Numerical studies for the scheduling of continuous annealing lines

Hönerloh, Hagen Alexander Article Numerical studies for the scheduling of continuous annealing lines Junior Management Science (JUMS) Provided in Cooperation with: Junior Management Science e. V. Suggested Citation: Hönerloh, Hagen Alexander (2025) : Numerical studies for the scheduling of continuous annealing lines, Junior Management Science (JUMS), ISSN 2942-1861, Junior Management Science e. V., Planegg, Vol. 10, Iss. 3, pp. 781-809, https://doi.org/10.5282/jums/v10i3pp781-809 This Version is available at: https://hdl.handle.net/10419/326973 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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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. https://creativecommons.org/licenses/by/4.0/ Junior Management Science 10(3) (2025) 781-809 Junior Management Science www.jums.academy ISSN: 2942-1861 Editor: DOMINIK VAN AAKEN Advisory Editorial Board: FREDERIK AHLEMANN JAN-PHILIPP AHRENS THOMAS BAHLINGER MARKUS BECKMANN SULEIKA BORT ROLF BRÜHL KATRIN BURMEISTER-LAMP CATHERINE CLEOPHAS NILS CRASSELT BENEDIKT DOWNAR KERSTIN FEHRE MATTHIAS FINK DAVID FLORYSIAK GUNTHER FRIEDL MARTIN FRIESL FRANZ FUERST WOLFGANG GÜTTEL NINA KATRIN HANSEN ANNE KATARINA HEIDER CHRISTIAN HOFMANN SVEN HÖRNER STEPHAN KAISER NADINE KAMMERLANDER ALFRED KIESER ALEKSANDRA KLEIN NATALIA KLIEWER DODO ZU KNYPHAUSEN-AUFSESS SABINE T. KÖSZEGI ARJAN KOZICA CHRISTIAN KOZIOL MARTIN KREEB WERNER KUNZ HANS-ULRICH KÜPPER MICHAEL MEYER JÜRGEN MÜHLBACHER GORDON MÜLLER-SEITZ J. 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This is an Open Access article distributed under the terms of the CC-BY-4.0 (Attribution 4.0 International). Open Access funding provided by ZBW. ISSN: 2942-1861 Numerical Studies for the Scheduling of Continuous Annealing Lines Hagen Alexander Hönerloh Leibniz University Hannover Abstract The continuous annealing of flat steel improves its properties for applications such as automotive manufacturing. Scheduling these processes on Parallel Heterogeneous Annealing Lines (PHALs) is complex due to diverse coil properties, incompatible process modes, and due date constraints. Introducing stringers to address incompatibilities between steel sheets raises costs, energy use, and CO2 emissions, highlighting the need for optimized scheduling. This thesis implements a mathematical model in Python using the Gurobi solver to optimize PHAL scheduling by minimizing stringer usage while meeting tardiness constraints. The model is extended to include coil-specific release dates and expanded to address trade-offs between stringer use, tardiness, and due date deviations, including earliness. A computational study evaluates the model under various scenarios, examining the effects of coil heterogeneity, urgency, process flexibility, and stringer processing times. Results show that optimized schedules reduce stringer use and delays, particularly under high process flexibility. These findings demonstrate the potential of optimization to improve efficiency and sustainability in steel production while guiding future research in dynamic scheduling approaches. Keywords: continuous annealing lines; Gurobi solver; scheduling optimization; steel industry; stringer minimization 1. Introduction 1.1. Subject and motivation Our modern economy thrives on digital transformation and its many facets. One key aspect is the computerization of processes using advanced digital technologies. Through computerization, companies can create and implement optimal schedules for their processes and thereby increase efficiency and productivity, as companies always sought to improve their decision-making through new scheduling and planning methods.1But the potential impact of computerization and digitalization could be far greater than anything before. The steel industry is one of the many industries that stand to benefit significantly from this development. By leveraging digital technologies, manufacturers are able to minimize production time, while also maximizing the use of resources and thus their profits.2The impact on this highly competitive industry, which supplies us with materials needed for 1Tang and Meng, 2021, p. 1 2Iannino et al., 2021, p. 620 everyday appliances, railways, or even buildings, is astonishing.3One process in the steel industry, that can benefit greatly from digitalization, is the continuous annealing of flat steel, a method of processing steel to change its physical and mechanical properties.4In this process, coils of cold rolled flat steel are processed in furnaces with different annealing temperatures and transport speeds that make up the process mode of an individual coil. Different process modes lead to different mechanical and physical properties of the steel.4Hence, the mode is chosen according to the desired characteristics. Cold rolled flat steel is essential for building cars, and household appliances and has many more areas of application.5Scheduling the continuous annealing process on continuous annealing lines (CALs) is a difficult task due to the different properties of the flat steel, processing modes, and other aspects like due dates. If the properties or processing modes of two successive coils are too different, scrap 3Zhao and Yang, 2016, p. 3417 and Terence, 2020 4Sarna, 2013 5Commodity-Inside, 2019 DOI: https://doi.org/10.5282/jums/v10i3pp781-809 © The Author(s) 2025. Published by Junior Management Science. This is an Open Access article distributed under the terms of the CC-BY-4.0 (Attribution 4.0 International). Open Access funding provided by ZBW. H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809782 coils, so-called stringers, must be added between the coils to bridge these differences, resulting in additional material costs and a loss of efficiency.6 The impact of this loss of efficiency cannot be overstated. Not only does the manufacturer lose valuable production time, but he also has to waste energy, increasing the amount of CO2emitted per kg of steel. With a share of seven % of the world’s annual CO2emissions, the steel industry is already one of the most energy-intensive industries worldwide.7In the meanwhile, in the European Union, the steel sector is facing an ever-increasing cost of energy, as well as an increase in CO2 price per ton of CO2 emitted, which some experts predict could reach 50% by the end of the decade.8 Hence, manufacturers should have a serious interest in optimizing their production schedules by minimizing the introduction of stringers and thus the costs caused by material and energy wastage, loss of efficiency, and CO2emissions, which can be achieved through the use of digital technologies. 1.2. Research question and structure of the work This thesis is based on a paper by Wegel et al. (2024), in which they propose a mathematical model to optimize the scheduling of the continuous annealing process. The objective of this model is to minimize the introduction of stringers in a schedule, while also considering a tardiness constraint that limits the number of delays. It is designed for shortterm planning and can therefore be used at the operational level of operations management.9So far, this model has only been implemented in the Julia programming language and used with a proprietary algorithm. Thus, the model has not yet been implemented in the popular Python programming language. This implementation forms the basis of this thesis. During the course of this work, a numerical study will be conducted on this model and on its expansions created during the work. It will consist of different sections, which study the impact of certain parameters and scenarios on the scheduling of CALs. The results will be thoroughly analyzed and discussed to derive emerging trends and formulate managerial insights. First, the CALs for flat steel will be explained in terms of their design and their scheduling, which will be followed by the current state of research. Afterwards, an explanation of the underlying problem and the mathematical model itself will be given. This will be continued by further extensions of the model with the aim of mimicking the real-world process. The numerical studies conducted will be the core of this thesis. Different trends that occur with increasing instance sizes will be explored first, followed by sections in which the impact of scenarios and the alteration of certain parameters on the scheduling process will be investigated. Furthermore, 6Besson, 1998, p. 29 7Joint-Research-Centre, 2022 8Twidale et al., 2021 and Krukowska, 2021 9Karakostas et al., 2020, p. 2 and Karakostas et al., 2019, p. 2 the base model will be compared with the extended model regarding its performance and solutions. The results of the computational study carried out on the Python implementation and the instances used are presented and discussed, leading to a summary of the work and an outlook for future research, further extensions, and managerial insights. 2. Continuous Annealing Lines for Flat Steel 2.1. Process description The industrial continuous annealing process is made possible by CALs, which consist of several sections. One such CAL is depicted in figure 1. CALs are not standalone, they are part of a greater complex consisting of different areas dedicated to different processing methods, like the cold rolling and galvanization of steel.10 This study will focus on the annealing of cold rolled steel, which is usually the bottleneck of steel processing.11 In some cases, it may also be possible and advantageous to anneal hot rolled steel.12 In comparison to hot rolling, cold rolling happens at far lower temperatures. But these processes are complementary and not substitutional, since the hot rolling process is an upstream process of cold rolling. While hot rolling rolls the steel to the desired width, cold rolling reduces its thickness by up to 90 %, increasing its strength and hardness but also severely diminishing its ductility and increasing its brittleness.13 To improve these mechanical and physical properties, the steel strips are annealed in CALs. Through recrystallization of microstructures and other processes, the steel regains some of its lost characteristics, especially its ductility.14 The process starts at the entry section with the coils of cold rolled flat steel and the so-called pay-off reel, as depicted in figure 1. These coils stem from the upstream cold rolling process. The pay-off reels fixate the coils of flat steel with a mandrel and rotate to continuously unwind them.16 Further downstream, a welding machine automatically welds the tail end to the head end of two consecutive coils together, thus providing continuous strip feeding to the succeeding sections.17 The welding process itself is of the utmost importance since a weld break could result in a complete line shutdown.18 Therefore, compatibility between consecutive coils regarding their thickness and width has to be ensured. If the two consecutive coils are incompatible with each other, a stringer has to be added between them to assure a continuous annealing process, resulting in a loss of efficiency and higher material and energy costs.19 Stringers can be reused, but only for a limited number of times.20 10 Zhao and Yang, 2016, p. 3418 11 Li et al., 2023, p. 1 12 Steel-Warehouse, n.d. 13 Zhao and Yang, 2016, p. 1 14 Sarna, 2013 15 Takurou et al. (2016), p. 113 16 Jiyuan-Shenzhou-Industry, n.d. 17 United-Enterprises, n.d. 18 Williamson, n.d., p. 3 19 Besson, 1998, p. 29 20 Mujawar et al., 2004, p. 1 H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 783 Figure 1: A Continuous Annealing Line as implemented in a Japanese factory.15 After degreasing by a degreasing unit, the steel strip enters the process section, where it first feeds into an entry looper.21 This looper counteracts interruptions in the continuous annealing process, such as the welding process, and maintains a continuous strip speed throughout the annealing process. It achieves this by moving its rolls apart from each other, thus increasing the length of strip steel it can hold and the distance the steel has to traverse. The continuous strip then enters several furnaces, that can reach temperatures of up to 850 °C, while maintaining a strip speed of up to 800 m/min.22 The steel strip cycles through the furnaces for several minutes, depending on the chosen processing mode. As previously described, a processing mode is a combination of annealing temperature and strip speed, which leads to certain steel properties.23 Since the strip speed and furnace temperature can only be adjusted in a certain range from the preceding coil to the succeeding one, compatibility between their processing modes is necessary or else a stringer has to be introduced between them.24 It should also be mentioned, that adjusting the furnace temperature costs a lot of energy and should therefore be minimized.25 After the heat treatment, the steel strip is cooled down in several steps, until it feeds into the delivery or exit looper.26 This looper works like the entry looper and can therefore compensate for interruptions like maintenance and the cutting process, which follows downstream.24 After the cutting, the tension reel recoils the steel strip into the previous coils.27 Compared to the batch annealing process, in which the steel is processed as a coil, the continuous annealing process has a higher efficiency and productivity, while also delivering a more uniform product, regarding the physical and mechanical properties of the strip steel. Some of its disadvantages, however, are the large amounts of space and capital needed to construct it.23 21 United-Enterprises, n.d. 22 United-Enterprises, n.d. 23 Sarna, 2013 24 Besson, 1998, p. 29 25 Zhao and Yang, 2016, p. 3417 26 Sarna, 2013 27 Jiyuan-Shenzhou-Industry, n.d. 2.2. Differentiation from literature The topic of scheduling has been a research subject since the early 20th century and is nowadays one of the most researched fields in operations research, with several hundred papers published each year.28 Scheduling in the steel industry particularly is one of the most difficult and complex problems, due to the complexity of the steel industry itself and therefore, there have been many attempts to optimize certain aspects of it.29 This section will feature different approaches concerning CALs and continuous galvanizing lines, a process further downstream of the annealing process that has many similarities with CALs regarding its scheduling.30 Li et al. (2023) aimed to minimize earliness and tardiness costs, as well as setup costs, which occur through changes in annealing temperature, on one processing line.31 The authors noted, that these objectives conflict with each other since minimizing the setup costs by constructing large batches of similar coils could increase the earliness and tardiness costs and vice versa. Stringers were not directly considered. To minimize the three objectives, they used an adaptive multi-objective differential evolutionary algorithm (MODE) based on deep reinforcement learning (DRL). Several other papers also include the use of MODEs for the optimization of CALs and for other processes as well, like for the hot rolling process.32 The usage of MODEs can be highly effective, as can be seen in a study by Dong et al. (2021), in which they were able to find optimal schedules for the color-coating process for up to 400 coils with three different objectives. During the color coating process, protective or decorative coatings are applied to the steel.33 Zhao and Yang (2016) tested a discrete differential evolution algorithm (DDE), which uses discrete job permutations to find optimal solutions, to average the line capacity and minimize the changeover costs. These occur through annealing temperature changes in the furnace, and Zhao and Yang (2016) concluded that their algorithm performed well for up to 90 coils on parallel processing lines.34 Even though the dif28 Potts and Strusevich, 2009, p. 1 29 Harjunkoski and Grossmann, 2001, p. 1649 30 Zhao and Yang, 2016, p. 3418 31 Li et al., 2023, p. 1-2 32 Tang and Wang, 2010, p. 104-116 and Pan et al., 2019, p. 327-348 33 Sarna, 2014 34 Tasgetiren et al., 2007, p. 271-273 H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809784 ferences between annealing temperatures of adjacent coils were considered, stringers and tardiness were not. Since DDEs are difficult to apply to real-life problems, the algorithms’ usefulness may be in question.35 Evolutionary algorithms also tend to focus on local optima, which can be less optimal than a global optimum.36 The so-called Tabu search solves this problem and was used for optimization in continuous galvanizing lines.37 Gao et al. (2008) claimed their respective approaches to be very effective as they were able to solve instances with up to 100 coils in only some seconds. But it has to be noted that some aspects like tardiness were not considered. A study by Pan et al. (2017) aimed to minimize the total weighted completion times on a parallel CAL with up to 18 different lines and 120 coils and achieved optimal solutions in under ten minutes.38 One similarity of the previously discussed studies is the deterministic characteristic of the used models. Therefore, some authors used dynamic approaches like a multi-agent system (MAS) to optimize dynamic scheduling problems, which consider randomness.39 A MAS uses artificial intelligence and multiple agents or perspectives to solve a problem to provide flexible solutions.40 Iannino et al. (2021) used both deterministic and dynamic models for scheduling. The authors used three different approaches to optimize a day’s schedule with around 2100 coils in several iterations, with the objective to improve scheduling flexibility. For short-term planning with unstable circumstances, they used a MAS. The second approach was a deterministic, mixed integer linear program (MILP), as is used in this study. The third and last approach utilized a continuous flow model (CFM) for longterm scheduling, which uses a simplified model to schedule the manufacturing process over various time periods.41 Iannino et al. (2021) summarized that all three approaches have their advantages and disadvantages and that they should be used complementarily rather than substitutionally to utilize each advantage in the right circumstances. The second most important study about scheduling for this thesis is by Mujawar et al. (2012). The authors proposed a MILP for the minimization of both stringers and tardiness in terms of total time overdue per coil. Due to technical limitations, they were only able to solve the model with up to 15 coils, with a solving time of close to three hours. Because of these limitations, they leveraged two different heuristics, which were able to yield feasible solutions for up to 150 coils, but could only minimize the number of stringers introduced in the schedule and disregarded the tardiness.42 Nonetheless, the proposed mathematical model mimicked the real-world process to a higher degree than others. Because of this, Wegel et al. (2024) decided to base their 35 Zhao and Yang, 2016, p. 3418 36 Mirjalili and Gandomi, 2023, p. 393 37 Gao et al., 2008, p. 1829-1833 38 Zhang and Yang, 2014, p. 800-802 39 Cowling et al., 2004, p. 178-188 and Ouelhadj et al., 2004, p. 161-172 40 Balaji and Srinivasan, 2010, p. 1-2 41 Iannino et al., 2021, p. 620-630 42 Mujawar et al., 2012, p. 440-444 study on the model proposed by Mujawar et al. (2012) and develop it further for better performance in the solving process. 3. Optimization model for Parallel Heterogeneous Annealing Line Scheduling 3.1. Problem description As mentioned in section 2.1, the continuous annealing process is very complex, and many parameters have to be accounted for. One such parameter, that was briefly mentioned before, is the individual due date of each coil. These are necessary, to schedule the further downstream production stages, like the galvanization.43 These schedules only have limited flexibility and to not put them at risk, the maximum number of delays has to be bound. This limitation is implemented through a service level, which value is relative to the number of coils processed. Thus, a greater instance is granted with a higher service level than a small or medium one. Until now, only the specific characteristics of the coils and their processing modes have been considered. But the lines they are processed on have different characteristics themselves. Some lines may only be able to process coils with certain thicknesses and widths, while others may be able to process all coils, regardless of their characteristics. When creating a schedule for a continuous annealing line, the manufacturer thus also has to consider which coil can be processed on which of the parallel heterogeneous lines, therefore decreasing the amount of planning flexibility. This results in a schedule that has to consider the different characteristics of the coils, their desired specifications in terms of strip speed and annealing temperature, the compatibility of said characteristics and processing modes of consecutive coils, as well as their compatibility with each processing line and their due dates, while it also has to comply with the service level and aims to minimize the number of used stringers. To be able to create such a complex schedule, Wegel et al. (2024) set four assumptions, regarding the release dates, internal service level, costs, and operating conditions. To reduce complexity, it is first assumed that every coil is available and waiting for processing at the beginning of the schedule. Therefore, each coil could be the first to be processed in the schedule since they have no release dates. The second assumption concerns the previously described internal service level. As already mentioned, the service level bounds the absolute number of delayed coils to a certain value that is relative to the instance size. But the internal service level also bounds the maximum delay that can occur in the schedule, since all coils have to be processed during it. This maximum delay depends on the maximum amount of time needed to process all coils in one schedule. 43 Zhao and Yang, 2016, p. 3418 H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 785 Table 1: Notation of the mathematical model. Symbol Meaning Indices and index quantities i,j∈1,...,NCoils k∈1,...,KProcessing lines m∈1,...,Mik Feasible processing modes of coil ion line k n∈1,...,Mjk Feasible processing modes of coil jon line k Parameters diDue date of coil i pikm Processing time of coil ion line kin mode m αService level MMaximum duration of the schedule ci jkmn Costs of adding a stringer between coils iin mode mand jin mode non line k ti jkmn Additional processing time of a stringer between coils iin mode mand jin mode non line k Decision variables yi jkmn ∈ {0,1}Binary variable with value 1, if coils iin mode mand jin mode nare processed in sequence on line k, else 0 δi jk ∈ {0,1}Binary variable with value 1, if coils iand coil jare processed in sequence on line k, else 0 xikm ∈ {0, 1}Binary variable with value 1, if coil iis processed in mode mon line k, else 0 si≥0 Start date of coil i zi∈ {0,1}Binary variable with value 1, if coil iis delayed, else 0 The third assumption made sets the costs of all coils to a fixed value. The reasoning behind this is that only the costs caused by the introduction of stringers should be considered since we cannot avoid the costs caused by the processing of coils. Due to CALs being quite stable and completely automated, Wegel et al. (2024) also assume that no randomness occurs during the process and therefore propose a deterministic model, in which all parameters are known a priori. This is the fourth and last assumption regarding the mathematical model. Based on these assumptions and the aspects mentioned previously, the authors build a deterministic optimization model that aims to create an optimal, cost minimizing, and tardiness-bound schedule for parallel heterogeneous annealing lines, which is based on the model by Mujawar et al. (2012)44 In the following sections, the notation of this mathematical model and the model itself will be explained, which will be followed by further extensions to it. 3.2. Notation This section presents the notation of the mathematical model proposed by Wegel et al. (2024). The coils of flat steel are denoted by N={1,..., N}and the parallel continuous annealing lines they are processed on by K={1,..., K}. Every coil i∈ N has a specific due date di>0. The maximum 44 Mujawar et al., 2012, p. 440 amount of acceptable delays, the service level, is denoted by α∈N. Each coil ialso has a set amount of different processing modes Mik on each line k∈ K . The feasible process mode mof coil ion line kincludes a specific annealing temperature and strip speed. If coil ican be processed in a specific mode on line kdepends on the characteristics of the coil iand of the line k, as well as on the parameters of the processing mode itself. It is therefore possible, that no mode for the processing of coil ion line kexists. If coil ican be processed on line k in mode m, the processing time pikm can be derived. Furthermore, coil ion line kcould be succeeded by coil j, with j∈ N , in the feasible processing mode n, with n∈ Mjk. In this case, the compatibility of coil iin mode mand coil jin mode nregarding their width, thickness, annealing temperatures, and strip speeds has to be reviewed. If the coils and modes are incompatible with each other, a stringer has to be added between them. The additional costs caused by introducing a stringer into the schedule between coil iin mode mand coil jin mode non line kare expressed by ci jkmn, while the additional processing time is contained in ti jkmn. If the coils and their respective modes are compatible, the additional costs and processing time will equal zero, since no stringer has to be introduced. The processing schedule will be defined by the following decision variables. Each coil’s start time is represented by si≥0, while its processing line and mode in the optimized H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809786 schedule are indicated by the binary variable xikm ∈ {0,1} taking the value of one if coil iis being processed on line kin mode mor zero, if not. If the completion of coil iis behind the scheduled due date di, the binary variable zi∈ {0,1}will equal one or zero, if coil i’s processing was finished in time. The sequence of the schedule is indicated by the binary variable δi jk ∈ {0,1}, with the value of one if coil i precedes coil jon line kor zero if this is not the case. Additional information regarding their respective modes mand nis provided by the binary variable yi jkmn ∈ {0, 1} with the value of one if coil iis being processed in mode m and is succeeded by coil jin mode non line kor zero, if not. 3.3. Mathematical model min z,s,x,δ,yZ=X i∈N X j∈N X k∈K X m∈Mik X n∈Mjk ci jkmn ·yi jkmn s.t. si+X k∈K X m∈Mik pikm ·xikm ≤di+zi·M∀i∈ N , (1) si+X k∈K X m∈Mik X n∈Mjk (pikm +ti jkmn)·yi jkmn ≤ sj+M·(1−X k∈K δi jk)∀i,j∈ N , (2) δi jk =X m∈Mik X n∈Mjk yi jkmn ∀i,j∈ N ,∀k∈ K , (3) X j∈N X n∈Mjk yi jkmn ≤xikm ∀i∈ N ,∀k∈ K ,∀m∈ Mik, (4) X i∈N X m∈Mik yi jkmn ≤xjkn ∀j∈ N ,∀k∈ K ,∀n∈ Mjk, (5) X k∈K N+1 X j=1 δi jk =1∀i∈ N , (6) X k∈K N X i=0 δi jk =1∀j∈ N , (7) N+1 X j=1 δ0jk =1∀k∈ K , (8) N X i=0 δi(N+1)k=1∀k∈ K , (9) N X j=0 δjik = N+1 X j=1 δi jk ∀i∈ N ,∀k∈ K , (10) X i∈N zi≤α, (11) si≥0∀i∈ N , (12) xikm,δi jk,yi jkmn,zi∈ {0, 1} ∀i,j∈ N ,∀k∈ K ,∀m∈ Mik,∀n∈ Mjk. (13) The main objective of this mathematical model is the minimization of costs caused by the introduction of stringers. Therefore, the objective function minimizes the sum of products of ci jkmn and yi jkmn. By multiplying ci jkmn and yi jkmn the cost of a stringer is only regarded if coil iin mode mand coil jin mode non line kare not compatible with each other and are processed in sequence in the optimized schedule, thus requiring a stringer. If they are not processed in sequence or if they are compatible with each other or both, the value of the product will equal zero since yi jkmn or ci jkmn or both will equal zero, respectively. By taking the sum over all coils, lines and modes, all introduced stringers in the schedule are considered. To mimic the real-world process and maintain consistency, several constraints are necessary and will be discussed in the following. Constraints (1) and (2) assure time consistency, while constraints (3)-(5) ensure the consistency of decision variables that indicate the same information. The consistency of the processing sequence in the schedule is established by the constraints (6)-(10). These are followed by three additional constraints, that regulate general aspects of the mathematical model and the schedule. The first constraint (1) takes the tardiness of the schedule into account. The time of completion of coil iis calculated by adding its processing time pikm in mode mon line kto its start date si. This completion date has to be smaller or equal to its due date difor the coil to be finished processing in time. In this case, ziwould equal zero due to the restriction of the number of delayed coils by the service level and the possibility that the delay could be used somewhere else in the schedule to further minimize the number of introduced stringers. But if the completion date is after the due date, zihas to equal one to not violate the equation, since the time of completion of coil iwill be greater than its due date di. Additionally, the product of ziand Massures that coil i has to be processed sometime in the schedule since Mis defined as maxk(Pi∈N ,Mik=;max j,m,n(pikm +ti jkmn)), which is the maximum amount of time necessary to process all coils in the schedule. This constraint has to be set for every coil, as every coil has a due date and could be delayed. Constraint (2) considers the time sequences of processing in the schedule. The left side represents the earliest time a succeeding coil jcould be processed after the processing of the preceding coil iwas finished. This is achieved by adding the processing time pikm of coil iin mode mon line kand the possible additional processing time of a stringer ti jkmn, in case of incompatibility of coil iin mode mand coil jin mode n, to the start date of coil i. Processing of the succeeding coil jcan only start after the processing of coil ior that of H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 787 the stringer is completed. Thus, sjhas to be greater or equal to that time of completion. In the case that coils iand jare processed in sequence on line k,δi jk would equal one, and thus, Mwould not be added to the right-hand side. But as this constraint holds for all coils, it could compare two coils with each other, that are not processed in sequence. To prevent this comparison to have an effect on the solution, Mis added to sj, since in this case, δi jk will equal zero. Thus, the right side will always be greater than or equal to the left side since M≤sj+Mand si+(pikm +ti jkmn)·yi jkmn ≤Min every scenario, because of the nature of Mdescribed previously. Constraint (3) assures consistency throughout the variables δi jk and yi jkmn. Both variables indicate the sequence of coils iand j, as well as the line kthey are processed on, and therefore should be equal to each other with the same coils i,jand line k. Since yi jkmn also indicates the processing modes of both coils and cannot represent a binary value without this information, the sum of yi jkmn over all modes has to be taken for this constraint to be effective. This constraint also reassures, that the coils are only processed in one mode and not in multiple, hence in this case, two yi jkmn would equal one with the same coils and line. Therefore, δi jk would have to equal two, which is impossible due to the binary nature of δi jk. Constraint (4) contributes to the decision variable consistency as well. Both variables yi jkmn and xikm indicate the processing line kand the processing mode mof coil i. Hence, they should be equal to each other for every coil, mode, and line. But in the case that coil iis the last coil that is being processed on line k,yi jkmn would equal zero since yi jkmn does not take the coils i∈ {0,N+1}into account and could therefore not be succeeded by another coil j. To not violate this restriction in this case, yi jkmn must only be less or equal to xikm. Therefore, with xikm equalling one, yi jkmn could be either one or zero. Furthermore, by taking the sum of yi jkmn over all coils and modes, the constraint prevents coil ifrom being succeeded by multiple different coils in multiple different modes and limiting this number to the value of xikm, which is either one or zero due to its binary nature. Therefore, coil ican only be succeeded by a single coil j in a single mode non line k, if iis being processed on line kin mode mor by none if it is not. This holds for all coils, lines, and modes. The following constraint (5) is essentially the same as (4) but for the succeeding coil j. Hence, it prevents coil jfrom succeeding multiple coils in multiple modes by limiting this number to xjkn. It has to be noted, that the indices of xjkn in this constraint differ from the indices of xikm in the other constraints since this constraint should only restrain the succeeding coil j. Additionally, yi jkmn must only be less or equal to xjkn, since all yi jkmn’s would be zero if jwould be the first coil to be processed on line k, which would violate the constraint. This is because this constraint only accounts for coils in Nand thus the first coil on line kcannot be preceded. This restriction applies to all succeeding coils on every line in every processing mode. Constraint (6) restricts the number of coils that succeed coil ion any line to one. This prevents coil ifrom being processed and preceded by several coils on different lines. If this would be the case, the sum of δi jk over all lines and coils in [1, N+1]would be greater for coil ithan one, thus violating the restraint. If coil iis the last coil to be processed on line k, it could not be succeeded by any coil, resulting in δi jk equalling zero. Therefore, a virtual coil N+1 has to be introduced so that in the case mentioned above, coil iwould not violate this constraint. This restriction applies to every coil. Constraint (7) is similar to constraint (6), as this constraint restricts the number of preceding coils of coil jon all lines to one and therefore prevents coil jfrom being processed and preceded more than once during the schedule. The extension of Nby zero can be explained with the constraint (8). Here, iis set to the value zero. This virtual coil zero marks the beginning of the schedule on every line k. Since this coil is not being processed, it is not included in most other constraints. But since coil zero is the first on every line k, it has to be succeeded by one coil j∈[1,N+1]. Therefore, constraint (8) ensures that one coil jis the first one to be processed on line k. No real coils are processed on line kduring the schedule if coil N+1 succeeds coil zero. This holds for every line because every line has to have a virtual first coil that is being succeeded by another coil, virtual or real. Constraint (9) on the other hand ensures that one coil i∈ {0,N } has to be the last real coil to be processed on line k. To mark this coil as the last coil to be processed on line k, it will be succeeded by another virtual coil N+1. As on every line kthere has to be a last coil to be processed, this holds for every line. The last sequence consistency constraint, constraint (10), assures that every real coil ihas to have a predecessor and successor coil jon the line kit is being processed on. But it also accounts for the extreme case, that a processing line kprocesses no real coil, which is why it considers coil zero as a predecessor on the left-hand side and coil N+1 as a successor on the right-hand side. Additionally, this considers that one real coil has to be the first and another has to be the last coil if real coils are processed on the line, as described in constraints (8) and (9). This restriction applies to every line. Hence, the introduction of the virtual coils zero and N+1 prevents δi jk from equalling zero in the cases that coil jis the first or coil iis the last coil to be processed on line k, which would violate the sequence consistency constraints. The eleventh constraint (11) restricts the maximum amount of delayed coils to the service level α. As this applies to the entire schedule, the sum of ziover all coils has to be taken. Since processing only starts at the beginning of the schedule, no start date sican take a value of less than zero. This is ensured by constraint (12). As every coil needs a start date, since every coil has to be processed at some point during the schedule, this holds for every coil. The last constraint (13) ensures that the binary variables xikm,δi jk, yi jkmn, and zionly equal the binary values of zero or one and also indicates which index is the element of what parameter. H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809788 3.4. Expansions of the model 3.4.1. Release dates The first expansion tackles the assumption of no coilspecific release dates made by Wegel et al. (2024), which was explained in section 3.1. To mimic a more realistic process, coil-specific release dates were added to the model with the new parameter ri. This parameter holds the specific release date of each coil i, with the release dates being distributed with the same distribution as the urgency of the coils, but with zero as the lower and aas the upper bound. Therefore, in the case of Ver y high UC, the release dates are distributed via the uniform distribution U(0, a). The UC and other factors will be explained in section 4.1 For the release dates to be considered in the model, the following constraints had to be added: si≥ri∀i∈ N (14) ri≥0∀i∈ N (15) Constraint (14) restricts the start time of coil i,si, from being earlier than its release date ri, while (15) prevents the release date from being earlier than the start of the schedule. These hold for all coils. This is only a minor extension of the model. The two major extensions will be described in the following sections. 3.4.2. Minimization of stringer use and tardiness Since both tardiness and costs caused by stringer introduction play a pivotal role in the scheduling of parallel heterogeneous annealing lines (PHALs), Wegel et al. (2024) proposed an expansion of the mathematical model. This expansion modifies the objective function in such a way, that the sum of introduced stringers and the absolute number of delays are minimized, as depicted below. Because of this, the service level αand constraint (11) from the mathematical model in section 3.3 are not necessary anymore and are removed. To account for different weights of tardiness and stringer costs, the two sub-objectives are multiplied with new parameters. While r_str sets the weight of the stringer costs, r_tar does the same for the tardiness, expressed by the sum of delayed coils, with r_str +r_tar =1. To achieve this, an additional restriction (16) was introduced. This constraint ensures that the total weight of tardiness and stringer costs does not differ from 100%. min z,s,x,δ,yZ=X i∈N X j∈N X k∈K X m∈Mik X n∈Mjk ci jkmn ·yi jkmn ·r_str +X i∈N zi·r_tar r_str +r_tar =1 (16) 3.4.3. Minimization of stringer use and due date deviation The second major extension of the model is an enhancement of the first extension by not only taking tardiness into account but also the earliness of the schedule. If products are produced before the due date, they have to be stored in warehouses, which leads to so-called inventory holding costs. They are not only caused by the storage and handling of the inventory, but also by insurance and other factors.45 Thus, there is also an incentive to minimize earliness in production processes. To minimize both the use of stringers and the deviation from the due date, the objective function had to be altered accordingly. min z,s,x,δ,yZ=X i∈N X j∈N X k∈K X m∈Mik X n∈Mjk ci jkmn ·yi jkmn ·r_str +X i∈N (vi·r_v+ei·r_e)·r_dd The first part of the objective function is the same as in the first major extension. But the second part introduces two new variables and three new parameters. The two new continuous variables eiand vican be explained by describing the new first constraint: si+X k∈K X m∈Mik pikm ·xikm +ei=di+vi∀i∈ N (17) These new continuous variables, with a lower bound of zero, work in such a way, that only one of them can have a value higher than zero for coil i. If coil iin mode mon line k was finished processing before its due date di,si+pikm·xikm < di. This would violate the constraint since both sides have to be equal to each other. Therefore, eiwill take the value of di−(si+pikm ·xikm), which represents the positive time left until the due date and therefore the earliness of coil i. In this case, viwill equal zero, since both eiand viare to be minimized. A vigreater than zero would lead to di+vi− (si+pikm ·xikm)>di−(si+pikm ·xikm), therefore increasing both viand eiand thus the objective value. The opposite case, in which coil iis delayed, is similar. Now, si+pikm ·xikm >di. To not increase this difference and hence the objective value, eiwill equal zero. Because of the necessary equality of both sides, viwill equal si+pikm ·xikm − di, representing the positive time that coil iis delayed. In the third and last case, in which the processing of coil iis finished processing exactly on time, si+pikm ·xikm =di. Since eiand viare to be minimized, both variables will equal their lower bound zero. In the altered objective function, ei and viare multiplied with the parameters r_vand r_e, to provide a possibility for weighing the costs of tardiness and earliness. The third and last parameter r_dd weighs the costs of due date deviation against the weight of the stringer costs 45 Durlinger, 2015 H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 795 10 20 30 40 50 60 70 0 2 4 6 8 10 12 # coils # stringers Default solver Low Medium High 10 20 30 40 50 60 70 0 2 4 6 8 10 12 # coils Tuned solver Low Medium High Figure 3: Average number of introduced stringers per Nand service level in optimizations with the default and tuned solver. therefore it is not a result of an outlier as it was in the case of 70 coils. Additionally, this suggests that this is not a random occurrence but a trend. Hence, it might be beneficial to observe the relative average number of stringers introduced, which are depicted in figure 4. 10 20 30 40 50 60 70 10 20 30 40 # coils Relative # stringers [%] Tuned solver Low Medium High Figure 4: Average number of stringers relative to the instance size of optimizations with the tuned solver in %. Figure 4portrays the average number of stringers relative to the instance size of the optimizations. To increase the visualization, only data from the tuned solver is shown. The figure depicts, that the absolute numbers of coils are deceiving in this regard and that in fact, the results from the optimization with 20 coils are in line with the observed downward trend. Now that the existence of a constant downward trend, except for the case with seed 100 mentioned above, is established, a possible explanation can be discussed. To recapitulate, a stringer has to be introduced between adjacent coils if the difference between the physical or modespecific or both characteristics of these coils is above the limit specified by the process flexibility PF. The introduction of a stringer can be avoided if the schedule introduces a third coil between these incompatible coils, but this is only possible if the third coil is compatible with the other two coils. An explanation for the high relative number of introduced stringers in optimizations with small instance sizes could therefore be the absence or an insufficient number of these third coils that can bridge the gap between two incompatible coils. By increasing the number of coils, these gaps could then be closed or narrowed by the new coils. The likelihood of this increases with the increasing number of coils, which results in fewer stringers that have to be introduced due to differences in characteristics. To give a simplified example, in which only the annealing temperature of the coils is considered as compatibility restriction: Three coils have to be scheduled in sequence, with coil one having an annealing temperature of 670 °C, coil two having an annealing temperature of 740 °C and coil three with an annealing temperature of 720 °C. The PF is 40 °C in a symmetric interval and the annealing temperature is only distributed in increments of ten °C, as explained in section 4.1. Thus, coil two and three are compatible with each other, but not with coil one. Since there is no other coil to bridge this gap, a stringer has to be added between them. Now, a fourth coil has been added to the schedule with an annealing temperature of 700 °C. It is therefore compatible with coil three, but not with coils one and two. Since no coil is compatible with coil one, a stringer still has to be added, but the relative number of stringers decreased from 33,3 % to 25 %. If another coil with an annealing temperature of 680 °C or 690 °C would be added, it would close the gap and lead to a stringer-free schedule. If it has a different annealing temperature though, it would still be compatible with one of the remaining coils and therefore, still only one stringer would H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809796 be necessary, decreasing the relative number of stringers even further. By adding more and more coils to this schedule, the probability that one of the coils eventually closes this gap increases. The reason for the increase in the number of absolute stringers from ten to 20 coils could therefore be that most of the existing gaps could not be closed by the new coils and that some coils may have characteristics which were incompatible with both ends of the gap, thus increasing the number of stringers that have to be introduced. To summarize, the probability of a single coil being compatible with the other coils in the schedule is higher for a higher number of coils. Thus, these schedules with a higher number of coils need fewer stringers, since there are fewer gaps and more possibilities to sequence these coils to avoid gaps. In the actual optimizations performed in this study, there are more compatibility restraints as well as due dates to consider, which is why in most cases there are still stringers introduced in the schedule, even with higher numbers of coils. The impact of the restriction by due dates can be observed by comparing the average of the optimizations with a high and low service level because, in fact, most optimizations do not utilize the entire service level. This is portrayed in figure 5. Since the service level is a parameter to limit the total amount of delayed coils relative to its instance size, the threshold for relative tardiness for a high service level is 20 % and so forth. As mentioned above, it can be observed that on average and with a high service level, at no instance size the full service level of 20 % is leveraged, with the highest average being 19,25 %. The data suggests, that for most cases a service level of 16 % to 17 % would suffice. This is supported by the fact, that optimizations with a service level of ten % always utilize the entire service level or almost all of it. This does not imply that if the service level would be increased to over 20 %, the solver would not leverage it. But it states, that to receive the solutions displayed in figure 3, a service level under 20 % would suffice, at least in most cases. As expected with a low service level, the solver does not allow a delay of coils. The data also suggests a slight decrease in delays in optimizations with a higher number of coils through the tuning of the solver. But, as previously described, this could be affected by the selectivity effect. When comparing the results of figure 3and figure 5, it can be observed that even though solutions found with a service level of 20 % always had a higher number of delays, they did not necessarily yield more optimal solutions in terms of stringers than optimizations with a service level of only ten %. In fact, the most significant difference in average stringers introduced between the two parameterizations was only 0,556 stringers, which occurred in the case of an instance size of 60 coils with the default solver and thus could be influenced by the selectivity effect. The question that could arise is, if not profoundly positively affecting the solutions in terms of introduced stringers, what is the benefit of using a higher service level? This question could be answered by taking a look at figure 6. The data of higher instance sizes, especially 60 and 70 coils, seems to be strongly affected by the aforementioned selectivity effect and will therefore be disregarded in the following discussion concerning the total optimization time. The top graphs portray, that a higher service level usually leads to a lower average total optimization time, which could save costs caused by the operation of the computer, but that in general, the total optimization time increases profoundly. This trend can be explained by the bottom graphs, which display the extraordinary increase in variables and constraints, and thus the size of the MILP. The stark increase in variables is mostly driven by binary variables, like yi jkmn and xikm. Since the number of variables and constraints are not influenced by the parameterization of the service level or tuning, they are only displayed by one graph each. When comparing the top with the bottom graphs, a positive correlation can clearly be observed. Therefore, the probable explanation for the increase in total optimization time is the growth of the model the solver is trying to solve. The size of the model has two different impacts on the total optimization time. The first and most obvious is, that it has to consider a growing number of constraints and variables when trying to find a feasible solution, which increases the time it takes to prove the feasibility of it. But before the model can be solved, it first has to be built, which is the second impact of the model size. The time it took the solver to build the model rose from only 4,5 seconds with ten coils to over 90 seconds with 50 coils, which does not seem to be affected by the service level or by tuning. Therefore, the model should be kept as small as possible to avoid wasting optimization time, which is especially important when certain circumstances severely limit it. What seems to be affected by the service level and by tuning is the solving time, which accounts for the majority of the total optimization time. Tuning can in fact increase the solving time through different parameterizations of PrePasses and Presolve, as an example.54 In the case of this study, the tuning thus seems to increase the success rate of optimizations, especially with greater instance sizes, at the cost of solving time. A higher service level, on the other hand, seems to decrease the solving time, which suggests that the solver can find optimal solutions faster if it is less restricted. The MIPGap also seems to benefit from an increased service level, as depicted in figure 7. A positive trend between the MIPGap and the instance size can be observed, as well as the trend that optimizations with more scheduling flexibility tend to have a higher grade of proven optimality, which can be explained by the lower solving times. The lower the service level, the more optimizations reach the solving time limit of 720 seconds and thus have to abort the solving process. Hence, the solver has no or less time to prove the optimality of the solution, resulting in a higher MIPGap. This does not necessarily result in a worse solution, as the solutions, in terms of stringers, between a service level of ten % and 20 % only differed slightly, as portrayed in figure 3. Thus, the general upwards trend of 54 Gurobi, n.d.-b H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 797 10 20 30 40 50 60 70 0 5 10 15 20 # coils Relative tardiness [%] Default solver Low Medium High 10 20 30 40 50 60 70 0 5 10 15 20 # coils Tuned solver Low Medium High Figure 5: Average tardiness relative to the different instance sizes per service level in optimizations with the default and tuned solver. 10 20 30 40 50 60 70 200 400 600 800 # coils Total time [sec] Default solver Low Medium High 10 20 30 40 50 60 70 200 400 600 800 # coils Tuned solver Low Medium High 10 20 30 40 50 60 70 1 2 3 ·107 # coils # of Variables 10 20 30 40 50 60 70 0.5 1 1.5 2 ·104 # coils # of linear constraints Figure 6: Average total optimization time (top) and number of variables (bottom-left), as well as average number of linear constraints (bottom-right) per service level and instant size in optimizations with the default and tuned solver. the MIPGap could be a consequence of the increasing solving times of each seed. Hence, it could be advantageous to choose a higher service level if the available optimization time is severely limited or if a higher proven grade of opti- H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809798 10 20 30 40 50 60 70 0 20 40 # coils MIPGap [%] Default solver Low Medium High 10 20 30 40 50 60 70 0 20 40 # coils Tuned solver Low Medium High Figure 7: Average MIPGap per instance size and service level with the default and tuned solver in %. mality of the feasible solution is preferred. The data from the tuned solver seems to be affected by the selectivity effect, but the MIPGaps of smaller instances seem to be similar to that of the default solver. The mentioned increase in solving times is displayed in figure 8with data from optimizations solved by the default solver due to its better visualization of this trend. All boxplots were created with the median and the highest and lowest value from the ten optimizations, as well as the 25 % and 75 % quartile. Figure 6already depicted the increasing average total time. But this data could suggest that this applies to every seed, when in fact it does not. The data in figure 8portrays that, starting from an instance size of 30, the differences in solving times increase drastically if they are solved with the two lower service levels. This increase can be observed from 40 coils onward, with the highest service level. The higher number of optimizations reaching the time limit combined with the higher average MIPGaps in optimization bound by the lowest service level indicates, that the solver has no time to prove the optimality of the feasible solution after finding it. With a medium service level, fewer optimizations have to be aborted due to the time limit and thus have time to prove the optimality of the feasible solution, which results in a lower average MIPGap. This is also true for the highest service level that displays the fewest number of time limit terminations. This concludes the first part of the numerical studies. Based on the presented data, an instance size of 40 coils was chosen for the following sections since it is the highest number of coils with a tolerable amount of failed optimizations. In addition, the optimizations were conducted with the tuned solver to increase the number of successful optimizations. 4.2.3. Bestand Worst-case scenario For the third part, a Bestand Worst-case scenario was derived from the different factors defined by Wegel et al. (2024), which were described in section 4.1. The results will be compared with the results achieved with the tuned solver from the previous section. In addition, the PF and HC were altered because it was suspected that these factors are the most influential among them. A more detailed analysis of the influence of each factor and level on the scheduling is provided by Wegel et al. (2024). Figure 9compares the number of stringers introduced between the Best-case scenario with a Ver y low PF and the Basecase scenario. The data from the Ver y high and Basecase PF is not portrayed, as no stringers are introduced at any point in these scenarios. As previously observed, the number of stringers decreases with higher service levels in the Best-case as well due to the higher scheduling flexibility. Compared with the results from the Basecase scenario, the Best-case yields better solutions, in terms of stringers, for every level of PF and for every service level. Even with Ver y low PF, a stringer reduction of over 50 % can be observed. Thus, a lower PF leads to an increase in stringer introduction, but in this case, this effect is probably reduced by the Ver y low HC. Since these effects could therefore counteract each other, the stringer reduction may be a result of the lower UC and SPT. If true, this could also result in fewer delayed coils, which are depicted in figure 10. By comparing the average tardiness in the Best-case scenario - Ver y low PF with the tardiness from the Basecase scenario, only a slight decrease can be observed. This difference increases with the B. PF, but does not with the highest PF. Thus, the Ver y low UC and SPT do not lead to severely less tardiness when HC and PF are parameterized as Ver y low. But the increased scheduling flexibility through more relaxed time constraints could still benefit the optimization, thus decreasing the number of introduced stringers. Figure 10 also displays the distribution solving times. The data suggests, that the level of PF, in the context of the Bestcase scenario, has an immense impact on the solving time. The optimizations with the lowest PF have the highest solving times, even higher than in the Basecase scenario. But by increasing the PF to the Basecase level, the median solving H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 799 10 20 30 40 50 60 70 200 400 600 800 Time limit # coils Solving time [sec] Low Medium High Figure 8: Distribution of solving times in seconds per instance size and service level with the tuned solver. Low Medium High 2 4 6 Service level # stringers Best-case - Vl. PF Basecase Figure 9: Average number of introduced stringers per service level in the Best-case scenario with Ver y low PF and Basecase scenario. times decrease to a fourth. Further increasing the PF to Ver y high, however, does decrease the solving times only slightly. That the solving times for the two higher investigated PF levels are remarkably lower than in the Basecase scenario can be explained by the parameterization of the HC, UC and SPT. As previously described, the Ver y low UC and SPT increase the scheduling flexibility by relaxing the time constraints. The Ver y low HC, on the other hand, increases the general compatibility between different coils, which leads to more feasible, stinger-free processing sequences. All of these factors could result in more solutions being feasible, thus decreasing the time it takes the solver to find an optimal, feasible solution. This could be applied to the lowest PF as well but in reverse. It could suggest that the Ver y low PF drastically limits the number of feasible, stringer-free processing sequences of adjacent coils, which increases the solving time since the use of stringers has to be minimized. In this regard, the other very beneficially parameterized factors cannot offset the Ver y low PF, which could indicate that the PF is the most impactful factor of the four. This statement could be confirmed by the data from the Worst-case scenario. Figure 11 portrays the impact of the HC on the Worst-case scenario in terms of introduced stringers and failed runs. As previously described, Ver y low HC is beneficial for stringerfree sequences. In this case, the HC seems to impact the average number of stringers more than the other factors, displayed by the fact that the lowest HC in combination with the Worst-case yields better solutions than the Basecase scenario. This would contradict the statement, that the PF is the H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809800 Low Medium High 0 2 4 6 Service level # delayed coils Best-case - Vl. PF Best-case - B. PF Best-case - Vh. PF Basecase Low Medium High 0 200 400 600 800 Time limit Service level Solving time [sec] Best-case Vl. PF Best-case B. PF Best-case Vh. PF Basecase Figure 10: Average number of delayed coils and distribution of solving times in seconds per service level in the Best-case instances and Basecase scenario. most influential factor. But the data from the two other HC parameterizations support this claim. While the optimizations with Worst-case and Basecase HC did still yield some but very high solutions, no optimization was able to calculate a feasible solution with the highest HC. In general, the Worst-case scenario seems to negatively impact the number of successful optimizations. As previously described, the reasons are twofold. While all runs with the lowest service level fail due to infeasibility, optimizations with a higher service level fail because of the solving time limit, with two exceptions with a medium service level and Ver y high HC. This demonstrates the great impact of the Worst-case and especially of the HC on the feasibility of the model. Before finalizing the second part of this numerical study, one last test was conducted to analyze which of the two factors, HC and PF, impact the solution the most. To accomplish this the case in table 10 was defined. Table 10: Case parameterization to identify the most impactful factor. Case HC UC PF SPT Impact Very low Basecase Very low Basecase The figure 12 displays the differences in stringers and delayed coils between the Basecase scenario and the Impact case, by subtracting the average number of stringers and delayed coils in the Impact case from the Basecase scenario. It can be concluded, that the PF has a greater impact on these aspects of the solution than the HC. This observation was confirmed by a second Impact case, which used the Basecase scenario and Ver y high PF and HC. But it has to be noted that this only applies to the definitions of the different levels of HC and PF by Wegel et al. (2024). Both factors indicate a strong influence on the characteristics of the solution and should therefore be optimized to achieve optimal solutions in real-world applications. After studying the drastic effects of the Bestand Worst-case, different parameterizations of the model will be explored in the following section. 4.2.4. Alteration of processing lines and due dates First, the number of processing lines in the model was altered. To recapitulate, in the first case, an additional line was added to the model, which can process every coil. In the second case, Removal I, the processing line that can only process wider lines was removed, while the processing line that can only process narrower coils was removed in case three. The line that can process every coil was not removed because it would lead to infeasibility since some coils could not be processed. Foremost, none of the three cases displayed in figure 13 seems to have a great impact on the number of introduced stringers. As expected, the number of stringers decreased in the addition case and increased in both removal cases, but all these effects were only minor. The reason the addition of one processing line generally reduces the number of stringers is that an incompatible coil or a sequence of coils, that is incompatible with another sequence of coils, can be processed on the additional line, thus reducing the number of stringers by one. H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 801 Low Medium High 0 5 10 15 20 25 Service level # stringers Worst-case Vl. HC Worst-case B. HC Basecase Low Medium High 0 2 4 6 8 10 Service level # failed runs Worst-case - Vl. HC Worst-case - B. HC Worst-case Vh. HC Basecase Figure 11: Average number of introduced stringers and failed runs per service level in the Worst-case instances and Basecase. Low Medium High 0 1 2 3 Service level # stringers Low Medium High 0 1 2 3 Service level # delayed coils Figure 12: Differences in stringers and delayed coils between the Impact case and Basecase per service level. To give an example, four sequences of coils have to be scheduled, with each coil in a sequence being compatible with every other coil in the same sequence. However, the sequences are incompatible with each other. If these four sequences are processed on one processing line, three stringers would have to be introduced to ensure compatibility. If a new processing line would be added, one of the sequences could be processed on the new line, reducing the number of stringers to two, but nothing more could be done to reduce the number of stringers further. Therefore, the maximum benefit of adding a processing line, in terms of stringer use, is the reduction of introduced stringers by one. This benefit could increase if the schedule introduced stringers to not violate the service level, but this was not the case for these optimizations. The reason the data does not show a reduction of the average introduced stringers by one is that two optimizations, that were unsuccessful in the Basecase scenario, were successfully solved in the addition case and have a stringer count of six, thus increasing the average number of stringers. In fact, all three cases had only one failed optimization each with a low service level. The explanation for this reduction could be the solving itself since the solver behaves differently even if only one parameter changed.55 Because the Addition case and the Removal cases have opposite effects, the addition of a processing line increases the number of feasible solutions and the removal decreases them, it is the most likely explanation. The removal of a processing line influences the number of stringers in the same way as an addition of a line does but in reverse. Thus, if no stringers are introduced to not violate the service level, the maximum increase in stringers is one. This is true for every optimization with a MIPGap of zero %. However, some optimizations with a MIPGap of zero % did not suffer any increase in stringer use, suggesting that they did not utilize the line in the schedule. But some optimizations with a MIPGap of over zero % had an increase of up to three stringers. Hence, the optimizations either had 55 Miltenberger, 2023b H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809802 Low Medium High 0 2 4 6 Service level # stringers Addition Removal I Removal II Basecase Figure 13: Average number of stringers in the three line-altering cases and Basecase scenario per service level. to introduce stringers to comply with the service level and the solver was unable to prove the optimality of the solution in time or the solver simply could not find the optimal feasible solution until it had to terminate the solving process. This termination happened more often with the addition of one line and is presented in figure 14. As portrayed, the majority of optimizations with an additional processing line and a low service level had to be terminated due to the time limit. This results in a higher MIPGap, as mentioned before. Higher solving times and MIPGaps than in the Basecase scenario are also observed with higher service levels but are not as severe with the highest one. This overall trend probably results from the increasing model complexity, which could also be observed in figure 6 from the first part of this numerical study. In this case, however, the increase results from the additional line and not from additional coils. The adverse effect can be examined in figure 15. The data indicates a decreasing solving time through complexity reduction when applying a medium or high service level, but not with a low one. A possible explanation could be, that the restrictions caused by the removal of a processing line in combination with the low service level result in many former solutions being infeasible and thus a longer search for a feasible solution. That Removal II displays lower solving times than Removal I could suggest that the line which can only process wider coils and was removed in Removal I, was utilized to a higher degree than the line that was removed in Removal II. The reason for this is the partially triangular distribution of the coil width due to the Basecase level of the HC, which results in more coils being wider than narrower. Hence, the effect of the removal of one line depends on the HC setting. The last parameter to investigate is tardiness, which indicates to not be severely affected by the addition or removal of a processing line. That the number of delays only complies with the service level and does not have to be minimized could explain why tardiness did not decrease with an additional line. Additionally, no increase in tardiness suggests that most optimizations in Removal I and Removal II had enough scheduling flexibility to find feasible solutions with similar tardiness, even with only two lines. In contrast, tardiness was the most important factor in the next part of the numerical study, in which the time horizon for the due dates was moved closer to the beginning of the schedule. These earlier due dates are expected to have a certain effect on the number of stringers and delayed coils. With earlier due dates and an imposed service level, the schedule has to implement more stringers to not exceed the allowed tardiness. This further reduces the scheduling flexibility due to the SPT and thus increases the difficulty of finishing the processing of other coils before their due date. Therefore, an increased number of failed optimizations and a higher stringer and tardiness count in successful runs were expected. The data displayed in figure 16 shows an increasing number of failed optimizations, which aligns with the expectations. In case 100 %, most optimizations are terminated and do not fail because of infeasibility, suggesting that they could be solved with a higher time limit. But this changes with case 50 %, in which no optimization with a low service level is feasible, while most unsuccessful optimizations with a higher service level are terminated. By reducing the values of aand bby 75 %, almost all optimizations with a low and medium service level are infeasible, but only one optimization with a high service level. Therefore, provided with a high service level and a higher time limit, most optimizations could still yield feasible solutions. However, the necessary solving time and the quality of the resulting solution cannot be determined without further studies. The second expectation was a rise in the number of introduced stringers. Due to the very low number of solutions, the following interpretations could be heavily influenced by the selectivity effect. Because of this, only the most important information will be presented in table 11 and briefly discussed. If a case is not included in a service level, then no optimizations for that case and service level yield a feasible solution. As indicated by the data, the number of introduced stringers increases through the shift of the due dates. In addition, almost all optimizations completely utilize the service level. Both of these observations represent an increase in comparison to the Basecase scenario and align with the expected results, thus concluding the third part of the numerical study. 4.2.5. Numerical studies on the extended models The last part of the numerical study consists of several tests conducted on the extended models described in sections 3.4.2 and 3.4.3. Both additionally incorporate the release dates of coils from section 3.4.1. The first extended model minimizes stringers and absolute tardiness. First, the average number of stringers and delayed coils in the H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809 803 Low Medium High 0 200 400 600 800 Time limit Service level Solving time [sec] Addition Basecase Low Medium High 0 5 10 15 20 Service level MIPGap [%] Addition Basecase Figure 14: Distribution of solving times in seconds and average MIPGaps in % per service level in the Addition case and Basecase scenario. Low Medium High 0 200 400 600 800 Time limit Service level Solving time [sec] Removal I Removal II Basecase Figure 15: Distribution of solving times in seconds per service level in the Removal I and II cases, as well as in the Basecase scenario. Basecase scenario with Ver y low and Ver y high UC will be examined. Figure 17 depicts an expected trend. The number of delayed coils is the highest if it has only a weight of 20 % of the objective value, which has to be minimized. Additionally, the number of stringers is the lowest at this point, since it has a weight of 80 %. Thus, the optimal solutions seem to delay some coils to minimize the number of stringers, since it has a much higher weight. This trend changes, with tardiness being reduced at the cost of stringers with higher weights of tardiness. As portrayed with Ver y low UC, the number of delayed coils is, even at its highest point, very low. Thus, the trade-off between stringers and tardiness cannot be obLow Medium High 0 2 4 6 8 10 Service level # failed runs 100 % 75 % 50 % 25 % Figure 16: Number of failed runs per service level in the four different due date altering cases. served as well as with the Ver y high UC on the right side of figure 17. In this scenario, an almost one-to-one trade-off between the number of delayed coils and stringers can be observed with the increasing weight of the former. The mechanism behind the trade-off is similar to the one discussed in the third part of the study, regarding the addition and removal of processing lines. If the tardiness has almost negligible weight, the coils are sequenced in a way that the number of stringers is minimized with no regard to the number of delayed coils. But if the tardiness weight increases, the objective value may benefit from introducing stringers to finish the processing of H. A. Hönerloh /Junior Management Science 10(3) (2025) 781-809804 80:20 65:35 50:50 35:65 20:80 0 1 2 3 4 5 6 Weight ratio Stringer:Tardiness # stringers Extended model I - Vl. UC 80:20 65:35 50:50 35:65 20:80 0 0.1 0.2 0.3 0.4 0.5 0.6 # delayed coils # delayed coils # stringers 80:20 65:35 50:50 35:65 20:80 0 2 4 6 8 Weight ratio Stringer:Tardiness # stringers Extended model I - Vh. UC 80:20 65:35 50:50 35:65 20:80 0 2 4 6 8 # delayed coils # delayed coils # stringers Figure 17: Average numbers of delayed coils and stringers with different UCs and weights in the first extended model. Table 11: Stringer use and tardiness in the cases 100 %, 75 % and 50 %. Service level Case # stringers # delayed coils Low 100 % 6 0 Medium 100 % 6.667 4 75 % 7 4 50 % 7 4 High 100 % 5 7.83 75 % 5.75 8 50 % 7 8 coils before their due date, thus decreasing the number of delayed coils at the cost of stringers. By further increasing the tardiness weight, more stringers are introduced to comply with due dates and therefore to minimize the objective value. With earlier due dates, this trend is better to observe since it is harder to comply with them, which increases the number of stringers that have to be introduced to decrease the tardiness by one unit and vice versa. Due to the different optimization objectives, the comparability between the extended model and the base model is limited. But the results of the extended model with a tardiness weight of 20 % are similar to the results yielded with a medium service level in the base model, presented in figure 3. The difference, however, is the UC, which is higher in the extended model and thus highlights its performance to minimize stringer use and tardiness at the same time. But this high performance comes with the cost of increased solving time, portrayed in figure 18. Generally, there is a wide disparity in solving time between different optimizations in the extended model, regardless of the weight ratio, with most median solving times being higher than that of the Basecase scenario with a low service level. The highest median solving times and terminations in the extended model can be observed with the two higher stringer weights of 80 % and 65 %. The median solving times of the other instances are significantly lower, which could suggest that the minimization of stringers is more complex than that of delayed coils. But this cannot explain, why the median of the solving time increased by increasing the tardiness weight further to 65 %. This effect mostly stems from one optimization, which suffered an increase of 510 seconds in solving time from 210 seconds to 720 seconds. Without this run, the median would be 283 seconds, which is similar to the instances with 50 % and 80 % tardiness weight and could indicate that this optimization is an outlier. The solving times of the extended model with a Ver y high UC are not depicted, since all of them had to be terminated due to the time limit. Nonetheless, every optimization yielded a feasible solution, which is a drastic increase over the number of failed runs in case 100 % of the third part of the numerical study, displayed in figure 16. Since every result with the extended model and a Ver y high UC also had a low relative tardiness and could therefore comply with the medium and high service levels implemented in the base model at almost every weight ratio, it suggests that it could be more performant to minimize both stringers and tardiness than to implement a service limit and only minimize