Scheduling method and scheduling system for steelmaking continuous casting with uncertain processing stage

Abstract

The invention provides a steelmaking continuous casting scheduling method and scheduling system with an uncertain processing stage, and the scheduling method comprises the steps: obtaining the production information of a steel plant; constructing a steelmaking continuous casting scheduling mixed integer programming model with an uncertain processing stage; the start time of each heat in the continuous casting stage is calculated according to the planned start time of each casting, the completion time of each heat in the primary refining stage is calculated, then the heat is sorted in a non-decreasing order according to the completion time of each heat in the primary refining stage, and a heat machining sequence is generated; according to the heat processing sequence, determining the heat distribution and the processing sequence of each stage, and calculating the start time of the heat of each stage; and recalculating the start-up time of each heat in each stage by using the start-up time of each heat in each stage and adopting a heat right shift strategy. According to the scheduling method and the scheduling system, uncertain steelmaking continuous casting scheduling in the processing stage can be optimized, the production efficiency is improved, and the production cost is reduced.

Description

Technical Field

[0001] The invention belongs to the technical field of steelmaking continuous casting scheduling, and mainly relates to a steelmaking continuous casting scheduling method and a scheduling system with uncertain processing stages. Background Art

[0002] Scheduling steelmaking and continuous casting involves the rational allocation of resources and optimization of process flows to complete production tasks within a limited timeframe, thereby improving multiple performance indicators. In modern steelmaking, the coordinated optimization of steelmaking and continuous casting has become a key approach to improving production efficiency and resource utilization. Its core focus lies in studying the allocation of steelmaking and continuous casting tasks, process integration, and resource utilization to achieve optimal production goals. A typical steelmaking and continuous casting process consists of three stages: steelmaking, refining, and continuous casting, each with multiple parallel machines. However, in real-world manufacturing, uncertainties and dynamic changes often occur, such as uncertainty in processing stages, processing time, and machine failures. To ensure that molten steel meets quality standards, some heats must undergo primary refining followed by secondary refining to remove impurities or heat the molten steel. Therefore, the traditional single-stage refining process is no longer sufficient for actual production needs, necessitating secondary refining. Specific objectives include shortening the maximum makespan (makespan), reducing the total waiting time (TWT), and minimizing the early penalty (EP) and late penalty (TP), thereby improving production efficiency and reducing overall costs. This optimization process not only helps improve the flexibility of steel production, but also provides important support for the industry to achieve green manufacturing. In steelmaking and continuous casting scheduling, production efficiency is mainly optimized through the following two methods: 1) Adopting high-efficiency equipment: By introducing advanced steelmaking and continuous casting equipment, production efficiency can be significantly improved, but this method requires a high capital investment. 2) Optimizing equipment operation strategies: Reasonable scheduling of tasks during equipment idle periods can reduce equipment idle time and improve equipment utilization. In steelmaking and continuous casting scheduling, there are contradictions between traditional optimization objectives (such as Makespan, TWT, EP, and TP). Among them, taking the maximum completion time as the optimization objective can shorten the overall production cycle, taking the total waiting time as the optimization objective can reduce inventory costs and improve resource utilization, and taking the early and late penalties as the optimization objectives can reduce the uncertainty of delivery time. However, these objectives still need to be weighed through scheduling decisions.

[0003] Therefore, how to balance the key production factors of maximum completion time (Makespan), total waiting time (TWT), early penalty (EP) and late penalty (TP) in the scheduling process to improve production line efficiency, reduce costs and realize steelmaking continuous casting scheduling with uncertainty in the processing stage has become an urgent problem to be solved. Summary of the Invention

[0004] In view of this, the present invention provides a steelmaking continuous casting scheduling method and scheduling system with uncertain processing stages to solve the problems existing in the prior art.

[0005] In one aspect, the present invention provides a method for scheduling steelmaking and continuous casting with uncertain processing stages, comprising:

[0006] S1: Obtaining production information of the steel plant, wherein the production information includes the number of machines in each stage, the number of casts processed by each machine in the continuous casting stage, the number of heats included in each cast, the planned start time of each cast on the continuous casting machine, the transportation time between processing stages, the start time of each cast, and the initial release time of each machine in each stage;

[0007] S2: Based on the production information, a mixed integer programming model for steelmaking and continuous casting scheduling with uncertain processing stages is constructed, wherein the model takes minimizing the maximum completion time, the total waiting time, the early penalty, and the late penalty as an optimization objective;

[0008] S3: Calculate the start time of each heat in the continuous casting stage based on the planned start time of each casting on the continuous casting machine, calculate the completion time of each heat in the primary refining stage, and then sort the heats in non-decreasing order according to the calculated completion time of each heat in the primary refining stage to generate a heat processing sequence;

[0009] S4: Based on the heat processing sequence, the heat allocation and processing sequence of each machine in each stage are determined according to the principle that the machine processing heat with the shorter release time has the higher priority, and the start time of the heat in each stage is calculated;

[0010] S5: Using the calculated start time of each heat in each stage, adopt the heat right shift strategy to recalculate the start time of each heat in each stage.

[0011] Preferably, S2 specifically includes the following steps:

[0012] S21: Establish an objective function with the optimization goal of minimizing the maximum completion time, the total waiting time, the early penalty, and the late penalty. The mathematical model of the objective function is as follows:

[0013] minF=ω1*TWT+ω2*EP+ω3*TP+ω4*Makespan (1)

[0014]

[0015] Makespan=max{C 4,j} (5)

[0016] Where TWT, EP, TP and Makespan represent the total waiting time, early penalty, late penalty and maximum completion time respectively; ω1, ω2, ω3 and ω4 are the weight coefficients of total waiting time, early penalty, late penalty and maximum completion time respectively; C 4,j is the completion time of heat j in the continuous casting stage, C 1,j is the completion time of heat j in the steelmaking stage, PT i,j is the processing time of heat j in the i-th stage, Ω is the set of n heats, Ω={1,2,...,n}; D l is the pretreatment time of pouring time l, is the start time of pouring time l, z l-1 +1 is the first heat of pouring batch l, and N is the total number of pouring batches;

[0017] S22: Establishing the constraint conditions of the objective function, wherein the constraint conditions are as follows:

[0018]

[0019] S 2,j -(S 1,j +PT 1,j +TT 1,2 )≥0, (10)

[0020] S 4,j -(S 2,j +PT 2,j +TT 2,4 )≥0, (11)

[0021] S i+1,j -(S i,j +PT i,j +TT i,i+1 )≥0,i=1,2,3, (12)

[0022]

[0023] x i,j,k ∈{0,1},j∈Ω,k∈M i ,i={1,2,3,4}, (17)

[0024]

[0025] Where Ω is a set of n heats, Ω = {1, 2, ..., n}; x i,j,k =1 means that the jth heat in the i-th stage is arranged on the k-th machine, otherwise, x i,j,k =0;x 3,j,k =1 means that in the third stage, heat j is arranged to be processed on the kth machine, otherwise, x3,j,k =0,Ar atr =2 means that heat j requires double refining; Ar atr =1 means that heat j does not require double refining; S i,j represents the start time of furnace j in the i-th stage, RT i,k represents the release time of machine k in the i-th stage; S 2,j and S 4,j are the start-up time of heat j in the primary refining stage and the continuous casting stage, S 1,j Indicates the start time of furnace j in the steelmaking stage, PT 1,j and PT 2,j are the processing time of heat j in the steelmaking stage and the primary refining stage, TT 1,2 Indicates the transportation time from the steelmaking stage to the primary refining stage, TT 2,4 represents the transportation time from the primary refining stage to the continuous casting stage; S i,j represents the start time of furnace j in the i-th stage, PT i,j represents the processing time of heat j in stage i, TT i,1+1 represents the transportation time from the i-th stage to the i+1-th stage; Indicates that heat j1 is processed earlier than heat j2. It means that heat j1 is processed later than heat j2. Indicates that j1 and j2 are processed simultaneously in stage i, but not on the same machine; U is a positive number greater than 30,000; Indicates pouring times b k-1 +1 adjustment time, Indicates pouring times b k-1 +1 for the first heat; Indicates the start time of pouring time 1, Indicates the processing time of pouring time l, ST l+1 Indicates the adjustment time of pouring time l+1.

[0026] Further preferably, in S3, the start time of each heat in the continuous casting stage is estimated based on the planned start time of each casting, and the completion time of each heat in the primary refining stage is calculated using the following formula:

[0027]

[0028] C 2,j =S 4,j -TT 3,4 -PT 3,j -TT 2,3 Ar atr =2 (20)

[0029] C2,j =S 4,j -TT 2,4 Ar atr =1 (21)

[0030] Where N represents the number of pouring times, D l is the planned start time of the lth pouring, z l-1 +1 indicates the first heat of the lth pouring, S 4,j represents the start time of furnace j in the continuous casting stage, S 4,j-1 Indicates the start time of heat j-1 in the continuous casting stage, PT 4,j-1 represents the processing time of heat j-1 in the continuous casting stage, Ar atr =2 means that heat j needs to be refining twice, Ar atr =1 means that heat j does not need double refining, C 2,j represents the completion time of heat j in primary refining, TT 3,4 is the transportation time from the secondary refining stage to the continuous casting stage, TT 2,4 is the transportation time from the primary refining stage to the continuous casting stage, TT 2,3 is the transportation time from primary refining to secondary refining, PT 3,j is the processing time of heat j in the secondary refining stage.

[0031] Further preferably, S4 specifically includes the following steps:

[0032] S41: Calculate the start time of each heat in the steelmaking stage: select heat π(j') from the heat processing sequence π and find the first available converter, where j'=1, 2, ..., n; let μ k is the release time of machine k∈M1, and the converter k* with the smallest release time is selected, that is, μ k* =min{μ k}, k∈M1, where M1 is the set of machines in the steelmaking stage; if machine k has not processed any heat, its release time μ k Equivalent to its release time RT 1,k ; Start-up time S of furnace π(j') 1,π(j') =μ k* , then, the next release time of converter k* is updated to μ k* =S 1,π(j') +PT 1,π(j') , where S 1,π(j') represents the start time of heat π(j') in the steelmaking stage, PT 1,π(j') represents the processing time of heat π(j') in the steelmaking stage;

[0033] S42: Calculate the start time of each heat in the primary refining stage: select heat π(j') from the heat processing sequence π and assign it to the first available primary refining furnace, where j'=1, 2, ..., n; select the primary refining furnace k* with the smallest release time, i.e. μ k* =min{μ k}, k∈M2, where M2 is the set of machines in the primary refining stage, and the start time of each heat π(j') is calculated as follows: S 2,π(j') =max{S 1,π(j') +PT 1,π(j') +TT 1,2 ,μ k*}, the next release time of machine k* is updated to μ k* =S 2,π(j') +PT 2,π(j') , where S 1,π(j') represents the start time of heat π(j') in the steelmaking stage, PT 1,π(j') represents the processing time of heat π(j') in the steelmaking stage, PT 2,π(j') represents the processing time of heat π(j') in the primary refining stage, TT 1,2 is the transportation time from the steelmaking stage to the primary refining stage, μ k* represents the release time of machine k*;

[0034] S43: Calculate the start time of each heat in the double refining stage: Determine the properties of each heat. If heat π(j') does not require double refining, skip this step. If heat π(j') undergoes double refining, select π(j') from the heat processing sequence π and assign it to the available double refining furnace. Select the primary refining furnace k* with the smallest release time, i.e. μ k* =min{μ k}, k∈M3, where M3 is the set of machines in the double refining stage, and the start time of each heat π(j') is calculated as follows: S 3,π(j') =max{S 2,π(j') +PT 2,π(j') +TT 2,3 ,μ k* ], the next release time of machine k* updates μ k* =S 3,π(j') +PT 3,π(j') , where S 3,π(j') represents the start-up time of heat π(j') in the secondary refining stage, S 2,π(j') represents the start-up time of heat π(j') in the primary refining stage, PT 2,π(j') represents the processing time of heat π(j') in the primary refining stage, TT 2,3represents the transportation time from the primary refining stage to the secondary refining stage, μ k* represents the release time of machine k*;

[0035] S44: Calculate the start time of each heat j in the continuous casting stage using formulas (22)-(23), where j = 1, 2, ..., n:

[0036] S 4,j =max{S 2,j +PT 2,j +TT 2,4 ,μ k}Ar atr =1 (22)

[0037] S 4,j =max{S 3,j +PT 3,j +TT 3,4 ,μ k}Ar atr =2 (23)

[0038] in, represents the release time of machine k, k∈M4, M4 is the set of machines in the continuous casting stage, if And j is processed on machine k, then, in, Indicates pouring times b k-1 +1 first heat, RT 4,k represents the release time of machine k, Indicates pouring times b k-1 +1 adjustment time, ST l+1 Indicates the adjustment time of pouring time l+1, S 4,j represents the start time of furnace j in the continuous casting stage, S 3,j represents the start-up time of heat j in the secondary refining stage, S 2,j represents the start time of heat j in the primary refining stage, PT 2,j represents the processing time of heat j in the primary refining stage, PT 3,j represents the processing time of heat j in the secondary refining stage, TT 2,4 Indicates the transportation time from the primary refining stage to the continuous casting stage, TT 3,4 represents the transportation time from the secondary refining stage to the continuous casting stage, Ar atr =1 means that heat j does not require double refining, Ar atr =2 means that heat j is to be subjected to double refining.

[0039] Further preferably, S5 specifically includes the following steps:

[0040] S51: Recalculate the start time of each heat in the continuous casting stage using formula (24):

[0041]

[0042] Among them, heat j and j+1 are in the same pouring Ω i In, S 4,j+1 represents the start time of heat j+1 in the continuous casting stage, S 4,j represents the start time of heat j in the continuous casting stage;

[0043] S52: Calculate the start time of the primary refining and secondary refining stages:

[0044] S521: If heat j does not undergo secondary refining, directly calculate the start time of each heat in primary refining. If heat j is the last heat of a machine k, its start time S 2,j =S 4,j -TT 2,4 -PT 2,j ;

[0045] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 2,j ;

[0046] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula:

[0047] For heat j':

[0048] Among them, S 2,j' is the start time of heat j' in primary refining, S 4,j' is the start time of heat j' in the continuous casting stage, PT 2,j' represents the processing time of heat j' in primary refining, μ k ' represents the release time of machine k, TT 2,4 is the transportation time from primary refining to the continuous casting stage;

[0049] S522: If heat j undergoes a secondary refining stage, first calculate the start time of each heat in the secondary refining stage, and then calculate the start time of each heat in the primary refining stage;

[0050] The calculation method for the start-up time of each heat in the secondary refining stage is as follows:

[0051] If the heat j is the last heat of a machine k, then its start time S 3,j =S 4,j-TT 3,4 -PT 3,j ;

[0052] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 3,j ;

[0053] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula:

[0054] For heat j':

[0055] Among them, S 3,j' is the start time of heat j' in secondary refining, S 4,j' is the start time of heat j' in the continuous casting stage, PT 3,j' represents the processing time of heat j' in double refining, μ k ' represents the release time of machine k, TT 3,4 is the transportation time from secondary refining to the continuous casting stage;

[0056] The calculation method for the start-up time of each heat in the primary refining stage is as follows:

[0057] If the heat j is the last heat of a machine k, then its start time S 2,j =S 3,j -TT 2,3 -PT 2,j ;

[0058] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 2,j ;

[0059] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula:

[0060] For heat j':

[0061] Among them, S 2,j' is the start time of heat j' in primary refining, S 3,j' is the start time of heat j' in the secondary refining stage, PT 2,j' represents the processing time of heat j' in primary refining, μ k ' represents the release time of machine k, TT 2,3 is the transportation time from primary refining to secondary refining;

[0062] S53: Calculate the start time of each heat in the steelmaking stage. If heat j is the last heat of a machine k, then its start time S 1,j =S 2,j -TT 1,2 -PT 1,j ;

[0063] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 1,j ;

[0064] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula:

[0065] For heat j':

[0066] Among them, S 1,j' is the start time of heat j' in the steelmaking stage, S 2,j' is the start time of heat j' in the primary refining stage, PT 1,j' represents the processing time of heat j' in the steelmaking stage, μ k ' represents the release time of machine k, TT 1,2 It is the transportation time from the steelmaking stage to the primary refining stage.

[0067] The present invention also proposes a steelmaking and continuous casting scheduling system with uncertain processing stages, which is used to execute the above-mentioned steelmaking and continuous casting scheduling method with uncertain processing stages.

[0068] The present invention provides a steelmaking and continuous casting scheduling method and scheduling system with uncertain processing stages, which can jointly optimize the maximum makespan (Makespan), total waiting time (TWT), early penalty (EP) and late penalty (TP) by using a heuristic algorithm. First, the production information of the steel plant is obtained. Secondly, based on the production information of the steel plant, a mixed integer programming model for steelmaking and continuous casting scheduling with uncertain processing stages is established. Then, the start time of each furnace in the continuous casting stage is estimated according to the planned start time of each casting, and the completion time of each furnace in the primary refining stage is calculated. After that, the furnaces are sorted in non-decreasing order according to the calculated completion time of each furnace in the primary refining stage to generate a furnace processing sequence. After that, the furnace allocation and processing sequence of the furnace in each stage are determined according to the obtained furnace processing sequence, and the start time of the furnace in each stage is calculated. Finally, the furnace right shift strategy is adopted to recalculate the start time of each furnace in each stage, and the scheduling scheme that minimizes the objective function can be obtained, including: 1) the processing sequence of all furnaces in each stage; 2) the start time of each furnace in each stage. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 A flow chart of a method for scheduling steelmaking and continuous casting with uncertain processing stages provided by the present invention;

[0070] Figure 2 95% confidence interval diagram of the average least significant difference between the IHeu heuristic algorithm and the Heupan heuristic algorithm adopted in the present invention. DETAILED DESCRIPTION

[0071] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further explained below with reference to specific embodiments, but the present invention is not limited thereto.

[0072] Scheduling for steelmaking and continuous casting plays a crucial role in the manufacturing process. With the advancement of energy conservation and emission reduction policies, its role in reducing energy consumption and economic penalties is becoming increasingly prominent. Optimizing scheduling decisions not only reduces energy waste during production but also significantly lowers penalty costs incurred due to early or late delivery. Specific goals include shortening the maximum make span (Makespan), reducing the total waiting time (TWT), and minimizing the early penalty (EP) and late penalty (TP), thereby improving production efficiency and reducing overall costs.

[0073] In view of this, if Figure 1 As shown, this embodiment provides a steelmaking and continuous casting scheduling method with uncertain processing stages, including:

[0074] S1: Obtaining production information of the steel plant, wherein the production information includes the number of machines in each stage, the number of casts processed by each machine in the continuous casting stage, the number of heats included in each cast, the planned start time of each cast on the continuous casting machine, the transportation time between processing stages, the start time of each cast, and the initial release time of each machine in each stage;

[0075] These parameters are shown in Table 1;

[0076] Table 1 Information of steel mills

[0077]

[0078]

[0079] in, And processed on machine k, Indicates pouring times The first heat of the same machine k is filled by D 1+1 =D1+ST l+1 +∑ j∈Ω PT 4,j Determine, where Ω l ,Ω l+1 ∈B k and k∈M4, M4 is the set of machines in the continuous casting stage, B k is the set of times of machine k, Ω l ,Ω l+1 are two adjacent pouring times on machine k, ST l is the adjustment time of pouring time l, D l Indicates the planned start time of pouring time l, RT i,k is the release time of machine k at stage i;

[0080] S2: Based on the production information, a mixed integer programming model for steelmaking and continuous casting scheduling with uncertain processing stages is constructed; wherein S2 specifically includes the following steps:

[0081] S21: Establish an objective function with the optimization goal of minimizing the maximum makespan (Makespan), the total waiting time (TWT), the early penalty (EP), and the late penalty (TP), wherein the mathematical model of the objective function is as follows:

[0082] minF=ω1*TWT+ω2*EP+ω3*TP+ω4*Makespan (1)

[0083]

[0084] Makespan=max{C4,j} (5)

[0085] Formula (1) represents the simultaneous minimization of TWT, EP, TP and Makespan, where ω1, ω2, ω3, ω4 are the weight coefficients of TWT, EP, TP and Makespan; Formula (2) is used to calculate the total waiting time, where C 4,j is the completion time of heat j in the continuous casting stage, C 1,j is the completion time of heat j in the steelmaking stage, PT i,j is the processing time of heat j in the i-th stage, Ω is the set of n heats, Ω = {1, 2, ..., n}; Equation (3) is used to determine the early penalty (EP), and Equation (4) is used to determine the late penalty (TP), where D l is the pretreatment time of pouring time l, is the start time of pouring time l, z l-1 +1 is the first heat of pouring batch l, N represents the total number of pouring batches, and formula (5) is used to calculate the maximum completion time;

[0086] S22: Establishing the constraint conditions of the objective function, wherein the constraint conditions are as follows:

[0087]

[0088] S 2,j -(S 1,j +PT 1,j +TT 1,2 )≥0, (10)

[0089] S 4,j -(S 2,j +PT 2,j +TT 2,4 )≥0, (11)

[0090] S i+1,j -(S i,j +PT i,j +TT i,i+1 )≥0,i=1,2,3, (12)

[0091]

[0092] x i,j,k ∈{0,1},j∈Ω,k∈M i ,i={1,2,3,4}, (17)

[0093]

[0094] Where Ω is a set of n heats, Ω = {1, 2, ..., n}. Equation (6) indicates that each heat must go through three processing stages, where x i,k,k =1 means that the jth heat in the i-th stage is arranged on the k-th machine, otherwise, x i,k,k = 0; Formula (7) stipulates that in the third stage, the heat j undergoing double refining needs to be processed on a single device, where x 3,j,k =1 means that in the third stage, heat j is arranged to be processed on the kth machine, otherwise, x 3,j,k =0,Ar atr =2 means that heat j needs double refining; Formula (8) means that heat j does not go through the double refining stage, Ar atr =1 means that heat j does not need to be refined twice; Formula (9) stipulates that the start time of any stage of the heat must be later than the release time of the machine, where S i,j represents the start time of furnace j in the i-th stage, RT i,k represents the release time of machine k in the i-th stage; Equations (10) and (11) stipulate that if heat j does not go through the double refining stage, the start time of the next stage must be greater than or equal to the sum of the completion time and transportation time of the previous stage, where S 2,j and S 4,j are the start-up time of heat j in the primary refining stage and the continuous casting stage, S 1,j Indicates the start time of furnace j in the steelmaking stage, PT 1,j and PT 2,j are the processing time of heat j in the steelmaking stage and the primary refining stage, TT 1,2 Indicates the transportation time from the steelmaking stage to the primary refining stage, TT 2,4 represents the transportation time from the primary refining stage to the continuous casting stage; Formula (12) stipulates that if heat j passes through all four stages, the start time of the next stage of heat j must be greater than or equal to the sum of the completion time and transportation time of the previous stage, where S i,j represents the start time of furnace j in the i-th stage, PT i,j represents the processing time of heat j in stage i, TT i,i+1 represents the transportation time from the i-th stage to the i+1-th stage; Formula (13) indicates that two heats j1 and j2 cannot be processed simultaneously on the same machine. Indicates that heat j1 is processed earlier than heat j2. It means that heat j1 is processed later than heat j2. = j1 and j2 are processed simultaneously in stage i, but not on the same machine. Formula (14) indicates that the current heat must be completed on a machine before the next heat can be processed. U is a large positive number, preferably greater than 30,000. Formula (15) stipulates that the start time of the first pouring on machine k must meet the adjustment time requirement. Indicates pouring times b k-1 +1 adjustment time, Indicates pouring times b k-1 +1 for the first heat; Formula (16) indicates that the start time of the next casting on the same continuous casting machine must be later than the completion time of the previous casting, Indicates the start time of pouring time 1, Indicates the processing time of pouring time l, ST l+1 represents the adjustment time of pouring time l+1. Equations (17) and (18) are used to establish the boundaries of the decision variable values;

[0095] S3: Based on the planned start time of each pour on the continuous casting machine, the start time of each heat in the continuous casting stage is estimated, and the completion time of each heat in the primary refining stage is calculated. Then, the heats are sorted in non-decreasing order according to the calculated completion time of each heat in the primary refining stage to generate a heat processing sequence (hereinafter referred to as the IHeu heuristic algorithm);

[0096] The formula for calculating the start time of each heat in the continuous casting stage based on the planned start time of each pour on the continuous casting machine and the completion time of each heat in the primary refining stage is as follows:

[0097]

[0098] C 2,j =S 4,j -TT 3,4 -PT 3,j -TT 2,3 Ar atr =2 (20)

[0099] C 2,j =S 4,j -TT 2,4 Ar atr =1 (21)

[0100] Where N represents the number of pouring times, D l is the planned start time of the lth pouring, z l-1 +1 indicates the first heat of the lth pouring, S 4,j represents the start time of furnace j in the continuous casting stage, S 4,j-1 Indicates the start time of heat j-1 in the continuous casting stage, PT4,j-1 represents the processing time of heat j-1 in the continuous casting stage, Ar atr =2 means that heat j needs to be refining twice, Ar atr =1 means that heat j does not need double refining, C 2,j represents the completion time of heat j in primary refining, TT 3,4 is the transportation time from the secondary refining stage to the continuous casting stage, TT 2,4 is the transportation time from the primary refining stage to the continuous casting stage, TT 2,3 is the transportation time from primary refining to secondary refining, PT 3,j is the processing time of heat j in the secondary refining stage;

[0101] Among them, the heats are sorted in non-decreasing order according to the calculated completion time of each heat in the primary refining stage, and the heat processing sequence π = {π(1),π(2),...,π(n)} is generated;

[0102] S4: Based on the heat processing sequence, the heat allocation and processing sequence of each machine in each stage are determined according to the principle that the machine processing heat with the shorter release time has the higher priority, and the start time of the heat in each stage is calculated;

[0103] S41: Calculate the start time of each heat in the steelmaking stage: select heat π(j') from the heat processing sequence π and find the first available converter, where j'=1, 2, ..., n; let μ k is the release time of machine k∈M1, and the converter k* with the smallest release time is selected, that is, μ k* =min{μ k}, k∈M1, where M1 is the set of machines in the steelmaking stage; it should be noted that if machine k has not processed any heat, its release time μ k Equivalent to its release time RT 1,k In order to arrange the heat π(j′) to the converter k* for processing as early as possible, the start time of the heat π(j′) is set to S 1,π(j') =μ k* , then, the next release time of converter k* is updated to μ k* =S 1,π(j′) +PT 1,π(j') , where S 1,π(j′) represents the start-up time of heat π(j′) in the steelmaking stage, PT 1,π(j') represents the processing time of heat π(j') in the steelmaking stage;

[0104] S42: Calculate the start time of each heat in the primary refining stage: select heat π(j') from the heat processing sequence π and assign it to the first available primary refining furnace, where j'=1, 2, ..., n, which means selecting the primary refining furnace k* with the smallest release time, i.e. μ k* =min{μ k}, k∈M2, where M2 is the set of machines in the primary refining stage. The start time of each heat π(j') can be calculated as follows: S2 ,π(j') =max{S 1,π(j') +PT 1,π(j′) +TT 1,2 ,μ k*}, the next release time of machine k* is updated to μ k* =S 2,π(j') +PT 2,π(j') , where S 1,π(j') represents the start time of heat π(j') in the steelmaking stage, PT 1,π(j') represents the processing time of heat π(j') in the steelmaking stage, PT 2,π(j') represents the processing time of heat π(j') in the primary refining stage, TT 1,2 is the transportation time from the steelmaking stage to the primary refining stage, μ k* represents the release time of machine k*;

[0105] S43: Calculate the start time of each heat in the double refining stage: Determine the properties of each heat. If heat π(j') does not require double refining, then skip this step. If heat π(j') undergoes double refining, then select π(j') from the heat processing sequence π and assign it to the available double refining furnace. This means selecting the primary refining furnace k* with the smallest release time, i.e. μ k* =min{u k}, k∈M3, where M3 is the set of machines in the double refining stage. The start time of each heat π(j') can be calculated as follows: S 3,π(j') =max{S 2,π(j') +PT 2,π(j') +TT 2,3 ,μ k*}, the next release time of machine k* updates μ k* =S 3,π(j') +PT 3,π(j') , where A 3,π(j') represents the start-up time of heat π(j') in the secondary refining stage, S 2,π(j') represents the start-up time of heat π(j') in the primary refining stage, PT 2,π(j') represents the processing time of heat π(j') in the primary refining stage, TT 2,3represents the transportation time from the primary refining stage to the secondary refining stage, μ k* represents the release time of machine k*;

[0106] S44: Calculate the start time of each heat j in the continuous casting stage using formulas (22)-(23), where j = 1, 2, ..., n:

[0107] S 4,j =max{S 2,j +PT 2,j +TT 2,4 ,μ k}Ar atr =1 (22)

[0108] S 4,j =max{S 3,j +PT 3,j +TT 3,4 ,μ k}Ar atr =2 (23)

[0109] in, represents the release time of machine k, k∈M4, M4 is the set of machines in the continuous casting stage, if And j is processed on machine k, then, in, Indicates pouring times b k-1 +1 first heat, RT 4,k represents the release time of machine k, Indicates pouring times b k-1 +1 adjustment time, ST l+1 Indicates the adjustment time of pouring time l+1, S 4,j represents the start time of furnace j in the continuous casting stage, S 3,j represents the start-up time of heat j in the secondary refining stage, S 2,j represents the start time of heat j in the primary refining stage, PT 2,j represents the processing time of heat j in the primary refining stage, PT 3,j represents the processing time of heat j in the secondary refining stage, TT 2,4 Indicates the transportation time from the primary refining stage to the continuous casting stage, TT 3,4 represents the transportation time from the secondary refining stage to the continuous casting stage, Ar atr =1 means that heat j does not require double refining, Ar atr =2 means that heat j is to be subjected to double refining;

[0110] S5: Using the calculated start time of each heat in each stage, adopt the heat right shift strategy to recalculate the start time of each heat in each stage;

[0111] S51: In order to ensure continuous casting and reduce the total waiting time, the start time of each furnace in the continuous casting stage is recalculated using formula (24):

[0112]

[0113] Among them, heat j and j+1 are in the same pouring Ω i In, S 4,j+1 represents the start time of heat j+1 in the continuous casting stage, S 4,j represents the start time of heat j in the continuous casting stage;

[0114] S52: Calculate the start time of the primary refining and secondary refining stages:

[0115] S521: If heat j does not undergo secondary refining, directly calculate the start time of each heat in primary refining. If heat j is the last heat of a machine k, its start time is S 2,j =S 4,j -TT 2,4 -PT 2,j ;

[0116] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 2,j ;

[0117] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position can be calculated by the following iterative formula:

[0118] For heat j':

[0119] Among them, S 2,j' is the start time of heat j' in primary refining, S 4,j' is the start time of heat j' in the continuous casting stage, PT 2,j' represents the processing time of heat j' in primary refining, μ k ' represents the release time of machine k, TT 2,4 is the transportation time from primary refining to the continuous casting stage;

[0120] S522: If heat j undergoes a secondary refining stage, first calculate the start time of each heat in the secondary refining stage, and then calculate the start time of each heat in the primary refining stage;

[0121] The calculation method for the start-up time of each heat in the secondary refining stage is as follows:

[0122] If the heat j is the last heat of a machine k, then its start time S 3,j =S 4,j -TT 3,4 -PT 3,j ;

[0123] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 3,j ;

[0124] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position can be calculated by the following iterative formula:

[0125] For heat j':

[0126] Among them, S 3,j' is the start time of heat j' in secondary refining, S 4,j' is the start time of heat j' in the continuous casting stage, PT 3,j' represents the processing time of heat j' in double refining, μ k ' represents the release time of machine k, TT 3,4 is the transportation time from secondary refining to the continuous casting stage;

[0127] The calculation method for the start-up time of each heat in the primary refining stage is as follows:

[0128] If the heat j is the last heat of a machine k, then its start time S 2,j =S 3,j -TT 2,3 -PT 2,j ;

[0129] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 2,j ;

[0130] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position can be calculated by the following iterative formula:

[0131] For heat j':

[0132] Among them, S 2,j' is the start time of heat j' in primary refining, S 3,j' is the start time of heat j' in the secondary refining stage, PT 2,j' represents the processing time of heat j' in primary refining, μk ' represents the release time of machine k, TT 2,3 is the transportation time from primary refining to secondary refining;

[0133] S53: Calculate the start time of each heat in the steelmaking stage. If heat j is the last heat of a machine k, then its start time S 1,j =S 2,j -TT 1,2 -PT 1,j ;

[0134] The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 1,j ;

[0135] Next, for each heat j' on machine k, the start time from the second-to-last position to the first position can be calculated by the following iterative formula:

[0136] For heat j':

[0137] Among them, S 1,j' is the start time of heat j' in the steelmaking stage, S 2,j' is the start time of heat j' in the primary refining stage, PT 1,j' represents the processing time of heat j' in the steelmaking stage, μ k ' represents the release time of machine k, TT 1,2 is the transportation time from the steelmaking stage to the primary refining stage;

[0138] In this embodiment, the IHeu heuristic algorithm is coded in C++, and all examples are run on a personal computer equipped with an Intel(R) Core(TM) i5-11400 CPU @ 2.6 GHz and 16.00 GB of memory under Microsoft Windows 10 system.

[0139] The present invention also provides a steelmaking and continuous casting scheduling system with uncertain processing stages, which is used to execute the above-mentioned steelmaking and continuous casting method with uncertain processing stages.

[0140] Next, the effectiveness of the present invention is verified by experiments. The experiment uses 20 examples, and each example is run independently five times to eliminate the influence of chance. The IHeu heuristic algorithm used in the present invention is compared with the existing Heupan heuristic algorithm. The smaller the ARPI value, the better the algorithm performance. Sol represents the solution of a single run of a specific example, Sol best The optimal solution among all solutions for the same instance. The calculation method of ARPI is shown in formula (25). The comparison results of the two heuristic algorithms are shown in Table 2.

[0141]

[0142] Table 2 Comparison of ARPI between IHeu and Heupan (optimal value is bolded)

[0143]

[0144] The present invention first establishes a mixed integer programming model based on the production information of the steel plant. Then, the start time of each furnace in the continuous casting stage is estimated according to the planned start time of each casting, and the completion time of each furnace in the primary refining stage is calculated. Then, the furnaces are sorted in non-decreasing order according to the calculated completion time of each furnace in the primary refining stage to generate a furnace processing sequence. Then, based on the obtained furnace processing sequence, the furnace allocation and processing sequence of the furnace in each stage are determined, and the start time of the furnace in each stage is calculated. Finally, the furnace right shift strategy is adopted to recalculate the start time of each furnace in each stage, so as to obtain a scheduling solution that minimizes the objective function.

[0145] According to the experimental results, Figure 2 Table 2 shows the 95% confidence intervals for the mean least significant difference (ALS) and the ARPI values ​​for each heuristic algorithm across different cases. IHeu achieves an ARPI of 0 in most cases, significantly outperforming the Heupan algorithm. This suggests that IHeu can generate more optimal scheduling solutions.

[0146] Those skilled in the art will appreciate that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the application can adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code. The scheme in the embodiment of the present application can be implemented in various computer languages, for example, object-oriented programming language Python and literal translation scripting language JavaScript, etc.

[0147] Other embodiments of the present invention will readily occur to those skilled in the art after considering the specification and practicing the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The description and examples are to be considered as exemplary only, with the true scope and spirit of the invention being indicated by the following claims.

[0148] It should be understood that the present invention is not limited to the above description and that various modifications and changes can be made without departing from the scope thereof. The scope of the present invention is limited only by the appended claims.

Claims

1. A method for scheduling steelmaking and continuous casting with uncertain processing stages, characterized in that: include: S1: Obtaining production information of the steel plant, wherein the production information includes the number of machines in each stage, the number of casts processed by each machine in the continuous casting stage, the number of heats included in each cast, the planned start time of each cast on the continuous casting machine, the transportation time between processing stages, the start time of each cast, and the initial release time of each machine in each stage; S2: Based on the production information, a mixed integer programming model for steelmaking and continuous casting scheduling with uncertain processing stages is constructed, wherein the model takes minimizing the maximum completion time, the total waiting time, the early penalty, and the late penalty as an optimization objective; S3: Calculate the start time of each heat in the continuous casting stage based on the planned start time of each casting on the continuous casting machine, calculate the completion time of each heat in the primary refining stage, and then sort the heats in non-decreasing order according to the calculated completion time of each heat in the primary refining stage to generate a heat processing sequence; S4: Based on the heat processing sequence, the heat allocation and processing sequence of each machine in each stage are determined according to the principle that the machine processing heat with the shorter release time has the higher priority, and the start time of the heat in each stage is calculated; S5: Using the calculated start time of each heat in each stage, adopt the heat right shift strategy to recalculate the start time of each heat in each stage.

2. The method for scheduling steelmaking and continuous casting with uncertain processing stages according to claim 1, characterized in that: S2 specifically includes the following steps: S21: Establish an objective function with the optimization goal of minimizing the maximum completion time, the total waiting time, the early penalty, and the late penalty. The mathematical model of the objective function is as follows: minF=ω1*TWT+ω2*EP+ω3*TP+ω4*Makespan (1) Makespan=max{C 4,j } (5) Where TWT, EP, TP and Makespan represent the total waiting time, early penalty, late penalty and maximum completion time respectively; ω1, ω2, ω3 and ω4 are the weight coefficients of total waiting time, early penalty, late penalty and maximum completion time respectively; C 4,j is the completion time of heat j in the continuous casting stage, C 1,j is the completion time of heat j in the steelmaking stage, PT i,j is the processing time of heat j in the i-th stage, Ω is the set of n heats, Ω={1,2,...,n}; D l is the pretreatment time of pouring time l, is the start time of pouring time l, z l-1 +1 is the first heat of pouring batch l, and N is the total number of pouring batches; S22: Establishing the constraint conditions of the objective function, wherein the constraint conditions are as follows: S 2,j -(S 1,j +PT 1,j +TT 1,2 )≥0, (10) S 4,j -(S 2,j +PT 2,j +TT 2,4 )≥0, (11) S i+1,j -(S i,j +PT i,j +TT i,i+1 )≥0,i=1,2,3, (12) x i,j,k ∈{0,1},j∈Ω,k∈M i ,i={1,2,3,4}, (17) Where Ω is a set of n heats, Ω = {1, 2, ..., n}; x i,j,k =1 means that the jth heat in the i-th stage is arranged on the k-th machine, otherwise, x i,j,k =0;x 3,j,k =1 means that in the third stage, heat j is arranged to be processed on the kth machine, otherwise, x 3,j,k =0,Ar atr =2 means that heat j requires double refining; Ar atr =1 means that heat j does not require double refining; S i,j represents the start time of furnace j in the i-th stage, RT i,k represents the release time of machine k in the i-th stage; S 2,j and S 4,j are the start-up time of heat j in the primary refining stage and the continuous casting stage, S 1,j Indicates the start time of furnace j in the steelmaking stage, PT 1,j and PT 2,j are the processing time of heat j in the steelmaking stage and the primary refining stage, TT 1,2 Indicates the transportation time from the steelmaking stage to the primary refining stage, TT 2,4 represents the transportation time from the primary refining stage to the continuous casting stage; S i,j represents the start time of furnace j in the i-th stage, PT i,j represents the processing time of heat j in stage i, TT i,i+1 represents the transportation time from the i-th stage to the i+1-th stage; Indicates that heat j1 is processed earlier than heat j2. It means that heat j1 is processed later than heat j2. Indicates that j1 and j2 are processed simultaneously in stage i, but not on the same machine; U is a positive number greater than 30,000; Indicates pouring times b k-1 +1 adjustment time, Indicates pouring times b k-1 +1 for the first heat; Indicates the start time of pouring time 1, Indicates the processing time of pouring time l, ST l+1 Indicates the adjustment time of pouring time l+1.

3. The method for scheduling steelmaking and continuous casting with uncertain processing stages according to claim 1, characterized in that: In S3, the start time of each heat in the continuous casting stage is estimated based on the planned start time of each casting, and the completion time of each heat in the primary refining stage is calculated as follows: C 2,j =S 4,j -TT 3,4 -PT 3,j -TT 2,3 Are atr =2 (20) C 2,j =S 4,j -TT 2,4 Are atr =1 (21) Where N represents the number of pouring times, D l is the planned start time of the lth pouring, z l-1 +1 indicates the first heat of the lth pouring, S 4,j represents the start time of furnace j in the continuous casting stage, S 4,j-1 Indicates the start time of heat j-1 in the continuous casting stage, PT 4,j-1 represents the processing time of heat j-1 in the continuous casting stage, Ar atr =2 means that heat j needs to be refining twice, Ar atr =1 means that heat j does not need double refining, C 2,j represents the completion time of heat j in primary refining, TT 3,4 is the transportation time from the secondary refining stage to the continuous casting stage, TT 2,4 is the transportation time from the primary refining stage to the continuous casting stage, TT 2,3 is the transportation time from primary refining to secondary refining, PT 3,j is the processing time of heat j in the secondary refining stage.

4. The method for scheduling steelmaking and continuous casting with uncertain processing stages according to claim 1, characterized in that: S4 specifically includes the following steps: S41: Calculate the start time of each heat in the steelmaking stage: select heat π(j') from the heat processing sequence π and find the first available converter, where j'=1, 2, ..., n; let μ k For the release time of machine k∈M1, select the converter k* with the smallest release time, that is, k∈M1, where M1 is the set of machines in the steelmaking stage; if machine k has not processed any heat yet, its release time μ k Equivalent to its release time RT 1,k ; Start time of furnace π(j') Subsequently, the next release time of converter k* is updated to Among them, S 1,π(j') represents the start time of heat π(j') in the steelmaking stage, PT 1,π(j') represents the processing time of heat π(j') in the steelmaking stage; S42: Calculate the start time of each heat in the primary refining stage: select heat π(j') from the heat processing sequence π and assign it to the first available primary refining furnace, where j'=1, 2, ..., n; select the primary refining furnace k* with the smallest release time, that is, k∈M2, where M2 is the set of machines in the primary refining stage, and the operating time of each heat π(j') is calculated as follows: The next release time of machine k* is updated to Among them, S 1,π(j') represents the start time of heat π(j') in the steelmaking stage, PT 1,π(j') represents the processing time of heat π(j') in the steelmaking stage, PT 2,π(j') represents the processing time of heat π(j') in the primary refining stage, TT 1,2 is the transportation time from the steelmaking stage to the primary refining stage, represents the release time of machine k*; S43: Calculate the start time of each heat in the double refining stage: Determine the properties of each heat. If heat π(j') does not require double refining, then skip this step. If heat π(j') undergoes double refining, then select π(j') from the heat processing sequence π and assign it to the available double refining furnace. Select the primary refining furnace k* with the smallest release time, i.e. k∈M3, where M3 is the set of machines in the double refining stage. The operating time of each heat π(j') is calculated as follows: Next release time update for machine k* Among them, S 3,π(j') represents the start-up time of heat π(j') in the secondary refining stage, S 2,π(j') represents the start-up time of heat π(j') in the primary refining stage, PT 2,π(j') represents the processing time of heat π(j') in the primary refining stage, TT 2,3 represents the transportation time from the primary refining stage to the secondary refining stage, represents the release time of machine k*; S44: Calculate the start time of each heat j in the continuous casting stage using formulas (22)-(23), where j = 1, 2, ..., n: S 4,j =max{S 2,j +PT 2,j +TT 2,4 ,μ k } Ar atr =1 (22) S 4,j =max{S 3,j +PT 3,j +TT 3,4 ,μ k }Ar atr =2 (23) in, represents the release time of machine k, k∈M4, M4 is the set of machines in the continuous casting stage, if And j is processed on machine k, then, in, Indicates pouring times b k-1 +1 first heat, RT 4,k represents the release time of machine k, Indicates pouring times b k-1 +1 adjustment time, ST l+1 Indicates the adjustment time of pouring time l+1, S 4,j represents the start time of furnace j in the continuous casting stage, S 3,j represents the start-up time of heat j in the secondary refining stage, S 2,j represents the start time of heat j in the primary refining stage, PT 2,j represents the processing time of heat j in the primary refining stage, PT 3,j represents the processing time of heat j in the secondary refining stage, TT 2,4 Indicates the transportation time from the primary refining stage to the continuous casting stage, TT 3,4 represents the transportation time from the secondary refining stage to the continuous casting stage, Ar atr =1 means that heat j does not require double refining, Ar atr =2 means that heat j is to be subjected to double refining.

5. The method for scheduling steelmaking and continuous casting with uncertain processing stages according to claim 1, characterized in that: S5 specifically includes the following steps: S51: Recalculate the start time of each heat in the continuous casting stage using formula (24): Among them, heat j and j+1 are in the same pouring Ω i In, S 4,j+1 represents the start time of heat j+1 in the continuous casting stage, S 4,j represents the start time of heat j in the continuous casting stage; S52: Calculate the start time of the primary refining and secondary refining stages: S521: If heat j does not undergo secondary refining, directly calculate the start time of each heat in primary refining. If heat j is the last heat of a machine k, its start time S 2,j =S 4,j -TT 2,4 -PT 2,j ; The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 2,j ; Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula: For heat j': Among them, S 2,j' is the start time of heat j' in primary refining, S 4,j' is the start time of heat j' in the continuous casting stage, PT 2,j' represents the processing time of heat j' in primary refining, μ k ' represents the release time of machine k, TT 2,4 is the transportation time from primary refining to the continuous casting stage; S522: If heat j undergoes a secondary refining stage, first calculate the start time of each heat in the secondary refining stage, and then calculate the start time of each heat in the primary refining stage; The calculation method for the start-up time of each heat in the secondary refining stage is as follows: If the heat j is the last heat of a machine k, then its start time S 3,j =S 4,j -TT 3,4 -PT 3,j ; The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 3,j ; Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula: For heat j': Among them, S 3,j' is the start time of heat j' in secondary refining, S 4,j' is the start time of heat j' in the continuous casting stage, PT 3,j' represents the processing time of heat j' in double refining, μ k ' represents the release time of machine k, TT 3,4 is the transportation time from secondary refining to the continuous casting stage; The calculation method for the start-up time of each heat in the primary refining stage is as follows: If the heat j is the last heat of a machine k, then its start time S 2,j =S 3,j -TT 2,3 -PT 2,j ; The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 2,j ; Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula: For heat j': Among them, S 2,j' is the start time of heat j' in primary refining, S 3,j' is the start time of heat j' in the secondary refining stage, PT 2,j' represents the processing time of heat j' in primary refining, μ k ' represents the release time of machine k, TT 2,3 is the transportation time from primary refining to secondary refining; S53: Calculate the start time of each heat in the steelmaking stage. If heat j is the last heat of a machine k, then its start time S 1,j =S 2,j -TT 1,2 -PT 1,j ; The latest release time μ of machine k when processing other heats k 'Determined by the following formula: μ k '=S 1,j ; Next, for each heat j' on machine k, the start time from the second-to-last position to the first position is calculated using the following iterative formula: For heat j': Among them, S 1,j' is the start time of heat j' in the steelmaking stage, S 2,j' is the start time of heat j' in the primary refining stage, PT 1,j' represents the processing time of heat j' in the steelmaking stage, μ k ' represents the release time of machine k, TT 1,2 It is the transportation time from the steelmaking stage to the primary refining stage.

6. A steelmaking and continuous casting scheduling system with uncertain processing stages, characterized by: A method for scheduling steelmaking and continuous casting with uncertain processing stages for executing any one of claims 1 to 5.

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