Optimization of an Integrated Lot Sizing and Cutting Stock Problem in the Paper Industry

Abstract

ABSTRACT.Two important optimization problems occur in the planning and production scheduling in paper industries: the lot sizing problem and the cutting stock problem. The lot sizing problem must determine the quantity of jumbos of different types of paper to be produced in each machine over a finite planning horizon. These jumbos are then cut in order to meet the demand of items for each period. In this paper, we deal with the integration of these two problems, aiming to minimize costs of production and inventory of jumbos, as well as the trim loss of paper generated during the cutting process. Two mathematical models for the integrated problem are considered, and these models are solved both heuristically and using an optimization package. Attempting to get lower bounds for the problem, relaxed versions of the models also have been solved. Finally, computational experiments are presented and discussed.

Classification and literature review of integrated lot-sizing and cutting stock problems

“…Multi-level and Levels 1 and 2 | T | > 1 (21), (22), (23), (24), (25) (2016) Multiple -Paper Poltroniere et al (2016Poltroniere et al ( , 2008 Multiple -Paper Reinders (1992) Multiple ( Table 3 also provides information on the type of industry in which the model is used. Once classified according to the two main axes of integration, i.e., between production levels and across time periods, other features such as capacity and setups are further analyzed in Section 3.3 .…”

“…In our generalized 3-level model, this implies that level 1 is modeled just using constraint (25) via an appropriate setting of the X o t variables, which in this case are considered as a parameter. On the other hand, Hendry, Fok, and Shek (1996) , Leão and Toledo (2016) , Poltroniere, de Araujo, and Poldi (2016) , Poltroniere, Poldi, Toledo, and Arenales (2008) and Campello, Oliveira, Ayres, and Ghidini (2017) proposed integrated models where at level 1 a complete capacitated lot-sizing problem with setup is modeled.…”

“…For most of the studies in the literature, the demand balance constraint of pieces is modeled at level 2. The inventory of pieces is modeled either by oversupply ( Farley, 1988 ), initial inventory ( Respício & Captivo, 2002 ) or inventory variables ( Agostinho et al, 2016;Arbib & Marinelli, 2005;Campello et al, 2017;Correia et al, 2004;Gramani & França, 2006;Gramani et al, 2011;Hendry et al, 1996;Krichagina et al, 1998;Leão & Toledo, 2016;Malik et al, 20 09;Ouhimmou et al, 20 08;Poltroniere et al, 2016;Poltroniere et al, 2008;Reinders, 1992;Santos et al, 2011;Silva et al, 2014;Suliman, 2012;Vanzela et al, 2017 ). Due to the specific environment which is present, some practical applications need additional constraints to model the inventory limits ( Arbib & Marinelli, 2005;Santos et al, 2011 ), upper and lower bound demand of pieces ( Silva et al, 2015 ), cumulative demand ( Respício & Captivo, 2002 ) or the transportation of the pieces ( Ouhimmou et al, 2008 ).…”

“…The resource consumption in the capacity constraint is stated mostly in terms of total time available. However, the capacity constraint can be imposed in different ways according to the specific application and the level that it is related to, such as, the total amount of produced material ( Campello et al, 2017;Poltroniere et al, 2016;Poltroniere et al, 2008 ), total amount or area of cut material ( Arbib & Marinelli, 2005;Gramani et al, 2009 ), total length and height of cut material ( Farley, 1988 ), saw cycles ( Santos et al, 2011;Vanzela et al, 2017 ) and number of cuts ( Suliman, 2012 ). Krichagina et al (1998) consider a capacity constraint that calculates the idleness of the machine with respect to the production of objects at level 1 and guarantees that it is always positive.…”

“…In papers where a capacity is considered at more than one level, the resource consumption is not necessarily modeled in the same way at each level ( Farley, 1988;Ouhimmou et al, 2008;Santos et al, 2011 ). In some studies ( Aktin & Özdemir, 2009;Alem & Morabito, 2012Reinders, 1992 ), overtime is allowed, whereas in others ( Campello et al, 2017;Farley, 1988;Leão & Toledo, 2016;Poltroniere et al, 2016;Poltroniere et al, 2008;Santos et al, 2011 ), several machines are used to produce the items.…”