An Improved Metaheuristic Algorithm for Maximizing Demand Satisfaction in the Population Harvest Cutting Stock Problem
چکیده
We present a greedy version of an existing metaheuristic algorithm for a special version of the Cutting Stock Problem (CSP). For this version, it is only possible to have indirect control over the patterns via a vector of continuous values which we refer to as a weights vector. Our algorithm iteratively generates new weights vectors by making local changes over the best weights vector computed so far. This allows us to achieve better solutions much faster than is possible with the original metaheuristic. Introduction In this paper, we deal with a variant of the Cutting Stock Problem CSP that arises in the forestry industry; this problem is a type of population harvesting, such as harvesting of plants or lumber. Viewed from an optimization perspective, forest harvesting is a bilevel optimization problem. The first lower level is the individual tree stem which must be cut, or “‘bucked” using forestry terminology, to optimize the total value of the log products produced; this is referred to as “bucking-to-value”. The second level is the stand or forest, which we consider as a unit. The objective is to minimize the difference between the global products pattern, i.e. the total amounts of products cut from the stand or forest, and the original customer demand; this is referred to as “buckingto-demand”. Bucking-to-value is a recursive problem, i.e. maximize value by cutting the first product and then maximize the value of the remainder. Therefore it can be solved by dynamic programming (DP) (Pnevmaticos and Mann 1972). The best current approach for the bucking-to-demand is to simulate the cutting of a sample of tree stems, starting with the original price (or weight) vector and adjusting it to better meet overall customer demand. After carrying out a simulation the chosen weights vector is used by harvesting machines to cut the real raw material elements, i.e. the full population of tree stems. In this paper we extend a well-known adaptive control heuristic (Murphy, Marshall, and Bolding 2004) by incorporating new greedy features. Copyright c © 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Population Harvesting CSP In this problem we have a fixed number r = 1 . . . |R| of raw material pieces each with its own dimensions σr ∈ S (e.g. tree stems of specific dimensions). Due to the different dimensions of the raw material pieces, the patterns differ for each piece. During search the metaheuristic produces a global pattern p = 〈a1, . . . , a|M|〉, where aj ∈ [0, 1] represents the percentage of units of product mj ∈ M cut from the set of raw material pieces R. The weight vector is v ∈ R+|M|, where each vj represents the value associated with the product mj ∈ M. The A algorithm, usually implemented as a DP procedure, simulates the cutting of each raw material piece so as to maximize its total value based on the values of the products (v). The mix of products that we obtain represents a global product pattern. A can be represented by the following mapping function: A(M, , v) → p. (1) The demands for products d ∈ [0, 1]|M| is a vector where dj represents the percentage of units of product mj ∈ M that are demanded. To measure the similarity of demanded and obtained amounts we use the Apportionment Degree (AD) (Kivinen, Uusitalo, and Nummi 2005). The AD function maps the difference between the global products pattern p and the demand d to [0, 100], where 100 means a perfect match. However, typically there is waste from the cutting process, therefore the optimal pattern cannot reach the AD of 100. The AD is defined as follows: AD(p, d) = 100(1− |M| ∑