ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Regular Article
A Prediction Method for Abnormal Condition of Scheduling Plan with Operation Time Delay in Steelmaking and Continuous Casting Production Process
Shengping Yu
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JOURNALS OPEN ACCESS FULL-TEXT HTML

2013 Volume 53 Issue 6 Pages 1028-1041

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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 Lij for the jth charge of the ith cast, and i = 1,..., N, j = 1,..., ni, where N indicates the total cast number and ni represents the total charge number of the ith cast. Each charge Lij consists of θij operations (oij1,..., oi,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 oijk (k = 1,..., θij). In this paper, we denote mgb as the bth machine in the gth machine group. The hg 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 Δt= 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 oijk. 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 Δt= 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 Li,j–1 and Lij, 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 Li,j–1 and Lij only by adjusting the processing time of Li,j–1 at continuous casting stage, or only by adjusting the processing time of Lij at each stage.

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

Three-level planned casting break. It can not resolve the planned casting break between charge Li,j–1 and Lij synchronously by adjusting the processing time of Li,j–1 at continuous casting stage and the processing time of Lij at each stage. But the planned casting break between charge Li,j–1 and Lij can be resolved synchronously by adjusting the processing time of Lij1 (j1j) at each stage.

Four-level planned casting break. It can not resolve the planned casting break between charge Li,j–1 and Lij synchronously by adjusting the processing time of Lij1 (j1j) at each stage. But the planned casting break between charge Li,j–1 and Lij can be resolved synchronously by adjusting the processing machines of Lij1 (j1j) at steelmaking stage and at refining stage, and by adjusting the processing time of Lij1 (j1j) 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 Li1j1 and Li2j2 on machine mgb, 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 Li1j1 and Li2j2 only by adjusting the processing time of Li1j1 on machine mgb, or only by adjusting the processing time of Li2j2 on machine mgb.

Two-level processing conflict. It can not resolve the processing conflict between charge Li1j1 and Li2j2 only by adjusting the processing time of Li1j1 on machine mgb, or only by adjusting the processing time of Li2j2 on machine mgb. But the processing conflict between charge Li1j1 and Li2j2 can be resolved synchronously by adjusting the processing time of Li1j1 and Li2j2.

Three-level processing conflict. It can not resolve the processing conflict between charge Li1j1 and Li2j2 synchronously by adjusting the processing time of Li1j1 and Li2j2 on machine mgb. But the processing conflict between charge Li1j1 and Li2j2 can be resolved synchronously by adjusting the processing time of Li1j1 and Li2j2 on each stage.

Four-level processing conflict. It can not resolve the processing conflict between charge Li1j1 and Li2j2 synchronously by adjusting the processing time of Li1j1 and Li2j2 on each stage. But the processing conflict between charge Li1j1 and Li2j2 can be resolved synchronously by adjusting the processing time of Li1j1, Li2j2 and other charges on each stage.

Five-level processing conflict. It can not resolve the processing conflict between charge Li1j1 and Li2j2 synchronously by adjusting the processing time of Li1j1, Li2j2 and other charges on each stage. But the processing conflict between charge Li1j1 and Li2j2 may be resolved synchronously by adjusting the processing machines and processing time of Li1j1, Li2j2 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(mg1b1,mg2b2) between machine mg1b1 and machine mg2b2. 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( m,o,χ,t, s ijk 0 , e ijk 0 | o ijk O ) (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 oijk from O, respectively. If operation oijk is not processed, then δ ijk 1 = 0. If operation oijk is being processed, then δ ijk 1 = 1. If operation oijk has not finished being processed, then δ ijk 2 = 0. If operation oijk has been processed, then δ 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 ={ z ijk 0 , s ijk 0 , e ijk 0 | o ijk L ij , L ij Ω } , where Ω is the set of unfinished casting charges.

Step2: Initialize the operations processing status: ∀oijk ∈ O, δ ijk 1 = 0, δ 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 oi1j1k1 from O which oi1j1k1 is equal to operation o. If χ = 1, then go to the next step, else go to Step5.

Step4: Let δ 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 < θij, τ > 0 then y = y3.

Step5: Let δ 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 < θij, 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 oijk and then it can be processed on the next operation oi,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= c l 1 l 2 l 1 , l 2 =1,,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 cl1l2 = 1, else cl1l2 = 0. For example, the adjacency matrix C in Fig. 2(b) is as follows:   
C=[ 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 ] (5)

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

R= r l 1 l 2 l 1 , l 2 =1,,N (6)

If there exists directed path from the node l1 to the node l2, then rl1l2 = 1, else rl1l2 = 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=[ 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 ] (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 oijk1 delays Δt minutes, that is Δt = s ij k 1 * - s ij k 1 0 , recalculate the starting time sijk1 and the completion time eijk1 of the operation oijk1 as follows:   

s ij k 1 = s ij k 1 * = s ij k 1 0 +Δt (8)
  
e ij k 1 = s ij k 1 + p ij k 1 0 = e ij k 1 0 +Δt (9)
where p ij k 1 0 is the processing time of operation oijk1 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 sijk1 and the completion time eijk1 of the operation oijk1 as follows:   
s ij k 1 = s ij k 1 0 (10)
  
e ij k 1 = e ij k 1 0 +Δt (11)
The new starting time sijk and the new completion time eijk of the subsequent operations of oijk1 are recalculated as follows:   
s ijk = e i,j,k-1 +u( z i,j,k-1 0 , z ijk 0 ) k= k 1 +1,, θ ij (12)
  
e ijk = s ijk + p ijk 0 k= k 1 +1,, θ ij (13)
where z i,j,k-1 0 and z ijk 0 are the processing machine of the operation oi,j,k–1 and the processing machine of the operation oijk, respectively. The u( z i,j,k-1 0 , z ijk 0 ) is the standard transportation time between oi,j,k–1 and oijk. We define the oI(ijk) as the subsequent operation of the operation oijk on machine z ijk 0 . The χ I( ijk ) ijk denotes the processing conflict time as follows:   
χ I( ijk ) ijk =min( e ijk , e I( ijk ) 0 ) -max( s ijk , s I( ijk ) 0 ) k= k 1 ,, θ ij (14)
The formula (14) can be rewritten as follows according to the formula (8) to formula (13):   
χ I( ijk ) ijk =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) k= k 1 ,, θ ij (15)
The formula (15) is the relational model between the processing conflict time between operation oijk and other operation o ijk ( k= k 1 ,, θ ij ) and the operation delay time Δt of operation oijk1 when the operation delay time of oijk1 is Δt. If χ I( ijk ) ijk ≤ 0, then the processing conflict does not occur. If χ I( ijk ) ijk > 0, then the processing conflict occurs and the conflict time is χ I( ijk ) ijk .

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

When the delay time of operation oijk1 is Δt, then the new starting time of oi,j,θij can be recalculated as follows according to the formula (8) to formula (13):   

s i,j, θ ij = e ij k 1 0 +Δt+ k 2 = k 1 +1 θ ij -1 p ij k 2 0 + k 3 = k 1 θ ij -1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) (16)
We define γ j-1,j i as the planned casting break time between charge Li,j–1 and charge Lij:   
γ j-1,j i = s i,j, θ ij - e i,j-1, θ i,j-1 0 (17)
The formula (17) can be rewritten as follows according to the formula (16):   
γ j-1,j i = e ij k 1 0 +Δt+ k 2 = k 1 +1 θ ij -1 p ij k 2 0 + k 3 = k 1 θ ij -1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) - e i,j-1, θ 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 oijk1 when the operation delay time of oijk1 is Δt. If γ j-1,j i 0 , then the planned casting break does not occur. If γ j-1,j i > 0, then the planned casting break occurs and the planned casting break time is γ j-1,j i .

4.3.3. Buffer Units of Scheduling System in SMCC Production Process

When the operation delay time of oijk1 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 ΔtT. If ΔtT < 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 ΔtT. If ΔtT < 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 oijk can be adjusted in [ p ijk min , p ijk max ] . Supposing that the processing time of oijk in the initial scheduling plan is p ijk 0 . If Δt-( p ijk 0 - p ijk min ) <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 oijk and oi,j,k+1 is u ijk 0 ( z ijk 0 , z i,j,k+1 0 ) . If Δt-( u 0 ( z ijk 0 , z i,j,k+1 0 ) -u( z ijk 0 , z i,j,k+1 0 ) ) <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:   

E= ε l l=1,,N (19)
where εl denotes the buffer time of the node l.

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

ε l =T (20)

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

ε 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   

ε l = u 0 ( z ijk 0 , z i,j,k+1 0 ) -u( z ijk 0 , z i,j,k+1 0 ) (22)
where u 0 ( z ijk 0 , z i,j,k+1 0 ) is the interval time between oijk and oi,j,k+1, and u( z ijk 0 , z i,j,k+1 0 ) is the standard transportation time between oijk and oi,j,k+1.

If εl > 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 – εl. If εl = 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 oijk1 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 oijk1 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:   

T 1 ( l 0 ,Δt ) = η l l=1,,N (23)
where the ηl denotes the final delay time of the starting time of the node l affected by the delay time Δt of the node l0. If rl0l = 0, then ηl = 0. If rl0l = 1 and there exists B(B ≥ 1) directed paths from the node l0 to the node l, then the ηl is equal to the maximum delay time of the starting time of all nodes in the directed paths:   
η l = max b=1,,B ( Δt- l 1 P b ( l 0 ,l ) l 1 l l 2 P b ( l 0 ,l ) \ l 2 l c l 1 l 2 ε l 1 ) r l 0 l =1 (24)
where Pb(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):   

T 2 ( l 0 ,Δt ) = λ l l=1,,N (25)
where the λl indicates the final delay time of the completion time of the node l affected by the delay time Δt of the node l0. If rl0l = 0, then λl = 0. If rl0l = 1, then λl = ηlεl.

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:   

χ ˜ I( ijk ) ijk =min( e ijk 0 + λ l 1 , e I( ijk ) 0 + λ l 2 ) -max( s ijk 0 + η l 1 , s I( ijk ) 0 + η l 2 ) (26)
where the oijk1 is the corresponding operation for the node l0, the oijk is the corresponding operation for the node l1 and the oI(ijk) is the corresponding operation for the node l2. If χ ˜ I( ijk ) ijk ≤ 0, then there is no processing conflict time between oijk and oI(ijk) through the buffer units. If χ ˜ I( ijk ) ijk > 0, it means that there is still processing conflict time between oijk and oI(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:   

γ ˜ j-1,j i =( s i,j, θ ij 0 + η l 2 ) -( e i,j-1, θ i,j-1 0 + λ l 1 ) = η l 2 - λ l 1 (27)
where the oi,j,θij is the corresponding operation for the node l2 and the oi,j–1,θi,j–1 is the corresponding operation for the node l1. If γ ˜ j-1,j i 0 , there is no planned casting break time between oi,j–1,θi,j–1 and oi,j,θij through the buffer units. If γ ˜ j-1,j i >0 , there is still planned casting break time between oi,j–1,θi,j–1 and oi,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 Lij) delay Δt, if γ j-1,j i >0 , then there is planned casting break time between Li,j–1 and Lij. The oi,j–1,θi,j–1 is the corresponding operation for the node l1 and the oi,j,θij is the corresponding operation for the node l2. We define the Vij as the all nodes set for charge Lij. The planned casting break decision is made as follows:

(1) If p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i 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 oijk1 the charge Lij) delay Δt, according to the formula (18), γ j-1,j i = e ij k 1 0 +Δt+ k 2 = k 1 +1 θ ij -1 p ij k 2 0 + k 3 = k 1 θ ij -1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) - e i,j-1, θ i,j-1 0 . e i,j-1, θ i,j-1 0 is the initial completion time of operation oi,j–1,θi,j–1 and the initial processing time of operation oi,j–1,θi,j–1 is p i,j-1, θ i,j-1 0 . If the processing time of operation oi,j–1,θi,j–1 is adjusted to p i,j-1, θ i,j-1 max , then the new completion time ei,j–1,θi,j–1 of operation oi,j–1,θi,j–1 can be recalculated as follows: e i,j-1, θ i,j-1 = e i,j-1, θ i,j-1 0 + p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 . Then according to the formula (18), the new planned casting break time (denoted as ( γ j-1,j i ) ' ) can be recalculated as follows:   

( γ j-1,j i ) = e ij k 1 0 +Δt+ k 2 = k 1 +1 θ ij -1 p ij k 2 0 + k 3 = k 1 θ ij -1 u( z ij k 3 0 , z i,j, k 3 +1 0 )    - e i,j-1, θ i,j-1 = e ij k 1 0 +Δt+ k 2 = k 1 +1 θ ij -1 p ij k 2 0 + k 3 = k 1 θ ij -1 u( z ij k 3 0 , z i,j, k 3 +1 0 )    - e i,j-1, θ i,j-1 0 - p i,j-1, θ i,j-1 max + p i,j-1, θ i,j-1 0 = γ j-1,j i - p i,j-1, θ i,j-1 max + p i,j-1, θ i,j-1 0 (28)
If p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i 0 , then ( γ j-1,j i ) ' 0 . It can be seen that the planned casting break between oi,j–1,θi,j–1 and oi,j,θij can be resolved only by adjusting the processing time of Li,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, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i <0 , then for the nodes l| lV,l V ij , let εl = 0. If γ ˜ j-1,j i 0 , then the abnormal condition of scheduling plan is one-level planned casting break ζ11.

Proof. If p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i <0 , then ( γ j-1,j i ) ' > 0 according to the formula (28). It can been seen that the planned casting break between oi,j–1,θi,j–1 and oi,j,θij can not be resolved only by adjusting the processing time of Li,j–1 at continuous casting stage. For the nodes l| lV,l V ij , let εl = 0, which means that only nodes belonging to Vij have buffer time. According to the formula (27), γ ˜ j-1,j i =( s i,j, θ ij 0 + η l 2 ) -( e i,j-1, θ i,j-1 0 + λ l 1 ) = η l 2 - λ l 1 . If γ ˜ j-1,j i 0 , then It can be seen that the planned casting break between oi,j–1,θi,j–1 and oi,j,θij can be resolved only by adjusting the processing time of Lij 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, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i <0 , then for the nodes l| lV,l V ij , let εl = 0. If γ ˜ j-1,j i >0 , but p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ ˜ j-1,j i 0, then the abnormal condition of scheduling plan is two-level planned casting break ζ12.

Proof. If p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i < 0, then ( γ j-1,j i ) ' > 0, it can been seen that the planned casting break between oi,j–1,θi,j–1 and oi,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| lV,l V ij , let εl = 0. According to the formula (27), γ ˜ j-1,j i =( s i,j, θ ij 0 + η l 2 ) -( e i,j-1, θ i,j-1 0 + λ l 1 ) = η l 2 - λ l 1 . If γ ˜ j-1,j i > 0, then It can be seen that the planned casting break between oi,j–1,θi,j–1 and oi,j,θij can not be resolved only by adjusting the processing time of Lij at each stage. If the processing time of operation oi,j–1,θi,j–1 is adjusted to p i,j-1, θ i,j-1 max , then the new completion time ei,j–1,θi,j–1 of operation oi,j–1,θi,j–1 can be recalculated as follows: e i,j-1, θ i,j-1 = e i,j-1, θ i,j-1 0 + p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 . Then according to the formula (27), the new planned casting break time (denoted as ( γ ˜ j-1,j i ) ' ) can be recalculated as follows:   

( γ ˜ j-1,j i ) ' =( s i,j, θ ij 0 + η l 2 )    -   ( e i,j-1, θ i,j-1 0 + λ l 1 ) =( s i,j, θ ij 0 + η l 2 )    -   ( e i,j-1, θ i,j-1 0 + p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 + λ l 1 ) =( s i,j, θ ij 0 + η l 2 )    -   ( e i,j-1, θ i,j-1 0 + λ l 1 ) - p i,j-1, θ i,j-1 max + p i,j-1, θ i,j-1 0 = γ ˜ j-1,j i - p i,j-1, θ i,j-1 max + p i,j-1, θ i,j-1 0 (29)
If p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ ˜ j-1,j i 0, then ( γ ˜ j-1,j i ) ' 0. It can be seen that the planned casting break between oi,j–1,θi,j–1 and oi,j,θij can be resolved synchronously by adjusting the processing time of Li,j–1 at continuous casting stage and the processing time of Lij 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 εl = 0 for the nodes l|lV, lVij, p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i < 0, γ ˜ j-1,j i > 0, p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ ˜ j-1,j i < 0. If let εl ≠ 0 for the nodes l| lV , γ ˜ j-1,j i 0, then the abnormal condition of scheduling plan is three-level planned casting break ζ13.

Proof. For the nodes l| lV,l V ij , let εl = 0. If p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ j-1,j i < 0, γ ˜ j-1,j i > 0, p i,j-1, θ i,j-1 max - p i,j-1, θ i,j-1 0 - γ ˜ j-1,j i < 0, it means that the planned casting break between oi,j–1,θi,j–1 and oi,j,θij can not be resolved synchronously by adjusting the processing time of Li,j–1 at continuous casting stage and the processing time of Lij at each stage according to the above analysis. If let εl ≠ 0 for the nodes l| lV , then all nodes have duffer time. If γ ˜ j-1,j i 0, then it means that the planned casting break between charge Li,j–1 and Lij can be resolved synchronously by adjusting the processing time of Lij1(j1j) 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| lV , if γ ˜ j-1,j i > 0, then the abnormal condition of scheduling plan is four-level planned casting break ζ14.

Proof. If let εl ≠ 0 for the nodes l| lV , then all nodes have duffer time. If γ ˜ j-1,j i >0 , then it can be known that the planned casting break between charge Li,j–1 and Lij can not be resolved synchronously by adjusting the processing time of Lij1(j1j) 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 Lij1(j1j) at steelmaking stage and refining stage and by adjusting the processing time of Lij1(j1j) 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 Lij) delay Δt, if χ I( ijk ) ijk > 0, then there is processing conflict time between oijk and oI(ijk). The oijk is the corresponding operation for the node l1 and the oI(ijk) is the corresponding operation for the node l2. We define the V1 as the all nodes set for charge Lij and the V2 as the all nodes set for charge including oI(ijk). The processing conflict decision is made as follows:

(1) If ε l 1 - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ21.

Proof. The operation oI(ijk) is the subsequent operation of the operation oijk on machine in the initial scheduling plan. When the operation time delay occurs, the processing conflict time χ I( ijk ) ijk can be calculated according to the formula (15). According to the new processing sequence between oijk and oI(ijk), the processing conflict can be divided into two cases.

■ Case 1: oijk is also processed before oI(ijk) in the new scheduling plan. In this case, the completion time of oijk is earlier than the completion time of oI(ijk) which means e ijk < e I( ijk ) 0 . So, it can be known that e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) < e I( ijk ) 0 . The processing conflict between oijk and oI(ijk) may be resolved by adjusting the processing time of oijk. According to the formula (21), ε l 1 = p ijk 0 - p ijk min . If the starting time of operation oijk is not changed and the processing time of oijk is adjusted to p ijk min , then according to the formula (15), the new processing conflict time (denoted as ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k-1 p ij k 2 0 + p ijk min + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) =min( e ij k 1 + k 2 = k 1 +1 k-1 p ij k 2 0 + p ijk 0 - ε l 1 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) (30)
Because e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) < e I( ijk ) 0 , e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) - ε l l < e I( ijk ) 0 . So the minimum value of first term in the formula (30) is e ij k 1 + k 2 = k 1 +1 k-1 p ij k 2 0 + p ijk 0 - ε l 1 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . Then the formula (30) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =( min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 )    - ε l 1 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - ε l 1 = χ I( ijk ) ijk - ε l 1 (31)
If ε l 1 - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) only by adjusting the processing time of oijk. 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: oijk is processed after oI(ijk) in the new scheduling plan. In this case, the starting time of oijk is later than the starting time of oI(ijk) which means s ijk > s I( ijk ) 0 . So, it can be known that s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) > s I( ijk ) 0 . According to the formula (21), ε l 1 = p ijk 0 - p ijk min . If the completion time of operation oijk is not changed and the processing time of oijk is adjusted to p ijk min , then according to the formula (15), the new processing conflict time (denoted as ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) + ε l 1 , s I( ijk ) 0 ) (32)
Because s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) > s I( ijk ) 0 , s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) + ε l 1 > s I( ijk ) 0 . So the maximum value of second term in the formula (32) is s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) + ε l 1 . Then the formula (32) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -( max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) + ε l 1 ) =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - ε l 1 = χ I( ijk ) ijk - ε l 1 (33)
If ε l 1 - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) only by adjusting the processing time of oijk. 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 ε l 1 - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ21.

(2) If ε l 2 - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ21.

Proof. The operation oI(ijk) is the subsequent operation of the operation oijk on machine in the initial scheduling plan. When the operation time delay occurs, the processing conflict time χ I( ijk ) ijk can be calculated according to the formula (15). According to the new processing sequence between oijk and oI(ijk), the processing conflict can be divided into two cases.

■ Case 1: oijk is also processed before oI(ijk) in the new scheduling plan. In this case, the starting time of oI(ijk) is later than the starting time of oijk which means s I( ijk ) 0 > s ijk . So, it can be known that s I( ijk ) 0 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . The processing conflict between oijk and oI(ijk) may be resolved by adjusting the processing time of oI(ijk). According to the formula (21), ε l 2 = p I( ijk ) 0 - p I( ijk ) min . If the completion time of operation oI(ijk) is not changed and the processing time of oI(ijk) is adjusted to p I( ijk ) min , then according to the formula (15), the new processing conflict time (denoted as ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 + ε l 2 ) (34)
Because s I( ijk ) 0 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 + ε l 2 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . So the maximum value of second term in the formula (34) is s I( ijk ) 0 + ε l 2 . Then the formula (34) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -( max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    + ε l 2 ) =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - ε l 2 = χ I( ijk ) ijk -    ε l 2 (35)
If ε l 2 - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) only by adjusting the processing time of oI(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: oijk is processed after oI(ijk) in the new scheduling plan. In this case, the completion time of oI(ijk) is earlier than the completion time of oijk which means e I( ijk ) 0 < e ijk . So, it can be known that e I( ijk ) 0 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . The processing conflict between oijk and oI(ijk) may be resolved by adjusting the processing time of oI(ijk). According to the formula (21), ε l 2 = p I( ijk ) 0 - p I( ijk ) min . If the starting time of operation oI(ijk) is not changed and the processing time of oI(ijk) is adjusted to p I( ijk ) min , then according to the formula (15), the new processing conflict time (denoted as ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 - ε l 2 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) (36)
Because e I( ijk ) 0 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 - ε l 2 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . So the minimum value of first term in the formula (36) is e I( ijk ) 0 - ε l 2 . Then the formula (36) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =( min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 )    - ε l 2 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - ε l 2 = χ I( ijk ) ijk - ε l 2 (37)
If ε l 2 - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) only by adjusting the processing time of oI(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 ε l 2 - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is one-level processing conflict ζ21.

(3) If ε l 1 - χ I( ijk ) ijk < 0, ε l 2 - χ I( ijk ) ijk < 0, ε l 1 + ε l 2 - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is two-level processing conflict ζ22.

Proof. The operation oI(ijk) is the subsequent operation of the operation oijk on machine in the initial scheduling plan. When the operation time delay occurs, the processing conflict time χ I( ijk ) ijk can be calculated according to the formula (15). According to above analysis, if ε l 1 - χ I( ijk ) ijk < 0, it can not resolve the processing conflict between oijk and oI(ijk) only by adjusting the processing time of oijk. If ε l 2 - χ I( ijk ) ijk < 0, it can not resolve the processing conflict between oijk and oI(ijk) only by adjusting the processing time of oI(ijk). According to the new processing sequence between oijk and oI(ijk), the processing conflict can be divided into two cases.

■ Case 1: oijk is also processed before oI(ijk) in the new scheduling plan. In this case, the completion time of oijk is earlier than the completion time of oI(ijk) which means e ijk < e I( ijk ) 0 . So, it can be known that e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) < e I( ijk ) 0 . The starting time of oI(ijk) is later than the starting time of oijk which means s I( ijk ) 0 > s ijk . So, it can be known that s I( ijk ) 0 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . The processing conflict may be resolved synchronously by adjusting the processing time of oijk and oI(ijk). According to the formula (21), ε l 1 = p ijk 0 - p ijk min and ε l 2 = p I( ijk ) 0 - p I( ijk ) min . If the starting time of operation oijk is not changed and the processing time of oijk is adjusted to, and if the completion time of operation oI(ijk) is not changed and the processing time of oI(ijk) is adjusted to p I( ijk ) min , then according to the formula (15), the new processing conflict time (denoted as ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k-1 p ij k 2 0 + p ijk 0 - ε l 1 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 + ε l 2 ) (38)
Because e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) < e I( ijk ) 0 and s I( ijk ) 0 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) - ε l 1 < e I( ijk ) 0 and s I( ijk ) 0 + ε l 2 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . Then the formula (38) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =( min( e ij k 1 + k 2 = k 1 +1 k-1 p ij k 2 0 + p ijk 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 )    - ε l 1 ) -( max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    + ε l 2 ) =min( e ij k 1 + k 2 = k 1 +1 k-1 p ij k 2 0 + p ijk 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - ε l 1 - ε l 2 = χ I( ijk ) ijk - ε l 1 - ε l 2 (39)
If ε l 1 + ε l 2 - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) synchronously by adjusting the processing time of oijk and oI(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: oijk is processed after oI(ijk) in the new scheduling plan. It can be known that s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) > s I( ijk ) 0 and e I( ijk ) 0 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . If the completion time of operation oijk is not changed and the processing time of oijk is adjusted to p ijk min , and if the starting time of operation oI(ijk) is not changed and the processing time of oI(ijk) is adjusted to p I( ijk ) min , then according to the formula (15), the new processing conflict time (denoted as ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 - ε l 2 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) + ε l 1 , s I( ijk ) 0 ) (40)
Because e I( ijk ) 0 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) and s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) > s I( ijk ) 0 , e I( ijk ) 0 - ε l 2 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) and s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) + ε l 1 > s I( ijk ) 0 . Then the formula (40) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =( min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 )    - ε l 2 ) -( max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    + ε l 1 ) =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 ) - ε l 1 - ε l 2 = χ I( ijk ) ijk - ε l 1 - ε l 2 (41)
If ε l 1 + ε l 2 - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) synchronously by adjusting the processing time of oijk and oI(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 ε l 1 - χ I( ijk ) ijk < 0, ε l 2 - χ I( ijk ) ijk < 0, ε l 1 + ε l 2 - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is two-level processing conflict ζ22.

(4) If ε l 1 + ε l 2 - χ I( ijk ) ijk < 0, then let εl = 0 for the nodes l| lV,l V 1 , l V 2 . If l 1 * V 1 * ε l 1 * + l 2 * V 2 * ε l 2 * - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is three-level processing conflict ζ23.

Proof. According to above analysis, if ε l 1 + ε l 2 - χ I( ijk ) ijk <0 , It can not resolve the processing conflict between oijk and oI(ijk) synchronously by adjusting the processing time of oijk and oI(ijk). We define oI(ijk) is the k1th operation of the charge L i 1 j 1 , so the oI(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 * k of Lij and the V 2 * as the all nodes set for o i 1 j 1 k 2 * | k 2 * 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 oijk and oI(ijk), the processing conflict can be divided into two cases.

■ Case 1: oijk is also processed before oI(ijk) in the new scheduling plan. It can be known that e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) < e I( ijk ) 0 and s I( ijk ) 0 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . If the starting time of oijk1 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 * -{ o i 1 j 1 k 1 } 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 ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 - l 1 * V 1 * ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 + l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 - l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 + l 2 * V 2 * ε l 2 * ) (42)
Because e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) < e I( ijk ) 0 and s I( ijk ) 0 > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 - l 1 * V 1 * ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 )          < e I( ijk ) 0 + l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * and s I( ijk ) 0 + l 2 * V 2 * ε l 2 * > s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 - l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . Then the formula (42) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =( min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 + l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * )    - l 1 * V 1 * ε l 1 * ) -( max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 - l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    + l 2 * V 2 * ε l 2 * ) =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 + l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 - l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - l 1 * V 1 * ε l 1 * - l 2 * V 2 * ε l 2 * =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - l 1 * V 1 * ε l 1 * - l 2 * V 2 * ε l 2 * = χ I( ijk ) ijk - l 1 * V 1 * ε l 1 * - l 2 * V 2 * ε l 2 * (43)
If l 1 * V 1 * ε l 1 * + l 2 * V 2 * ε l 2 * - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) synchronously by adjusting the processing time of operations of Lij 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: oijk is processed after oI(ijk) in the new scheduling plan. It can be known that s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) > s I( ijk ) 0 and e I( ijk ) 0 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . If the starting time of oijk1 is not changed, the processing time of operation o ij k 4 * in V 1 * -{ o ijk } is adjusted to p ij k 4 * max , the processing time of operation oijk 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 ( χ I( ijk ) ijk ) ' ) can be recalculated as follows (k = k1,...,θij):   

( χ I( ijk ) ijk ) ' =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 - l 2 * V 2 * ε l 2 * ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + l 1 * V 1 * ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 - l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * ) (44)
Because s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) > s I( ijk ) 0 and e I( ijk ) 0 < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + l 1 * V 1 * ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) > s I( ijk ) 0 - l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * and e I( ijk ) 0 - l 2 * V 2 * ε l 2 * < e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) . Then the formula (44) can be rewritten as follows:   
( χ I( ijk ) ijk ) ' =( min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 )    - l 2 * V 2 * ε l 2 * ) -( max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 - l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * )    + l 1 * V 1 * ε l 1 * ) =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + l 1 * V 1 * -{ o ijk } ε l 1 * + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 - l 2 * V 2 * -{ o i 1 j 1 k 1 } ε l 2 * )    - l 1 * V 1 * ε l 1 * - l 2 * V 2 * ε l 2 * =min( e ij k 1 + k 2 = k 1 +1 k p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , e I( ijk ) 0 ) -max( s ij k 1 + k 2 = k 1 k-1 p ij k 2 0 + k 3 = k 1 k-1 u( z ij k 3 0 , z i,j, k 3 +1 0 ) , s I( ijk ) 0 )    - l 1 * V 1 * ε l 1 * - l 2 * V 2 * ε l 2 * = χ I( ijk ) ijk - l 1 * V 1 * ε l 1 * - l 2 * V 2 * ε l 2 * (45)
If l 1 * V 1 * ε l 1 * + l 2 * V 2 * ε l 2 * - χ I( ijk ) ijk 0, then ( χ I( ijk ) ijk ) ' 0. It can resolve the processing conflict between oijk and oI(ijk) synchronously by adjusting the processing time of operations of Lij 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 ε l 1 + ε l 2 - χ I( ijk ) ijk < 0, then let εl = 0 for the nodes l| lV,l V 1 , l V 2 . If l 1 * V 1 * ε l 1 * + l 2 * V 2 * ε l 2 * - χ I( ijk ) ijk 0, then the abnormal condition of scheduling plan is three-level processing conflict ζ23.

(5) If l 1 * V 1 * ε l 1 * + l 2 * V 2 * ε l 2 * - χ I( ijk ) ijk < 0 when εl = 0 for the nodes l| lV,l V 1 , l V 2 and χ ˜ I( ijk ) ijk 0 for the nodes l| lV , 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 oijk and oI(ijk) synchronously by adjusting the processing time of operations of Lij and the charge including o I(ijk) if l 1 * V 1 * ε l 1 * + l 2 * V 2 * ε l 2 * - χ I( ijk ) ijk < 0. The processing conflict may be resolved synchronously by adjusting the processing time of operations of Lij, the charge including oI(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 χ ˜ I( ijk ) 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 oijk and oI(ijk) synchronously by adjusting the processing time of operations of Lij, charge including oI(ijk) and other charges if χ ˜ I( ijk ) ijk 0 when εl ≠ 0 for the nodes l| lV . 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| lV , if χ ˜ I( ijk ) ijk > 0, then the abnormal condition of scheduling plan is five-level processing conflict ζ25.

Proof. According to above analysis, if χ ˜ I( ijk ) ijk > 0 when εl ≠ 0 for the nodes l| lV , it can not resolve the processing conflict between oijk and oI(ijk) only by adjusting the processing time of operations of Lij, charge including oI(ijk) and other charges. Because χ ˜ I( ijk ) ijk > 0 for the nodes l| lV , 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 oijk and oI(ijk). It can change the buffer units which affect the oijk and oI(ijk) by adjusting the processing machines of operations at steelmaking stage and refining stage. Then it may resolve the processing conflict between oijk and oI(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 ( χ ˜ I( ijk ) ijk ) ' ) according to the formula (26) when the new buffer units are considered. If ( χ ˜ I( ijk ) ijk ) ' 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 Ω 1 ={ L 11 , L 12 , L 13 , L 14 , L 15 , L 16 , L 17 } . The cast two is processed on 5CC and Ω 2 ={ L 21 , L 22 , L 23 , L 24 , L 25 , L 26 , L 27 , L 28 } . The cast three is processed on 6CC and Ω 3 ={ L 31 , L 32 , L 33 , L 34 , L 35 , L 36 } . 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.
MachineLDRHLFIR_UT
StandardIntervalStandardIntervalStandardIntervalStandardInterval
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 Lij is the first chare in the cast, then the processing time of the charge Lij on continuous casting machine is calculated as follows:   

p i,j, θ ij = ω ij × 10 6 α ij × ζ ij 1 + ζ ij 2 2 ×7.8×2×( υ ij -0.2 ) (46)
where ωij is the total weight of the molten steel in charge Lij, αij is the average thickness of the slab of charge Lij and υij is the pulling speed. The ζ ij 1 and ζ ij 2 are the average width of the odd strand slab and the dual slab of charge Lij, respectively. If the charge Lij is not the first chare in the cast, then the processing time of the charge Lij on continuous casting machine is calculated as follows:   
p i,j, θ ij = ω ij × 10 6 - γ ij ×31.2×( ξ ij 1 + ξ ij 2 + ξ i,j-1 1 + ξ i,j-1 2 ) α ij × ( ξ ij 1 + ξ ij 2 + ξ i,j-1 1 + ξ i,j-1 2 ) 4 ×7.8×2×( υ ij -0.2 ) + 8 υ ij -0.2 (47)
where ωij is the total weight of the molten steel in charge Lij, αij is the average thickness of the slab of charge Lij and υij is the pulling speed. The ζ ij 1 and ζ ij 2 are the average width of the odd strand slab and the dual slab of charge Lij, respectively. The ζ i,j-1 1 and ζ 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.
Operationso113o123o133o143o153o163o173o213o223o233o243
Minimal processing time (m)4545455055454045455045
Standard processing time (m)4848495860464246465050
Maximum processing time (m)6060617075606060606565
Table 3.The processing time of charges at casting stage.
Operationso253o263o273o283o313o323o333o343o353o363
Minimal processing time (m)45504545606060606060
Standard processing time (m)50555254747476767677
Maximum processing time (m)65706770858585858585
Table 4.The transportation time between machines.
Transportation time4LD5LD6LD5RH-15RH-23RHLF-1LF-2IR_UT4CC5CC6CC
4LD01213101013151513161514
5LD012151513151513161514
6LD0151513151513161615
5RH-101821201823222525
5RH-2021201823222525
3RH0202118252222
LF-101821222525
LF-2023222525
IR_UT0252222
4CC01616
5CC013
6CC0

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 , χ=1 . τ= s 241 * - s 241 0 =20>0 because s 241 0 =14:01 and s 241 * =14:21 , so y = y1, and Y={ o 241 ,2LD, y 1 ,20 } .

(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.
Node1234567891011121314151617181920
Operation o 151 κ 161 151 o 161 κ 171 161 o 171 o 241 κ 251 241 o 251 κ 261 251 o 261 κ 271 261 o 271 κ 281 271 o 281 o 331 κ 341 331 o 341 κ 351 341 o 351 κ 361 351
Node2122232425262728293031323334353637383940
Operation o 361 μ 152 151 μ 162 161 μ 172 171 μ 242 241 μ 252 251 μ 262 261 μ 272 271 μ 282 281 μ 332 331 μ 342 341 μ 352 351 μ 362 361 o 142 κ 152 142 o 152 κ 162 152 o 162 κ 172 162 o 172
Node4142434445464748495051525354555657585960
Operation o 242 κ 252 242 o 252 κ 262 252 o 262 κ 272 262 o 272 κ 282 272 o 282 o 332 κ 342 332 o 342 κ 352 342 o 352 κ 362 352 o 362 μ 143 142 μ 153 152 μ 163 162 μ 173 172
Node6162636465666768697071727374757677787980
Operation μ 243 242 μ 253 252 μ 263 262 μ 273 272 μ 283 282 μ 333 332 μ 343 342 μ 353 352 μ 363 362 o 133 o 143 o 153 o 163 o 173 o 233 o 243 o 253 o 263 o 273 o 283
Node8182838485
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 γ 3,4 2 between charge L23 and charge L24 is calculated as follows according to the formula (18): γ 3,4 2 = e 241 0 +Δt+ p 242 0 +u( z 241, 0 z 242 0 ) +u( z 242, 0 z 2423 0 ) - e 233 0 =20 . Because γ 3,4 2 >0 , the planned casting break between L23 and L24 occurs. Because p 233 max - p 233 0 - γ 3,4 2 =65-20-20=-5<0 , let ε l =0| l V 24 , then the planned casting break time γ ˜ 3,4 2 between L23 and L24 is calculated as follows according to the formula (27): γ ˜ 3,4 2 = η 76 - λ 75 =12>0 and p 233 max - p 233 0 - γ ˜ 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 χ 251 241 of charge L24 on 5LD is calculated as follows according to the formula (15): χ 251 241 =min( e 241 0 +Δt, e 251 0 ) -max( s 241 0 +Δt, s 251 0 ) =5>0 , so the processing conflict between L24 and L25 on 5LD occurs. Because ε 6 - χ 251 241 =-2<0 , ε 8 - χ 251 241 =-2<0 and ε 6 + ε 8 - χ 251 241 =1>0 , then the abnormal condition of processing conflict is two-level processing conflict ζ22.

(5) The processing conflict time χ 252 242 is calculated as follows according to the formula (15): χ 252 242 =min( e 241 0 +Δt+u( z 241 0 , z 242 0 ) + p 242 0 , e 252 0 ) -max( e 241 0 +Δt+u( z 241 0 , z 242 0 ) , s 252 0 ) =-10<0 , so there is no processing conflict between L24 and L25 on 5RH-2.

(6) The processing conflict time χ 253 243 is calculated as follows according to the formula (15): χ 253 243 =min( e 241 0 +Δt+u( z 241 0 , z 242 0 ) + p 242 0 +u( z 242 0 , z 243 0 ) + p 243 0 , e 253 0 ) -max( e 241 0 +Δt+u( z 241 0 , z 242 0 ) + p 242 0 +u( z 242 0 , z 243 0 ) , s 253 0 ) =20>0 , so the processing conflict between L24 and L25 on 5CC occurs. ε 76 - χ 253 243 =-15<0 , ε 77 - χ 251 241 =-15<0 and ε 76 + ε 77 - χ 253 243 =-10<0 . When let ε l =0| l V 24 ,| l V 25 , χ ˜ 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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