Development and evaluation of risk treatment paths within energy-oriented production planning and control

Keller et al. [6] suggest a medium- and short-term add-on module for the enterprise resource planning system to implement energy-oriented production planning. The result is an energy demand plan that is coordinated with the production plan. Mitra et al. [9] investigate the optimal production planning and control with time-changing energy prices for continuous, energy-intensive chemical industry processes. To do this, they formulate a deterministic, mixed-integer optimisation problem as a model of temporally discrete transitions between states to solve a flexible job-shop scheduling problem. Moon et al. [10] utilise a non-linear mixed-integer optimisation, considering distributed energy resources and energy storage. Rager et al. [11] present an energy-oriented scheduling approach for parallel machine environments, which is based on evolutionary algorithms. The approach focuses on allocating and sequencing production orders on identical parallel machines within a fixed planning horizon. A method for integrated scheduling of batch process and combined heat and power plant is presented by Agha [12]. The discrete time modelling and mixed-integer linear programming-based approach reduce energy costs and harmful gas emissions by optimising the use of cogeneration and more efficient exploitation of utility resources. Schulz et al. [13] developed an approach for considering the three strategies for reducing energy costs: energy efficiency, energy-flexibility and avoiding peak loads. It is based on constructing a mathematical model that considers volatile energy prices and an iterated local search algorithm. Possible additional costs due to the deviation from an originally intended plan are taken into account by Schultz et al. [14] and Rösch et al. [15]. The deviations are responded to with situational measures when they occur, this means that there is no risk treatment with sufficient lead time to avoid risks or to define countermeasures for events. This may lead to actions based on incomplete situation overviews and improper handling with undesired results.

Risks are basically described as the product of the likelihood of occurrence and the extent of the damage [16]. There are various approaches to identify and manage risks in the production system environment. Chauhan et al. [17] show to how risks within the development process of new products can be estimated and compensated with an integrated approach. Niesen et al. [18] provide a tool for online simulation of production processes to identify critical situations before they occur. Geiger and Reinhart [19] show an approach for machine scheduling under consideration of uncertainties. Simon et al. [20] apply four strategies for the risk evaluation of energy-flexibility. The strategy for evaluating the influence of EFM on production risks complements classic risk management with all possible influences of EFM on the identified risks. Klöber-Koch et al. [21] integrate production risks into a classic PPC system to make predictions about the risk situation during planning and take these into account in planning. Therefore, risks and the production system are modelled. The risks are evaluated regarding their mutual effects and their effects on logistical target values. Finally, suitable preventive measures are incorporated into production planning.

In energy-flexible production systems, the general modelling of energy-flexibility is useful for modelling measures. In Schott et al. [22], the aim is to build a generally applicable data model for modelling performance-related flexibility, which allows transferability between different industries and offers the possibility of standardising the description of energy-flexibility.

It is essential to record the structure between risk and measure interactions in complex systems to quantify them. Klöber-Koch et al. [23] use interpretive structural modelling (ISM) to structure risks in production systems. A structural self-interaction matrix (SSIM) is created based on an existing risk inventory. This assigns mutual dependencies and effects to the system elements, i.e., risks, based on expert knowledge in a pairwise comparison. Pfohl et al. [24] and Shakya and Chauhan [25] also use the ISM methodology to model risk interdependencies. Pereira et al. [26] and Daultani et al. [27] use Bayesian Networks to structure risk interdependencies, interactions and dependencies. Bayesian Networks are acyclic graphs made up of nodes and directed arrow connectors, which indicate their dependencies. Pereira et al. [26] also use what is known as an analytic hierarchy process (AHP) to analyse the influence of individual risks. Wu et al. [28] integrate the Bayesian Networks approach into the ISM methodology. The system structure of the identified risks is first generated according to the ISM method and then expanded to include the calculation of conditional probabilities of risk occurrence.

According to the current state of the art, there are several approaches with which a production plan can be drawn up that consider variable prices from electricity exchanges. Approaches to production control show the possibilities for initiating measures to reduce costs depending on the situation in the event of a malfunction. The work on risk management in the production area shows how the risk aversion of planners and possible uncertainties in the planning process can be considered during the planning process. However, the complex origins of energy costs from load peaks and absolute demand quantity deviations and the use of predefined energy-flexibility measures are not subject to the presented approaches for risk treatment in production planning. This is necessary to be able to integrate all the experts involved. In addition to PPC experts, these are, for example, electricity purchasing and energy management. It is another requirement to weigh up the effects on various additional energy-related criteria transparently, such as penalties and increased network charges. Therefore, a need for research exists into approaches to risk treatment that operative PPC can apply in the context of energy-flexible production systems.

Risks and measures must be known and inventoried (compare Fig. 2) before starting the hereby presented approach, which commences with evaluating and structuring known risks and already identified possible measures derived from them. Risks, in general, might be associated with specific schedule deviations, disruptions in the dimensions of time and all logistic target measures, as well as all sorts of undesired impacts. Measures are generally aimed at the prevention and compensation of risk-induced damage. The identification of risks and RTM can be accomplished by applying creative methods, such as morphological analysis and brainstorming or via collective methods, such as checklists, strengths, weaknesses, opportunities and threats (SWOT) analysis. Furthermore, analytic search methods like bow-tie-analysis, empirical data analysis, fault tree analysis, failure mode and effects analysis (FMEA) and root cause analysis (RCA) can be used [30]. However, these methods are not explained in more detail here, as the approach is applied after identifying individual risks.

As mentioned before, the impact of the identified possible risk events can be quantified by modelling event-related extents of damage and associated probabilities of occurrence. For this, specific information is to be obtained. Based on Refs. [20, 22, 31], risk and measure data from the identified inventories must contain identifiers and variables for specifying the process steps to which the specified elements refer. Also, time-related information, such as activation, deactivation, minimal effect and regeneration durations of elements are required. For quantifying the actual effects on energy-oriented PPC target parameters, risk and measure-induced negative and positive deviations of wait time, transport time, lead time and process start times must be stated for every individual element. PPC target parameters are for example, lead time, capacity, stock or delivery reliability.

Furthermore, impacts of elements on quality, setup and intralogistics processes must be specified via respective variables. Finally, risk or measure-induced maximum and average load change capabilities complete the dataset. In addition to the risk- and measure-specific information, basic data on the production and energy procurement situations are required. For example, the sequence and dependency of processes, order data, and the systems’ electrical energy demand and price forecasts are relevant.

With this information, it is possible to calculate the identified risks’ extent of damage and measure impacts. In this approach, risk and measure effects are quantified in the dimensions of selected target variables of energy-flexible production systems. The underlying principle is that individual elements’ values can be added to calculate combined extents of damage for risk- measure combinations. Table 1 lists the parameters selected for this purpose:

The time unit (TU) mentioned in Table 1 refers to the smallest specified time slice and prescribes the approach’s temporal resolution. For the consideration of energy-oriented production systems, the spot-market trading time-slots of 15 min are an appropriate choice for one TU. Not all parameters that are presented directly in the discrete and absolute dimensions of time, electrical power or energy consumption are mathematically related to target values or overall values from the production system to ensure summability.

Next, the probability of occurrence for single risk events and measures is modelled to conclude a complete risk quantification. According to Ref. [34], damage distributions for individual damages can be described by a random variable with an associated probability distribution. The damage amount can, for example, be related to the maximum damage and thus lies between 0 and 1. Several distribution functions are available and commonly used for this task. For modelling specific damage cases, these distribution functions can be selected and in part be tailored to the distributions of actual use cases by adjusting function parameters. If only the full damage is relevant or even possible in the case of a specific risk occurring, the assignment of a simple probability value can be sufficient. The targeted selection and parameterisation of a distribution can generate a higher number of evaluation options and increase the observation’s granularity. However, this also leads to a correspondingly higher effort and knowledge requirement.

As well as risks, RTM should hereafter be modelled as the product of the probability of their occurrence and their effect. While risk events are regarded as uncertain, the application of RTM and EFM is always considered to be planned and therefore certain in this approach. Hence their probability of occurrence is generally set to the value of 1 (100%). This seemingly unrequired step is justified by the system’s feature that states that the probabilities of occurrence of individual elements not only serve to consider the respective singular risk or measure situation, but are also used as starting values for the calculation of probabilities of occurrence of paths that emerge from the considered individual elements. The assignment of a dummy probability of occurrence of 100% is necessary to create compatible structures in the modelling of risks and measures and enable the uniform presentation of risks and measures in the standardised procedure for calculating the probability of occurrence for paths is presented in the following.

After the evaluation of individual elements has been carried out, a systematic quantification of system structures and causality between elements must be provided. The structural aspect of this requirement will be presented in the following section.

The methodology of interpretive structural modelling (ISM) is used to structure risk dependencies and interactions by several state-of-the-art-sources in Sect. 2.3. Its advantages are the structured approach, the relatively simple provision of input data and the derivation of complex logical and graphical structures from simple logical relationships with defined tools. Thus, the hereby presented method also applies parts of the ISM methodology for modelling causal structures of risks and measures. The approach aims to generate a graphical representation of dependencies and interactions of objects and to identify relationships [24]. For this, ISM methodology involves a series of defined steps. Firstly, all relevant elements in the system under consideration must be listed. When applying the ISM methodology to systems of risks, RTMs and EFMs, this step identifies risks and measures. A structural self-interaction matrix (SSIM) is generated based on the recorded portfolio. Here, the identification letters V, A, X and O are entered in cells of pairwise element comparison and thus formalise influence between the system elements. Influence in this step of the method means any form of dependence, mutual exclusion or targeted and un-targeted effect relationship between the system elements of risks RTM and EFM. An example SSIM and the meanings of the appropriated code letters according to Refs. [24, 25, 35] are listed in Fig. 3.

From the basic SSIM, the binary initial reachability matrix (IRM) is generated using the transformations rules depicted in Fig. 4.

The IRM is then transferred to the final reachability matrix (FRM) via transitivity conditions [24, 25, 35]. Transitivity in this context is the consideration of connections of indirectly linked logical elements. Subsequently, nodes A and C are to be equally connected by the value of one in the respective FRM cell if, in the IRM, a node A is directly connected to another node B, and node B is directly connected to a third node C [35, 36].

At this stage of the approach, further ISM-associated steps are omitted. The underlying goal of structuring the system of interactions between risks, RTM and EFM is to derive complex paths of element combinations from the simple, paired indication of dependencies and interactions in the SSIM. The further course of the approach presented here is based on the FRM. It can be understood as an adjacency matrix extended by indirect element connections. Adjacency matrices are matrices that formalise the topology of direct relations of elements within systems of complex interactions by binary variables [37]. By applying the transitivity concept to the IRM, the adjacency matrix is extended by information on indirect interactions. The FRM then contains all information that is necessary to derive paths by specifying paired connections. The matrix representation is transformed into a directed graph by representing the system elements as nodes and connecting the nodes with directed edges for each “1” entry in the FRM. This digraph is cyclic and therefore contains loops. Paths that are extracted directly from this representation are endless and repetitive. The graph must be reduced to the essential compounds to eliminate these properties. For this purpose, all the connectors that do not increase the accessibility of individual elements are removed by transitive reduction. The repeated appearance of elements in extracted paths is thus eliminated. The result of the transitive reduction is a graph that only shows the most direct paths of the original cyclic graph without repetitions of elements and loops, but with the same accessibility.

Figure 5 displays the graphs for an exemplary system of 14 risks and 15 measures in cyclic (top) and acyclic form (bottom):

This form of visualisation considerably simplifies the understanding of the path term, compared to a binary matrix [37] and makes it possible to visually track paths.

By transferring simple, paired information into a complex visual model of paths, the framework for continuing risk and measure modelling by applying it to combinations of individual elements has been provided. Below, the generated structural framework is used for modelling risk-measure paths. It is worth mentioning that the presented procedure considers all possible combinations of elements according to the SSIM specifications to ensure complete consideration of the possible interactions down to the last detail.

After Sect. 4.3, the procedure of setting up the required system elements and their structure provides a framework for modelling element interactions in the following.

Based on the structuring of element interrelations, the probability of occurrence and the indirect damage impact of the path occurrence are modelled in this section. The Bayesian Network theory is applied for path occurrence probabilities.

In addition to the presentation of the causal structure of interactions and interdependencies in the system of risks and measures, their effects must be classified in terms of time, since the specific deviations of the elements were determined independently of the specific point in time in the previous steps of the approach. To locate risk- or measure-induced effects in the production plan, risks and measures must be related to a particular process step. If the process step affected by a risk is known, the earliest time at which the risk may occur can be determined. This is also necessary for measures, because the time of the processes affected by the measures must be considered. In general, the discretisation is always based on a planning period whose length t is defined in the input. Figure 8 depicts an exemplary planning period and its subdivision for one machine. The orders are scheduled for the machine with buffers. Every order contains a specified number m of products that are displayed as units. For every unit, the lead time is split according to Ref. [2]. Time-related effects of system elements are generally considered as effects on these time components.

When specifying points in time, the entire event always refers to the timeline specified in the input process plan.

The start time of the planning period enables the conversion of periods of time, specified for the effects of measures and risks into absolute times by acting as a reference point. The predicted timestamp of risks and measures introduced in this way precisely determines the timing of all possible damage or measure effects. Risks in particular and production engineering interventions for risk compensation or EFM, can be added directly to the planned load profile at the appropriate point of time.

EFMs that do not consist of changes in process parameters or manipulated variables of the production system are referred to as the energy-flexibility of the production system’s periphery according to the corresponding assumption in Sect. 4.2. These measures are not to be included in the general term and catalogue of measures, as they are not based on adaptations inside the production system, such as changes in order sequence or processing flows. Peripheral energy-flexibility includes the use of energy storage and in-house generation facilities, as well as the short-term procurement of electricity through intraday spot market trades. Their possible compensatory effects on the changed load profile of the process and the company’s overall load profile depend on the input parameters provided. The compensation of load curve deviations by the means mentioned is then realised automatically. Figure 9 depicts the process of time-discrete consideration of risk and measure impact and the effects of third-order periphery EFMs on the load profile that is carried out for every single possible path identified:

After modelling risk, RTM and EFM, as well as paths of their possible combinations within the considered system, the respectively present overall risk situation is to be evaluated. During the following evaluation, the risk and measure-induced changes of cost and the corresponding changes in energy-flexibility and overall risk situation are to be assessed and quantified. Conflicting objectives between the three target figures must be considered by production managers when deciding on preventive risk compensation during production planning.

The method and its software implementation are demonstrated using the example of a foundry. The foundry industry is generally energy-intensive due to the physical conditions required for melting metals. The process being considered, depicted in Fig. 13, consists of the process steps of melting plan preparation (1), mould preparation (2), core preparation (3), raw material loading (4), four separately considered process steps for melting the raw material in four furnaces (5–8), additional loading (9), the addition of additives in the treatment ladle (10), the transfer of the melt into the ladle (11), forklift transport of the casting ladle to the casting fields (12), casting (13), cooling (14), demoulding (15), transporting the castings to finishing (16) and post-processing (17). The small furnaces 1 and 4 are used for direct feeding of the melt treatment ladle (10) or for pre-melting sump material for furnaces 2 (6) and 3 (7). The sump can be supplemented by additional recharging. The production plan presented is transferred to a table of process step times and loads to generate a resulting planned load schedule. The duration of the associated planning period is 65 TUs, which equals 16.25 h.

For the casting process, ten identified risks were pre-prioritised in advance and described with the necessary information for automated utilisation by the software, e.g., the risk of a defective mould core or casting damage while demoulding. The risks are always listed in relation to the process step for classification in the production plan. In addition to the risks, the system considers one preventive RTM per risk and three additional EFMs for the use case. Measures include adjusting furnace operation parameters or changing the casting order, for example. Further input data sets derived from the use case are the overall electricity procurement schedule of the considered company, calculatory cost factors, basic information and the SSIM.

A total of 7544 possible paths for the interactions and dependencies of the 23 individual elements specified in the SSIM have been identified by the method. As an example, the Risk Dashboard for path P4968 is presented in Fig. 12.

The randomly selected exemplary path P4968 consists of the elements R5, R6, R7, M1 and M2. Thus, a planning period is modelled in which the occurrence of R5, R6 and R7 is assumed before M1 and M2 are applied. The three risks considered are the possibilities of crane failure at the casting field, failure of the forklift for transport to the casting fields and the failure of furnace number 4. To counteract them measures M1 and M2 are deployed. The composition of the path results from the specified element interactions and dependencies in the SSIM.

Due to the number of coinciding elements (five), the path’s extent of damage is greater than for an individual risk, but the probability of occurrence is very low. To directly compare the different measure and consequence risk constellations, starting from certain risks, the corresponding paths can be selected under the tab “Path comparison” and then be matched in pairs regarding the characteristics of costs and deviations from target values.

Non-relevant constellations can be filtered out by applying a certain minimum value of risk to reduce the large number of paths.

To decide the specific measures to be taken, two almost similar paths that are only distinguished by these measures must be considered. Between the paths, the preferred relationship between risk reduction and costs can be selected. In addition, flexibility can be considered to ensure that a chosen path offers sufficient flexibility for possible interventions that are not considered in risk management in production planning.

For example, path P4969 can be compared to the above-mentioned path P4968, as it is identical to P4968 but includes RTM M3 as an additional element. Measure M3 (change of process parameters in furnace 3) lowers the average power consumption of melting in furnace 3 by 400 kW. The application of M3 on P4968 leads to a decrease of the energy-flexibility trilemma parameter from 0.563 to 0.7382. Path-induced additional overall costs \({C}_{tot}\) increase from €2449 to €2686 in absolute numbers, including risk and measure related cost deviations. The relative cost trilemma parameter, which only considers measure-induced cost deviations, ranges from − 0.02673 to − 0.0111. Risk decreases from 2.396E−08 to 1.737E−08. The overall results indicate that the application of RTM M3 may reduce the relative risk by roughly − 27.5% at the absolute price of €237. This exemplary constellation can now be interpreted and compared to all the alternative paths and the respective calculation results by users according to their respective risk adversity.

The results show how the structuring of risk and measure interdependencies and interactions, the construction of a general format for the quantification of risk-induced damage and measure impact, the discretisation of risk and measure effects in the dimensions of time and energy and the presented approach to the triple evaluation work together to enable risk-aware decision support in the planning of energy-flexible production systems. The presented software implementation carries out these steps automatically and visualises the results fully and directly. Only the required input data and information must be provided in the given structure on the user side. Similar paths are listed subsequently and are visually displayed as node-arc-combinations to ensure easy comparability for the user. However, considering the targeted search for specific elements or their combinations, the user interface could be improved in the coming versions.