# Boosted Arc Flow Formulation Using Graph Compression for the Two-Dimensional Strip Cutting Problem

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## Abstract

**:**

## 1. Introduction

- RF: Rotation of items by 90° is allowed (R) and no guillotine restriction is imposed (F)
- RG: Rotation of items by 90° is allowed (R) and guillotine restriction is imposed (G)
- OF: All items have a fixed orientation (O) and no guillotine restriction is imposed (F)
- OG: All items have a fixed orientation (O) and guillotine restriction is imposed (G)

## 2. Literature Review

## 3. State-of-the-Art Mathematical Models

#### 3.1. The Model Based on Lodi et al. [15]

_{j}, that may include any item with height less than or equal to h

_{j}(including item j itself). Note that not all the defined shelves need to be used. The decision variables are therefore defined as follows:

#### 3.2. The Model Based on Furini et al. [31]

_{jk}, a continuous decision variable y

_{k}is included that represents the height of shelf k ($k=1,\dots ,n)$. The model is formulated as follows:

#### 3.3. The Model Based on Silva et al. [21]

**Example 1.**

#### 3.4. The Model of Mrad [28]

- The items are ordered according to a decreasing order of their widths.
- In each shelf, any path must include an item with the same height as the shelf.
- The number of occurrences of an item in a path cannot exceed its own demand.

Algorithm 1. Graph Construction Algorithm |

Let M be a matrix with W + 1 rows and ${n}_{k}$ columns. Where$\hspace{1em}{n}_{k}$ is the number of items in the set ${S}_{k}$. M[i][j] takes 0 if it is allowed to build an arc representing item j and starting from node i and 1 otherwise. |

1: M[i][j] $\leftarrow $0 $\forall \hspace{1em}i=0,\hspace{1em}\dots ,\hspace{1em}W\hspace{1em}\forall \hspace{1em}j\in {S}_{k}$ |

2: ${V}_{k}\hspace{1em}\leftarrow \hspace{1em}\left\{0,\hspace{1em}W\right\}$ |

3: $\hspace{1em}for\hspace{1em}j\hspace{1em}\in \hspace{1em}{S}_{k}/\left\{k\right\}\hspace{1em}do$ |

4: $for\hspace{1em}i=0,\dots ,{w}_{k}-1$ |

5: M[i][j] $\leftarrow $1 |

6: end for |

7: end for |

8: $for\hspace{1em}i\hspace{1em}\in \hspace{1em}{S}_{k}\hspace{1em}do$ |

9: $SetOfNewNodes\leftarrow \hspace{1em}\varnothing $ |

10: $for\hspace{1em}j\hspace{1em}\in \hspace{1em}{V}_{k}/\left\{W\right\}\hspace{1em}do$ |

11: $\alpha \leftarrow j$ |

12: r $\leftarrow $ 0 |

13: $while\hspace{1em}\left(\alpha +{w}_{i}\hspace{1em}\u2a7d\hspace{1em}W\right)\hspace{1em}and\hspace{1em}\left(r\hspace{1em}\u2a7d\hspace{1em}{d}_{i}\right)$ do |

14: if M[$\alpha $][i] = 0 do |

15: ${A}_{k}\hspace{1em}\leftarrow \hspace{1em}{A}_{k}\hspace{1em}{{\displaystyle \cup}}^{\hspace{1em}}\hspace{1em}\left(\alpha ,\hspace{1em}\alpha +{w}_{i}\right)$ |

16: M[$\alpha $][i]=1 |

17: $SetOfNewNodes\hspace{1em}\leftarrow \hspace{1em}SetOfNewNodes\hspace{1em}{{\displaystyle \cup}}^{\hspace{1em}}\hspace{1em}\left\{\alpha +{w}_{i}\right\}$ |

18: end if |

19: $r\hspace{1em}\leftarrow \hspace{1em}r+1$ |

20: $\alpha \hspace{1em}\leftarrow \alpha +{w}_{i}$ |

21: end while |

22: $end\hspace{1em}for$ |

23: ${V}_{k}\leftarrow \hspace{1em}{V}_{k}\hspace{1em}{{\displaystyle \cup}}^{\hspace{1em}}\hspace{1em}SetOfNewNodes$ |

24: $end\hspace{1em}for$ |

25: $for\hspace{1em}j\hspace{1em}\in \hspace{1em}{V}_{k}/\left\{W\right\}\hspace{1em}do$ |

26: $if\hspace{1em}\nexists \left(a,b\right)\in {A}_{k}\hspace{1em}such\hspace{1em}that\hspace{1em}a=j$ |

27: ${A}_{k}\hspace{1em}\leftarrow \hspace{1em}{A}_{k}\hspace{1em}{{\displaystyle \cup}}^{\hspace{1em}}\hspace{1em}\left(j,\hspace{1em}W\right)$ |

28: end if |

29: end for |

**Example 2.**

_{1}(7, 5, 2), I

_{2}(6, 4, 1), I

_{3}(5, 3, 2) and I

_{4}(4, 2, 2) (where I (h, w, d) denotes an item I of height h, width w and demand d). Four shelves $\pi $

_{1}, $\pi $

_{2}, $\pi $

_{3}and $\pi $

_{4}are then required to be used in the second stage. Figure 4 depicts the corresponding four graphs G

_{1}, G

_{2}, G

_{3}and G

_{4}, generated by Algorithm 1. In each graph, each non-dashed arc represents an associated item. Dashed arcs represent dummy items (waste material) and are added in order to respect the path between the source node and the target node.

_{k}, denote by ${H}_{k}$ the set of different item heights in ${S}_{k}$. Let ${x}_{abh}^{k}$ be an integer variable associated to each arc (a,b) ∈ ${A}_{k}$ that takes the number of items of width (b−a) and height $h$ ∈ ${H}_{k}$ placed at position $a$ from the beginning of shelf $\pi $

_{k}. This variable represents the flow on the arc $\left(a,b\right)$ associated to the item of height $h$ in the graph ${G}_{k}$. For example, for shelf $\pi $

_{3}, we have ${S}_{3}=\left\{3,4\right\}$ and ${H}_{3}=\left\{5,4\right\}.$ Consider the arc (5,7) in graph ${G}_{3}$, then the variable ${x}_{685}^{3}$ (resp. ${x}_{684}^{3}$) denotes the number of items of width 8 − 6 = 2 and height 5 (resp. 4) placed at position 6 from the beginning of shelf $\pi $

_{3.}

## 4. Application of the Graph Compression Technique to the 2D-SCP

**Example 3.**

_{1}(5, 4, 1), I

_{2}(5, 3, 3) and I

_{3}(3, 2, 1) (using the same notations of Example 2). Figure 5 shows the graph corresponding to h = 5 (without dummy arcs). Clearly, the two paths 0-4-6-9 and 0-4-7-9 represent the same cutting pattern.

**Example 4.**

## 5. Computational Experiments

#### 5.1. The Benchmark Instances

- Set 3 consists of 500 instances that are divided into 10 classes. Each class includes 50 instances where for each value of $n\u03f5\left\{20,40,60,80,100\right\}$ there are 10 generated instances. Berkey and Wang [16] introduced the first six classes, while the remaining four classes were presented in Ref. [38]. The structure of instances and values of ${w}_{i}$ and ${h}_{i}$ are uniformly distributed in the listed intervals as shown in Table 4. All the item demands are equal to one.
- Set 4 contains 20 instances (ATP30,…, ATP49) that are used by Ref. [28].

#### 5.2. Impact of the Graph Compression

**Size reduction**: it reflects the impact of the graph compression on the number of variables of the generated mathematical model. It is computed as $100\left({n}_{1}-{n}_{2}\right)/{n}_{1}$, where n_{1}and n_{2}represent the number of variables corresponding to the non-compressed and compressed graphs, respectively.**Time ratio**: it represents the ratio of the CPU time of the non-compressed model over the one of the compressed model. It indicates the impact of the graph compression on increasing/decreasing the running time.**Gap improvement**: it represents the percentage gap improvement for unsolved instances. The gap is equal to $100\left(UB-LB\right)/UB$, where UB and LB denote the best found upper and lower bounds, respectively. The gap improvement is computed as $100\left(ga{p}_{1}-ga{p}_{2}\right)/ga{p}_{1}$, where gap_{1}and gap_{2}represent the gaps obtained by the non-compressed and compressed models, respectively.

#### 5.3. Comparison with the State-of-the-Art Mathematical Models

#### 5.4. New Results for Open Benchmark Instances

_{lit}(and, respectively, LB

_{lit}) denotes the best obtained upper (and, respectively, lower) bound in the literature [11], and UB

_{new}(and, respectively, LB

_{new}) denotes the upper (and, respectively, lower) bound obtained by the proposed compressed model. The gap improvement is computed for each instance by $100\left(De{v}_{lit}-De{v}_{new}\right)/De{v}_{lit}$, where Dev

_{lit}= UB

_{lit}−LB

_{lit}and Dev

_{new}= UB

_{new}−LB

_{new}. The asterisk symbol indicates that the new obtained upper/lower bound reaches the optimum.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Valério De Carvalho, J.V. LP models for bin packing and cutting stock problems. Eur. J. Oper. Res.
**2002**, 141, 253–273. [Google Scholar] [CrossRef] - Aktin, T.; Özdemir, R.G. An integrated approach to the one-dimensional cutting stock problem in coronary stent manufacturing. Eur. J. Oper. Res.
**2009**, 196, 737–743. [Google Scholar] [CrossRef] - Parreño, F.; Alonso, M.; Alvarez-Valdes, R. Solving a large cutting problem in the glass manufacturing industry. Eur. J. Oper. Res.
**2020**, 287, 378–388. [Google Scholar] [CrossRef] - Li, F.; Chen, Y.; Hu, X. Manufacturing-oriented silicon steel coil lengthwise cutting stock problem with useable leftover. Eng. Comput.
**2021**, 39, 477–492. [Google Scholar] [CrossRef] - Pierini, L.M.; Poldi, K.C. Lot Sizing and cutting stock problems in a paper production process. Pesqui. Oper.
**2021**, 41. [Google Scholar] [CrossRef] - Wattanasiriseth, P.; Krairit, A. An Application of Cutting-Stock Problem in Green Manufacturing: A Case Study of Wooden Pallet Industry. IOP Conf. Ser. Mater. Sci. Eng.
**2019**, 530, 012005. [Google Scholar] [CrossRef] - Varela, R.; Vela, M.D.C.R.; Puente, J.; Sierra-Sanchez, M.R.; González-Rodríguez, I. An effective solution for a real cutting stock problem in manufacturing plastic rolls. Ann. Oper. Res.
**2008**, 166, 125–146. [Google Scholar] [CrossRef][Green Version] - Lemos, F.K.; Cherri, A.C.; de Araujo, S.A. The cutting stock problem with multiple manufacturing modes applied to a construction industry. Int. J. Prod. Res.
**2020**, 59, 1088–1106. [Google Scholar] [CrossRef] - Huang, Y.-H.; Lu, H.-C.; Wang, Y.-C.; Chang, Y.-F.; Gao, C.-K. A Global Method for a Two-Dimensional Cutting Stock Problem in the Manufacturing Industry. In Application of Decision Science in Business and Management; IntechOpen: London, UK, 2020. [Google Scholar] [CrossRef][Green Version]
- Lodi, A.; Martello, S.; Vigo, D. Heuristic and Metaheuristic Approaches for a Class of Two-Dimensional Bin Packing Problems. INFORMS J. Comput.
**1999**, 11, 345–357. [Google Scholar] [CrossRef] - Bezerra, V.M.R.; Leao, A.A.S.; Oliveira, J.F.; Santos, M.O. Models for the two-dimensional level strip packing problem—A review and a computational evaluation. J. Oper. Res. Soc.
**2019**, 71, 606–627. [Google Scholar] [CrossRef] - Gilmore, P.C.; Gomory, R.E. Multistage Cutting Stock Problems of Two and More Dimensions. Oper. Res.
**1965**, 13, 94–120. [Google Scholar] [CrossRef][Green Version] - Hifi, M. An improvement of viswanathan and bagchi's exact algorithm for constrained two-dimensional cutting stock. Comput. Oper. Res.
**1997**, 24, 727–736. [Google Scholar] [CrossRef] - Hifi, M. Exact algorithms for the guillotine strip cutting/packing problem. Comput. Oper. Res.
**1998**, 25, 925–940. [Google Scholar] [CrossRef] - Lodi, A.; Martello, S.; Vigo, D. Models and Bounds for Two-Dimensional Level Packing Problems. J. Comb. Optim.
**2004**, 8, 363–379. [Google Scholar] [CrossRef][Green Version] - Berkey, J.O.; Wang, P.Y. Two-Dimensional Finite Bin-Packing Algorithms. J. Oper. Res. Soc.
**1987**, 38, 423. [Google Scholar] [CrossRef] - Belov, G.; Scheithauer, G. A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting. Eur. J. Oper. Res.
**2006**, 171, 85–106. [Google Scholar] [CrossRef] - Pisinger, D.; Sigurd, M. Using Decomposition Techniques and Constraint Programming for Solving the Two-Dimensional Bin-Packing Problem. INFORMS J. Comput.
**2007**, 19, 36–51. [Google Scholar] [CrossRef] - Bekrar, A.; Kacem, I. An Exact Method for the 2D Guillotine Strip Packing Problem. Adv. Oper. Res.
**2009**, 2009, 1–20. [Google Scholar] [CrossRef][Green Version] - Dyckhoff, H. A New Linear Programming Approach to the Cutting Stock Problem. Oper. Res.
**1981**, 29, 1092–1104. [Google Scholar] [CrossRef] - Silva, E.; Alvelos, F.; Valério de Carvalho, J.M.V. An integer programming model for two- and three-stage two-dimensional cutting stock problems. Eur. J. Oper. Res.
**2010**, 205, 699–708. [Google Scholar] [CrossRef] - Macedo, R.; Alves, C.; de Carvalho, J.V. Arc-flow model for the two-dimensional guillotine cutting stock problem. Comput. Oper. Res.
**2010**, 37, 991–1001. [Google Scholar] [CrossRef] - Carvalho, J.V. Exact Solution of Cutting Stock Problems Using Column Generation and Branch-and-Bound. Int. Trans. Oper. Res.
**1998**, 5, 35–44. [Google Scholar] [CrossRef] - Mrad, M.; Meftahi, I.; Haouari, M. A branch-and-price algorithm for the two-stage guillotine cutting stock problem. J. Oper. Res. Soc.
**2013**, 64, 629–637. [Google Scholar] [CrossRef] - Rinaldi, F.; Franz, A. A two-dimensional strip cutting problem with sequencing constraint. Eur. J. Oper. Res.
**2007**, 183, 1371–1384. [Google Scholar] [CrossRef] - Cintra, G.; Miyazawa, F.; Wakabayashi, Y.; Xavier, E. Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation. Eur. J. Oper. Res.
**2008**, 191, 61–85. [Google Scholar] [CrossRef] - Bettinelli, A.; Ceselli, A.; Righini, G. A branch-and-price algorithm for the two-dimensional level strip packing problem. 4OR
**2007**, 6, 361–374. [Google Scholar] [CrossRef] - Mrad, M. An arc flow-based optimization approach for the two-stage guillotine strip cutting problem. J. Oper. Res. Soc.
**2015**, 66, 1850–1859. [Google Scholar] [CrossRef] - Zhu, K.; Ji, N.; Li, X.D. Hybrid Heuristic Algorithm Based on Improved Rules & Reinforcement Learning for 2D Strip Packing Problem. IEEE Access
**2020**, 8, 226784–226796. [Google Scholar] [CrossRef] - Iori, M.; de Lima, V.L.; Martello, S.; Miyazawa, F.K.; Monaci, M. Exact solution techniques for two-dimensional cutting and packing. Eur. J. Oper. Res.
**2020**, 289, 399–415. [Google Scholar] [CrossRef] - Furini, F.; Malaguti, E.; Durán, R.M.; Persiani, A.; Toth, P. A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size. Eur. J. Oper. Res.
**2012**, 218, 251–260. [Google Scholar] [CrossRef] - Lodi, A.; Monaci, M. Integer linear programming models for 2-staged two-dimensional Knapsack problems. Math. Program.
**2003**, 94, 257–278. [Google Scholar] [CrossRef] - Brandão, F.; Pedroso, J.P. Bin packing and related problems: General arc-flow formulation with graph compression. Comput. Oper. Res.
**2016**, 69, 56–67. [Google Scholar] [CrossRef][Green Version] - Scholl, A.; Klein, R.; Jürgens, C. Bison: A fast hybrid procedure for exactly solving the one-dimensional bin packing problem. Comput. Oper. Res.
**1997**, 24, 627–645. [Google Scholar] [CrossRef] - Hopper, E.; Turton, B. An empirical investigation of meta-heuristic and heuristic algorithms for a 2D packing problem. Eur. J. Oper. Res.
**2001**, 128, 34–57. [Google Scholar] [CrossRef] - Christofides, N.; Whitlock, C. An Algorithm for Two-Dimensional Cutting Problems. Oper. Res.
**1977**, 25, 30–44. [Google Scholar] [CrossRef][Green Version] - Beasley, J.E. An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure. Oper. Res.
**1985**, 33, 49–64. [Google Scholar] [CrossRef] - Martello, S.; Vigo, D. Exact Solution of the Two-Dimensional Finite Bin Packing Problem. Manag. Sci.
**1998**, 44, 388–399. [Google Scholar] [CrossRef] - Ali, T. Impact of graph compression on the two stage cutting stock problems. Ph.D. Thesis, Department of Industrial Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia, 2023. [Google Scholar]

$\mathbf{Item}\mathit{j}$ | $\mathbf{Height}{\mathit{h}}_{\mathit{j}}$ | $\mathbf{Width}{\mathit{w}}_{\mathit{j}}$ |
---|---|---|

1 | 40 | 20 |

2 | 30 | 10 |

3 | 20 | 40 |

Item i | ${\mathit{h}}_{\mathit{i}}$ | ${\mathit{w}}_{\mathit{i}}$ | ${\mathit{d}}_{\mathit{i}}$ |
---|---|---|---|

0 | 3 | 7 | 2 |

1 | 5 | 6 | 2 |

2 | 4 | 5 | 2 |

Categories | Number of Items | Strip Width | ||
---|---|---|---|---|

P1 | P2 | P3 | ||

C1 | 16 | 17 | 16 | 20 |

C2 | 25 | 25 | 25 | 40 |

C3 | 28 | 29 | 28 | 60 |

C4 | 49 | 49 | 49 | 60 |

C5 | 73 | 73 | 73 | 60 |

C6 | 97 | 97 | 97 | 80 |

C7 | 196 | 197 | 196 | 160 |

Class | $\mathit{W}$ | ${\mathit{w}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{i}}$ |
---|---|---|---|

1 | 10 | [1, 10] | [1, 10] |

2 | 30 | [1, 10] | [1, 10] |

3 | 40 | [1, 35] | [1, 35] |

4 | 100 | [1, 35] | [1, 35] |

5 | 100 | [1, 100] | [1, 100] |

6 | 300 | [1, 100] | [1, 100] |

7 | 100 | [$\frac{2W}{3},W$] | [$1,\frac{H}{2}$] |

8 | 100 | [1,$\frac{W}{2}$] | [$\frac{2H}{3},H$] |

9 | 100 | [$\frac{w}{2},W$] | [$\frac{H}{2},H$] |

10 | 100 | [1,$\frac{W}{2}$] | [$1,\frac{H}{2}$] |

Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | |
---|---|---|---|---|---|

Average size reduction | 5.22% | 34.70% | 12.27% | 30.96% | 60.32% |

Maximum size reduction | 23.44% | 58.17% | 54.22% | 59.55% | 81.14% |

Average time ratio | 0.95 | 1.79 | 1.32 | 1.85 | 2.11 |

Maximum time ratio | 2.12 | 5.79 | 17.53 | 9.85 | 17.27 |

Average gap improvement | 20.53% | 7.86% | 7.36% | 31.12% | - |

Maximum gap improvement | 56.09% | 7.86% | 100.00% | 100.00% | - |

Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | |
---|---|---|---|---|---|

Compressed Model | 57.14% | 93.75% | 74.20% | 20.00% | 100.00% |

M1ineq | 71.43% | 93.75% | 85.60% | 0.00% | 76.74% |

M2ineq | 42.86% | 62.50% | 21.20% | 0.00% | 27.91% |

M3 | 42.86% | 93.75% | 63.40% | 0.00% | 72.09% |

Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | |
---|---|---|---|---|---|

Compressed Model | 1617.43 | 225.08 | 962.63 | 2981.18 | 24.72 |

M1ineq | 1035.99 | 225.58 | 627.89 | 3600.00 | 1019.47 |

M2ineq | 2059.22 | 1395.06 | 2842.01 | 3600.00 | 2597.23 |

M3 | 2085.10 | 225.78 | 1363.38 | 3600.00 | 1106.27 |

Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | |
---|---|---|---|---|---|

Compressed Model | 11.15% | 7.57% | 5.79% | 0.53% | 0.00% |

M1ineq | 1.83% | 2.00% | 1.81% | 1.70% | 0.53% |

M2ineq | 9.83% | 7.57% | 9.59% | 16.54% | 35.01% |

M3 | 20.48% | 7.11% | 7.33% | 9.58% | 0.21% |

Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | |
---|---|---|---|---|---|

Compressed Model | 100.00% | 100.00% | 100.00% | 100.00% | 86.05% |

M1ineq | 0.00% | 6.25% | 0.00% | 0.00% | 0.00% |

M2ineq | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

M3 | 0.00% | 25.00% | 3.40% | 0.00% | 46.51% |

Instance | UB_{lit} | LB_{lit} | UB_{new} | LB_{new} | Gap Improvement |
---|---|---|---|---|---|

ATP30 | 1262 | 1242 | 1255 * | 1255 * | 100.00% |

ATP31 | 13,068 | 12,866 | 12,964 | 12,893 | 64.85% |

ATP32 | 1504 | 1491 | 1500 | 1496 | 69.23% |

ATP33 | 11,802 | 11,742 | 11,771 | 11,770 | 98.33% |

ATP34 | 2432 | 2421 | 2432 | 2426 | 45.45% |

ATP35 | 4743 | 4707 | 4742 * | 4742 * | 100.00% |

ATP36 | 1794 | 1778 | 1788 | 1783 | 68.75% |

ATP37 | 9764 | 9683 | 9743 | 9722 | 74.07% |

ATP38 | 3635 | 3611 | 3630 * | 3630 * | 100.00% |

ATP39 | 5566 | 5481 | 5509 | 5506 | 96.47% |

ATP40 | 2084 | 2037 | 2056 | 2041 | 68.09% |

ATP41 | 4170 | 4123 | 4143 | 4137 | 87.23% |

ATP42 | 4296 | 4227 | 4260 | 4231 | 57.97% |

ATP43 | 11,235 | 11,001 | 11,118 | 11,030 | 62.39% |

ATP44 | 4210 | 4150 | 4204 | 4165 | 35.00% |

ATP45 | 4329 | 4270 | 4326 | 4293 | 44.07% |

ATP46 | 5650 | 5534 | 5581 | 5549 | 72.41% |

ATP47 | 6757 | 6698 | 6749 | 6708 | 30.51% |

ATP48 | 1987 | 1950 | 1988 | 1958 | 21.62% |

ATP49 | 2214 | 2195 | 2211 * | 2211 * | 100.00% |

A11 | 94,890 | 94,598 | 94,873 * | 94,873 * | 100.00% |

A14 | 139,959 | 139,938 | 139,959 | 139 959 * | 100.00% |

A21 | 57,348 | 57,340 | 57,348 | 57,348 * | 100.00% |

A35 | 34,554 | 34,525 | 34,554 | 34,554 * | 100.00% |

Average gap improvement | 69.82% |

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## Share and Cite

**MDPI and ACS Style**

Ali, T.G.; Mrad, M.; Balma, A.; Gharbi, A.; Samhan, A.; Louly, M.A.
Boosted Arc Flow Formulation Using Graph Compression for the Two-Dimensional Strip Cutting Problem. *Processes* **2023**, *11*, 790.
https://doi.org/10.3390/pr11030790

**AMA Style**

Ali TG, Mrad M, Balma A, Gharbi A, Samhan A, Louly MA.
Boosted Arc Flow Formulation Using Graph Compression for the Two-Dimensional Strip Cutting Problem. *Processes*. 2023; 11(3):790.
https://doi.org/10.3390/pr11030790

**Chicago/Turabian Style**

Ali, Tamer G., Mehdi Mrad, Ali Balma, Anis Gharbi, Ali Samhan, and Mohammed A. Louly.
2023. "Boosted Arc Flow Formulation Using Graph Compression for the Two-Dimensional Strip Cutting Problem" *Processes* 11, no. 3: 790.
https://doi.org/10.3390/pr11030790