2021 National Higher Education Cup Mathematical Modeling
On-line Optimization of Continuous Casting Cutting in Problem D
Reproduce the original title
Continuous casting is the production process of turning molten steel into billets. The specific process is as follows (Figure 1):
The molten steel is continuously poured into the crystallizer from the tundish, and pulled out from the crystallizer at a certain speed, and enters the secondary cooling section. When the molten steel passes through the mold, a solid billet shell is formed where it contacts the surface of the mold. In the secondary cooling section, the billet shell gradually thickens and finally solidifies to form a billet. Then, the steel billet is cut according to certain size requirements.
When the continuous casting is stopped, tail billets will be produced, and the length of the tail billets is related to the amount of molten steel remaining in the tundish and other factors. Therefore, the cutting of the tail blank is also an integral part of the continuous casting cutting.
When the cutting machine cuts the steel billet, there is a fixed working starting point, and the cutting of the billet must start from the working starting point. During the cutting process, the cutting machine rides on the billet and moves synchronously with the billet to ensure that the cutting line is perpendicular to the direction of the billet. After the cutting is finished, return to the starting point of the work and wait for the next cutting.
In the cutting plan, the cutting loss is given priority, and the cutting loss is required to be as small as possible. Here, the cutting loss is defined as the length of the scrapped billet; secondly, the user's requirements are considered. Under the same cutting loss, the cut billet should meet the user's target value as much as possible.
In the process of pouring steel, the crystallizer will appear abnormal. At this time, a section of steel billet located inside the crystallizer needs to be scrapped, and this section of steel billet is called a scrap section (Figure 2). When there is an abnormality in the crystallizer, the cutting process will know immediately, so that the cutting plan can be adjusted immediately.
The billet after cutting cannot contain scrap section when it enters the next process. When there is a scrap section of the billet, first cut off the billet with the scrap section attached by the cutting machine, and then make the remaining billet meet the length required by the next process through offline secondary cutting; other billets entering the next process must also meet the requirements of the next process. The length requirement of the process.
Now invite your team to establish a mathematical model or design an algorithm to solve the following problems:
Problem 1 Under the condition of meeting the basic requirements and normal requirements, formulate the optimal cutting plan according to the length of the tail blank. Assume that the user's target value is 9.5 meters, and the target range is 9.0~10.0 meters. For the following tail blank lengths: 109.0, 93.4, 80.9, 72.0, 62.7, 52.5, 44.9, 42.7, 31.6, 22.7, 14.5 and 13.7 (unit: meter) , according to the list of "tail length, cutting plan, cutting loss" and other content list to give the specific optimal cutting plan.
Problem 2. When the crystallizer is abnormal, give the optimal cutting plan in real time: (1) When the scrapped section of the billet appears for the first time, give the cutting plan for this section of the billet; (2) After a new scrapped section appears (as shown in Figure 2), give the cutting plan of the new section of billet and the adjustment plan of the cutting of the current section of billet, or declare that no adjustment will be made.
Assume that the abnormal moments of the crystallizer are 0.0, 45.6, 98.6, 131.5, 190.8, 233.3, 266.0, 270.7 and 327.9 (unit: minute), the user’s target value is 9.5 meters, and the target range is 9.0~10.0 meters. Under the condition of meeting the basic requirements and normal requirements, the specific optimal cutting plan at these times is given according to the list of "initial cutting plan, adjusted cutting plan, cutting loss" and other contents.
Question 3 Assuming that the real-time optimal cutting plan and the abnormal moment of the crystallizer are the same as Question 2, under the condition of meeting the basic requirements and normal requirements, for (1) the user's target value is 8.5 meters, and the target range is 8.0~9.0 meters , (2) The user's target value is 11.1 meters, and the target range is 10.6~11.6 meters. The specific optimal cutting plan is given according to the contents of "initial cutting plan, adjusted cutting plan, cutting loss" and other contents respectively.
Appendix: Parameters and Requirements
Process Parameters: It takes 3 minutes for the cutting machine to cut off a billet, and it takes 1 minute to return to the starting point after cutting. The length of the slab from the center of the crystallizer to the starting point of the cutting machine is 60.0 meters, and the speed of continuous casting and casting is 1.0 m/min. When the crystallizer is abnormal, the length of the scrap section is 0.8 meters.
Basic requirements: The length of the billet after cutting must be between 4.8 and 12.6 meters, otherwise it cannot be transported away and hinders production. The acceptable billet length for the next process is 8.0~11.6 meters. If it is not within this range, the billet can be transported away for secondary off-line cutting, but the cut part will be scrapped, resulting in losses. For example, if a billet of 12.6 meters is cut off by 1.0 meters to become 11.6 meters, the cut 1.0 meter is scrapped; while billets smaller than 8.0 meters can only be scrapped entirely.
Normal requirements: Normal cutting is to cut according to the length required by the user. User requirements include the target value and target range. The cutting length of the billet should meet the target value as much as possible, and the length within the target range is also acceptable. For example, if the target value is 9.5 meters and the target range is 9.0~10.0 meters, then the cutting length should be 9.5 meters as much as possible, and the length between 9.0~10.0 meters is allowed. Losses occur when the billet length is not within the target range. For example, the billet length is 11.6 meters, and the extra 1.6 meters are scrapped.
Overview of the overall solution process (abstract)
This paper mainly aims at the problem of continuous casting cutting, based on the parameters given in the appendix, the requirements of billet length and different target values of users, a mathematical model is established to formulate a specific optimal cutting plan.
Aiming at problem 1, firstly, the multi-objective programming model is established with the minimum cutting loss of tail damage, the largest number of roots with a cutting length of 9.5 meters, and the largest number of roots with a cutting length of 9.0-10.0 meters. Then, through the sequential solution method, the optimal solution is solved with the minimum cutting loss as the objective function and the tail blank length as the constraint, and the obtained optimal solution is regarded as the constraint condition and put into the first goal (that is, the root with a cutting length of 9.5 meters). maximum number) to find the optimal solution. Then, the optimal solutions of the first two objectives are regarded as constraints and put into the third objective to solve the optimal solution, and the optimal solution of the multi-objective programming model is obtained. Finally, the lengths of the 12 tail blanks were brought into the model, and the optimal cutting scheme and cutting loss of different tail blanks were obtained by using matlab.
For question two. For the first question, the multi-objective programming model is established based on the three objectives of the minimum cutting loss of steel damage, the maximum number of cutting machines with a cutting length of 9.5 meters, and the maximum number of cutting lengths of 90-100 meters. Then, analyze the remaining 44.8-meter billet after removing the 0.8-meter scrap section, and use matlab to give a specific cutting plan, see Table 2. On this basis, attach the 0.8-meter scrap section to the 4.8-meter minimum loss section Excision, given the specific cutting scheme, see Table 3. Finally, combine the 60.0m or 64.0m steel billet before the scrap section with the required cutting length (45.6m) to get the final optimal cutting plan. For the second small question, first use the model of question 2 to get the cutting scheme of the new section of steel failure after the emergence of a new scrapped section, see Table 5. Based on this scheme and related process parameters, a real-time optimal cutting scheme is further obtained. See Table 6. Then, in order to obtain the optimal adjustment scheme of the current steel damage cutting system, two methods are used to find the initial damage cutting scheme. Method 1: Assuming the nth time window (n≥3), only the n-1th time window has no abnormalities, and all other time windows have abnormalities. Get the specific cutting plan after removing the scrapped section of 0.8m from the 3rd to the 9th window, each window is 2 abnormal moments before, and then add the cutting plan of 7 time periods to each time period before this time period The cutting scheme of the unit, the initial cutting scheme is obtained, see Table 7. Method 2: Assume that all time windows are normal. Calculate the sum of the cutting schemes of all time units before the last 7 windows respectively to obtain the initial cutting scheme, see Table 9. Finally, through comparative analysis, it is found that the cutting scheme obtained by method 1 is closer to the adjusted cutting scheme. Accordingly, the initial cutting plan obtained by method 1, the adjusted cutting plan and the cutting loss are integrated to obtain the specific optimal cutting plan at all times.
To solve the third problem, firstly, according to the requirements of users, the multi-objective programming model is established with the three goals of small cutting loss, maximum number of cutting lengths of 8.5 meters, and maximum number of cutting lengths of 8.0-9.0 meters. . Then use the better method obtained from question 2 (i.e. method 1) to obtain the initial cutting plan, the adjusted cutting plan and the final cutting plan and calculate the cutting loss, see Table 14. In the same way, the user's target value is 11.1 meters and the target range is 10.6-11.6 meters. The optimal cutting plan.
Model assumptions:
1. When there is an abnormality in the crystallizer, the scrap section will appear immediately.
2. When the crystallizer is abnormal at time 0.0, the cutting machine waits for cutting at the starting point of work.
3. Only one decimal place is reserved for billet length.
problem analysis:
The analysis of problem 1
is to formulate the optimal cutting plan according to the length of the tail blank under the condition of meeting the basic requirements and normal requirements, because the cutting loss needs to be given priority in the cutting plan, and the cutting loss (the length of the steel billet) is required to be as far as possible Second, consider the user's requirements. Under the same cutting loss, the user's target value should be satisfied as much as possible. Therefore, a multi-objective programming model can be established. Through the sequential solution method, the targets are sequentially transformed into multiple linear For the planning model, taking the 109.0-meter long house as an example, the preliminary cutting plan is obtained with the goal of minimizing the cutting loss. Then take the steel billet length of 9.5 meters as the target to obtain a further cutting plan, and finally draw the final cutting plan with the steel billet length in the range of 9.0-10.0 meters as the goal.
Analysis of Question 2
First of all, it is necessary to obtain the cutting plan of the steel billet when the scrapped section appears for the first time. It can be considered that when the mold is abnormal, a section of the steel section inside the mold will be scrapped immediately, and the steel section after cutting will be scrapped when it enters the next section. The scrap section cannot be included in the first process. If there is a scrap section in the billet, the remaining billet can meet the length required by the next process through off-line secondary cutting. Therefore, we can consider attaching the scrap section of 0.8 meters to other billets. , for the remaining 44.8 meters of steel damage, a multi-objective programming model can be established to obtain a specific cutting plan. If there is a cutting loss in the specific cutting plan, the scrapped section is attached to the minimum loss section and cut off together. When there is no cutting loss, Attach it to the longest section and cut it, and then separate it from the steel that meets the requirements by offline secondary cutting. In addition, when the crystallizer is abnormal, the scrap section is produced in the crystallizer, and steel damage has already occurred in the secondary cooling section before this scrap section. Because the length of the broken steel from the center of the crystallizer to the working starting point of the cutting machine is 60.0 meters, and it takes 3 minutes for the cutting machine to cut a billet, and it takes 1 minute to return to the working starting point after cutting, the cutting machine and the continuous casting pull back The speed of the synchronous motion is 1.0 m/min. It is impossible to judge the position of the cutting machine when there is a scrapping section. Here, we can only consider the two situations where the cutting machine is waiting for cutting at the starting point and just starting to cut the steel billet. Therefore, there is still a steel billet with a length of 60.0 meters or 64.0 meters before the scrapping section. , and then use the model to find out the specific cut-off plan for 60.0m and 64.0m steel return, and then add the specific cut-off plan for 60.0m and 64.0m steel return to the specific cut-off plan for 44.8m, that is to say, the scrapped section where steel failure first occurs , the cutting scheme of this section of billet.
Secondly, to obtain the cutting plan of a new section of steel billet after a new scrapped section appears, it is similar to solving the model established by the cutting plan of this section of steel billet when a scrapped section appears for the first time. Consider attaching a 0.8-meter scrapped section For other steel defects, a specific cutting plan is given for the remaining steel defects.
Next, it is required to adjust the billet adjustment plan for the current section after the scrapped section. The adjustment of the plan can be understood as the result of some of the nine time points changing from no abnormality to abnormality. Two methods can be used to solve the problem. First, The definition takes each abnormal moment as the time window, then there are 9 windows, and two adjacent abnormal moments as the time unit, then there are 8 time units. For the understanding of the initial cutting plan, the first method is considered to be only in the current section of billet Under the condition that there is no abnormality in the penultimate time window, and all previous time windows have abnormal conditions, from the 3rd window to the 9th window, each window is calculated from the first 2 abnormal moments to remove the 0.8m scrapped section The length of the steel defect is obtained by obtaining the length of 7 steel defects, and then the optimal cutting plan is obtained according to the model established by solving the cutting plan of the steel defect when the first scrapped section of the steel defect occurs in the second problem. Then, the cutting plan of the seven time periods is added to the cutting plan of each time unit before the time period, and the second method considers that under the condition that there is no abnormality in all time windows, the total of the third window to the ninth window is 7 windows, respectively calculate the sum of the cutting schemes of all time units before the 7 windows. As for the understanding of the adjusted cutting plan, both methods consider that under the condition that all time windows are normal, there are 7 windows in total from the 3rd window to the 9th window. The sum of the cutting schemes of all previous time units. Then according to the established model, the initial cutting scheme and the adjusted cutting scheme of method 1 and method 2 are obtained respectively, and finally the adjustment scheme of the current billet is obtained by comparison.
Finally, it is necessary to give the real-time optimal cutting plan when the crystallizer is abnormal. Since the position of the cutting machine cannot be judged when the crystallizer is abnormal from time 0.0, only the cutting machine is waiting for cutting at the starting point. The length of the steel slab from the center of the crystallizer to the working starting point of the cutting machine is 60.0 meters, and it is known that the casting speed of the continuous casting billet is 1.0 meters per minute, so it takes 60.0 minutes for the scrap section that appears at 0.0 to reach the working starting point, so you can first The steel with a length of 60.0 meters from the center of the crystallizer to the working starting point of the cutting machine is also cut, and then a real-time optimal cutting plan is formulated according to the optimal cutting plan for each subsequent time unit.
Analysis of Question Three
Under the condition that the real-time optimal cutting plan and the moment when the crystallizer is abnormal are the same as the second question, and the basic requirements and normal requirements are met, the corresponding solutions are given respectively according to "initial cutting plan, adjusted cutting plan, and cutting loss". The target value is 8.5 meters to 11.1 meters, and the target range is 8.0-9.0 meters and 10.6-11.6 meters. The specific optimal cutting plan is the same as problem 2. The user's target value is 9.5 meters, and the target range is 9.0-10 meters. The model established by the specific optimal cutting plan. First of all, according to the user's requirements, a multi-objective programming model is established with the three goals of minimum cutting loss of steel damage, maximum number of roots with a cutting length of 8.5 meters, and maximum number of roots with a cutting length of 8.0-9.0 meters. Then use the better method obtained in question 2 to solve the initial cutting plan and the adjusted cutting plan according to the established model and calculate the cutting loss. Similarly, the user's target value is 11.1 meters, and the target range is 10.6-11.6 meters optimal cutting scheme.
Model establishment and solution Overall paper thumbnail
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Program code: (code and documentation not free)
k=[44.8 52.2 32.1 58.5 41.7 31.9 56.4];
k=k(1);%现在求的是 0.0-45.6 这个时间段中除去开头 0.8 米报废段的钢还的最优切割方案，如果要求其它时间段钢坯长度的最优切割方案只需要更改k 中括号内的数字
f1=zeros(1,79)
f1(1,1:32)=[4.8:0.1:7.9]
f1(1,44:79)=[9.1:0.1:12.6]
intcon=[1:79];
a=[];
b=[];
aeq1=[4.8:0.1:12.6];
beq1=k;
lb=zeros(79,1);
ub=[];
[x1,y1]=intlinprog(f1,intcon,a,b,aeq1,beq1,lb,ub);
x1=x1',y1;
f2=zeros(1,79);
f2(1,38)=-1;
c2=zeros(1,79);
c2(1,1:32)=[4.8:0.1:7.9];
c2(1,44:79)=[9.1:0.1:12.6];
aeq2=[4.8:0.1:12.6;c2];
beq2=[k,y1];
[x2,y2]=intlinprog(f2,intcon,a,b,aeq2,beq2,lb,ub);
x2=x2';
y2=-y2
f3=zeros(1,79);
f3(1,33:43)=-1;
c3=zeros(1,79);
c3(1,1:32)=[4.8:0.1:7.9];
c3(1,44:79)=[9.1:0.1:12.6];
d(1,38)=1;
aeq3=[4.8:0.1:12.6;c3;d]
beq3=[k,y1,y2];
[x,y]=intlinprog(f3,intcon,a,b,aeq3,beq3,lb,ub);
x=x';
y=-y