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Some mathematical programming models in the design and manufacture of plywood Raghavendra, Bangalore Gururajachar


One factor of wood loss in the manufacture of plywood is implicit in the form of excess thickness in plywood due to the choice of veneer thicknesses and plywood designs used in assembly. The thickness and designs currently in use appear to have come largely from tradition and there is no evidence in the literature to show what constitutes the most economical veneer thicknesses and plywood designs for a mill. The problem of determining them is very complex since many types of plywood are assembled in each mill as some integral multiple combination of a few veneers satisfying the 'balanced design' and other structural specifications. The consumption of logs is dependent on the excess thickness in plywood and the economics of the mill further depend on how efficiently a given set of veneers and designs are used to satisfy the orderfile requirements. In this dissertation, these aspects of the Plywood Design and Manufacturing (PDM) problem are addressed using a mathematical programming approach. The problem of finding the optimal veneer thicknesses, associated plywood designs and product mix is formulated as a non-linear mixed integer mathematical programming model. Utilizing the structure of the constraints and by selecting appropriate variables to branch on, it is demonstrated that the PDM problem can be solved efficiently through an implicit enumeration algorithm involving a tree search procedure. The subproblem to be solved at each feasible node of the tree is a Linear Multiple Choice Knapsack (LMCK) problem whose solution can be obtained explicitly following its coefficient structure. A computer code is written in FORTRAN for the implicit enumeration algorithm. Data obtained from a plywood mill in B.C. is analysed using the PDM model and this code. It is demonstrated that the annual net revenue of the mill can be substantially increased through the use of the PDM model. The PDM model is further extended to mill situations involving more than one species and varying orderfile requirements. The model is reformulated in each case and it is demonstrated that essentially the same tree search procedure can be used to solve all these models. When the orderfile is independent of species, the subproblem to be solved at each node of the tree is a Generalized Network problem. It is shown that this Generalized Network problem can be reduced to a Generalized Transportation problem utilizing the structure of the coefficients and solved as an ordinary Transportation problem. When the orderfile is dependent on species, the subproblem decomposes into several Linear Multiple Choice Knapsack problems. If more than one species of veneer can be mixed within a plywood panel, the subproblem is a linear programming problem. The PDM model is further shown to be a special case of a disjunctive programming problem. Following the development of the PDM model, methods to determine the efficiency of plywood designs and the optimum number of veneer thicknesses for a plywood mill are developed.

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