Uncertainty and stochasticity compromise real-life production systems in many ways. For instance, customer demand, customer order changes, operator absences (Merten et al., 2022a), machine breakdowns, or supply delays are very difficult to predict, and, thus, they pose a difficult challenge for the managers and operators of such production systems (Koh et al., 2002). Failure to cope with uncertainty might result in reduced customer order punctuality (Koh and Saad, 2002), quality issues, increased production costs, and diminished performance in general. Additionally, production systems often entail nonlinear transformations or correlations. One nonlinear transformation of this sort (Pervozvanskii, 1965) is portrayed by a system that has a precisely defined upper threshold on the input, above which production is forced to stop and stays idle for a fixed amount of time. The disruptive interplay of a threshold and uncertain inputs described here is also of relevance in the context of predictable production (Cho and Erkoc, 2009).

While the threshold value is usually beyond the system operator’s control, the average production level (i.e., the input) can frequently be selected at will. Therefore, such systems display a fundamental optimization problem that covers the interface of production planning and control as well process design: How to choose the average input level of threshold-impeded stochastic production (TISP) systems–given the input fluctuation type, the threshold value or the duration of the idle time–so that the system’s output is maximized? Figure 1 offers an intuitive graphical representation of this question: By altering the mean of the input stochastic distribution (e.g., 0.45 , 0.55 , or 0.65 input units per unit time) and trimming values that exceed the upper threshold of 0.70 input units per unit time, the authors provide evidence that the system’s output indeed has a maximal value.

The goals of this investigation are: 1) The authors introduce the basic notion of threshold-impeded stochastic production systems. 2) The authors show that these systems display a non-monotonous relationship between the production output and the average input level (representing the resource load at which the production system is run). This relationship is a universal feature of such TISP systems. 3) The authors offer a method to assess the output sensitivity of TISP systems in response to varying system and input parameters. 4) The authors illustrate applicability of this theoretical concept to real-life systems through two application scenarios which both fall into the category of TISP systems.

TISP systems can be found in diverse industrial contexts such as the continuous casting of steel and the generation of electric power through wind turbines. For example, if a continuous caster operates so fast that the mass flow of manufactured steel outruns the supply of liquid steel, the entire casting procedure needs to be interrupted for maintenance (Merten et al., 2022b). Analogously, wind turbines have to be switched off during stormy weather, or else they will possibly suffer structural damages (Klimstra and Hotakainen, 2011). The idea behind choosing two vastly different application scenarios is to emphasise the wide-ranging applicability of TISP concepts.

The authors intend to approach the TISP-inherent optimization problem by abstracting real-life nonlinear systems into minimal models (Batterman and Rice, 2014), capturing the “stylized facts” of the production mechanisms and then revealing, via careful numerical analysis, the influencing factors that allow an accurate control of the system. The study of nonlinear dynamics has a long tradition of contributing to a better understanding of phenomena in industrial production, assembly and supply systems (Chankov et al., 2016; Alkan et al., 2018; Chankov et al., 2018; Lin and Naim, 2019).

The stylized representation of TISP systems allows one to define TISP subtypes, which lead to various input-output relationships and sensitivities with respect to noise types. To this end, the authors consider the output of TISP systems that exhibit typical nonlinear transformations, as sketched in Figure 2. Depending on the application scenario, three TISP subtypes are distinguished (i) standard TISP: The nonlinearity of the system (i.e., the threshold): brings excess input values down to zero (see Figure 2B); (ii) penalized TISP: The nonlinearity brings excess input values to a value below zero (see Figure 2C); (iii) lagged recovery TISP: After excess input values are reduced, the system is unable to produce further output for a certain number of time steps (see Figure 2D).

In order to provide a quantitative relationship between the generic TISP systems (i.e., standard, penalized, or lagged recovery) and the application scenarios outlined above, the authors estimate the systems’ output (i.e., steel mass flow and electric power) by simulating and transforming random input values from several probability models (e.g., uniform or Weibull noise) as well as computing expected values numerically. Since these expected values originate in output integrals of the minimal models, it is shown how the models compare against the simulated outputs. Then, varying the average input level enables one to determine the maximally achievable output and its corresponding parameter values. This procedure is repeated numerous times subject to changing statistical features of the noisy input (e.g., distribution width) and other experimental parameters (e.g., penalty and lag size) which reveals how sensitive the system output is to such adjustments. For the continuous casting application, the authors also examined time-correlated input from an Ornstein-Uhlenbeck process that experiences a lagged recovery nonlinearity (see Figure 2D). Contrasting the mean-reversing behavior of an Ornstein-Uhlenbeck process with uncorrelated Gaussian input will demonstrate the impact of noise correlations on the production performance.

To the authors’ best knowledge, the category of TISP systems has not been described before. Understanding such high-level categories can be helpful for production planning and control, where the resource load either can be selected or the system can be set up to perform optimally at typical input levels, and for a more detailed modeling of specific systems. For such modeling efforts, the authors argue that in any TISP system, non-monotonous input-output relationships will be encountered independent of the intricacies of the mathematical or computational model.

The remainder of this article is organized in the following way: First, the authors present selected works revolving around uncertainty in production systems and they point out in what sense this study fits into the research landscape (Section 2). Section 3 (“Application scenarios”) explains the production mechanisms and input distributions behind the two scenarios, i.e., steel continuous casting and wind power generation. Once the underlying production mechanism has been understood, the authors establish the expected value integrals that describe the output of the various TISP system subtypes (see Section 4 “Minimal models”). In Section 5, the Ornstein-Uhlenbeck process for time-correlated input values is introduced which becomes useful in the case of the lagged recorvery TISP system (see Figure 2D). Section 6 (“Methods”) discloses the simulation framework and it reveals how the minimal models as well as the respective system or input parameters are deployed to assess the output sensitivity of TISP systems. Throughout Section 7, the numeric and simulation results are presented and discussed before, finally, in Section 8 the authors conclude the findings and give an outlook on promising future research directions.

In response to uncertainty, Correa (1992) offers two remedies, namely, control and flexibility. “Control” includes all efforts that aim at proactively reducing uncertainty before it arises, whereas “flexibility” stands for reactively coping with uncertainty after it has arisen. Angkiriwang et al. (2014) compare the usage of reactive uncertainty strategies such as buffering and proactive uncertainty strategies such as redesigning. However, buffering will not have the desired effect if the interplay between uncertain production inputs and TISP thresholds complicates the determination of appropriate buffer strategies. Similarly, redesigning the production process will not reduce uncertainty if the production process cannot be redesigned further due to technology and cost restrictions or saturation effects.

Sreedevi & Saranga, (2017) state that flexibility helps “in reducing […] supply and manufacturing process risks” but the “effect is context-dependent.” As a solution to this, the authors offer a generalized description of uncertainty in threshold-impeded production systems that can be adapted to different contexts (the only condition being that the production mechanism itself is understood and can be modelled mathematically). Regardless of whether common control or flexibility strategies are applicable, this description deepens the understanding of production outputs and their sensitivity to uncertain inputs.

Production uncertainty can be combatted reactively through rescheduling (Vieira et al., 2003; Psarommatis et al., 2021). Rescheduling has a long history in the steel industry; especially multi-agent systems were often used for this purpose (Cowling et al., 2003; Cowling et al., 2004; Ouelhadj et al., 2004). More recently, evolutionary algorithms (Guo and Tang, 2019; Merten et al., 2024) and machine learning (Iglesias-Escudero et al., 2019; Li et al., 2020) have become the preferred solutions. However, all of these solutions are extremely context-dependent and they are invalid with respect to other application scenarios such as wind turbines. As stated above, this is something the authors try to avoid with the TISP approach.

In the context of wind turbines, a common “buffering” strategy comprises changing the orientation of the wind turbine rotor blades with respect to the direction from which the wind is blowing. So, one can either harness more energy by aligning the rotor blades with the wind or protect the wind turbine from strong winds by turning the rotor blades away from the wind input direction (Castellani et al., 2015; Yan, 2015). Nevertheless, this is only feasible for rather small variations in the wind speed and, again, this strategy is highly context-dependent.

On top of that, to reduce the power output variability of wind farms, diversification methods could be applied. For instance, spatially distributing wind turbines over a given area (Cassola et al., 2008) or combining different types of wind turbines (e.g., smaller turbines for weaker winds and larger turbines for stronger winds) helps to reduce variability. One a higher level, it is even feasible to interconnect entire wind farms (Archer and Jacobson, 2007; Katzenstein et al., 2010). While these diversification methods are to some extent adaptable to other application scenarios such as steel continuous casting, they primarily aim at only reducing the output variability but not at maximizing the overall output. Nevertheless, output maximization is “usually one of the most important objectives for any [wind farm] designer” because it is closely linked to the revenue of a wind farm company (Feng and Shen, 2017). On the contrary, maximizing the overall output is the central purpose of the TISP approach as explained in the introduction of this article.

In steel continuous casting, the tundish (which is a reservoir of molten steel; see Section 3) may serve as a buffer containment. Therefore, it can be used to slightly slow down or ramp up the production process. Yet, the tundish’s ability to combat uncertainty is limited because changing the flow pattern of the molten steel has profound effects on the steel quality (Zhong et al., 2007). Besides, one has to consider a multitude of constraints when adjusting the casting speed or else the resulting steel strand might be torn apart or it might not be sufficiently solidified before leaving the production machine (Merten et al., 2022b). Both these occurrences would cause the production to halt for cost- and time-intensive maintenances.

A proactive way to reduce uncertainty is to modernize the production environment (Gerwin and Tarondeau, 1982; Ettlie, 1990; Groover, 2006; Bertsimas and Thiele, 2014; Dotoli, et al., 2019; Ghobakhloo, 2020). For this purpose, Industry 4.0 technologies such as cyber-physical systems (CPS), internet of things (IoT), artificial intelligence (AI), and cloud computing (Zhong et al., 2017; Xu et al., 2018; Oztemel and Gursev, 2020) are routinely implemented. In particular, AI applications have proven valuable as means to combat production uncertainty (Iglesias-Escudero et al., 2019; Arinez et al., 2020). For example, Roy et al. (2004) describe an inference model that is capable of handling schedule disturbances in steel production. With respect to wind turbine control and optimization, neural networks are frequently utilized (Chatterjee and Dethlefs, 2021). But what is to be done in production environments where access to advanced technologies is limited or the necessary input data does not exist (Lee et al., 2013)? In fact, many steel factories still fall under this category. For these situations, the authors offer a technique that, instead of focusing on individual input events or threshold violations, yields universal recommendations which work very well on average.

On a more theoretical level, Peidro et al. (2009) suggested analytical models and simulations (Aouam et al., 2018; Jamalnia et al., 2019; Gupta and Maravelias, 2020; Tordecilla et al., 2021). Simulation tools have been successfully deployed to tackle uncertainty in production systems (Negahban and Smith, 2014; Jeon and Kim, 2016; Zhang et al., 2019) because “due to [their] low cost, quick analysis, low risk and meaningful insight that [they] may provide” (Mourtzis, 2020) simulation tools allow “for the experimentation and validation of […] process and system design” (Mourtzis et al., 2014) and show “unique advantages in solving practical problems” (Zhang et al., 2019). Nevertheless, a few research gaps concerning state-of-the-art simulation tools have been identified: (i) Capable tools only exist for a selective subset of application scenarios (Mourtzis et al., 2014) and (ii) “unified approaches and terminology” are missing (Mourtzis, 2020). As a solution to this, the methodology can be viewed as a simulation toolbox that is adaptable to diverse application scenarios and, regardless of the scenario, always follows the same underlying recipe or approach.

Over the past 70 years, continuous casting has widely replaced ingot casting as the primary fabrication method for steel slabs (Santos et al., 2003). The working principle of a continuous caster is shown in Figure 3. First, the steel alloy is poured into the tundish from which it passes through a mold. Here, the tundish serves as a funnel and buffer containment that ideally ensures a constant delivery of liquid metal. The design of the mold basically dictates the shape and proportions of the emerging steel slabs, while the amount of steel flowing across the mold can be modified with the help of a nozzle. At the mold exit, the nascent steel slabs are still largely molten, and secondary cooling in the form of water sprays has to be carried out for further shell solidification (Irving, 1993).

Extensive attempts have been made to increase the operating speed of casters and, therefore, their output. Faced with the ubiquitous compromise between output maximization (Li and Thomas, 2000) and process quality/steadiness, numerous researchers have examined likely limitations of the operating speed (Merten et al., 2022b) which, among others, encompass several customer-dependent slab properties (e.g., slab dimensions). For instance, to ensure sufficient shell solidification the operating speed has to be chosen so that the slabs have enough time to solidify. This solidification time is tightly linked to one of the most important planning parameters in steel production (Özgür et al., 2021), namely, the slab thickness (Thomas, 2002). In practice, it is often not possible to maintain a constant operating speed which would be beneficial for the slab quality (Zhang et al., 2006; Wang and Zhang, 2010; Zhang and Wang, 2010) since subsequently manufactured customer orders typically exhibit divergent thickness values.

Another example shedding light onto the output maximization trade-off is discussed in Merten et al. (2022b). The authors inspected historical production data from an industrial casting machine which revealed an abrupt adjustment of the casting speed strategy and, subsequently, they traced this regime transition back to the existence of a theoretical limit for the maximum achievable steel mass flow. In essence, this maximum achievable steel mass flow is motivated by the fact that the tundish (see Figure 3) must not run out of liquid steel. As soon as the steel mass flow (and, hence, the operating speed) exceeds this maximum limit, the continuous casting process has to be stopped for cost-intensive and time-consuming maintenance during which no output can be generated.

In this application scenario, the system’s input uncertainty originates from complex planning constraints (Merten et al., 2022a) as well as unstable customer demand entailing fluctuations of the casting speed-relevant variables. At the same time, a threshold/nonlinear transformation is imposed by the maximum achievable steel mass flow beyond which the steel slab might be torn apart (Merten et al., 2022b). These systemic features turn the continuous casting of steel into a TISP device.

Wind turbines play an important role in today’s renewable energy market (Liu et al., 2013; Wood et al., 2013). By means of rotor blades and an electric generator, they convert wind kinetic energy into electric energy, where the relationship of the wind speed v (unit: m / s ) and the generated electrical power P (unit: kg × m 2 / s 3 ) is demonstrated to be cubic (Bergey, 1979; Salameh and Safari, 1992): P = ρ A v 3 2 , where ρ and A denote the air density (unit: kg / m 3 ) and the rotor blade swept area (unit: m 2 ). Typically, wind turbines are characterized by three different thresholds on the wind speed (i.e., cut-in, rated and cut-out speeds) that govern the transformation from wind to energy. At speeds below the cut-in threshold v i n , no electric energy is built up because of static friction between the mechanical components of the wind turbine and negligible torque on the rotor blades (Wood et al., 2013). Above the cut-in speed, the electric energy production grows substantially with the wind speed; however, wind turbines are constructed to trim any energy surpluses beyond a rated threshold v r corresponding to the maximum capacity of the integrated electric generator (Manwell et al., 2010; Wood et al., 2013). Moreover, wind turbines need to be protected against storms as pointed out earlier. Thus, they are shut down as soon as the wind speed surpasses a specified cut-out threshold v o u t (Klimstra and Hotakainen, 2011). The impact of the wind intensity on the electric power output is commonly documented in power curves (Carrillo et al., 2013; Hau, 2013). A schematic illustration of such a power curve is shown in Figure 4 (Salameh and Safari, 1992). In the theoretical framework, the wind corresponds to the stochastic input, whereas the wind turbine and its speed thresholds act as a nonlinear transformation. Based on these technological aspects, the authors stylize the functioning of a wind turbine as a TISP system and they utilize various minimal models (see Section 4) to study how wind turbine parameters as well as the wind speed pattern affect the turbine performance.

From the principles of fluid flow, Lanchester and Betz have deduced a physical boundary for the maximum energy that can be obtained through wind turbines (Bergey, 1979). More recently, the literature concerning wind energy mainly addressed two distinct research topics—(i) the fitting of stochastic distributions to wind speed data (Morgan et al., 2011) and (ii) the analysis of technical details of wind turbines (Carrillo et al., 2013)—with some articles considering both jointly (Kwon, 2010; Liu et al., 2013). While Morgan et al. (2011) explore the suitability of theoretical wind speed models with regards to offshore wind measurements, Carillo et al. (2013) evaluate a series of generic equations to approximate the conversion of wind to energy via wind turbines. Stevens and Smoulders (1979) observe the energy production as a function of the Weibull distribution shape parameter (whereby the Weibull distribution is the most commonly used distribution to characterize wind speeds). Furthermore, frameworks that enable one to match wind turbines to wind speed distributions/wind sites are provided by Salameh and Safari (1992)/Ritter and Deckert (2017). Here, the authors intend to expand previous methodologies through additional TISP features such as penalties and lagged recoveries.

For both application scenarios–the continuous casting of steel and the generation of power via wind turbines–the authors choose the input probability distributions according to the pertaining literature and industrial data. A summary of all distributions is presented in Table 1 and an explanation for their parameters is enclosed in Table 2. Moreover, Table 2 indicates the range of the noise parameter values in the numerical experiments.

As can be seen in Table 1, the authors characterize the mass flow data (Merten et al., 2022b) by uniform and symmetric triangular distributions, while wind speeds are most commonly reproduced by Weibull distributions (Stevens and Smulders, 1979). The mean steel mass flow as well as the scale of the wind speed distribution are varied between 0.055 and 0.070 tons per second as well as 1.00 and 25.00 meters per second, respectively (see Table 2). Also, the authors select the threshold wind speeds v i n , v r , v o u t as reported in Carillo et al. (2013) and the maximum possible steel mass flow T according to Merten et al. (2022b). The cut-off thresholds T and v o u t are fixed at 0.070 meters per second and 27.00 meters per second (see Table 3), respectively.

After studying the discrepancy between the numerical integration and the simulations, the authors determine, by means of regression, the position of the performance maximum as a function of the noise parameters w / k , the penalty p , and the lag l . For this purpose, the “combination of penalized and lagged recovery TISP” system (see Section 4) is implemented as a minimal model because it embeds the “standard TISP” system ( p = l = 0 ), the “penalized TISP” system ( l = 0 ), and the “lagged recovery TISP” system ( p = 0 ) in its limits.

Next, the effects of input correlations on the output productivity of a lagged recovery TISP system (lag l = 1 ) are examined (see Section 4) with respect to the continuous casting application scenario. To this end, the authors use correlated noise from an Ornstein-Uhlenbeck process and white Gaussian noise as a baseline comparison. The distribution mean μ varies from 0.055 to 0.070 tons/second, whereas the standard variation σ changes between 0.001 and 0.030 tons/second. The OU-specific parameters are located between 2 and 10 (for the stiffness θ ) as well as 0.055 and 0.070 tons/second (for the theoretical mean μ /start value x 0 ), respectively. For both types of input fluctuations, the authors assume the same threshold parameter as in the previous subsection (see Subsection 6.2).

For the simulation of the OU-process starting at a start value X 0 equal to its theoretical mean μ , the process’ mean hitting time H (Cerbone et al., 1981) is adopted as the simulation length. Through Bland-Altman diagrams (Bland and Altman, 1986), the authors assess the validity of the hitting time formula (see Section 5) by comparing it with the average time span that a simulated OU-process needs to exceed a given distance from its equilibrium mean. Bland-Altman diagrams plot the difference between two measurement series against the (arithmetic) average of the two series; they visualize to what degree the series deviate from each other through the indication of confidence intervals. Afterwards, the minimal model for time-correlated input (see Section 4) is held against the output simulations (obtained by transforming a random OU-sample according to the cut-off threshold T ). Finally, the authors measure the impact of the noise parameters (i.e., standard deviation and stiffness) on the position of the performance maximum by using regression methods.

Earlier, the authors have established the mean hitting time of an Ornstein-Uhlenbeck process (see Section 5). In Figure 7, the analytical formula from Cerbone et al. (1981) is compared with the average time span that a simulated OU-process needs to surpass a given threshold, by looking at how much the simulation deviates from the analytical hitting time. If the cut-off threshold is chosen to be 0.070 tons per second and change theoretical mean μ (and with it the starting value x 0 ) from 0.055 to 0.070 tons per second, the formula predicts the hitting time fairly well for values up to 40 time steps. This is confirmed by the corresponding Bland-Altman diagram (Bland and Altman, 1986; see Supplementary Material A: Figure A.5) as the deviation between the two series of measurements exceeds the confidence interval consistently only after 40 time steps.

Figure 8 displays the comparison between the minimal models and the output simulations for the lagged recovery TISP model (lag l = 1 ), in the case of an Ornstein-Uhlenbeck process and an uncorrelated Gaussian input. Both series coincide seamlessly for the uncorrelated input type. On the contrary, the discrepancies for the OU-process tend to increase along the x-axis which is confirmed by the Bland-Altman diagram (see Supplementary Material A: Figure A.6). The Authors exclude a systematic divergence between the minimal model and the output simulation, since the mean difference between them is close to zero regardless. The inherent random error presumably originates in the fact that for larger starting values/theoretical mean values (i.e., very short average hitting times/simulation lengths) an OU-process may spend a considerable period above or below the threshold and, therefore, the arithmetic mean of the simulated output is distorted by chance. Selecting a smaller simulation step would resolve this problem but is computationally very expensive.

Apart from this, Figure 8 also suggests that a production device powered by OU-noise input could be less susceptible to the disruptive threshold, as the curve associated with the uncorrelated Gaussian noise sinks much quicker. For a starting value/theoretical mean equal to the size of the disruptive threshold, the curves differ by already 50 percent. This is due to the mean-reverting behavior of the OU-process or the system operator that meticulously attempts to counterbalance performance fluctuations. Lastly, the authors monitored what consequences a change in the input distribution standard deviation or stiffness has on the location of the maximum performance. The position of the maximum appears to move logarithmically with the stiffness and inversely proportional with the standard deviation (see Supplementary Material A: Figure A.7). Thus, the larger the stiffness (or the more the input fluctuations are regulated by the system operator), the better the performance.

In this article, the authors have examined the output characteristics of several threshold-impeded stochastic production (TISP) systems using minimal models and simulations. The output experiments differ in terms of their input fluctuations (e.g., Gaussian or Weibull noise), application scenarios (i.e., steel continuous casting and wind turbines), and nonlinear features of the transformation (e.g., number of thresholds, magnitude of the penalties/lags). The influence of multiple input characteristics (e.g., noise distribution width) on the position of the maximum output performance in the parameter space has been explored. It turns out that for the continuous casting application scenario, neither the size of the penalty nor the duration of the lag have a significant effect on the maximum output, while they are essential for the calculation of the maximum power that can be generated by wind turbines. Ultimately, time-correlated and time-uncorrelated inputs have been compared by applying a simple nonlinear transformation to a mean-reversing Ornstein-Uhlenbeck process and uncorrelated Gaussian noise. The authors showed that a hypothetical production apparatus fuelled by OU-noise input would be superior to an equivalent apparatus fuelled by uncorrelated Gaussian noise (provided with the same system and noise parameters). Hence, a machine operator that constantly tries to dampen the input fluctuations of a TISP-type system improves the output performance.

Our work has further highlighted and developed the work of Merten et al. (2022b) as they essentially describe the existence of a TISP system in the continuous casting of steel (see Section 3). Clearly, in the case of a maximum possible steel flow around 0.070 tons per second, the choice of the mean production level is immensely important because, for example, overshooting the optimal level by less than eight percent can theoretically lead to a production loss that is greater than 23 percent (see Figure 5 (a)). In fact, since the introduction of the novel production strategy at the steel factory investigated in Merten et al. (2022b) the ratio of devaluated casting products has gone down by more than 50 percent. If the value of devaluated products depreciated by just 20 percent on average and if one assumes a mean mass flow of 0.05 tons per second as well as a steel price of 1000 US dollars per ton, this decrease (5.5 percent to roughly 2.7 percent) in downgrading percentage would entail an optimization potential of almost 9 million US dollars per year. Analogously, selecting a suitable location and thereby the right wind speed pattern is crucial for the construction of new wind farms (see Figure 6). In a way, the authors have extended the approach of Salameh and Safari (1992) who developed a framework to determine the correct wind turbine parameters for a specific wind site by including additional degrees of freedom related to the amount of energy necessary to restart a wind turbine after it has been shut down as well as the length of its idle time following a threshold violation.

With this work the authors want to draw attention to production situations, where a balance is required between the process stability and the strive for increased production–due to both input stochasticity and the presence of disruptive thresholds. Often such a balance is hidden within the intricacies of the production process (e.g., disruptive thresholds masked as load-dependent errors or sudden declines in product quality when reaching critical load levels in the production system). Identification of such situations requires a dialog between different divisions of a production facility that are responsible for maximizing production (e.g., operations management) or surveying the occurrence of component failures and errors, as well as quality standards (e.g., quality control). The authors believe that the procedure can contribute to this dialog as it is applicable to any production system that a) relies on stochastic inputs and b) is subject to nonlinear transformations involving disruptive thresholds. Accordingly, the authors would like to encourage researchers to test this methodology with regards to a wider range of applications scenarios and extra technical details such as different noise correlations and flavors. One potential scenario for this endeavour is portrayed by a situation where a company wants to sell a product through its website. Obviously, more website visitors should lead to greater sales; however, once a critical number of visitors is reached the website server might crash resulting in extended downtimes. The respective company should adjust its marketing activities to account for this possibility.

In earlier publications (Merten et al., 2022a; Merten et al., 2022b) we have studied the technical details and algorithmic challenges of steel production in some detail. Hence, the description here has a stronger emphasis on this example. With our second example, wind farms, we wish to emphasize that the TISP systems are not just confined to this one application domain. We hope that other researchers are encouraged to think about their production systems from a TISP perspective.