Optimization of two-dimensional irregular bin packing problem considering slit distance and free rotation of pieces

In this paper, we present a two-dimensional irregular bin packing problem (2DIBPP) that takes into account the slit distance and allows the pieces to rotate freely. The target is to arrange a specified collection of pieces with irregular shapes into a minimal number of bins. Firstly, we develop a mathematical model for the 2DIBPP that considers slit distance and free rotation of the pieces, and an equidistant edge expansion approach is then proposed to handle the slit distance. Secondly, a two-stage method is implemented to get a finite collection of promising rotation angles, effectively decreasing the search neighbourhood. Thirdly, we decompose the 2DIBPP into two sub-problems: piece assignment and packing. The Partial Bin Packing (PBP) strategy is employed in the allocation stage, and we adopt an overlap minimization method to pack the pieces into an individual bin. Finally, we use a local search (LS) algorithm to advance the quality of the solutions by adjusting the piece assignment across bins. Experimental evidence exhibits that our approach is competitive in most instances of the literature, with four better results in five benchmark instances.


Introduction
Steel production causes about 6% of manufactured CO2 emissions yearly, and one ton of steel production emits about 1.8 tons of CO2 (Quader et al., 2016). Improving material utilisation can help reduce steel production and CO2 emission, which in turn do a favour to combat global climate change. Additionally, recently there has been a high level of steel waste and low material utilisation in manufacturing, which is detrimental to the sustainable development of enterprises and the environment.
Manufacturing enterprises use various pieces in production, mainly produced through the packing and cutting process. As is well known, piece packing and cutting problems are widespread in industries such as shipbuilding, textiles and glass, where packing is the basis for cutting, and the generation of highly utilisable packing solutions is the key to material saving. Therefore, it is of great importance to research the piece packing problem. The two-dimensional irregular piece packing problem can be divided into the strip packing problem (2DISPP) (Umetani & Murakami, 2022;Elkeran, 2013;Pinheiro et al., 2016) and the bin packing problem (2DIBPP) (Martinez-Sykora et al., 2017;Abeysooriya et al., 2018;Zhang et al., 2022;Liu et al., 2020). Although the packing problem has received considerable attention, most studies have been carried out on the 2DISPP. This problem aims to pack pieces within a strip stock sheet accompanying infinite length and fixed width to obtain the shortest packing length. During the packing process, the following two constraints should be met together: (1) The pieces do not overlap; (2) The pieces cannot exceed the contour of the stock sheet. Given this problem, scholars have made numerous research achievements. Umetani and Murakami (2022) proposed a dual scanline representation to solve the 2DISPP of rasterized patterns and developed coordinate descent heuristics for the raster model. An optimization method incorporating the cuckoo search and guided local search was proposed by Elkeran (2013). A pairwise clustering approach was introduced to group identical polygons together. Pinheiro et al. (2016) proposed a random key genetic algorithm (RKGA) combined with the Bottom-Left (BL) (Jakobs, 1996) algorithm to solve the packing problem. Meanwhile, a compression algorithm running in RKGA was proposed to improve the local solution quality. In addition, some scholars have attempted to use mathematical programming methods to solve the 2DISPP. These models were composed of mixed integer programming (MIP) models with linear objective functions and mixed integer constraints (Alvarez-Valdes et al., 2013;Cherri et al., 2016;Leao et al., 2016;Rodrigues & Toledo, 2017;Bennell et al., 2018) and non-linear programming models with non-linear objective functions (Cherri et al., 2016;Leao et al., 2020). However, the limitation of these models was obvious, as these models could only solve small-scale piece packing problems and were not able to help with a larger number of pieces. Moreover, solving the programming models was costly in terms of time. For more detailed information, please look up the literature (Leao et al., 2020), which describes various programming models for solving the packing problem.
As to the 2DIBPP, pieces need to be packed into several bins of fixed length and width, satisfying the two constraints like 2DISPP, aiming to obtain the minimum number of bins and the optimal solution. This problem is common in industrial production and has applications in several enterprises. Bennell et al. (2018) proposed a beam search algorithm and successfully applied it to multi-bin and single-bin size instances. Still, the limitation was evident because it could only handle convex pieces and was solely adapted to the guillotine cutting process. However, this study did not allow the pieces to rotate. Some studies allowed pieces to rotate by a limited number of special angles (Zhang et al., 2022;Liu et al., 2020), such as 90º, 180º and 270º. Zhang et al. (2022) introduced a waste least first decreasing (WLFD) scheme to allot pieces and used a greedy method to exchange pieces between two bins. In addition, a hot start along with an iterative doubling search strategy was submitted to accelerate the packing. A heuristic algorithm was used by Liu et al. (2020) to solve the 2DIBPP, in which a First Fit Decreasing (FFD) strategy was utilised to distribute pieces to bins. As far as we know, up to now, only two studies (Martinez-Sykora et al., 2017;Abeysooriya et al., 2018) allowed pieces to rotate freely. Martinez-Sykora et al. (2017) compared five methods of assigning pieces and used the MIP model to pack the pieces into bins. Abeysooriya et al. (2018) considered piece allocation and packing together and introduced the Jostle strategy to the 2DIBPP for the first time. What is more, a diversification mechanism was introduced to improve the Jostle method's performance.
The above literature has achieved many achievements based on heuristic algorithms and mathematical programming models. However, most of these have been done based on arranging pieces as closely as possible to produce a solution without considering actual manufacturing parameters. When we study the packing problem, it is valuable to consider the actual manufacturing parameters rather than only the geometry of pieces (Anand & Babu, 2015). In the piece cutting process, as the cutting machine's cutter has a definite width, to escape damaging the pieces, a specified distance needs to be reserved between pieces and between pieces and the edges of the plate, which is referred to as slit distance in this paper. All the studies of this paper are based on considering slit distance. This paper studies the 2DIBPP that considers slit distance and allows the pieces to rotate freely. As far as we know, this is the first research that considers the two elements simultaneously. We have addressed and solved a common and pragmatic problem for many industries. Specifically, we have carried out the following work. Firstly, we propose a mathematical model considering the two elements, and an equidistant edge expanding method is introduced to deal with the slit distance. Secondly, we decompose the 2DIBPP into two sub-problems: piece allocation and piece packing. In the piece allocation stage, Partial Bin Packing (PBP) method is used, and the superiority of this method is proved by comparison with the Direct Constructive Heuristic (DCH) strategy. In the piece packing stage, we use the BL and overlap minimization algorithms. Thirdly, a twostage method is proposed to obtain a finite set of promising rotation angles, effectively narrowing the search neighbourhood. By comparing the results obtained by free rotation and limited angles strategy, the effectiveness of the former is demonstrated. Finally, we use a local search algorithm to progress the final solution quality by adjusting piece allocation across bins.
The paper's structure is established along these lines. We first execute a detailed problem description and define some notations in Section 2. In section 3, we introduce some essential geometric pre-processing methods and tools. After that, we solve the piece assignment and packing problems in Section 4 and Section 5, respectively. Then, we introduce a local search strategy in Section 6. In Section 7, we analyse the experimental design and computational results. Finally, conclusions are drawn in Section 8.

Problem description
Following the terminological conventions of previous studies, we will refer to the plates for packing pieces as bins in the following, and the pieces are represented by polygons. Before formally describing the 2DIBPP, some applicable definitions are firstly given.
Definition 1 (translational operator ⊕ ). Given a polygon i p and a translation vector Definition 2 (rotation operator ( ) i i p θ ). Let the set of allowable rotation angles for piece i p be i ϑ , then the rotation angle set , ,..., , 1,..., In addition, let the quantity of pieces packed into the th j bin be j n . Let the reference point As mentioned, piece packing is the basis for cutting operations. According to the requirements of the cutting process, a defined slit distance should be reserved between different pieces and between the pieces and the boundaries of the bin. Let the slit distance between any pieces a p and b p be 1 ( , ) Based on the above contents, we then defined the problem as follows.
min N (2) subject to and a n θ ϑ θ π ∈ ∈ ≤ ≤ (6) 2 , 1 a p r a n ∈ ≤ ≤  (7) N Z ∈ + As can be expected, it is possible to get solutions with the same quantity of bins. Considering the reuse of the residuals, we cut the least utilised bins horizontally or vertically to disconnect the unused portion of the bins for future use. Eqs.
* P is the percentage of utilisation corresponding to the lowest utilised bin after the bin has been disconnected vertically or horizontally. Clearly, in results with an equal quantity of bins, from the point of material reuse, we want to keep the solution with F as large as possible and K as small as possible, i.e., a solution with the smaller * P will be selected. It is clear that the solution corresponding to Fig. 1(b) has a smaller * P than Fig. 1(a), so we tend to retain the result of Fig. 1

Equidistant edge expansion caused by considering slit distance
During the practical cutting process, slit distances 1 d and 2 d should be reserved. According to the relevant industrial production guidelines, the relationship between 1 d and 2 d could satisfy the following expression: 2 1 1 2 d d = .
In the packing process, edges of irregular pieces are expanded by 1 1 2d outwards in translation. The expanded piece is extended so that each set of adjacent edges intersects at a point, completing the equidistant edge expansion of the piece.
As to the equidistant edge expansion of a piece, Hansen & Arbab (1992) proposed a proven method called the pair-wise offset method. The method is generally divided into three stages. First, offset each side of the polygon in the equidistant direction, check each equidistant line's self-intersection, and finally remove the invalid intersection loops. This method can satisfy the general equidistant offset problem.
In the process of piece equidistant offset, the subsequent cutting process determines that the distance of outward expansion of pieces is small. Therefore, an offset algorithm is proposed for the equidistant edge expansion of pieces. The coordinate values of each vertex of the pieces are known, and the coordinate values after the offset of each side can be found using the principle of geometric calculation. In this process, the vector cross product in geometry is used, which can be used to judge the relative position of two line segments. We can see from Fig. 2(a) that the cross product of vector 1 α  and 2 α  is ( ) α  lies in the counterclockwise direction of 2 α  , as is shown in Fig. 2(a). When the equidistant offset of a piece is carried out, firstly, all the vertex coordinates of the piece will be traversed, and then each pair of adjacent edges will be regarded as a vector, and the vector cross product will be performed. At the same time, the bisector vector of the angle between adjacent edges should be calculated. After the trigonometric operation of the angle and the cross product sign judgement of the adjacent edge vectors, the coordinates of the offset point of each vertex can be gotten.
To make the best use of space during the packing process, we hope that b p and a p are arranged tangentially after the equidistant offset. Therefore, points on the outline of

Penetration depth and penetration vector
In order to measure the overlap created during the piece packing process, the penetration depth and penetration vector (Zhang et al., 2022;Leung et al., 2012) are introduced in this paper. When pieces a p and b p intersect, we illustrate the penetration PD p p as the minimal distance travelled to dissociate a p and b p .
Definition 3 (penetration depth and penetration vector). The penetration depth between two intersecting pieces a p and b p can be defined as This research utilises the square of the penetration distance for measuring overlap. Let the set of pieces packed into bin j b be j P , the set of piece reference point placement positions be j V and the set of piece rotation angles be j R . We denote ( ) , , ab j i i h P V R as the overlap between a p and b p , which can be denoted using the penetration depth as In the same way, the overlap between a p and j b can be described as . Then the expression of the overlap is shown in Eq. (13).

Assignment strategy
In studying the 2DIBPP, such an approach is often adopted: packing pieces into a bin until it is full and cannot contain more pieces, then closing it and opening a new one to carry on the packing process (Parreno et al., 2010). In this paper, we first assign pieces to bins and then pack the pieces into every single bin. That is, the assignment and packing are independent of each other, and the assignment process is executed before packing. As the assignment is mainly based on piece area, and pieces vary in shape, especially when concave surfaces are present, inevitably, one or more pieces cannot be packed into the specified bin. Given this situation, we also design corresponding modified measures.
The general framework of our method is adapted from Martinez-Sykora et al. (2017).

1DBPP
Firstly, each piece is approximately represented by its area i s , then the First Fit Decreasing (FFD) (Kang & Park, 2003) algorithm is used to obtain the maximal bin number needed to pack every piece, designated as . If piece i p is packed into j b , the value of ij w is one, otherwise zero. Then the integer programming (IP) model for distributing pieces to different bins can be expressed as 1DBPP, described as follows.
The formula (14) minimizes the bin volume used to pack all the pieces. Constraint (15) ensures that the pieces' total area packed into a bin will not exceed the bin's space. Constraint (16) ensures that if a piece is packed into a specified bin, that bin will appear in the eventual solution. Constraint (17) makes sure that every piece will be packed into a bin, i.e., all pieces will be involved in the packing process. Moreover, constraint (18) ensures that j e and ij w are binary variables. It is worth noting that this solution would be optimal if we find an optimal solution to this IP model and successfully pack the pieces allocated to each bin into it.

Piece assignment strategy
In this section, we use the PBP (Martinez-Sykora et al., 2017) strategy to assign pieces bins, which allocates pieces to bins through working out an IP model of the 1DBBP.
In the assignment process, the greedy method tends to assign smaller pieces first, which allows for a larger number of pieces to be assigned to a bin but leaves larger pieces behind, ultimately resulting in an overall poor solution and secondary allocation. As a result, the PBP averts this circumstance by concentrating on allocating pieces to only one bin and employs an objective function which tends to assign bigger pieces first, as is shown in formula (19). max s is the piece area with the largest area in the piece set P . i w is a two-valued variable that equals one when i p is allocated to the bin and zero otherwise. In fact, piece allocation is a knapsack problem, aiming to maximize the piece area assigned to a bin.
This section also briefly describes another direct construction heuristic (DCH) strategy. Unlike PBP, DCH does not pre-assign pieces but solves piece allocation in the packing process. DCH directly packs pieces into bins in a given sorting order according to these pieces' area, using the packing approach in Section 5. The steps of the DCH are described below.
(1) Arrange all pieces in a non-increasing sequence of area, and open a bin.
(2) For each piece i p , pack it into the open bin using the packing algorithms. If a bin is filled, close it and open a new one. (3) Repeat procedure (2) until all pieces are arranged into the bins.

Getting a definite collection of promising rotation angles
In this paper, we consider two cases regarding the rotation of pieces. Firstly, we let the pieces rotate by some limited rotation angles, denoted as limit ϑ , and the detailed information of these angles can be seen in Table1 from Martinez-Sykora et al.
(2017). Next, we consider the case where the pieces can rotate freely.
In the first case, we generate a feasible placement position set using the packing algorithms and record the best results for each rotation angle set.
There are many possible rotation angles in the free rotation case, and only a subset can be assessed. Based on the mechanism introduced by Abeysooriya et al. (2018), we propose a two-stage method to reduce infinite rotation angles to a finite collection, whose principle is to identify a promising angle subset in accord with the piece distribution in the present partial solution.
In the first stage, pack the first piece a p . Place the longest side of the piece parallel to the bin's edge. Choose a random type of discharge if there is more than one longest edge.
In the second stage, pack the other pieces. We start this stage by packing piece b p in a momentary position and angle using the method in the first case, ensuring that b p touches the boundary of the pieces placed earlier and the best rotation angle is selected. As b p touches the boundary of other pieces, two new orientation angles will be defined by each touching point or edge.

Fig. 4. (a) An edge-vertex combination, (b) A vertex-vertex combination, (c) An edge-edge combination, (d) Angles generated by the bin's edge
An edge-vertex combination, a vertex-vertex combination and an edge-edge combination can be seen in Fig. 4. Piece a p consists of counter-clockwise direction of edges, while b p is composed of clockwise direction of edges. In each case, b p is rotated by an angle β in the counter-clockwise direction and an angle γ in the clockwise direction. In other words, if the direction of edges directs away from the point or edge of contact, the piece is rotated counter-clockwise by an angle of β. If the direction of edges directs towards the point or edge of contact, the piece is rotated clockwise by an angle γ. Two independent touching points and consequent four new orientations can be seen in the second example of Fig. 4(a). The above constitute new candidate rotation angles, and according to the placement program, the best placement angles and positions for pieces of these angles are obtained. These are compared with the short-lived placement positions and rotation angles. If new angles fail to provide a superior position, the packing algorithms take the momentary position and rotation angle as the final result. If b p contacts the bin's edge, the angles generated by the bin's edge will also be considered, as displayed in Fig.  4(d).
Through this method, a definite collection of promising rotation angles free ϑ is obtained, narrowing the scope of the search neighbourhood.

Placement of a single bin
This paper uses the overlap minimization method (Elkeran, 2013;Zhang et al., 2022;Liu et al., 2020;Leung et al., 2012) to pack assigned pieces into bins, which is widely used in solving the 2DISPP.

Initial solution generation using BL algorithm
We first use the BL algorithm in Algorithm 3 to produce an original layout for every piece. Algorithm 3 attempts to place the allocated pieces into one of the bins, and the candidate placement positions of the pieces are obtained using NFP . When the BL algorithm is called, the length of the bin is regarded as infinite, so it is possible to produce an initial layout where pieces are discharged beyond the boundary of the bin. Under this condition, we consider that the pieces overlap the bin. We use Overlap to measure the overlap. To achieve a reasonable layout, we first randomly select two pieces from the pieces packed in the bin, then switch their positions by invoking the Swap algorithm and minimize the Overlap using the Separation algorithm. If and only if Overlap equals zero, the solution will be accepted. At this time, a constant True will be returned, otherwise False . Additionally, the iterations of random search are controlled by number . Algorithm 2 describes the overlap minimization method in detail.
Algorithm 2: Pack the pieces in the piece set j P assigned to bin j b to obtain the final position j V and rotation angle j R of the pieces Input: The assigned piece set j P and the bin j b Output: The final position j V and rotation angle j R Randomly select p and p from P return True 12 return False Algorithm 3 details the process of generating initial positions and angles for the pieces using the BL algorithm. Before this, the pieces are first arranged in a non-increasing order of area to obtain a corresponding order and then placed in the bin one by one in this order. For each piece in the piece set j P , the set 1 Q of vertices and intersections of the

Swap algorithm
The initial solution generated by the BL algorithm has a non-zero overlap in many cases, which requires using the Swap algorithm (Leung et al., 2012) The method Move (Leung et al., 2012) In equation (22) Get the vertices and midpoint of edges add to Q of IFP

Separation algorithm
This part employs the Separation algorithm to detach the overlapping pieces after the Swap operation, which is proposed by Leung et al. (2012). By fixing the rotation j R in Eq. (13), we can obtain Eq. (23), in which we omit j P and j b for the sake of expression.
An unconstrained non-linear programming model is utilised to deal with this problem. The following are the detailed steps.
(1) Express the ( ) ab j h V and ( ) a j k V in terms of penetration depth as follows. (2) The penetration vector v is used to replace the penetration depth in step 1 and is denoted as Thus,

( )
ab j h V and its gradient can be expressed in terms of Similarly, ( ) a j k V and its gradient can be expressed as follows.
( ) (3) The limited memory BFGS (L-BFGS) (Boggs & Byrd, 2019) method is invoked in steps 1 and 2 to solve the unrestricted nonlinear programming problem (23) to get the minimum value.

Local search algorithm for solution improvement
In this section, we use a local search (LS) algorithm to better the final solution, modified from LS2 by Martinez-Sykora et al. (2017). Combining Eq. (9) to Eq. (11), we can see that the F value increases when either of the two cases occurs. First, the number of bins used is reduced. Second, the number of bins remains unchanged, but the utilisation of one of the bins increases. In both cases, the new solution is accepted.
Given a solution containing N bins, sort these bins in a non-decreasing sequence of use ratio, i.e., 1 In the LS algorithm, we first select the least utilised bin c b and then try to move pieces from c b into a collection of bins Array . Once all the pieces in c b have been considered, we consider moving the pieces removed from each bin back to their original place. Again, if the movement is unsuccessful, the piece is also temporarily stored in 2 Array .
In this case, we start to check if any bin in B′ has increased in utilisation. If none of these bins has been utilised more, the move is invalid. If 2 Array is empty, one bin has been reduced, and the new solution is accepted. Otherwise, the pieces in 2 Array are loaded into a new bin, and the above process is repeated, i.e., one piece is loaded into one bin at a time in the nonincreasing sequence of its area. If a bin cannot maintain all the pieces from 2 Array , the new solution requires one more bin than the original one and is discarded. Meanwhile, a False is returned. If a bin can hold all the pieces from 2 Array and the final F value increases, the new solution is accepted, and a True is returned. Algorithm 6 shows the procedure of the LS algorithm.
with a 2.90 GHz processor and 8 GB of RAM to implement them.

Data
We have used two benchmark instances, the nesting instances that are widely used in the 2DISPP, and the instances offered by López-Camacho et al. (2013). They are available from the ESICUP (EURO Special Interest Group on Cutting and Packing) website https://www.euro-online.org/websites/esicup/.
There are 23 datasets in the nesting instances, and the jigsaw puzzle instances consist of two subsets, JP1 and JP2. Subset JP1 is made up of 540 instances, divided into 18 classes with 30 problems, and all the pieces are convex. JP2 has a collection of 480 instances (18 classes with 30 problems), and there are convex and non-convex pieces. In addition, the number of pieces in each class is different. In the jigsaw puzzle instances, the bins' length and width are set to 1000 fixedly. In contrast, the sizes of the nesting instance bins are needed to be defined.
Referring to the paper of Martinez-Sykora et al. (2017), for each instance, three kinds of square bins are defined depending on the maximal dimension of the enclosing rectangle of the pieces in their initial orientation, represented as max d . In addition, pieces and bins are dimensioned in uniform units.
Due to the three kinds of bins, 69 instances of bin packing are produced from the 23 nesting instances. Table 1 from Martinez-Sykora et al. (2017) exhibits the detailed information of these instances, including the piece number, the dimension of three bins and the limited rotation angles limit ϑ for each instance.

Comparison between PBP and DCH
This section compares the results obtained by the PBP and DCH algorithms in Table 1. We present the average bin amount ( N ), the mean value of F and K , and the mean execution time T with the unit in second. As far as indicator F is concerned, all the results obtained by the PBP algorithm are larger than those obtained by the DCH, although the gap in some instances is not very obvious. For N and K , PBP also gets better results, except for a few N values that agree with the results obtained by the DCH. As for the computation time T , compared with the DCH, most of which the PBP takes is shorter, except for the JP1 instance. The PBP outperforms the DCH in all parameters regarding the average value. PBP implements piece assignment by solving the 1DBPP one bin at a time and reduces the method's greediness by modifying the objective function. DCH does not pre-assign pieces and only places the pieces in the first suitable bin, which results in more effort to find a bin to arrange a piece, thus boosting the computing time. As a result, we come to the conclusion that it is beneficial to allocate pieces to bins before packing. In the process of actual industrial production, we can also consider using this strategy to improve the utilisation rate of plates. Because the experimental results have proved that the PBP is superior to the DCH, it is used in our subsequent studies and analyses.

Comparison between limited rotation angles and free rotation
Through the approach in Section 5.1, we get a finite set of promising rotation angles for each piece. To evaluate the effect of allowing pieces to rotate freely, the pieces are firstly rotated at limited angles of 90°, 180° and 270°, and the results are compared with those obtained by free rotation. Table 2 displays the results. The nesting instances are used in the comparison, as they provide angles at which each piece set can be rotated. In the Limited angles column, the angles allowed for each piece set are exhibited in Table 1 from Martinez-Sykora et al. (2017). The results obtained by free rotation are displayed in the free rotation column. The superior results are displayed in bold. As can be observed from Table 2, out of the 23 instances, better results are obtained for 16 sets in the free rotation column.
The F values for the fu and swim instances are the same for both conditions. These demonstrate that free rotation can improve bin utilisation by giving pieces more viable placement options. Because of the specific nature of the production process, companies such as garments have strict requirements on piece placement angles. While for shipyards and other enterprises, because the primary purpose of piece packing is to improve the plate utilisation rate, there is no special requirement for the placement angles. We can conclude that when there is no practical reason to set a particular rotation angle, the free rotation will eliminate any deviation in creating the piece set to optimize the packing results. It is worth paying special attention to dighe1 and dighe2. The F value obtained in the Limited angles column is much higher than that of the free rotation column.
According to Table 1 from Martinez-Sykora et al. (2017), the two instances have a permissible rotation angle of 0°, i.e., the pieces are allowed to be packed in their original orientation, which will develop the perfect or near-perfect solution. Furthermore, to avoid repetitive work, only medium-sized bins are selected by us, as this is sufficient to illustrate the superiority of the free rotation strategy.

Comparison between PBP and LS-PBP
In the following, we will compare the results of the two cases: only using the PBP and introducing the LS strategy depicted in Section 6.
( ) % Inc F represents the percentage of improvement in indicator F for LS-PBP compared with PBP.
Comparing the results in Table 3, it is clear that the LS strategy effectively improves the results. Inevitably, the computational time increases exponentially due to the additional computational effort associated with introducing this strategy. We note that processing the Nest-SB and Nest-MB datasets takes much longer than processing the Nest-LB, and this is because for the same set, the larger the bin size, the fewer bins are required, so there are fewer opportunities to exchange pieces across bins, and thus the processing time will be relatively less. In addition, we can also observe that for ( ) % Inc F , the percentage of Nest-SB and Nest-MB increase are much higher than that of Nest-LB, which means the smaller the bin size, the better the effect of crossing bin to distribute pieces. Because the smaller-size bin can accommodate fewer pieces, the pieces' bin-cross movement will significantly change the final results.

Table 3
Comparison between PBP and LS-PBP.  Table 4 because they were the best results in separate studies. As the above algorithms were written in different languages on different platforms, the experimental results are not comparable in terms of time. In addition, since the values of N and K were not explicitly listed in other literature, we only compare the F value obtained by different algorithms. As can be observed, the LS-PBP (slit) column represents the F value obtained when the cut distance is considered, and the results obtained by us are better than those of some of the above literature, but this advantage is not prominent. This is because, in our study, the pieces are not allowed to be packed as closely as possible as in all other papers. Instead, slit distance is reserved. Although the slit distance is small in relation to piece and bin dimensions, the effect of slit distance on the utilisation of the individual bin and the F value cannot be ignored when abundant pieces are packed. We also add the column LS-PBP (no slit), which describes the F value obtained when the slit distance is not taken into account, so we can compare the results with other papers under the same standard. In this case, the comparison results are more convincing. Our method achieves the optimal results in JP1, nest-SB, nest-MB as well as Nest-LB, and the gap with the optimal results in JP2 is very small. Thus, we draw the conclusion that our algorithms are somewhat competitive.

Conclusions
In this paper, we investigate a 2DIBPP that considers slit distance and allows pieces to rotate freely. Although this problem often arises in industrial production, there is no good approach to solve it. Better results are obtained using our methods, and practical industrial production could benefit from them. Specifically, we carry out the following work.
Firstly, we propose a mathematical model that considers the above two elements, and an equidistant edge expansion algorithm is introduced to handle the slit distance. Secondly, the 2DIBPP is divided into two sub-problems: piece allocation and packing. We also describe a DCH algorithm that handles the two sub-problems simultaneously and demonstrates the superiority of the PBP through comparative experiments. We then investigate the effect of free rotation versus limited angles on the nesting instances. Except for a few piece datasets, better solutions are found by allowing pieces to rotate, as the free rotation strategy eliminates any deviations in the creation of the piece dataset and thus optimises the packing results. In the packing process, a two-stage method is presented to obtain promising rotation angles for pieces, which effectively decreases the search neighbourhood and betters the results. Finally, we use the LS algorithm to improve the solution by adjusting piece allocation across bins. The data show that this strategy's introduction effectively enhances the packing results.
For future work, we plan to apply the methods in this paper to bins with irregular shapes. We also plan to investigate the problem of packing irregular pieces into bins with some defective regions.