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UBC Theses and Dissertations

Log allocation by dynamic programming Bailey, Gordon Raymond

Abstract

This thesis describes an optimization model for the allocation of logs from alternative sources for a series of successive time periods. The model was formulated as a multi-stage decision process and is solved by dynamic programming. The analytic framework consists of two connected recurrence equations, each incorporating two decision-variables. These two relationships, together with the feasibility regions defined for a hypothetical problem, describe the optimization process. The hypothetical allocation problem, used as a vehicle for model development, requires mill demands for quantities of pulplogs and sawlogs to be satisfied for three time periods. Logs are delivered from five available sources, four log-producing areas and an open log market, and temporary log surpluses are allowed. Only a limited quantity of logs is available from each source in each period, two of the four forest areas supply only pulplogs and a third area is inaccessible in one period. The variable unit costs of delivered logs differ not only between each source and period but are also dependent upon the magnitude of an allocation. In addition to satisfying mill demands for specified quantities of logs, there is a further requirement with regard to sawlogs. Average lumber prices are assumed to be dependent upon the tree species processed and variable log conversion costs are assumed to decrease with increase in log diameter. Consequently, the comparison of alternative allocation policies involves not only the sum of the variable delivered log costs but also a measure of the value of delivered sawlogs. In the thesis "sawlog net worth" is evaluated and combined with log production and log transportation costs to give a composite cost term, "net delivered log cost." This is the measure used to evaluate each allocation and is incorporated in the first of the two recurrence equations. This equation is used to derive minimum cost allocation policies for all possible quantities that may be allocated from each period. In the second equation the minimum costs derived from the first allocation process are combined with a second term to evaluate alternative allocations between periods. This second cost component is incurred when surplus logs are "cold-decked" for subsequent mill conversion. To ensure a sufficient flow of logs, and to take advantage of seasonal differences in "net delivered log costs," log surpluses are permitted. When log surpluses are "cold-decked" additional log handling costs are incurred which must be considered when alternative allocations between periods are evaluated. This requirement is satisfied by introducing the additional cost component into the second equation. The different derivations given for the "net delivered log costs" for five sources of logs demonstrate an important feature of the formulation: there is no requirement that costs must be linearly related to the quantity allocated. This freedom is well illustrated by the introduction of additional "fixed" costs which are dependent upon the magnitudes of the quantities allocated. In a demonstration of the flexibility of the formulation a complex log production system was assumed for source two. For each allocation from this source two optimal quantities were derived. The first was the quantity of sawlogs selected from specified log classes, the second was the optimal portion of the quantity of peeler logs developed that should be traded. An exchange could be made either for sawlogs, or for pulplogs, or both. With the development of this log allocation model a fresh approach to log production planning is now possible. The analytic framework is capable of extensive adaptation and the model itself can be readily modified to suit a variety of conditions. Whether used as described, or as part of a larger analytical system, the computational advantages of dynamic programming are now available to the planner.

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