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Integer Programming
Published in Albert G. Holzman, Mathematical Programming, 2020
The categorization of integer models as direct, coded, and transformed is due to Taha [69]. Application of "either-or" constraints in the scheduling problem is due to Marine [53]. Other applications in scheduling theory may be found in Bowman [10], Wagner [74], Conway et al. [13], Pritsker et al. [59], and Taha [66] . The capital budgeting problem was first proposed by Weingartner [76]. Hillier [44] and Peterson and Launghhunn [58] treat the probabilistic version of the problem. Gilmore and Gomory [29, 30] use the knapsack model to solve the cutting-stock problem. The use of the knapsack model in connection with the solution of the general integer problem was proposed by Gomory [40] and later in a completely different context by Elmaghraby and Wig [20]. Other applications of the covering problem include political districting [24], information retrieval [17], line balancing [63], switching theory [46, 60], and capital investment [73]. Piece-wise linear approximation is used in solving separable programming problems [12]. Other approximations are given by Markowitz and Manne [55] and Healey [43]. A closely related problem to the fixed-charge problem is the plant location problem. An excellent exposition of the plant location model is given by Elshafei [21].
Models for two- and three-stage two-dimensional cutting stock problems with a limited number of open stacks
Published in International Journal of Production Research, 2023
Mateus Martin, Horacio Hideki Yanasse, Maristela O. Santos, Reinaldo Morabito
The two-dimensional cutting stock problem (2D-CSP) deals with the cutting of a set of rectangular item types with a pre-determined length, width, and demand out of a minimum number of rectangular large objects. Practical applications include the cutting of paper reels (Matsumoto, Umetani, and Nagamochi 2011), wooden boards (Morabito and Arenales 2000), glass panels (Durak and Aksu 2017; Parreño and Alvarez-Valdes 2021), defective materials (Martin et al. 2020; Martin, Morabito, and Munari 2021a), and concrete poles (Lemos, Cherri, and de Araujo 2021). The applications have motivated the development of different solution strategies to tackle problems that vary according to the special requirements of each application field and cutting device. Most of these solution strategies are surveyed and categorised in the works of Dyckhoff (1990), Lodi, Martello, and Monaci (2002), Wäscher, Haußner, and Schumann (2007), Bennell and Oliveira (2008), and Scheithauer (2018), among others. The cutting stock problem and most of its variants are known to be NP-hard (Garey and Johnson 1979).
Integrated optimisation on flow-shop production with cutting stock
Published in International Journal of Production Research, 2019
Weihao Wang, Zhongshun Shi, Leyuan Shi, Qingbin Zhao
The cutting-stock problem is to find the best way of cutting large stock materials into smaller ones so as to satisfy the customer demand for these small items (Amor and de Carvalho 2005; Reinertsen and Vossen 2010). It is very common in various industries such as glass, steel, paper (Dyson and Gregory 1974; Armbruster 2002; Kallrath et al. 2014) and has earned pretty much attention. Many settings and requirements in industry have also been discussed. Yanasse and Lamosa (2007) used a lagrangian approach to solve an integrated cutting stock and pattern sequencing problem in the wood hardboard industry where the maximum number of open stacks is limited. Poldi and Arenales (2009) dealt with a case where there were a set of different stock lengths available in limited quantities. Reinertsen and Vossen (2010) took into account the due date of orders in cutting-stock problem so as to meet customer demand while minimising waste. Cui et al. (2017) proposed a heuristic algorithm for the one-dimensional cutting-stock problem with leftovers usable to meet future demands. Wuttke and Heese (2018) studied a two-dimensional cutting-stock problem with sequence dependent setup time in the technical textile industry. Reviews of the important mathematical models and algorithms for the cutting-stock problem and related bin packing problem could be found in Lodi et al. (2013) and Delorme, Iori, and Martello (2016). Some of these problems involve the decision of the cutting-stock sequence, but don't consider the subsequent procedures after the cutting-stock process.
One-dimensional multi-period cutting stock problems in the concrete industry
Published in International Journal of Production Research, 2022
Caroline de Arruda Signorini, Silvio Alexandre de Araujo, Gislaine Mara Melega
The cutting stock problem arises in many industries as a fundamental subproblem of production planning, and establishes the best way to cut standard length objects (material units available in stock, such as aluminium or paper coils, metal or wood sheets, and steel bars, among others) into smaller items with specified dimensions, minimising material waste while meeting the demand. It can be seen in a variety of papers with uses in different industries: corrugated box factory (Savsar and Cogun 1994), furniture industry (Morabito and Arenales 2000; Vanzela et al. 2017), sheet metal operations (Verlinden, Cattrysse, and Oudheusden 2007), window frame manufacturer (Kim et al. 2016), mattress industry (Christofoletti, de Araujo, and Cherri 2020), wood products (Kokten and Sel 2020) and in more general contexts (Gilmore and Gomory 1961, 1963, 1965; Dyckhoff 1981; Foerster and Wascher 2000; Yanasse and Morabito 2006; Wang et al. 2019). Correia, Oliveira, and Ferreira (2012) proposed an integrated assignment and cutting stock model to paper production planning to meet a diverse demand using multiple paper machines. Also, in a general context, Delorme, Iori, and Martello (2016) reviewed the main mathematical formulations of the one-dimensional bin packing and cutting stock problems, appraising the exact algorithms and the performance of the most important software tools. Considering the production process of automotive springs, Andrade et al. (2021) proposed an integrated lot sizing and cutting stock model to reduce inventory costs and losses in the steel bar cutting process, taking into account parallel machines and assembly of final products in multiple time periods. Nascimento, Cherri, and de Araujo (2020) consider the integrated lot sizing and one-dimensional cutting stock problem with usable leftovers.