A Prediction Method for Abnormal Condition of Scheduling Plan with Operation Time Delay in Steelmaking and Continuous Casting Production Process

Abstract

In the steelmaking and continuous casting (SMCC) production process, the operation time delay often occurs which may lead to planned casting break or processing conflict so that the initial scheduling plan becomes unrealizable. Existing rescheduling methods with disturbances firstly classify the disturbances according to the disturbance type and disturbance quantity only by artificial experience or rules, and then directly adjust initial scheduling plan with corresponding rescheduling method. Those methods don’t analyze the influence degree of disturbances to the initial scheduling plan in detail, so the adjustment degree of initial scheduling plan is always too greater, which leads to the poor continuity and stability of initial scheduling plan. In this paper, the relation among operation time delay, planned casting break and processing conflict is deeply analyzed. Then a novel prediction method for abnormal condition of scheduling plan with operation time delay disturbance in SMCC production process is proposed including disturbance identification of operating time delay based on event-driven mechanism, analysis on charges based on reachability matrix, analysis on influence degree of disturbance and abnormal condition decision of initial scheduling plan. As a result, the real-time application shows that the proposed prediction method can timely and accurately predict the abnormal condition of the scheduling plan with operation time delay disturbance in SMCC production process, which can only adjust the affected charges that must to be rescheduled in the initial scheduling plan and reduce the frequency of complete rescheduling. The initial scheduling plan can also maintain the good continuity and stability.

1. Introduction

The SMCC production process is the core working procedure in modern large steel plant. Due to the process complexity, a large number of machines and frequent changes of the production environment, operation time delay often occurs which may lead to planned casting break or processing conflict so that the initial scheduling plan becomes unrealizable. It is of great significance to quickly and effectively adjust scheduling plan to ensure steel quality and production stability.

Nowadays the studies of rescheduling method for abnormal condition of scheduling plan with disturbances mainly focus on schedule repair and complete rescheduling according to the disturbance type and disturbance quantity. Right-shift rescheduling is adopted for smaller time delay disturbance, which means to reschedule by globally shifting the remaining operations schedule forwards in time. For the quality disturbance, equipment failure or excessive time delay, total rescheduling is used to adjust scheduling plan without consideration of the initial schedule.¹⁾ If disturbance event is very large impact on production, complete rescheduling is needed which includes production path planning and production time scheduling.²⁾ An algorithm for rescheduling the affected operations in a job shop is presented in order to preserve as much as possible the robustness of the initial schedule, which is different from the right-shift rescheduling and complete rescheduling.³⁾ However, the research on scheduling method for SMCC production process focuses on static scheduling problem,^4,5,6) and rescheduling methods are rarely addressed. The development of a knowledge model including task, inference and domain, which describes the reasoning process in managing schedule disturbance in steel-making, is presented.⁷⁾ A constraint-based approach for steelmaking-continuous casting rescheduling problem is presented for machine failure.^8,9,10) A real-time scheduling method is advanced with disturbances event considering four major categories, and a rescheduling algorithm of backward method and hybrid intelligent method is proposed.¹¹⁾ A general framework for using real time information to improve scheduling decisions is developed, and general measures of utility and stability are defined to evaluate strategies for deal with the deal time information.¹²⁾

Existing rescheduling methods with disturbances firstly classify the disturbances according to the disturbance type and disturbance quantity only by artificial experience or rules, and then directly adjust initial scheduling plan with corresponding rescheduling method. Those methods include schedule repair for part of charges, complete time rescheduling only adjusting operation time and complete rescheduling including production path planning and production time scheduling. Above methods don’t analyze the influence degree of disturbances to the initial scheduling plan in detail, so the adjustment degree of initial scheduling plan is always too greater, which leads to the poor continuity and stability of initial scheduling plan.

In this paper, the disturbance type, disturbance point and disturbance value are firstly determined according to the disturbance event information. Secondly, affected charges are analyzed based on reachability matrix. Then the prediction models of abnormal condition decision of initial scheduling plan without considering buffers and with considering buffers are established to analyze influence degree of operation time delay disturbance to planned casting break and processing conflict, respectively. Finally, abnormal condition of initial scheduling plan is predicted. As a result, the real-time application shows that the proposed prediction method can timely and accurately predict the abnormal condition of the scheduling plan with operation time delay disturbance in SMCC production process, which can only adjust the affected charges in the initial scheduling plan and reduce the frequency of complete rescheduling. The initial scheduling plan can also maintain the good continuity and stability.

The paper is organized as follows. In Section 2, we describe the SMCC production process and the scheduling plan of SMCC production process. The abnormal condition of planned casting break and processing conflict is defined. Section 3 presents a novel prediction strategy for abnormal condition of scheduling plan with operation time delay disturbance in SMCC production process. In Section 4, a novel prediction method for abnormal condition of scheduling plan with operation time delay disturbance in SMCC production process is proposed, which includes disturbance identification of operating time delay based on event-driven mechanism, analysis on charges based on reachability matrix, analysis on influence degree of disturbance and abnormal condition decision of initial scheduling plan. Section 5 presents industrial application in Baosteel factory of China. Finally, conclusions are outlined in Section 6.

2. Prediction Problem Description for Abnormal Condition of Scheduling Plan with Operation Time Delay Disturbance in SMCC Production Process

2.1. Scheduling Plan for SMCC Production Process

Molten steel is firstly smelt in the converter. Then one ladle carries molten steel from one converter to the refining process. Molten steel in the ladle is transported to the continuous caster after refined. One ladle carrying molten steel on the processing of all processes, and the transportation of the ladle among the processes are called a Charge. Molten steel in one charge is exactly from one converter and is exactly carried by one ladle. Each charge processed on each machine is regarded as an Operation at the processing stage. The operation type and operation number of charge must meet the process requirement. All charges which are continuously drained into the same tundish in a continuous caster are called a Cast.

We denote L_ij for the jth charge of the ith cast, and i = 1,..., N, j = 1,..., n_i, where N indicates the total cast number and n_i represents the total charge number of the ith cast. Each charge L_ij consists of θ_ij operations (o_ij1,..., o_i,j,θij). Initial SMCC scheduling plan solution S⁰ includes the initial processing machine $z_{ijk}^0$ , the initial starting time $s_{ijk}^0$ , the initial processing time $p_{ijk}^0$ , the initial completion time $e_{ijk}^0$ of operation o_ijk (k = 1,..., θ_ij). In this paper, we denote m_gb as the bth machine in the gth machine group. The h_g represents the total number of the gth machine group.

When operation time delay at time t leads to casting break or operation time conflict, the original scheduling plan S⁰ becomes unrealizable. It is important to analyze the influence degree of disturbances to the initial scheduling plan in detail in order to maintain the good continuity and stability.

2.2. Operation Time Delay

The actual operation time of charges will often deviate from the initial operation time of charges due to random factors, such as proficiency of operators, environmental parameters, etc. Such deviation is called operation time delay which includes starting time delay and completion time delay according the different time points.

2.2.1. Starting Time Delay

As one of raw materials for converters, molten irons of high temperature are transported by torpedo car from the blast furnace to the steelmaking plant and are poured into the ladles, and are finally poured into converters from ladles. During the transportation, molten irons need to be processed by the former slag, desulfurization, posterior slag. In those processes, the actual starting time of charges will be often later than the initial starting time because of proficiency of operators, environmental parameters, etc.

The production scheduling in SMCC production process mainly considers main equipment (converters, refining furnaces, and continuous casting machines). But in the actual production process, incoordination between the scheduling of auxiliary equipment and the scheduling of main equipment will result that crane and trolley can not be in place in time, which leads to starting time delay at refining stage.

We denote the $s_{ijk}^{*}$ for the actual starting time of operation $o_{ijk}$ . If $s_{ijk}^{*}$ is later than the initial starting time $s_{ijk}^0$ ,   

$$s_{ijk}^{*}>s_{ijk}^0$$(1)

then starting time delay disturbance occurs. The Δt indicates the time delay, and $\mathrm{\Delta }t=s_{ijk}^{*}-s_{ijk}^0$ .

2.2.2. Completion Time Delay

Due to a variety of random factors, such as the impact of machine failure, the proficiency of the operation of workers, environmental parameters, etc, it is hard to get accurate processing time and can only get an approximate data or data range for processing time of the molten steel in the equipment. In the actual production process, the processing time of charge is closely related to the molten steel temperature and components. With the changes of these factors, the actual processing time is also changing and is often different from initial processing time.

We denote the $e_{ijk}^{*}$ for the actual completion time of the operation o_ijk. If $e_{ijk}^{*}$ is later than the initial completion time $e_{ijk}^0$ ,   

$$e_{ijk}^{*}>e_{ijk}^0$$(2)

then completion time delay disturbance occurs. The Δt indicates the time delay, and $\mathrm{\Delta }t=e_{ijk}^{*}-e_{ijk}^0$ .

2.3. Classification in Abnormal Conditions of Scheduling Plan in SMCC Production Process

Abnormal conditions of scheduling plan in SMCC production process mainly include planned casting break and processing conflict.

(1) Planned casting break

Casting break means that continuous caster can not pull out the slab through the mold because of a variety of reasons which results that continuous casting process is interrupted. In this paper, we divide casting break into “unplanned casting break” and “planned casting break”. Unplanned casting break is that molten steel can not enter the mold for condensation because of nozzle clogging for low temperature of molten steel, nozzle clogging of ladle, nozzle clogging of tundish, etc. Unplanned casting break actually occurs in the production process. The following preventive measures can be adopted to reduce the number of occurrences of unplanned casting break: improving qualified rate of molten steel temperature in tundish, improving the cleanliness of molten steel, calcium treatment in molten steel, and etc. Planned casting break is that other factors affects the initial scheduling plan which results that some molten steel can not reach steel ladle turret in time according to initial scheduling plan and continuous casting is interrupted in the future because of molten steel supply interruption. Planned casting break is not what actually happened in the production process. It means that some factors, such as operation time delay, affect the initial scheduling plan, and if production is still going on according to the initial scheduling plan, then casting break will happen in the future.

If planned casting break occurs between charge L_i,j–1 and L_ij, then abnormal conditions of scheduling plan can be divided into the following four conditions:

■ One-level planned casting break. It can resolve the planned casting break between charge L_i,j–1 and L_ij only by adjusting the processing time of L_i,j–1 at continuous casting stage, or only by adjusting the processing time of L_ij at each stage.

■ Two-level planned casting break. It can not resolve the planned casting break between charge L_i,j–1 and L_ij only by adjusting the processing time of L_i,j–1 at continuous casting stage, or only by adjusting the processing time of L_ij at each stage. But the planned casting break between charge L_i,j–1 and L_ij can be resolved synchronously by adjusting the processing time of L_i,j–1 at continuous casting stage and the processing time of L_ij at each stage.

■ Three-level planned casting break. It can not resolve the planned casting break between charge L_i,j–1 and L_ij synchronously by adjusting the processing time of L_i,j–1 at continuous casting stage and the processing time of L_ij at each stage. But the planned casting break between charge L_i,j–1 and L_ij can be resolved synchronously by adjusting the processing time of L_ij1 (j₁ ≤ j) at each stage.

■ Four-level planned casting break. It can not resolve the planned casting break between charge L_i,j–1 and L_ij synchronously by adjusting the processing time of L_ij1 (j₁ ≤ j) at each stage. But the planned casting break between charge L_i,j–1 and L_ij can be resolved synchronously by adjusting the processing machines of L_ij1 (j₁ ≤ j) at steelmaking stage and at refining stage, and by adjusting the processing time of L_ij1 (j₁ ≤ j) at each stage.

(2) Processing conflict

SMCC process is a non-preemptive processing, which means that when one operation is being processed on the machine, other operations are prohibited to preempt the same machine. Once one operation begins to be processed, it is not allowed to interrupt until the processing finishes. If processing conflict occurs between charge L_i1j1 and L_i2j2 on machine m_gb, then abnormal conditions of processing conflict can be divided into the following five conditions:

■ One-level processing conflict. It can resolve the processing conflict between charge L_i1j1 and L_i2j2 only by adjusting the processing time of L_i1j1 on machine m_gb, or only by adjusting the processing time of L_i2j2 on machine m_gb.

■ Two-level processing conflict. It can not resolve the processing conflict between charge L_i1j1 and L_i2j2 only by adjusting the processing time of L_i1j1 on machine m_gb, or only by adjusting the processing time of L_i2j2 on machine m_gb. But the processing conflict between charge L_i1j1 and L_i2j2 can be resolved synchronously by adjusting the processing time of L_i1j1 and L_i2j2.

■ Three-level processing conflict. It can not resolve the processing conflict between charge L_i1j1 and L_i2j2 synchronously by adjusting the processing time of L_i1j1 and L_i2j2 on machine m_gb. But the processing conflict between charge L_i1j1 and L_i2j2 can be resolved synchronously by adjusting the processing time of L_i1j1 and L_i2j2 on each stage.

■ Four-level processing conflict. It can not resolve the processing conflict between charge L_i1j1 and L_i2j2 synchronously by adjusting the processing time of L_i1j1 and L_i2j2 on each stage. But the processing conflict between charge L_i1j1 and L_i2j2 can be resolved synchronously by adjusting the processing time of L_i1j1, L_i2j2 and other charges on each stage.

■ Five-level processing conflict. It can not resolve the processing conflict between charge L_i1j1 and L_i2j2 synchronously by adjusting the processing time of L_i1j1, L_i2j2 and other charges on each stage. But the processing conflict between charge L_i1j1 and L_i2j2 may be resolved synchronously by adjusting the processing machines and processing time of L_i1j1, L_i2j2 and other charges on each stage.

3. Prediction Strategy for Abnormal Condition of Scheduling Plan with Operation Time Delay Disturbance in SMCC Production Process

A four-stage prediction strategy for abnormal condition of scheduling plan with operation time delay disturbance in SMCC production process is proposed as shown in Fig. 1, which includes: disturbance identification of operating time delay based on event-driven mechanism, analysis on charges based on reachability matrix, analysis on influence degree of disturbance and abnormal condition decision of initial scheduling plan.

Fig. 1.

Prediction strategy for abnormal condition of scheduling plan with operation time delay disturbance in SMCC production process.

(1) Disturbance identification of operating time delay based on event-driven mechanism

Disturbance identification of operating time delay based on event-driven mechanism compares the actual starting time $s_{ijk}^{*}$ and the initial starting time $s_{ijk}^0$ , or compares the actual completion time $e_{ijk}^{*}$ and the initial completion time $e_{ijk}^0$ according to the actual starting time $s_{ijk}^{*}$ or the actual completion time $e_{ijk}^{*}$ . It can obtain the disturbance identification result according to the comparison result, which is denoted as Y, and Y = {o,m,y,τ}. The o is defined as the charge with operating time disturbance. The m denotes the machine on which disturbance occurs. The τ indicates disturbance value. We define the y as the disturbance type, y ∈ {y₀,y₁,y₂,y₃,y₄}. The y₀ represents that there is no disturbance occurs. The y₁ represents that starting time delay occurs on the converter. The y₂ represents that completion time delay occurs on the converter. The y₃ represents that starting time delay occurs on the refining furnace. The y₄ represents that completion time delay occurs on the refining furnace.

(2) Analysis on charges based on reachability matrix

Analysis on charges based on reachability matrix is to analyze the charges which are affected by the disturbance identification result Y according to the initial processing machine $z_{ijk}^0$ , the initial starting time $s_{ijk}^0$ and the initial completion time $e_{ijk}^0$ . The O denotes those charges set.

(3) Analysis on influence degree of disturbance

Analysis on influence degree of disturbance analyzes the influence degree of disturbance on charges from O according to the disturbance identification result Y, initial scheduling plan S⁰, the minimum processing time $p_{ijk}^{min}$ , the standard processing time $p_{ijk}^{nor}$ , the maximum processing time $p_{ijk}^{max}$ , the transportation time u(m_g1b1,m_g2b2) between machine m_g1b1 and machine m_g2b2. We define T as the influence degree result.

(4) Abnormal condition decision of initial scheduling plan

Abnormal condition decision of initial scheduling plan predicts the abnormal condition according to the influence degree result T. The H indicates the abnormal condition result, and H = {m,o₁,o₂,ζ}. The m denotes the machine on which disturbance occurs. The o₁ and o₂ indicate operations with disturbance, respectively. We define the ζ as the abnormal condition type, and ζ ∈ {ζ₁₁,ζ₁₂,ζ₁₃,ζ₁₄,ζ₂₁,ζ₂₂,ζ₂₃,ζ₂₄,ζ₂₅}. The ζ₁₁, ζ₁₂, ζ₁₃ and ζ₁₄ denote one-level planned casting break, two-level planned casting break, three-level planned casting break and four-level planned casting break, respectively. The ζ₂₁, ζ₂₂, ζ₂₃, ζ₂₄ and ζ₂₅ represent one-level processing conflict, two-level processing conflict, three-level processing conflict, four-level processing conflict and five-level processing conflict, respectively.

4. A Novel Prediction Method for Abnormal Condition of Scheduling Plan with Operation Time Delay Disturbance in SMCC Production Process

4.1. Disturbance Identification of Operating Time Delay Based on Event-driven Mechanism

In the SMCC production process, once the operation is being processed or finished, dynamic event information collected from actual production includes the machine code m, operation o, processing status type χ, occurrence time of processing status t. If processing status is starting, then χ = 1. If processing status has been finished, then χ = 2. Disturbance identification model is established according above dynamic event information as follows:   

$$Y=f\left(m,o,\chi ,t,s_{ijk}^0,e_{ijk}^0|o_{ijk}\in \mathrm{O}\right)$$(3)

where O denotes the unfinished operations set. The $s_{ijk}^0$ and $e_{ijk}^0$ indicate the initial starting time and the initial completion time of operation o_ijk from O, respectively. If operation o_ijk is not processed, then ${\delta }_{ijk}^1$ = 0. If operation o_ijk is being processed, then ${\delta }_{ijk}^1$ = 1. If operation o_ijk has not finished being processed, then ${\delta }_{ijk}^2$ = 0. If operation o_ijk has been processed, then ${\delta }_{ijk}^2$ = 1. The disturbance identification algorithm of operating time delay based on event-driven mechanism is described in detail as follows:

Step1: Initialize the scheduling information: $S^0=\left\{z_{ijk}^0,s_{ijk}^0,e_{ijk}^0|o_{ijk}\in L_{ij},L_{ij}\in \mathrm{\Omega }\right\}$ , where Ω is the set of unfinished casting charges.

Step2: Initialize the operations processing status: ∀o_ijk ∈ O, ${\delta }_{ijk}^1$ = 0, ${\delta }_{ijk}^2$ = 0.

Step3: If event occurs, then get event information: machine code m, operation o, processing status type χ, occurrence time of processing status t. Select the operation o_i1j1k1 from O which o_i1j1k1 is equal to operation o. If χ = 1, then go to the next step, else go to Step5.

Step4: Let ${\delta }_{i_1{j}_1{k}_1}^1$ = 1, $s_{i_1{j}_1{k}_1}^{*}$ = t, τ = $s_{i_1{j}_1{k}_1}^{*}$ – $s_{i_1{j}_1{k}_1}^0$ . If τ = 0, then y = y₀. If k = 1, τ > 0, then y = y₁. If 1 < k < θ_ij, τ > 0 then y = y₃.

Step5: Let ${\delta }_{i_1{j}_1{k}_1}^2$ = 1, $e_{i_1{j}_1{k}_1}^{*}$ = t, τ = $e_{i_1{j}_1{k}_1}^{*}$ – $e_{i_1{j}_1{k}_1}^0$ . If τ = 0, then y = y₀. If k = 1, τ > 0, then y = y₂. If 1 < k < θ_ij, τ > 0, then y = y₄.

Step6: Get the final disturbance identification result Y, and Y = {o,m,y,τ}.

4.2. Analysis on Charges Based on Reachability Matrix

In the SMCC production, it must have been processed on the former operation o_ijk and then it can be processed on the next operation o_i,j,k+1 for each charge. The adjacent charges in the same machine must be processed one by one. So, when operation time delay of one operation occurs, it will affects other operations and scheduling performance, even make the initial scheduling plan become unrealizable. It firstly needs to analyze the charges which are affected by the disturbance identification in order to analyze the influence degree of disturbances to the initial scheduling plan S⁰. The working status of each machine is divided into idle, processing the first operation or middle operation of one charge and processing the last operation of one charge. We define the □, ○, △ as those three status, respectively. The interval time between adjacent operations from the same charge is the sum of the transportation time and waiting time, which is indicated as ellipse. The scheduling plan including two charges is as shown in Fig. 2(a). L₁₁ = {o₁₁₁,o₁₁₂,o₁₁₃} and L₁₂ = {o₁₂₁,o₁₂₂,o₁₂₃}. Operations of each charge are processed by 1LD, 1RH and 1CC. The time constraints relationship between operations of charge L₁₁ and L₁₂ is shown in Fig. 2(b). The node 2 and node 4 denote operation o₁₁₁ and operation o₁₂₁ on machine 1LD, respectively. The node 8 and node 10 denote operation o₁₁₂ and operation o₁₂₂ on machine 1RH, respectively. The node 14 and node 15 denote operation o₁₁₃ and operation o₁₂₃ on machine 1CC, respectively. The node 5 denotes the interval time between operation o₁₁₁ and operation o₁₁₂. The node 6 denotes the interval time between operation o₁₂₁ and operation o₁₂₂. The node 11 and node 12 denote the interval time between operation o₁₁₂ and operation o₁₁₃ and the interval time between operation o₁₂₂ and operation o₁₂₃, respectively. The nodes 1, 3, 7, 9, 13 indicate the idle time.

Fig. 2.

The time constraints relationship between operations of charges.

The arrows between operations indicate the influence direction of operating time delay as shown in Fig. 2(b). For example, the arrows between operations on the same machine represent that the operating time delay of former operation will cause the operating time delay of the subsequent other operation. The arrows between operations on the different machine denote that the operating time delay of former operation will cause the operating time delay of the subsequent operation from the same charge.

The reachability matrix R, which reflects the system node connectivity in graph theory, is introduced to represent the interaction relationship between operations in order to analyze the influence degree of disturbances to the initial scheduling plan S⁰. R can be obtained by the adjacency matrix C which indicates the adjacency relationship of nodes. The adjacency matrix C is defined as follows:   

$$C=\Vert c_{l_1{l}_2}\Vert \begin{array}{cc}& \end{array}{l}_1,l_2=1,\dots ,N$$(4)

where l₁ and l₂ denote the nodes, respectively. N indicates the total number of nodes. If node l₁ and node l₂ are adjacent, and the arrow points to the node l₂ from the node l₁, then c_l1l2 = 1, else c_l1l2 = 0. For example, the adjacency matrix C in Fig. 2(b) is as follows:   

$$C=\left[\begin{array}{cccccccccccccccc}& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15\\ 1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 2& 0& 0& 1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 3& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 4& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 5& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 6& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 7& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 8& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& 0\\ 9& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 11& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 12& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 13& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 14& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 15& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$(5)

The reachability matrix R can be obtained by the adjacency matrix C as follows:   

$$R=\Vert r_{l_1{l}_2}\Vert \begin{array}{cc}& \end{array}{l}_1,l_2=1,\dots ,N$$(6)

If there exists directed path from the node l₁ to the node l₂, then r_l1l2 = 1, else r_l1l2 = 0. When one node l task represented by ○ in Fig. 2(b) delays, it will only make other node task delay which is connected with the node l and does not make other node task delay which is not connected with the node l. It can clearly identify the operations which are affected by operation time delay by the reachability matrix. The reachability matrix in Fig. 2(b) is as follows:   

$$R=\left[\begin{array}{cccccccccccccccc}& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15\\ 1& 1& 1& 1& 1& 1& 1& 0& 1& 1& 1& 1& 1& 0& 1& 1\\ 2& 0& 1& 1& 1& 1& 1& 0& 1& 1& 1& 1& 1& 0& 1& 1\\ 3& 0& 0& 1& 1& 0& 1& 0& 0& 0& 1& 0& 1& 0& 0& 1\\ 4& 0& 0& 0& 1& 0& 1& 0& 0& 0& 1& 0& 0& 0& 0& 1\\ 5& 0& 0& 0& 0& 1& 0& 0& 1& 1& 1& 1& 1& 0& 1& 1\\ 6& 0& 0& 0& 0& 0& 1& 0& 0& 0& 1& 0& 1& 0& 0& 1\\ 7& 0& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 1& 0& 1& 1\\ 8& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 0& 1& 1\\ 9& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 1& 0& 0& 1\\ 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 1\\ 11& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1& 1\\ 12& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1\\ 13& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1\\ 14& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1\\ 15& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$(7)

4.3. Analysis on Influence Degree of Disturbance

4.3.1. Analysis on Processing Conflict Caused by Operation Time Delay

It can find all operations which are affected by operation time delay by the reachability matrix. When node l delays Δt minutes, then the node which is connected with the node l will also delay Δt minutes.

When the starting time of the node l₀ denoting the operation o_ijk1 delays Δt minutes, that is Δt = $s_{ij{k}_1}^{*}-s_{ij{k}_1}^0$ , recalculate the starting time s_ijk1 and the completion time e_ijk1 of the operation o_ijk1 as follows:   

$$s_{ij{k}_1}=s_{ij{k}_1}^{*}=s_{ij{k}_1}^0+\mathrm{\Delta }t$$(8)

  

$$e_{ij{k}_1}=s_{ij{k}_1}+p_{ij{k}_1}^0=e_{ij{k}_1}^0+\mathrm{\Delta }t$$(9)

where $p_{ij{k}_1}^0$ is the processing time of operation o_ijk1 in the initial scheduling plan. When the completion time of the node l₀ delays Δt minutes, that is Δt = $e_{ij{k}_1}^{*}-e_{ij{k}_1}^0$ , recalculate the starting time s_ijk1 and the completion time e_ijk1 of the operation o_ijk1 as follows:   

$$s_{ij{k}_1}=s_{ij{k}_1}^0$$(10)

  

$$e_{ij{k}_1}=e_{ij{k}_1}^0+\mathrm{\Delta }t$$(11)

The new starting time s_ijk and the new completion time e_ijk of the subsequent operations of o_ijk1 are recalculated as follows:   

$$s_{ijk}=e_{i,j,k-1}+u\left(z_{i,j,k-1}^0,z_{ijk}^0\right)\begin{array}{cc}& \end{array}k=k_1+1,\dots ,{\theta }_{ij}$$(12)

  

$$e_{ijk}=s_{ijk}+p_{ijk}^0\begin{array}{cc}& \end{array}k=k_1+1,\dots ,{\theta }_{ij}$$(13)

where $z_{i,j,k-1}^0$ and $z_{ijk}^0$ are the processing machine of the operation o_i,j,k–1 and the processing machine of the operation o_ijk, respectively. The $u\left(z_{i,j,k-1}^0,z_{ijk}^0\right)$ is the standard transportation time between o_i,j,k–1 and o_ijk. We define the o_I(ijk) as the subsequent operation of the operation o_ijk on machine $z_{ijk}^0$ . The ${\chi }_{I\left(ijk\right)}^{ijk}$ denotes the processing conflict time as follows:   

$${\chi }_{I\left(ijk\right)}^{ijk}=min\left(e_{ijk},e_{I\left(ijk\right)}^0\right)-max\left(s_{ijk},s_{I\left(ijk\right)}^0\right)\begin{array}{cc}& \end{array}k=k_1,\dots ,{\theta }_{ij}$$(14)

The formula (14) can be rewritten as follows according to the formula (8) to formula (13):   

$$\begin{array}{c}{\chi }_{I\left(ijk\right)}^{ijk}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\\ k=k_1,\dots ,{\theta }_{ij}\end{array}$$(15)

The formula (15) is the relational model between the processing conflict time between operation o_ijk and other operation $o_{ijk}\left(k=k_1,\dots ,{\theta }_{ij}\right)$ and the operation delay time Δt of operation o_ijk1 when the operation delay time of o_ijk1 is Δt. If ${\chi }_{I\left(ijk\right)}^{ijk}$ ≤ 0, then the processing conflict does not occur. If ${\chi }_{I\left(ijk\right)}^{ijk}$ > 0, then the processing conflict occurs and the conflict time is ${\chi }_{I\left(ijk\right)}^{ijk}$ .

4.3.2. Analysis on Planned Casting Break Caused by Operation Time Delay

When the delay time of operation o_ijk1 is Δt, then the new starting time of o_i,j,θij can be recalculated as follows according to the formula (8) to formula (13):   

$$s_{i,j,{\theta }_{ij}}=e_{ij{k}_1}^0+\mathrm{\Delta }t+\sum _{k_2=k_1+1}^{{\theta }_{ij}-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{{\theta }_{ij}-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$$(16)

We define ${\gamma }_{j-1,j}^i$ as the planned casting break time between charge L_i,j–1 and charge L_ij:   

$${\gamma }_{j-1,j}^i=s_{i,j,{\theta }_{ij}}-e_{i,j-1,{\theta }_{i,j-1}}^0$$(17)

The formula (17) can be rewritten as follows according to the formula (16):   

$${\gamma }_{j-1,j}^i=e_{ij{k}_1}^0+\mathrm{\Delta }t+\sum _{k_2=k_1+1}^{{\theta }_{ij}-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{{\theta }_{ij}-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)-e_{i,j-1,{\theta }_{i,j-1}}^0$$(18)

The formula (18) is the relational model between the planned casting break time and the operation delay time Δt of operation o_ijk1 when the operation delay time of o_ijk1 is Δt. If ${\gamma }_{j-1,j}^i\le 0$ , then the planned casting break does not occur. If ${\gamma }_{j-1,j}^i$ > 0, then the planned casting break occurs and the planned casting break time is ${\gamma }_{j-1,j}^i$ .

4.3.3. Buffer Units of Scheduling System in SMCC Production Process

When the operation delay time of o_ijk1 is Δt, we can obtain the processing conflict time and the planned casting break time according to the formula (15) and (18). If the node l₀ has the buffer function, then the operation time delay of the subsequent connected node can be reduced through the buffer function. Supposing that the buffer time of the node l₀ is T, then the operation time delay of the subsequent connected node is Δt – T. If Δt – T < 0, then the operation time of the subsequent connected node will not delay. When the operation delay time of node l₁ is Δt, the buffer units of scheduling system in SMCC production process are as follows:

(1) The idle time on machines. If the node l₁ connects to the node of the idle time, then the node of the idle time will play the role of time compensation. Supposing that the idle time is T, then the operation time delay of the subsequent connected node is Δt – T. If Δt – T < 0, then the operation time of the subsequent connected node will not delay.

(2) Adjustment range of processing time of charge on the machine. We can only get a range of processing time of operations because the processing time of operations is affected by randomness. The processing time includes the minimum processing time $p_{ijk}^{min}$ , the standard processing time $p_{ijk}^{nor}$ , the maximum processing time $p_{ijk}^{max}$ . The processing time of operation o_ijk can be adjusted in $\left[p_{ijk}^{min},p_{ijk}^{max}\right]$ . Supposing that the processing time of o_ijk in the initial scheduling plan is $p_{ijk}^0$ . If $\mathrm{\Delta }t-\left(p_{ijk}^0-p_{ijk}^{min}\right)<0$ , then the operation time of the subsequent connected node will not delay.

(3) The interval time between adjacent operations of the same charge. SMCC process is a non-preemptive processing, which means that when one operation is being processed on the machine, other operations are prohibited to preempt the same machine. Once one operation begins to be processed, it is not allowed to interrupt until the processing finishes. If the interval time between adjacent operations of the same charge includes the waiting time, then it has the buffer function. Supposing that the interval time between o_ijk and o_i,j,k+1 is $u_{ijk}^0\left(z_{ijk}^0,z_{i,j,k+1}^0\right)$ . If $\mathrm{\Delta }t-\left(u^0\left(z_{ijk}^0,z_{i,j,k+1}^0\right)-u\left(z_{ijk}^0,z_{i,j,k+1}^0\right)\right)<0$ , then the operation time of the subsequent connected node will not delay.

The buffer matrix is defined as follows according to the above analysis:   

$$\mathrm{E}=\Vert {\mathit{\epsilon }}_l\Vert \begin{array}{cc}& \end{array}l=1,\dots ,N$$(19)

where ε_l denotes the buffer time of the node l.

(1) If the node l is the node of the idle time T, then   

$${\mathit{\epsilon }}_l=T$$(20)

(2) If the node l indicates the operation o_ijk of the charge, then   

$${\mathit{\epsilon }}_l=p_{ijk}^0-p_{ijk}^{min}$$(21)

(3) If the node l denotes the interval time between adjacent operations of the same charge, then   

$${\mathit{\epsilon }}_l=u^0\left(z_{ijk}^0,z_{i,j,k+1}^0\right)-u\left(z_{ijk}^0,z_{i,j,k+1}^0\right)$$(22)

where $u^0\left(z_{ijk}^0,z_{i,j,k+1}^0\right)$ is the interval time between o_ijk and o_i,j,k+1, and $u\left(z_{ijk}^0,z_{i,j,k+1}^0\right)$ is the standard transportation time between o_ijk and o_i,j,k+1.

If ε_l > 0 in the buffer matrix E, it means that the node l has the buffer time. The operation time delay of the subsequent connected node can be reduced through the buffer time. If the operation delay time of the node l is Δt, then the operation time delay of the subsequent connected node can be reduced to Δt – ε_l. If ε_l = 0, then the operation time delay of the subsequent connected node will also be Δt.

It can be known that we can obtain the processing conflict time and the planned casting break time according to the formula (15) and (18) when the operation delay time of o_ijk1 is Δt. If there have buffer units, then the processing conflict time and the planned casting break time can not be directly obtained according to the formula (15) and formula (18). These are closely related with the buffer units. So, it is needed to deeply analyze the relationship between delay time Δt and buffer units in order to find out the really impact on the scheduling system by delay time Δt.

4.3.4. Analysis on Influence Degree of Disturbance by Operation Time Delay under Buffer Units

When the operation delay time of o_ijk1 is Δt, the analysis model of the impact on the starting time of operations by operation time delay under buffer units is established as follows according to the reachability matrix R and the buffer matrix E:   

$${\mathrm{T}}_1\left(l_0,\mathrm{\Delta }t\right)=\Vert {\eta }_l\Vert \begin{array}{cc}& \end{array}l=1,\dots ,N$$(23)

where the η_l denotes the final delay time of the starting time of the node l affected by the delay time Δt of the node l₀. If r_l0l = 0, then η_l = 0. If r_l0l = 1 and there exists B(B ≥ 1) directed paths from the node l₀ to the node l, then the η_l is equal to the maximum delay time of the starting time of all nodes in the directed paths:   

$${\eta }_l=\underset{b=1,\dots ,B}{max}\left(\mathrm{\Delta }t-\sum _{\begin{array}{c}{l}_1\in P_b\left(l_0,l\right)\\ l_1\ne l\end{array}}\sum _{\begin{array}{c}{l}_2\in P_b\left(l_0,l\right)\backslash \\ l_2\ne l\end{array}}{c}_{l_1{l}_2}{\mathit{\epsilon }}_{l_1}\right)\begin{array}{cc}& \end{array}{r}_{l_{0}l}=1$$(24)

where P_b(l₀,l) is the bth directed path in B directed paths.

The analysis model of the impact on the completion time of operations by operation time delay under buffer units is established as follows according to the formula (19) and the formula (23):   

$${\mathrm{T}}_2\left(l_0,\mathrm{\Delta }t\right)=\Vert {\lambda }_l\Vert \begin{array}{cc}& \end{array}l=1,\dots ,N$$(25)

where the λ_l indicates the final delay time of the completion time of the node l affected by the delay time Δt of the node l₀. If r_l0l = 0, then λ_l = 0. If r_l0l = 1, then λ_l = η_l–ε_l.

According to the above analysis, the analysis model of the processing conflict time between adjacent operations is established as follows when the operation delay time is Δt:   

$${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}=min\left(e_{ijk}^0+{\lambda }_{l_1},e_{I\left(ijk\right)}^0+{\lambda }_{l_2}\right)-max\left(s_{ijk}^0+{\eta }_{l_1},s_{I\left(ijk\right)}^0+{\eta }_{l_2}\right)$$(26)

where the o_ijk1 is the corresponding operation for the node l₀, the o_ijk is the corresponding operation for the node l₁ and the o_I(ijk) is the corresponding operation for the node l₂. If ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}$ ≤ 0, then there is no processing conflict time between o_ijk and o_I(ijk) through the buffer units. If ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}$ > 0, it means that there is still processing conflict time between o_ijk and o_I(ijk) through the buffer units.

The analysis model of the planned casting break time between adjacent charges in the same cast is established as follows when the operation delay time is Δt:   

$${\tilde{\gamma }}_{j-1,j}^i=\left(s_{i,j,{\theta }_{ij}}^0+{\eta }_{l_2}\right)-\left(e_{i,j-1,{\theta }_{i,j-1}}^0+{\lambda }_{l_1}\right)={\eta }_{l_2}-{\lambda }_{l_1}$$(27)

where the o_i,j,θij is the corresponding operation for the node l₂ and the o_{i,j–1,θi,j–1} is the corresponding operation for the node l₁. If ${\tilde{\gamma }}_{j-1,j}^i\le 0$ , there is no planned casting break time between o_{i,j–1,θi,j–1} and o_i,j,θij through the buffer units. If ${\tilde{\gamma }}_{j-1,j}^i>0$ , there is still planned casting break time between o_{i,j–1,θi,j–1} and o_i,j,θij through the buffer units.

4.4. Abnormal Condition Decision of Initial Scheduling Plan

The formula (26) and the formula (27) denote the final the processing conflict time and the final planned casting break time under buffer units, respectively. The processing conflict time and the planned casting break time are resolved by under buffer units in fact, which is the adjustment method. The different abnormal condition of the scheduling plan by the operation time delay can lead to the different adjustment method to resolve the abnormal condition. So, prediction for abnormal condition of scheduling plan with operation time delay disturbance in SMCC production process is very important.

4.4.1. The Planned Casting Break Decision of Initial Scheduling Plan by Operation Time Delay under Buffer Units

The G denotes the time constraint graph between operations. The V indicates the all nodes set in G. When the node l₀ (l₀ is correspond to the charge L_ij) delay Δt, if ${\gamma }_{j-1,j}^i>0$ , then there is planned casting break time between L_i,j–1 and L_ij. The o_{i,j–1,θi,j–1} is the corresponding operation for the node l₁ and the o_i,j,θij is the corresponding operation for the node l₂. We define the V_ij as the all nodes set for charge L_ij. The planned casting break decision is made as follows:

(1) If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i\ge 0$ , then the abnormal condition of scheduling plan is one-level planned casting break ζ₁₁.

Proof. When the node l₀ (l₀ is correspond to operation o_ijk1 the charge L_ij) delay Δt, according to the formula (18), ${\gamma }_{j-1,j}^i=e_{ij{k}_1}^0+\mathrm{\Delta }t+\mathrm{\sum }_{k_2=k_1+1}^{{\theta }_{ij}-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{{\theta }_{ij}-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)-e_{i,j-1,{\theta }_{i,j-1}}^0$ . $e_{i,j-1,{\theta }_{i,j-1}}^0$ is the initial completion time of operation o_{i,j–1,θi,j–1} and the initial processing time of operation o_{i,j–1,θi,j–1} is $p_{i,j-1,{\theta }_{i,j-1}}^0$ . If the processing time of operation o_{i,j–1,θi,j–1} is adjusted to $p_{i,j-1,{\theta }_{i,j-1}}^{max}$ , then the new completion time e_{i,j–1,θi,j–1} of operation o_{i,j–1,θi,j–1} can be recalculated as follows: $e_{i,j-1,{\theta }_{i,j-1}}=e_{i,j-1,{\theta }_{i,j-1}}^0+p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0$ . Then according to the formula (18), the new planned casting break time (denoted as ${\left({\gamma }_{j-1,j}^i\right)}^{\text{'}}$ ) can be recalculated as follows:   

$$\begin{array}{c}\left({\gamma }_{j-1,j}^i\right)\prime =e_{ij{k}_1}^0+\mathrm{\Delta }t+\sum _{k_2=k_1+1}^{{\theta }_{ij}-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{{\theta }_{ij}-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)\mathrm{\hspace{0.17em} }-e_{i,j-1,{\theta }_{i,j-1}}\hfill \\ =e_{ij{k}_1}^0+\mathrm{\Delta }t+\sum _{k_2=k_1+1}^{{\theta }_{ij}-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{{\theta }_{ij}-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)\mathrm{\hspace{0.17em} }-e_{i,j-1,{\theta }_{i,j-1}}^0\hfill \\ -p_{i,j-1,{\theta }_{i,j-1}}^{max}+p_{i,j-1,{\theta }_{i,j-1}}^0\hfill \\ ={\gamma }_{j-1,j}^i-p_{i,j-1,{\theta }_{i,j-1}}^{max}+p_{i,j-1,{\theta }_{i,j-1}}^0\hfill \end{array}$$(28)

If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i\ge 0$ , then ${\left({\gamma }_{j-1,j}^i\right)}^{\text{'}}\le 0$ . It can be seen that the planned casting break between o_{i,j–1,θi,j–1} and o_i,j,θij can be resolved only by adjusting the processing time of L_i,j–1 at continuous casting stage. So, according to the classification of planned casting break of scheduling plan (section 2.3), the abnormal condition is one-level planned casting break ζ₁₁.

(2) If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i<0$ , then for the nodes $l|l\in V,l\notin V_{ij}$ , let ε_l = 0. If ${\tilde{\gamma }}_{j-1,j}^i\le 0$ , then the abnormal condition of scheduling plan is one-level planned casting break ζ₁₁.

Proof. If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i<0$ , then ${\left({\gamma }_{j-1,j}^i\right)}^{\text{'}}$ > 0 according to the formula (28). It can been seen that the planned casting break between o_{i,j–1,θi,j–1} and o_i,j,θij can not be resolved only by adjusting the processing time of L_i,j–1 at continuous casting stage. For the nodes $l|l\in V,l\notin V_{ij}$ , let ε_l = 0, which means that only nodes belonging to V_ij have buffer time. According to the formula (27), ${\tilde{\gamma }}_{j-1,j}^i=\left(s_{i,j,{\theta }_{ij}}^0+{\eta }_{l_2}\right)-\left(e_{i,j-1,{\theta }_{i,j-1}}^0+{\lambda }_{l_1}\right)={\eta }_{l_2}-{\lambda }_{l_1}$ . If ${\tilde{\gamma }}_{j-1,j}^i\le 0$ , then It can be seen that the planned casting break between o_{i,j–1,θi,j–1} and o_i,j,θij can be resolved only by adjusting the processing time of L_ij at each stage. So, according to the classification of planned casting break of scheduling plan (section 2.3), the abnormal condition is one-level planned casting break ζ₁₁.

(3) If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i<0$ , then for the nodes $l|l\in V,l\notin V_{ij}$ , let ε_l = 0. If ${\tilde{\gamma }}_{j-1,j}^i>0$ , but $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\tilde{\gamma }}_{j-1,j}^i\ge$ 0, then the abnormal condition of scheduling plan is two-level planned casting break ζ₁₂.

Proof. If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i<$ 0, then ${\left({\gamma }_{j-1,j}^i\right)}^{\text{'}}>$ 0, it can been seen that the planned casting break between o_{i,j–1,θi,j–1} and o_i,j,θij can not be resolved only by adjusting the processing time of $L_{i,j-1}$ at continuous casting stage. For the nodes $l|l\in V,l\notin V_{ij}$ , let ε_l = 0. According to the formula (27), ${\tilde{\gamma }}_{j-1,j}^i=\left(s_{i,j,{\theta }_{ij}}^0+{\eta }_{l_2}\right)-\left(e_{i,j-1,{\theta }_{i,j-1}}^0+{\lambda }_{l_1}\right)={\eta }_{l_2}-{\lambda }_{l_1}$ . If ${\tilde{\gamma }}_{j-1,j}^i>$ 0, then It can be seen that the planned casting break between o_{i,j–1,θi,j–1} and o_i,j,θij can not be resolved only by adjusting the processing time of L_ij at each stage. If the processing time of operation o_{i,j–1,θi,j–1} is adjusted to $p_{i,j-1,{\theta }_{i,j-1}}^{max}$ , then the new completion time e_{i,j–1,θi,j–1} of operation o_{i,j–1,θi,j–1} can be recalculated as follows: $e_{i,j-1,{\theta }_{i,j-1}}=e_{i,j-1,{\theta }_{i,j-1}}^0+p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0$ . Then according to the formula (27), the new planned casting break time (denoted as ${\left({\tilde{\gamma }}_{j-1,j}^i\right)}^{\text{'}}$ ) can be recalculated as follows:   

$$\begin{array}{c}{\left({\tilde{\gamma }}_{j-1,j}^i\right)}^{\text{'}}=\left(s_{i,j,{\theta }_{ij}}^0+{\eta }_{l_2}\right)\mathrm{\hspace{0.17em} }-\mathrm{\hspace{0.17em} }\left(e_{i,j-1,{\theta }_{i,j-1}}^0+{\lambda }_{l_1}\right)\hfill \\ =\left(s_{i,j,{\theta }_{ij}}^0+{\eta }_{l_2}\right)\mathrm{\hspace{0.17em} }-\mathrm{\hspace{0.17em} }\left(e_{i,j-1,{\theta }_{i,j-1}}^0+p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0+{\lambda }_{l_1}\right)\hfill \\ =\left(s_{i,j,{\theta }_{ij}}^0+{\eta }_{l_2}\right)\mathrm{\hspace{0.17em} }-\mathrm{\hspace{0.17em} }\left(e_{i,j-1,{\theta }_{i,j-1}}^0+{\lambda }_{l_1}\right)-p_{i,j-1,{\theta }_{i,j-1}}^{max}+p_{i,j-1,{\theta }_{i,j-1}}^0\hfill \\ ={\tilde{\gamma }}_{j-1,j}^i-p_{i,j-1,{\theta }_{i,j-1}}^{max}+p_{i,j-1,{\theta }_{i,j-1}}^0\hfill \end{array}$$(29)

If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\tilde{\gamma }}_{j-1,j}^i\ge$ 0, then ${\left({\tilde{\gamma }}_{j-1,j}^i\right)}^{\text{'}}\le$ 0. It can be seen that the planned casting break between o_{i,j–1,θi,j–1} and o_i,j,θij can be resolved synchronously by adjusting the processing time of L_i,j–1 at continuous casting stage and the processing time of L_ij at each stage. So, according to the classification of planned casting break of scheduling plan (section 2.3), the abnormal condition is two-level planned casting break ζ₁₂.

(4) If let ε_l = 0 for the nodes l|l∈V, l∉V_ij, $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i<$ 0, ${\tilde{\gamma }}_{j-1,j}^i>$ 0, $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\tilde{\gamma }}_{j-1,j}^i<$ 0. If let ε_l ≠ 0 for the nodes $l|l\in V$ , ${\tilde{\gamma }}_{j-1,j}^i\le$ 0, then the abnormal condition of scheduling plan is three-level planned casting break ζ₁₃.

Proof. For the nodes $l|l\in V,l\notin V_{ij}$ , let ε_l = 0. If $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\gamma }_{j-1,j}^i<$ 0, ${\tilde{\gamma }}_{j-1,j}^i>$ 0, $p_{i,j-1,{\theta }_{i,j-1}}^{max}-p_{i,j-1,{\theta }_{i,j-1}}^0-{\tilde{\gamma }}_{j-1,j}^i<$ 0, it means that the planned casting break between o_{i,j–1,θi,j–1} and o_i,j,θij can not be resolved synchronously by adjusting the processing time of L_i,j–1 at continuous casting stage and the processing time of L_ij at each stage according to the above analysis. If let ε_l ≠ 0 for the nodes $l|l\in V$ , then all nodes have duffer time. If ${\tilde{\gamma }}_{j-1,j}^i\le$ 0, then it means that the planned casting break between charge L_i,j–1 and L_ij can be resolved synchronously by adjusting the processing time of L_ij1(j₁ ≤ j) at each stage. So, according to the classification of planned casting break of scheduling plan (section 2.3), the abnormal condition is three-level planned casting break ζ₁₃.

(5) For the nodes $l|l\in V$ , if ${\tilde{\gamma }}_{j-1,j}^i>$ 0, then the abnormal condition of scheduling plan is four-level planned casting break ζ₁₄.

Proof. If let ε_l ≠ 0 for the nodes $l|l\in V$ , then all nodes have duffer time. If ${\tilde{\gamma }}_{j-1,j}^i>0$ , then it can be known that the planned casting break between charge L_i,j–1 and L_ij can not be resolved synchronously by adjusting the processing time of L_ij1(j₁ ≤ j) at each stage according to the above analysis. In this case, the planned casting break must be resolved synchronously by adjusting the processing machines of L_ij1(j₁ ≤ j) at steelmaking stage and refining stage and by adjusting the processing time of L_ij1(j₁ ≤ j) at each stage. So, according to the classification of planned casting break of scheduling plan (section 2.3), the abnormal condition is four-level planned casting break ζ₁₄.

4.4.2. The Processing Conflict Decision of Initial Scheduling Plan by Operation Time Delay under Buffer Units

When the node l₀ (l₀ is correspond to the charge L_ij) delay Δt, if ${\chi }_{I\left(ijk\right)}^{ijk}>$ 0, then there is processing conflict time between o_ijk and o_I(ijk). The o_ijk is the corresponding operation for the node l₁ and the o_I(ijk) is the corresponding operation for the node l₂. We define the V₁ as the all nodes set for charge L_ij and the V₂ as the all nodes set for charge including o_I(ijk). The processing conflict decision is made as follows:

(1) If ${\mathit{\epsilon }}_{l_1}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ₂₁.

Proof. The operation o_I(ijk) is the subsequent operation of the operation o_ijk on machine in the initial scheduling plan. When the operation time delay occurs, the processing conflict time ${\chi }_{I\left(ijk\right)}^{ijk}$ can be calculated according to the formula (15). According to the new processing sequence between o_ijk and o_I(ijk), the processing conflict can be divided into two cases.

■ Case 1: o_ijk is also processed before o_I(ijk) in the new scheduling plan. In this case, the completion time of o_ijk is earlier than the completion time of o_I(ijk) which means $e_{ijk}<e_{I\left(ijk\right)}^0$ . So, it can be known that $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)<e_{I\left(ijk\right)}^0$ . The processing conflict between o_ijk and o_I(ijk) may be resolved by adjusting the processing time of o_ijk. According to the formula (21), ${\mathit{\epsilon }}_{l_1}=p_{ijk}^0-p_{ijk}^{min}$ . If the starting time of operation o_ijk is not changed and the processing time of o_ijk is adjusted to $p_{ijk}^{min}$ , then according to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^{k-1}{p}_{ij{k}_2}^0+p_{ijk}^{min}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^{k-1}{p}_{ij{k}_2}^0+p_{ijk}^0-{\mathit{\epsilon }}_{l_1}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\hfill \end{array}$$(30)

Because $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)<e_{I\left(ijk\right)}^0$ , $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)-{\mathit{\epsilon }}_{l_l}<e_{I\left(ijk\right)}^0$ . So the minimum value of first term in the formula (30) is $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^{k-1}{p}_{ij{k}_2}^0+p_{ijk}^0-{\mathit{\epsilon }}_{l_1}+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . Then the formula (30) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=\left(min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_1}\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\epsilon }_{l_1}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-{\mathit{\epsilon }}_{l_1}\hfill \end{array}$$(31)

If ${\mathit{\epsilon }}_{l_1}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) only by adjusting the processing time of o_ijk. So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is one-level processing conflict ζ₂₁.

■ Case 2: o_ijk is processed after o_I(ijk) in the new scheduling plan. In this case, the starting time of o_ijk is later than the starting time of o_I(ijk) which means $s_{ijk}>s_{I\left(ijk\right)}^0$ . So, it can be known that $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)>s_{I\left(ijk\right)}^0$ . According to the formula (21), ${\mathit{\epsilon }}_{l_1}=p_{ijk}^0-p_{ijk}^{min}$ . If the completion time of operation o_ijk is not changed and the processing time of o_ijk is adjusted to $p_{ijk}^{min}$ , then according to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)+{\mathit{\epsilon }}_{l_1},s_{I\left(ijk\right)}^0\right)\hfill \end{array}$$(32)

Because $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)>s_{I\left(ijk\right)}^0$ , $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)+{\mathit{\epsilon }}_{l_1}>s_{I\left(ijk\right)}^0$ . So the maximum value of second term in the formula (32) is $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)+{\mathit{\epsilon }}_{l_1}$ . Then the formula (32) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -\left(max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)+{\mathit{\epsilon }}_{l_1}\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_1}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-{\mathit{\epsilon }}_{l_1}\hfill \end{array}$$(33)

If ${\mathit{\epsilon }}_{l_1}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) only by adjusting the processing time of o_ijk. So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is one-level processing conflict ζ₂₁.

To sum up, if ${\mathit{\epsilon }}_{l_1}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ₂₁.

(2) If ${\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ₂₁.

Proof. The operation o_I(ijk) is the subsequent operation of the operation o_ijk on machine in the initial scheduling plan. When the operation time delay occurs, the processing conflict time ${\chi }_{I\left(ijk\right)}^{ijk}$ can be calculated according to the formula (15). According to the new processing sequence between o_ijk and o_I(ijk), the processing conflict can be divided into two cases.

■ Case 1: o_ijk is also processed before o_I(ijk) in the new scheduling plan. In this case, the starting time of o_I(ijk) is later than the starting time of o_ijk which means $s_{I\left(ijk\right)}^0>s_{ijk}$ . So, it can be known that $s_{I\left(ijk\right)}^0>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . The processing conflict between o_ijk and o_I(ijk) may be resolved by adjusting the processing time of o_I(ijk). According to the formula (21), ${\mathit{\epsilon }}_{l_2}=p_{I\left(ijk\right)}^0-p_{I\left(ijk\right)}^{min}$ . If the completion time of operation o_I(ijk) is not changed and the processing time of o_I(ijk) is adjusted to $p_{I\left(ijk\right)}^{min}$ , then according to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{l}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0+{\mathit{\epsilon }}_{l_2}\right)\hfill \end{array}$$(34)

Because $s_{I\left(ijk\right)}^0>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ , $s_{I\left(ijk\right)}^0+{\mathit{\epsilon }}_{l_2}>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . So the maximum value of second term in the formula (34) is $s_{I\left(ijk\right)}^0+{\mathit{\epsilon }}_{l_2}$ . Then the formula (34) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -\left(max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }+{\mathit{\epsilon }}_{l_2}\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_2}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-\mathrm{\hspace{0.17em} }{\mathit{\epsilon }}_{l_2}\hfill \end{array}$$(35)

If ${\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) only by adjusting the processing time of o_I(ijk). So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is one-level processing conflict ζ₂₁.

■ Case 2: o_ijk is processed after o_I(ijk) in the new scheduling plan. In this case, the completion time of o_I(ijk) is earlier than the completion time of o_ijk which means $e_{I\left(ijk\right)}^0<e_{ijk}$ . So, it can be known that $e_{I\left(ijk\right)}^0<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . The processing conflict between o_ijk and o_I(ijk) may be resolved by adjusting the processing time of o_I(ijk). According to the formula (21), ${\mathit{\epsilon }}_{l_2}=p_{I\left(ijk\right)}^0-p_{I\left(ijk\right)}^{min}$ . If the starting time of operation o_I(ijk) is not changed and the processing time of o_I(ijk) is adjusted to $p_{I\left(ijk\right)}^{min}$ , then according to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{l}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0-{\mathit{\epsilon }}_{l_2}\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\hfill \end{array}$$(36)

Because $e_{I\left(ijk\right)}^0<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ , $e_{I\left(ijk\right)}^0-{\mathit{\epsilon }}_{l_2}<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . So the minimum value of first term in the formula (36) is $e_{I\left(ijk\right)}^0-{\mathit{\epsilon }}_{l_2}$ . Then the formula (36) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=\left(min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_2}\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_2}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-{\mathit{\epsilon }}_{l_2}\hfill \end{array}$$(37)

If ${\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) only by adjusting the processing time of o_I(ijk). So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is one-level processing conflict ζ₂₁.

To sum up, if ${\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ₂₁.

(3) If ${\mathit{\epsilon }}_{l_1}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, ${\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, ${\mathit{\epsilon }}_{l_1}+{\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is two-level processing conflict ζ₂₂.

Proof. The operation o_I(ijk) is the subsequent operation of the operation o_ijk on machine in the initial scheduling plan. When the operation time delay occurs, the processing conflict time ${\chi }_{I\left(ijk\right)}^{ijk}$ can be calculated according to the formula (15). According to above analysis, if ${\mathit{\epsilon }}_{l_1}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, it can not resolve the processing conflict between o_ijk and o_I(ijk) only by adjusting the processing time of o_ijk. If ${\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, it can not resolve the processing conflict between o_ijk and o_I(ijk) only by adjusting the processing time of o_I(ijk). According to the new processing sequence between o_ijk and o_I(ijk), the processing conflict can be divided into two cases.

■ Case 1: o_ijk is also processed before o_I(ijk) in the new scheduling plan. In this case, the completion time of o_ijk is earlier than the completion time of o_I(ijk) which means $e_{ijk}<e_{I\left(ijk\right)}^0$ . So, it can be known that $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)<e_{I\left(ijk\right)}^0$ . The starting time of o_I(ijk) is later than the starting time of o_ijk which means $s_{I\left(ijk\right)}^0>s_{ijk}$ . So, it can be known that $s_{I\left(ijk\right)}^0>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . The processing conflict may be resolved synchronously by adjusting the processing time of o_ijk and o_I(ijk). According to the formula (21), ${\mathit{\epsilon }}_{l_1}=p_{ijk}^0-p_{ijk}^{min}$ and ${\mathit{\epsilon }}_{l_2}=p_{I\left(ijk\right)}^0-p_{I\left(ijk\right)}^{min}$ . If the starting time of operation o_ijk is not changed and the processing time of o_ijk is adjusted to, and if the completion time of operation o_I(ijk) is not changed and the processing time of o_I(ijk) is adjusted to $p_{I\left(ijk\right)}^{min}$ , then according to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{l}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^{k-1}{p}_{ij{k}_2}^0+p_{ijk}^0-{\mathit{\epsilon }}_{l_1}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0+{\mathit{\epsilon }}_{l_2}\right)\hfill \end{array}$$(38)

Because $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)<e_{I\left(ijk\right)}^0$ and $s_{I\left(ijk\right)}^0>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ , $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)-{\mathit{\epsilon }}_{l_1}<e_{I\left(ijk\right)}^0$ and $s_{I\left(ijk\right)}^0+{\mathit{\epsilon }}_{l_2}>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . Then the formula (38) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=\left(min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^{k-1}{p}_{ij{k}_2}^0+p_{ijk}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_1}\right)\hfill \\ -\left(max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }+{\mathit{\epsilon }}_{l_2}\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^{k-1}{p}_{ij{k}_2}^0+p_{ijk}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_1}-{\mathit{\epsilon }}_{l_2}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-{\mathit{\epsilon }}_{l_1}-{\mathit{\epsilon }}_{l_2}\hfill \end{array}$$(39)

If ${\mathit{\epsilon }}_{l_1}+{\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) synchronously by adjusting the processing time of o_ijk and o_I(ijk). So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is one-level processing conflict ζ₂₂.

■ Case 2: o_ijk is processed after o_I(ijk) in the new scheduling plan. It can be known that $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)>s_{I\left(ijk\right)}^0$ and $e_{I\left(ijk\right)}^0<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . If the completion time of operation o_ijk is not changed and the processing time of o_ijk is adjusted to $p_{ijk}^{min}$ , and if the starting time of operation o_I(ijk) is not changed and the processing time of o_I(ijk) is adjusted to $p_{I\left(ijk\right)}^{min}$ , then according to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0-{\mathit{\epsilon }}_{l_2}\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)+{\mathit{\epsilon }}_{l_1},s_{I\left(ijk\right)}^0\right)\hfill \end{array}$$(40)

Because $e_{I\left(ijk\right)}^0<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ and $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)>s_{I\left(ijk\right)}^0$ , $e_{I\left(ijk\right)}^0-{\mathit{\epsilon }}_{l_2}<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ and $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)+{\mathit{\epsilon }}_{l_1}>s_{I\left(ijk\right)}^0$ . Then the formula (40) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=\left(min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-{\mathit{\epsilon }}_{l_2}\right)\hfill \\ -\left(max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }+{\mathit{\epsilon }}_{l_1}\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)-{\mathit{\epsilon }}_{l_1}-{\mathit{\epsilon }}_{l_2}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-{\mathit{\epsilon }}_{l_1}-{\mathit{\epsilon }}_{l_2}\hfill \end{array}$$(41)

If ${\mathit{\epsilon }}_{l_1}+{\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) synchronously by adjusting the processing time of o_ijk and o_I(ijk). So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is one-level processing conflict ζ₂₂.

To sum up, If ${\mathit{\epsilon }}_{l_1}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, ${\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, ${\mathit{\epsilon }}_{l_1}+{\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is two-level processing conflict ζ₂₂.

(4) If ${\mathit{\epsilon }}_{l_1}+{\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, then let ε_l = 0 for the nodes $l|l\in V,l\notin V_1,l\notin V_2$ . If $\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is three-level processing conflict ζ₂₃.

Proof. According to above analysis, if ${\mathit{\epsilon }}_{l_1}+{\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}<0$ , It can not resolve the processing conflict between o_ijk and o_I(ijk) synchronously by adjusting the processing time of o_ijk and o_I(ijk). We define o_I(ijk) is the k¹th operation of the charge $L_{i^1{j}^1}$ , so the o_I(ijk) can also be denoted as $o_{i^1{j}^1{k}^1}$ . We define the $V_1^{*}$ as the all nodes set for $o_{ij{k}_1^{*}}|k_1^{*}\le k$ of L_ij and the $V_2^{*}$ as the all nodes set for $o_{i^1{j}^1{k}_2^{*}}{|k_2^{*}\le k}^1$ of $L_{i^1{j}^1}$ . The processing conflict may be resolved synchronously by adjusting the processing time of operations in $V_1^{*}$ and $V_2^{*}$ . According to the new processing sequence between o_ijk and o_I(ijk), the processing conflict can be divided into two cases.

■ Case 1: o_ijk is also processed before o_I(ijk) in the new scheduling plan. It can be known that $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)<e_{I\left(ijk\right)}^0$ and $s_{I\left(ijk\right)}^0>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . If the starting time of o_ijk1 is not changed, the processing time of operation $o_{ij{k}_1^{*}}$ in $V_1^{*}$ is adjusted to $p_{ij{k}_1^{*}}^{min}$ , the earliest starting time of operation in $V_2^{*}$ is not changed, the processing time of operation $o_{i^1{j}^1{k}_3^{*}}$ in $V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}$ is adjusted to $p_{i^1{j}^1{k}_3^{*}}^{max}$ , and the processing time of operation $o_{i^1{j}^1{k}^1}$ is adjusted to $p_{i^1{j}^1{k}^1}^{min}$ , the processing conflict may be resolved. According to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0+\sum _{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0-\sum _{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0+\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\right)\hfill \end{array}$$(42)

Because $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)<e_{I\left(ijk\right)}^0$ and $s_{I\left(ijk\right)}^0>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ , $e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0-\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }<e_{I\left(ijk\right)}^0+\mathrm{\sum }_{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}$ and $s_{I\left(ijk\right)}^0+\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}>s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0-\mathrm{\sum }_{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . Then the formula (42) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=\left(min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0+\sum _{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}\right)\mathrm{\hspace{0.17em} }-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}\right)\hfill \\ -\left(max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0-\sum _{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }+\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0+\sum _{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0-\sum _{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\hfill \end{array}$$(43)

If $\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) synchronously by adjusting the processing time of operations of L_ij and charge $L_{i^1{j}^1}$ . So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is three-level processing conflict ζ₂₃.

■ Case 2: o_ijk is processed after o_I(ijk) in the new scheduling plan. It can be known that $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)>s_{I\left(ijk\right)}^0$ and $e_{I\left(ijk\right)}^0<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . If the starting time of o_ijk1 is not changed, the processing time of operation $o_{ij{k}_4^{*}}$ in $V_1^{*}-\left\{o_{ijk}\right\}$ is adjusted to $p_{ij{k}_4^{*}}^{max}$ , the processing time of operation o_ijk is adjusted to $p_{ijk}^{min}$ , the earliest starting time of operation in $V_2^{*}$ is not changed, the processing time of operation $o_{i^1{j}^1{k}_2^{*}}$ in $V_2^{*}$ is adjusted to $p_{i^1{j}^1{k}_2^{*}}^{min}$ , the processing conflict may be resolved. According to the formula (15), the new processing conflict time (denoted as ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) can be recalculated as follows (k = k₁,...,θ_ij):   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0-\sum _{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}\right)\hfill \end{array}$$(44)

Because $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)>s_{I\left(ijk\right)}^0$ and $e_{I\left(ijk\right)}^0<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ , $s_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)>s_{I\left(ijk\right)}^0-\mathrm{\sum }_{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}$ and $e_{I\left(ijk\right)}^0-\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}<e_{ij{k}_1}+\mathrm{\sum }_{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\mathrm{\sum }_{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right)$ . Then the formula (44) can be rewritten as follows:   

$$\begin{array}{c}{\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}=\left(min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\right)\hfill \\ -\left(max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0-\sum _{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}\right)\mathrm{\hspace{0.17em} }+\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}\right)\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{l_1^{*}\in V_1^{*}-\left\{o_{ijk}\right\}}{\mathit{\epsilon }}_{l_1^{*}}+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0-\sum _{l_2^{*}\in V_2^{*}-\left\{o_{i^1{j}^1{k}^1}\right\}}{\mathit{\epsilon }}_{l_2^{*}}\right)\mathrm{\hspace{0.17em} }-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\hfill \\ =min\left(e_{ij{k}_1}+\sum _{k_2=k_1+1}^k{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),e_{I\left(ijk\right)}^0\right)\hfill \\ -max\left(s_{ij{k}_1}+\sum _{k_2=k_1}^{k-1}{p}_{ij{k}_2}^0+\sum _{k_3=k_1}^{k-1}u\left(z_{ij{k}_3}^0,z_{i,j,k_3+1}^0\right),s_{I\left(ijk\right)}^0\right)\mathrm{\hspace{0.17em} }-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\hfill \\ ={\chi }_{I\left(ijk\right)}^{ijk}-\sum _{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}-\sum _{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}\hfill \end{array}$$(45)

If $\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then ${\left({\chi }_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0. It can resolve the processing conflict between o_ijk and o_I(ijk) synchronously by adjusting the processing time of operations of L_ij and charge $L_{i^1{j}^1}$ . So, according to the classification of processing conflict of scheduling plan (section 2.3), the abnormal condition is three-level processing conflict ζ₂₃.

To sum up, if ${\mathit{\epsilon }}_{l_1}+{\mathit{\epsilon }}_{l_2}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0, then let ε_l = 0 for the nodes $l|l\in V,l\notin V_1,l\notin V_2$ . If $\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}-{\chi }_{I\left(ijk\right)}^{ijk}\ge$ 0, then the abnormal condition of scheduling plan is three-level processing conflict ζ₂₃.

(5) If $\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0 when ε_l = 0 for the nodes $l|l\in V,l\notin V_1,l\notin V_2$ and ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}\le$ 0 for the nodes $l|l\in V$ , then the abnormal condition of scheduling plan is four-level processing conflict ζ₂₄.

Proof. According to above analysis, it can not resolve the processing conflict between o_ijk and o_I(ijk) synchronously by adjusting the processing time of operations of L_ij and the charge including $o_{I(ijk)}$ if $\mathrm{\sum }_{l_1^{*}\in V_1^{*}}{\mathit{\epsilon }}_{l_1^{*}}+\mathrm{\sum }_{l_2^{*}\in V_2^{*}}{\mathit{\epsilon }}_{l_2^{*}}-{\chi }_{I\left(ijk\right)}^{ijk}<$ 0. The processing conflict may be resolved synchronously by adjusting the processing time of operations of L_ij, the charge including o_I(ijk) and other charges. Section 4.3.4 analyzes the new processing conflict time in detail when buffer units are considered and the new processing conflict time ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}$ is obtained according to the formula (26). Moreover, according to the analysis of formula (26) in section 4.3.4, it can resolve the processing conflict between o_ijk and o_I(ijk) synchronously by adjusting the processing time of operations of L_ij, charge including o_I(ijk) and other charges if ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}\le$ 0 when ε_l ≠ 0 for the nodes $l|l\in V$ . So, the abnormal condition is four-level processing conflict ζ₂₄ according to the classification of processing conflict of scheduling plan (section 2.3).

(6) For the nodes $l|l\in V$ , if ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}>$ 0, then the abnormal condition of scheduling plan is five-level processing conflict ζ₂₅.

Proof. According to above analysis, if ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}>$ 0 when ε_l ≠ 0 for the nodes $l|l\in V$ , it can not resolve the processing conflict between o_ijk and o_I(ijk) only by adjusting the processing time of operations of L_ij, charge including o_I(ijk) and other charges. Because ${\tilde{\chi }}_{I\left(ijk\right)}^{ijk}>$ 0 for the nodes $l|l\in V$ , the original buffer capacity can not satisfy the demand for resolving the processing conflict. The only way is to increase the buffer capacity which can affect the o_ijk and o_I(ijk). It can change the buffer units which affect the o_ijk and o_I(ijk) by adjusting the processing machines of operations at steelmaking stage and refining stage. Then it may resolve the processing conflict between o_ijk and o_I(ijk) by the new buffer units, namely adjusting the processing time of operations at each stage. We can obtained the new processing conflict time (denoted as ${\left({\tilde{\chi }}_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}$ ) according to the formula (26) when the new buffer units are considered. If ${\left({\tilde{\chi }}_{I\left(ijk\right)}^{ijk}\right)}^{\text{'}}\le$ 0, the processing conflict can be resolved by the new buffer units. So, according to the classification of the processing conflict of scheduling plan (section 2.3), the abnormal condition is five-level the processing conflict ζ₂₅.

5. Industrial Application

5.1. Application Background

The Baosteel factory of China can produce 1000 kinds steel grade. There are three parallel converters of 250 t at steel making stage (4LD, 5LD, 6LD), three kinds of refining machines (5RH-1, 5RH-2, 3RH, LF-1, LF-2, IR_UT) at the refining stage, and three continuous casters (4CC, 5CC, 6CC). The number of refining processing ranging is from 1 to 4, and refining routes are more than 20. The computer systems have the level 2 computer system of process control and the level 3 computer system of region management. In the SMCC production process, operation time delay often occurs which may lead to planned casting break or processing conflict so that the initial scheduling plan becomes unrealizable. The rescheduling for the operation time delay is mainly relies on manual adjustment. The rescheduling methods are simply divided into four categories by the artificial experience: (1) if the operation time delay is within five minutes, then the scheduling plan is not adjusted. (2) If the operation time delay is between five minutes and ten minutes, then adjusting the starting time and the completion time of all operations. (3) If the operation time delay is between ten minutes and thirty minutes, then adjusting the processing machine, the starting time and the completion time of all operations. (4) If the operation time delay is greater than thirty minutes, then using other methods. Existing manual rescheduling methods with disturbances don’t analyze the influence degree of disturbances to the initial scheduling plan in detail, so the adjustment degree of initial scheduling plan is always too greater, which leads to the poor continuity and stability of initial scheduling plan.

5.2. Application Example

This paper takes the actual rescheduling problem in BaoSteel plant for example. Figure 3 shows the scheduling plan information at time t₁. The cast one is processed on 4CC and ${\mathrm{\Omega }}_1=\left\{L_{11},L_{12},L_{13},L_{14},L_{15},L_{16},L_{17}\right\}$ . The cast two is processed on 5CC and ${\mathrm{\Omega }}_2=\left\{L_{21},L_{22},L_{23},L_{24},L_{25},L_{26},L_{27},L_{28}\right\}$ . The cast three is processed on 6CC and ${\mathrm{\Omega }}_3=\left\{L_{31},L_{32},L_{33},L_{34},L_{35},L_{36}\right\}$ . The standard processing time and processing time range are as shown in Table 1.

Fig. 3.

The scheduling plan at time t₁.

Table 1.The processing time of operations at steelmaking and refining stage.

Machine	LD		RH		LF		IR_UT
	Standard	Interval	Standard	Interval	Standard	Interval	Standard	Interval
Processing time (m)	35	[32,38]	20	[15,30]	30	[22,35]	30	[25,35]

The processing time of charge on continuous casting machine is related to following facts: the total weight of the molten steel in charge, the average thickness of the slab, the average width of the odd strand slab and the dual slab of charge and pulling speed of the continuous casting machine. If the charge L_ij is the first chare in the cast, then the processing time of the charge L_ij on continuous casting machine is calculated as follows:   

$$p_{i,j,{\theta }_{ij}}=\frac{{\omega }_{ij}\times {10}^6}{{\alpha }_{ij}\times \frac{{\zeta }_{ij}^1+{\zeta }_{ij}^2}{2}\times 7.8\times 2\times \left({\upsilon }_{ij}-0.2\right)}$$(46)

where ω_ij is the total weight of the molten steel in charge L_ij, α_ij is the average thickness of the slab of charge L_ij and υ_ij is the pulling speed. The ${\zeta }_{ij}^1$ and ${\zeta }_{ij}^2$ are the average width of the odd strand slab and the dual slab of charge L_ij, respectively. If the charge L_ij is not the first chare in the cast, then the processing time of the charge L_ij on continuous casting machine is calculated as follows:   

$$\begin{array}{c}{p}_{i,j,{\theta }_{ij}}=\frac{{\omega }_{ij}\times {10}^6-{\gamma }_{ij}\times 31.2\times \left({\xi }_{ij}^1+{\xi }_{ij}^2+{\xi }_{i,j-1}^1+{\xi }_{i,j-1}^2\right)}{{\alpha }_{ij}\times \frac{\left({\xi }_{ij}^1+{\xi }_{ij}^2+{\xi }_{i,j-1}^1+{\xi }_{i,j-1}^2\right)}{4}\times 7.8\times 2\times \left({\upsilon }_{ij}-0.2\right)}\hfill \\ +\frac{8}{{\upsilon }_{ij}-0.2}\hfill \end{array}$$(47)

where ω_ij is the total weight of the molten steel in charge L_ij, α_ij is the average thickness of the slab of charge L_ij and υ_ij is the pulling speed. The ${\zeta }_{ij}^1$ and ${\zeta }_{ij}^2$ are the average width of the odd strand slab and the dual slab of charge L_ij, respectively. The ${\zeta }_{i,j-1}^1$ and ${\zeta }_{i,j-1}^2$ are the average width of the odd strand slab and the dual slab of charge $L_{i,j-1}$ , respectively. The pulling speed includes the minimum pulling speed, standard pulling speed, the maximum pulling speed. The processing time of charges on continuous casting machine is shown in Tables 2 and 3 according to the formula (46) and formula (47). The transportation time is as shown in Table 4.

Table 2.The processing time of charges at casting stage.

Operations	o113	o123	o133	o143	o153	o163	o173	o213	o223	o233	o243
Minimal processing time (m)	45	45	45	50	55	45	40	45	45	50	45
Standard processing time (m)	48	48	49	58	60	46	42	46	46	50	50
Maximum processing time (m)	60	60	61	70	75	60	60	60	60	65	65

Table 3.The processing time of charges at casting stage.

Operations	o253	o263	o273	o283	o313	o323	o333	o343	o353	o363
Minimal processing time (m)	45	50	45	45	60	60	60	60	60	60
Standard processing time (m)	50	55	52	54	74	74	76	76	76	77
Maximum processing time (m)	65	70	67	70	85	85	85	85	85	85

Table 4.The transportation time between machines.

Transportation time	4LD	5LD	6LD	5RH-1	5RH-2	3RH	LF-1	LF-2	IR_UT	4CC	5CC	6CC
4LD	0	12	13	10	10	13	15	15	13	16	15	14
5LD		0	12	15	15	13	15	15	13	16	15	14
6LD			0	15	15	13	15	15	13	16	16	15
5RH-1				0	18	21	20	18	23	22	25	25
5RH-2					0	21	20	18	23	22	25	25
3RH						0	20	21	18	25	22	22
LF-1							0	18	21	22	25	25
LF-2								0	23	22	25	25
IR_UT									0	25	22	22
4CC										0	16	16
5CC											0	13
6CC												0

The scheduling plan at time t₂(t₂ = 14:21) is shown in Fig. 4. The operation $o_{241}$ on 5LD is starting at time 14:21, that is $s_{241}^{*}=14:21$ . The abnormal condition prediction of the scheduling plan is made as follows according to the proposed method:

Fig. 4.

The scheduling plan at time t₂.

(1) Get event information: m = 2LD, $o=o_{241}$ , $\chi =1$ . $\tau =s_{241}^{*}-s_{241}^0=20>0$ because $s_{241}^0=14:01$ and $s_{241}^{*}=14:21$ , so y = y₁, and $Y=\left\{o_{241},2LD,y_1,20\right\}$ .

(2) The processing time of operations which are being processed or have been processed can not be changed. So, the operations not be processed are only considered when the reachability matrix is established. The correspondence relationship between nodes and operations is shows in Table 5. The κ denotes the idle state, and the μ indicates the interval state between adjacent operations of the same charge.

Table 5.The correspondence relationship between nodes and operations.

Node	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Operation	$o_{151}$	${\kappa }_{161}^{151}$	$o_{161}$	${\kappa }_{171}^{161}$	$o_{171}$	$o_{241}$	${\kappa }_{251}^{241}$	$o_{251}$	${\kappa }_{261}^{251}$	$o_{261}$	${\kappa }_{271}^{261}$	$o_{271}$	${\kappa }_{281}^{271}$	$o_{281}$	$o_{331}$	${\kappa }_{341}^{331}$	$o_{341}$	${\kappa }_{351}^{341}$	$o_{351}$	${\kappa }_{361}^{351}$
Node	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
Operation	$o_{361}$	${\mu }_{152}^{151}$	${\mu }_{162}^{161}$	${\mu }_{172}^{171}$	${\mu }_{242}^{241}$	${\mu }_{252}^{251}$	${\mu }_{262}^{261}$	${\mu }_{272}^{271}$	${\mu }_{282}^{281}$	${\mu }_{332}^{331}$	${\mu }_{342}^{341}$	${\mu }_{352}^{351}$	${\mu }_{362}^{361}$	$o_{142}$	${\kappa }_{152}^{142}$	$o_{152}$	${\kappa }_{162}^{152}$	$o_{162}$	${\kappa }_{172}^{162}$	$o_{172}$
Node	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
Operation	$o_{242}$	${\kappa }_{252}^{242}$	$o_{252}$	${\kappa }_{262}^{252}$	$o_{262}$	${\kappa }_{272}^{262}$	$o_{272}$	${\kappa }_{282}^{272}$	$o_{282}$	$o_{332}$	${\kappa }_{342}^{332}$	$o_{342}$	${\kappa }_{352}^{342}$	$o_{352}$	${\kappa }_{362}^{352}$	$o_{362}$	${\mu }_{143}^{142}$	${\mu }_{153}^{152}$	${\mu }_{163}^{162}$	${\mu }_{173}^{172}$
Node	61	62	63	64	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80
Operation	${\mu }_{243}^{242}$	${\mu }_{253}^{252}$	${\mu }_{263}^{262}$	${\mu }_{273}^{272}$	${\mu }_{283}^{282}$	${\mu }_{333}^{332}$	${\mu }_{343}^{342}$	${\mu }_{353}^{352}$	${\mu }_{363}^{362}$	$o_{133}$	$o_{143}$	$o_{153}$	$o_{163}$	$o_{173}$	$o_{233}$	$o_{243}$	$o_{253}$	$o_{263}$	$o_{273}$	$o_{283}$
Node	81	82	83	84	85
Operation	$o_{323}$	$o_{333}$	$o_{343}$	$o_{353}$	$o_{363}$

(3) It can be known that the o₂₄₁ is the corresponding operation for the node 6 according to the Table 5. The planned casting break time ${\gamma }_{3,4}^2$ between charge L₂₃ and charge L₂₄ is calculated as follows according to the formula (18): ${\gamma }_{3,4}^2=e_{241}^0+\mathrm{\Delta }t+p_{242}^0+u\left(z_{241,}^0{z}_{242}^0\right)+u\left(z_{242,}^0{z}_{2423}^0\right)-e_{233}^0=20$ . Because ${\gamma }_{3,4}^2>0$ , the planned casting break between L₂₃ and L₂₄ occurs. Because $p_{233}^{max}-p_{233}^0-{\gamma }_{3,4}^2=65-20-20=-5<0$ , let ${\mathit{\epsilon }}_l=0|l\notin V_{24}$ , then the planned casting break time ${\tilde{\gamma }}_{3,4}^2$ between L₂₃ and L₂₄ is calculated as follows according to the formula (27): ${\tilde{\gamma }}_{3,4}^2={\eta }_{76}-{\lambda }_{75}=12>0$ and $p_{233}^{max}-p_{233}^0-{\tilde{\gamma }}_{3,4}^2=3>0$ , so the abnormal condition of planned casting break is two-level planned casting break ζ₁₂.

(4) The processing conflict time ${\chi }_{251}^{241}$ of charge L₂₄ on 5LD is calculated as follows according to the formula (15): ${\chi }_{251}^{241}=min\left(e_{241}^0+\mathrm{\Delta }t,e_{251}^0\right)-max\left(s_{241}^0+\mathrm{\Delta }t,s_{251}^0\right)=5>0$ , so the processing conflict between L₂₄ and L₂₅ on 5LD occurs. Because ${\mathit{\epsilon }}_6-{\chi }_{251}^{241}=-2<0$ , ${\mathit{\epsilon }}_8-{\chi }_{251}^{241}=-2<0$ and ${\mathit{\epsilon }}_6+{\mathit{\epsilon }}_8-{\chi }_{251}^{241}=1>0$ , then the abnormal condition of processing conflict is two-level processing conflict ζ₂₂.

(5) The processing conflict time ${\chi }_{252}^{242}$ is calculated as follows according to the formula (15): ${\chi }_{252}^{242}=min\left(e_{241}^0+\mathrm{\Delta }t+u\left(z_{241}^0,z_{242}^0\right)+p_{242}^0,e_{252}^0\right)-max\left(e_{241}^0+\mathrm{\Delta }t+u\left(z_{241}^0,z_{242}^0\right),s_{252}^0\right)=-10<0$ , so there is no processing conflict between L₂₄ and L₂₅ on 5RH-2.

(6) The processing conflict time ${\chi }_{253}^{243}$ is calculated as follows according to the formula (15): ${\chi }_{253}^{243}=min\left(e_{241}^0+\mathrm{\Delta }t+u\left(z_{241}^0,z_{242}^0\right)+p_{242}^0+u\left(z_{242}^0,z_{243}^0\right)+p_{243}^0,e_{253}^0\right)-max\left(e_{241}^0+\mathrm{\Delta }t+u\left(z_{241}^0,z_{242}^0\right)+p_{242}^0+u\left(z_{242}^0,z_{243}^0\right),s_{253}^0\right)=20>0$ , so the processing conflict between L₂₄ and L₂₅ on 5CC occurs. ${\mathit{\epsilon }}_{76}-{\chi }_{253}^{243}=-15<0$ , ${\mathit{\epsilon }}_{77}-{\chi }_{251}^{241}=-15<0$ and ${\mathit{\epsilon }}_{76}+{\mathit{\epsilon }}_{77}-{\chi }_{253}^{243}=-10<0$ . When let ${\mathit{\epsilon }}_l=0|l\notin V_{24},|l\notin V_{25}$ , ${\tilde{\chi }}_{253}^{243}=-1<0$ , so the abnormal condition of processing conflict is three-level processing conflict ζ₂₃.

5.3. Application Effect

According to the above application example, it can be known that abnormal condition of scheduling plan with operation time delay disturbance are determined according the proposed method when the operation delay time of o₂₄₁ is twenty minutes.(1) The abnormal condition of planned casting break between the charge L₂₃ and the charge L₂₄ on 5CC is two-level planned casting break ζ₁₂, so the planned casting break between charge can be resolved synchronously by adjusting the processing time of L₂₃ at continuous casting stage and the processing time of L₂₄ at each stage.(2) The abnormal condition of processing conflict between the charge L₂₄ and the charge L₂₅ on 5LD is two-level processing conflict ζ₂₂, so the processing conflict between charge L₂₄ and L₂₅ can be resolved synchronously by adjusting the processing time of L₂₄ and L₂₅ on 5LD.(3) The abnormal condition of processing conflict between the charge L₂₄ and the charge L₂₅ on 5CC is three-level processing conflict ζ₂₃, so the processing conflict between charge L₂₄ and L₂₅ can be resolved synchronously by adjusting the processing time of L₂₄ and L₂₅ on each stage. Because the operation delay time of o₂₄₁ is twenty minutes which is between ten minutes and thirty minutes, the abnormal condition of the scheduling plan will be resolved by adjusting the processing machine, the starting time and the completion time of all operations according to the manual method. The proposed method can solve the abnormal condition only by adjusting the processing time of some charges, which can maintain the good continuity and stability.

6. Conclusions

In the steelmaking and continuous casting (SMCC) production process, operation time delay often occurs which may lead to planned casting break or processing conflict so that the initial scheduling plan becomes unrealizable. Existing rescheduling methods don’t analyze the influence degree of disturbances to the initial scheduling plan in detail, so the adjustment degree of initial scheduling plan is always too greater, which leads to the poor continuity and stability of initial scheduling plan. In this paper, the relation among operation time delay, planned casting break and processing conflict is firstly deeply analyzed. Then a novel prediction method for abnormal condition of scheduling plan with operation time delay disturbance in SMCC production process is proposed including disturbance identification of operating time delay based on event-driven mechanism, analysis on charges based on reachability matrix, analysis on influence degree of disturbance and abnormal condition decision of initial scheduling plan. The real-time application shows that the proposed prediction method can timely and accurately predict the abnormal condition of the scheduling plan with operation time delay disturbance, which can only adjust the affected charges that must to be rescheduled in the initial scheduling plan and reduce the frequency of complete rescheduling. The initial scheduling plan can also maintain the good continuity and stability.

Acknowledgements

This research is partly supported by National Basic Research Program of China (2009CB32060l), National Natural Science Foundation of China (61104174, 61174187); 111 Project (No. B08015), Basic Scientific Research Foundation of Northeast University under Grant N110208001, Starting Foundation of Northeast University under Grant 29321006.

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