In the context of forest products, a cutting order is a list of dimension parts along with demanded quantities. The cutting-order problem is to minimize the total cost of filling the cutting order from a given lumber grade (or grades). Lumber of a given grade is supplied to the production line in a random sequence, and each board is cut in a way that maximizes the total value of dimension parts produced, based on a value (or price) specified for each dimension part. Hence, the problem boils down to specifying suitable dimension-part prices for each board to be cut. The method we propose is adapted from Gilmore and Gomory's linear programming approach to the cutting stock problem. The main differences are the use of a random sample to construct the linear program and the use of prices rather than cutting patterns to specify a solution. The primary result of this thesis is that the expected cost of filling an order under the proposed method is approximately equal to the minimum possible expected cost, in the sense that the ratio (expected cost divided by the minimum expected cost) approaches one as the size of the order (e.g., in board feet) and the size of the random sample grow large. A secondary result is a lower bound on the minimum possible expected cost. The actual minimum is usually impractical to calculate, but the lower bound can be used in computer simulations to provide an absolute standard against which to compare costs. It applies only to independent sequences, whereas the convergence property above applies to a large class of dependent sequences, called alpha-mixing sequences. Experimental results (in the form of computer simulations) suggest that the proposed method is capable of attaining nearly minimal expected costs in moderately large orders. The main drawbacks are that the method is computationally expensive and of questionable value in smaller orders. / Graduation date: 2001

Thesis (MEng)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: For wood manufacturers around the world, the single biggest cost factor is known to be its raw material. Thus maximum utilisation, specifically volume recovery of this raw material, is of key importance for the industry. The wood products industry consists of several interrelated manufacturing steps for converting trees into logs and logs into finished lumber. At most primary and secondary wood processors the different manufacturing steps are optimised in isolation or based on operator experience. This can lead to suboptimal decisions and a substantial waste of raw material. The objective of this study was to determine the optimal machine settings for two interrelated operations, namely the sawing and ripping operations which have traditionally been optimised individually. A model, having two decision variables, was developed which aims to satisfy market demand at a minimal cost. The first decision was how to saw the log supply into different thicknesses by choosing specific sawing patterns. The second was to decide on a rip saw’s settings, namely part priority values, which determines how the products from the primary sawing operation are ripped into products of a certain thickness and width. The techniques used to determine the machine settings included static simulation with the SIMSAW software to represent the sawing operation and mixed integer programming to model the ripping operation. A metaheuristic, namely the Population Based Incremental Learning algorithm, was the link between the two operations and determined the optimal settings for the combined process. The model’s objective function was formulated to minimise the cost of production. This cost included the raw material waste cost and the over or under production cost. The over production cost was estimated to include the stock keeping costs. The under production cost was estimated as the buy-in cost of purchasing the under supplied products from another wood supplier. The model performed well against current decision software available in South Africa, namely the Sawmill Production Planning System package, which combines simulation (SIMSAW) and mixed integer programming techniques to maximise profit. The model added further value in modelling and determining the ripping priority settings in addition to the primary sawing patterns. / AFRIKAANSE OPSOMMING: Die grootste enkele koste vir houtprodukvervaardigers wêreldwyd is dié van hulle roumateriaal. Die maksimale gebruik van rou materiaal, of volume herwinning, is dus van primêre belang vir hierdie industrie. Die vervaardigingsproses in die houtprodukte-industrie bestaan uit ‘n verskeidenheid interafhanklike stappe om bome na stompe te verwerk en stompe na eindprodukte. By meeste primêre -en sekondêre houtvervaardigers word die verskillende vervaardigingsstappe in isolasie ge-optimeer. Hierdie praktyk lei tot sub-optimale besluite en ‘n vermorsing van roumateriale. Die doelwit van hierdie studie was om die optimale masjienverstellings vir twee interafhanklike prosesse, die primêre -en kloofsaag prosesse, te bepaal. Tradisioneel word hierdie twee prosesse individueel optimeer. ‘n Model met twee besluitnemingsveranderlikes is ontwikkel wat poog om die markaanvraag te bevredig teen ‘n minimum koste. Die eerste besluit was watter saagpatroon gekies moet word om die stompe in die regte dikte produkte te saag. Die tweede besluit was wat die kloofsaagstellings, ook bekend as prioriteitswaardes, moet wees sodat die regte wydte produkte gesaag word. Die tegnieke wat gebruik is sluit statiese simulasie met SIMSAW sagteware in om die primêre saagproses te modelleer en gemengde heelgetalprogammering (“mixed integer programming”) om die kloofsaagproses te modelleer. ‘n Metaheuristiek genaamd die “Population Based Incremental Learning” algoritme,was die skakel tussen die twee operasies om die optimale masjienstellings vir die proses te bepaal. Die model se doelfunksie was geformuleer om die koste van produksie te minimeer. Hierdie koste sluit die roumateriaal afvalkoste en die kostes van oor -en onderproduksie in. Die oorproduksiekoste was ‘n skatting van die voorraadkostes. Die onderproduksiekoste was ‘n skatting van die koste om voorraad van ‘n ander verskaffer aan te koop. Die model het goed opgeweeg teen die beskikbare besluitnemingssagteware in Suid Afrika, die “Sawmill Production Planning System”, wat ‘n kombinasie van SIMSAW en ‘n gemengde heelgetalprogrammeringstegniek is. Die model het verder waarde toegevoeg deur die kloofsaag se prioriteitswaardes te modelleer saam met die primêre saagpatrone.