The Normalized Direct Trigonometry Model for the Two-Dimensional Irregular Strip Packing Problem

Abstract

Background: The Irregular Strip Packing Problem (ISPP) involves packing a set of irregularly shaped items within a strip while minimizing its length. Methods: This study introduces the Normalized Direct Trigonometry Model (NDTM), an innovative enhancement of the Direct Trigonometry Model (DTM). The NDTM incorporates a distance function that supports the integration of the separation constraint, which mandates a minimum separation distance between items. Additionally, the paper proposes a new set of constraints based on the bounding boxes of the pieces aimed at improving the non-overlapping condition. Results: Comparative computational experiments were performed using a comprehensive set of 90 instances. Results show that the NDTM finds more feasible and optimal solutions than the DTM. While the NDTM allows for the implementation of the separation constraint, the number of feasible and optimal solutions tends to decrease as more separation among the items is considered, despite not increasing the number of variables or constraints. Conclusions: The NDTM outperforms the DTM. Moreover, the results indicate that the new set of non-overlapping constraints facilitates the exploration of feasible solutions at the expense of optimality in some cases.

1. Introduction

The Irregular Strip Packing Problem (ISPP) is a real-world challenge with significant implications. Its goal is to optimally arrange a set of irregular items on a strip of material with a fixed width and infinite length. By minimizing the strip length required, the ISPP directly contributes to reducing waste and cost, making it a critical issue in industries ranging from textile and garment manufacturing to sheet metal cutting [1,2,3], where material efficiency directly translates to economic and environmental benefits.

Acknowledged as an NP-hard problem [4] the ISPP has attracted substantial interest due to its inherent computational complexity and significant real-world implications. Researchers have frequently employed metaheuristic approaches to tackle this complexity, balancing the quest for optimal solutions and manageable computational time. However, with the evolution of computational technologies and capacities, there is an increasing focus on developing more precise and efficient mathematical models [5].

In this context, this paper introduces the Normalized Direct Trigonometry Model (NDTM), an innovative adaptation of the Direct Trigonometry Model (DTM) proposed by [6]. The NDTM stands out by utilizing a distance function between a line and a point rather than relying on the D-function. The NDTM also introduces flexibility in the search space by allowing multiple separating lines between convex sub-polygons. Utilizing the distance function allows the incorporation of a separation constraint among the pieces, i.e., a minimum separation distance between pieces. Therefore, this study corresponds to the first study that involves this constraint within a MILP model based on direct trigonometry. Additionally, this study explores the incorporation of additional non-overlapping constraints that use the bounding boxes of the pieces.

The main findings of this study are listed below:

• Introduction of the NDTM: This study presents the NDTM, which has proven to be a more flexible and efficient framework than the DTM, generating more feasible and optimal solutions.
• Integration of the separation constraint: This study successfully incorporates the separation constraint into the trigonometry-based MILP model (NDTM), which is relevant for various industrial applications. Computational experiments show that this constraint decreases the number of feasible and optimal solutions despite not increasing the number of variables or constraints.
• Novel non-overlapping constraints: A new set of non-overlapping constraints based on bounding boxes is introduced, allowing the generation of more feasible solutions at the cost of optimality of some instances.

The paper is structured as follows: Section 2 provides an in-depth description of the ISPP. Section 3 reviews related literature and research. Section 4 details the NDTM, including its mathematical foundation, the implementation of the separation constraint, and the model’s modifications. Section 5 presents the results of computational experiments comparing the NDTM and the DTM, the impacts of the separation constraint, and the effects of the NDTM’s modifications. Finally, Section 6 concludes the paper with a summary of findings and implications for future research.

2. Problem Description

The Irregular Strip Packing Problem (ISPP) addresses how to pack a set of irregular 2D items into a rectangular container such that the length of the container is minimized [7]. In this problem, all items must be packed, satisfying the containment and the non-overlapping constraints. The former indicates that all items must lay within the container’s limits. The latter constraint states that the packed items must not overlap but may touch. Moreover, in this study, the items have several fixed rotations, broadening the NDTM’s applications in several scenarios. For example, the fabric may have a pattern or stamps with valid clothing orientations in the garment industry.

This study also incorporates the separation constraint, also known as the distance constraint, which mandates that at least a specified distance must separate the packed items within the container [8,9,10]. This constraint is related to the necessary space for cutting tools to extract the shapes once placed within the container. The separation between items depends on the material of the container from which pieces are extracted and the technology of the cutting tool. For example, wood may be cut with a saw and metal with a laser. Both materials/technologies require different separations among the items to extract them without altering their dimensions or shapes.

The distance constraint can be interpreted in three scenarios based on how the separation $s$ among pieces is measured. First, the edge-to-edge separation measures the distance between the edges of the two pieces (Figure 1a). This scenario is relevant when the edges are parallel, as the distance between them remains constant. Second, the edge-to-vertex separation measures the distance between the edge of one item and the vertex of the other item (Figure 1b). It is worth noting that this scenario encompasses the previous one, as the separation $s$ between parallel edges is equivalent to the separation between an edge and any vertex of the other edge. Finally, the vertex-to-vertex separation measures the distance between the vertices of both items (Figure 1c).

This study considers the edge-to-vertex separation (which also includes the edge-to-edge separation) but not the vertex-to-vertex separation. The edge-to-vertex separation effectively separates convex items and results in linear constraints, as will be shown in the methodology section. In contrast, the vertex-to-vertex separation leads to quadratic constraints based on the Euclidean distance between independent points.

It is worth noting that considering the separation constraint is not equivalent to enlarging the items to be packed. This misconception is common in some industries. While there are cases where it might be true, when dealing with pieces with sharp edges, enlarging items may imply a larger separation between pieces than the desired one. Figure 2a–c illustrate cases where an edge-to-vertex separation ($s$), as considered in this study, is achieved. In scenarios where the approach is to increase the size of the pieces, each edge moves away by half the desired separation ($s/2$). Figure 2d shows a case where this separation is appropriate and achieves the same effect as considering the edge-to-vertex separation. However, Figure 2e,f demonstrate cases where this strategy of enlarging the pieces would not be appropriate because there would be overlap between the enlarged pieces. This situation occurs because, by considering a separation ($s/2$) between the edges of the original and the enlarged piece, some vertices would be at a distance greater than ($s/2$) from the corresponding vertices. As seen in Figure 2e,f, this phenomenon is more evident when the internal angle between the edges of a piece is smaller.

Notably, the separation enforced by the distance constraint applies only between items, i.e., $ite{m}_i-ite{m}_j$, and not between items and the container. The item-container separation is not considered because, in many industrial applications, the proximity of items to the container’s edge is not a significant issue, as the edges can be seen as areas where cutting precision is not critical.

Moreover, the item-container separation can easily be managed by adjusting the container’s dimensions. For instance, if a $\delta$ separation is desired between the items and the container, the container’s width should be reduced by $2\delta$ to account for this constraint on the top and bottom of the container. After obtaining the solution with the reduced width, $\delta$ must be added to the position of each item along the width axis (y-axis). The same process of adding $\delta$ would apply to the length axis (x-axis) to account for separating the items and the container along its length. Consequently, the container’s length would be increased by $2\delta$ to include this constraint.

Another valuable constraint for industries is considering containers of different shapes other than a rectangle or creating no-packaging zones, such as damaged or inappropriate container areas. These constraints are not directly considered in this study. However, it is worth noting that these constraints can be addressed by packing items without rotation (with only one valid rotation) in fixed positions within the container.

3. Related Work

Cutting and Packing Problems (C&PPs) have gained significant attention from researchers who have developed numerous algorithms, software solutions, and practical applications [11]. These problems are especially challenging when addressing irregular shapes. The literature on two-dimensional C&PPs handling irregular shapes has focused on the IBPP and the ISPP. The former involves packing irregular items into the fewest bins possible. The latter consists of packing irregular items into a strip with minimal length. The IBPP has been solved with various approaches [1,12,13]. However, the researchers on C&PPs have concentrated on the ISPP (focus of this research). Consequently, most two-dimensional irregular packing instances are tailored to ISPP, allowing straightforward comparisons of different approaches.

ISPP is an NP-hard problem that has been addressed through various strategies. Due to its complexity, many researchers have focused on developing heuristics and metaheuristics [14,15,16,17]. Novel reinforcement learning techniques have also been employed, showing promising results [18]. Recent advancements in computing power and optimization software have enabled researchers to explore matheuristic approaches that integrate metaheuristics with mathematical models, enhancing the speed and accuracy of solving subproblems [19,20,21]. In addition, various modeling paradigms have been utilized, including constraint programming [22], quadratically constrained programming [23], and mixed integer quadratically constrained programming [24].

This study focuses on mixed-integer linear programming (MILP) for the ISPP. Some formulations are reviewed next. The Dotted-Board Model (DBM) proposed by [25] uses a raster representation method, and the solution quality depends on the grid size [26]. Efforts have been made to reduce the size of the DBM by clustering pieces [27] or employing a clique-covering model [3]. It is worth noting that the DBM’s dependence on the grid size makes it a non-exact model for the problem, i.e., an optimal solution of the DBM is not necessarily an optimal solution to the problem.

Some researchers have explored alternative geometric representations to avoid the limitations of grid dependence. These geometric representations are primarily employed to model the non-overlapping constraint. [28] introduced a semi-raster representation that uses strips continuous on the x-axis and discretized on the y-axis, allowing for a semi-continuous formulation. Alternative representations to raster ones include phi-functions [29], which offer a flexible yet complex representation of item proximity. Another geometric tool is finding separating lines among pieces [30,31]. The Direct Trigonometry Model (DTM) [6] uses separating lines based on the hyperplane separating theorem. These authors also proposed the NoFit Polygon Covering Model (NFP-CM), which was later improved with the introduction of Vertical Slices (NFP-CM-VS) by Lastra-Díaz & Ortuño (2024).

Recent literature has also explored the integration of the ISPP with the Cutting Path Determination (CPD), i.e., finding the shortest path a cutting tool must follow to cut the pieces. The CPD is generally solved after an ISPP solution is obtained. This integration has been explored through mathematical models [2] and matheuristics [32]. Additionally, new variants of the ISPP have considered uncertain demand for pieces [33] and industry-specific constraints such as defective areas within the strip, where no items can be placed [34], and a distance constraint requiring a minimum distance between pieces. The distance constraint has only been incorporated in models based on phi-functions [8,9,10]. This study is the first model incorporating this constraint in a trigonometry-based model.

4. Methodology

One approach to tackle the ISPP is the Direct Trigonometry Model (DTM) proposed by [6]. This model requires the irregular items to be represented as a union of one or several convex polygons for the non-overlapping constraint. This way, two irregular items are separated if each pair of convex polygons (one of each item) is separated. The DTM uses the D-function to ensure that two convex polygons are separated.

The D-function (Equation (1)) is used to identify the relative position of a point $p:(p_x, p_y)$ to an oriented line segment that starts at point $a:(a_x, a_y)$ and ends at point $b:(b_x, b_y)$. If $D_{abp}<0$, $p$ is on the negative side of the line $\overline{ab}$. If $D_{abp}>0$, $p$ is on the positive side of the line $\overline{ab}$. If $D_{abp}=0$, $p$ is on the line $\overline{ab}$.

$$D_{abp}=\left(a_x-b_x\right)\left(a_y-p_y\right)-\left(a_y-b_y\right)\left(a_x-p_x\right)$$

(1)

The proposed model uses the Signed Euclidean Distance function (SED), i.e., without the absolute value, between an oriented line $l: Ax+By+C=0$ and a point $p:(p_x, p_y)$ (Equation (2)). The lines associated with the edges of the convex polygons are oriented such that the polygon’s interior lies on the positive side of the lines (see Figure 3). Consequently, these lines are correctly oriented when their (SED) to the centroid of the convex polygon is positive. The centroid can be calculated as the average of all vertices of the convex polygon.

$$E_{lp}={\displaystyle \frac{A\cdot p_x+B\cdot p_y+C}{\sqrt{A^2+B^2}}}$$

(2)

The difference between the Euclidean distance function and the D-function is the interpretation and comparability of the returning value. In the former, the value is a distance and can be compared to another distance value. Therefore, two pairs of line-vertex can be compared with the Euclidean distance to determine which pair is more separated. This comparison is valid even if the lines and points are different. In the D-function, the value does not have an interpretation, and the comparisons are limited to the same line. The returning value of the D-function is related to the distance. However, comparisons are only valid for the same line. A point associated with a larger value indicates that it is farther than one associated with a smaller value. However, the actual distances are unknown.

The Euclidean distance is the normalization of the D-function. For this reason, the proposed model has been called the Normalized Direct Trigonometry Model (NDTM).

4.1. Separation of Convex Polygons

Two convex polygons are separated if a line separates them, i.e., one polygon is on the line’s negative side, and the other is on the line’s positive side. When two convex polygons are separated by a distance $ϵ>0$, infinite lines can separate them (Figure 4a). However, to limit the number of considered lines, this study focuses on the separating lines that are collinear with the edges of the polygons (Figure 4b). This section describes the mathematical fundamentals of separating a pair of convex polygons.

Polygons do not change their shape when translated (without rotation), i.e., the axis distances between the vertices of a polygon are fixed. Therefore, if the location of a polygon vertex is known, the rest of the vertices are found by adding the fixed distances to the known vertex. Thus, any vertex $v:(v_x, v_y)$ of a polygon can be related to a single reference point $q:(q_x, q_y)$ with the fixed distances in each axis ($\alpha$ and $\beta$). If $\alpha =v_x-q_x$ and $\beta =v_y-q_y$, $\mathrm{v}$ can be represented as $v:\left(v_x, v_y\right)\to v:(q_x+\alpha , q_y+\beta )$.

The components of the line $l:Ax+By+C=0$ formed with two adjacent vertices of the same convex polygon ($a:\left(q_x+{\alpha }_a,q_y+{\beta }_a\right)$ and $b:\left(q_x+{\alpha }_b,q_y+{\beta }_b\right)$) are computed as follows: $A=\left({\beta }_b-{\beta }_a\right)/n$, $B=\left({\alpha }_a-{\alpha }_b\right)/n$, and $C=-A\left(q_x+{\alpha }_a\right)-B\left(q_y+{\beta }_a\right)$, where $n={\left({\beta }_b-{\beta }_a\right)}^2+{\left({\alpha }_a-{\alpha }_b\right)}^2$. Applying Equation (2) with the line of a convex polygon and a point $c:(p_x+{\alpha }_c,p_y+{\beta }_c)$ of another polygon results in Equation (3), where $C_2=A\left({\alpha }_c-{\alpha }_a\right)+B\left({\beta }_c-{\beta }_a\right)$.

$$E_{lc}=A\left(p_x-q_x\right)+B\left(p_y-q_y\right)+C_2$$

(3)

Equation (3) shows the Signed Euclidean Distance (SED) between a line and a vertex. Line $\mathrm{l}$ is anchored to the reference point $\mathrm{q}$ of a convex polygon. The vertex $\mathrm{c}$ is anchored to the reference point $\mathrm{p}$ of another convex polygon. Note that the term ${\mathrm{C}}_2$ does not depend on the reference vertices. Therefore, ${\mathrm{C}}_2$ is constant regardless of the positions of the polygons (reference points).

Two convex polygons are separated if all vertices of the first polygon lay on one line’s side and all vertices of the second polygon lay on the other line’s side. In this study, the edges are oriented such that all vertices of the same polygon are on the positive edges’ side. Therefore, if an arbitrary point lies on the positive side of all polygon edges, it is on the polygon. Otherwise, if the arbitrary point lies on the negative side of at least one polygon edge, it is outside the polygon. Moreover, the non-overlapping constraint allows touching; therefore, the vertices of both polygons can be colinear to the separating line (Figure 5).

An arbitrary point is on a line’s positive (negative) side if the respective SED is zero or greater (less) than zero. Therefore, two convex polygons are separated if there is a line where the SED for the vertices of one polygon is zero or positive and for the vertices of the other polygon is zero or negative. Furthermore, since this study uses the convention that all edges are oriented, such as the interior of the polygon lays on the positive side, two convex polygons are separated if there exists a line of a polygon whose SED for all vertices of the other polygon is zero or negative.

Computing the SED of all lines (edges) of a polygon with all vertices of the other and vice versa implies many computations. However, the computations can be significantly reduced if the SED associated with a line is calculated for only one vertex instead of all vertices of the other convex polygon. This vertex is the closest to the line on the negative line’s side when the line separates the two convex polygons (Figure 5). Identifying the appropriate vertex for each line requires a preprocessing procedure of determining the SED with all vertices and identifying the one with the highest SED. These calculations can be done with arbitrary reference points for the polygons; however, if both reference points are set to the origin $\left(\mathrm{0,0}\right)$, Equation (3) is simplified to $C_2$. Therefore, the appropriate vertex for a line is the one with the highest $C_2$.

The separation of a pair of convex polygons can be applied to non-convex polygons if the last ones can be represented as unions of convex sub-polygons. Therefore, two non-convex polygons are separated if each pair of convex polygons (one of each non-convex polygons) are separated.

4.2. The Basic Normalized Direct Trigonometry Model (NDTM)

The basic formulation for the NDTM considers no rotation for the pieces. This model consists of a set of pieces $I:\{Piec{e}_1, \dots , Piec{e}_n\}$, each piece $i$ has a subset of convex parts (sub-polygons) $P_i:\{Par{t}_{1i},\dots ,Par{t}_{mi}\}$, and each part $p$ has a subset of lines $L_p:\{Lin{e}_{1p}, \dots , Lin{e}_{rp}\}$ (see Figure 6). The parameters and variables for the NDTM are shown in

Table 1. Parameters of the basic formulation for the NDTM.

Parameter	Description
$W$	Strip’s width.
$x_i^{-}$	x-axis distance between the reference vertex and the furthest left vertex of piece $i\in I$.
$x_i^{+}$	x-axis distance between the reference vertex and the furthest right vertex of piece $i\in I$.
$y_i^{-}$	y-axis distance between the reference vertex and the furthest down vertex of piece $i\in I$.
$y_i^{+}$	y-axis distance between the reference vertex and the furthest up vertex of piece $i\in I$.
$A_{lpi}$	x coefficient of the line $l\in L_p$ of part $p\in P_i$ of piece $i\in I$.
$B_{lpi}$	y coefficient of the line $l\in L_p$ of part $p\in P_i$ of piece $i\in I$.
$C_{lpi}^{qj}$	Relative position constant between the line $l\in L_p$ of part $p\in P_i$ of piece $i\in I$ and the part $q\in P_j$ of piece $j\in I$.
$M_{lpi}^{qj}$	Big-M constant for the relative position between the line $l\in L_p$ of part $p\in P_i$ of piece $i\in I$ and the part $q\in P_j$ of piece $j\in I$.

and

Table 2. Variables of the basic formulation for the NDTM.

Variable	Description
$L$	Strip’s length.
$x_i$	x-axis position of the reference vertex of piece $i\in I$.
$y_i$	y-axis position of the reference vertex of piece $i\in I$.
$u_{lpi}^{qj}$	Indicator variable. It is 1 if the line $l\in L_p$ separates part $p\in P_i$ of piece $i\in I$ and the part $q\in P_j$ of piece $j\in I$. It is 0 otherwise.

, respectively. Additionally, Figure 7 illustrates the notation of the parameters related to a piece $i\in I$ listed in Table 2.

The formulation of the NDTM is presented in Equations (4)–(11).

$$\min L \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l} \end{array}$$

(4)

$$\mathrm{s}.\mathrm{t}. \hspace{1em}\hspace{1em}{x}_i^{-}\le x_i\le L-x_i^{+} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l}\forall i\in I\end{array}$$

(5)

$$y_i^{-}\le y_i\le W-y_i^{+} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l}\forall i\in I\end{array}$$

(6)

$$A_{lpi}\left(x_j-x_i\right)+B_{lpi}\left(y_j-y_i\right)+C_{lpi}^{qj}\le M_{lpi}^{qj}(1-u_{lpi}^{qj}) \hspace{1em}\hspace{1em}\begin{array}{l}\forall l\in L_p, \forall p\in P_i, \forall q\in P_j,\\ \forall i,j\in I|i\ne j\end{array}$$

(7)

$$\sum _{l\in L_p}{u}_{lpi}^{qj}+\sum _{l\in L_q}{u}_{lqj}^{pi}\ge 1 \hspace{1em}\hspace{1em}\begin{array}{l}\forall p\in P_i, \forall q\in P_j,\\ \forall i,j\in I|i<j\end{array}$$

(8)

$$L\ge 0 \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l} \end{array}$$

(9)

$$x_i, y_i\ge 0 \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l}\forall i\in I\end{array}$$

(10)

$$u_{lpi}^{qj}\in \{\mathrm{0,1}\} \hspace{1em}\hspace{1em}\begin{array}{l}\forall l\in L_p, \forall p\in P_i, \forall q\in P_j,\\ \forall i,j\in I|i\ne j\end{array}$$

(11)

Constraints (5) and (6) ensure the containment constraint in the x-axis and the y-axis. These inequalities limit the value for the reference points on the strip based on their distance from the polygon’s bounding box. The non-overlapping constraint is ensured by Constraints (7) and (8). Constraint (7) is based on Equation (3), which indicates that the SED must be negative for the separating line. Otherwise, the value of the SED is limited by a Big-$M_{lpi}^{qj}$ value. Constraint (8) ensures that at least one separating line exists for each pair of convex polygons. The separating line may come from any convex polygon. It is worth mentioning that another difference between the NDTM and the DTM is the inequality in Constraint (8). The NDTM considers that at least one separating line must exist for each pair of convex polygons, in contrast with the DTM, which states that there must be precisely one separating line for each pair of convex polygons. Constraints (9)–(11) indicate the nature of the variables, i.e., the possible values or ranges for the variables.

4.2.1. Big-M Calculation

The parameter $M_{lpi}^{qj}$ is analogically calculated, as [6] proposed. $M_{lpi}^{qj}$ is used in Constraint (7) and corresponds to the maximum SED between a line and a reference point when the line does not separate a pair of convex polygons. This value must be large enough to avoid eliminating feasible solutions but should not be enormous to avoid numeric instability. Since $M_{lpi}^{qj}$ is a SED, Equation (3) holds, but some manipulations are required to obtain a valid value (Equation (12)).

$$M_{lpi}^{qj}=\left|A_{lpi}\right|L^{max}+\left|B_{lpi}\right|W+C_{lpi}^{qj} \hspace{1em}\hspace{1em}\begin{array}{l}\forall l\in L_p, \forall p\in P_i, \forall q\in P_j,\\ \forall i,j\in I|i\ne j\end{array}$$

(12)

Equation (12) indicates that the difference between two reference points (Equation (3)) is the biggest possible in both axes. Since the strip’s width is fixed, the reference points can be separated at most by $\mathrm{W}$ in the y-axis. On the other hand, the strip’s length is estimated. [6] used the sum of the pieces’ lengths ($L^{max}$). In this case, pieces are arranged side by side from the strip start. Therefore, the reference points can be separated at most by $L^{max}$ in the x-axis. It is worth noting that $W$ and $L^{max}$ are positive values; therefore, the relative position between the reference points is controlled by applying absolute value to the line’s components $A$ and $B$.

4.2.2. Bounds

Ref. [6] proposed using bounds for the strip’s length (Constraint (13)). The upper bound for $L$ ($L^{max}$) was detailed in Section 4.2.1. The lower bound for $L$ ($L^{min}$) corresponds to the maximum between the sum of the pieces’ area divided by $W$ and the maximum length of all pieces to be packed.

$$L^{min}\le L\le L^{max}$$

(13)

4.2.3. Reduction in the Number of Constraints and Variables

This study uses the simplification proposed by [6] related to parallel edges. A pair of convex polygons may have parallel edges with inverse orientation. These edges generate redundant, non-overlapping constraints. Therefore, these cases are identified, and one of the constraints is dropped. Since the lines’ components $A$ and $B$ are normalized, the sum of the components of parallel edges with inverse orientation is zero, i.e., $A_i+A_j=0$ and $B_i+B_j=0$.

4.2.4. Symmetry Breaking

This study also adopts the symmetry-breaking proposed by [6]. An item can appear more than once in a packing pattern. If two equal items are interchanged, the solution is the same. This symmetry is solved by indicating that $x_i\le x_j, \forall i<j$, where pieces $i$ and $j$ are the same type. It is worth noting that this symmetry breaking can be applied to the y-axis instead of the x-axis. However, this study chooses the x-axis since $L$ may be greater than $W$. One way to select the appropriate axis is to compare $W$ with $L^{max}$. If $L^{max}$ is greater than W, the x-axis is chosen; otherwise, the y-axis would be used.

4.3. The General NDTM

This section indicates the changes to the basic NDTM that have to be performed to include two or several fixed orientations of the items. Moreover, this section considers that items have several duplicates. This formulation requires more subsets and more variables.

This formulation consists of a set of pieces types $I:\{Piec{eType}_1,\dots ,Piec{eType}_n\}$, each piece type $i$ has a subset of orientations $F_i:\{Orientatio{n}_{1i},\dots ,Orientatio{n}_{qi}\}$, each orientation $f$ has a subset of convex parts (sub-polygons) $P_f:\{Par{t}_{1f},\dots ,Par{t}_{mf}\}$, and each part $p$ has a subset of lines $L_p:\{Lin{e}_{1p},\dots ,Lin{e}_{rp}\}$. Besides these sets, an additional one is used to distinguish between the duplicates. Therefore, each piece type $i$ has a subset of duplicates $D_i:\{Duplicat{e}_{1i},\dots ,Duplicat{e}_{ti}\}$ and each duplicate $\mathrm{d}$ has a subset of orientations. The incorporation of more sets implies that parameters and variables have additional indices. The parameters and variables whose indices change are shown in

Table 3. Parameters of the formulation for the general NDTM with modified indices.

Parameter	Description
$x_{fi}^{-}$	x-axis distance between the reference vertex and the further left vertex of the orientation $f\in F_i$ of piece type $i\in I$.
$x_{fi}^{+}$	x-axis distance between the reference vertex and the further right vertex of the orientation $f\in F_i$ of piece type $i\in I$.
$y_{fi}^{-}$	y-axis distance between the reference vertex and the further down vertex of the orientation $f\in F_i$ of piece type $i\in I$.
$y_{fi}^{+}$	y-axis distance between the reference vertex and the further up vertex of the orientation $f\in F_i$ of piece type $i\in I$.
$A_{lpfi}$	x coefficient of the line $l\in L_p$ of part $p\in P_f$ of orientation $f\in F_i$ of piece type $i\in I$.
$B_{lpfi}$	y coefficient of the line $l\in L_p$ of part $p\in P_f$ of orientation $f\in F_i$ of piece type $i\in I$.
$C_{lpfi}^{qgj}$	Relative position constant between the line $l\in L_p$ of part $p\in P_f$ of orientation $f\in F_i$ of piece type $i\in I$ and the part $q\in P_g$ of orientation $g\in F_j$ of piece type $j\in I$.
$M_{lpfi}^{qgj}$	Big-M constant for the relative position between the line $l\in L_p$ of part $p\in P_f$ of orientation $f\in F_i$ of piece type $i\in I$ and the part $q\in P_g$ of orientation $g\in F_j$ of piece type $j\in I$.

and

Table 4. Variables of the formulation for the general NDTM with modified indices.

Variable	Description
$x_{fdi}$	x-axis position of the reference vertex of the piece with orientation $f\in F_i$ of the duplicate $d\in D_i$ of piece type $i\in I$.
$y_{fdi}$	y-axis position of the reference vertex of the piece with orientation $f\in F_i$ of the duplicate $d\in D_i$ of piece type $i\in I$.
$v_{fdi}$	Indicator variable. It is 1 if the piece with orientation $f\in F_i$ of the duplicate $d\in D_i$ of piece type $i\in I$ is used. It is 0 otherwise.
$u_{lpfdi}^{qgej}$	Indicator variable. It is 1 if the line $l\in L_p$ separates part $p\in P_f$ of orientation $f\in F_i$ of duplicate $d\in D_i$ of piece type $i\in I$ and the part $q\in P_g$ of orientation $g\in F_j$ of duplicate $e\in D_j$ of piece type $j\in I$. It is 0 otherwise.

.

The formulation of the general NDTM is presented in Equations (14)–(26).

$$\min L \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l} \end{array}$$

(14)

$$\mathrm{s}.\mathrm{t}. \hspace{1em}\hspace{1em}{L}^{min}\le L\le L^{max} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l} \end{array}$$

(15)

$$\sum _{f\in F_i}{v}_{fdi}=1 \hspace{1em}\hspace{1em}\begin{array}{l}\forall d\in D_i,\forall i\in I\end{array}$$

(16)

$$x_{fi}^{-}\le x_{fdi}\le L-x_{fi}^{+} \hspace{1em}\hspace{1em}\begin{array}{l}\forall f\in F_i, \forall d\in D_i,\forall i\in I\end{array}$$

(17)

$$y_{fi}^{-}\le y_{fdi}\le W-y_{fi}^{+} \hspace{1em}\hspace{1em}\begin{array}{l}\forall f\in F_i, \forall d\in D_i,\forall i\in I\end{array}$$

(18)

$$\begin{array}{cc}\hfill & f\in F_i,\forall g\in F_j, \forall d\in D_i,\hfill \\ \hfill A_{lpfi}\left(x_{gej}-x_{fdi}\right)+B_{lpfi}\left(y_{gej}-y_{fdi}\right)+C_{lpfi}^{qgj}& \forall e\in D_j,\forall i,j\in I,\hfill \\ \hspace{1em}\hspace{1em}\le M_{lpfi}^{qgj}(1-u_{lpfdi}^{qgej})\hfill & |i\ne j \bigvee d\ne e \bigvee f\ne g\hfill \end{array}$$

(19)

$$\sum _{l\in L_p}{u}_{lpfdi}^{qgej}+\sum _{l\in L_q}{u}_{lqgej}^{pfdi}\ge v_{fdi}+v_{gej}-1 \hspace{1em}\hspace{1em}\begin{array}{l}\forall p\in P_f, \forall q\in P_g, \forall f\in F_i\\ \forall g\in F_j, \forall d\in D_i,\forall e\in D_j\\ \forall i,j\in I, |i<j\\ \bigvee (i=j \bigwedge d<e) \\ \bigvee (i=j \bigwedge d=e \bigwedge f<g)\end{array}$$

(20)

$$\sum _{l\in L_p}{u}_{lpfdi}^{qgej}+\sum _{l\in L_q}{u}_{lqgej}^{pfdi}\le \left(\left|L_p\right|+\left|L_q\right|\right)v_{fdi} \hspace{1em}\hspace{1em}\begin{array}{l}\forall p\in P_f, \forall q\in P_g, \forall f\in F_i\\ \forall g\in F_j, \forall d\in D_i,\forall e\in D_j\\ \forall i,j\in I, |i<j\\ \bigvee (i=j \bigwedge d<e) \\ \bigvee (i=j \bigwedge d=e \bigwedge f<g)\end{array}$$

(21)

$$\sum _{l\in L_p}{u}_{lpfdi}^{qgej}+\sum _{l\in L_q}{u}_{lqgej}^{pfdi}\le \left(\left|L_p\right|+\left|L_q\right|\right)v_{gej} \hspace{1em}\hspace{1em}\begin{array}{l}\forall p\in P_f, \forall q\in P_g, \forall f\in F_i\\ \forall g\in F_j, \forall d\in D_i,\forall e\in D_j\\ \forall i,j\in I, |i<j\\ \bigvee (i=j \bigwedge d<e) \\ \bigvee (i=j \bigwedge d=e \bigwedge f<g)\end{array}$$

(22)

$$x_{fdi}+L^{max}\left(1-v_{fdi}\right)\ge x_{gei}-L^{max}\left(1-v_{gei}\right) \hspace{1em}\hspace{1em}\begin{array}{l}\forall f,g\in F_i,\forall d,e\in D_i, \forall i\in I\\ |e=d-1 \bigwedge d>1\end{array}$$

(23)

$$L\ge 0 \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}$$

(24)

$$x_{fdi}, y_{fdi}\ge 0 \hspace{1em}\hspace{1em}\begin{array}{l}\forall f\in F_i, \forall d\in D_i,\forall i\in I\end{array}$$

(25)

$$u_{lpfdi}^{qgej}\in \{\mathrm{0,1}\} \hspace{1em}\hspace{1em}\begin{array}{l}\forall l\in L_p, \forall p\in P_f, \forall q\in P_g,\\ \forall f\in F_i,\forall g\in F_j, \forall d\in D_i,\\ \forall e\in D_j,\forall i,j\in I,\\ |i\ne j \bigvee d\ne e \bigvee f\ne g\end{array}$$

(26)

Constraint (15) indicates that $L^{min}$ and $L^{max}$ bound the strip’s length ($L$). These parameters are estimated differently when rotation and duplicates of the pieces are considered. $L^{min}$ is determined as the maximum value of the strip’s length considering two extreme cases. First, suppose the strip’s width is infinite. In that case, the minimum strip’s length is the largest among the pieces when considering only the orientation that yields the minimum length. The second extreme case assumes a perfect fit, which yields a 100% utilization. Therefore, the minimum strip’s length is the sum of the area of all items to be packed divided by the strip’s width.

$L^{max}$ continues to be the required strip’s length to arrange all items side by side; however, it is computed differently. $L^{max}$ is obtained by considering the shortest length of the valid rotations of each piece type. Next, this length is multiplied by the number of copies for the piece type. $L^{max}$ is the sum of all these products with all piece types.

Constraint (16) ensures that the quantity of each piece type is packed within the container. Constraints (17) and (18) ensure that all packed pieces lay within the container (containment constraint). Constraints (19) and (20) ensure non-overlapping constraints. Constraint (19) is based on the SED like Constraint 7 with the difference in the indices for parameters and variables. Constraints (21) and (22) break symmetry regarding separating lines when a piece is not placed within the container. These last two constraints use $\left|L_p\right|$ and $\left|L_q\right|$, which indicate the number of lines that may separate the convex parts $p$ and $q$. This cardinality allows choosing more than one separating line. Constraint (23) ensures symmetry breaking regarding positioning several duplicates of a piece type along the x-axis. This constraint is only active with the actually packed items, i.e., $v_{fdi}=v_{gei}=1.$ Finally, Constraints (24)–(26) indicate the nature of the variables, i.e., the ranges of values of the problem variables.

4.4. Separation or Distance Constraint

As mentioned above, the distance constraint indicates that a minimum distance must separate all pairs of pieces. Incorporating this constraint in the general NDTM requires the parameter $s$, which indicates the minimum separation among items. This parameter increases $L^{max}$ by $s(n-1)$, where $n$ is the total number of pieces packed within the strip. The minus one is because the separation is only considered between the pieces and not between the pieces and the strip.

The parameter $s$ is used in Constraint (27), which replaces Constraint (19) in the general NDTM. It is worth noting that a positive value for $s$ indicates a separation of the pieces, and a negative value indicates overlapping.

$$\begin{array}{cc}\hfill & \forall l\in L_p, \forall p\in P_f, \forall q\in P_g,\hfill \\ \hfill A_{lpfi}\left(x_{gej}-x_{fdi}\right)+B_{lpfi}\left(y_{gej}-y_{fdi}\right)+C_{lpfi}^{qgj}& \forall f\in F_i,\forall g\in F_j, \forall d\in D_i,\hfill \\ \hspace{1em}\hspace{1em}\le M_{lpfi}^{qgj}\left(1-u_{lpfdi}^{qgej}\right)-s\hfill & \forall e\in D_j,\forall i,j\in I,\hfill \\ \hfill & |i\ne j \bigvee d\ne e \bigvee f\ne g\hfill \end{array}$$

(27)

Since the separation parameter is considered in Constraint (27) and this constraint is based on the SED, the separation considered in this study is limited to edge-vertex separation, which does not include vertex-vertex separation. Including the latter separation would require nonlinear constraints outside the scope of this study. Furthermore, vertex-vertex separations become significant only if the closest distance between two convex polygons occurs between their vertices at an angle between the angles formed by their respective edges.

4.5. New Valid Inequalities

This study proposes new valid inequalities regarding the non-overlapping constraint by implementing a non-overlapping constraint considering the bounding boxes of the original pieces. The non-overlapping constraint states that each pair of packed pieces must not overlap. This constraint can be accomplished by ensuring a separating line for each pair of convex sub-pieces. However, suppose a pair of pieces is far enough, so their bounding boxes do not overlap. In that case, estimating separation lines between each pair of convex sub-pieces seems unnecessary. Therefore, the modifications are oriented to allow the model to be flexible in determining separating lines if the bounding boxes of the pieces are separated.

This study considers two modifications to ease the non-overlapping constraint with the bounding boxes of the pieces. These modifications require new variables indicated in Table 5. The first modification replaces Constraints (20)–(22) of the general NDTM formulation with Constraints (28)–(32). Constraints (28) and (29) define variables ${\dot{x}}_{fdi}^{gej}$ and ${\dot{y}}_{fdi}^{gej}$, respectively, i.e., separating two pieces according to their bounding boxes. Constraints (30)–(32) ensure that at least one separation criterion exists for each pair of convex sub-pieces. Two convex sub-pieces are separated when there is a separating line among them or the bounding boxes of the respective pieces are separated. Therefore, the first modification aims to make the problem more flexible by allowing more alternatives to indicate the separation of pieces.

Table 5. Variables for the incorporation of the new non-overlapping constraints based on bounding boxes.

Variable	Description
${\dot{x}}_{fdi}^{gej}$	Indicator variable. It is 1 if the bounding box of the duplicate piece $d\in D_i$ with orientation $f\in F_i$ of piece type $i\in I$ is separated and on the right of the bounding box of the duplicate piece $e\in D_i$ with orientation $g\in F_i$ of piece type $j\in I$. It is 0 otherwise.
${\dot{y}}_{fdi}^{gej}$	Indicator variable. It is 1 if the bounding box of the duplicate piece $d\in D_i$ with orientation $f\in F_i$ of piece type $i\in I$ is separated and above of the bounding box of the duplicate piece $e\in D_i$ with orientation $g\in F_i$ of piece type $j\in I$. It is 0 otherwise.
$s_{fdi}^{gej}$	Indicator variable. It is 1 if the bounding box of the duplicate piece $d\in D_i$ with orientation $f\in F_i$ of piece type $i\in I$ is separated from the bounding box of the duplicate piece $e\in D_i$ with orientation $g\in F_i$ of piece type $j\in I$. It is 0 otherwise.

Table 5. Variables for the incorporation of the new non-overlapping constraints based on bounding boxes.

Variable	Description
${\dot{x}}_{fdi}^{gej}$	Indicator variable. It is 1 if the bounding box of the duplicate piece $d\in D_i$ with orientation $f\in F_i$ of piece type $i\in I$ is separated and on the right of the bounding box of the duplicate piece $e\in D_i$ with orientation $g\in F_i$ of piece type $j\in I$. It is 0 otherwise.
${\dot{y}}_{fdi}^{gej}$	Indicator variable. It is 1 if the bounding box of the duplicate piece $d\in D_i$ with orientation $f\in F_i$ of piece type $i\in I$ is separated and above of the bounding box of the duplicate piece $e\in D_i$ with orientation $g\in F_i$ of piece type $j\in I$. It is 0 otherwise.
$s_{fdi}^{gej}$	Indicator variable. It is 1 if the bounding box of the duplicate piece $d\in D_i$ with orientation $f\in F_i$ of piece type $i\in I$ is separated from the bounding box of the duplicate piece $e\in D_i$ with orientation $g\in F_i$ of piece type $j\in I$. It is 0 otherwise.

$$L^{max}\left(1-{\dot{x}}_{fdi}^{gej}\right)+x_{fdi}-x_{fi}^{-}\ge x_{gej}+x_{gj}^{+} \begin{array}{l}\forall f\in F_i,\forall g\in F_j, \forall d\in D_i,\\ \forall e\in D_j,\forall i,j\in I,\\ |i\ne j \bigvee d\ne e \bigvee f\ne g\end{array}$$

(28)

$$W\left(1-{\dot{y}}_{fdi}^{gej}\right)+y_{fdi}-y_{fi}^{-}\ge y_{gej}+y_{gj}^{+} \begin{array}{l}\forall f\in F_i,\forall g\in F_j, \forall d\in D_i,\\ \forall e\in D_j,\forall i,j\in I,\\ |i\ne j \bigvee d\ne e \bigvee f\ne g\end{array}$$

(29)

$$\begin{array}{cc}\hfill & \forall p\in P_f, \forall q\in P_g,\forall f\in F_i,\hfill \\ \hfill & \forall g\in F_j, \forall d\in D_i,\forall e\in D_j,\hfill \\ \hfill {\displaystyle \sum _{l\in L_p}}{u}_{lpfdi}^{qgej}+{\displaystyle \sum _{l\in L_q}}{u}_{lqgej}^{pfdi}+{\dot{x}}_{fdi}^{gej}+{\dot{x}}_{gej}^{fdi}+{\dot{y}}_{fdi}^{gej}+{\dot{y}}_{gej}^{fdi}& \forall i,j\in I,|i<j \hfill \\ \hspace{1em}\hspace{1em}\ge v_{fdi}+v_{gej}-1\hfill & \bigvee (i=j \bigwedge d<e)\hfill \\ \hfill & \bigvee (i=j \bigwedge d=e \bigwedge f<g)\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \forall p\in P_f, \forall q\in P_g,\forall f\in F_i,\hfill \\ \hfill & \forall g\in F_j, \forall d\in D_i,\forall e\in D_j,\hfill \\ \hfill {\displaystyle \sum _{l\in L_p}}{u}_{lpfdi}^{qgej}+{\displaystyle \sum _{l\in L_q}}{u}_{lqgej}^{pfdi}+{\dot{x}}_{fdi}^{gej}+{\dot{x}}_{gej}^{fdi}+{\dot{y}}_{fdi}^{gej}+{\dot{y}}_{gej}^{fdi}& \forall i,j\in I,|i<j \hfill \\ \hspace{1em}\hspace{1em}\le \left(\left|L_p\right|+\left|L_q\right|\right)v_{fdi}\hfill & \bigvee (i=j \bigwedge d<e)\hfill \\ \hfill & \bigvee (i=j \bigwedge d=e \bigwedge f<g)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \forall p\in P_f, \forall q\in P_g,\forall f\in F_i,\hfill \\ \hfill & \forall g\in F_j, \forall d\in D_i,\forall e\in D_j,\hfill \\ \hfill {\displaystyle \sum _{l\in L_p}}{u}_{lpfdi}^{qgej}+{\displaystyle \sum _{l\in L_q}}{u}_{lqgej}^{pfdi}+{\dot{x}}_{fdi}^{gej}+{\dot{x}}_{gej}^{fdi}+{\dot{y}}_{fdi}^{gej}+{\dot{y}}_{gej}^{fdi}& \forall i,j\in I,|i<j \hfill \\ \hspace{1em}\hspace{1em}\le \left(\left|L_p\right|+\left|L_q\right|\right)v_{gej}\hfill & \bigvee (i=j \bigwedge d<e)\hfill \\ \hfill & \bigvee (i=j \bigwedge d=e \bigwedge f<g)\hfill \end{array}$$

(32)

The second modification includes the first one and Constraints (33)–(35). Constraints (33) and (34) determine if two pieces are separated through their bounding boxes by defining the value of variable $s_{fdi}^{gej}$. Constraint (35) dictates that no separating line must be determined among any pair of convex sub-pieces if the optimizer separates the respective pieces by their bounding boxes. Therefore, the second modification is more aggressive than the first one, aiming to eliminate symmetries considering the separating lines in pieces separated by their bounding boxes.

$$s_{fdi}^{gej}\le {\dot{x}}_{fdi}^{gej}+{\dot{x}}_{gej}^{fdi}+{\dot{y}}_{fdi}^{gej}+{\dot{y}}_{gej}^{fdi} \begin{array}{l}\forall f\in F_i,\forall g\in F_j, \forall d\in D_i,\\ \forall e\in D_j,\forall i,j\in I,\\ |i<j \bigvee (i=j \bigwedge d<e)\\ \bigvee (i=j \bigwedge d=e \bigwedge f<g)\end{array}$$

(33)

$$2s_{fdi}^{gej}\ge {\dot{x}}_{fdi}^{gej}+{\dot{x}}_{gej}^{fdi}+{\dot{y}}_{fdi}^{gej}+{\dot{y}}_{gej}^{fdi} \begin{array}{l}\forall f\in F_i,\forall g\in F_j, \forall d\in D_i,\\ \forall e\in D_j,\forall i,j\in I,\\ |i<j \bigvee (i=j \bigwedge d<e)\\ \bigvee (i=j \bigwedge d=e \bigwedge f<g)\end{array}$$

(34)

$$\sum _{l\in L_p}{u}_{lpfdi}^{qgej}+\sum _{l\in L_q}{u}_{lqgej}^{pfdi}\le \left(\left|L_p\right|+\left|L_q\right|\right)\left(1-s_{fdi}^{gej}\right) \begin{array}{l}\forall f\in F_i,\forall g\in F_j, \forall d\in D_i,\\ \forall e\in D_j,\forall i,j\in I,\\ |i<j \bigvee (i=j \bigwedge d<e)\\ \bigvee (i=j \bigwedge d=e \bigwedge f<g)\end{array}$$

(35)

5. Results

This section details the results and analysis of the computational experiments. This study has employed a comprehensive set of 90 instances (see Appendix A) to ensure a robust comparison across the models. Considerable effort was made to compile instances from various authors, which enhances the dataset’s diversity and improves this study’s rigor. Therefore, the findings are less biased towards specific types of instances and apply to various practical scenarios.

These instances typically specify a range of permissible rotations or orientations for the pieces. However, this study establishes three scenarios regarding rotations resulting in 270 evaluated instances:

• No rotation: Pieces are placed as provided, i.e., the only allowed rotation is 0°.
• Two rotations: Pieces may rotate 180°, allowing placements at 0° and 180°.
• Four rotations: Pieces may rotation by 90° increments, enabling placements at 0°, 90°, 180° and 270°.

The pieces of all instances were pre-decomposed into convex sub-polygons. Therefore, the time required for decomposition is not factored into our analysis. Applying the NDTM with arbitrary polygons would require preprocessing them with a convex decomposition algorithm [35,36,37].

The results presented in Table 6, Table 7, Table 8 and Table 9 are quantified using the following metrics:

• Util. [%]: This represents the average strip utilization percentage, calculated from the optimal solution when available or based on the best feasible solution found (upper bound).
• Time [s]: This shows the average time to find the optimal solution or until the computation time exceeds the time limit.
• #Optimals: Counts the optimal solutions found across the instances.
• #Feasibles: Tally of all feasible solutions found, including optimal ones.
• Objective Function: This indicates the value of the strip’s length.

The results are presented in two formats: individual results and consensus results. The former shows the performance of each model across the entire set of instances. The latter shows the performance of each model considering only the instances where all models in the respective table found a solution (feasible or optimal).

All tests were conducted using multithreading (8 threads) on an Intel^® Xeon^® Gold 6242R processor 3.1 GHz with 32 GB of RAM. The models were implemented in C++, and the optimizer used was CPLEX 22.11. Each model was run with a time limit of 3600 s. The results are directly comparable since all models and test cases were implemented and executed under the same conditions.

The implementation of the models, instance data, solution files, and visuals can be found at the following repository: https://github.com/gfpantoja/NDTM.

5.1. Comparison between the DTM and the NDTM

There are two main differences between the DTM and the NDTM. First, NDTM1 and NDTM substitute the traditional D-function used in DTM with a signed distance in the separation constraint. Second, NDTM enhances the model’s flexibility by allowing the identification of multiple separating lines between convex sub-pieces. In contrast, DTM restricts this to exactly one separating line.

Table 6. Comparison between the DTM and the NDTM on the 270 instances.

Orientations	Metric	Individual Results			Consensus Results
		DTM	NDTM1	NDTM	DTM	NDTM1	NDTM
0°	Util. [%]	69.05	68.24	68.52	69.01	68.52	69.66
	Time [s]	2386.71	2442.37	2499.79	2351.52	2391.90	2324.23
	GAP [%]	17.46	18.64	19.60	17.11	17.78	16.56
	#Optimals	25	24	26	25	24	26
	#Feasibles	71	72	80	69	69	69
0°, 180°	Util. [%]	68.89	68.94	68.97	68.89	68.94	70.05
	Time [s]	2381.51	2459.94	2527.74	2345.06	2408.83	2335.33
	GAP [%]	18.23	18.04	19.76	17.84	17.45	16.67
	#Optimals	24	25	24	24	25	24
	#Feasibles	69	70	79	67	67	67
0°, 90°, 180°, 270°	Util. [%]	71.53	70.29	69.55	71.60	71.18	71.27
	Time [s]	2494.79	2426.52	2485.64	2454.49	2340.94	2262.15
	GAP [%]	15.83	17.32	19.73	15.29	15.54	15.87
	#Optimals	21	21	22	21	21	22
	#Feasibles	57	59	66	55	55	55
Aggregate	Util. [%]	69.71	69.08	68.98	69.72	69.43	70.26
	Time [s]	2416.16	2443.84	2505.46	2378.90	2383.16	2310.25
	GAP [%]	17.26	18.04	19.69	16.85	17.02	16.40
	#Optimals	70	70	72	70	70	72
	#Feasibles	197	201	225	191	191	191

The values in bold indicate the best metric value.

Table 6. Comparison between the DTM and the NDTM on the 270 instances.

Orientations	Metric	Individual Results			Consensus Results
		DTM	NDTM1	NDTM	DTM	NDTM1	NDTM
0°	Util. [%]	69.05	68.24	68.52	69.01	68.52	69.66
	Time [s]	2386.71	2442.37	2499.79	2351.52	2391.90	2324.23
	GAP [%]	17.46	18.64	19.60	17.11	17.78	16.56
	#Optimals	25	24	26	25	24	26
	#Feasibles	71	72	80	69	69	69
0°, 180°	Util. [%]	68.89	68.94	68.97	68.89	68.94	70.05
	Time [s]	2381.51	2459.94	2527.74	2345.06	2408.83	2335.33
	GAP [%]	18.23	18.04	19.76	17.84	17.45	16.67
	#Optimals	24	25	24	24	25	24
	#Feasibles	69	70	79	67	67	67
0°, 90°, 180°, 270°	Util. [%]	71.53	70.29	69.55	71.60	71.18	71.27
	Time [s]	2494.79	2426.52	2485.64	2454.49	2340.94	2262.15
	GAP [%]	15.83	17.32	19.73	15.29	15.54	15.87
	#Optimals	21	21	22	21	21	22
	#Feasibles	57	59	66	55	55	55
Aggregate	Util. [%]	69.71	69.08	68.98	69.72	69.43	70.26
	Time [s]	2416.16	2443.84	2505.46	2378.90	2383.16	2310.25
	GAP [%]	17.26	18.04	19.69	16.85	17.02	16.40
	#Optimals	70	70	72	70	70	72
	#Feasibles	197	201	225	191	191	191

The values in bold indicate the best metric value.

Table 6 shows that NDTM1 performs like DTM regarding the number of optimal solutions produced. However, NDTM1 finds more feasible solutions than the DTM. When evaluating NDTM against DTM, the results consistently favor NDTM, highlighting superior performance metrics. This improvement underscores the effectiveness of the modifications introduced in NDTM. Figure 8 shows the two additional optimal solutions the NDTM found compared to the DTM.

The first modification, implemented in NDTM1, enhances solution interpretability while not yielding significant advancements in optimal solution outcomes. The NDTM1 (and NDTM) indicate which lines are separating ones (as DTM does) and quantify the distance by which the convex pieces are separated. The second modification, exclusive to NDTM, substantially increases both the quantity of optimal and feasible solutions. These enhancements are attributed to the increased model flexibility by allowing multiple separation lines.

Table 6 demonstrates the expected behavior of optimization models, where increasing the problem size yields fewer optimal and feasible solutions. In this context, the problem size is closely related to the complexity of the pieces (e.g., if a piece is composed of more convex sub-polygons and these sub-polygons have more edges), the number of pieces, and the valid rotations for each piece. Therefore, a detailed analysis of the model’s behavior based on the complexity of the instances is challenging. However, some insights can be drawn from the data in Table 6, considering only the valid rotations. Specifically, Table 6 shows that as the problem size grows (i.e., more valid rotations for each piece), both models, DTM and NDTM, tend to decrease the number of optimal and feasible solutions. This phenomenon occurs because, in combinatorial problems like this, expanding the solution space makes it increasingly difficult for the optimizer to find even a feasible solution. However, it is worth noting that the flexibility of the NDTM over the DTM results in more feasible solutions than the DTM.

An in-depth analysis of the DTM and NDTM models can be conducted by examining their relaxed versions, i.e., assuming that the binary variables are continuous. For this analysis, 25 instances were selected in a no-rotation scenario to minimize the noise associated with multiple rotations. These instances ensured that both models returned an optimal solution. Table 7 presents the objective function values obtained from both the relaxed and non-relaxed versions of the models. This data indicates that both models achieve the same objective function values. This outcome is expected as both models are founded on the same mathematical principles and apply identical bounds to the objective function. These bounds ensure the relaxed solution does not deviate significantly from the non-relaxed solution. Table 7 also shows that, in some cases, the non-relaxed solution is identical to the relaxed solution. Moreover, in no case does the objective function of the non-relaxed solutions exceed twice that of the relaxed solutions.

Table 7. Comparison between the relaxed and non-relaxed formulations of the DTM and the NDTM on the 25 instances without rotations.

Instance	Relaxed Formulations			Non-Relaxed Formulations
	Objective Function	Time [s]		Objective Function	Time [s]
		DTM	NDTM		DTM	NDTM
Blaz1_1	5.4	0.05	0.07	7.4	492.58	378.07
Dagli_1	23	0.04	0.07	23	2.83	5.10
Dighe2	100	0.05	0.06	100	335.19	14.83
Fu_10	25.4474	0.07	0.06	28.6875	172.33	399.34
Fu_5	14	0.07	0.07	17.8889	0.45	0.27
Fu_6	16.6842	0.05	0.08	23	0.10	0.23
Fu_7	18.5263	0.06	0.03	24	0.14	0.44
Fu_8	19.7105	0.06	0.06	24	0.30	0.52
Fu_9	22.2895	0.07	0.06	25	1.61	5.86
Metal0_3	294.576	0.06	0.06	501	0.12	0.15
Metal0_4	346.096	0.06	0.06	501	0.15	0.14
Metal0_5	359.16	0.06	0.06	501	0.25	0.14
Metal0_6	461.064	0.06	0.05	785	0.48	0.76
Metal0_7	284	0.06	0.07	501	81.26	3.34
Metal0_8	328.844	0.07	0.07	529	24.39	14.20
Metal0_9	364.044	0.06	0.07	529	57.79	32.61
Metal1_1	286	0.07	0.07	286	195.11	15.88
Poly1c	13	0.07	0.08	13	125.38	34.63
Shapes_4	14	0.06	0.06	24	0.58	0.24
Shapes2	14	0.07	0.06	14	4.90	10.48
Shirts1_2	13	0.06	0.07	13	3.38	3.63
Three	4	0.06	0.06	6	1.95	0.06
Threep2	6.57143	0.06	0.02	9.33333	2.09	0.63
Threep2w9	5.11111	0.06	0.06	8	5.50	2.32
Threep3	9.85714	0.06	0.06	13.5333	2290.33	2498.72

Table 7. Comparison between the relaxed and non-relaxed formulations of the DTM and the NDTM on the 25 instances without rotations.

Instance	Relaxed Formulations			Non-Relaxed Formulations
	Objective Function	Time [s]		Objective Function	Time [s]
		DTM	NDTM		DTM	NDTM
Blaz1_1	5.4	0.05	0.07	7.4	492.58	378.07
Dagli_1	23	0.04	0.07	23	2.83	5.10
Dighe2	100	0.05	0.06	100	335.19	14.83
Fu_10	25.4474	0.07	0.06	28.6875	172.33	399.34
Fu_5	14	0.07	0.07	17.8889	0.45	0.27
Fu_6	16.6842	0.05	0.08	23	0.10	0.23
Fu_7	18.5263	0.06	0.03	24	0.14	0.44
Fu_8	19.7105	0.06	0.06	24	0.30	0.52
Fu_9	22.2895	0.07	0.06	25	1.61	5.86
Metal0_3	294.576	0.06	0.06	501	0.12	0.15
Metal0_4	346.096	0.06	0.06	501	0.15	0.14
Metal0_5	359.16	0.06	0.06	501	0.25	0.14
Metal0_6	461.064	0.06	0.05	785	0.48	0.76
Metal0_7	284	0.06	0.07	501	81.26	3.34
Metal0_8	328.844	0.07	0.07	529	24.39	14.20
Metal0_9	364.044	0.06	0.07	529	57.79	32.61
Metal1_1	286	0.07	0.07	286	195.11	15.88
Poly1c	13	0.07	0.08	13	125.38	34.63
Shapes_4	14	0.06	0.06	24	0.58	0.24
Shapes2	14	0.07	0.06	14	4.90	10.48
Shirts1_2	13	0.06	0.07	13	3.38	3.63
Three	4	0.06	0.06	6	1.95	0.06
Threep2	6.57143	0.06	0.02	9.33333	2.09	0.63
Threep2w9	5.11111	0.06	0.06	8	5.50	2.32
Threep3	9.85714	0.06	0.06	13.5333	2290.33	2498.72

It is important to note that the only integer (binary) variables in both models (DTM and NDTM) are associated with determining the separating lines between each pair of convex polygons. Thus, relaxing these variables affects the non-overlapping constraint. Consequently, this analysis evaluates the impact of this constraint on the model. The results indicate that this constraint significantly influences the objective function.

5.2. Results of the NDTM with the Separation Constraint

The modification introduced within NDTM1 and NDTM facilitates the interpretation of the results and allows the incorporation of the separation constraint. This constraint is absent in classic problem instances, i.e., no separating values are indicated. Therefore, this study used two separation values corresponding to 1% and 10% of the strip’s width (see Figure 9), with the detailed outcomes presented in Table 7.

Table 8. Results of the NDTM with the separation constraint on the 270 instances.

Orientations	Metric	Individual Results			Consensus Results
		$\mathit{s}=0$	$\mathit{s}\leftarrow 1\%$	$\mathit{s}\leftarrow 10\%$	$\mathit{s}=0$	$\mathit{s}\leftarrow 1\%$	$\mathit{s}\leftarrow 10\%$
0°	Util. [%]	68.52	62.78	36.98	68.50	63.08	37.20
	Time [s]	2499.79	2617.84	2923.81	2471.55	2554.83	2880.42
	GAP [%]	19.60	24.31	45.30	19.30	23.18	43.94
	#Optimals	26	25	19	26	25	19
	#Feasibles	80	83	83	78	78	78
0°, 180°	Util. [%]	68.97	63.48	37.97	69.31	63.90	38.36
	Time [s]	2527.74	2737.94	2891.33	2470.39	2691.88	2862.96
	GAP [%]	19.76	24.31	46.33	18.82	23.24	45.31
	#Optimals	24	20	17	24	20	17
	#Feasibles	79	79	78	75	75	75
0°, 90°, 180°, 270°	Util. [%]	69.55	65.35	39.71	70.61	66.11	40.61
	Time [s]	2485.64	2668.16	2883.57	2393.70	2649.75	2809.00
	GAP [%]	19.73	22.16	46.10	17.88	21.02	43.72
	#Optimals	22	17	14	21	16	14
	#Feasibles	66	63	64	58	58	58
Aggregate	Util. [%]	68.98	63.74	38.10	69.37	64.20	38.55
	Time [s]	2505.46	2674.10	2901.10	2449.74	2629.63	2854.58
	GAP [%]	19.69	23.71	45.88	18.74	22.61	44.37
	#Optimals	72	62	50	71	61	50
	#Feasibles	225	225	225	211	211	211

Table 8. Results of the NDTM with the separation constraint on the 270 instances.

Orientations	Metric	Individual Results			Consensus Results
		$\mathit{s}=0$	$\mathit{s}\leftarrow 1\%$	$\mathit{s}\leftarrow 10\%$	$\mathit{s}=0$	$\mathit{s}\leftarrow 1\%$	$\mathit{s}\leftarrow 10\%$
0°	Util. [%]	68.52	62.78	36.98	68.50	63.08	37.20
	Time [s]	2499.79	2617.84	2923.81	2471.55	2554.83	2880.42
	GAP [%]	19.60	24.31	45.30	19.30	23.18	43.94
	#Optimals	26	25	19	26	25	19
	#Feasibles	80	83	83	78	78	78
0°, 180°	Util. [%]	68.97	63.48	37.97	69.31	63.90	38.36
	Time [s]	2527.74	2737.94	2891.33	2470.39	2691.88	2862.96
	GAP [%]	19.76	24.31	46.33	18.82	23.24	45.31
	#Optimals	24	20	17	24	20	17
	#Feasibles	79	79	78	75	75	75
0°, 90°, 180°, 270°	Util. [%]	69.55	65.35	39.71	70.61	66.11	40.61
	Time [s]	2485.64	2668.16	2883.57	2393.70	2649.75	2809.00
	GAP [%]	19.73	22.16	46.10	17.88	21.02	43.72
	#Optimals	22	17	14	21	16	14
	#Feasibles	66	63	64	58	58	58
Aggregate	Util. [%]	68.98	63.74	38.10	69.37	64.20	38.55
	Time [s]	2505.46	2674.10	2901.10	2449.74	2629.63	2854.58
	GAP [%]	19.69	23.71	45.88	18.74	22.61	44.37
	#Optimals	72	62	50	71	61	50
	#Feasibles	225	225	225	211	211	211

Incorporating the separation (or distance) constraint does not increase the number of variables or the constraints within the model. However, as shown in Table 8, incorporating the separation parameter into the non-overlapping constraint significantly complicates the problem-solving process as the minimum separation between pieces increases. The distance parameter does not significantly affect the overall count of feasible solutions. However, the optimizer’s performance varies with different separation settings, as evidenced by the discrepancies between the number of feasible solutions shown in the individual and consensus results. This variation underscores the model’s sensitivity to the defined separation, impacting how effectively the optimizer can find feasible solutions under varying constraints.

5.3. Results of the NDTM with the Non-Overlapping Constraints Based on Bound Boxes

In addition to the primary modifications made to the DTM that resulted in the NDTM, this study introduces two further modifications, the impacts of which are documented in Table 9. These modifications, referred to as Mod1 and Mod2, facilitate the separation of distant pieces using their bounding boxes. However, they differ in the strictness with which these separations are implemented. Mod1 allows the model to determine that a pair of sub-pieces are separated because at least one separating line exists between them, the bounding boxes of the respective pieces are separated, or by both methods. Mod2, however, strictly uses the separating lines when a pair of pieces is not indicated to be separated by their bounding boxes.

Table 9. Results of the NDTM with additional valid constraints on the 270 instances.

Orientations	Metric	Individual Results			Consensus Results
		NDTM	Mod1	Mod2	NDTM	Mod1	Mod2
0°	Util. [%]	68.52	68.45	66.94	68.52	69.04	68.38
	Time [s]	2499.79	2693.33	2802.64	2499.79	2602.05	2702.47
	GAP [%]	19.60	21.49	23.58	19.60	19.90	20.96
	#Optimals	26	24	21	26	24	21
	#Feasibles	80	88	90	80	80	80
0°, 180°	Util. [%]	68.97	67.72	68.66	68.97	68.41	69.23
	Time [s]	2527.74	2799.61	2705.75	2527.74	2748.73	2637.60
	GAP [%]	19.76	22.55	21.73	19.76	21.25	20.43
	#Optimals	24	19	22	24	19	22
	#Feasibles	79	84	85	79	79	79
0°, 90°, 180°, 270°	Util. [%]	69.55	68.11	67.54	69.55	70.08	70.08
	Time [s]	2485.64	2738.75	2903.64	2485.64	2607.60	2765.49
	GAP [%]	19.73	22.76	23.74	19.73	19.40	19.48
	#Optimals	22	19	19	22	19	19
	#Feasibles	66	76	79	66	66	66
Aggregate	Util. [%]	68.98	68.10	67.70	68.98	69.12	69.18
	Time [s]	2505.46	2743.25	2801.63	2505.46	2655.18	2698.18
	GAP [%]	19.69	22.24	23.01	19.69	20.23	20.34
	#Optimals	72	62	62	72	62	62
	#Feasibles	225	248	254	225	225	225

The values in bold indicate the best metric value.

Table 9. Results of the NDTM with additional valid constraints on the 270 instances.

Orientations	Metric	Individual Results			Consensus Results
		NDTM	Mod1	Mod2	NDTM	Mod1	Mod2
0°	Util. [%]	68.52	68.45	66.94	68.52	69.04	68.38
	Time [s]	2499.79	2693.33	2802.64	2499.79	2602.05	2702.47
	GAP [%]	19.60	21.49	23.58	19.60	19.90	20.96
	#Optimals	26	24	21	26	24	21
	#Feasibles	80	88	90	80	80	80
0°, 180°	Util. [%]	68.97	67.72	68.66	68.97	68.41	69.23
	Time [s]	2527.74	2799.61	2705.75	2527.74	2748.73	2637.60
	GAP [%]	19.76	22.55	21.73	19.76	21.25	20.43
	#Optimals	24	19	22	24	19	22
	#Feasibles	79	84	85	79	79	79
0°, 90°, 180°, 270°	Util. [%]	69.55	68.11	67.54	69.55	70.08	70.08
	Time [s]	2485.64	2738.75	2903.64	2485.64	2607.60	2765.49
	GAP [%]	19.73	22.76	23.74	19.73	19.40	19.48
	#Optimals	22	19	19	22	19	19
	#Feasibles	66	76	79	66	66	66
Aggregate	Util. [%]	68.98	68.10	67.70	68.98	69.12	69.18
	Time [s]	2505.46	2743.25	2801.63	2505.46	2655.18	2698.18
	GAP [%]	19.69	22.24	23.01	19.69	20.23	20.34
	#Optimals	72	62	62	72	62	62
	#Feasibles	225	248	254	225	225	225

The values in bold indicate the best metric value.

The results presented in Table 9 indicate that while both modifications reduce the number of optimal solutions found, they significantly increase the number of feasible solutions. The increase in feasible solutions can be attributed to bounding boxes, simplifying finding feasible arrangements by dealing with less complex shapes (boxes) than irregular items. The reduction in optimal solutions is due to the additional constraints and variables introduced by these modifications, which increase the complexity and size of the model, thereby making it more challenging to solve.

6. Conclusions

The NDTM proposed in this study marks a significant advancement in solving the Irregular Strip Packing Problem (ISPP) by integrating the separation (or distance) constraint that mandates a minimum separation distance between items. This novel approach not only facilitates compliance with industry-specific requirements but also enhances the packing model’s practical utility. The findings indicate that the NDTM provides a more flexible and efficient framework than the DTM because it allows multiple separation lines.

Moreover, the study’s computational experiments demonstrate that incorporating non-overlapping constraints based on bounding boxes within the NDTM delivers superior performance in generating feasible solutions at the cost of optimality. This trade-off may be advantageous in practical applications.

Including the distance constraint within a trigonometry-based MILP framework (NDTM) opens new avenues for further research to integrate more industry-related constraints to expand packing problem applications in two and three dimensions.