This study aims to meet the requirements of engineering embryo materials, while minimizing cost. For the setting of the objective function, two aspects are considered: minimization of raw material number and loss.

From the perspective of the number of raw materials, the ultimate goal is to minimize the total number of raw materials. To minimize Z (which denotes the total number of raw materials) based on the previously set decision variables, the objective function can be expressed follows:

(1)MinZ=x1+x2+x3+⋯+xn

Z≥0,x≥0where Z is the total number of steel bars.

The mathematical model for minimizing raw material demand uses the total number of raw materials required for the final calculation as the objective function to evaluate the blanking plan. The mathematical method for minimizing loss summarizes all the remaining material lengths obtained from cutting, sets the remaining material length as an index, and evaluates the cutting plan. The corresponding objective function is as follows:

(2)MinS=LZ−∑i=1n(libi)where S is the total length of remaining material.

The analysis shows that, if both reach the optimal solution, the material utilization rates corresponding to the two different objective functions are equal. Therefore, the number of raw steel bars used is equal to the material loss. Because the calculation amount of the objective function set by the minimization of raw materials is smaller, in the follow-up study, the number of raw materials is selected to minimize the objective function.

Although there exist several objective functions, the final model constraints are mainly based on the number of roots that meet the engineering requirements. Steel bar blanking can be regarded as a combination of multiple effective cutting methods. Each effective cutting method involves blanks of different lengths. The combination of effective cutting methods ensures that the number of blanks with different lengths is equal to the number of blanks required by the project.

For the calculation of effective cutting, the following methods are used [30].

S1: Processing of the blank length set li:{l1,l2,⋯,lm} in descending order, make l1>l2>⋯>lm;

S2: n=1, where n is the number of cutting methods;

an1=min{b1,[L/l1]},

an2=min{b2,[(L−an1l1)/l2]},…

anm=min{bm,[(L−∑i=1m−1anili)/lm]},

The function represents Gaussian rounding, that is, rounding down.

S3: Integrated cutting plan: [an1,an2,⋯,anm];

S4: Iteration based on S3 cutting plan. If there is ani>0 at 1≤i≤m−1, let k be equal to the maximum value of all i that meet the condition ani. If there is no ani>0 at 1≤i≤m−1, the algorithm ends;

S5: n=n+1an1=a(n−1)1,an2=a(n−1)2,⋯,an(k−1)=a(n−1)(k−1),

ank=a(n−1)k−1

an(k+1)=min{bk+1,[(L−∑i=1kanili)/lk+1]}…

anm=min{bm,[(L−∑i=1n−1anili)/ln]]} to S2

Numbered lists can be added as follows.

It is assumed that bi steel bar blanks with a length of li m are required. In the first effective cutting method, a1i steel bars with a length of li meters can be cut. In the second effective cutting method, a2i steel bars with a length of li meters can be cut, and so on. In the n effective cutting method, ani steel bars with a length of li meters can be cut. The following constraints can be obtained.

(3)a1ix1+a2ix2+…+anixn=bi

Assuming that m different lengths of steel bar blanks are required, there are n effective cutting methods, and x1,x2,…,xn represents the number of raw steel bars in the corresponding plan. For all steel bars with different lengths, the following constraints can be obtained.

(4){a11x1+a21x2+…+an1xn=b1a12x1+a22x2+…+an2xn=b2…a1mx1+a2mx2+…+anmxn=bm

x1,x2,…,xn≥0where a is the number of steel bars of a certain length that can be cut by various schemes.

Under the above objective function and its constraints, a mathematical programming algorithm was used to determine the optimal solution. Mathematical planning is an important branch of operations research, and an important branch of mathematical programming is integer programming. Integer linear programming limits the decision variables in linear programming to integers, and its standard form can be expressed as follows:

(5)Minf=cTx

(6)s.t.{Ax=bxj≥0(j=1,2,3,⋯n)xj(j=1,2,3,⋯n;xjis an integer.)where c=(c1,c2,⋯,cn)T, x=(x1,x2,⋯,xn)T.

For the mathematical model of rebar cutting, the decision variable represents the number of raw materials to be cut, which must be an integer. This exactly matches the principle and idea of integer linear programming, so the integer linear programming algorithm can be used to solve the scheme first.

At this stage, there are mainly exhaustive methods and branch-and-bound methods for solving integer linear programming. Although the exhaustive method is theoretically feasible, it is far greater than the other two algorithms under the same calculation scale. A new method for solving the fully fuzzy system of linear equations using the linear programming problem approach is used.

The branch-and-bound method is one of the main methods for solving integer linear programming problems. It has been widely used in the fields of mixed-integer programming and pure integer programming. The branch-and-bound method first relaxes the original problem into a linear programming problem. Then, it iteratively corrects situations wherein the optimal solution is not all integers. When the algorithm starts to search, the optimal solution currently found is ∅. During solving, the solution space is gradually reduced. At any time, the solution space comprises a subset of the solution space that has not been explored and the optimal solution found thus far. The unsearched subspace is represented as a node in the dynamically generated search tree. The search tree includes only the root node before the algorithm starts. Each iteration of the branch-and-bound method processes a node. The iteration has three parts: branching, boundary calculation, and screening. The branch is generally based on the principle of selecting variables that do not meet the integer constraint, and pick a non-integer variable xn=bn. Let [bn] be an integer rounded down to bn, and the original feasible region is R. R is divided into R1 and R2 by xn= [bn] and xn= [bn] + 1. The non-integer area between [bn] and [bn] + 1 in R is eliminated to reduce feasible domain. The current value divides it into two subproblems for the solution. The boundary calculation aims to solve the subproblems generated by the branch. According to its optimal objective function value, the upper and lower bounds before branching are updated. Screening is based on the relationship between the optimal function value of the branch subproblem and the preset upper and lower bounds. It deletes or rebranches the branch until the optimal objective function value is obtained.

The solution process is as follows.

IL is an integer programming problem that needs to be solved. The linear programming problem corresponding to this problem is called L.

The problem is solved in two stages.

First, to solve problem L, one of the following situations can be obtained. (1)

If there is no feasible solution for L, and there is no feasible solution for the IL, then stop.

Second, iterative operations are performed for the (3) situation in the first stage. (1)

Branch. In the optimal solution of L, a variable is chosen that does not satisfy the integer condition. Assuming that its value is lj, two constraints, xj≤[lj] and xj≥[lj]+1, are established. These conditions are added to question L, and L is divided into two subproblems, L1 and L2. Regardless of the integer condition requirements, L1 and L2 are solved.

The algorithms were tested. Considering a sample project as an example, the representative data of different scales in the steel bar schedule were extracted. The length of the steel bar raw material was 12 m; there were 23 types of blank that needed to be cut; the length was set li {10.772, 9.888, 9.36, 8.14, 7.66, 6.56, 6.05, 5.82, 5.73, 5.722, 5.684, 3.165, 2.96, 2.885, 2.67, 2.59, 2.565, 2.25, 2.22, 2.146, 1.946, 1.726, 1.44}; and the corresponding quantity set was ai {2, 2, 4, 8, 8, 16, 21, 18, 21, 2, 16, 34, 912, 17, 76, 8, 17, 18, 912, 68, 68, 66, 36}.

An integer linear programming algorithm is used to build the mathematical model. First, it is necessary to calculate all feasible blanking schemes. According to the calculation in Section 2.2.2, there are 3226 feasible blanking methods.

A mathematical model is established based on the cutting method. Solving it by an integer linear programming algorithm took 15.138 s. Finally, 539 12-m-long raw materials were used, with a total length of 12 × 539 = 6468 m. The total length of the blank material required by the project was ∑i=123liai=6397.22 m. The overall utilization rate of the materials was 98.906%. The specific blanking plan is presented in Table 3.

The test calculations revealed that the integer linear programming algorithm could obtain the global optimal solution for small-scale optimization problems. To determine the global optimal solution, it is necessary to enumerate all possible cutting solutions. Thus, the number of cutting plans is equal to the dimensions of the corresponding mathematical model. Numerous blanking options for large-scale optimization problems lead to several mathematical models. The algorithm cannot be solved in polynomial time. Therefore, one of the problems to be solved is how to ensure the calculation effect of the integer programming algorithm as much as possible, while considering the calculation efficiency.

The cutting-stock problem is an NP problem. It has been shown [31] that no algorithm guarantees the solution of problems of different scales [32]. Approximation algorithms are more suitable for determining acceptable solutions within a reasonable time. Here, it is proposed that the problem of steel bar blanking can be solved by introducing a greedy strategy. A greedy strategy is introduced to solve this problem. Unlike integer programming calculations, this method does not need to calculate all the blanking plans. First, the current optimal blanking plan is calculated. The plan is then reused to the greatest extent, and the completed workload is removed from the engineering requirements. Finally, iteration is performed based on the updated demand cycle. An iteration comprises three parts: calculating the optimal plan, reusing the plan to the greatest extent, and updating the engineering requirements. The original problem is solved when the updated demand is empty. In each sequence, the optimal solution to the original problem is the set of optimal solutions.

Because the number of blanks for each blanking is limited, there is no exponential increase in the number of solutions. The optimal blanking method for each subsequence can be solved using an integer linear programming algorithm. The mathematical model is as follows:

(7)max∑i=1mailisubject to{0≤ai≤biai is an integer∑i=1maili≤L(i=1,2,⋯,m)

The optimal cutting method is calculated as a subroutine of the multilevel sequential linear optimization method, which can be used repeatedly. The overall process is as follows:

S1: Calculate the optimal cutting method.

S2: Calculate the number of repetitions of the cutting method d=Min[b1a1,b2a2,⋯,bmam].

S3: Update the remaining engineering requirements.

S4: If the updated engineering requirement is empty, the calculation is completed; if it is not empty, then cycle steps S1 to S3.

S5: Summarize the optimal cutting method and number of repetitions for each sequence, which is the optimal solution to the original problem.

Introducing a greedy strategy enables the original optimization problem to be divided into multiple sequences, and optimization calculations can be performed sequentially. In the calculation, the scale of the problem is reduced. As the scale of the problem shrinks, the advantages of the greedy strategy are reduced accordingly. Fig. 1 shows the calculation of this case. The integer programming algorithm and introduction of the greedy strategy are used to compare the remaining materials of all the solutions solved by the algorithm. The figure shows that the results based on the greedy strategy are mostly better than the integer programming solution in the initial stage. However, as the data size shrinks, after the 11th solution, the disadvantages of the greedy strategy gradually become apparent. The difference between the quality of the calculation result of the greedy strategy and that of integer programming increase continuously. The main reason for this is analyzed, and the solution is based on the greedy strategy. Although it seems that the optimal solution is selected at each stage, owing to the lack of overall consideration in the early stage to improve the quality of the solution, several short embryos that can be used to fill the gap are employed. Most remaining blanks in the middle and late stages can be optimized to a low degree, resulting in the poor quality of the blanking methods calculated in the middle and later stages, and more surplus materials are generated. However, the integer linear programming algorithm calculates all possible blanking methods in the mathematical model establishment stage. In the solution, one can comprehensively consider the combination of all blanking methods, and the quality of the overall blanking method is relatively average. As shown in Fig. 1, the integer programming algorithm uses a blanking method with a remaining material length of less than 0.2. The only solution with a residual material of 1.228 is when the longest blank is 10.772 and the shortest is 1.44 in the calculation example, which cannot be optimized further. The integer programming algorithm uses less leftover material, which ensures that the quality of the final optimization result of the algorithm is better.

When solving large-scale problems using the greedy strategy, there are serious losses in the later stages of the calculation, as shown in Fig. 1. To consider the efficiency and quality of calculation, a new approach is proposed, combining an integer linear programming algorithm with a greedy strategy. For small-scale problems, the integer programming algorithm is used; for large-scale problems, the greedy strategy is employed first. When the problem size is reduced to a certain extent, an integer programming algorithm is used to solve the cutting problem of the remaining blanks. Finally, the two components are integrated. The phase solution results are the final result.

An integer linear programming algorithm is used later. This can compensate for the serious loss in the later calculation of the blanking plan based on the greedy strategy design.

As mentioned above, the quality of the cutting method is directly related to the quality of the final calculation result of the algorithm. The calculation is based on the greedy strategy, and the loss of the scheme used in the early stage is low. The loss of the calculation scheme of this algorithm mainly occurs in the middle and later stages, such as the stage after the 14th scheme in the above calculation example. The greedy algorithm provides a solution every iteration, so the loss is serious after 14 iterations. However, the integer programming algorithm cannot be used to solve the remaining blanks directly after the 14th iteration. The reason for the serious loss is that, in the early stage, to improve the quality of the plan, several short blanks that can be used to fill the gaps are used, and most remaining blanks are in the middle and late stages. The degree of optimization is not high. Therefore, it is necessary to start using the integer linear programming algorithm to solve the problem a few rounds in advance to ensure that the short blanks are included in the blanks that must be cut at that time.

Through calculation, the loss gradually increases after 14 iterations. An integer programming algorithm is used after 6–16 iterations. The raw material consumption of the steel bars is shown in Table 4.

Fig. 2 shows that, after the 14th iteration, if the integer linear programming algorithm is used to solve the problem, 551 raw steel bars are used. Compared with the 552 bars of the greedy algorithm, the optimization effect is not apparent. If seven rounds in advance, after the seventh round, the integer linear programming algorithm is used to solve the problem, and only 539 are required. Consistent with the direct use of the integer programming algorithm, the global optimal solution is calculated, and the calculation efficiency is much higher than that of the integer programming algorithm.

From the actual engineering, 300 groups of steel engineering data of different scales were selected. For the case wherein a cutting method with a serious loss occurs in the Nth round under a different number of advance iterations, the optimal data volume achieved in 300 sets of test data is shown in Fig. 3. The number of early iterations concentrated in interval [7,21] has the most data, accounting for approximately 90%. Therefore, the value range of the final threshold T can be selected in interval [N−20, N−6]; that is, the integer linear programming algorithm is used to solve the problem after the Tth iteration, which can achieve better results.

Based on the above research and considering the characteristics of steel engineering data, an optimization algorithm for steel blanking is proposed. The specific steps are as follows:

S1: Judge the scale of the problem of rebar blanking.

S2: If it is a small-scale problem, directly call the integer linear programming algorithm to solve it, output the optimization result, and end the algorithm.

S3: If it is a large-scale problem, solve it based on a greedy strategy, and record all the cutting methods used and the number of repetitions, as well as the remaining blank length li and the number of required roots ai after each iteration.

S4: Comprehensively analyze all blanking methods, screening the number of iterations N, where serious losses begin to appear.

S5: Based on the critical threshold setting method, select an appropriate critical threshold T within the threshold interval.

S6: Based on the remaining blank length lis and the number of demand roots ais after the Tth iteration in S3, call the integer linear programming algorithm.

S7: Base synthesis S3 on the greedy strategy to solve the blanking plan before the Tth iteration; S6 uses the integer linear programming algorithm to solve the blanking plan, which is the final blanking plan of the original steel-bar-blanking problem.

The specific algorithm flow chart is shown in Fig. 4.