorithm to assess the SCU.

Application

Estimating SCU is crucial for OECD countries because it allows them to run their economies at peak efficiency without depleting natural resources or harming the environment. This importance becomes even clearer when we consider the three key aspects of sustainability: environmental, social, and economic. From an environmental standpoint, sustainable capacity utilization means that industries and businesses can operate in ways that minimize their impact on the planet. By effectively estimating and managing resource use, countries can reduce pollution, conserve natural resources, and mitigate the effects of climate change. Striking a balance between economic growth and environmental protection is essential for the long-term well-being of both people and the Earth. On the social side, it's vital that SCU promotes social equity and enhances the overall well-being of the population. By optimizing resource use and productivity, economies can create more jobs, improve living standards, and elevate the quality of life for their citizens. Estimating SCU also helps ensure that resources are accessible to all, helping to close income gaps and resource access.

Economically, estimating SCU is key for maintaining stability. By avoiding overexploitation of resources and preventing production bottlenecks, countries can achieve steady economic performance and resilience against external shocks. SCU fosters increased competitiveness, innovation, and overall economic prosperity among OECD nations.

In summary, to foster a comprehensive approach to development in OECD countries, estimating SCU must address environmental, social, and economic aspects. Balancing these three dimensions will enable nations to integrate economic advancement, social welfare, and environmental stewardship for a sustainable future.

The SCU of the 30 OECD countries presented in Table 2 is addressed. This information comes from refs.,^29,30 and random measurements follow a normal distribution. We have analyzed the performance metrics and detailed them in Table 3 after reviewing existing research and studies. With access to data on input and output indicators for these countries from 2019 to 2021, we have evaluated their performance and SCU. Other countries were excluded from this analysis due to a lack of complete and reliable data. The choice of input and output parameters was based on their availability, reliability, and comparability across different nations and regions. Average values and standard deviations of the performance parameters are included in the Appendix (Table A1 and Table A2).

Table 2.

Countries under consideration.

Number	Country	Year of accession	Number	Country	Year of accession
1	Austria	1961	16	Lithuania	2018
2	Belgium	1961	17	Luxembourg	1961
3	Canada	1961	18	Netherlands	1961
4	Czechia	1995	19	New Zealand	1973
5	Denmark	1961	20	Norway	1961
6	Estonia	2010	21	Poland	1996
7	Finland	1969	22	Portugal	1961
8	France	1961	23	Slovak Republic	2000
9	Germany	1961	24	Slovenia	2010
10	Greece	1961	25	Spain	1961
11	Hungary	1996	26	Sweden	1961
12	Ireland	1961	27	Switzerland	1961
13	Italy	1962	28	United Kingdom	1961
14	Japan	1964	29	United States	1961
15	Latvia	2016	30	Australia	1971

Table 3.

Explanation of variables.

Variables	Definitions	Unit	Dimension	Reference
Labor force (LF)	Workforce comprises the total of employed individuals and job seekers by the end of the year.	Total, thousand persons	Economic	31–38
Fossil fuel consumption (FFC)	The term “fossil fuels” refers to finite resources such as oil, coal, and natural gas.	TWh	Environmental	31–34,38,39
Renewable energy consumption (REC)	In this research, REC predominantly derives from sustainable and environmentally sound power sources: solar, wind, and hydropower.	TWh	Environmental	31–34,38,39
Investment stock	Investment stocks refer to the procurement of a fabricated item (which may encompass the acquisition of a pre-owned item), subtracting any selling transactions, encompassing the creation of an item intended for the manufacturer's utilization.	Million US $	Economic	38,40
Gross domestic product (GDP)	GDP measures the overall economic performance of a nation based on the total expenditure on final goods and services, excluding imports.	US $/capita	Economic	31–39
CO2	CO2 is linked to energy use and primarily results from the consumption of fossil fuels.	Tones/capita	Environmental	31,33–35,38,39,41
Life expectancy	Period life expectancy is a measure that condenses mortality rates from all age groups into a single value for a specific year.	Years	Social	41,42

Natural disposable inputs in production include the labor force, fossil fuel consumption, and investment stock. The labor force and fossil fuel consumption are flexible inputs that can be adjusted based on production needs. In contrast, investment stock is a fixed input that isn’t easily changed in the short term. From a managerial standpoint, renewable energy consumption is viewed as a resource that can be managed. When we look at indicators, gross domestic product (GDP) and life expectancy stand out as two key outputs we want to achieve, while CO2 emissions are seen as negative outputs. Also, renewable energy consumption is something that managers can control and allocate within their organizations. GDP reflects a country's economic health, while life expectancy gives us an idea of how long people are expected to live on average. Together, these factors offer insights into the overall well-being and prosperity of a nation.

By boosting renewable energy use, managers can potentially enhance GDP and improve life expectancy. For example, cost savings, better energy efficiency, and reduced environmental impacts that come with renewable energy can contribute to economic growth and healthier populations. However, these benefits can be overshadowed by ongoing CO2 emissions, which are harmful to our environment and contribute to global warming. Reducing these emissions is crucial for mitigating the negative effects of greenhouse gases on both the planet and human health.

This framework emphasizes the significance of managing renewable energy consumption to foster economic growth and well-being while also minimizing harmful environmental effects like CO2 emissions.

In Table 3, we outline the performance parameters, with the fourth column detailing sustainability dimensions and the last column listing relevant studies. Economic indicators include the labor force, investment stock, and GDP, while environmental measures consist of fossil fuel consumption, renewable energy consumption, and CO2 emissions. Life expectancy is the sole social factor included.

Accordingly, the stochastic performance and SCU of the 30 OECD countries are analyzed. By considering the risk levels 0.01, 0.1, 0.3, and 0.5, the results achieved from expressions (6)-(8) and also statements (10)-(12) for including natural disposable variable inputs and managerial disposable inputs and ignoring them are presented in Table 4. As can be seen in Figures 2 and 3, the output-oriented efficiency scores achieved from statements (8) and (12) are non-increasing when the risk level increases. This means that the efficiency scores decrease or are without change when the risk levels increase. In all cases under examination, Poland gained the least efficiency score for all risk levels considered. Furthermore, the number of efficient countries is non-increasing as the risk level increases. The comparison of $E_{\xi }^A$ and $E_{\xi }^F$ for different risk levels shows that some countries such as Austria, Japan, and Spain worsened the stochastic performance by meaning that these countries can increase GDP and life expectancy and decrease CO2 emissions provided that they provide changes in the number of labor force, fossil fuel consumption, and renewable energy consumption.

Figure 2.

Stochastic efficiency for different risk levels.

Figure 3.

Biased stochastic efficiency for different risk levels.

Table 4.

Output-oriented efficiency scores.

#	$\xi =0.01$				$\xi =0.1$				$\xi =0.3$				$\xi =0.5$
	$E_{\xi }^A$	$E_{\xi }^{\mathrm{\prime }A}$	$E_{\xi }^F$	$E_{\xi }^{\mathrm{\prime }F}$	$E_{\xi }^A$	$E_{\xi }^{\mathrm{\prime }A}$	$E_{\xi }^F$	$E_{\xi }^{\mathrm{\prime }F}$	$E_{\xi }^A$	$E_{\xi }^{\mathrm{\prime }A}$	$E_{\xi }^F$	$E_{\xi }^{\mathrm{\prime }F}$	$E_{\xi }^A$	$E_{\xi }^{\mathrm{\prime }A}$	$E_{\xi }^F$	$E_{\xi }^{\mathrm{\prime }F}$
1	1.157	0.864	1.167	0.857	1.166	0.858	1.175	0.851	1.171	0.854	1.18	0.847	1.179	0.848	1.19	0.84
2	1.245	0.803	1.245	0.803	1.253	0.798	1.253	0.798	1.26	0.794	1.26	0.794	1.265	0.79	1.266	0.79
3	1	1	1.473	0.679	1	1	1.489	0.672	1	1	1.502	0.666	1	1	1.514	0.66
4	1.371	0.729	1.371	0.729	1.398	0.715	1.398	0.715	1.437	0.696	1.442	0.694	1.472	0.679	1.475	0.678
5	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
6	1	1	1	1	1	1	1.292	0.774	1.327	0.754	1.329	0.753	1.348	0.742	1.349	0.741
7	1.101	0.908	1.121	0.892	1.133	0.882	1.147	0.872	1.158	0.864	1.172	0.853	1.175	0.851	1.19	0.841
8	1.061	0.942	1.163	0.859	1.086	0.921	1.17	0.854	1.102	0.908	1.175	0.851	1.112	0.9	1.178	0.849
9	1	1	1.239	0.807	1.016	0.985	1.248	0.801	1.033	0.968	1.254	0.797	1.045	0.957	1.267	0.789
10	1.411	0.709	1.442	0.694	1.436	0.696	1.457	0.686	1.45	0.69	1.47	0.68	1.46	0.685	1.483	0.674
11	1	1	1.264	0.791	1	1	1.294	0.773	1.335	0.749	1.335	0.749	1.364	0.733	1.364	0.733
12	1	1	1	1	1	1	1.001	0.999	1	1	1.045	0.957	1	1	1.061	0.942
13	1.162	0.861	1.267	0.789	1.185	0.844	1.271	0.787	1.198	0.835	1.274	0.785	1.206	0.829	1.277	0.783
14	1.21	0.826	1.365	0.732	1.259	0.794	1.395	0.717	1.314	0.761	1.444	0.693	1.358	0.737	1.477	0.677
15	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1.034	0.967
16	1	1	1	1	1	1	1	1	1	1	1	1	1.003	0.997	1.003	0.997
17	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
18	1.219	0.82	1.23	0.813	1.231	0.812	1.235	0.81	1.238	0.808	1.238	0.808	1.241	0.806	1.248	0.802
19	1.106	0.904	1.139	0.878	1.167	0.857	1.187	0.843	1.208	0.828	1.227	0.815	1.235	0.81	1.257	0.795
20	1	1	1.021	0.979	1	1	1.041	0.961	1	1	1.058	0.945	1	1	1.072	0.933
21	1.545	0.647	1.546	0.647	1.55	0.645	1.571	0.637	1.575	0.635	1.604	0.624	1.604	0.624	1.627	0.615
22	1.041	0.961	1.048	0.954	1.069	0.935	1.077	0.929	1.093	0.915	1.102	0.908	1.108	0.902	1.119	0.894
23	1.274	0.785	1.274	0.785	1.359	0.736	1.359	0.736	1.422	0.703	1.423	0.703	1.467	0.681	1.474	0.679
24	1.119	0.894	1.125	0.889	1.176	0.85	1.18	0.847	1.218	0.821	1.222	0.819	1.248	0.801	1.252	0.799
25	1.067	0.938	1.136	0.881	1.124	0.89	1.192	0.839	1.167	0.857	1.237	0.809	1.2	0.833	1.271	0.787
26	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
27	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
28	1.137	0.879	1.229	0.814	1.15	0.87	1.235	0.81	1.159	0.863	1.241	0.806	1.164	0.859	1.245	0.803
29	1	1	1.233	0.811	1	1	1.296	0.772	1	1	1.36	0.735	1	1	1.423	0.703
30	1.344	0.744	1.399	0.715	1.362	0.734	1.406	0.711	1.377	0.726	1.411	0.709	1.4	0.714	1.428	0.7
Average	1.119	0.907133	1.183233	0.859933	1.137333	0.894067	1.2123	0.8398	1.174733	0.867633	1.2335	0.826667	1.188467	0.859267	1.251467	0.8157
Max	1.545	1	1.546	1	1.55	1	1.571	1	1.575	1	1.604	1	1.604	1	1.627	1
Min	1	0.647	1	0.647	1	0.645	1	0.637	1	0.635	1	0.624	1	0.624	1	0.615
% stochastic efficient countries	43.3%	43.3%	26.7%	26.7%	40%	40%	20%	20%	33.3%	33.3%	20%	20%	30%	30%	13.3%	13.3%

Next, the output-oriented short-run stochastic SCU for levels 0.01, 0.1, 0.3, and 0.5 is assessed by using expression (13). The findings are presented in Table 5. As can be seen, the short-run capacity is fully utilized in some countries, including Denmark, Lithuania, Luxembourg, Sweden, and Switzerland for risk levels under examination with $SCU=1$ . $SCU<1$ demonstrates that the average rate of expansion may be heightened by eliminating limitations on natural disposable variable inputs and managerial disposable inputs. The short-run capacity of Belgium is fully utilized at levels 0.01, 0.1, and 0.3 while the utilization of labor force, fossil fuel consumption, and renewable energy consumption should be addressed at risk level 0.5. Also, Czechia has applied full capacity for levels 0.01 and 0.1, while the usage of labor force, fossil fuel consumption, and renewable energy consumption should be investigated for levels 0.3 and 0.5. Among countries, Canada has had the least output-oriented short-run stochastic SCU for different risk levels that shows labor force, fossil fuel consumption and renewable energy consumption need to be dealt with. As can be seen in the last raw of Table 5, the percentage of stochastic-efficient countries is non-increasing as the risk levels increase. It is clear that uncertainty and risk levels may change SCU ratios. By applying statement (15), the utilization rates of natural disposable variable inputs, labor force, and fossil fuel consumption for different risk levels, 0.01, 0.1, 0.3, and 0.5, are measured. The outcomes are provided in Table 6. ${\kappa }_{io}*>1$ shows the shortfalls of labor force and fossil fuel consumption and ${\kappa }_{io}*<1$ indicates the their excesses. The utilization rate of natural disposable variable inputs, labor force, and fossil fuel consumption for all risk levels under examination is optimal in some countries, i.e., Denmark, Luxembourg, Sweden, and Switzerland. In Estonia, the utilization rate of the labor force and fossil fuel consumption for the risk level 0.01 is optimal, but excesses of the labor force and fossil fuel consumption for the risk levels 0.1 and 0.3 are observed. For the risk level 0.5, a shortfall in the labor force and the excess in fossil fuel consumption appear. In this case, the importance of risk levels is clearer. Similarly, the utilization rate of the labor force and fossil fuel consumption for other countries can be analyzed. Moreover, statement (17) is used to measure the utilization rates of the managerial disposable input, renewable energy consumption, for different risk levels, 0.01, 0.1, 0.3, and 0.5. The findings are shown in Table 7. ${\overline{\kappa }}_{qo}*>1$ exhibits the shortfalls in renewable energy consumption and ${\overline{\kappa }}_{qo}*<1$ indicates the excesses of them. In Poland, the shortfall of renewable energy consumption is observed for $\xi =0.01$ while the excesses of them are found for other risk levels under examinations. Two countries Sweden and Switzerland operate optimally for all risk levels under examination. Also, as can be seen, ratios may change when the risk level changes.

Table 5.

Output-oriented SCU measures for different risk levels.

Country	$SC{U}_{\xi }^O$
	$\xi =0.01$	$\xi =0.1$	$\xi =0.3$	$\xi =0.5$
1	0.991431	0.99234	0.992373	0.990756
2	1	1	1	0.99921
3	0.678887	0.671592	0.665779	0.660502
4	1	1	0.996533	0.997966
5	1	1	1	1
6	1	0.773994	0.998495	0.999259
7	0.982159	0.987794	0.988055	0.987395
8	0.912296	0.928205	0.937872	0.943973
9	0.807103	0.814103	0.823764	0.824783
10	0.978502	0.985587	0.986395	0.984491
11	0.791139	0.772798	1	1
12	1	0.999001	0.956938	0.942507
13	0.917127	0.932337	0.940345	0.944401
14	0.886447	0.902509	0.909972	0.919431
15	1	1	1	0.967118
16	1	1	1	1
17	1	1	1	1
18	0.991057	0.996761	1	0.994391
19	0.971027	0.983151	0.984515	0.982498
20	0.979432	0.960615	0.94518	0.932836
21	0.999353	0.986633	0.98192	0.985864
22	0.993321	0.992572	0.991833	0.99017
23	1	1	0.999297	0.995251
24	0.994667	0.99661	0.996727	0.996805
25	0.939261	0.942953	0.943411	0.944138
26	1	1	1	1
27	1	1	1	1
28	0.925142	0.931174	0.933924	0.93494
29	0.81103	0.771605	0.735294	0.702741
30	0.960686	0.968706	0.975904	0.980392
Average	0.950336	0.943035	0.956151	0.953394
Max	1	1	1	1
Min	0.678887	0.671592	0.665779	0.660502
% stochastic efficient countries	36.7%	30%	30%	20%

Table 6.

Random natural disposable variable input utilization rate using expression (15).

Country	$\xi =0.01$		$\xi =0.1$		$\xi =0.3$		$\xi =0.5$
	LF	FFC	LF	FFC	LF	FFC	LF	FFC
1	0.909934	0.512715	0.909934	0.512715	0.918188	0.512581	0.942448	0.467827
2	0.899876	0.266439	0.900607	0.249194	0.900996	0.249277	0.457194	0.228492
3	0.026691	0.019854	0.026691	0.019854	0.01554	0.015493	0.582234	0.015493
4	0.649387	0.318905	0.660524	0.322824	0.279166	0.275454	0.478161	0.302775
5	1	1	1	1	1	1	1	1
6	1	1	0.47178	0.736944	0.484271	0.744833	1.036774	0.747889
7	0.857018	0.611739	0.708141	0.487087	0.718156	0.491117	1.024426	0.494047
8	0.166081	0.111768	0.166081	0.111768	0.166081	0.111768	1	0.111768
9	0.1135	0.054294	0.1135	0.054294	0.1135	0.054294	0.501704	0.053919
10	0.195022	0.229735	0.195022	0.229735	0.130334	0.219264	0.69726	0.22453
11	0.166012	0.502119	0.136635	0.430169	0.136635	0.430169	0.49495	0.401982
12	1	1	1.829509	1.076146	2.103128	1.129569	1.089477	1.006944
13	0.187672	0.102159	0.187672	0.102159	0.194436	0.102209	1.036038	0.102209
14	0.071383	0.03399	0.035813	0.033755	0.035813	0.033755	1	0.033755
15	1	1	1	1	1	1	0.391384	1.272469
16	1	1.000005	1	1.000005	1	1.000005	0.8695	0.728027
17	1	1	1	1	1	1	1	1
18	0.522239	0.169327	0.522239	0.169327	0.522239	0.169327	0.501704	0.168159
19	0.684642	0.567642	0.503521	0.446113	0.503521	0.446113	0.584826	0.42794
20	1.149523	0.736879	1.185251	0.755061	1.200637	0.75942	1.022112	0.762489
21	0.262936	0.132216	0.137977	0.126255	0.137977	0.126255	1.049824	0.131467
22	0.383896	0.410411	0.470002	0.492922	0.520209	0.550042	1.114374	0.552719
23	0.363122	0.431334	0.363122	0.431334	0.209203	0.40352	0.613121	0.417509
24	0.408296	0.89017	0.408296	0.89017	0.408296	0.89017	0.879061	0.885234
25	0.214488	0.135725	0.214488	0.135725	0.214488	0.135725	1	0.135725
26	1	1	1	1	1	1	1	1
27	1	1	1	1	1	1	1	1
28	0.136048	0.09347	0.144883	0.093548	0.144883	0.093548	1	0.093548
29	0.015286	0.006872	0.015286	0.006872	0.015286	0.006872	0.127178	0.001861
30	0.151449	0.085781	0.151449	0.085781	0.151449	0.085781	0.152609	0.02699

Table 7.

Random managerial disposable variable input utilization rate.

Country	${\overline{\kappa }}_{qo}*$ related to REC
	$\xi =0.01$	$\xi =0.1$	$\xi =0.3$	$\xi =0.5$
1	0.605915	0.605915	0.613908	0.569412
2	2.257419	1.530458	1.531153	0.412034
3	0.005015	0.005015	0.002291	0.002291
4	2.606157	2.653584	0.652483	0.745955
5	0.999995	0.999995	0.999995	0.999995
6	1.000043	0.377739	0.393783	0.400043
7	0.495598	0.396752	0.403003	0.407547
8	0.314448	0.314448	0.314448	0.314448
9	0.158421	0.158421	0.158421	0.047098
10	0.317342	0.317342	0.13913	0.146712
11	3.882367	3.466776	3.466776	1.011551
12	1	6.63597	5.999	3.363636
13	0.324452	0.324452	0.341189	0.341189
14	0.209962	0.062421	0.062421	0.062421
15	1.000034	1.000034	1.000034	0.427138
16	1	1	1	2.660857
17	1.000125	1.000125	1.000125	1.000125
18	1.140411	1.140411	1.140411	0.339041
19	0.436035	0.296968	0.296968	0.106352
20	0.190419	0.196756	0.199398	0.201258
21	1.090124	0.352258	0.352258	0.370787
22	0.50279	0.640282	0.717847	0.723145
23	0.985821	0.985821	0.336536	0.364179
24	0.353571	0.353571	0.353571	0.244214
25	0.353503	0.353503	0.353503	0.353503
26	1	1	1	1
27	1	1	1	1
28	0.274946	0.300813	0.300813	0.300813
29	0.012665	0.012665	0.012665	0.001024
30	0.155184	0.155184	0.155184	0.015211

In this stage, by considering the risk levels 0.01, 0.1, 0.3, and 0.5, the input-oriented stochastic performance using expressions (19)-(20) and also statements (22)-(23) is assessed in two cases, containing all measures and also without undesirable outputs such that the levels of output observed on the right side of the output constraints are adjusted to be zero. The consequences are provided in Table 8. As shown, the input-oriented efficiency scores resulted from statements (20) and (23) are non-increasing when the risk level increases. This implies that performance metrics decline or remain constant with increased risk levels. In all instances under consideration, Poland obtained the lowest input-oriented efficiency score for all risk levels taken into account. Furthermore, the number of efficient countries does not increase as the risk level rises. The analysis of $E_{\xi }^I$ and $E_{\xi }^0$ for varying risk levels reveals that certain countries such as Austria, Belgium, and Finland have deteriorated their stochastic performance, implying that these nations have the potential to change labor force, fossil fuel consumption and renewable energy consumption. Some countries, such as Canada, Latvia, Luxembourg, Norway, and United States, are determined to be stochastic efficient in all risk levels under investigation and both cases, containing every possible measure and ensuring no undesirable outputs while the levels of output that fall beyond the constraints on the right side are adjusted to zero.

Table 8.

Input-oriented efficiency scores.

Country	$\xi =0.01$		$\xi =0.1$		$\xi =0.3$		$\xi =0.5$
	$E_{\xi }^I$	$E_{\xi }^0$	$E_{\xi }^I$	$E_{\xi }^0$	$E_{\xi }^I$	$E_{\xi }^0$	$E_{\xi }^I$	$E_{\xi }^0$
1	0.501	0.432	0.39	0.387	0.365	0.36	0.349	0.344
2	0.163	0.132	0.152	0.13	0.149	0.128	0.147	0.127
3	1	1	1	1	1	1	1	1
4	0.121	0.11	0.115	0.104	0.111	0.1	0.108	0.097
5	1	0.422	1	0.393	1	0.376	1	0.366
6	1	1	1	1	1	0.68	1	0.553
7	0.655	0.465	0.515	0.444	0.468	0.431	0.458	0.423
8	0.605	0.416	0.49	0.404	0.478	0.388	0.47	0.378
9	1	0.553	1	0.551	0.791	0.55	0.738	0.549
10	1	0.329	1	0.215	1	0.167	1	0.163
11	1	1	1	1	0.121	0.106	0.08	0.08
12	1	0.243	1	0.24	1	0.238	1	0.236
13	1	0.403	0.748	0.377	0.507	0.357	0.412	0.343
14	1	0.32	1	0.287	1	0.269	1	0.258
15	1	1	1	1	1	1	1	1
16	1	0.386	1	0.355	1	0.333	1	0.318
17	1	1	1	1	1	1	1	1
18	0.122	0.119	0.113	0.109	0.106	0.101	0.101	0.095
19	1	0.58	1	0.501	1	0.44	1	0.399
20	1	1	1	1	1	1	1	1
21	0.063	0.06	0.062	0.059	0.061	0.058	0.06	0.058
22	1	0.294	1	0.27	1	0.256	1	0.247
23	0.262	0.217	0.248	0.208	0.238	0.202	0.234	0.197
24	1	0.773	1	0.699	1	0.648	1	0.613
25	1	0.382	1	0.369	0.563	0.358	0.433	0.351
26	1	0.764	1	0.74	1	0.722	1	0.711
27	1	0.472	1	0.419	1	0.388	1	0.37
28	0.463	0.393	0.455	0.386	0.449	0.381	0.445	0.37
29	1	1	1	1	1	1	1	1
30	1	0.142	1	0.109	1	0.103	1	0.099
Average	0.7985	0.513567	0.776267	0.491867	0.713567	0.438	0.701167	0.424833
Max	1	1	1	1	1	1	1	1
Min	0.063	0.06	0.062	0.059	0.061	0.058	0.06	0.058
% stochastic efficient countries	70%	23.3%	66.7%	23.3%	56.7%	16.7%	56.7%	16.7%

Subsequently, the input-oriented short-run stochastic SCU for levels 0.01, 0.1, 0.3, and 0.5 is evaluated using expression (24). The results are depicted in Table 9. It is evident that certain countries, such as Canada, Latvia, Luxembourg, Norway, and the United States, have fully optimized their short-run capacity for the examined risk levels with $SCU=1$ . $SCU>1$ suggests that removing constraints on natural disposable variable inputs and managerial disposable inputs could potentially enhance the average growth rate. Estonia, on the other hand, has fully utilized its short-run capacity for levels 0.01 and 0.1, but the utilization of labor, fossil fuel consumption, and renewable energy consumption requires attention at risk levels of 0.3 and 0.5. Australia, among all countries, displays the most input-oriented short-run stochastic SCU for various risk levels, indicating the need to essential change in the variable inputs, i.e., labor, fossil fuel consumption, and renewable energy consumption. The utilization rates of natural disposable variable inputs, labor, and fossil fuel consumption are then assessed for risk levels of 0.01, 0.1, 0.3, and 0.5 using equation (26), with the outcomes presented in Table 10. The table highlights deficiencies and excesses in labor and fossil fuel consumption, with certain countries, such as Canada, Latvia, Luxembourg and the United States, achieving optimal utilization rates across all risk levels. In Estonia, an optimal utilization rate is observed for labor and fossil fuel consumption at risk levels of 0.01 and 0.1, but moderations are noted at higher risk levels of 0.3 and 0.5. The significance of risk levels becomes more pronounced in this context. Similar analyses can be conducted for other countries’ utilization rates of labor and fossil fuel consumption.

Table 9.

Input-oriented SCU measures for different risk levels.

Country	$SC{U}_{\xi }^I$
	$\xi =0.01$	$\xi =0.1$	$\xi =0.3$	$\xi =0.5$
1	1.159722	1.007752	1.013889	1.014535
2	1.234848	1.169231	1.164063	1.15748
3	1	1	1	1
4	1.1	1.105769	1.11	1.113402
5	2.369668	2.544529	2.659574	2.73224
6	1	1	1.470588	1.808318
7	1.408602	1.15991	1.085847	1.082742
8	1.454327	1.212871	1.231959	1.243386
9	1.808318	1.814882	1.438182	1.344262
10	3.039514	4.651163	5.988024	6.134969
11	1	1	1.141509	1
12	4.115226	4.166667	4.201681	4.237288
13	2.48139	1.984085	1.420168	1.201166
14	3.125	3.484321	3.717472	3.875969
15	1	1	1	1
16	2.590674	2.816901	3.003003	3.144654
17	1	1	1	1
18	1.02521	1.036697	1.049505	1.063158
19	1.724138	1.996008	2.272727	2.506266
20	1	1	1	1
21	1.05	1.050847	1.051724	1.034483
22	3.401361	3.703704	3.90625	4.048583
23	1.207373	1.192308	1.178218	1.187817
24	1.293661	1.430615	1.54321	1.631321
25	2.617801	2.710027	1.572626	1.233618
26	1.308901	1.351351	1.385042	1.40647
27	2.118644	2.386635	2.57732	2.702703
28	1.178117	1.178756	1.178478	1.202703
29	1	1	1	1
30	7.042254	9.174312	9.708738	10.10101
Average	1.895158	2.044311	2.102327	2.140285
Maximum	7.042254	9.174312	9.708738	10.10101
Minimum	1	1	1	1
% efficient countries	23.3%	23.3%	16.7%	20%

Table 10.

Random natural disposable variable input utilization rate using expression (26).

Country	$\xi =0.01$		$\xi =0.1$		$\xi =0.3$		$\xi =0.5$
	LF	FFC	LF	FFC	LF	FFC	LF	FFC
1	0.615482	0.605185	0.563886	0.562212	0.532398	0.535985	0.512986	0.519819
2	0.238298	0.148375	0.236718	0.14768	0.235404	0.147101	0.234398	0.14666
3	1	1	1	1	1	1	1	1
4	0.143883	0.17312	0.136276	0.16759	0.131633	0.164214	0.12877	0.162134
5	0.405923	0.717783	0.399972	0.71033	0.39634	0.705783	0.3941	0.702974
6	1	1	1	1	0.519259	0.765852	0.544841	0.781611
7	0.629571	0.676831	0.61953	0.668479	0.613402	0.663375	0.609624	0.660235
8	0.291587	0.729238	0.096246	0.123073	0.096246	0.123073	0.096246	0.123073
9	0.229538	0.421002	0.229075	0.419981	0.22872	0.419198	0.228465	0.418635
10	0.17879	0.26196	0.202571	0.283611	0.207855	0.28842	0.20341	0.284374
11	1	0.999998	1	0.999998	0.027108	0.225652	0.056031	0.184243
12	0.303722	0.414375	0.30122	0.41234	0.299132	0.411014	0.298196	0.41025
13	0.390281	0.781416	0.370478	0.737424	0.357092	0.70498	0.347467	0.681648
14	0.293083	0.590092	0.284351	0.57092	0.268924	0.53705	0.25869	0.514579
15	1	1	1	1	1	1	1	1
16	0.284538	0.713514	0.27442	0.702032	0.268246	0.695011	0.264439	0.690697
17	1	1	1	1	1	1	1	1
18	0.211606	0.138698	0.194491	0.129761	0.180039	0.122306	0.169162	0.116616
19	0.473472	0.584576	0.507884	0.615483	0.531506	0.636695	0.546479	0.650139
20	1	1.000002	1	1.000002	1	1.000002	1	1.000002
21	0.089567	0.088244	0.088276	0.087289	0.087488	0.086706	0.087002	0.086347
22	0.285114	0.491031	0.277521	0.481292	0.272887	0.475349	0.270031	0.471682
23	0.216381	0.408446	0.208016	0.399948	0.202912	0.394755	0.199764	0.391559
24	0.397082	0.923404	0.412684	0.939574	0.423393	0.95066	0.430515	0.958043
25	0.124298	0.149454	0.124298	0.149454	0.124298	0.149454	0.124298	0.149454
26	0.445811	0.7644	0.515157	0.863065	0.515157	0.863065	0.515157	0.863065
27	0.427988	0.856641	0.394239	0.802179	0.373643	0.768945	0.360946	0.748455
28	0.157721	0.322614	0.157721	0.322614	0.157721	0.322614	0.316068	0.794054
29	1	1	1	1	1	1	1	1
30	0.249088	0.16053	0.208839	0.110014	0.200373	0.106229	0.193887	0.10333

The rates of using the managerial disposable input, renewable energy consumption, are evaluated for diverse degrees of risk (0.01, 0.1, 0.3, and 0.5) using equation (28). The consequences are shown in Table 11, which indicates the shortages and surpluses in renewable energy consumption. In most countries under examination, shortages of renewable energy consumption have been observed. Some countries, including Canada and the United States, show optimal utilization rates across all risk levels.

Table 11.

Random managerial disposable input utilization rate.

Country	${\overline{\kappa }}_{qo}^{\mathrm{\prime }}$ related to (REC)
	$\xi =0.01$	$\xi =0.1$	$\xi =0.3$	$\xi =0.5$
1	2.457634	2.227261	2.086673	2
2	2.044212	2.0263	2.011404	1.999995
3	1	1	1	1
4	2.410517	2.203854	2.077753	1.999989
5	2.079787	2.039629	2.015109	2.000005
6	1.000043	1.000043	1.317261	1.667609
7	2.079463	2.03946	2.015045	1.999997
8	1.823581	1.085929	1.085929	1.085929
9	1.002986	1.001697	1.00071	1
10	1.647432	1.987973	2.063651	1.999993
11	0.99998	0.99998	2.059102	2.00002
12	2.055515	2.027576	2.010515	2
13	2.140936	2.080534	2.03369	2.000001
14	2.201765	2.150538	2.060041	1.999999
15	1.000034	1.000034	1.000034	1.000034
16	2.634429	2.315143	2.120143	2
17	1.000125	1.000125	1.000125	1.000125
18	2.612086	2.364144	2.157966	2.000003
19	1.670102	1.825606	1.932338	1.999996
20	0.999999	0.999999	0.999999	0.999999
21	2.073685	2.036596	2.013955	2
22	2.141746	2.070387	2.026839	1.999996
23	2.364375	2.180946	2.068982	1.999982
24	1.182071	1.353214	1.470643	1.548786
25	1.220806	1.220806	1.220806	1.220806
26	1.057633	1.244588	1.244588	1.244588
27	2.445838	2.221405	2.084432	2
28	1.344285	1.344285	1.344285	2
29	1	1	1	1
30	2.32285	2.173774	2.075384	2.000002

Notice that when the predetermined level $\xi$ is equal to 0.5, regardless of the variations in inputs and outputs, the stochastic efficiencies determined by the deterministic form are equivalent to those derived from DEA models based solely on the means of inputs and outputs. As can be found, the risk levels may affect performance scores and SCU.

It should be noted that using data from 2019 to 2021 allows us to capture the responses of entities during a period marked by significant economic fluctuations, particularly due to the COVID-19 pandemic. This choice was primarily driven by the availability of complete and reliable data for parameters required for our CCDEA models. A critical lens to understand immediate adjustments in capacity utilization is provided. Also, the analysis of different risk levels gives insights into how SCU is affected by uncertainty. This is crucial for long-term planning, as it allows policymakers and managers to understand the potential vulnerabilities of their economies and organizations to future shocks.

Discussion and managerial implications

The concept of SCU is crucial for organizations aiming to maximize productivity and efficiency while also considering environmental sustainability and resource limitations. By exploring SCU, organizations can improve their resource allocation, enhance planning, and develop effective operational strategies. Non-radial DDF models, supported by CCDEA methods, provide a solid framework for assessing SCU in systems influenced by random factors and undesirable outputs. This approach allows for a comprehensive evaluation of SCU, taking into account the uncertainties and variations in input-output measures.

Analyzing SCUs for 30 OECD countries offers valuable insights for policymakers and managers, helping them grasp efficiency and sustainability. The countries can find ways to improve their performance by comparing their SCU with other countries under managerial as well as natural disposability. These findings suggest that varying risk levels can affect performance and SCU. For instance, Canada has shown the lowest output-oriented short-run stochastic SCU compared to its peers, indicating a need for adjustments in labor availability, fossil fuel use, and renewable energy consumption at different risk levels. On the other hand, Australia stands out with the highest input-oriented short-run stochastic SCU across various risk scenarios, suggesting significant changes in variable inputs, specifically labor, fossil fuel consumption, and the utilization of renewable energy sources. It is important to note that when the predetermined threshold is set at 0.5, the stochastic efficiencies calculated using a deterministic approach align closely with those derived from DEA models that use average inputs and outputs, regardless of fluctuations in these variables. This highlights how different risk levels can influence both performance scores and SCU ratios. Our analysis reveals that the relationship between risk level and SCU differs significantly across countries. As demonstrated in Tables 4 and 8, the output and input-oriented efficiency scores generally decline as risk levels increase, implying that higher uncertainty can negatively impact performance. This underscores the need for robust risk management strategies.

The implications of this research are significant for organizations. By gaining insights into their SCU, organizations can discover ways to optimize resource use, enhance operational efficiency, and reduce waste and environmental impact. These findings can also guide strategic decision-making, resource planning, and performance evaluations. By focusing on improving efficiency and reducing negative outputs, organizations can work towards better sustainability outcomes. Comparing the performance of countries at various risk levels can illuminate how uncertainties affect sustainability, aiding organizations in decision-making and risk management. Identifying less efficient countries and understanding the reasons behind their inefficiencies can lead to targeted strategies for improvement, such as addressing low labor force participation or high CO2 emissions, ultimately enhancing overall sustainability performance. For example, consider Australia. Australia consistently demonstrates the highest input-oriented short-run stochastic SCU across all risk levels (Table 9), suggesting effective change of labor, fossil fuel consumption, and renewable energy consumption. Managers of this country should benchmark against other sustainable practices such as Canada, Latvia, Luxembourg, Norway and United States. Specifically, a deeper dive into sustainable countries’ policies regarding energy mix, labor market flexibility, and resource allocation could yield valuable lessons. As another example, consider Poland. Poland consistently shows the lowest efficiency scores across all risk levels (Tables 4 and 8). This suggests significant room for improvement. Poland should focus on strategies to improve GDP and life expectancy and decrease CO2 emissions by making changes in the number of labor force, fossil fuel consumption, and renewable energy consumption. This highlights the need for targeted interventions to address specific inefficiencies. Analyzing the reasons behind Poland's lower performance, such as its reliance on fossil fuels or labor force participation, can inform the development of country-specific strategies.

Several countries show suboptimal utilization of labor force, fossil fuel consumption, and renewable energy consumption (Tables 5–7 and 9–11). For instance, Estonia's performance fluctuates significantly with risk levels, highlighting the importance of implementing adaptable strategies. Managers in these countries should investigate policies to optimize the use of labor, fossil fuel consumption, and renewable energy consumption. Luxembourg consistently demonstrates optimal performance across multiple metrics (Tables 5–7 and 9–11). By analyzing the policies and practices of this country, other nations can identify best practices to improve their SCU.

We have measured how effectively countries utilized their existing sustainable capacity during a period of significant demand shocks, rather than conflating utilization with long-term structural change. It should be noted that CCDEA is suitable for long-term efficiency analysis, provided that the model accounts for temporal changes, data consistency, and the evolving nature of uncertainties. When properly implemented, it can offer valuable insights into performance under uncertainty over extended periods.

In summary, exploring SCU with innovative analytical techniques can support organizations in achieving sustainable growth and competitiveness in today's resource-limited business landscape. Prioritizing environmental sustainability, efficiency, and effective capacity utilization is essential for long-term success, which can be accomplished through a comprehensive approach to capacity utilization.

Conclusion and policy implications

Analyzing sustainable capacity utilization is crucial for effective planning and resource management. In this study, we introduced two approaches to assess sustainable capacity utilization, focusing on both output and input. These approaches are based on chance-constrained DEA and consider three dimensions of sustainability in the short term. By applying these chance-constrained DEA methods, along with non-radial DDF models that take into account managerial and natural disposability, a solid framework was offered for estimating the sustainable capacity utilization of systems that deal with random factors and undesirable outputs.

Our research specifically evaluated the sustainable capacity utilization of OECD countries, enhancing our insights into this concept in the short run. The performance and sustainable capacity utilization of these countries were estimated while accounting for random and undesirable outputs. Overall, the methodologies we proposed provide valuable information that can improve decision-making and support sustainable development across various sectors. The findings indicate that uncertainty and the level of risk can significantly affect performance and sustainable capacity utilization ratios.

An important consideration in our empirical analysis was the temporal scope of the data. This choice was primarily driven by the availability of data for the parameters required for introduced CCDEA models. While this represents a short-term horizon, the period is not arbitrary; it encompasses the significant global macroeconomic shock of the COVID-19 pandemic. This makes it a unique and valuable case study for testing the robustness of our stochastic evaluation method under extreme conditions. The specific SCU scores presented are a snapshot of this volatile period. By explicitly modeling inputs and outputs as random variables, our approach can be developed for longer time horizons where economic fluctuations, policy changes, and technological uncertainties are more pronounced.

Future research can expand this framework to a broader dataset to empirically validate its long-term robustness. Exploring the sustainable capacity utilization of complex networks and processes with random measures is another promising avenue for subsequent studies. There is also potential for examining more areas and incorporating additional performance measures.