FORMULATIONS AND SOLUTION APPROACHES FOR THE INTEGRATED LOT-SIZING AND CUTTING STOCK PROBLEM APPLIED TO LATTICE JOIST PRODUCTION

ABSTRACT

In the construction industry, lattice joists (final items) are precast structures that combine a steel truss and a concrete base. Their production is based on a client’s order that specifies the quantity, length, and deadline for delivery. A process manager uses this information to create a weekly production plan that determines the number and type of final items to produce in each period. The primary concerns of the factory include minimizing the waste of steel bars (objects) and optimizing setups and inventory costs. In this paper, decisions regarding purchasing steel bars and managing stocks in different sizes are considered extending the literature by adding an extra level of decision. We propose two alternative mathematical formulations for this integrated lot-sizing and cutting stock problem. Further, heuristic solution strategies are proposed to address the resulting mixed-integer programming formulations. Computational tests are conducted to evaluate the effectiveness of the proposed solution approaches.

Keywords:
production planning; column generation; precast structures

1 INTRODUCTION

Manufacturing companies encounter challenges in planning their production processes, particularly when determining the appropriate timing and quantities of products to meet demand over multiple periods. In this research, we consider a complex production process, in which small items with customized length need to be cut from bigger objects of various standard lengths.

Such cutting processes are typically found in the steel, paper, and wood processing industries. Given an expanding competitive edge, environmental liability, and natural resources scarcity, an efficient plan for this cutting process may significantly reduce the waste of materials and minimize total costs.

This research paper considers an application in the construction industry in which lattice joists (final items) are produced by combining steel truss and a concrete base. Considering the client’s orders that specify the quantities, lengths, and deadlines for delivery, a process manager uses this information to create a weekly production plan that determines the number and type of final items to produce in each period, aiming to minimize the waste of steel bars (objects) and optimizing mold utilization, setups, and inventory. It also incorporates decisions regarding purchasing steel bars and managing stocks in different sizes. To contribute to this context, we developed novel mathematical formulations and heuristic solution approaches for the studied problem. The proposed methodology combines mathematical programming and heuristic optimization techniques to generate high-quality production plans within acceptable computational time.

Integrated lot-sizing and cutting-stock problems were recently classified in an extensive review by Melega et al. (2018). The authors identify three distinct levels of production: the first level (L1) involves purchasing or manufacturing the raw objects; the second (L2) entails cutting these objects into smaller pieces; and the third (L3) consists of assembling those pieces into final products. According to this classification, the present paper addresses the first level, associated with the purchasing and/or manufacturing of objects, and the second level, related to cutting these objects into items. Therefore, the problem is classified as L1/L2/ −/M and can be characterized as an Integrated Lot-Sizing and Cutting Stock Problem, since it involves the integration of multiple production levels. This extends the study of Signorini et al. (2023), which was classified as −/L2/ −/M and characterized as a Multi-Period Cutting Stock Problem (Melega et al. (2018)), as it considers only the cutting level without integrating multiple production levels.

Instead of manufacturing objects on-site at the production plant, they are purchased from different suppliers in a varying of lengths and costs. These procurement choices determine the quantity of objects sourced from each supplier, considering the inventory of these objects at the production plant, to fulfill the internal demand for objects in the cutting process. Purchasing decisions significantly impact the cutting process at the production plant. It is important to clarify why Level L1 should be integrated into the well-studied multi-period cutting stock problem (−/L2/ −/M), rather than treated as a separate optimization problem. Incorporating L1 (sourcing) increases the realism and consistency of the overall decision-making framework across the supply chain. By explicitly modeling sourcing alternatives, the integrated approach enables coordination between upstream procurement decisions and downstream cutting and production activities, resulting in more cost-efficient and synchronized operations.

Therefore, the contributions of this work are as follows. First, we extend the formulation proposed by Signorini et al. (2023) by incorporating multiple suppliers for raw material acquisition. From an application perspective, this extension introduces new practical features, including purchasing and inventory decisions for objects of different sizes. Additionally, we introduce symmetry-breaking constraints for the mold setup variables to eliminate equivalent solutions generated by mold-label renumbering, thereby improving computational efficiency. Second, we propose an alternative mathematical formulation designed to provide tighter lower bounds and improve solution quality. Third, we develop novel solution methodologies associated with this formulation, including approaches for generating both lower and upper bounds. For the lower bound, different column generation-based strategies are proposed with innovative aspects arising from the two types of subproblem that have to be solved. Regarding upper bounds, based on the columns generated when solving the linear relaxation, different decomposition approaches based rounding strategies are used to obtain feasible solutions. Finally, we present a computational study using the proposed heuristics and compare the obtained results against a benchmark heuristic from the literature when solving the integrated problem.

The remainder of the paper is organized as follows. In Section 2, the production process of lattice joists is described based on information obtained from a Brazilian specialized factory. A literature review is presented in Section 3. Then, Section 4 presents two mathematical formulations representing the production planning of lattice joists. The proposed formulations are heuristically solved by several solutions’ strategies derived from the well-known Column Generation procedure, which are explained in Section 5. Then, in Section 6, the proposed formulations and solution strategies are evaluated via computational tests using data based on real information obtained from the factory. Section 7 presents some conclusions and indicates possible extensions of this research.

2 PROBLEM DESCRIPTION

The studied factory produces lattice joists (final items), which combine steel truss with a concrete base, as illustrated in the top of Figure 1. In this paper, we consider a one-to-one correspondence between the length of each final item and the length of the required steel item. Therefore, only the steel item at the top of the lattice joist (Figure 1) is considered, disregarding the cuts required for the triangular web members that efficiently support loads. The production planning process for lattice joists is driven by customer orders detailing quantities, lengths, and due dates. The process manager consolidates this data into a weekly production plan, mapping specific final items and volumes to designated periods across the planning horizon.

Figure 1
Representations of a lattice joist (top) and the production mold sections with separators (bottom).

As illustrated in Figure 2, the first stage of the production process consists of cutting steel bars into the required steel item lengths that compose the ordered final items. The steel bars are purchased from multiple suppliers in various standard lengths and are cut using one-dimensional cutting patterns. Therefore, in the proposed problem, multiple suppliers are available for the procurement of objects, and the production system jointly addresses inventory management and cutting planning of these objects within the production plant. The objective related to this first stage is to minimize the total cost, which includes the purchasing cost of objects, as well as the inventory holding cost associated with the cut steel items.

Figure 2
Process flow diagram of the manufacturing of steel trusses.

The second stage is manufacturing concrete planks using molds divided into sections of identical dimensions according to the bottom of Figure 1. The first activity is the mold setup, which consists of preparing an entire mold through cleaning and release-agent application. Since the use of a mold in a given production period requires prior preparation, this activity consumes both processing time and workforce resources. The second activity is cutting-pattern setup, which involves positioning separators in mold sections to define the arrangement of items within each section. This operation also requires significant time and labor, particularly when multiple cutting pattern configurations are used within the same mold.

After preparing the mold, a mixture of cement and other materials is poured mechanically into it to create the concrete bases of the lattice joists. Next, the cut steel trusses are inserted into the liquid concrete bases. After the concrete curing process, the lattice joists are removed from the mold (Figure 1). The manufactured final items are then stored in inventory. Therefore, the objective related to this second stage is to minimize the mold setup cost, the cutting-pattern setup cost, which is related to positioning separators in mold sections, and the inventory cost of final items.

3 LITERATURE REVIEW

In the context of civil construction, several studies have incorporated aspects of the cutting stock problem (CSP), as reviewed by Signorini et al. (2023), including the works of Prata et al. (2015), Pitombeira-Neto & Prata (2020), Lemos et al. (2021), Araújo et al. (2021a), Araújo et al. (2021b), Signorini et al. (2022), and da Silva et al. (2023). More recently, Bang et al. (2025) consider a company that produces stainless-steel pipes for plumbing applications. The authors propose a mathematical model for the multi-period cutting stock problem including raw material-product eligibility and inventory level restrictions. To tackle practical instances a large neighborhood search algorithm is proposed achieving a 24.8% reduction in cost and a 14% improvement in inventory optimization relative to current company practices. Ghafari et al. (2025) consider a case study in a major Iranian copper manufacturing company specializing in semi-finished copper and copper-alloy products used in buildings and infrastructure. The authors propose a cutting stock-based mathematical model whose main objectives are to minimize the number of setups resulting from machine blade changes and to reduce the deviation between the produced quantities and customer demand for each order. A comparison between the model output and the company’s current practice reveals that trim loss is reduced by 4% when using the cutting patterns generated by the proposed model. Furthermore, a reduction of up to 46.7% in setup time is observed for certain order sets.

Since the present paper extends the literature through the incorporation of multiple suppliers for raw material acquisition, studies that integrate Level L1 (object production/acquisition) decisions into multi-period cutting stock problems are also relevant, even when they are not explicitly applied to the construction sector. Few studies incorporate decisions regarding object production within the multi-period cutting stock problem. Melega et al. (2018) classified 12 papers that consider decisions on the production of objects (Level L1) integrated with the cutting stage (Level L2) in multi-period settings (M). These papers are related to the production of objects in practical applications found in the paper industry (Campello et al. (2020); Correia et al. (2004); Leão et al. (2017); Malik et al. (2009); Poldi & de Araujo (2016); Poltroniere et al. (2016, 2008)), copper industry (Hendry et al. (1996)), wood and furniture industry (Reinders (1992); Silva et al. (2014)), steel industry (Viegas et al. (2016)), as well as in general applications (Nascimento et al. (2022); Trkman & Gradisar (2007)). However, as said before, these papers consider only the decisions related to the production of objects, not including decisions about purchasing objects from different suppliers. Therefore, assuming supplier selection for raw material procurement from multiple suppliers, as addressed in our study, has not been explored sufficiently in the integrated lot-sizing and cutting stock problem literature.

For the integrated lot-sizing (Akartunalı et al. (2021)) and supplier selection problem, the primary goal is to decide which supplier(s) should be selected and how much of the final product(s) should be ordered from the chosen supplier(s) to meet the client’s demand and minimize purchasing costs (variable purchasing costs associated with the acquired quantities), ordering costs (fixed ordering costs incurred when placing orders), and holding costs. Therefore, most of the studies concentrate on managing the supply of final products, without taking into account a multi-level structure or any processes performed after the acquisition of the product(s) from the supplier(s). In contrast, in our integrated problem, due to the multi-level structure, the supplies (objects) are the inputs for the downstream levels, i.e., there are operations (cutting process) performed using the products provided by the suppliers.

A comprehensive literature review on supplier selection and lot-sizing problems can be found in Aissaoui et al. (2007); a classification under different quantity discounts can be found in Benton & Park (1996); and a literature review on multi-criteria decision-making approaches for supplier evaluation and selection is presented in Ho et al. (2010). We could find only three papers tackling supplier selection in multi-level structures yet in different contexts. Senyigit & Soylemez (2012) consider the supply selection of components used to produce the final product. Cunha et al. (2018) consider an application in the chemical industry minimizing the costs of acquisition and inventory of raw materials, together with the production of final products. Mohammadi et al. (2020) address the supplier selection for the raw materials, which are used to manufacture products on the production lines.

4 MATHEMATICAL FORMULATIONS

This section develops two integrated lot-sizing and cutting-stock formulations for the production planning of lattice joists in a Brazilian manufacturing facility. The models are based on a two-stage, one-dimensional cutting-stock structure and aim to minimize total system costs, en-compassing mold and cutting-pattern setup costs, steel bar procurement costs, production costs, and inventory holding costs for both steel items and final items. The formulations include inventory balance constraints (with no backorders), capacity limitations, and symmetry-breaking constraints. The mathematical formulations below use the following sets, parameters, and decision variables.

Sets:

• I: set of final items and steel items (index i);

• P: set of time periods (index p);

• L: set of steel bars with different standard length (index ℓ);

• $K_p^C$: set of available molds in period p (index k);

• $J_p^C$: set of cutting patterns for concreting the mold sections in period p (index j^C );

• $J_{\ell p}^S$: set of cutting patterns for the steel bars of length ℓ in period p (index $j_{\ell }^S$);

Parameters:

• W^C: length of molds (objects);

• $W_{\ell }^S$: length of steel bar ℓ (object);

• N: number of sections in a mold;

• w_i: length of final item and steel item i;

• d_ip: demand of final item i in period p;

• $a_{i{j}^{C}p}$: number of final items of length w_i in the cutting pattern j ^C , which composes a mold section configuration, in period p;

• $b_{i{j}_{\ell }^{S}p}$: number of cut steel items of length w_i in the cutting pattern $j_{\ell }^S$ in period p;

• $s{c}_p^{mold}$: setup cost of a mold in period p;

• $s{c}_p^{pat}$: setup cost for using a cutting pattern in a mold section in period p;

• $c_p^C$: mold section production cost per unit length in period p;

• $c_p^S$: steel item cutting cost per unit length in period p;

• $p{c}_{\ell p}^S$: purchase cost per meter of steel bar of length ℓ in period p;

• $h_{ip}^C$: inventory cost of final item i in period p;

• $h_{ip}^S$: inventory cost of steel item of length w_i in period p.

Variables

• $Y_{kp}^{mold}$: binary variable that takes value 1 if mold k is used in period p, or 0 otherwise;

• $Y_{j^{C}kp}^{pat}$: binary variable that takes value 1 if the cutting pattern (mold section configuration) j^C is used in mold k in period p, or 0 otherwise;

• $X_{j^{C}kp}^C$: frequency of cutting pattern j^C in mold k in period p;

• $X_{j_{\ell }^{S}p}^S$: frequency of cutting pattern $j_{\ell }^S$ in period p;

• $S_{ip}^C$: units of inventoried final item i at the end of period p;

• $S_{ip}^S$: units of inventoried cut steel item of length w_i at the end of period p;

• Z_ℓp: number of steel bars of type ℓ (with length $W_{\ell }^S$) purchased in period p;

• H_ℓp: number of steel bars of type ℓ (with length $W_{\ell }^S$) available in stock at the end of period p.

4.1 Formulation based on Gilmore & Gomory (GG)

The first mathematical formulation presented is based on Gilmore & Gomory (1961, 1963) built upon the concept of cutting patterns, which refers to the various arrangements of items in the mold while adhering to length constraints.

Formulation GG

$$\begin{gathered} \text{minimize}\sum _{p\in P}\sum _{k\in K_p^C}s{c}_p^{mold}{Y}_{kp}^{mold}+\sum _{p\in P}\sum _{k\in K_p^C}\sum _{j^C\in J_p^C}s{c}_p^{pat}{Y}_{j^{C}kp}^{pat}+\sum _{p\in P}\sum _{\ell \in L}p{c}_{\ell p}^S{W}_{\ell }^S{Z}_{\ell p} \\ +\sum _{p\in P}\sum _{k\in K_p^C}\sum _{j^C\in J_p^C}{c}_p^C{W}^C{X}_{j^{C}kp}^C+\sum _{p\in P}\sum _{\ell \in L}\sum _{j_{\ell }^S\in J_{\ell p}^S}{c}_p^S{W}_{\ell }^S{X}_{j_{\ell }^{S}p}^S \\ +\sum _{p\in P}\sum _{i\in I}{h}_{ip}^C{S}_{ip}^C+\sum _{p\in P}\sum _{i\in I}{h}_{ip}^S{S}_{ip}^S \end{gathered}$$ (1)

$$\begin{gathered} \text{subject to:} \\ \sum _{k\in K_p^C}\sum _{j^C\in J_p^C}{a}_{i{j}^{C}p}{X}_{j^{C}kp}^C+S_{i,(p-1)}^C-S_{i,p}^C=d_{ip},i\in I,p\in P \end{gathered}$$ (2)

$$\begin{gathered} \sum _{\ell \in L}\sum _{j_{\ell }^S\in J_{\ell p}^S}{b}_{i{j}_{\ell }^{S}p}{X}_{j_{\ell }^{S}p}^S+S_{i,(p-1)}^S-S_{i,p}^S \\ =\sum _{k\in K_p^C}\sum _{j^C\in J_p^C}{a}_{i{j}^{C}p}{X}_{j^{C}kp}^C,i\in I,p\in P \end{gathered}$$ (3)

$\sum _{j^C\in J_p^C}{X}_{j^{C}kp}^C⩽N{Y}_{kp}^{mold},p\in P,k\in K_p^C$ (4)

$Y_{(k+1)p}^{mold}⩽Y_{kp}^{mold},p\in P,k\in K_p^C\setminus \left\{\right|K_p^C\left|\right\}$ (5)

$X_{j^{C}kp}^C⩽N{Y}_{j^{C}kp}^{pat},p\in P,j^C\in J_p^C,k\in K_p^C$ (6)

$H_{\ell p}-H_{\ell ,(p-1)}+\sum _{j_{\ell }^S\in J_{\ell p}^S}{X}_{j_{\ell }^{S}p}^S-Z_{\ell p}=0\ell \in L,p\in P$ (7)

$H_{\ell 0}=0,\ell \in L$ (8)

$S_{i0}^C=0,S_{i0}^S=0,i\in I$ (9)

$S_{ip}^C,S_{ip}^S\in {\mathrm{ℝ}}_{+},i\in I,p\in P$ (10)

$Y_{kp}^{mold}\in \mathbb{B},p\in P,k\in K_p^C$ (11)

$Y_{j^{C}kp}^{pat}\in \mathbb{B},X_{j^{C}kp}^C\in {\mathrm{\mathbb{Z}}}_{+},p\in P,j^C\in J_p^C,k\in K_p^C$ (12)

$X_{j_{\ell }^{S}p}^S\in {\mathrm{\mathbb{Z}}}_{+},\ell \in L,p\in P,j_{\ell }^S\in J_{\ell p}^S$ (13)

$Z_{\ell p},H_{\ell p}\in {\mathrm{\mathbb{Z}}}_{+},\ell \in L,p\in P$ (14)

The objective function (1) correspondingly minimizes the costs of the mold and cutting pattern setup; the purchase of steel bars of standard lengths; the use of mold sections and the cut of steel bars of standard length; inventory of final items and cut steel items. The cost associated with preparing the mold is known as the mold setup cost, which takes into account the time and labor required for this task. Additionally, the cutting pattern setup cost is imposed for using distinct cutting patterns in different sections of the same mold during each period.

The final items inventory balance is imposed by constraints (2). Constraints (3) set the relationship between the cutting of steel items and the production of the concrete planks: depending on the lattice joists’ production, the steel items are cut and carried as inventory to be utilized in the following periods when necessary. Observe that, in these constraints, the parameters for the cutting patterns $a_{i{j}^{C}p}$ and $b_{i{j}_{\ell }^{S}p}$ depend on the period p because the underlying problem is dynamic, integrating time-indexed production and inventory optimization. This temporal dependence directly influences the solution approach described in Section 5.1. Within the column generation framework, the pricing subproblems are solved independently for each period, tailoring the generated patterns to that specific period’s demand, capacity, and cost structure. The mold setup constraints (4) ensure that the mold is prepared only when required and limit the frequency of cutting patterns used in each mold to the number of sections. Following the approach of Jans (2009), we break the symmetry associated with alternative optimal mold-labeling solutions using constraints (5). By eliminating equivalent solutions arising from mold label renumbering, these constraints significantly improve computational efficiency. In the production process of lattice joists, the arrangement of separators in mold sections requires a significant amount of time and workforce, which are associated with the number of different configurations (cutting patterns) used in the same mold. Therefore, using fewer cutting patterns in the same mold results in a faster process. Constraints (6) combined with the second term of the objective function (1) address the use of different cutting patterns in the same mold. Constraints (7) regulate the availability of steel bars of various lengths in stock during each period. Constraints (8)-(9) set the initial inventory of steel bars, cut steel items and final items to zero without loss of generality. The domain of the variables is specified by constraints (10)-(14).

As an extension of the research reported by Signorini et al. (2023), this formulation introduces the procurement of steel bars of different standard lengths to the production planning problem. In mathematical terms, we define a set L of different types of steel bars ℓ with a different standard length $\left(W_{\ell }^S\right)$. For all periods, the set of cutting patterns was extended to each length of steel bar through the addition of the index ℓ to the set $J_{\ell p}^S$ and to the associated index $j_{\ell }^S$. Finally, constraints (7) are added to the formulation and consider the Level L1 of decisions integrated with Level L2.

4.2 Formulation based on Vanderbeck (VB)

This section presents an alternative formulation for the lattice slab production planning problem based on the variables used in Vanderbeck (2000) and Ma et al. (2019). We define a new cutting pattern vector that includes not only a feasible configuration of the final items (i.e., the $a_{i{j}^{C}p}$ parameters) but also its frequency $\left(X_{j^{C}kp}^C\right)$. Let $x_{j^{C}p}$ be the number of times that the cutting pattern j ^C is used in a period p. The new cutting pattern vector is then defined as follows:

$\left(x_{j^{C}p},a_{1j^{C}p},\dots ,a_{\left|I\right|j^{C}p}\right)\in {\mathrm{\mathbb{Z}}}_{+}^{\left|I\right|+1}$ (15)

where

$\sum _{i\in I}{w}_i{a}_{i{j}^{C}p}⩽W^C,x_{j^{C}p}⩽N,\text{and}{x}_{j^{C}p}{a}_{i{j}^{C}p}⩽\sum _{r=p}^{\left|P\right|}{d}_{ir},i\in I.$ (16)

We also define a new set for these cutting patterns (15). Let ${\overline{J}}_p^C$ be the set of cutting patterns for concreting the mold sections in period p (index j ^C ). As a consequence, we define a new variable ${\overline{Y}}_{j^{C}kp}^{pat}$ as a binary variable that takes the value 1 if the combination of final items $\left(a_{1j^{C}p},\dots ,a_{\left|I\right|j^{C}p}\right)$ is used $x_{j^{C}p}$ times in mold k in period p and assumes the value 0 otherwise. Using these new variables, we obtain the following mathematical formulation:

Formulation VB

$$\begin{gathered} \text{minimize}\sum _{p\in P}\sum _{k\in K_p^C}s{c}_p^{mold}{Y}_{kp}^{mold}+\sum _{p\in P}\sum _{k\in K_p^C}\sum _{j^C\in {\overline{J}}_p^C}\left(s{c}_p^{pat}+c_p^C{W}^C{x}_{j^{C}p}\right){\overline{Y}}_{j^{C}kp}^{pat} \\ +\sum _{p\in P}\sum _{\ell \in L}p{c}_{\ell p}^S{W}_{\ell }^S{Z}_{\ell p}+\sum _{p\in P}\sum _{\ell \in L}{c}_p^S{W}_{\ell }^S\sum _{j_{\ell }^S\in J_{\ell p}^S}{X}_{j_{\ell }^{S}p}^S \\ +\sum _{p\in P}\sum _{i\in I}{h}_{ip}^C{S}_{ip}^C+\sum _{p\in P}\sum _{i\in I}{h}_{ip}^S{S}_{ip}^S \end{gathered}$$ (17)

subject to: (5), (7), (8), (9), (10), (11), (13), (14)

$\sum _{k\in K_p^C}\sum _{j^C\in {\overline{J}}_p^C}{x}_{j^{C}p}{a}_{i{j}^{C}p}{\overline{Y}}_{j^{C}kp}^{pat}+S_{i,(p-1)}^C-S_{i,p}^C=d_{ip},i\in I,p\in P$ (18)

$$\begin{gathered} \sum _{\ell \in L}\sum _{j_{\ell }^S\in J_{\ell p}^S}{b}_{i{j}_{\ell }^{S}p}{X}_{j_{\ell }^{S}p}^S+S_{i,(p-1)}^S-S_{i,p}^S \\ =\sum _{k\in K_p^C}\sum _{j^C\in {\overline{J}}_p^C}{x}_{j^{C}p}{a}_{i{j}^{C}p}{\overline{Y}}_{j^{C}kp}^{pat},i\in I,p\in P \end{gathered}$$ (19)

$\sum _{j^C\in {\overline{J}}_p^C}{x}_{j^{C}p}{\overline{Y}}_{j^{C}kp}^{pat}⩽N{Y}_{kp}^{mold},p\in P,k\in K_p^C$ (20)

${\overline{Y}}_{j^{C}kp}^{pat}\in \mathbb{B},p\in P,j^C\in {\overline{J}}_p^C,k\in K_p^C$ (21)

Note that this model is very similar to the one described in Section 4.1. The differences lie in the ${\overline{Y}}_{j^{C}kp}^{pat}$ variables. In this formulation, they carry the cost of using the mold sections in addition to the cost of cutting pattern setup they already had (represented by the second term of the objective function (17)). These variables are also present in the demand balance constraints (18), in the equations that establish the integration between the two cutting stages (19), and in the mold setup constraints (20). In addition, the second group of inequalities in (16), which limit the parameter of the cutting pattern frequency in this formulation, combined with constraints (20) ensure the mold section cutting pattern assignment restrictions described by the inequalities (6) of the GG formulation.

The choice of using this approach in the second cutting stage (i.e., for the molds) is due to the tailing-off effect that occurred in this cutting problem in the CG formulation during the computational tests for column generation.

5 SOLUTION STRATEGIES

In this section, different column generation-based strategies for solving the proposed formulations are described. Firstly, in Section 5.1 we describe the methods used to obtain lower bounds. Based on column generation, these methods feature innovative aspects arising from the two-stage problem, where two different groups of subproblems are solved. Thereafter, to obtain upper bounds, in Section 5.2 we explore decomposition by stages.

5.1 Lower bounds (LB)

Formulations for the cutting stock problem, based on the seminal works of Gilmore and Gomory (1961; 1963), often involve a substantial number of variables corresponding to all feasible cutting patterns (de Lara Andrade et al. (2021)). To efficiently manage this large number of variables, column generation procedures are commonly employed to solve the linear relaxations of these formulations, thereby obtaining valid lower bounds for the problem. In this approach, a limited set of cutting patterns (columns) is initially selected to construct the basis of the linear relaxation. The dual solution of this Restricted Master Problem (RMP) is then used to formulate a pricing problem that identifies potential columns with negative reduced costs to be added to the relaxed problem. This process is repeated iteratively until no further promising columns can be found, or another user-defined stopping criterion is met.

The proposed method involves initially defining the RMP by considering only homogeneous cutting patterns, provided that

$a_{i{j}^{C}p}=\left\{\begin{array}{cl}\min \left\{⌊\frac{W^C}{w_i}⌋;\max \left\{1;{\gamma }_{ip}\right\}\right\}& \text{if}i=j^C\\ 0& \text{otherwise,}\end{array}\right.$

and

$b_{i{j}_{\ell }^S\ell p}=\left\{\begin{array}{cl}\min \left\{⌊\frac{W_{\ell }^S}{w_i}⌋;\max \left\{1;{\gamma }_{ip}\right\}\right\}& \text{if}i=j_{\ell }^S\\ 0& \text{otherwise,}\end{array}\right.$

where γ _ip represents the cumulative demand for final item i from period p to the last period, that is,

${\gamma }_{ip}=\sum _{r=p}^{\left|P\right|}{d}_{ir}i\in I,p\in P.$

5.1.1 Subproblems for the GG Formulation

Upon solving the RMP, the corresponding optimal dual values λ _ip , σ _ip , µ _kp , and δ _ℓp are obtained from constraints (2), (3), (4), and (7)-(8), respectively. Subsequently, for each period p ∈ P and each steel bar type ℓ ∈ L, the knapsack subproblems (22)-(24) and (25)-(27) are solved. These subproblems are used to identify cutting patterns with negative reduced cost that can improve the current RMP solution.

• Stage 1: Steel bar cutting subproblem

${\mathrm{OF}}_{\mathrm{SUB}}^{\mathrm{S}}(\mathrm{\ell },\mathrm{p})=\min c_p^S{W}_{\ell }^S-\sum _{i\in I}{\sigma }_{ip}{\beta }_i-{\delta }_{\ell p}$ (22)

$$\begin{gathered} \text{s.t.} \\ \sum _{i\in I}{w}_i{\beta }_i⩽W_{\ell }^S \end{gathered}$$ (23)

${\beta }_i\in {\mathrm{\mathbb{Z}}}_{+}i\in I.$ (24)

• Stage 2: Mold section concrete filling subproblem

${\mathrm{OF}}_{\mathrm{SUB}}^{\mathrm{C}}\left(\mathrm{p}\right)=\underset{k\in K_p^C}{\min }{c}_p^C{W}^C-\sum _{i\in I}({\lambda }_{ip}-{\sigma }_{ip}){\alpha }_i-{\mu }_{kp}$ (25)

$$\begin{gathered} \text{s.t.} \\ \sum _{i\in I}{w}_i{\alpha }_i⩽W^C \end{gathered}$$ (26)

${\alpha }_i\in {\mathrm{\mathbb{Z}}}_{+},i\in I$ (27)

Since all molds are identical, assigning different sets of columns to each mold would be inconsistent. Therefore, in the Stage 2 subproblems, the minimum is taken over all molds k ∈ K _p and only the column with the lowest reduced cost is added to the RMP. This ensures that the corresponding cutting pattern is applicable to all molds. Once the $\sum _{p\in P}(1+|K_p\left|\right)$ subproblems are solved, a new iteration of the column generation process begins. The procedure continues until no attractive columns are identified in either subproblem or the improvement in the RMP objective value falls below the tolerance threshold (0.001) for three consecutive iterations, a condition known as the tailing-off criterion. At this point, the method is terminated, and integrality constraints are enforced on the variables $X_{j^{C}kp}^C,X_{j_{\ell }^{S}p}^S,Y_{kp}^{mold}$ and $Y_{j^{C}kp}^{pat}$ in the final RMP formulation.

In the column generation procedure applied to the GG formulation, solution degeneracy was encountered in all tested instances due to the second cutting stage, leading to a tailing-off effect. Since the method was discontinued by an outer criterion, the linear relaxation obtained cannot be used as a precise lower bound. To address it, one may consider the linear relaxation solution OF ^r at iteration r (RMP at iteration r), to calculate a lower bound (LB) following the formula presented by Degraeve & Jans (2007) (see also Desaulniers et al., 2005):

$LB=O{F}^r+\sum _{p\in P}\left(O{F}_{SUB}^S\left(p\right)+O{F}_{SUB}^C\left(p\right)\right).$ (28)

5.1.2 Subproblems for the VB formulation

Similar to the procedure described in Section 5.1.1, after solving the RMP, the dual optimal solution [λ _ip , σ _ip , µ _kp , δ _ℓp ], for which the variables are associated to constraints (18), (19), (20), and (7)-(8), is recovered and used to build the subproblems (29)-(31) and (39)-(42).

• Stage 1: Steel bar cutting subproblem

$O{F}_{SUB}^S(\ell ,p)=\min c_p^S{W}_{\ell }^S-\sum _{i\in I}{\sigma }_{ip}{\beta }_i-{\delta }_{\ell p}$ (29)

$$\begin{gathered} \text{s.t.} \\ \sum _{i\in I}{w}_i{\beta }_i⩽W_{\ell }^S \end{gathered}$$ (30)

${\beta }_i\in {\mathrm{\mathbb{Z}}}_{+}i\in I.$ (31)

• Stage 2: Mold section concrete filling subproblem: for each p ∈ P,

$O{F}_{SUB}^C\left(p\right)=\min s{c}_p^{pat}+\left(c_p^C{W}^C-\sum _{i\in I}({\lambda }_{ip}-{\sigma }_{ip}){\alpha }_i-u_p\right)x$ (32)

$$\begin{gathered} \text{s. t.} \\ \sum _{i\in I}{w}_i{\alpha }_i⩽W^C \end{gathered}$$ (33)

$x⩽N$ (34)

$x{\alpha }_i⩽\sum _{r=p}^{\left|P\right|}{d}_{ir},i\in I$ (35)

$x\in {\mathrm{\mathbb{Z}}}_{+}$ (36)

${\alpha }_i\in {\mathrm{\mathbb{Z}}}_{+},i\in I$ (37)

where $u_p=\underset{k\in K_p^C}{\max }{\mu }_{kp},x=x_{j^{C}p}\in \{1,\dots ,N\}$ and N is the number of sections in a mold.

Note that the set of subproblems for Stage 2 involves nonlinearities. To address this, we observe that the parameter $x_{j^{C}p}$ is bounded:

$x_{j^{C}p}⩽\left\{N;\underset{i\in I}{\max }\sum _{r=p}^{\left|P\right|}{d}_{ir}\right\},p\in P,j^C\in {\overline{J}}_p^C.$ (38)

As a consequence, one can enumerate values of the cutting pattern frequencies. Therefore, for each value of $\overline{x}={\overline{x}}_{j^{C}p}\in \{1,\dots ,N\}$, the following linear subproblem can be solved:

$\overline{O{F}_{SUB}^C(p,\overline{x})}=\max \sum _{i\in I}({\lambda }_{ip}-{\sigma }_{ip}){\alpha }_i$ (39)

$$\begin{gathered} \text{s. t.} \\ \sum _{i\in I}{w}_i{\alpha }_i⩽W^C \end{gathered}$$ (40)

${\alpha }_i⩽\lfloor \frac{{\displaystyle \sum _{r=p}^{\left|P\right|}{d}_{ir}}}{\overline{x}}\rfloor ,i\in I$ (41)

${\alpha }_i\in {\mathrm{\mathbb{Z}}}_{+},i\in I.$ (42)

If the reduced cost is lower than the tolerance −0.001, the associated solution (column) is added to the RMP, otherwise, if the reduced costs of the subproblems from both stages through all periods p ∈ P are greater than the tolerance, the column generation procedure ends.

As described previously, both GG and VB formulations use the traditional column generation approach, where each period is associated with a matrix composed of the initial homogeneous columns and the generated columns for the respective period. Hence, the production in each period is limited to employing only the columns present in the respective cutting pattern matrix. It is worth mentioning that we have tried an alternative approach with a unique matrix composed of all different cutting patterns distributed across the periods, considering all initial columns and the ones that were generated. This strategy provides all generated columns for every period. Since this alternative strategy did not obtain good results, we will omit its detailed computational results.

5.2 Upper bounds (UB)

Multiple alternatives to obtain upper bounds assessing the strengths of the proposed formulations are described below.

• GG: After solving the linear relaxation of the GG formulation using the column generation procedure described in Section 5.1, the integrality constraints (11)-(14) for the GG formulation are added to the associated RMP built with all generated columns, and the optimization package CPLEX solves the resulting MIP, obtaining a feasible solution to the GG mixed-integer formulation.

• VB: After solving the linear relaxation of the VB formulation using the column generation procedure described in Section 5.1, the integrality constraints (11)-(14) for the VB formulation are added to the associated RMP built with all generated columns, and the optimization package CPLEX solves the resulting MIP, obtaining a feasible solution to the VB mixed-integer formulation.

• VB Fix: After solving the linear relaxation of the VB formulation using the column generation procedure described in Section 5.1, the setup variables for which the linear relaxation are close to zero (smaller than 0.0001) are fixed at zero. After fixing some setup variables, the integrality constraints for the remaining variables on (11)-(14) are added to the associated RMP built with all generated columns, and the optimization package CPLEX solves the resulting MIP, obtaining a feasible solution to the VB mixed-integer formulation.

• VB GG: A hybrid approach is also applied to experiment with the possibility of combining the strengths of each formulation. In this strategy, the columns generated by the VB formulation are used to build and solve the MIP of the GG formulation. This is based on the understanding that the lower bound provided by the VB model is stronger than the one obtained by the GG formulation, whereas the latter produces a better upper bound. So, after solving the linear relaxation of the VB formulation using the column generation procedure described in Section 5.1, the integrality constraints, (11)-(14) for the GG formulation, are added to the associated RMP built with all generated columns for the VB formulation, and the optimization package CPLEX solves the resulting MIP for the GG formulation, obtaining a feasible solution.

• VB+SCG: Based on the results from Signorini et al. (2023), the two stages of the problem are solved separately. First, the column generation is applied to the linear relaxation of the Stage 2 sub-model, which includes the first two and the fifth parcels of the objective function (17), the constraints (18), (20) and (5), and the variables $S_{ip}^C\in {\mathrm{ℝ}}_{+}(S_{i0}^C=0),Y_{kp}^{mold}\in \{0,1\}$ and $Y_{j^{C}kp}^{pat}\in \{0,1\}$. After that, the integrality constraints are added ($Y_{kp}^{mold}\in \mathbb{B}$ and $Y_{j_{C}kp}^{pat}\in \mathbb{B}$), and the resultant mixed-integer RMP is solved. Subsequently, the integer solution of the Stage 2 is fixed and used as parameters to solve via column generation the linear relaxation of the Stage 1 sub-model, which includes the third, fourth and the sixth parcels of the objective function (17), the constraints (19) and (7), and the variables $S_{ip}^S\in {\mathrm{ℝ}}_{+}(S_{i0}^S=0),X_{j_{\ell }^{S}p}^S\in {\mathrm{ℝ}}_{+},Z_{lp}\in {\mathrm{ℝ}}_{+}$ and $H_{lp}\in {\mathrm{ℝ}}_{+}(H_{l0}=0)$. Finally, the integrality constraints are added to the last RMP ($X_{j_{\ell }^{S}p}^S\in {\mathrm{\mathbb{Z}}}_{+},Z_{lp}\in {\mathrm{\mathbb{Z}}}_{+}$ and $H_{lp}\in {\mathrm{\mathbb{Z}}}_{+}$), and the resultant mixed-integer RMP is solved. The solutions of both stages compose a feasible solution to the original problem.

In total, we are proposing five alternative approaches, the first four based on the Integrated Column Generation and the last one based on the Separated Column Generation. It is worth mentioning that the last three strategies were proposed based on the initial results obtained by the second one, meaning the high-quality lower bound and low-quality upper bound obtained by the VB approach. So, the last three strategies try to take advantage of the lower bounds and improve the upper bounds of the VB strategy.

6 COMPUTATIONAL RESULTS

The evaluation of the different solution strategies applied to the considered problem is presented in this section. The experiments were performed on a Dell 64-bit desktop with Intel Core i7-8700 CPU @ 3.20 GHz and 16 GB RAM. Microsoft Visual Studio Express 2017 for Windows Desktop was used to run the IBM Concert technology in a C++ project and CPLEX version 20.1 with default settings. For each instance, a maximum time limit of 1,800 seconds was set. The solution approaches give lower bounds (LB) and upper bound (UB), and the ‘gap’ is calculated by the formula below:

$gap=100\frac{UB-LB}{UB}\%.$ (43)

6.1 Parameter settings

The data set presented in Table 1 is organized into 27 classes characterized according to the number of different final item (and steel item) lengths (|I| = 5, 10, or 15), the number of periods considered in the planning horizon (|P| = 5, 10, or 15), and the number of different steel bars lengths for the cutting process (|L| = 3, 5 or 8). These combinations define instances ranging from small-scale settings (|I| = 5, |P| = 5, |L| = 3) to larger-scale settings (|I| = 15, |P| = 15, |L| = 8). In addition, 5 instances were generated for each class, resulting in a total of 135 instances.

Table 1
Data set parameters.

Class	No. of final items	No. of periods	No. of different standard lengths of steel bars
1			3
2		5	5
3			8
4			3
5	5	10	5
6			8
7			3
8		15	5
9			8
10			3
11		5	5
12			8
13			3
14	10	10	5
15			8
16			3
17		15	5
18			8
19			3
20		5	5
21			8
22			3
23	15	10	5
24			8
25			3
26		15	5
27			8

In all instances: the final item (and stell item) length (w _i ) ranges from 0.6 to 5.0 meters; the final item demand (d _ip ) is estimated to be an integer number between 50 and 150 in each period; the standard length of a steel bar $\left(W_{\ell }^S\right)$ is selected from even numbers ranging between 8 and 22. In each period p ∈ P, the number of available molds is equivalent and is calculated by (44):

$K_p^C=\underset{p\in P}{\max }⌈1.5\times \frac{{\displaystyle \sum _{i\in I}{w}_i{d}_{i,p}}}{{\displaystyle N{W}^C}}⌉.$ (44)

Namely, the number of available molds in each period is one and a half times the greatest number of molds needed for producing items in any given period throughout the planning horizon. So, $K_p^C$ identical molds divided into N = 10 identical sections measuring W ^C = 10 meters long are considered.

The costs used in the instances were simulated based on real data acquired in a search over Brazilian specialized websites and online stores (see Signorini et al. (2023), Section 6.1). Hence, for each period p ∈ P: the cost of preparing a mold is $s{c}_p^{mold}=200$; the cost of preparing the mold sections using separators to indicate the cutting patterns in each section is fixed in $s{c}_p^{pat}=20$; the cost to purchase a meter of steel bar of length ℓ ∈ L is $p{c}_{\ell p}^S=10$; the cost of filling a mold section with concrete (mold section production cost) is $c_p^C=1.6$ per meter; and the steel item cutting cost is $c_p^S=1.1$ per meter. The cost for holding inventory is calculated as a percentage of the total length of each produced final item or cut steel item respectively: $h_{ip}^C=0.5w_i$ and $h_{ip}^S=0.25w_i$ for i ∈ I and p ∈ P.

6.2 Analysis of the results

In this section, detailed information about the solutions for each class is addressed in Figure 3 and Tables 2, 3 and 4. The values presented in these tables represent the average values for each class, considering only instances with feasible solutions across GG, VB, VB Fix, and VB + SCG strategies.

Figure 3
Representation of the solution status distribution for each solution strategy.

Table 2
Comparison of Lower Bounds and Upper Bounds.

Lower Bounds Comparison			Upper Bounds Comparison
Class	GG	VB	GG	VB	VB Fix	VB+SCG
1	99.75%	100%	100%	100.39%	101.99%	100.17%
2	99.76%	100%	100%	100.30%	102.00%	100.19%
3	99.76%	100%	100%	100.41%	102.08%	100.18%
4	99.75%	100%	100%	100.58%	100.67%	100.07%
5	99.75%	100%	100%	100.62%	100.80%	99.98%
6	99.76%	100%	100%	100.63%	100.76%	99.97%
7	99.75%	100%	100%	100.49%	100.64%	100.00%
8	99.75%	100%	100%	100.84%	100.56%	99.96%
9	99.76%	100%	100%	100.55%	100.55%	99.82%
10	99.77%	100%	100%	101.13%	101.24%	99.94%
11	99.77%	100%	100%	101.62%	101.18%	99.93%
12	99.77%	100%	100%	101.98%	101.56%	100.07%
13	99.75%	100%	100%	103.46%	99.79%	99.47%
14	99.76%	100%	100%	101.16%	99.75%	99.61%
15	99.76%	100%	100%	101.04%	100.36%	100.11%
16	99.77%	100%	100%	102.40%	100.37%	100.05%
17	99.76%	100%	100%	100.59%	99.93%	99.39%
18	99.77%	100%	100%	99.87%	99.59%	99.51%
19	99.78%	100%	100%	102.07%	100.75%	100.10%
20	99.77%	100%	100%	102.06%	100.55%	99.73%
21	99.77%	100%	100%	103.37%	100.45%	99.93%
22	99.77%	100%	100%	100.52%	99.17%	99.27%
23	99.77%	100%	100%	100.80%	99.61%	99.78%
24	99.77%	100%	100%	100.28%	99.40%	98.97%
25	99.77%	100%	100%	98.84%	99.18%	98.75%
26	99.76%	100%	100%	98.95%	98.09%	98.25%
27	99.77%	100%	100%	97.67%	97.59%	97.15%
Total Average	99.76%	100%	100%	100.77%	100.07%	99.58%

Table 3
Gap (in percentage).

Class	GG	VB	VB Fix	VB+SCG
1	0,71	0.84	2.39	0.63
2	0.75	0.82	2.47	0.70
3	0.75	0.92	2.54	0.69
4	0.86	1.20	1.27	0.68
5	0.88	1.22	1.38	0.60
6	0.91	1.27	1.38	0.63
7	0.91	1.16	1.27	0.64
8	0.93	1.52	1.23	0.64
9	1.10	1.37	1.36	0.64
10	1.39	2.25	2.28	1.02
11	1.23	2.57	2.10	0.91
12	1.12	2.74	2.39	0.95
13	2.14	5.18	1.69	1.37
14	1.74	2.66	1.23	1.10
15	1.38	2.11	1.47	1.21
16	1.32	3.42	1.47	1.13
17	2.04	2.20	1.65	1.13
18	2.07	1.72	1.40	1.29
19	1.52	3.27	2.07	1.42
20	2.01	3.74	2.33	1.54
21	1.87	5.51	2.13	1.61
22	2.55	2.81	1.50	1.61
23	2.11	2.66	1.48	1.67
24	2.62	2.66	1.81	1.37
25	3.34	1.96	2.32	1.88
26	3.78	2.55	1.68	1.83
27	4.68	2.18	2.10	1.66
Total Average	1.48	2.17	1.79	1.04

Table 4
Processing time (in seconds).

Class	GG	VB	VB Fix	VB+SCG
1	1,800.00	1,780.89	439.86	899.76
2	1,800.00	1,799.81	730.37	899.70
3	1,800.00	1,799.68	1,154.13	899.80
4	1,800.00	1,800.00	1,799.99	898.71
5	1,800.00	1,800.00	1,799.99	875.20
6	1,800.00	1,799.52	1,798.48	900.16
7	1,800.00	1,800.00	1,792.24	900.32
8	1,800.00	1,800.00	1,766.40	900.25
9	1,800.00	1,800.00	1,799.09	900.56
10	1,800.00	1,800.00	1,452.52	902.38
11	1,800.00	1,800.00	1,579.80	902.27
12	1,800.00	1,800.00	1,579.40	900.86
13	1,800.00	1,800.00	1,739.46	906.79
14	1,800.00	1,800.00	1,771.17	906.95
15	1,800.00	1,800.00	1,758.96	906.53
16	1,800.00	1,800.00	1,770.94	911.60
17	1,800.00	1,800.00	1,800.00	910.31
18	1,800.00	1,800.00	1,800.00	910.57
19	1,800.00	1,800.00	1,797.20	912.45
20	1,800.00	1,800.00	1,765.62	911.08
21	1,800.00	1,800.00	1,793.34	912.50
22	1,800.00	1,800.00	1,800.00	936.09
23	1,800.00	1,800.00	1,800.00	935.40
24	1,800.00	1,800.00	1,800.00	928.59
25	1,800.00	1,800.00	1,800.00	957.09
26	1,800.00	1,800.00	1,800.00	963.75
27	1,800.00	1,800.00	1,800.00	962.67
Total Average	1,800.00	1,799.22	1,620.59	908.44

Figure 3 provides the number of instances in each class for which a feasible solution is achieved. Additionally, an unknown solution status refers to cases where the solver could not find a feasible solution within the allotted time, while infeasible status means that the instances are not feasible for the related approach. The GG approach produced feasible solutions for 127 instances out of 135. The VB approach achieved feasible solutions for 122 instances, while the combined version of the two formulations, referred to as VB GG, only obtained 81 feasible solutions. When fixing some setup variables to zero (VB Fix approach) 129 feasible solutions were found and 6 instances became infeasible. Finally, when separating the column generation procedure (VB+SCG) 129 feasible solutions were found.

In Tables 2 and 3 we present the indicators (in percentage) regarding the solution quality: lower and upper bounds, and also the gap values. The background colour for each result gives the comparative intensity between the set of results on a grey scale. Given the weak performance of the VB GG approach in obtaining a feasible solution, we are not considering this approach in the next tables. In Table 2, the reported values are normalized relative to reference solutions. Specifically, the lower bound obtained by the VB formulation was used as the baseline for lower-bound comparisons (set to 100%), while the upper bound obtained by the GG formulation was used as the baseline for upper-bound comparisons (set to 100%). The remaining values are presented as percentages relative to these references.

As previously mentioned, the lower bounds obtained by the VB formulation are tighter than those obtained by the GG formulation. Conversely, the upper bounds obtained by the GG formulation are better than the upper bounds from the VB formulation. Meanwhile, the strategies (VB Fix and VB+SCG) proposed to improve the upper bounds of the VB strategy proved to be effective and presented enhanced results. For each class, the number highlighted in bold in Table 3 represents the lowest average gap between the four solution strategies. Considering the integrated approaches, GG obtained the lowest average of gaps, 1.48% on average. However, the separated approach (VB+SCG) reduced the gap value to 1.04%.

Table 4 presents the average total processing time for each approach. On average, all solution strategies considering the integrated approach (GG and VB) reached the time limit of 1,800 seconds, indicating the high difficulty of the problem. When fixing some variables (VB Fix), some instances could be solved before the time limit, and the average was reduced to 1,620.59 seconds. The separated approach was shown to be efficient, having an average computational time of 908.44 seconds.

Further insights yielded from the computational tests are presented as follows. Table 5 presents the percentage of each cost component in the total cost. The last column is the general average of all four methods. In this table, the values in each line represent the average of the 27 classes for each solution strategy. The difference between the solution approaches is not substantial and the values indicate the concentration of a general average of 66.19% of the total cost in the purchase cost of the steel bars in contrast to only 0.05% in the steel item inventory cost.

Table 5
Allocation of costs in the objective function (in percentage of the total).

	Solution Method				Averages
Type of cost	GG	VB	VB Fix	VB + SCG
Mold setup	13.76	14.17	13.53	13.69	13.79
Cutting patterns setup	1.83	1.52	1.34	1.41	1.53
Mold section production	10.70	10.77	10.73	10.77	10.74
Steel bar purchase	66.20	65.73	66.40	66.44	66.19
Steel item cutting	7.28	7.23	7.30	7.31	7.28
Steel item inventory	0.04	0.12	0.02	0.00	0.05
Final item inventory	0.18	0.46	0.67	0.38	0.42

7 CONCLUSIONS AND RESEARCH DIRECTIONS

In this manuscript, as an extension of the research reported by Signorini et al. (2023), we incorporated the procurement of objects (steel bars of standard lengths) into the model, which makes this model view closer to the real scenario. Two different mathematical formulations are employed for the multi-period cutting stock problem with two-stage problems representing the production planning of lattice slabs in a Brazilian specialized factory. The proposition of alternative formulations to the same problem contributes to analyzing their advantages and disadvantages for the studied problem. In this case, the alternative and new formulation (VB) proved more promising in obtaining tighter lower bounds while the classic formulation (GG) yielded better upper bounds. Additionally, this paper proposes five strategies for finding high-quality upper bounds, one of which proved highly effective.

To extend this research, opportunities regarding the mathematical model include considering different steel bar suppliers, acquisition or rental of molds, and incorporation of lead times, which yields a more flexible production planning. Other possibilities are the integration of the cutting and procurement decisions with other production processes, e.g., packing and distribution of final products, and the use of alternative mathematical models.

Data Availability

The data that support this research are publicly available at the following repository: https://github.com/carolinesignorini-unesp/production-planning-PO-data.

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• Funding
  This research was funded by the National Council for the Improvement of Higher Education (CAPES)- Finance Code 001, by the National Council for Scientific and Technological Development (CNPq) under Grants 302998/2022-5, 402240/2023-5 and 402592/2024-7; the São Paulo Research Foundation (FAPESP) under Grants 2018/10284-0, 2022/05803-3 and 2024/01409-4.

Edited by

• Editor responsible for the review
  Editor-in-Chief: Antônio Augusto Chaves.

Publication Dates

• Publication in this collection
  28 Aug 2026
• Date of issue
  2026

History

• Received
  27 Jan 2026
• Accepted
  11 May 2026