Discrete OptimizationThe stochastic trim-loss problem☆
Introduction
The cutting stock problem (CSP) is one of the oldest and most studied problems in the field of combinatorial optimization. Its relevance from an applicative standpoint stems from the large number of manufacturing problems that can be framed as CSP. For a recent classification of cutting problems, readers are referred to [22], [57].
The goal is to determine the optimal plan to cut an inventory of standard size materials (rolls of paper, wire) to satisfy a set of customers’ demands. Cutting stock problems may involve cuts being made in one or more dimensions and may be solved with a variety of objectives (e.g., minimizing trim loss, maximizing profit and so on). The typical approach aims at determining a set of cutting patterns that will produce the required collection of items with a minimal waste production or trim loss.
Cutting stock problems in all their variants have been extensively treated in the literature in the last decades. Due to their complexity [38], [39], most approaches found in the literature rely on heuristics. At least three different heuristic methods for solving the CSP can be identified in the literature: sequential heuristic procedures [29], [30], linear programming based methods [21], [31] and metaheuristics techniques, especially for the two-dimensional case. Among these metaheuristics, it is worth mentioning the application of tabu search [7], simulated annealing [20], [41], genetic algorithms [37], [56], hybrid genetic algorithms [49], ant colony [23], evolutionary algorithms [42], grasp methaheuristic [8]. Another stream of research entails to solve the traditional CSP by solving exactly the transformed integer linear problem obtained from its integer nonlinear formulation. Various authors [18], [52], [55] have developed exact solution approaches based on the branch and bound framework. Cutting plane algorithms were investigated in [9], [53], whereas a branch and cut approach has been recently proposed in [10]. Harjunkoski et al. [32], [33], [34], [35], [59], [60] presented several linear transformations for the originally non-convex nonlinear CSP formulation. They also presented convex transformations resulting in a convex nonlinear formulation which they solved with an extended cutting plane method [61].
The deterministic CSP relies on the strong assumption that all the problem parameters are known in advance. Such an assumption is rather restrictive for a sector as the manufacturing one, where different sources of uncertainty (e.g., processing times, product demands, resource availability, etc.) may affect the production system [6], [51]. In particular, in the case of CSP it is evident that the customers demands may be affected by significant uncertainty. A trivial approach to cope with uncertainty consists in replacing the uncertain parameters of the deterministic model by either mean or worst case estimations. However, the recommendations provided by the new deterministic formulations may have little value resulting in over production plans or profit losses.
Very few research papers investigated randomness in cutting stock problems. An exact solution approach for the stochastic one-dimensional cutting stock problem was proposed by Sculli [54] for the case in which the dimension of the original roll to be cut is uncertain due to possible defects caused by the winding process. The solution approach relies on the assumption that the portions of the roll damaged by the production process, respectively, at the two extremes of the roll are Normal random variables independently distributed. A cutting stock problem for a paper plant that services random customer demands has been investigated in [40]. Rather than presenting an exact approach for this problem, the authors propose a heuristic two-step procedure. In the first step, average demand is embedded into a linear programming problem which aims at finding the best average activity rates in order to minimize the average paper waste. In the second step, the linear programming solution is used to approximate the problem as a control problem for Brownian motion.
Randomness in products demand is addressed by means of safety buffers in [19]. Buffers are created by stocking rolls, since the reels are too huge to stock and finished products may have very high inventory holding costs. Under the assumption that finished products demand follows independent stationary stochastic processes with known mean and variance, safety buffers can be easily computed as well as the mean trim loss. The objective of the resulting integer nonlinear problem is to define assignments of products to rolls which minimize total inventory and trim loss costs. Thanks to a set partitioning reformulation of the problem, a tailored solution approach based on branch and price is proposed.
In this paper, we address uncertainty in customers demand by means of the stochastic programming paradigm and propose a two-stage stochastic formulation. The choice of the modeling paradigm associates our paper with two recent contributions [5], [16]. In particular, a two-stage stochastic problem with recourse was proposed and solved in [5]. The same stochastic formulation appears independently as a test problem for assessing the efficiency of column generation within the L-shaped framework in [16]. Besides the choice of the modeling paradigm, our model differs from the cited papers in some important aspects. First, the variable definition is completely different. In [5], [16], the first-stage decisions involve the determination of the number of repeat of patterns prior to observing the exact demands, while the second-stage decisions constitute a simple recourse action of paying a penalty on the shortage or surplus once demand information becomes available. Since the penalty is paid on integer shortfall values, the recourse action is integer. Our problem is an example of a two-stage stochastic program with general integer recourse. Here, deciding the number of repetition of patterns constitute the second-stage or recourse action taken on the basis of the cutting patterns generated in the first-stage and actual demand realized. In this respect, our model can be viewed as a generalization of previous work on the stochastic cutting stock problem. The paper unfolds as follows. In Section 2, we formulate a two-stage stochastic programming model where the uncertain customers orders are represented via a set of scenarios. The resulting model is a nonlinear integer program whose non-convex nature together with its huge size prevent the application of standard software. In Section 3, we describe a decomposition coordination approach designed to solve the problem to global optimality. Preliminary computational results are reported in Section 4.
Section snippets
Problem formulation
In this section, we formulate a mathematical model for the CSP under uncertainty. We first describe a deterministic formulation, and later we extend this formulation to a stochastic setting by introducing a set of scenarios.
Consider the problem of cutting different product paper rolls from raw paper roll. There are different types of product rolls to be cut. Each type of product roll has associated a certain width. The length of the product paper rolls is assumed to be equal to the length of
The solution approach
Integer requirements in stochastic programming problems have serious consequences on structural properties and algorithm design. Although in the last decades significant effort has been devoted toward theoretical and algorithmic studies of stochastic linear integer programming, the nonlinear case is an ongoing field of research. In [58], two algorithms (the optimality gap and the confidence level method) have been proposed to effectively solve stochastic nonlinear convex mixed integer problems.
Computational experiments
In this section we describe numerical experiments carried out to validate the proposed mathematical model and evaluate the performance of the solution strategy. We first introduce a toy example which serves as test for assessing the quality of the stochastic programming solution in comparison to those obtained using a deterministic approach. Later, we report on the performance of the solution method by presenting numerical results carried out on a set of test problems.
Concluding remarks
In this paper, we have proposed a stochastic version of the well known CSP. In particular, we dealt with the inherent uncertainty of the products demand by formulating the problem within the framework of stochastic two-stage recourse model. This makes the resulting model more suitable and versatile in terms of better handling the real cases.
Moreover, the specific structure of the resulting stochastic mixed integer nonlinear problem has been effectively and efficiently faced by the development
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2023, European Journal of Operational ResearchCitation Excerpt :The solution strategy considered that the parts of the object damaged at the two extremes of the object were normal random variables that were independently distributed. Beraldi, Bruni, & Conforti (2009) considered uncertainties in customer demand and proposed a two-stage stochastic formulation to represent the CSP. The authors solved this problem using a two-phase-based algorithm.
A reinforcement learning approach to the stochastic cutting stock problem
2022, EURO Journal on Computational OptimizationCitation Excerpt :The objective is to minimize the total expected cost in both stages. Beraldi et al. [39] propose a similar two-stage stochastic program with the difference that only the patterns to be used are decided upon before seeing the demand, while the actual amounts of objects cut according to each pattern are decided only after seeing the demand. Zanarini [40] also proposes a two-stage stochastic programming model in which demands are uncertain.
Simulation of the stochastic one-dimensional cutting stock problem to minimize the total inventory cost
2019, Procedia ManufacturingClassification and literature review of integrated lot-sizing and cutting stock problems
2018, European Journal of Operational ResearchCitation Excerpt :Alem and Morabito (2013) proposed a two-stage stochastic optimization model under stochastic demand and setup times. Beraldi, Bruni, and Conforti (2009) consider the case of demand uncertainty for a cutting stock problem. Beyond the various objectives discussed previously for both problems, an alternative approach is a multi-criteria optimization (Wäscher, 1990).
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This work has been partially supported by Università della Calabria under the special project “Modelli e Metodi per l’Ottimizzazione di Sistemi Complessi”, 2004.