Managing product-inherent constraints with artificial intelligence: production control for time constraints in semiconductor manufacturing

May, Marvin Carl; Oberst, Jan; Lanza, Gisela Article — Published Version Managing product-inherent constraints with artificial intelligence: production control for time constraints in semiconductor manufacturing Journal of Intelligent Manufacturing Provided in Cooperation with: Springer Nature Suggested Citation: May, Marvin Carl; Oberst, Jan; Lanza, Gisela (2024) : Managing product-inherent constraints with artificial intelligence: production control for time constraints in semiconductor manufacturing, Journal of Intelligent Manufacturing, ISSN 1572-8145, Springer US, New York, NY, Vol. 35, Iss. 8, pp. 4259-4276, https://doi.org/10.1007/s10845-024-02472-6 This Version is available at: https://hdl.handle.net/10419/315310 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Journal of Intelligent Manufacturing (2024) 35:4259–4276 https://doi.org/10.1007/s10845-024-02472-6 Managing product-inherent constraints with artificial intelligence: production control for time constraints in semiconductor manufacturing Marvin Carl May1·Jan Oberst1·Gisela Lanza1 Received: 17 May 2023 / Accepted: 19 July 2024 / Published online: 3 August 2024 © The Author(s) 2024 Abstract Continuous product individualization and customization led to the advent of lot size one in production and ultimately to product-inherent uniqueness. As complexities in individualization and processes grow, production systems need to adapt to unique, product-inherent constraints by advancing production control beyond predictive, rigid schedules. While complex processes, production systems and production constraints are not a novelty per se, modern production control approaches fall short of simultaneously regarding the flexibility of complex job shops and product unique constraints imposed on production control. To close this gap, this paper develops a novel, data driven, artificial intelligence based production control approach for complex job shops. For this purpose, product-inherent constraints are resolved by restricting the solution space of the production control according to a prediction based decision model. The approach validation is performed in a real semiconductor fab as a job shop that includes transitional time constraints as product-inherent constraints. Not violating these time constraints is essential to avoid scrap and similarly increase quality-based yield. To that end, transition times are forecasted and the adherence to these product-inherent constraints is evaluated based on one-sided prediction intervals and point estimators. The inclusion of product-inherent constraints leads to significant adherence improvements in the production system as indicated in the real-world semiconductor manufacturing case study and, hence, contributes a novel, data driven approach for production control. As a conclusion, the ability to avoid a large majority of violations of time constraints shows the approaches effectiveness and the future requirement to more accurately integrate such product-inherent constraints into production control. Keywords Complex job shop ·Time constraint ·Semiconductor manufacturing ·Production control Introduction The trend towards individualization is advancing at breakneck speed and enforces ever increasing requirements on production. Customization and individualization lead to manufacturing systems with unique products. Each unique product exhibits traits that differentiate it from similar, yet not identical, unique products. Examples can be due to personalization or in a re-manufacturing setting where each core that is returned from the usage phase exhibits unique charBMarvin Carl May [email protected] 1wbk Institute of Production Science, Karlsruhe Institute of Technology (KIT), Kaiserstraße 12, 76131 Karlsruhe, Germany acteristics (Wurster et al., 2022). This uniqueness increases the complexity in manufacturing and feeds the trend towards complex job shops, in which each job has unique characteristics and thus, needs unique processing, transportation or setup times (Waschneck et al., 2016). These require more flexible production networks (Yin et al., 2021) and a much more flexible manufacturing system with an in turn fast, real-time production control system that is capable of handling unique products and all their individual constraints (Yuan et al., 2021). The manufacturing of computer chips, in other words semiconductors, takes place in complex job shops and presents a great example for such product-inherent constraints: Time constraints that limit the waiting and transportation time between two or more processing steps. Semiconductor products play a key role in the fourth industrial revolution. With digitization taking place in almost 123 4260 Journal of Intelligent Manufacturing (2024) 35:4259–4276 every industry, the demand for semiconductors continues to grow, especially driven by booming markets like artificial intelligence, 5G and the Internet of Things (IoT) (May et al., 2024). This leads to strong competition within the industry and manufacturers having to produce cost-effectively. To realize necessary cost reductions, improvement of operational processes offers the best opportunities (Mönch et al., 2009). Manufacturing of semiconductors takes place in so-called fabs – short for semiconductor fabrication plants. Re-entrant product flows, re-routing due to machine failures, arrival of urgent jobs and stringent quality requirements make it one of the most complex and dynamic manufacturing environments (Mönch et al., 2013). Several hundred process steps in different work areas are required to turn a wafer into a chip, associated with many process-related challenges. During the time between one process step and another, natural phenomena such as oxidation, crystal formation and ion migration can cause wafers’ surfaces to change their properties (Lima et al., 2021). There is a plethora of research on the physical and chemical property changes during theses processes. Kinetic models play an important role here (Markowich et al., 2012) yet they are focused solely on a product variant level and do not regard the interplay with fab operations. Thus, operations management in a semiconductor fab has to abstract these restrictions into product-inherent constraints for high quality. To ensure this quality is not negatively affected, time limits between various operations are installed. If a lot of wafers, signifying the product, exceeds the time limit, it either has to be scrapped or expensive rework is required (Klemmt & Monch, 2012). When a lot reaches the starting equipment of a time constraint, a decision whether a lot is released and a time constraint is started has to be taken. This time consuming and stressful task is usually performed manually based on heuristics and the experience of the operators (Lima et al., 2017b), giving rise to the need of an automated approach that improves time constraint adherence. To that end, this study presents a novel approach for production control with product-inherent constraints on the example for time constraints in semiconductor manufacturing. The proposed model is based on uncertainty informed artificial intelligence and provides production control with material flow by inhibiting the onset of processing certain lots if the risk of violating the time constraint exceeds a determined threshold. The AI model is used to forecast expected transition times, that cannot exceed the time constraint, so that lots are withhold if their estimated probability of time constraint adherence suggests that potential scrapping or rework will become necessary. Thus, the model provides a novel approach to utilize the uncertainty in manufacturing related artificial intelligence models to distill knowledge on product individual level into an intelligent production control. The model is validated with real world industrial semiconductor manufacturing data from an entire fab over several months. The paper is organized as follows. The literature review is presented in “Literature review” section and focuses on product-inherent constraints in manufacturing and the stateof-the-art of manufacturing under such time constraints in semiconductor fabs. Furthermore, the main contribution of this work is highlighted. “Decision model” section gives an overview of the manufacturing setting at hand and defines the problem treated in this paper. Additionally, the specifications of the constructed models are presented. “Numerical experiments” section presents the results of the numerical study carried out in the real-world manufacturing setting. In “Experimental validation of constrained actions” section, the results are discussed and the benefits of the modeling approaches are presented. Finally, “Conclusions and further developments” section concludes and summarizes the paper and discourses possible relevant further developments. Literature review To that end, this paper presents a holistic literature review on the state-of-the-art for adequately considering productinherent constraints for production control in general in “Product-inherent production control constraints” section. In the concrete context of semiconductor manufacturing, the state-of-the-art techniques are reviewed in “State-of-the-art time constraint control in semiconductor manufacturing” section. The contribution of this paper is outlined in “Contribution” section. Product-inherent production control constraints Generally, production control consists of scheduling at a higher level and dispatching at a lower level. It can therefore be distinguished from production planning which consists of making strategic decisions on a quarterly base and releasing orders at a monthly scale (Mönch et al., 2013). Product-inherent constraints on the production control can be of different type and are found across various industries. Production processes can be constrained for yield, safety, environmental or quality reasons. Another example can be found in the chemical industry in polymerization reactors. Several process variables like reactor temperature and feed flow rate have to be constrained in order to comply with quality and safety requirements (Abel et al., 2000). Highly energy intensive industries like steel manufacturing are forced to constrain their processes and energy consumption in order to reduce their environmental impact (Somboonwiwat et al., 2018). In order to accomplish quality goals and increase throughput, manufacturers may also 123 Journal of Intelligent Manufacturing (2024) 35:4259–4276 4261 impose time constraints within production processes. Time constraints can be required when scheduling because one process step may depend on the output of a previous step forcing it to wait. For example in oncology clinics, treatments may sometimes only start after a specific start time because the patient may only be treated after an oncologist appointment (Liang et al., 2015). Analogically, a process step may require an intermediate product forcing it to be scheduled after the process step of which the output is the intermediate product (Sundaramoorthy & Karimi, 2004). Time constraints can also be imposed to limit the time between multiple process steps in order to prevent the product from degrading. For example, when using cold cure bonding, the time between the anodizing and the bond lay-up process step must not exceed a certain time limit (Higgins, 2000). When handling perishable products in the food industry (e.g. yogurt), each process step in the production has to be performed within a certain time limit for the product not to perish and lose its value (Amorim et al., 2013). Also, holding times of intermediate products can be limited due to sterility reasons in pharmaceutical manufacturing (Eberle et al., 2016). This phenomenon can also be observed in semiconductor manufacturing. Time constraints are frequently imposed to mitigate negative effects on the wafers surface caused by oxidation and contamination (Wang et al., 2018). Given the increasingly available data and availability of high performance algorithms Artificial Intelligence and Machine Learning techniques are perfectly fit to counter such product-inherent constraints. As the field is large and evolving fast the reader shall be referred to the respective state-of-the-art literature. Due the frequent occurrence of time constraints in semiconductor manufacturing and a high availability of quality data, semiconductor manufacturing is investigated in this paper in order to verify the proposed production control model. A state-of-the-art literature review of time constraint control in semiconductor manufacturing is conducted which is presented in the following. State-of-the-art time constraint control in semiconductor manufacturing In semiconductor manufacturing, equipment, denoting the manufacturing equipment which is used interchangeably to the commonly used machines, is the biggest cost driver (Hong et al., 2023). Thus, using capacities with opportunistic behavior and ensuring zero defect manufacturing is central (Valet et al., 2022) and accordingly, fabs are run for 24h on every single day. In order to minimize setups and improve coordination, wafers, with several ICs each, are stacked to lots, each containing up to 50 or typically 25 wafers (Ziarnetzky et al., 2017). This technologically intense manufacturing, with complicated chip design, is seen in the presence of abrasive processes, chemical and electro-physical processes, forming and cutting processes. As ICs are manufactured layer by layer metrology and advanced surface engineering that are required need complicated machines or equipment as well as complex control and organization of production systems (Mönch et al., 2013). Aside from technical and technological complexity,the greatest challenge in producing semiconductors is to coordinate this complex job shop (Mönch et al., 2011). Each wafer requires 1000 or more processing steps, that need several minutes up to many hours each (Valet et al., 2022). Recurrent material flow gradually builds these integrated chips, layer by layer. Factories in semiconductor manufacturing are operated on the verge of the physical and technological boundaries resulting in only a part of the ICs from manufacturing being able to reach the highest usability level (Mönch et al., 2009). Yield denotes this share of functional chips over total manufactured and should be kept as high as possible. Besides errors, yield loss can also be made up of wafers contamination. The contamination often stems from ion migration, crystal formation, native oxidation or the deposition of dust (Lima et al., 2021). Such a contamionation, also called impurity, alters the surface of the chips and inhibits the electrical flow from following the designed patters. To minimize contamination equipment in manufacturing of semiconductors is, hence, located in a so called clean room (Klemmt & Monch, 2012). Still, each wafer can only remain in the clean room for several hours, because otherwise the wafers often have to be scrapped due to the contamination which cannot always be cleaned (Altenmüller et al., 2020). Not violating these time constraints is, thus, crucial to the success of semiconductor manufacturers (Arima et al., 2015). Production and business activities of companies are crucially impacted by production planning and control (PPC) (Wang & Liu, 2013). The main goals of PPC in semiconductor manufacturing are the minimization of costs and an increase in productivity, while continuously improving quality and due date performance (Uzsoy et al., 1992). In order to identify relevant literature treating time constraint management in semiconductor manufacturing, a systematic literature review using grounded theory is conducted using the approach proposed by Wolfswinkel et al. (2013). The review was performed by querying Scopus about publications dealing with time constraints or time coupling and relevant alternative descriptions in semiconductor manufacturing. The literature is filtered by title, abstract and keywords to preserve production system based approaches leading to 28 publications. These are extended with a forward and backward search. The results are clustered into capacity planning, scheduling and dispatching according to the PPC hierarchy as outlined in the following. Time constraints have to be taken into account at all the different stages of PPC. According to Mönch et al. (2013), PPC can be divided into three different levels: capacity planning, scheduling and dispatching. The literature treating time 123 4262 Journal of Intelligent Manufacturing (2024) 35:4259–4276 constraint control at a capacity planning and scheduling level is summarized in Table 1. Overall, only few papers regard time constraints at a capacity planning level. The majority of publications rely on queuing systems, assuming a general independent distribution of interarrival times (GI) and a general distribution for service times (G) with a number of m machines in order to form a GI/G/m queuing network, like Tu and Liou (2006) and Kitamura et al. (2006). Furthermore, Pappert et al. (2016) perform production simulations and conclude that time constraints significantly reduced production capacity. Additionally, Kuo et al. (2011) introduce a modeling approach based on neural networks with the goal of reducing the cycle time of a wafer fab. Lastly, Mastrangelo et al. (2024) build a policy to modulate a two-staged manufacturing system’s capacity with time constraints. They apply a markovian system representation and validate the model in real environment in diffusion and cleaning stages to show a significantly pareto-improvement on throughput and quality yield. The biggest share of the identified literature, however, considers time constraints at a scheduling level. As displayed in Table 1, the scheduling problem is predominantly modeled with mixed integer programming (MIP), mixed integer linear programming (MILP) or mixed integer non-linear programming (MINLP). Few exceptions are a disjunctive graph, a Kanban system, an analytical model or constraint programming (CP), also in combination with MIP. To minimize the waiting time variation and thus reduce time constraint violations, Yu et al. (2013) develop a MIPbased scheduler for up to 25 jobs that can exactly solve this limited problem set. As the MIP model is not able to solve problems with more than 25 jobs, an additional approximate solution is provided, showcasing the disadvantage of the state-of-the-art approaches. While An et al. (2016) are able to find optimal solutions for problems with up to 30 jobs within a reasonable computation time, Kim and Lee (2017) was only able to solve 20 jobs with branch and bound techniques, due to the consideration of complex time constraints. The idea behind decomposition-based approaches is to divide the problem into multiple sub-problems inspired by the divide and conquer paradigm. For example, Sun et al. (2005) and Maleck et al. (2019) divide the scheduling problem into three levels and propose a solution for each problem individually. Klemmt and Monch (2012) and Jung et al. (2014) break down the problems recursively to find near-optimal solutions and optimize KPIs such as total tardiness and time constraint violation rates. Genetic algorithms (GA), on the contrary, are inspired by natural selection and belong to the class of evolutionary algorithms. They are commonly used to find solutions to complex problems that are impossible to solve exactly within a reasonable time frame. Klemmt et al. (2008) find that exact approaches are limited in terms of problem dimension and therefore apply a GA to find near-optimal solutions to four representative oven batching problems. Finally, Han and Lee (2023) consider a three-machine flow shop with missing operations and provide various types of heuristic algorithms to solve the scheduling problem heuristically. Two metaheuristic algorithm—iterated greedy and simulated annealing—have shown to be effective and efficient. Only few studies deviate from a MIP-based modeling approach. A disjunctive graph representation is used by Yugma et al. (2012) to group lots in batches and apply a simulated annealing algorithm to optimize cycle time and machine capacity. A decision support system is developed by Perraudat et al. (2019) using a kanban system model. Wu et al. (2016b) build an analytical model to quantify the trade-off between a higher capacity and a lower rework rate requiring a higher and a lower WIP-level respectively. At a dispatching level, individual lots have to be managed. Due to the high uncertainty of production control in semiconductor manufacturing, the non-linear material flow, lots re-entering the production flow and sudden machine breakdowns, managing time constraints ultimately comes down to dispatching the right lots at the right time. The respective literature is shown in table 2. Lima et al. (2017a,2019) and Sadeghi et al. (2015) apply a method based on sampling for predicting if a lot will adhere to time constraints upon entry. Building on this, Lima et al. (2021) present an improved algorithm compared to the original by Lima et al. (2017a), along with a problem modeling approach that aligns more closely with real industrial conditions. Additionally, Lima et al. (2017b) develop a decision support system to identify tool interruptions by grouping machines based on shared recipes and aggregating time constraint transitions by their ending equipment. In contrast, Altenmüller et al. (2020) use reinforcement learning (RL) and train a RL agent that is rewarded for reducing time constraint violations, successfully outperforming dispatching rules such as the First In, First Out (FIFO) heuristic. Several studies focus on implementing rule-based systems, often validated through simulations. For instance, Tu et al. (2010) introduce dynamic job control in the furnace area, allowing managers to dynamically manage work in progress (WIP) and ensure adherence to time constraints. Arima et al. (2015) maximize throughput for lots adhering to time constraints by combining dispatching and loading rules. Similarly, Kobayashi et al. (2013) aim to optimize dispatching rules for a re-entrant flow shop. Kopp et al. (2020) apply a rule-based approach that considers lot priority, setup time, and a time constraint criticality factor. Ciccullo et al. (2014) and Pirovano et al. (2020) examine the batching process before a time constraint between cleaning and diffusion processes, proposing heuristic algorithms to avoid scrapped lots. Furthermore, Zhang et al. (2016) develop a plan-based control system to handle the dynamic and stochastic envi123 Journal of Intelligent Manufacturing (2024) 35:4259–4276 4263 Table 1 Classification of the relevant literature based on modeling, approach and objective at a capacity planning and scheduling level Level Modeling Approach Source Capacity planning Experiments Simulations Pappert et al. (2016), Huang et al. (2011) Queueing system Queueing theory Tu and Chen (2009a,2009b,2010,2011), Tu and Liou (2006), Kitamura et al. (2006), Ono et al. (2006) Neural network Regression Kuo et al. (2011) Scheduling MIP/MILP/ MINLP Branch and bound An et al. (2016), Kim and Lee (2017) Cuckoo search algorithm Zhou et al. (2019) Decomposition Jung et al. (2014), Klemmt and Monch (2012) Estimation of distribution algorithm Wang et al. (2014) Exact solution Maleck et al. (2017), Yu et al. (2013), Cho et al. (2014), Kao et al. (2011), Klemmt et al. (2008) Genetic algorithm Lee (2020), Wang et al. (2015), Chien and Chen (2007), Klemmt et al. (2008) Simulated annealing Nattaf et al. (2019), Zhou and Wu (2017), Han and Lee (2023) Heuristic control policy Su (2003), Yurtsever et al. (2009), Yu et al. (2017) MIP + CP Decomposition Sun et al. (2005), Bixby et al. (2006) Exact solution Maleck et al. (2018) CP Decomposition Maleck et al. (2019) Disjunctive graph Heuristic control policy Yugma et al. (2012) Kanban system Simulation Perraudat et al. (2019) Analytical model Heuristic control policy Wu et al. (2016b) 123 4264 Journal of Intelligent Manufacturing (2024) 35:4259–4276 Table 2 Classification of the relevant literature based on modeling, approach and objective at a dispatching level Modeling Approach Source Disjunctive graph Sampling-based heuristic Lima et al. (2017a,2017b,2019,2021), Sadeghi et al. (2015) Feedforward ANN Heuristic control policy Chakravorty and Nagarur (2020) MDP Numerical calculation algorithm Wu et al. (2010,2012a,2012b,2016a) RL Altenmüller et al. (2020) Experiments Heuristic control policy Tu et al. (2010), Arima et al. (2015), Kopp et al. (2020), Ciccullo et al. (2014), Kobayashi et al. (2013), Pirovano et al. (2020) Plan-based heuristic Zhang et al. (2016) BIP Heuristics Ham et al. (2011) MIP/MILP Exact solution Maleck and Eckert (2017) Heuristic control policy Chang and Chang (2012), Wang et al. (2018) Neural network + heuristic control policy Li et al. (2012) Genetic algorithm Jia et al. (2013) ARMA PE + PI May et al. (2021c) ML & ARIMA May et al. (2021b) Queueing model Heuristic control policy Yang et al. (2015) ronment in semiconductor manufacturing, though it lacks real-time capabilities and practical application in a real semiconductor setting. Though the following studies share a common foundation in integer programming models, their objectives, work areas, and utilized parameters differ significantly. Ham et al. (2011) focus on minimizing computation time to provide a real-time dispatching heuristic using binary integer programming (BIP) in a two-machine flow shop. In contrast, Chang and Chang (2012) integrate dispatching rules into a three-stage approach to reduce cycle time. Jia et al. (2013) combine a pull-pull-push-push strategy with a genetic algorithm to develop a closed-loop dispatching heuristic. Multiple dispatching rules incorporating risk factors are proposed by Maleck and Eckert (2017) to account for tool failure probabilities. Li et al. (2012) use a learning-based approach, training a neural network to set the weighted parameters of a dispatching rule based on job due dates and machine workloads. Wang et al. (2018) explore time-link area constraints and develop a control policy for initiating the first process of a time constraint. May et al. (2021c) and May et al. (2021b) apply autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models, respectively. May et al. (2021b)also construct a recurrent neural network (RNN), specifically a Long-Short-Term-Memory (LSTM) network, to compare the performance of different approaches. Both studies combine a single point estimator (PE) with an extended prediction interval to estimate the probability of a lot adhering to a time constraint. Other approaches include using a neural network to predict cycle time by Chakravorty and Nagarur (2020) or queue lengths (May et al., 2021a), starting processing only if the predicted cycle time is within the time limit. Finally, Yang et al. (2015) address the challenge of long setup times for implantation equipment compared to their process times. To prevent continuous production runs from being interrupted by arriving lots with time limits, they propose a novel dispatching algorithm that combines recipe changes with processing arriving time constraint lots. Contribution As explained by the literature review, time constraints in semiconductor manufacturing pose a serious trouble to operations and currently are predominantly modeled using mathematical models and heuristics. Many of these approaches make strong simplifications regarding the number of machines or the heterogeneity of the transitions in a wafer fab. Thus, applying them in an existing semiconductor fab is hardly conceivable for this theoretical work. As a result, solutions are oftentimes not verified in real-world sized semiconductor systems let alone in an existing semiconductor manufacturing environment and in particular not in an entire fab. Thus, they fall short of any applicability in real-world environments. An important contribution is hence the demonstration of the proposed model in a real-world semiconductor fab with practical implications. Additionally, machine learning approaches are significantly underrepresented despite their so far promising results. Most notably, the inclusion of uncertainty in artificial intelligence has been neglected. In contrast to alternative approaches, AI based approaches can be enhanced to deal with industrial size problems and replace industrial, human or priority rule based approaches. To that end, the proposed models must signifi123 Journal of Intelligent Manufacturing (2024) 35:4259–4276 4265 Fig. 1 Simple time constraint: Transition time tbetween two operations Onand On+1limited by an upper limit du cantly outperform state-of-the-art approaches. Therefore, the need to regard time constraints holistically using real-world data from semiconductor manufacturing is derived and a modeling approach is developed accordingly. The model is validated with a real-world wafer fab. Decision model We develop a decision model based on real-time data to improve the production control in a complex job shop manufacturing setting. This section defines the formal manufacturing setting, gives a detailed problem description and introduces the model specification. Manufacturing setting A complex job shop is a manufacturing setting where m∈M machines, herein also denoted as equipment, manufacture j∈Jjobs that can involve several complexities. Each job jcontains a process flow f∈Fwith f={o1,o2,...,ok} with o∈Osignifying possible operations. Each machine can perform Om⊂Ooperations. First, a re-entrant flow of jobs (i.e. a job is re-entering the manufacturing setting after one pass through) is possible, as for instance a job’s process flow fjcontains operations that are performed outside of the manufacturing system such as quality assurance. Furthermore, the setup times are sequence dependent, meaning that the setup time tsetup o1,o2can differ significantly from the setup times tsetup o3,o2and tsetup o2,o1for o1,o2,o3∈Om. Likewise, the processing time tprocessing ok,m1varies for different jobs while lot sizes for any lot l1={j1,j2,...}are heterogeneous. Due to frequent and hard to control machine breakdowns, decisions in a complex job shop have to be taken under a high degree of uncertainty (Waschneck et al., 2016). Prescribed due dates and time constraints, that limit the time between operations oaand obwith oa,ob∈fare a major concern. Failing to hold due dates can result in high costs and low customer satisfaction, while failing to adhere to time constraints can cause insufficient quality and may require to restart manufacturing of this lot from scratch. Whenever an equipment has to select the next lot to be processed, a decision in terms of gate has to be made. Regarding a time constrained lot, this decision includes the initialization of the time constraint as there is no turning back after the processing is started. The latest possible decision is therefore right before starting processing and targets the minimizing the number of violations of time constraints. This decision is often based on heuristics or the experience of the technicians in charge and is a time consuming and stressful task (Lima et al., 2017b). The objective of this approach is to predict the probability of a lot adhering to its given time constraint to assist the technicians at a dispatching level by using the advanced machine learning techniques (Chen et al., 2023) without using simulation or digital twin based approaches (May et al., 2022). The modeling approach presented in this paper specifically targets simple time constraints. As depicted in Fig. 1,the transition time stands for the total time that elapses between the end of operation Onand the beginning of operation On+1 including transportation, handling and queue time. When the processing of the lot arriving at the first machine mnis started, the time constraint is initialized and there is no turning back. Only when processing at the subsequent machine mn+1is started within the given time limit du, the time constraint is fulfilled and the lot is prevented from exceeding its time constraint. Therefore, time constraint violations can be identified in retrospect. Hence, any improvement from true positive and true negative predictions have a large effect on the semiconductor fab and can save precious value, energy and time. Problem description In semiconductor manufacturing, time constraints can occur limiting the transition time between two or more machines by an upper limit. Time constraints are classified into different levels of complexity according to Klemmt et al. (2008) and Wang et al. (2018) as follows: Simple time constraints put a limit to the transition time between two consecutive machines, i.e. without any intermediate steps, while transitions spanning multiple machines are called timelink area constraints. Time constraints are considered complex when they consist of multiple overlapping or directly successive time constraints. Simple time constraints are specifically regarded in this paper, due to them occurring most frequently in practice. 123 4266 Journal of Intelligent Manufacturing (2024) 35:4259–4276 Fig. 2 Decision model using the prediction interval for estimating the probability of adhering to the time constraint At the moment, dispatching decisions are performed manually by wafer fab operators mostly based on their experience and defined heuristics. In order to provide the operators with meaningful information, the estimated adherence probability of a time constraint transition is predicted. As outlined in “State-of-the-art time constraint control in semiconductor manufacturing” section, machine learning approaches are considerably underrepresented in the literature of time constraint management in semiconductor manufacturing settings. Due to their strong ability to cope with dynamic and complex manufacturing environments, two models are constructed and presented in the following. Model specification Generally, neural networks can be treated as black boxes, where a defined input provides a defined output. How or why specific results are produced oftentimes can not be determined leading to a lack of interpretability of the result. Therefore, the point estimators of the neural networks are supplemented by an uncertainty quantification method in order to construct a prediction interval as introduced in May et al. (2021b). Therefore, Monte Carlo dropout is introduced in order to estimate the model uncertainty. Given a specific time limit and a prescribed confidence level, the uncertainty is in turn used to calculate the estimated time constraint adherence probability. Based on this concept, two models are constructed: For the resource-based modeling approach, the time series data is aggregated based on equipment groups to exploit equipment group-specific characteristics. The transitional modeling approach is fit to the entire data set. Each transition between any two machines ma,mb∈M can potentially contain time constraints. A transition is only interesting and regarded if there is more than one transition of a job j∈Jthat transitions from machine mato mb. Transitions without any job transitions or with only one job transition do not provide enough data to base the decision on previous observations. The transitions can contain time-constrained jobs that have an upper limit dufor the transition time and non-constrained transitions. In the transitional modeling approach, each possible transition is treated individually, whereas the resource-based modeling approach pools transitions based on temporal-spatial data. For each model of a transition, or pooled resource transition, a prediction model is formed as time series data on the transition times is available. Each next potential transition uses the currently available (past) data to predict the next transition time. Based on the prediction interval fed by the Monte Carlo dropout, this point predictor of an individual transition is transformed into an estimated probability. The approach is presented in Fig.2, which shows an exemplary case of four total jobs, the second and forth being time constrained. The second has successfully adhered to the time constraint. Using the past three observations, the estimation for job 4 is performed. As indicated, the time constraint is lower than the expected transition time and also has a low estimated adherence probability (below the bar) compared to a greater estimated violation probability. This can be regarded as the confidence about the transition time being smaller than the time constraint. Selecting this confidence is paramount for a good model and based on operator and expert discussions we selected 90% and 95% as the prescribed confidence. Note, that the confidence is not to be mistaken with the expected share of jobs that violate the time constraint despite being predicted to adhere with the given confidence. Prediction intervals A prediction interval is defined as a future value lying between an upper and a lower bound with a given probability (Chatfield, 2001). Since we are only interested in a transition possibly exceeding its defined time limit, the lower bound of the prediction interval is set to −∞. The upper bound can than be calculated by ˆy+zα/2Var(e)(1) where ˆydenotes the point estimation, zα/2the appropriate percentage point of a standard normal distribution and Var(e)the variance of the models prediction error (Chatfield, 2001). The first two parameters are directly available from the models prediction and the prescribed confidence interval respectively. The variance of the prediction error consists of the sum of the model-specific variance σ2 ˆyirepresenting the model’s uncertainty and the variance of the noise σ2 ˆin the underlying data (Khosravi et al., 2011). The variance present in the noise can be calculated by division of the sum of the squared differences between the observed values 123 Journal of Intelligent Manufacturing (2024) 35:4259–4276 4273 Author contributions All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript. 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