Yield Frontier Analysis of Forest Inventory

Transcription

1 Yield Frontier Analysis of Forest Inventory S.A. Mitchell & K. Kim Department of Engineering Science University of Auckland New Zealand Abstract Standing inventory analysis allows a forester to predict the yield (volume by logtype) of a section of forest before the trees have been felled. Current standing inventory tools such as MARVL and ATLAS Cruiser predict yield. They do this by simulating the bucking that will take place when the trees are harvested. These tools produce a point estimate of the yield for given input data. In practice, when the trees are bucked at harvest, the harvesting crew can make different decisions than those predicted by the analysis. This paper presents a method of standing inventory analysis that seeks to characterise all possible yields for a section of forest. 1 Introduction Standing inventory analysis allows a forester to predict the yield (volume by logtype) of a section of forest (henceforth called a stand) before the trees have been felled. This analysis is used in three ways. It is used to determine the overall profitability of the forest, by determining the potential value of the forest. It is used to direct operational harvesting operations; to enable customers orders to be fulfilled, and to anticipate logistical requirements. It is used to audit the efficiency of the harvesting crews. The techniques in this paper will be mainly applicable in the operational planning and auditing uses. Current standing inventory tools (such as MARVL (Deadman and Goulding 1979), ATLAS Cruiser (West 2000) and YTGen) predict yield by, among other things, simulating bucking of single stems. Bucking is the process of dividing a fallen stem (the trunk of a tree) into the logs that are required by the customers. Bucking is made difficult by the physical variation of the stems, and the different specifications for the various log-types required by customers (including minimum and maximum length and diameter requirements, and wood quality specifications).

2 In practice, when the trees are bucked at harvest, the harvesting crew can make different decisions than those predicted by the bucking simulation. Often times these differences are so great that the forester using the tool may complain that the tool is inaccurate. In this paper, we will argue that bucking simulation can be used to model the yields of a forest even if the decision making process is different from that used in the simulator. This paper presents a method of analysis for standing inventory that seeks to characterise all possible yields for a section of forest. A yield frontier will represent the most efficient yields (under different price lists) from the stand. All other possible yields will lie within this frontier. This paper will show that the yield frontier can be produced by a convex combination of different single yield estimates. Given the convex hull of the yield estimates an LP (Linear Program) can define the relationship between the volumes for each log-type in the stand. The feasible region of the LP will contain all possible yields by log-type. The solution of the LP with a specific price list (objective function) will generate the single estimates generated by current tools but with much less effort. In this paper, we will discuss how potential yield is estimated and the related technique of bucking optimisation. Then the properties of a yield frontier will be investigated and techniques used to generate a yield frontier will be detailed. Some results a small test data set will be shown. 2 Yield Estimation A forestry company will use a specialist inventory software package to collate, calculate and present standing inventory estimates. The base data for these estimates are found by cruising an area of forest. Statistical functions in the inventory package, along with sampling techniques used while cruising, allow a cruised sample of trees to estimate the yield of the whole stand. This practice reduces the expense of collecting inventory measurements. The inventory software will also contain components that model tree growth and damage while felling. The most important component in the inventory analysis software, for the purpose of this paper, is the bucking simulator, which takes each sampled stem and determines what logs will be bucked. The input data to the bucking simulator are: Physical descriptions of sampled standing trees; Log-type specifications; A list of prices or priorities to be used in the simulator. Yield estimates are found by simulating bucking based on this data for each stem. If these data are altered, a new yield estimate can be generated. There are four main types of bucking simulation algorithm. Fixed yield estimates may be altered by log conversions. A bucker can use a priority list heuristic. A bucker can optimise value with a DP (Dynamic Program) recursion. A bucker can optimise value by a NFP (Network Flow Problem) formulation. The last two algorithms are guaranteed to find the optimal value of each stem given a stem description and a description and price for each log product. Common implementations of these two algorithms divide the stem into stages along its length.

3 A shortest path problem is then formulated using possible logs as arcs and 1 times their price for arc costs. The solution to this problem will allocate logs along the stem. Figure 1 shows one possible way to buck a stem into three different logs. See Deadman and Goulding 1979 for a complete reference for DP bucking and Kim 2004 for a description of a NFP formulation. Large End First Log High quality Prunned butt Second Log Medium quality Sawlog Third Log Low quality Pulp log Small End Figure 1: A bucked stem 3 Bucking Optimisation Bucking optimisation is a class of problem where a heuristic controls the bucking simulation and drives it achieve some goal. The need for bucking optimisation was driven by the realisation that earlier methods for bucking stems optimised the value of each stem and did not consider the overall demand for logs. In Eng, Daellenbach, and Whyte 1986, the following comment is made. A stem-by-stem optimisation may thus result in a serious mismatch of volumes of logs supplied and end-use product requirements, thereby reducing the value derived from harvesting the forest resource. Bucking optimisation is related to yield frontier analysis because more than one yield estimate is used for each stand. The methods used to consider demand are varied. Mendoza and Bare 1986 and Eng, Daellenbach, and Whyte 1986 both use iterative methods based on stem-class classification. A stem class is a grouping of identical stems that are not necessarily found in the same stand. Thus, a specific bucking pattern is developed for each stem-class in the input data. The usefulness of a stem-class based result is questionable, however, as it requires the harvesting crews to classify each of the stems harvested, then use a specific bucking pattern on each. Sessions, Olsen, and Garland 1989 and Cossens 1996 both use iterative methods based on cutting strategies. Cutting strategies are sets of values applied to individual stems in the stand. From a cutting strategy and the stem description, the bucking pattern for each stem can be found. Cossens 1996 presents a forest wide, multi-period, bucking optimisation model that uses a decomposition method similar to Eng, Daellenbach, and Whyte The yield estimates are generated by MARVL (Method of Assessment of Recoverable Volume by Log-type) (Deadman and Goulding 1979), a New Zealand inventory system that uses a DP bucking algorithm. Laroze and Greber 1993 and Laroze and Greber 1997 describe methods that generate priority list bucking instructions for a stand. In each of these papers, a different method is used. These papers do not use an iterative strategy, however, all instructions are generated a-priori. In Laroze 1999, these methods are applied to a forest-wide problem.

4 Bucking Optimisation can be incorporated into OHS (Operational Harvest Scheduling) problems, which include crew allocation. Both Epstein et al and Mitchell 2004 use iterative generation of yields in their solution algorithms. However, Epstein et al uses priority list based systems while Mitchell 2004 uses MARVL. 4 Properties of the Yield Frontier A single yield estimate can be thought of as a single choice or snapshot from all the possible yields that can be obtained from a stand. If a different list of prices is used in a bucking simulator, a different snapshot will be returned. Imagine the yield estimates are points in a space where each axis represents the volume of a log-type. As the prices are changed, more points are added to this space until the yield estimates define the region that includes all possible yields that can be obtained from a stand. A diagram of this concept is shown in Figure 2. In the arguments that follow, three assumptions are made: That a DP or NFP bucking simulator will give the optimal single yield estimate for a section of forest, for a given price list. All feasible yield estimates lie within the envelope of optimal estimates given by a continuously varying price list. Since the sampled stems are representative of a section of a forest, a convex combination of two yield estimates gives a feasible yield estimate. From here onwards the term optimal bucker, may be used to describe both DP and NFP bucking simulators. 4.1 Relative Prices Yield estimates represent single solutions from an optimal bucker. These estimates are discrete points in this space and do not change continuously with a changing price vector. Firstly, consider two input price vectors (p 1, p 1 ) representing the prices for each log-type. If one vector is the scalar multiple of the other p 2 = cp 1. The yield estimates obtained from these vectors (the optimal solution) will be identical. As an optimal solution (x ) for the first price vector x arg max x R n {z 1 = p 1 x} z 1 = max x R n {z 1 = p 1 x} is also optimal for the second x arg max x R n {z 2 = p 2 x} z 2 = max x R n {z 2 = p 2 x}. However, the optimal objective value for each price vector will be scaled so that z2 = cz1. Therefore, the decisions made in the optimal bucker are based on the comparison of the relative prices between competing log-types. In recognition of this feature, input prices to an optimal bucker are known as relative prices because the relativity between the log-types is important.

5 4.2 Movement along the frontier If the relativity between log prices is changed, the yield estimates will change in discrete steps. These changes in the yield estimates will only occur when the prices change to make alternative decisions viable within the optimal bucker. To illustrate this, consider a yield estimate with two log-types: a single length of sawlog; and a single length of pulp. A diagram of a possible frontier is illustrated in Figure 2. Pulp Production / Hectare Yield Prediction Frontier Production Possibilities Figure 2: The theoretical yield frontier Sawlog Production/ Hectare There is a point on each axis that represents the estimate for a relative price vector where one log-type is at zero value and the other is at a positive value. Therefore, the estimate is one log-type at zero volume and the other at its maximum obtainable volume. If we follow the frontier from the bottom right to the top left, we observe discrete shifts in the volume by log-type of the yield estimate. When the price vector is changed so that both log-types have a value, but sawlogs are favoured over pulp, the yield estimate abruptly changes. The yield estimate then shows the maximum volume of sawlogs possible with the left over volume converted to pulp if possible. The yield estimate will remain at this point, as the price vector changes, until some new relativity is reached between pulp and sawlog prices. At this stage, another solution becomes optimal. Eventually when the price vector has a zero value for sawlogs and pulp at a positive value the estimate maximises pulp volume with no sawlogs produced. 4.3 Joining Extreme Points and Network Flow Representation This yield estimate frontier is directly equivalent to the convex hull extreme points produced by a parametric analysis of changing objective function coefficients in an LP. This equivalence can be established by formulating the bucking problem as a shortest path NFP. This technique is described in Kim The solution to the NFP will be naturally integer and therefore equivalent to the solutions from a DP recursion. The space illustrated in Figure 2 is therefore a projection of the variable space in the NFP where volumes of the individual logs are summed by log-type.

6 For practical reasons, it can be assumed that the yield estimates can be linked by a piecewise linear convex curve (the lines in Figure 2). Though the bucking simulator does not produce continuous estimates, the yields can be assumed continuous in a practical sense. In practice, if stands are assumed homogeneous, a harvesting crew could buck using one price vector, then change over to another price vector, giving a linear combination of the outputs from both vectors. The complete curve is a frontier that encloses all possible solutions to the bucking problem. 4.4 Convexity As the DP or NFP buckers are guaranteed to produce optimal solutions, all other feasible solutions will lie within the frontier. All yields must lie within this frontier including yields generated by priority lists, log volume conversions or stem-class techniques, such as those found in Eng, Daellenbach, and Whyte These other methods may produce solutions that are on the frontier, but since they are not guaranteed optimal, they cannot be used to produce extreme points. 5 Generation of a Extreme Points The yield frontier is the convex hull of possible yield estimates from an optimal bucker. These points may be generated by iteratively obtaining these extreme points by changing the input price lists. The price lists required for these iterations may be generated by three different methods. The input prices are changed by fixed steps. The relative (proportional) prices are changed. The results from the parametric analysis of the optimal basis is used to determine the next iteration. Changing the prices given to the bucking simulator by fixed steps is the simplest method to generate the extreme points. In the results section, the points were generated by iteratively changing the prices of each log from $-10 to $80 by steps of $10. As three log-types were used, 1,000 iterations were required. One thousand yield estimates were obtained but many of them were identical. This was the method used to generate the results as it was the easiest to implement. The identical solutions obtained in the fixed step method are a result of the nature of the bucking problem described in Section 4.1. If price list A has identical relative (proportional) prices to price list B the yield estimates produced will also be identical. The relative prices should therefore be altered in each iteration to prevent the generation of repeated solutions. In the two methods above, there is no mention of a way to ensure that all possible extreme points are generated. If we change the prices by fixed steps we cannot ensure that we have not stepped over parts of the yield frontier that have many extreme points close together. There is an argument we do not need to represent these rapidly varying regions accurately because the imprecision in the stem measurements will not give accurate answers. However, if we use a NFP formulation of the bucking problem and a parametric analysis of the cost coefficients, we can capture these rapid changes while not requiring unnecessary yield estimations where the solution will not change.

7 6 Construction of LP representation When all the yield estimates are available, the yield frontier can be generated. The yield frontier may be represented as a piece-wise linear equation with L variables where L is the number of log-types. Alternatively, the frontier could be represented as a LP (Linear Program) with L variables and a number of inequalities. The LP formulation is particularly interesting as it represents a significant reduction in problem size compared to the original problem. This representation will also allow the addition of log-volume constraints to the inventory problem, and it will significantly decrease the solution time of iterative bucking optimisation methods. The LP formulation given in the results is obtained by entering the extreme points into the PORTA software that is under a GPL license and available on 7 Some Results A simple sample dataset from ATLAS Cruiser was used to generate these results. The data set contained 40 stems and only three different log-types pruned logs, saw logs and pulp logs. Figure 3: The Sample Yield Frontier The wedge shape of Figure 3 is due to the log-type specifications. In this sample, all pruned logs could be sold as saw logs and all saw logs could be sold as pulp. For more log-types and complex specifications, shapes that are more complicated may result. The PORTA output describes the feasible region of possible yields from the sampled stems as:

8 where: x pr x s x p x pr < 5 x pr + x s < 23 x pr + x s + x p < 35 5x pr + 4x s < 93 and x pr, x s, x p 0 is the volume of pruned logs; is the volume of saw logs; is the volume of pulp logs. This feasible region is outlined in Figure 3. 8 Conclusion We have shown in this paper that it is possible to produce a yield frontier that contains all possible yield estimates for a stand. Furthermore, this yield frontier may be represented by a LP (Linear Program) containing one variable for each log-type considered. This LP can then be used in several ways. The feasible region can be presented to foresters to show them the different ways resources may be harvested from a stand. The LP can be solved with additional volume constraints to determine the best way to harvest a stand with some demand constraint. The LP can be embedded within a bucking optimisation iteration to increase the speed of each iteration. References Cossens, G Paul Optimisation of Short Term Log Allocation. Master of Applied Science Thesis, Lincoln University. Deadman, M. W., and C. J. Goulding A Method for the Assessment of Recoverable Volume by Log Types. New Zealand Journal of Forest Science 9: Eng, G, HG Daellenbach, and A G D Whyte Bucking Tree-Length Stems Optimally. Canadian Journal of Forest Research 16: Epstein, R., E. Nieto, A. Weintraub, J. Gabarro, and P. Chevalier A System for the Design of a Short Term Harvesting Strategy. European Journal of Operational Research 119: Kim, Ki-Young LP Formulation of Forestry Inventory Assessment Yield Frontier Analysis. A 4th year project, Dept of Engineering Science, School of Engineering, University of Auckland. Laroze, Andre A Linear Programming, Tabu Search Method for Solving Forest-Level Bucking Optimisation Problems. Forest Science 45 (1):

9 Laroze, Andre, and Brian Greber. 1993, March. Using Monte-Carlo Simulation to Generate Rule-Based Bucking Patterns. Edited by V Paredes, Proceedings International Symposium on System Analysis and Management Decisions in Forestry. Universidad Austral de Chile,, Valdivia, Chile, Laroze, Andre J, and Brian Greber Using Tabu Search to Generate Stand- Level, Rule-Based Bucking Patterns. Forest Science 43 (2): Mendoza, G A, and B Bare A Two-Stage Decision Model for Log Bucking and Allocation. Forest Products Journal 36 (10): Mitchell, Stuart A Operational Forest Harvest Scheduling Optimisation. Ph.D. diss., School of Engineering, University of Auckland. Sessions, J, E Olsen, and J Garland Tree Bucking for Optimal Stand Value with Log Allocation Constraints. Forest Science 35 (1): West, Graham Development of Integrated Computer Modelling Systems at Forest Research. New Zealand Journal of Forestery 45 (1): (May).