A linear optimization approach to the combined production planning model

Two fundamental processes usually arise in the production planning of many industries. The first one consists of deciding how many final products of each type have to be produced in each period of a planning horizon, the well-known lot sizing problem. The other process consists of cutting raw materials in stock in order to produce smaller parts used in the assembly of final products, the well-studied cutting stock problem. In this paper the decision variables of these two problems are dependent of each other in order to obtain a global optimum solution. Setups that are typically present in lot sizing problems are relaxed together with integer frequencies of cutting patterns in the cutting problem. Therefore, a large scale linear optimizations problem arises, which is exactly solved by a column generated technique. It is worth noting that this new combined problem still takes the trade-off between storage costs (for final products and the parts) and trim losses (in the cutting process). We present some sets of computational tests, analyzed over three different scenarios. These results show that, by combining the problems and using an exact method, it is possible to obtain significant gains when compared to the usual industrial practice, which solve them in sequence. (C) 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Assessing the uncertainties of model estimates of primary productivity in the tropical Pacific Ocean

Depth-integrated primary productivity (PP) estimates obtained from satellite ocean color-based models (SatPPMs) and those generated from biogeochemical ocean general circulation models (BCGCMs) represent a key resource for biogeochemical and ecological studies at global as well as regional scales. Calibration and validation of these PP models are not straightforward, however, and comparative studies show large differences between model estimates. The goal of this paper is to compare PP estimates obtained from 30 different models (21 SatPPMs and 9 BOGCMs) to a tropical Pacific PP database consisting of similar to 1000 C-14 measurements spanning more than a decade (1983-1996). Primary findings include: skill varied significantly between models, but performance was not a function of model complexity or type (i.e. SatPPM vs. BOGCM); nearly all models underestimated the observed variance of PR specifically yielding too few low PP (< 0.2 g Cm-2 d(-1)) values; more than half of the total root-mean-squared model-data differences associated with the satellite-based PP models might be accounted for by uncertainties in the input variables and/or the PP data; and the tropical Pacific database captures a broad scale shift from low biomassnormalized productivity in the 1980s to higher biomass-normalized productivity in the 1990s, which was not successfully captured by any of the models. This latter result suggests that interdecadal and global changes will be a significant challenge for both SatPPMs and BOGCMs. Finally, average root-mean-squared differences between in situ PP data on the equator at 140 degrees W and PP estimates from the satellite-based productivity models were 58% lower than analogous values computed in a previous PP model comparison 6 years ago. The success of these types of comparison exercises is illustrated by the continual modification and improvement of the participating models and the resulting increase in model skill. (C) 2008 Elsevier BY. All rights reserved.