As a key and bottle neck process in steel manufacturing, steelmaking and continuous casting (SCC) production play an important role in maintaining production continuity and product quality. SCC production includes three stages: steelmaking, refining, and continuous casting. In each stage, there are several parallel machines, respectively named converters, finery furnaces, and continuous casters. So far, there have been many studies on SCC static scheduling [1-3]. Nevertheless, during production, there exist many random and uncertain factors to affect the actual execution of the pre-decided schedule. Once the monitoring data indicates that the pre-decided schedule will be infeasible in practical production, rescheduling should be carried out to keep the stability and continuity of production. Therefore, the study on SCC monitoring and rescheduling is significant and has gradually attracted the attention of scholars in recent years.

Existing studies on SCC rescheduling can be mainly divided into two classes. One class explores a targeted rescheduling method for a specific disturbance, and the other class finds a general or integrated method for most disturbances.

In terms of rescheduling for one specific disturbance, Zhang et al. [4] mapped SCC rescheduling for quality disturbance to a dynamic constraint satisfaction problem (DCSP) and built a DCSP model to minimize the differences in machine assignment, starting times, and processing times between the new and original schedules. Yu and Pan [5] considered an operation time delay disturbance and presented a three-stage rescheduling method including batches splitting, forward scheduling, and backward scheduling. Mao et al. [6] studied a SCC rescheduling problem with machine breakdown and processing time variations with two objectives in terms of efficiency and stability and developed a time-index formulation and an effective Lagrangian relaxation approach. Some scholars have focused on rescheduling strategies for a variety of disturbances, trying to find some similarities among disturbance influences and rescheduling strategies. Ouelhadj et al. [7] described a new model for robust predictive/reactive SCC scheduling based on multi-agents, tabu search, and heuristics, and they presented three measures including robustness, utility, and stability. Hou and Li [8] analyzed typical disturbances and their effects on shop floor, put forward repair procedures for these disturbances, and then proposed a repair strategy for machine failure. These studies have laid the foundation for the study of general/integrated rescheduling methods.

A general/integrated rescheduling method takes one algorithm or integrates a few algorithms against a vast majority of disturbances. Chen et al. [9] presented a directed network model to describe the SCC process, in which rescheduling is driven by disturbances mechanism and recalculated by a backward and hybrid intelligent algorithm; furthermore, a real-time scheduling system was established. Tang et al. [10] proposed an improved differential evolution algorithm with a real-coded matrix representation for each individual of the population, presented a two-step method to generate an initial population, and put forward a new mutation strategy. Hao et al. [11] classified all unexpected events into two categories, critical events and noncritical events, and developed a soft-decision-based two-layered approach with a particle swarm optimization for offline layer and a heuristic for online layer. Zheng et al. [12] indicated that the key points of SCC dynamic scheduling are the connection problems between two adjacent schedules, the matching problem between the schedule and the production material, and the deviation elimination problem between the executing schedule and the practical production data.

The theoretical research on SCC rescheduling is in its initial exploration stage, especially in terms of general/integrated rescheduling methods. To explore the integrated rescheduling models and algorithms, this paper present a system model and rescheduling algorithms to monitor and reschedule SCC production. They can infer production situation in real time, detect disturbances and give alarms in time, and apply an appropriate rescheduling algorithm to eliminate disturbance effects, thereby maintaining the stability and continuity of SCC production. The rest of this paper is organized as follows. Section 2 presents a system framework. Section 3 details the modularity knowledge management approach in this system. In Section 4, the reasoning process of this monitoring and rescheduling system is proposed. Section 5 designs two algorithms according to the disturbance type and influencing degree. In Section 6, simulation experiments are carried out. Section 7 concludes the paper with a short summary.

SCC is a semi-continuous manufacturing, and a heat in steelmaking corresponds to a job in the scheduling problem. Heats (Jobs) are discretely produced in the steelmaking and refining stages and continuously produced in the casting stage by composing several batches called casts in steel production (Figure 1). Generally, containment relationships between casts and heats are already determined in a casting plan, and SCC scheduling arranges machines and time variables for each heat in accordance with the casting plan. SCC production is under the environment of high temperature with complex physical and chemical changes of steel, and it has no buffer between any two consecutive stages. There are many unexpected and uncertainty factors in the production, such as time variations, machine fault, and quality fluctuations. Therefore, monitoring and rescheduling (MRS) for SCC is essential during the production period. In MRS for SCC, once a disturbance happens, a quick response to repair the original schedule must be offered to ensure the stability and continuity of production.

Based on the above analysis, the monitoring and rescheduling system for SCC (MRS-SCC) in this paper contains three main functions:

$\cdot$ Simulation function: dynamically simulate production situations in workshops;

$\cdot$ Monitoring function: monitor the real-time status of jobs and machines and give an alarm when a disturbance occurs;

$\cdot$ Rescheduling function: exploit rescheduling algorithms for disturbances and select an appropriate algorithm while an emergency happens.

Three typical kinds of disturbances in SCC production are considered in this system: (a) time variation, including the earliness or tardiness of starting times and the shortening or extension of processing times; (b) machine fault, such as machine breakdowns and temporary maintenance; (c) quality fluctuations, such as abnormal temperatures or components of molten steel.

The system MRS-SCC is designed as an expert system and developed on a G2 real-time intelligent expert system platform. In the G2 platform, an object-oriented modeling mechanism and rule-based reasoning technique are applied to build a wide range of knowledge models that can be dealt with automatically [13]. At present, this system platform has been widely used in control and monitoring, fault analysis, optimization simulation, and so on [14].

To classify the knowledge in MRS-SCC and enhance knowledge reusability, modularity management is introduced to this system to organize monitoring and rescheduling knowledge into a two-level tree construction, as shown in Figure 2.

The knowledge in MRS-SCC is divided into three types: (a) workshop knowledge focusing on processing characteristics of SCC; (b) monitor knowledge for uninterruptedly monitoring production situation; (c) rescheduling knowledge with rescheduling strategies and algorithms. These types of knowledge represent three abstract conceptions of the problem, and they are independent of each other to form three knowledge modules in the bottom level of the system. The module called MRS Module in the top level integrates the three types of knowledge to completely describe and simulate SCC dynamic production environment.

To represent the knowledge effectively, the object-oriented and production-rule-based representations are combined to build a knowledge network for monitoring and rescheduling.

The Workshop Module is composed of the knowledge about production process features in the SCC workshop. Object-oriented representation is applied in this module to describe production entities and their production technology knowledge.

There are mainly two kinds of entities as machines and heats. Machines can be further divided into converters, finery furnaces, and casters. Therefore, four classes are defined in this module, respectively named CONVERTER, FINERY, CASTER, and HEAT. Attributes of each class representing technical parameters of the entity are defined based on disturbance types and flexible processes in SCC production.

Considering the three disturbances as time variation, machine fault, and quality fluctuation, HEAT concerns parameters of temperature, component, weight, and steel grade, and the focus of three machine classes is their processing statuses with the three values of busy, free, and breakdown. During SCC production, processing times of a heat in refining and casting stages are flexible, and they canbe exploited as flexible factors on rescheduling. The maximum and minimum processing times in the refining stage are related to the processed heat, and those in the casting stage are determined by the casting speed of the caster. Thus, the upper bound and lower bound of the refining time are set as attributes of HEAT, and the casting speed is set as an attribute of CASTER.

Definitions of classes and their attributes are listed in Table 1. In Table 1, the icon of each entity is designed for visual simulation.

To enhance independence and reusability, a monitoring function is separately abstracted to form a monitor module. In this module, a conceptual class called MONITOR is defined to represent general characteristics and actions of monitored entity. There are six attributes of MONITOR: Description, Mon-States (status of the monitored entity), Expect (expect value), Actual (actual value), Variation (difference between the actual value and expect value), and Threshold. In the above attributes, values of Describe, Expect, and Threshold are known in advance; Actual values are timely updated by production data from PCS; and Variation and Mon-States values are inferred from Actual, Expect, and Threshold values. Consequently, there are some procedures for MONITOR, and the main ones are shown below.

$\cdot$ Mon-Calculate-Variation(class MONITOR): Calculate the Variation value as the difference between the Actual and Expect values.

$\cdot$ Mon-Calculate-Status(class MONITOR): Assign a value belonging to {normal, high, low} to Mon-States according to the Variation and Threshold values.

$\cdot$ Monitor-Events(class MONITOR): Deal with the disturbance according to the Mon-State value. This procedure is empty in MONITOR and would be overridden in its child classes to meet different requirements of the corresponding monitored entity.

Executions of the above procedures are triggered by production rules, which are based on an event-driven based reasoning mechanism. For example, the update of the Mon-Status value of any MONITORinstance by Rule1 will trigger Rule2.

$\cdot$ Rule1: If the Actual of any MONITOR M receives a value

Then in order start Mon-Calculate-Variation(M) and start Mon-Calculate- Status(M)

$\cdot$ Rule2: If the Mon-Status of any MONITOR M is not “normal”

Then start Monitor-Events(M)

The most important knowledge in this system is the rescheduling knowledge, including two aspects: scheduling problem description and rescheduling models. The SCC scheduling problem can be abstracted as a special hybrid flowshop scheduling (HFS); hence, the Rescheduling Module is developed for a general HFS for reusability.

In the Rescheduling Module, the following classes are defined: SCHEDULE, JOB, OPERATION, MACHINE, and DISTURBANCE. Considering the three main types of disturbances, three subclasses of DISTURBANCE are further defined. The class diagram is shown in Figure 3.

In this system, independent pieces of knowledge in the three bottom modules are integrated in the top module, the MRS Module, by multiple inheritances of classes in bottom modules, as Figure 4 shows.

In the MRS Module, according to disturbance characteristics in SCC production, monitoring approaches are defined by the inheritance of the class Monitor in the Monitor Module.

$\cdot$ To monitor steel temperatures, in the class SCH-HEAT inherited from MONITOR, the attribute Description is set as “steel temperature”; Expect values are updated automatically by values of Steel-Grade and Job-Status; Actual values are the real-time data collected from PCS.

$\cdot$ Starting times of heats are monitored by the time at which Job-Num value of SCH-MACHINE is changed.

$\cdot$ To monitor processing times, in the class SCH-MACHINE inherited from MONITOR, Description is set as the “processing time for the current job”; Expect and Actual values are respectively the anticipated and actual completion times of operations on this machine.

$\cdot$ To monitoring machine status, whether the machine is busy or available is inferred by the tracking information of ladles from PCS; the fault status is acquired by defining a variable Breakdown-Var to receive machine fault conditions from PCS.

Furthermore, a dynamic simulation interface of the SCC production is designed in the MRS Module (see Section 6). Rules for simulation are defined to infer logistics only by one type of external data, ladle position, thereby decreasing data transmission with external systems.

Consequently, the MRS Module integrates function knowledge in bottom modules by defining and inheriting classes and production rules, so that the functions of simulation, monitoring, and rescheduling are realized in this system.

Rule-based reasoning (RBR) is used in this system with a G2 inference engine. Two reasoning strategies are applied as forward reasoning and backward reasoning. The forward reasoning responds to updated data and real-time events, and the backward reasoning calls other rules or procedures. The reasoning process of this system is shown in Figure 5.

The SCC scheduling problem is a special hybrid flowshop problem. Denoting the operation of Heat i in Stage j by O_ij, the processing time and starting time of O_ij by pt_ij and st_ij respectively, the ordered sequence of operations on Machine k in Stage j by S_jk, and the i^th operation in S_jk by S_jk(i), the following constraints should be satisfied for SCC scheduling, where the decision variables are st_ij and S_jk, and for rescheduling, pt_ij is also a variable.

Temporal constraints. A heat cannot be processed in a stage until the processing of this heat in the precedent stage is finished, as shown in Equation (1).

Machine resource constraints. One operation must be assigned and only assigned to one machine in every stage. On each machine, only if one heat is completed can the next operation be started, as Equations (2)-(4) show.

Continuous casting constraints. Heats in the same cast must be continuous processing in the casting stage. Defining s as the last stage, i.e. casting stage, CST as the cast set, and CST_l as the operation set in Cast l, where $l\in CST$, Equation (5) can be obtained.

There are two kinds of decision variables: time variables and machine assignment variables. Accordingly, SCC rescheduling mainly repairs time and machines for heats, and for different types of disturbances, repairing approaches should be different.

In this system, while the effect of a disturbance is weak, such as the delay of processing times and a little fault time of a machine, only starting times and processing times need to be modified; conversely, for the large effect of a disturbance such as a serious machine failure, machine reassignment will be carried out to ensure continuous processing. In addition, disturbances of quality fluctuations including temperature and steel grade can be eliminated by adjusting the refining time, so that those disturbances can be transformed into processing time variations in the refining stage.

Therefore, we present two rescheduling algorithms: Time Adjustment (TA) and Machine Reassignment (MR). Table 2 shows the correspondence between disturbances and rescheduling algorithms in MRS-SCC.

The algorithm Time Adjustment (TA) takes full account of reserved time of operations to modify starting times and processing times of directly or indirectly affected heats and takes advantage of flexible processes in SCC production to ensure continuous casting. The details are shown below.

(1) Modifying starting times

Modification of starting times begins with the directly affected operation, and then recursively delay starting times of succeed operations in temporal constraints and machine resource constraints by the utility of reserved time.

Step 1 Set the directly affected operation as the conflict operation ${{O}_{ij}}$, and repair its starting time or processing time according to the disturbance.

Step 2 Calculate the latest starting times of the conflict operation ${{O}_{ij}}$ as $ls{{t}_{i}}=s{{t}_{i,j+1}}-{{p}_{ij}}$, which is relative to ${{O}_{i,j+1}}$, and $ls{{t}_{j}}=s{{t}_{kj}}-{{p}_{ij}}$ to ${{O}_{kj}}$, where ${{O}_{kj}}$ is the operation processed after ${{O}_{ij}}$ in the same machine.

Step 3 If $s{{{t}'}_{ij}}\le ls{{t}_{i}}$, the reversed time between ${{O}_{ij}}$ and ${{O}_{i,j+1}}$ is large enough to eliminate the influence of time delay, and thus no modification is required for the starting time of ${{O}_{i,j+1}}$; otherwise, mark ${{O}_{i,j+1}}$ as the conflict operation and set the delay of starting time as $\Delta t=s{{{t}'}_{ij}}-ls{{t}_{i}}$ and $s{{{t}'}_{i\text{,}j+1}}=s{{t}_{i\text{,}j+1}}+\Delta t$. In the same way, determine the relationship between $s{{{t}'}_{ij}}$ and $ls{{t}_{j}}$, and then calculate the starting time of ${{O}_{kj}}$.

Step 4 Repeat Step 2 and Step 3 until there is no conflict operation.

(2) Adjusting process parameters

SCC production has a special requirement of group processing in the casting stage. The above procedure to modify starting times may go against continuous casting constraints; hence, flexible factors in refining and casting stages can be exploited to maintain the continuous processing in the casting stage: in the refining stage, the processing times is flexible with restrictions of maximum and minimum values; in the casting stages, the casting speed of casters can be adjusted to change processing times. Therefore, the procedure of adjusting process parameters is designed to satisfy continuous casting constraints, shown as follows.

Step 1 Detect whether a cast in the new schedule satisfies continuous casting constraints. If it does not, then go to Step 2;otherwise, go to Step 4.

Step 2 If the starting time of the cast is later than the disturbance time, then delay its starting time; otherwise, shorten the refining time of the interrupted heat and go to Step 3.

Step 3 If the continuous casting constraints of this cast are still violated, then reduce the casting speed to extend casting times of precede heats in the same cast. Denote the total processing time of this cast by PT, the duration of interruption time by $\Delta t$, and the current casting speed by v. The speed of the casting machine should be reduced to ${v}'=\left( PT/PT+\Delta t \right)\times v$.

Step 4 Repeat Steps 1-3 until all the casts satisfy the continuous casting constraints.

For serious disturbances, Machine Reassignment (MR) for affected operations is needed. In this system, the rescheduling strategy of local repair is applied to reassign machines for operations in the disturbance stage.

Step 1 Generate rescheduling operation set A.

Define ${{t}_{b}}$ and ${{t}_{r}}$ as the starting time and finishing time of the disturbance. The rescheduling operation set A includes all the operations that are planned to be processed in the disturbed stage ${{j}_{b}}$ during $\left[ {{t}_{b}},{{t}_{r}} \right]$, shown in Equation(6).

Step 2 Reassign machines in the stage ${{j}_{b}}$ to operations in the set A.

(1) Select the operation ${{O}_{ij}}$ that has the smallest starting time in the set A.

(2) ${{t}_{1}}=\max \left\{ s{{t}_{ij}},{{t}_{b}} \right\}$; ${{t}_{2}}=\min \left\{ s{{t}_{ij}}+p{{t}_{ij}},{{t}_{r}} \right\}$.

(3) Calculate unused capacities of all the running machines in the stage ${{j}_{b}}$ during $\left[ {{t}_{1}},{{t}_{2}} \right]$ by Equation(7), where $ca{{p}_{k}}$ is the unused capacity of the machine k.

(4) Assign the machine with the maximum unused capacity to ${{O}_{ij}}$.

(5) Remove ${{O}_{ij}}$ from the set A. If $A\ne \varnothing $, then go to Step 2(1); otherwise, go to Step 3.

Step 3 Detecting and resolving conflicts.

(1) Detect whether starting times of operations on a running machine in the stage ${{j}_{b}}$ satisfy machine resource constraints.

(2) If they do not, find the conflict operation that has the earliest starting time and delay its starting time, and then call the algorithm TA to repair succeed operations.

(3) Repeat this step until all the machines in the stage ${{j}_{b}}$ have been detected.

This system is developed on the G2 platform, and experiments are carried out on the data in Guo and Li [15]. They considered a steel mill with three converters, three finery furnaces, and three casters, and they gave a schedule for 18 heats and 3 casts (Table 3).

In the experiments, processing time, machine status, and steel temperature are monitored. As Figure 6 shows, in this monitoring and rescheduling system, the simulation interface SCHEDULING-GUI dynamically simulates production situation, and values of important attributes are shown in real time. While a disturbance occurs, an alarm is given in the message board MESSAGE-BOARD. Moreover, dynamical curve graphs of machine statuses are provided to indicate the utilization conditions of machines. These interfaces can assist schedulers in mastering the production situation timely.

Consider the disturbance in which pt₇₂ increases by 15min. An instance of the class TIME-FLCT is generated, and then the algorithm TA is carried out. Results are shown in Table 4, and parts of attribute values are shown in Figure 7.

It can be seen from Table 4 that when this disturbance occurs, there is a total of 33 uncompleted operations, and 19 out of those 33 operations have directly or indirectly constraint relationships with O₇₂. In TA, only one processing time and six starting times are repaired, which is much less than the number of succeed operations. This indicates that TA can effectively decrease the number of rescheduled operations and reduce unnecessary delays of starting times.

Now, consider the breakdown of the machine Converter-2 at 125min. The estimated maintenance time is 35min. It is clear that the production of Heat 13 on that fault machine is interrupted, so that the algorithm MR is selected for rescheduling. Results are shown in Table 5, and parts of attribute values of the disturbance instance are shown in Figure 8.

In MR, the machine assignment approach on unused machine capacities is applied to changing the converter of Heat 13 and delaying the starting times of operations that have constraint relationships with this heat. After that, there is a 30min time lag between O₅₃ and O₆₃ in Cast 1 that started its casting at the disturbance time; thus, the refining time of Heat 6 is reduced to the minimum value of 20min, and the starting time of the casting is setas$s{{t}_{63}}=275$. However, a 5min time lag still exists, hence the casting speed of Caster-1 needs to be adjusted. The original and required total processing times of Cast-1 are 145min and 150min respectively; hence, the casting speed of Caster-1 should be reduced to 0.97 times of the original speed. From the new schedule, it can be seen that only one operation changes its converter, the starting times of 19 operations are adjusted, and the processing times of six operations are repaired. The above results indicate that MR can reasonably reassign machines and take full advantage of reserved times in the original schedule and flexible factors in the production process.

Consequently, the monitoring and rescheduling system for SCC production in this paper can perform optimally in simulation, monitoring, and rescheduling, and the proposed rescheduling algorithms are feasible and effective for multiple types of disturbances.

SCC is a high continuity process with complex chemical reactions and physical changes. Therefore, the quick response capability of its monitoring and rescheduling (MRS) system is significant. In this paper, a monitoring and rescheduling system for SCC production is exploited on a G2 expert system platform to implement three functions as simulation, monitoring, and rescheduling. This system is comprised of knowledge base, inference engine, and interface. In the knowledge base, to classify production knowledge and enhance knowledge reusability, modularity management is taken to organize MRS knowledge of SCC into a two-level tree construction, where the three modules workshop, monitor, and rescheduling are built in the bottom level, and integrated knowledge forms the MRS Module in the top level. To express the knowledge flexibly, a combination of object-oriented and production-rule-based representations is applied. In the inference engine, rule-based reasoning is used to respond to updated data and real-time events and trigger other rules or procedures. Moreover, three typical kinds of disturbances, which include time variations, machine fault, and quality fluctuations, are considered and classified into weak disturbances and serious disturbances, and two rescheduling algorithms are presented. One algorithm for weak disturbances only repairs time variables by utilizing reserved times and flexible factors in SCC production. The other algorithm is designed for serious disturbances to reassign machines based on unused machine capacities. Simulation experiments illustrated that this system can dynamically monitor and simulate production situation and reschedule in time while a disturbance occurs, thereby ensuring the stability and continuity of SCC production.

Due to the modularity management of knowledge, this system has good flexibility and scalability. In the practical application, new types of disturbances and rescheduling algorithms can be constantly expanded to the knowledge base of this system, thereby enhancing quick response capability to different types of emergency events in the production process. Thiswill be further explored in future works.

This research was supported by the National Natural Science Foundation of China (No. 71701016, 71471015), the Beijing Natural Science Foundation (No. 9174038), the Humanity and Social Science Youth Foundation of Ministry of Education of China (No. 17YJC630143), and the Fundamental Research Funds for Central Universities (No. FRF-BD-17-009A).