2009 2nd Conference on Data Mining and Optimization 27-28 October 2009, Selangor, Malaysia

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Optimization Model in Timber Harvest Planning Based on An Incremental Solution Approach

Munaisyah Abdullah *1, Salwani Abdullah*2, Abdul Razak Hamdan *3 , Roslan Ismail #4 * Faculty of Information Science and Technology

University Kebangsaan Malaysia

43600 Bangi, Selangor Darul Ehsan 1munaisyah@ftsm.ukm.my,

2salwani@ftsm.ukm.my,

3arh@ftsm.ukm.my

# Malaysia Institute of Information Technology

University Kuala Lumpur

50250 Kuala Lumpur 4drroslan@miit.unikl.edu.my

Abstract— Timber harvest planning deals with selecting a

forest area to harvest in a particular time period take into

consideration both economic and environmental issues. The

problem has been modelled using unit restriction approach and

area restriction approach. An optimal or a feasible harvest

schedules subject to adjacency constraints is generated using

exact methods or meta-heuristic optimization techniques

respectively. The adjacency constraints are used to prevent a

large opening size in a forest area. However, at the same it also

affects the timber productivity by limiting the size of an

available area to harvest. This paper proposes a timber harvest

planning model based on selective cutting and generates an

optimal harvest schedule subject to harvest volume restriction.

A priority and deviation optimization based technique is

proposed to solve the problem in stages.

Keywords— Timber harvest planning, optimization, selective

cutting, unit restriction model, area restriction model, constraint

I. INTRODUCTION

Planning can be defined as a process of providing

information for decision making. The process involved is generally influenced by the complexity of the problem. Numerous methods have been proposed to solve planning problems and can be classified into three categories include optimization, expert system (ES) and multi-criteria decision making (MCDA). Optimization approach involves finding the best solution either to minimize or maximize the objective function subject to a set of constraints [5, 1]. Expert (or knowledge-based) system uses knowledge to mimic the reasoning processes of experts in decision making [25, 26, 27], while MCDA is used to solve planning problems which involve multiple criteria in the decision process [7].

Timber harvest planning (THP) is an important part of forest resources management which aims to identify and schedule forest areas to be harvested in a particular time period [15]. It is a complex problem in which the decision process not only deals with numerous harvest blocks with several planning periods, but also involves evaluation of conflicting in multiple management objectives. These

objectives include economical objective for example to maximize the timber harvest productivity or net present value (NPV), ecological and environmental objectives such as ecosystem sustainability, habitat protection, biodiversity conservation, water quality, and erosion reduction [11, 17, 21].

Most of the reported studies address the environmental issues by limiting the size of area to harvest at a particular time period and using optimization approach to solve the problem [6, 14, 19].This restriction however would affect a timber productivity due to a limited area to harvest [21]. Therefore, in this paper, a new model that allows selective cutting with damage control analysis is designed to deal with environmental requirements. A priority and deviation optimization based approach is proposed to solve the problem.

This paper is organized as follows; section II discusses the modelling approaches and optimization techniques that have been employed in timber harvest planning, section III describes the proposed model. Section IV explains the optimization technique used in this study and conclusion and future research directions are presented in section five.

II. STUDY BACKGROUND

A. THP Modelling

THP model is used to determine which forest areas to be harvested in a particular time period. The forest is divided into small blocks. The harvest area is formed by selecting the most profitable blocks with minimal impacts to the environment [15]. This problem has been modelled using two approaches which are known as unit restriction model (URM) and area restriction model (ARM) [3].

The harvest area in URM is constructed based on human expert’s justification to manually form the harvest block in which the size of each block is near the maximum harvest area limitation. So that, the total size of combination of two blocks will exceed the maximum harvest area limitation, thus, harvesting one block will preclude neighbouring blocks from being harvested simultaneously [15, 20].

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In ARM, the size of each block is smaller than the maximum harvest area limitation and harvesting neighbouring blocks is permitted as long as the total size of the combined blocks is less than the maximum opening restriction. The decision process to form the harvest area in ARM is integrated in the harvest model [15, 20]. Overall, the size of each harvest area for both models is restricted to the maximum opening size limitation to avoid large clear cut area in the forest.

B. THP Optimization Techniques

Optimization in THP aims to generate optimal or feasible harvest schedules in specified planning period [6, 9, 14, 16, 22, 23]. The optimization techniques are used to maximize profits generated in each harvest schedule while the negative impact to the environmental is minimized [15]. Various optimization techniques such as Linear programming (LP), Mixed Integer programming (MIP), Tabu Search (TS), Simulated Annealing (SA) and Genetic Algorithm (GA) have been proposed to solve THP problems.

1) Mathematical Programming Technique: Earlier research works reported in the literature have employed an optimization technique based on mathematical programming such as LP to find optimal solutions in large forest areas without spatial elements. The examples of LP-based models include FORPLAN by Johnson (1986), SPECTRUM by Grear and Meneghin (1997) and FOLPI by Gracia (1984). However, the need to incorporate spatial requirements such as adjacency constraints to fulfil the rise concern in environmental aspects has restricted the use of this technique. This is because LP formulation requires the decision variable to be linear [12], whereas spatially constrained in THP problem is non-linear in natural [10].

Integer or Mixed Integer Programming (MIP) is another mathematical programming that has been applied in THP problem [2, 8, 15, 16, 19, 23]. However, the performance of this technique is very much governed by the size of the problem [12, 18]. Nevertheless, in a most recent published study by Constantino [22] demonstrated that the proposed new MIP formulation has successfully increased the efficiency of finding within or slightly above 1% optimal solution within a reasonable computation time.

2) Meta-heuristic Optimization techniques: The problem complexity will increase as the problem size increases. As an alternative to handle large problem size, meta-heuristic optimization techniques including TS, SA and GA have been used in THP problem [4, 9, 13, 14, 24]. A meta-heuristic method has the ability to solve large-scale and multi-objectives planning problems. Nevertheless, as compared to mathematical programming techniques, these methods do not produce an optimal solution. It generates a number of feasible harvest schedules within a reasonable time. [17].

SA approach was first introduced in THP problem by Lockwood and Moore [6] to generate a harvest schedule with adjacency constraints. Richard and Gun [9] solved a THP problem with a road construction using TS approach to

maximize net present value (NPV) using a trade-off analysis with an objective function to minimize negative impacts on a forest productivity due to a cost of road construction.

Bettinger [24] used TS in landscape-level optimization problem that uses a set of predefined optimal stand management prescriptions to maximize the even-flow of timber harvest volume over a 100-year planning horizon. Lu and Eriksson [13] used GA to form harvest units with an objective function is to maximize NPV, deducted by the entry costs of accessing the harvest unit with harvesting equipment.

III. PROPOSED MODEL

A. Terminology

We define a harvest block as a management unit that comprises of different trees species and sizes. A combination of a number of harvest blocks will form an available area for harvesting called a harvest area. A planning horizon is a time frame of one harvest cycle and can be divided into several planning periods. For an instance, in this study we consider one planning horizon for 30 years and divided into three planning period in which each planning period is 10 years in length.

B. Model Formulation

The proposed model is based on selective cutting with damage analysis where a tree that conforms to specified criteria will be selected for harvesting provided with minimal damage to the residual trees. The harvest area will be formed subject to the allowable harvest volume restriction in one planning period. The model formulation is shown as follows:

Maximize:

Q = ∑ ∑ HV (1)

subject to: t i it

TD ≥ RTD

it

(2)

∑ PVB ≥ RVB i it

(3)

VB ≥ RVB it

(4)

∑ HV ≤ AVP

i

it

(5)

Where i = harvest block t = planning period TD = tree diameter PVB = potential harvest volume in each block VB = volume harvested in each block HV = harvest volume RTD = required tree diameter restriction RVB = required volume harvested in each block AVP = allowable volume harvested in each planning

period The objective function (Q) as shown in equation (1) is to

maximize the total harvest volume in each planning period where HV denotes the harvest volume that is greater or equal

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to a required volume requirement in each block. The first set of the constraints, constraint (2) and constraint (3) are used to identify potential trees to be harvested in each block. Constraint (2) is a minimum tree diameter limitation for harvesting in which a diameter of a potential tree to cut must be equal or greater than the minimum requirement. However, if the total potential harvest volume for the following planning horizon is less than the minimum harvest volume constraint in each block, some of the trees may be saved from the harvest (see constraint 3).

Constraint (4) and constraint (5) are used to form a harvest area. Constraint (4) indicates that a block will be scheduled for harvesting if the total harvest volume is greater or equal to the minimum harvest volume requirement in each block to avoid uneconomic harvesting. The enforcement of this constraint would reduce the solution space by reducing the number of blocks for finding an optimal solution. Constraint (5) identifies and selects blocks derived from constraint (4) to construct an optimal harvest area based on a given maximum harvest volume limitation in each planning period in order to ensure the continuity of timber supply.

IV. PROPOSED MODEL DEVELOPMENT

A. Optimization with Incremental Solution Approach

In this study, we reduce the problem complexity by dividing the task into smaller subtasks where each subtasks focuses only on a specific problem and solves the problem in stages. Thus, the harvest planning problem will be divided into three stages as illustrated in Fig. I.

Fig. 1 Proposed framework

1) Stage I: In the first stage, the aim is to determine trees to harvest in each block that take into consideration constrains (2) and (3). Detail inventory of each block such as tree location, tree species, tree size and age, tree status as well as tree filling direction will be stored in a database (d/b). The trees are classified based on their size and species as shown in TABLE I. Only a mature tree is considered for harvesting and marked as a “Potential tree to harvest”. A tree with diameter greater than 20 cm dbh (diameter at breast height)

will be marked as a “Protected” tree, other are considered as “Common” category.

TABLE I TREE SIZE CLASSIFICATION

Tree diameter Species Status

TD ≥ 50 Non-dipterocarp Potential tree to harvest TD ≥ 45 Dipterocarp Potential tree to harvest 20 <=TD < 45 Both Protected TD < 20 Both Common For each “Potential tree to harvest”, damage analysis to

estimate the number of affected trees will be done based on the tree filling direction and the distance of the neighbouring trees. The total number of affected trees is grouped according to their category. Highest priority is assigned to the “Potential tree to harvest” with highest damage to the “Protected” trees.

Residual trees growth based on a diameter increment is then predicted to estimate the total potential trees to be harvested in each block (PVB) for the next planning horizon. If the PVB is less than the minimum volume requirement in each block, the “Potential tree to harvest” with highest priority will be saved from the harvest. The process will continue until PVB meets the harvest volume requirement to ensure the sustainability of wood supply.

The overall process in Stage I is known as tree cutting selection analysis and can be demonstrated in Fig. 2. Assume that, we have defined sixteen blocks to be harvested which are labelled as B1, B2,…,B16. The trees are marked using colours; green represents “Potential tree to harvest”, blue represents “Protected” tree, white represents “Common” trees and red represents affected or damaged trees.

Fig. 2 Tree cutting selection

This analysis will be performed in each harvest block. In

this example, we examine B6. From the block inventory in the database, we can estimate the damage and predict the PVB. In this case, suppose that we have two scenarios; first scenario is used to demonstrate the PVB which is greater than the minimum harvest volume which is required in B6 for the next planning horizon. In this situation, all the trees that have been marked to be cut are allowed to be harvested. In second scenario, the value of PVB is less than the minimum harvest volume requirement for the next planning horizon. Thus, “Potential tree to harvest” with highest priority is saved from the harvest. The process is continued until PVB meets the requirement. The total harvest volume of each block based on

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number of trees to cut is calculated after the process is completed.

2) Stage II: In stage II, the aim is to determine blocks to harvest subject to the minimum harvest volume requirement or RVB in each block for each planning period to ensure an economic harvesting (constraint 4). For example, as a continuation from Stage I, suppose that the total volume of each block for first planning period (t=1) is as shown in Fig. 3 (total volume (VB) of B1=20, B2 (VB) = 30,…, B6 (VB) = 60,…, B16 (VB) = 30). The same process is used to calculate the total volume for subsequent planning period (t=2 and t=3).

Fig. 3 Total volume in each block

In this example where t=1 and assume that RVB is set to

50 in which only blocks with VB which is equal or greater than 50 are allowed to harvest. The blocks that meet the requirement are selected and marked as potential blocks to be scheduled for harvest (PHBs) in planning period t =1 as illustrated in Fig 4.

Fig. 4 Potential blocks to harvest (PHBs)

3) Stage III: In the last stage, the aim is to form a harvest area subject to constraint (5) to avoid fluctuation in harvest volume throughout the planning horizon. In this stage, only those identified blocks for harvest from the previous stage, PHBs, are considered for further analysis.

The method used in optimizing blocks to be harvested in each planning period is based on deviation approach. The proposed procedure can be illustrated as below:

1. Calculate total current harvest volume (CVP) in period t. 2. Find the deviation (DHV1) between AVP and CVP; 2.1. If DHV1 is equal or less than zero (0), form the harvest

area by selecting all PHBs.

2.2. If DHV1 is greater than zero (0) but less than AVP, compare DHV1 with the smallest harvest volume (SHV) in period t,

2.2.1. If DHV1 is smaller than SHV, take away the SHV from PHBs to form an optimal (or near optimal) harvest area in period t.

2.2.2. If DHV1 is greater than SHV, calculate accumulative harvest volumes (THVa) in ascending order (SHV1+SHV2+ … + SHVn).

2.2.2.1. If THVa is equal or greater than DHV1, deduct the THVa from PHBs to form the harvest area.

2.2.2.2. If THVa is greater than AVP, deduct the THVa with SHVn and form the harvest area by selecting the remaining PHBs.

2.3. If DHV1 is greater than AVP, calculate accumulative harvest volumes (THVd) in descending order (LHV1+LHV2+ … + LHVn) and compare the value with AVP;

2.3.1. If THVd is equal with AVP, the harvest area is constructed by selecting LHV1 until LHVn from PHBs.

2.3.2. If THVd is less than AVP, calculate the deviation (DHV2) of THVd from AVP and add the THVd with SHV(1…n) then compare the value with AVP,

2.3.2.1. If DHV2 is less or equal with AVP, form the harvest area by adding SHV (1…n) with (2.3.1).

2.3.2.2. If DHV2 is greater than AVP, maintain the selected harvest area from (2.3.1).

To demonstrate the proposed procedure, we create two test cases where in the first test case (Case I), we assign AVP to 450 and in the second test case (Case II) the value of AVP is assigned to 300 as shown in Fig. 5. The value of CVP is 450 for both test cases. In Case I, the value of DHV1 is equal to negative 25 which is less than zero (0). Thus, all PHBs will be selected to form the harvest area (see procedure 2.1). In Case II, the value of DHV1 is equal to 125, which is greater than zero (0). The next step is to compare DHV1 with SHV. In this example, the value of SHV is 50. Since DHV1 is greater than SHV, THVa is calculated until it is equal or greater than DHV1 (THVa = (B5) 50 + (B15) 50 + (B11)55 = 155). Lastly, the harvest area is formed by deducting THVa from PHBs (see procedure 2.2, 2.2.2, 2.2.2.1)

Fig. 5 Harvest area formation

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V. CONCLUSIONS

In this paper, we proposed an optimization model of THP based on an incremental solution approach. A priority and deviation optimization based technique is proposed to solve the problem. The proposed method is expected to generate an optimal (or near optimal) harvest area with minimal damage to the residual trees for environmental protection. Lastly, our future work will focus on model development and an assessment to test the effectiveness of the proposed approach as compared to the adjacency approach that is currently available in the literature.

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