Comparison of Mip Models for the Integrated Lot-Sizing and One-Dimensional Cutting Stock Problem

Abstract

ABSTRACT. Production processes comprising both the lot-sizing problem and the cutting stock problem are frequent in various industrial sectors. However these problems are usually treated separately, which can generates suboptimal overall solution and consequently causes production losses. In this paper, we propose different mathematical models for the integrated problem combining alternative models for the lot-sizing and the cutting stock problem, in order to evaluate and indicate the impact of these changes on the models' performance. An extensive computational study is done using randomly generated data and as a solution strategy we used a commercial optimization package and the application of a column generation technique.

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

“…In Melega, de Araujo, and Jans (2016) , several models with a general application that integrate the lot-sizing problem and cutting stock problem at different levels are proposed. One of the models is composed of three levels, where at level 1, a demand balance constraint of objects is modeled with a parameter that limits the number of objects available in each period for each type of object, i.e., the production of objects is assumed as given and the decisions are only related to the planning of the objects in stock.…”

“…Analyzing the features of the objects, we can see from Table 4 that less than half of the studies consider only one type of object, whereas most of the studies address multiple types, which correspond to different materials, lengths, colors, classes, diameters, families, grades, thicknesses or qualities. In some studies, the objects can be available in a limited number ( Agostinho et al, 2016;Aktin & Özdemir, 2009;Melega et al, 2016;Reinders, 1992;Suliman, 2012;Viegas et al, 2016 ). In other studies that consider usable leftover, there are also residual objects available, in addition to the limited number of standard objects ( Agostinho et al, 2016;Silva et al, 2014;Trkman & Gradisar, 2007;Viegas et al, 2016 ).…”

“…Viegas et al (2016) also consider a variable, in the demand balance constraints of objects, which is related to the number of objects obtained from the suppliers. As discussed before, in some of the papers that include level 1 ( Agostinho et al, 2016;Melega et al, 2016;Reinders, 1992;Viegas et al, 2016 ), it is assumed that the production of objects is already decided up front. As such, the demand balance constraints only relate to the planning of objects in stock.…”

“…Only a few papers consider independent demand of pieces at level 2 ( Correia et al, 2004;Reinders, 1992 ). In the studies where the inventory of pieces is not incorporated at level 2 ( Aktin & Özdemir, 2009;Alem & Morabito, 2012Ghidini et al, 2007;Gramani et al, 2009;Melega et al, 2016;Prata et al, 2015;Trkman & Gradisar, 2007;Wu et al, 2017 ), a piece necessary in the assembly or further processing must be cut in the same time period as the production of the corresponding final product.…”

“…As mentioned before, in some applications (textile, wood processing furniture), there is a one-to-one relationship between a cut piece and a final product. After being cut, the pieces undergo some transformation process to become final products, but there is no assembly process ( Farley, 1988;Melega et al, 2016;Ouhimmou et al, 2008 ). However, most of the papers that model level 3 consider an assembly structure, in which cut pieces correspond to components which are assembled into a final product.…”