Dataset used in article "A 2-dimensional guillotine cutting stock problem with variable-sized stock for the honeycomb cardboard industry"

This document is the accepted version of the research paper: •Research article: Paula Ter´an-Viadero, Antonio Alonso-Ayuso, F. Javier Mart´ın-Campo (2023) A 2-dimensional guillotine cutting stock problem with variablesized stock for the honeycomb cardboard industry, International Journal of Production Research, DOI: 10.1080/00207543.2023.2279129 •Data used in computational experiments are available at: Paula Ter´an-Viadero, Antonio Alonso-Ayuso, F. Javier Mart´ın-Campo (2023) Dataset used in article “A 2-dimensional guillotine cutting stock problem with variable-sized stock for the honeycomb cardboard industry” (version 1.2), Zenodo, DOI: 10.5281/zenodo.10185247 Note: A copy of this document is also available here. A 2-dimensional guillotine cutting stock problem with variable-sized stock for the honeycomb cardboard industry Paula Ter´an-Viaderoa, Antonio Alonso-Ayusoband F. Javier Mart´ın-Campoc aFacultad de CC. Matem´aticas, Universidad Complutense de Madrid, Spain bDSLAB - CETINIA, Universidad Rey Juan Carlos, Spain cDepartamento de Estad´ıstica e Investigaci´on Operativa, Instituto de Matem´atica Interdisciplinar, Universidad Complutense de Madrid, Spain ABSTRACT This paper introduces novel mathematical optimisation models for the 2Dimensional guillotine Cutting Stock Problem with Variable-Sized Stock that appears in a Spanish company in the honeycomb cardboard industry. This problem mainly differs from the classical cutting stock problems in the stock, which is considered variable-sized, i.e., we have to decide the panel dimensions, width, and length. This approach is helpful in industries where the stock is produced simultaneously with the cutting process. The stock is then cut into smaller rectangular pieces that must meet the customers’ requirements, such as the type of item, dimensions, demands, and technical specifications. Furthermore, in the problem tackled in this paper, the cuts are guillotine, performed side to side. The proposed mathematical models are validated using real data from the company, obtaining results that drastically reduce the produced material and leftovers, reducing operation times and economic costs. KEYWORDS Cutting Stock Problem; 2-Dimensional cutting; Variable-sized stock; Mixed Integer Linear Optimisation; Cardboard industry. 1. Introduction The industrial world is increasingly facing complex processes that require expert systems to support decision-making therefore it is necessary to develop useful methods and techniques. This area of research has been extensively studied by Jean-Marie Proth. See Dolgui and Proth (2010), which describes several problems in supply chain management, such as pricing, outsourcing, inventory, manufacturing, etc. On the contrary, Proth and Hillion (1990) present different mathematical tools to solve them such 1 as simulation, linear or dynamic programming, and queueing theory, among others. Finally, Govil and Proth (2002), introduce a general perspective of the supply chain design taking into account the different levels of strategies: strategic, tactical, and operational. One of the problems found in many industries is the Cutting Stock Problem (CSP), introduced by Kantorovich (1960) and Gilmore and Gomory (1961), is one of the optimisation problems that has been deeply explored in the literature due to its many applications in the industry. The CSP is present in different sectors such as glass (Parre˜no and Alvarez-Valdes (2021)), stone (Baykaso˘glu and ¨ Ozbel (2021)), wood (Kokten and Sel (2020)), steel (Sierra-Paradinas et al. (2021) and Antonio et al. (1999)), concrete (Signorini, de Araujo, and Melega (2021)), construction (Lemos, Cherri, and de Araujo (2020)), paper (Kallrath et al. (2014)), and textile (Salem et al. (2023)). The cutting process is crucial for companies producing pieces cut from a previous stock. The companies are especially interested in minimising the leftovers and production times, although they can consider other goals, for example in Antonio et al. (1999), reducing computing times as much as possible while achieving a reasonable cost is used by the sales department to respond in ‘real time’ to the customers’ demands. Due to the different industrial processes and their specific restrictions, several versions of the CSP have been developed based on specifications such as dimensions, pattern characteristics, and cutting restrictions, among others. Both exact and nonexact (heuristics, metaheuristics, and/or matheuristics) algorithms have been explored in terms of problem resolution. The problem studied in this paper is motivated by the collaboration with a mediumsized Spanish company in the honeycomb cardboard sector whose aim for the future medium term is to automatise the operations to gain efficiency and effectiveness. It is part of the 2-Dimensional Cutting Stock Problems (2DCSP). The 2DCSP is concerned with obtaining a set of different rectangular items cut from one or more rectangular panels in stock. In our case, the size of panels is not predefined in advance and is determined by the optimisation model. The cutting process must be defined to produce enough pieces of each item to cover a previously known demand. Additionally, as detailed below, different specifications from the cardboard industry must be considered. The main contributions of this paper are: •A 0–1 linear optimisation model is introduced for the Multi-Stock 2-Dimensional CSP. The proposal improves the current operation in the factory. •A novel mixed 0–1 linear optimisation model for the 2DCSP with variable-sized stock and guillotine cuts, able to decide the dimensions of the panels produced, is also introduced. •A real problem from a Spanish company in the cardboard industry is tackled. Data from the company are used and extensively analysed, comparing the current operation with the proposals. •As a result, new, simple and straightforward strategies are proposed for the operation of the company, providing up to a 50% leftover reduction. The remaining part of this paper is organised as follows: Section 2 reviews the different variants of the Cutting Stock Problem, as well as the specifications to meet in the problem to study; Section 3 describes the problem to deal with and the current operation managed by the company; Section 4 introduces two mathematical optimisation models to solve the problem; Section 5 presents an extensive computational experiment based on real-world data provided by the company; finally, Section 6 concludes and presents the future research lines. 2 Panel Length Panel Width (a) 1DCSP Panel Length Panel Width (b) 2DCSP Figure 1. Caption: 1DCSP vs 2DCSP Figure 1. Alt Text: Two diagrams with a rectangular shape. On the left, caption (a) 1DCSP: Diagram with a rectangular panel divided into different horizontal strips where the top strip in dashed is leftover. On the right, caption (b) 2DCSP: Diagram with a rectangular panel divided into three horizontal strips where the top strip in dashed is leftover. The other two strips are divided in rectangular pieces. 2. The Cutting Stock Problem The CSP was introduced by Kantorovich (1960) (first published in Russian in 1939), followed by the seminal work presented by Gilmore and Gomory (1965), which extends the works by the same authors for the 1-Dimensional CSP (1DCSP), Gilmore and Gomory (1961) and Gilmore and Gomory (1963). In the latter, a column generation method is presented for tightening the lower bound. However, that method does not obtain good results for the 2-Dimensional CSP (2DCSP). Dyckhoff (1990) presents a consistent and systematic approach for a comprehensive typology integrating the various kinds of problems for the cutting and packing problems. A deeper classification is presented in Dyckhoff and Finke (1992). One of the most important characteristics is the number of dimensions to consider for the cuts. Then, the CSP is classified into 1-Dimensional Cutting Stock Problems (1DCSP), where the stock is cut into strips (performing only lengthwise cuts on the stock), and 2-Dimensional Cutting Stock Problems (2DCSP), where the stock is width and lengthwise cut. Fig. 1 shows the differences between both problems. Fig. 1(a) shows a panel cut lengthwise into different strips whereas Fig. 1(b) shows a panel cut into rectangular pieces. In both cases, the leftovers are represented with the diagonal striped pattern. Some other dimensional problems have been studied in the literature, such as the 1.5-Dimensional Cutting Stock Problem (1.5DCSP). In this variation, the ordered pieces are given by their lengths and widths but fix one dimension and leave the other variable depending on other factors, such as weight (see Sierra-Paradinas et al. (2021)). Haessler and Sweeney (1991) presents a good review of the 1DCSP, 2DCSP and 1.5DCSP and their main differences. The 2DCSP is related with two classical packing problems: the Bin Packing Problem (BPP) and the Strip Packing Problem (SPP). The 2-Dimensional BPP (2DBPP) is a particular case of the 2DCSP where the demand for each item is equal to one. The SPP considers a strip of fix width and infinite length and consists in cutting all the items from the strip by minimising the used strip length. Lodi, Martello, and Monaci (2002) review the mathematical models, lower bounds, heuristics, exact and approximation algorithms for 2DSPP. Most of the methods proposed are heuristics. Oliveira et al. (2016) review the heuristics methods and classify them according to their type: constructive heuristics, improvement heuristics over sequences, and improvement heuristics over layouts. Lodi, Martello, and Vigo (2004) consider the 2DBPP and SPP, where items must be packed by levels. For further information on other variants of the CSP, together with the evaluation of the different CSPs, see the recent review Bezerra 3 (a) Non-guillotine cuts (b) Guillotine cuts Figure 2. Caption: 2DCSP: Non-guillotine vs Guillotine cuts Figure 2. Alt text: Two diagrams with a rectangular shape. On the left, caption (a) Non-guillotine cuts: Diagram with a rectangular panel divided with non-guillotine cuts forming different pieces. On the right, caption (b) Guillotine cuts: Diagram with a rectangular panel divided into three horizontal strips. Each strip is divided into different rectangular pieces of the same or less width than the strip. 2 2 222 1 1 (a) Exact 2-staged cuts Trimming Trimming Trimming 2 2 2 2 2 2 2 1 1 (b) Non-exact 2-staged cuts 1 2 2 2 3 3 3 (c) Exact 3-staged cuts Figure 3. Caption: Exact and non-exact staged cuts Figure 3. Alt text: Three diagrams with a rectangular shape: caption (a) Exact 2-staged cuts, caption (b) Non-exact 2-staged cuts and caption (c) Exact 3-staged cuts. Figure 3. Long description: On the left, caption (a) Exact 2-staged cuts: Diagram with a rectangular panel divided in three horizontal strips. Each strip is divided into rectangular pieces of the strip width. The dashed area at the end of the strips is leftover. In the middle, caption (b) Non-exact 2-staged cuts: Diagram with a rectangular panel divided in three horizontal strips. Each strip is divided into rectangular pieces. In the first strip some pieces’ widths are smaller than the strip generating some leftover on the top of them, trimming in dashed. On the rigth, caption (c) Exact 3-staged cuts: Diagram with a rectangular panel divided first into two horizontal strips. Each strip, is divided into two rectangular pieces and, third, the two rectangular pieces generated (right hand side of the original strips) are cut into horizontal strips. The dashed area is leftover. et al. (2019), focused on the 2-Dimensional level strip packing problem. An essential characteristic widespread in the industrial sector is the type of cuts. Often, the cutting equipment can produce only guillotine cuts performed from side to side parallel to the edges producing two new rectangles from a larger one. In the example shown in Fig. 1(a), all the cuts are guillotine. In the example shown in Fig. 1(b), all the cuts are also guillotine but in sequential order. First, lengthwise cuts are performed, generating two types of strips. Each one is then guillotine cut to produce the smaller rectangles. Fig. 2 shows an example of non-guillotine and guillotine cuts. The 2DCSP is classified as 2-staged (2SCSP) or 3-staged (3SCSP) if two or three sequential cuts are needed, respectively. The first stage consists in performing parallel lengthwise guillotine cuts that produce a set of rectangular strips. Each strip is individually cut in the second stage with the remaining parallel guillotine crosscuts. If there is no need for an additional cut, i.e., all the pieces’ widths equal the ordered dimensions, the pattern is called exact 2-staged guillotine. Otherwise, it is called non-exact since the pieces need a third stage, adding a cut to move away some scrap to meet the requested dimensions (see Salem et al. (2023) and Andrade et al. (2014)). Fig. 3 shows three different cases: exact 2-staged (Fig. 3(a)), non-exact 2-staged (Fig. 3(b)), and exact 3-staged (Fig. 3(c)). Observe in Fig. 3(b) that a third cut is needed for the smaller pieces in the first strip to meet the dimensions whereas in the case showed in Fig. 3(c), even being exact, the third cut is needed to obtain the final pieces. See 4 Vanderbeck (2001), Alvarez-Valdes, Parajon, and Tamarit (2002), Yanasse and Morabito (2008) Silva, Alvelos, and de Carvalho (2010), Macedo, Alves, and de Carvalho (2010), among others, for the 2SCSP and 3SCSP. Cintra et al. (2008) study the 2DCSP with guillotine cuts and their variants. They study the SPP with two, three, and four staged patterns. Dolatabadi, Lodi, and Monaci (2012) introduce an exact recursive procedure that constructs the set of guillotine packing associated with a given set of items as part of the 2-Dimensional Knapsack Problem (2DKP). Furini, Malaguti, and Thomopulos (2016) present a framework to model general guillotine constraints in 2DCSP formulated as mixed-integer linear programming, based on the formulation presented in Dyckhoff (1981). Furthermore, as a variant of the 2SCSP, Martin et al. (2022) propose probably for the first time models for 2SCSP and 3SCSP with a limited number of open stacks. An element to consider in the Cutting problems is the treatment of leftovers. They can be classified as reusable leftovers or scrap. The reusable leftovers are considered when the waste generated can be used again if it meets some specifications (typically minimum dimensions). Then, those reusable leftovers are considered stock in future processes (see do Nascimento, Cherri, and Oliveira (2022)). On the contrary, the scrap is directly discarded. The stock is also used to classify the CSP. Typically, the CSP considers a stock formed by an infinite number of panels with dimensions W×L. However, it is also common to consider a heterogeneous stock, formed by panels with varied dimensions. This problem is known as the Multi-Stock 2-Dimensional CSP (MS2DCSP). A particular case with heterogeneous stock is the problem with usable leftovers, where there is a high variability of stock dimensions and often only one piece of stock of each size. Furini et al. (2012) introduce a column generation heuristic for the MS2DCSP with 2-Staged guillotine cutting. Lately, Furini and Malaguti (2013) present some mathematical models for MS2DCSP with 2-Staged guillotine cutting. Previously, Pisinger and Sigurd (2005) consider a closely related problem, the 2DBPP with variable bin sizes. In the literature, the CSP considers that the stock dimensions are known in advance, even if it is heterogeneous. In Mosallaeipour (2017) the problem of supplier material selection and production planning in the carton box production industries is presented. In this work, the panel sizes considered are those offered by the suppliers and the cutting patterns are given by a software. Another problem related to the cardboard sector is presented in Sipahi (2022), where, in the first step, the panel sizes are determined by a simulated annealing algorithm, and in the second step, the products are assigned using an integer linear programming model. Nevertheless, the stock dimensions could be a decision as part of the production process. This variant, known as the 2DCSP with Variable-Sized stock (2D-VSCSP), has been recently presented in Salem et al. (2023). To the best of our knowledge, and as the authors mention in the paper, this is the only work dealing with the 2D-VSCSP. They present two mathematical models. The first is based on the work by Lodi and Monaci (2003), while the second is based on a Bin Packing Problem (BPP). As the authors remark in the paper, the problem can be approximated as a BPP since, in their practical application (textile sector), the demand is not very high, even though the demand for each item is greater than one unit. The problem we are dealing with fits into the 2D-VSCSP. Note that the 2D-VSCSP could be also treated as a 3-Staged Strip Packing Problem (3S-SPP). The first stage produces the different panels, while the second and third stages produce the strips and the final items, respectively. If more than one width is available in stock, the problem can be considered a Multi-Strip (3S) Packing Problem 5 (a) Honeycomb structure (b) Produced panels Figure 4. Caption: Honeycomb cardboard panels Figure 4. Alt text: Two pictures. On the left, caption (a) Honeycomb structure: Picture of the honeycomb structure, sandwich of two layers of grey paper with the net in the middle. Each cell is hexagonal. On the right, caption (b) Produced panels: Picture of a collection of honeycomb panels piled. (MSPP). 3. Problem description The problem considered takes place in the honeycomb cardboard industry. The product’s name comes from the honeycomb structure inside the carton panel. The honeycomb cardboard panels are made of paper, with a sandwich structure of two paper layers and a paper net inside (the honeycomb structure). Fig. 4 shows a graphical representation of a honeycomb cardboard panel and some panels produced by the company. The success of the honeycomb panels is due to their characteristics: green, light, resistant, economical, and easy to manipulate. Green: it is made with recycled and recyclable materials. Light: compared with wood panels, commonly used in transportation, its weight is around 1/6 the wood panels. Resistant: it resists 5kg/cm2to a compression of 50t/m2. Economical: compared to wood, or other Extended Polystyrene (EPS) products, the price is significantly lower. Finally, it is easy to manipulate and cut thanks to its low weight. In its beginnings, the honeycomb panels were used for transportation replacing the EPS. They are very resistant and light, making them perfect for protecting the transported products by absorbing the blows. Recently, this material has gained presence in the advertising and decorating sectors as it is possible to print on it. The company receives every day different orders that must be served. An order consists of a list of different items (rectangular pieces) given by their length (ℓi), width (wi), and the number of units to serve (di). The pieces are obtained by producing carton panels and cutting them into smaller items. Therefore, the company has to decide: (1) how many carton panels to produce, (2) their dimensions, and (3) how to cut them aiming to reduce the leftovers. It is worth pointing out that the pieces cannot be rotated as the material is weaker depending on how the pieces are positioned. The factory layout encloses two different areas, each devoted to a phase in the production process. The first one contains a machine, called line, to produce the honeycomb cardboard panels (see Figs. 5(a) and 5(b)). The second area comprises two cutting machines that cut the panels into smaller pieces (see Fig. 5(c)). The line unrolls the paper rolls used as covers and introduces in between them the honeycomb net, that is glued to the two covers. The paper roll width defines the panel’s width, while the length can be decided and adjusted. The cutting machinery 6 (a) Supply module for the line (b) Line machine (c) Cutting Machine Figure 5. Caption: Line and cutting machines Figure 5. Alt text: Three pictures. On the left, caption (a) Supply module for the line: Picture of the machine where the panels are produced. The paper rolls are allocated to feed the machine in four shafts. In the middle, caption (b) Line machine: Picture of the machine producing a continuous panel. On the right, caption (c) Cutting Machine: Picture of a cutting machine. The blades are allocated along a shaft to cut the carton panels. imposes minimum and maximum lengths to the panels, say ℓand ℓ, respectively. A width modification implies paper rolls replacement, whereas a length adjustment only needs to set the new value in the panel control. Both actions force the line to stop. Once the panels are produced, they are moved to the cutting area. The cutting machines have a set of circular blades distributed along a shaft that performs guillotine cuts. The blades can (1) fully cross the panel obtaining separated pieces or (2) partially cross the panel keeping it together. In the latter, the pieces are manually separated by the customer. This option is frequently chosen since packing on pallets is easier. In addition, for operation purposes, many customers prefer to receive pallets with panels containing only one reference, making their manipulation easier. Therefore, each panel can only contain one item reference. The operation produces leftovers but never trimming as all the obtained pieces have the same width as the strip generated. Therefore, the problem leads to the exact 2-Staged 2DCSP. Currently, the company works with panels of 1200×2400 mm2, not taking the advantage of the flexibility given by the line to better adjust the panels’ dimensions to the orders. As the company serves other products, they also work with different widths (paper rolls) that could be used to produce panels with different widths. In an analogous way, they could work with different lengths that are easily configured in the line. The aim of this contribution is to propose mathematical models allowing the company to explore new strategies and even decide on a better panel configuration to serve its demand. Then, the problem we are tackling fits into the 2DCSP family with variable-sized stock, 2D-VSCSP. As far as we know, 2D-VSCSP was recently introduced in Salem et al. (2023). Models presented in that work only solve cases with low total demand (up to 30 units) while our company manages many different items (up to 90) with high demands (frequently, thousands of pieces per item). These models’ size depends on the square of the total number of pieces ordered, making it impossible to apply them to our problem. Summarising, the main hypotheses considered are: •Panels are produced in the factory and have a rectangular shape (CSP). •The panel dimensions (width and length) have to be decided (2D-VSCSP). •Orders are known before planning the panel production. •Pieces have a rectangular shape (2D-CSP) and cannot be rotated. •Only guillotine cuts are allowed. 7 •Final pieces are supplied in pre-cut panels (they are not separated individually) and each panel includes only one item. •The objective is to reduce leftovers. 4. Mathematical optimisation models In this section, two Mixed Integer Linear Optimisation Models to solve the 2D-VSCSP with guillotine constraints are presented. Both models aim to determine the panels’ configuration such that the leftovers are minimised. A configuration is given by its length and width. In the first model, a set of potential configurations (width and length) is predefined by the company, and a subset of them is selected. On the contrary, in the second one, the configurations are proposed by the model: the paper roll selected defines the width, and the length is a variable in [ℓ, ℓ]. 4.1. Model 1: Selection Model (SM) The SM considers a set of predefined configurations, selects a subset and assigns each item to one of them. Each configuration jis given by its width Wjand length Lj. Upper bounds on the total number of configurations, widths and lengths can be imposed, nc, nwand nℓ, respectively. This model provides more options than the company’s current operation, which only uses the configuration 1200×2400 mm2. Note that if only one configuration is used, there is no place for optimisation since it is easy to calculate the leftover generated. If configuration jis selected the number of panels needed to serve the demand of item iis: npij =ldi nij m, where nij =jWj wik×jLj ℓikis the maximum number of pieces of item ithat can be produced with one panel of configuration j. However, when more than one configuration is selected, it is necessary to decide the assignment of each item to a configuration and, then, there is a combinatorial number of feasible solutions. 4.1.1. Sets I={1, . . . , I},set of items, being Ithe total number of different items. J={1, . . . , J},set of available configurations, being Jthe total number of configurations considered. W={ˆw1,ˆw2,..., ˆwn},set of available configuration widths. L={ˆ ℓ1,ˆ ℓ2,...,ˆ ℓm},set of available configuration lengths. Jˆw={j∈ J :Wj= ˆw} ⊂ J ,set of configurations of width ˆw, for ˆw∈ W. Jˆ ℓ={j∈ J :Lj=ˆ ℓ}⊂J,set of configurations of length ˆ ℓ, for ˆ ℓ∈ L. 4.1.2. Parameters Wj,width of configuration j, for j∈ J . 8 Lj,length of configuration j, for j∈ J . wi,width of item i, for i∈ I. ℓi,length of item i, for i∈ I. di,demand of item i, for i∈ I. sij,Leftover generated if item iis assigned to configuration j, for i∈ I,j∈ J . It can be calculated as sij =npij ·Wj·Lj−di·wi·ℓi. nc, nw, nℓ,maximum number of different configurations, widths and lengths that can be selected. 4.1.3. Decision variables xij = 1 if item iis assigned to configuration j, 0 otherwise, for i∈ I, j ∈ J . yj= 1 if configuration jis selected, 0 otherwise, for j∈ J . uˆw= 1 if at least one configuration of Jˆwis selected, 0 otherwise, for ˆw∈ W. vˆ ℓ= 1 if at least one configuration of Jˆ ℓis selected, 0 otherwise, for ˆ ℓ∈ L. 4.1.4. SM mathematical formulation min X i∈I X j∈J sijxij (SM.1) subject to X j∈J xij = 1 ∀i∈ I (SM.2) xij ⩽yj⩽X i∈I xij ∀i∈ I, j ∈ J (SM.3) yj⩽uˆw⩽X j∈J ˆw yj∀ˆw∈ W, j ∈ J ˆw(SM.4) yj⩽vˆ ℓ⩽X j∈J ˆ ℓ yj∀ˆ ℓ∈ L, j ∈ J ˆ ℓ(SM.5) X j∈J yj⩽nc(SM.6) X ˆw∈W uˆw⩽nw(SM.7) X ˆ ℓ∈L vˆ ℓ⩽nℓ(SM.8) xij, yj∈ {0,1} ∀i∈I, j ∈ J (SM.9) uˆw∈ {0,1} ∀ ˆw∈ W (SM.10) vˆ ℓ∈ {0,1} ∀ˆ ℓ∈ L (SM.11) The objective function (SM.1) minimises the total leftovers generated in the pro9 SM - |W| = 1 - nc= 1 VSM - |W| = 1 - nc= 1 VSM - |W| = 4 - nc= 1 Cons. Vars. 0–1 Vars. NonZero Cons. Vars. 0–1 Vars. NonZero Cons. Vars. 0–1 Vars. NonZero I1 56 53 52 217 783 507 259 2529 2655 1729 890 8633 I2 34 53 52 145 410 269 140 1319 1277 845 448 4133 I3 91 89 88 357 756 474 247 2373 2715 1727 907 8659 I4 167 185 184 685 1407 884 464 4417 4485 2847 1515 14235 I5 81 65 64 309 3641 2402 1208 12017 11396 7525 3794 37665 I6 80 77 76 305 521 317 167 1595 1907 1185 630 5977 I7 73 81 80 293 1985 1302 660 6501 7151 4711 2395 23531 I8 103 109 108 405 1847 1193 609 5973 6320 4107 2107 20579 I9 97 101 100 377 982 617 320 3096 3310 2101 1100 10561 I10 126 137 136 502 7417 4900 2466 24505 24838 16437 8286 82217 I11 89 89 88 349 4251 2804 1412 14023 14433 9539 4813 47719 I12 355 349 348 1429 8875 5789 2937 28993 30826 20199 10273 101199 I13 133 113 112 515 13427 8906 4466 44549 45035 29895 15003 149563 I14 218 197 196 863 14178 9377 4712 46915 47727 31609 15902 158177 I15 301 281 280 1203 18424 12178 6123 60931 62137 41135 20707 205851 I16 133 113 112 515 24287 16146 8086 80749 83591 55599 27855 278083 I17 301 281 280 1203 26062 17270 8669 86391 88135 58467 29373 292511 I18 148 133 132 576 20716 13757 6894 68809 71047 47213 23672 236173 I19 383 389 388 1552 22634 14953 7524 74808 77393 51223 25805 256299 I20 383 389 388 1552 30524 20213 10154 101108 104213 69103 34745 345699 Table 3. Models dimensions 16 nc Sce. 1 2 3 4 1 2 21 26 26 2 7 32 39 42 3 11 33 37 43 4 – 49 55 62 Table 5. Leftover reduction % nc Sce. 1 2 3 4 1 99 93 91 91 2 98 89 86 85 3 96 88 87 85 4 – 83 81 78 Table 6. Area improvement % 1234 0 20 40 60 Maximum configurations (nc) Leftover reduction (%) Scenario 1 Scenario 2 Scenario 3 Scenario 4 Figure 7. Caption: Leftover lines for each scenario Figure 7. Alt Text: Lines chart for each scenario showing the percentage of leftover reduction for nc= 1,2,3,4 1234 80 85 90 95 100 Maximum configurations (nc) Area improvement (%) Scenario 1 Scenario 2 Scenario 3 Scenario 4 Figure 8. Caption: Area lines for each scenario Figure 8. Alt Text: Lines chart for each scenario showing the percentage of area improvement for nc= 1,2,3,4 variables (0–1 Vars.), and non-zero elements (NonZero). It is worth pointing out that for nc>1, the number of variables and constraints is approximately proportional to the value of nc. To evaluate the results, as the total area of the demanded items is known, a key element is the percentage of material produced that is considered leftover. These percentages are reported in Table 4 for each scenario and each value of nc. For each instance, a colour scale is used, where the worst percentage is dark, and the best percentage is light. It can be observed that the company operation always reports the worst percentage compared to the scenarios tested. However, for Scenario 1 (model SM) and nc= 1, in most of the instances, the selected configuration is the one used by the company. As expected, for each scenario, the higher the value of nc, the better the results obtained. Observe that the largest improvement is from nc= 1 to nc= 2. In some specific instances, in particular instances I5 and I14 (scenario 3) and I5, I10, and I13 (scenario 4), it can be observed that the percentage of leftovers does not improve when the value of ¯ncis increased from 3 to 4. This is because, due to the increased complexity of the model, the optimiser does not achieve the optimal solution and only provides a (good) feasible solution within 1800 seconds. Finally, the last row of this table provides the average percentage that is considered leftover. Currently, the company discards on average 36% of the material, while with the extreme Scenario 5, the leftover is reduced to a third (13%). To better illustrate these results, Tables 5 and 6 report the average percentage of leftover reduction and material used improvement, respectively, using SM and VSM against the factory’s current operation. As mentioned, with Scenario 5, it is possible to reduce the leftover to only 13% 17 Sce. 0 Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 company SM - W1VSM - W1VSM - W2-nw= 1 VSM - W2-nw= 2 VSM - W2 1.2×2.4 nc= 1 nc= 2 nc= 3 nc= 4 nc= 1 nc= 2 nc= 3 nc= 4 nc= 1 nc= 2 nc= 3 nc= 4 nc= 2 nc= 3 nc= 4 nw= 4-nc= 8 I1 25.2 19.4 18.0 17.6 17.6 15.8 13.7 12.8 12.6 14.3 13.7 12.8 12.6 10.3 9.1 8.7 6.3 I2 27.7 27.7 24.1 24.1 24.1 27.1 21.7 19.9 18.9 18.2 12.3 9.9 8.6 8.9 6.3 4.7 4.1 I3 32.9 31.7 23.1 21.4 21.3 21.7 16.8 14.2 12.9 21.7 16.8 14.2 12.9 16.8 13.5 11.0 7.4 I4 34.1 34.1 27.5 25.6 25.3 29.0 23.3 21.9 21.1 29.0 23.3 21.9 21.2 19.8 15.5 14.5 10.5 I5 33.3 19.2 18.7 18.6 18.6 19.2 18.1 17.5 17.3 7.1 4.1 3.7 5.5 4.6 2.8 3.2 1.4 I6 16.1 16.1 14.7 14.3 14.1 13.9 6.7 3.8 2.9 13.9 6.7 3.8 2.9 6.7 3.8 2.9 1.8 I7 17.6 17.6 15.7 15.3 15.2 17.2 8.5 5.4 4.1 17.2 8.5 5.4 4.1 8.5 5.8 4.3 3.0 I8 42.8 42.8 37.9 35.6 35.6 38.8 28.9 26.5 25.7 38.8 28.9 26.7 25.4 25.6 23.5 21.4 17.0 I9 28.5 28.5 19.5 19.3 19.3 23.1 18.7 16.1 14.5 23.1 18.7 16.1 14.5 15.1 10.9 9.5 6.3 I10 35.3 35.3 34.3 34.1 34.1 35.3 27.3 26.3 26.1 24.0 15.2 13.8 13.6 13.0 8.0 11.2 4.7 I11 35.5 35.5 33.6 33.6 33.6 29.1 23.0 22.0 21.5 29.1 23.0 22.0 21.4 20.3 19.0 18.1 16.8 I12 40.7 40.7 33.5 31.9 31.9 34.8 28.2 26.5 24.8 34.8 28.2 26.5 24.8 24.5 21.2 19.6 15.9 I13 39.7 39.7 37.6 37.2 37.2 36.8 29.7 29.1 28.3 32.2 26.9 23.9 23.9 22.1 19.0 23.6 11.8 I14 39.3 39.3 36.9 36.4 36.4 37.6 30.1 28.8 27.6 33.8 27.9 24.9 26.1 23.1 19.3 18.2 13.6 I15 38.0 38.0 36.0 35.4 35.4 36.0 28.4 26.9 25.9 36.0 28.4 27.1 26.3 22.8 20.6 17.4 15.6 I16 27.2 22.4 14.5 11.2 11.2 20.9 10.0 6.7 4.5 21.2 10.0 6.7 4.4 9.7 5.3 3.0 2.0 I17 38.3 38.3 36.2 35.7 35.7 36.4 28.8 27.1 27.7 35.9 28.8 31.1 27.0 22.8 20.6 19.7 16.1 I18 35.9 35.9 29.8 28.2 28.2 35.9 29.8 27.6 26.3 34.9 29.4 27.3 26.3 25.2 22.8 19.2 15.3 I19 35.4 35.4 29.5 27.7 27.7 35.4 29.6 26.9 26.2 34.8 32.0 29.2 26.4 24.5 22.6 19.5 13.7 I20 34.9 34.9 29.2 27.5 27.5 34.9 29.2 27.1 26.2 34.2 28.9 29.5 27.3 23.7 23.2 18.9 13.1 av. 35.5 35.1 30.4 29.1 29.1 34 27.2 25.2 24.3 32.8 27 25.8 24 22.1 19.9 17.4 12.9 W1={1200} W2={1200,1400,1550,1600} Table 4. Percentage of leftover 18 (73% reduction with respect to the company’s operation) resulting in a total material used of 178,827 m2instead of 241,735 m2to serve the 20 orders. However, this strategy is not an option for the company as it supposes to work with eight different paper rolls (two per width). In these tables, it can be observed that some strategies using not more than four paper rolls improve its current operation significantly. From the analysis of these results, it is possible to identify simple strategies that can even halve the leftover amount. If only one configuration can be used, the best strategy would be to consider Scenario 3 with nc= 1. For each instance, one width is selected from the four available in the stock, and only one length is fixed (defined by the VSM model) for all the produced panels. The leftover reduction is 11% on average (for instance I5, the reduction is from 33% to 7%), obtaining in global total material saving of 9,674 m2. Nevertheless, considering the use of two configurations can lead to an important progress. If ncis set to 2, Scenario 2 (only 1200 mm width is available) and Scenario 3 (four widths are available and just one can be selected), obtain a big reduction in the percentage of leftover, 32% and 33%, respectively. It could be observed that both scenarios gave very similar improvements, therefore keeping the use of 1200 mm width (Scenario 2) is more convenient for operation purposes. With this operation, the company will have to adjust the machines only to fix the panels’ length to the two best lengths defined by the model. In this case, the 32% in leftover reduction turns into 27,586 m2of material savings. Notice that this saving is equivalent to 2×23 = 46 km of paper rolls with 1200 mm width. Finally, the best results are obtained when the company considers the option of selecting two different widths per order (Scenario 4). Different to the previous operation presented, in this case, the line needs to stop to change the paper rolls, which is not too time-consuming as four rolls can be allocated. If nc= 2, the percentage of leftovers is halved with respect to the amount generated with the current operative (49%). With this operation, the total amount of material used is reduced by 42,244 m2. Notice that when nc= 4, the leftover reduction is 62%; however, manipulating four different configurations makes the production process more complicated. Summarising, it can be observed that there are three strategies that can improve drastically the leftover reduction without implying significant changes in the company’s current operation: •Strategy 1 (Scenario 3 - nc= 1): 11% leftover reduction / 96% area. •Strategy 2 (Scenario 2 - nc= 2): 32% leftover reduction / 89% area. •Strategy 3 (Scenario 4 - nc= 2): 49% leftover reduction / 83% area. Finally, Table 7 reports some statistics on the computational performance of the selected strategies. The following information is provided: the value for the incumbent solution obtained (zIP ), the optimality gap (in %), and the computing time in seconds (Time). For comparison purposes, the last column reports the total area of the company’s current operation. Notice, that the total area of the demanded items is a lower bound for the VSM optimal solution (zIP ) and, therefore, it can be considered to compute the optimality gap. In Table 7, the optimality gap reported is the minimum between the one obtained by Gurobi (GAPG) and the one calculated with the total demanded area (a) as lower bound: GAP (%) = minGAPG,100zIP −a zIP . Results using aas a lower bound of zIP in the VSM model are not reported since they are worse in terms of computing time and/or incumbent solution. 19 Strategy 1 Strategy 2 Strategy 3 Company zIP Gap Time zIP Gap Time zIP Gap Time zIP I1 196 0.0 7.6 195 0.0 35.9 187 10.3 1800∗225 I2 407 0.0 0.8 425 0.0 2.1 366 0.0 1.9 461 I3 526 0.0 1.5 495 0.0 4.8 495 0.0 44.7 613 I4 799 0.0 2.1 740 0.0 13.3 708 0.0 124.9 861 I5 864 7.1 1800∗980 18.1 1800∗841 4.6 1800∗1204 I6 994 0.0 0.9 917 0.0 1.2 917 0.0 15.0 1020 I7 1652 0.0 30.7 1494 0.0 96.5 1494 8.5 1800∗1659 I8 2297 0.0 1.9 1975 0.0 56.1 1889 5.9 1800∗2454 I9 1881 0.0 1.1 1779 0.0 6.5 1703 0.0 25.8 2022 I10 4795 0.0 8.5 5015 27.3 1800∗4190 13.0 1800∗5630 I11 5691 0.0 64.0 5245 0.0 733.7 5063 10.2 1800∗6261 I12 8663 0.0 781.8 7861 4.4 1800∗7473 4.4 1800∗9521 I13 9392 0.0 23.2 9059 13.4 1800∗8173 22.1 1800∗10555 I14 10242 0.0 6.2 9697 13.0 1800∗8818 23.1 1800∗11169 I15 16902 36.0 1800∗15111 12.2 1800∗14011 22.8 1800∗17430 I16 17800 21.2 1800∗15569 10.0 1800∗15523 9.7 1800∗19267 I17 22520 35.9 1800∗20281 21.2 1800∗18709 22.8 1800∗23394 I18 36138 26.9 1800∗33537 27.7 1800∗31480 25.2 1800∗36746 I19 38266 31.9 1800∗35441 27.6 1800∗33039 24.5 1800∗38627 I20 52036 31.7 1800∗48333 29.2 1800∗44864 23.7 1800∗52618 ∗time limit exceeded Table 7. Model results for the three best operation strategies It can be observed in Table 7 that the areas do not increase from Strategy 1 to Strategy 3. Notice that in general, the more complex the strategy is, the higher the computing times. Furthermore, there is a relation between the size of the instance and the solution times/GAPS. Certainly, there are other factors that increase the difficulty of finding the optimal solution in a reasonable time, such as in instance I5, which corresponds to an order with small items (narrow and short) and a large variability in the quantity demanded for each of them (see Fig. 6). Regardless, although some of the GAPs are high, the solution obtained clearly improves the current operation of the company. 6. Conclusions The results reveal that using expert systems based on mathematical optimisation to support decision-making processes provides significant advantages compared to the operational processes designed based on the operator’s experience, even if they have a deep knowledge of the problem. We have presented two novel linear optimisation models for a cutting stock problem in the honeycomb cardboard industry suggested by a Spanish company. This problem belongs to the family of 2-Dimensional Cutting Stock Problems with Variable-Sized stock (2D-VSCSP) recently introduced by Salem et al. (2023). The models have been validated using real data from the company. The results improve the current operation in the factory by reducing leftovers and panel production. Furthermore, three straightforward strategies have been proposed to the company that slightly modify its 20 current operation. Concerning the number of configurations, the obtained results have shown that it is necessary to consider two configurations at most for a notable leftover reduction (until 49%). Taking into account the line specifications, the impact of the proposed strategies is not relevant in terms of time consumption and operation. Finally, we propose different research lines: (1) tight the VSM model with new valid cuts, trying to reduce the GAP and the computing times; (2) develop (meta)heuristics to obtain good feasible solutions in a short time, avoiding the use of commercial optimisers that are expensive for the company; (3) as the company has two cutting machines, consider job sequencing to minimise the production times. Acknowledgement(s) The authors would like to thank the company managers for providing us with real data and for giving us insight into the company’s current operation. Disclosure of interest The authors report there are no competing interests to declare. 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Mart´ın-Campo. 2023. “Dataset used in article A 2-dimensional guillotine cutting stock problem with variable-sized stock for the honeycomb 23 cardboard industry.” https://doi.org/10.5281/zenodo.8300923. Vanderbeck, F. 2001. “A Nested Decomposition Approach to a Three-Stage, Two-Dimensional Cutting-Stock Problem.” Management Science 47 (6): 864–879. https://doi.org/10. 1287/mnsc.47.6.864.9809. Yanasse, H.H., and R. Morabito. 2008. “A note on linear models for two-group and three-group two-dimensional guillotine cutting problems.” International Journal of Production Research 46 (21): 6189–6206. https://doi.org/10.1080/00207540601011543. Appendix A. Illustrative small-scale case study This section presents a small-case study for illustrative purposes. The order includes the following items: Items wi(mm)li(mm)di(mm) i160 500 340 i2265 600 72 i3320 700 24 For the sake of clarity, we assume that only 1200 mm wide rolls are available and that only one configuration can be defined. In addition, the company imposes a lower and upper limit of ℓ= 1800 mm and ℓ= 3100 mm for the panels’ length. Then, using the notation defined above, we have: I={i1, i2, i3},J={j1},W= {1200},J1200 ={j1},nc= 1, ℓ= 1800 and ℓ= 3100. The first step is to calculate rij, the number of rows of each item ithat fits in configuration j: ri1j1=jWj1 wi1k=j1200 60 k= 20, ri2j1=j1200 265 k= 4, ri3j1=j1200 320 k= 3 Then, it is possible to calculate cij, the number of columns needed to meet the demand of item iusing configuration j: ci1j1=ldi1 ri1j1m=l340 20 m= 17, ci2j1=l72 4m= 18, ci3j1=l24 3m= 8 The next step is to calculate a lower and an upper bound for the number of panels needed to produce this number of columns for each item. These bounds depend on ℓ= 1800 mm and ℓ= 3100 mm, the minimum and maximum lengths allowed for the configuration, respectively: ki1j1=lci1j1 ⌊ℓ ℓi1 ⌋m=l17 6m= 3, ki2j1=l18 5m= 4, ki3j1=l8 4m= 2, ki1j1=lci1j1 ⌊max{ℓi1,ℓ} ℓi1 ⌋m=l17 3m= 6, ki2j1=l18 3m= 6, ki3j1=l8 2m= 4. Therefore, Ki1j1={3,4,5,6},Ki2j1={4,5,6}and Ki3j1={2,3,4}. 24 Finally, the values of ℓijk are calculated. For item i1: ℓi1j13= maxnℓ, ℓi1lci1j1 3mo = maxn1800,500l17 3mo = 3000, ℓi1j14= maxnℓ, ℓi1lci1j1 4mo = maxn1800,500l17 4mo = 2500, ℓi1j15= maxnℓ, ℓi1lci1j1 5mo = maxn1800,500l17 5mo = 2000, ℓi1j16= maxnℓ, ℓi1lci1j1 6mo = maxn1800,500l17 6⌉o= 1800. Analogously: •Item i2:ℓi2j14= 3000, ℓi2j15= 2400 and ℓi2j16= 1800. •Item i3:ℓi3j12= 2800, ℓi3j13= 2100 and ℓi3j14= 1800. The objective function for the VSM model results: min 12003δi1j13+ 4δi1j14+ 5δi1j15+ 6δi1j16+ 4δi2j14+ 5δi2j15+ 6δi2j16+ 2δi3j12+ 3δi3j13+ 4δi3j14 Constraints (VSM.2), all items must be assigned to a configuration: xi1j1= 1, xi2j1= 1, xi3j1= 1. Constraints (VSM.3), a configuration is selected if and only if at least one item is assigned to it: xi1j1⩽yj1⩽xi1j1+xi2j1+xi3j1, xi2j1⩽yj1⩽xi1j1+xi2j1+xi3j1, xi3j1⩽yj1⩽xi1j1+xi2j1+xi3j1. Constraints (VSM.4) assure that a width is used if and only if a configuration of that width is selected. yj1⩽u1200 ⩽yj1. Constraints (VSM.5), a number of panels to be produced must be selected for each item assigned to a configuration: zi1j13+zi1j14+zi1j15+zi1j16=xi1j1, zi2j14+zi2j15+zi2j16=xi2j1, zi3j12+zi3j13+zi3j14=xi3j1. Constraints (VSM.6), the minimum configurations’ length is imposed by the ℓijk calculated: 3000zi1j13+ 2500zi1j14+ 2000zi1j15+ 1800zi1j16⩽Lj1, 3000zi2j14+ 2400zi2j15+ 1800zi2j16⩽Lj1, 2800zi3j12+ 2100zi3j13+ 1800zi3j14⩽Lj1. 25