Heuristic solution approaches to operational forest planning problems

OR Spektrum (1995) 17:193-203

�9 Springer-Verlag 1995

Heuristic solution approaches to operational forest planning problems Alan T. Murray, Richard L. Church

National Center for Geographic Information and Analysis, Department of Geography, University of California at Santa Barbara, Santa Barbara, CA 93106, USA (Fax: (805) 893-4217, e-mail: [email protected], e-mail: [email protected])

Received: 23 September 1993/Accepted in revised form: 9 September 1994

Abstract. Operational forest planning problems are typically very difficult problems to solve due to problem size and constraint structure. This paper presents three heuristic solution approaches to operational forest planning problems. We develop solution procedures based on In- terchange, Simulated Annealing and Tabu search. These approaches represent new and unique solution strategies to this problem. Results are provided for applications to two actual forest planning problems and indicate that these approaches provide near optimal solutions in relatively short amounts of computer time.

Zusammenfassung. Operationale Forstplanungsproble- me sind typischerweise sehr schwierige Probleme, was durch die Problemgr613e und durch die Struktur der ,,constraints" gegeben ist. Dieser Artikel zeigt drei heuri- stische L6sungsans/~tze f/Jr operationale Forstplanungs- probleme auf. Wir haben L6sungsprozeduren entwiekelt, die auf interchange, simulated annealing und Tabu-Suche basieren. Diese Ans/itze stellen neue und andersartige L6- sungsstrategien ftir dieses Problem dar. Ergebnisse bei Anwendung auf zwei tats~ichliche Forstplanungsproble- me werden vorgestellt. Sic zeigen, dab diese Ans~itze nahe- zu optimale L6sungen bei relativ kurzer Bereehnungszeit liefern.

Key words: Operational forest planning, interchange, simulated annealing, Tabu search

Schliisselwiirter: Operationale Forstplanung, interchange, simulated annealing, Tabu-Suche

Introduction

The United States Forest Service manages 191 million acres (over 77 million hectares) of national forest land. This management domain is approximately twice the

Correspondence to: A.T. Murray

area of Germany. Management planning of forest lands must take numerous concerns into account before imple- mentation of any management scheme is initiated. To assist in the development and regulation of plans, forest analysts of the U.S. Forest Service have been required by U.S. Public Law 94-588 (National Forest Management Act of 1976) to provide analysis that includes mathematical modeling. The purpose of this regulation is to help provide fair and equitable management plans in a multiple-use forest environment, which must take into account the concerns of the timber industry, environmentalists, recreationalists, and concerns for the ecosystem in general. Further, budgetary difficulties have dictated the need of more economically sound planning policies, which require detailed and comprehensive analysis.

Forest planning has been accomplished by a decision making process that is decidedly hierarchical in nature (Church and Barber 1992). This hierarchy has been com- prised of three principal levels. The top level, called the strategic level (i.e. National Forest), involves the planning of forest-wide goals, where large scale programming problems are optimized to determine activites, like harvesting or wilderness preservation, over decades. The second level, called the tactical level (i.e. Ranger Districts), is associated with the translation of the activities specified in the strategic level to the more geographically specific land units defined at the tactical level. The third level, called the operational level (i.e. Analysis Areas), involves the determination of a land use plan for an area of the forest, whereby, stands are selected for treatment, road building and maintenance is scheduled, etc. Each planning level has been the subject of mathematical optimization research. Examples of model development applied to these various levels include FORPLAN (Johnson and Stuart 1987), VIP (Church et al. 1992), and SNAP (Sessions and Sessions 1991). As greater concern has been expressed for the viability of species like the spotted owl and salmon, the hierarchical planning process has since added a global-bioregional analysis level. This level involves large scale planning which spans across a number of national forests and encompasses one or more major habitats. As

194 A.T. Murray, R.L. Church:

one moves from the top level of analysis to the operational level, there is increasing spatial detail, which necessi- tates more complex planning.

The focus of this research is on operational planning. Past operational analysis has been limited by computa- tional resources and the size and tractability of model formulations. The reason for this is that the operational forest planning problem is inherently difficult. What leads to the difficulty is the size of a typical model formulation and the relatively large number of binary decision variables involved. These variables represent decisions on harvest units and road construction in operational plans. That is, the harvest quantities have been decided upon, now the decision lies in determining which units and when, as well as which roads to build and maintain and when to build and maintain them. Optimizing harvest and road scheduling requires, in large part, knowledge or tracking of compartmental unit interaction, often represented by the use of adjacency restrictions. Consequently, operational forest planning problems characteristically have many constraints and many integer decision variables. It is not surprising that no one particular solution approach to this problem has been generally accepted, since the problem is quite formidable.

There have been a number of different approaches used to solve this problem. Linear programming (LP) was one of the first methods used to approximate solutions (Thompson et al. 1973) and other LP based approaches have been developed as well (Kirby et al. 1986; Jones et al. 1991 a). Integer programming (IP) and mixed integer programming (MIP) have been applied, even though problem size is a limiting factor (Kirby et al. 1986; Nelson and Brodie 1990). Given the need for spatial detail and the limitations in model size that can be solved optimally by traditional MIP approaches, researchers have turned to Lagrangian relaxation (Torres et al. 1991), forms of random sampling and random search (Bullard et al. 1985; O'Hara et al. 1989; Nelson and Brodie 1990; Clements et al. 1990; Nelson et al. 1991 ; Nelson and Howard 1991), artificial intelligence (Hokans 1983), simulated annealing (Lockwood and Moore 1993; Dahlin and Sallnas 1993), and dynamic or nonlinear programming (Roise 1986, 1990; Hof and Joyce 1992). These approaches allow for the incorporation of large modeling formulations, however, this is often done at the expense of not finding the "optimal" solution or in some cases not even a feasible solution. Additionally, with the exception of certain random sampling techniques (Nelson and Brodie 1990) and MIP based approaches (Jones et al. 1991 a), roading decisions have not been incorporated into these alternative approaches. Processes capable of incorporating all of the necessary details as well as generating feasible alternatives are needed.

This paper investigates three methods for generating operational forest plans, based on common optimization approaches: Interchange, Simulated Annealing and Tabu search. The use of interchange and Tabu search represents new and unique approaches to the operational forest planning problem. Our application of Simulated An- nealing also differs markedly from past work. Further, we provide comparisons of all three approaches, as well

Heuristic solution approaches to operational forest planning problems

as comment on the efficacy of Monte Carlo sampling as applied to this complex planning problem. These solution approaches traditionally provide a wide range of solution alternatives that are close to optimal. We will begin with a brief problem background and give a formulation of the operational forest planning problem. Following this, the heuristic processes will be described. Finally, results will be given and conclusions will be made.

Background

The operational forest planning problem revolves around three items: compartmental unit activity scheduling, road building, and road maintenance. The defining feature of operational planning is managing neighboring compartment interaction. The interaction of activities between adjacent units prevents two or more adjacent units from being simultaneously harvested. The need to deal with adjacency concerns has been demonstrated in many papers (Thompson et al. 1973; Meneghin et al. 1988; Covington et al. 1988; Torres and Brodie 1990; Clements etal. 1990; Jamnick and Walters 1991; Jones etal. 1991 b). Mealey et al. (1982) discuss that the National Forest Service is mandated to ensure habitat dispersion (see also Jones et al. 1991 b; Torres et al. 1991; Barahona et al. 1992). This means that large areas cannot be harvested simultaneously, since this generates large open areas which are not particularly beneficial to elk and other habitat nor is it visually appealing. Given this, models which help generate operational forest plan alternatives must limit large contiguous areas from being harvested. Within optimization models, this has been accomplished by the use of special inequalities called adjacency constraints. Such constraints, which involve the use of 0-1 integer variables, prevent adjacent units from being harvested at the same time. Jamnick et al. (1990) found that this increased spatial detail also allows for a more precise measure of net present value, which is needed for accurate model representations. The need to incorporate greater spatial detail to prevent simultaneous harvest of adjacent units, and the need for 0-1 integer decision variables to track unit harvesting in a given period, results in even larger mathematical formulations. Coupled with this is the necessity to include roading decisions (e.g. maintaining, building and scheduling road linkages), which are also represented by 0-1 integer decision variables. Such operational conditions have the burden of making integer programming a necessity as compared to using large scale linear programming.

Problem formulation

Detailed mathematical formulations for forest planning that include adjacency and roading constraints can be found in Kirby et al. (1986), Nelson and Brodie (1990), and Nelson and Finn (1991). These formulations involve many integer variables and their difficulty in solving is well documented (Kirby et al. 1986; Nelson and Brodie 1990; Jones et al. 1991 a). Below, we present a general

A.T. Murray, R .L . Church : H e u r i s t i c solution approaches to operat ional forest p lanning problems ~95

formulation of the operational forest plannig problem which is based in large part on the work of Nelson and Brodie (1990). Before presenting the mathematical formulation, we define the following notation:

i j =

t =

ait =

~lit

~ t = H t = L t =

LR t =

p N, =

H i

m j =

S~ =

index of harvest units (i = 1, 2, . . . , I); index of road links (j = 1, 2 . . . . . J); index of time periods (t = 1, 2 . . . . . T); discounted revenue generated from harvesting unit i in period t; undiscounted revenue generated from harvesting unit i in period t; discounted cost to build road link j in period t; undiscounted cost to build road link j in period t; upper bound on total volume harvest in period t; lower bound on total volume harvest in period t; lower bound on total undiscounted revenue generated in period t; harvest exclusion period length; set of harvest units adjacent to unit i; coefficient necessary to impose adjacency restrictions around unit i; set of road links for which one must be built in order to build road link j; set of links for which one must be built in order to access unit i.

Decision variaNes:

1 if unit i is harvest in time t x . = 0 otherwise

{ ~ if road link j is built in time t r j, = otherwise

(5) Cannot harvest a unit unless necessary access roads are built to unit.

xi, <- ~, rjt forM/ i , j ~g i , t; / = 1

(6) Upper and lower bounds on harvest volume in each time period.

(7)

(a) ~ vit x~t >__ L , for all t;

(b) ~ vi, x , <_ H, . i

Undiscounted revenue bound requirement in each time period.

Z ~. x , - Z ~j, rj, >~ LRt for all t; i j

(8) Integer requirements.

xlt=O, 1 for alli, t; rj, = 0 , 1 for allj , t.

The objective of this formulation is to maximize discounted net revenue. Constraint (1) limits treatment activity in the appropriate time intervals. Notice that the p value in constraint (1) specifies that a harvest unit can be harvested at most once every 2p + 1 time periods. Constraint (1) defines the t - p through t + p interval based on a year t. Constraint (2) restricts the construction of a road to one time period. Constraint (3) imposes connectivity requirements for constructed road linkages. Constraint (4) imposes adjacency restrictions for harvested compartments. Simple adjacency conditions for two neighboring units i and k can be written as:

Xit "~- Xkt <~ 1 �9

Operational forest planning problem formulation

Maximize Z = Z ~ % x . - Z ~2 % r~t. i t j t

Subject to:

(i) Limit unit harvest to at most once in planning interval t - p to t+p. t+e

x u < l foral l i , t r l = t - p

(2) Limit construction of each road link to at most one time period.

~2rj~<l for a l l j ; t

(3) Cannot build road link j unless link f h a s been built.

rj, <_ ~ r]l for all j, t, f e Ms; 1 = I

(4) Adjacency restrictions to prevent simultaneous harvest of neighboring units.

n i x . + Z xh<n~ for alli, t;

Contraint (4) represents a special aggregated version, where one adjacency constraint is constructed for each compartmental unit in each time period. The structure of various forms of constraint (4) is the subject of several papers (Torres and Brodie 1990; Jones et al. 1991b; Yoshimoto and Brodie 1992; Murray and Church 1993 a). Constraint (5) establishes road link access to a harvest unit before treatment can be carried out. Constraint (6) places lower and upper harvest volume bounds on each time period. Constraint (7) requires that undiscounted revenue totals in each time period be greater than or equal to a lower limit. Constraint (8) imposes integer restrictions on all decision variables. It is worth noting that extensions to this basic formulation focusing more on the transportation planning involved can be found in Kirby et al. (1986).

This formulation has roughly O(JT + IT) constraints. The notation O() indicates that the number is on the order of the given quantity. The size of this formulation is relatively compact due to the form of the adjacency constraints (4). Alternative formulations may have a significantly larger number of constraints (4) or slightly fewer (Murray and Church 1994), but we find this particular form most beneficial for the solution procedures that will be described later in this paper. Added to the number of

196 A.T. Murray, R.L. Church: Heuristic solution approaches to operational forest planning problems

constraints is the use of I T + J T binary (0-1) integer variables. These integer requirements necessitate the use of a branch and bound procedure in practical application. Due to the limitations of general integer programming solution procedures, the size bounds for problems that can be optimally solved are very restrictive and alternative solution approaches have become necessary.

It is not surprising that the difficulty in solving these problems has resulted in efforts directed towards heuristic solution development (Nelson and Brodie 1990; Dahlin and Sallnas 1993). As eluded to earlier, the main goal of a heuristic approach is to be able to incorporate all of the necessary aspects of the planning problem, as well as be capable of providing good feasible solutions. In order to be successful, heuristic solution approaches must also require less computer resources, in terms of storage requirements and processing times, and be capable of producing near optimal or high quality solutions. Given this, heuristic solution approaches can be ideal for generating a range of solution alternatives. This is typical of the need in most decision making environments, since one "optimal" solution may not be acceptable to a politically charged decision making body (Walker and Priess 1988). This paper examines various heuristic approaches for solving the operational forest planning problem, which shares many common characteristics with other spatial optimization problems.

Heuristic solution approaches

There have been numerous attempts at solving the general operational forest planning problem. One of the more promising is the sampling heuristic approach called Monte Carlo Integer Programming (MCIP) (Nelson and Brodie 1990). This approach is basically a biased sam- piing scheme designed to generate feasible solution alternatives. The success of MCIP, however, is directly related to the number of sample solutions generated. If the sample size of generated solutions is very large, the likelihood of obtaining near optimal solutions increases. Unfortu- nately, larger sample sizes require longer solution times, which tie up computing resources. The most striking feature of the MCIP approach in solving the operational forest planning problem is its ability to generate a feasible solution in less time. This paper presents methods which improve upon the Monte Carlo feasible solutions generated in the MCIP process. That is, we develop methods which begin with a feasible Monte Carlo generated solution and successively improve upon it. We demonstrate that these improvement methods lead to near optimal solutions, without needing to generate a large sample, as is the case with the MCIP approach. Additionally, the time needed to improve upon an initial Monte Carlo solution is significantly less than the time needed to generate a large number of MCIP solutions, so these improvement methods provide high quality solutions in a shorter amount of time. Three improvement methods will be examined in this paper: Interchange, Simulated An- nealing and Tabu search.

Interchange

There have been many successful applications of interchange procedures for 0-1 integer programming problems (Teitz and Bart 1968; Lin and Kernighan 1973; Goldberg and Paz 1991; Densham and Rushton 1992). The basic idea is to transform a current solution into an improved solution. That is, move from one feasible solution to another feasible solution by interchanging decisions, however, doing so only when such an interchange improves the objective function measure. This has been referred to as a hill climbing procedure, since only improved solution transformations are accepted. The interchange process continues until no improved solution can be identified. The success of the interchange procedure depends primarily on how the transformation from one solution to the next takes place. Specifically, how many alternative transformations are evaluated and how good are the ending or convergent solutions. Success basically depends on the speed in converging to the "best" solution, the quality of the "best" solution, and the ability to apply the solution process to large-scale problems.

The interchange process begins with an initial feasible solution. The Monte Carlo sampling approach to generate an initial feasible solution is employed, although other procedures could be utilized. The interchange process maintains feasibility at all times throughout solution transformations. This is done by evaluating each non- harvested unit in all time periods for possible harvest. That is, we evaluate the resulting inclusion of all xlt = 0. Given the set of non-harvested units, x~t = 0, each unit is successively selected for harvest, e.g. xlt = 1. If there is an adjacency or exclusion period violation, then all units that present a violation with respect to the current unit being evaluated (e.g. xit = 1 j ~ Ni) are then set to non-harvest (x j, = 0). Following this, necessary roads are constructed. If this new solution generated by setting x~t = 1 maintains feasibility for all other problem constraints, then its objective function is evaluated. Other problem constraints consist of upper and lower volume bounds and revenue requirements (see constraints (6) and (7)). If the transformation results in an improvement, then it is a candidate for replacing the current solution. The best improvement is accepted after all non-harvested units in all time periods have been evaluated. This is a global search strategy, since all non-harvested units are evaluated before an improvement is accepted. If there are no improvements, then a local optima has been found and the process stops.

Thus far, the roading decision variables have not been discussed in detail. The roading decision variables are determined so that harvesting can be supported at least cost. That is, road segments are based on the harvesting decisions for a given solution. In other words, given harvest unit variables, a road network is determined. For many applications the potential road network is a tree structure. If this is the case, selecting only the road segments necessary to support/reach harvest units at the time of harvest represents an optimal roading solution to that specific harvest plan. Optimality of this approach is based on the tree structure of the potential road network,

A.T. Murray, R. L. Church: Heuristic s o l u t i o n approaches to operational forest planning problems 1 9 7

where only one potential path can be used to reach a particular harvest unit. If the potential road network is not a spanning tree, more effort must be invested to either optimally solve or approximate the required roading network. All of the problems that we have solved thus far have had a roading network conforming to the spanning tree structure.

A common criticism of the interchange process is that it is very likely to get trapped in local optima, hence not capable of finding the true optimal solution. This process is relatively fast, however, so that it may be repeated for a number of initial starting solutions. This is a form of diversification that assists the process in finding alternative optima or near optimal solutions. Still, the inherent nature of the process is what leads it to local optima, since only improved solutions are accepted. For this reason, we explored alternative approaches which maintain the basic transition strategy of Interchange, however, these alternative approaches incorporate methods which allow these processes to avoid local optima in an attempt to identify solutions closer to the true optimal solution. We begin with Simulated Annealing.

Simulated annealing

Simulated annealing is structured after the process of annealing solids. Annealing is the process of moving matter from a high-energy state to low-energy state. In a high- energy state (or high temperature state), particles are able to move about or rearrange themselves, hence transitioning into different configurations. The main issue in the annealing process is how best to lower (or cool) the process in order to bring the solid to a low-energy state, ffthe process is cooled too quickly, then the matter will be arranged in a metastable or suboptimal configuration. Given this, the two important elements to an annealing process are:

1. Reaching an optimum state at each temperature. 2. The rate at which the process is cooled.

In the development of alternative solution approaches to optimization problems, researchers have found that these basic strategies may be incorporated into a general solution strategy. The simulated annealing approach can be thought of as a procedure where a solution represents a set of atoms moving among energy states. The available energy states are a function of the current temperature. When the temperature is high, the atoms are able to move in space as well as in energy states. When the temperature is low, both position and energy state become frozen. The inherent controlling condition in a simulated annealing process is the probability of accepting a posifiona! change based on temperature.

Kirkpatrick et al. (1983) applied this concept to combi- natorial optimization problems based on the work of Me- tropolis et al. (1953). Since then, simulated annealing has enjoyed a number of successful applications in various optimization problems (van Laarhoven and Aarts 1987; Lockwood and Moore 1993; Murray and Church 1993 b). Such success is due to representing the solution of an optimization problem as an annealing process.

As was indicated earlier, simulated annealing applications have been applied to this general problem (Lock- wood and Moore 1993; Dahlin and Sallnas 1993), however, compared to the simulated annealing application that this paper presents, these approaches are dramatically different. Both of these approaches begin their processes with an infeasible solution and proceed towards feasibility and improvement. Thus, they may terminate with an infeasible solution. This is not the case with the approach we describe here.

The simulated annealing process begins with an initial feasible solution. Following this, a non-harvested unit and time period are randomly selected (a variable where xi~ = 0). This unit is then set in for harvest and adjacency and/or temporal violation adjustments are made. Road- ing decisions are adjusted and the new solution is then evaluated for feasibility with respect to volume and revenue constraints. If the solution is not feasible, another non-harvested unit in some time period is selected. If the solution is feasible, then the objective function value of this solution is calculated. If the new solution is an improvement over the previous solution objective value, 0_ old, the new solution is selected as the current solution. If the new solution is not an improvement, Count is evaluated. If Count exceeds a specified limit, N, the temperature, T, is cooled by a factor p and Count is reset to zero. If Count is less then N, the new solution objective value (O_new) is evaluated as a function of the previous solution (O_old) and the temperature (T). Specifically, this is the exponential function of these values: e[(O_ new - O_ old)IT].

If a randomly chosen number, y, is less than this value, then the new solution is accepted. Otherwise, the current best solution is kept. This allows for a probability of accepting a change which results in a poorer solution. The probability of accepting a non-improvement interchange decreases with the temperature and decreases with the relative increase in the objective function. Simulated annealing is continued until the temperature is cooled below the specified tolerance value, tol.

What the Simulated Annealing process accomplishes is that it can avoid potentially local optima encountered in the interchange process, since not all accepted moves are strictly improving. Parameter adjustments can play a critical role in the success of not only getting trapped in local optima, but the speed of the process as well. In our applications, we have found the computation time of simulated annealing to be very competitive, since it does not evaluate all x , = 0. Thus, simulated annealing may utilize the diversification strategy of multiple random solution starts, thereby increasing the likelihood that "good" solutions are identified.

Tabu search

An alternative metaheuristic approach to simulated annealing is Tabu search. Tabu search has enjoyed numerous successful applications in a wide variety of problem areas (Glover 1989; Hertz and de Werra 1990). Tabu


search is devised to overcome local optimality in a more orderly fashion than simulated annealing (de Werra and Hertz 1989). Rather than relying on a functional probability of accepting non-improvement solutions, Tabu search systematically forces the process into new regions of the solution space. This is accomplished by employing short-term and long-term memory search strategies (Glover 1989, 1990). These strategies help the process to search the solution space for potentially better solutions, while avoiding unproductive cycling (movement back and forth between the same solutions). There are three important elements of Tabu search: defining the neighborhood, intensification and diversification (Glover 1989, 1990; Hertz and de Werra 1990). The neighborhood is the set of feasible solution transitions, which have been defined in the two previous heuristic processes (i.e. the interchange mechanism for this forest planning problem). Thus, the neighborhood is the set ofxit = 0 such that their inclusion into the solution (e.g. x , = 1 followed by appropriate violation adjustments) is a feasible solution. Inten- sification can be thought of as the acceptance of positive or hill-climbing moves. That is, focus is placed on pro- gressing towards a local optima. Diversification may be thought of as a means of guiding the solution into new areas of the solution space, specifically, moving out of points of local optima and into new regions for explo- ration. Diversification is imposed through the use of short and long term memory in the search process. Short-term memory keeps the process from cycling back into a locally optimal solution that has already been identified, and long-term memory is used to boost the process into a solution region that has not been previously encountered. In other words, long-term memory enables the process to move away from a locally optimal solution. The purpose of these concepts is to allow the process to transition out of local optima without transitioning directly back.

Figure 1 depicts the Tabu search solution process for the operational forest planning problem. As with the other two processes, the Tabu process begins with an initial feasible solution. Also, Count, TABU_short and TABU_long are initialized to zero. TABU_short is an array used to keep track of deleted elements, both the specific compartment and the time period, and is used as a short-term memory device. Specifically, it allows the process to remember what it just discarded, so that it does not turn around and place it back into the solution. TABU_long and Count are used in the sense of a long- term memory. TABU_long is an array that keeps track of the deleted elements of accepted "lesser" solutions (again, both the specific compartment and time period) used to drive the process into new regions of the solution space. Thus, TABU_short keeps track of deleted elements from improvement solutions and TABU_long keeps track of deleted elements from accepted non-improvement solutions. Count is used as a stopping criteria to ensure that the process terminates in a finite number of iterations. What makes the TABU_short and TABU_long arrays useful is that, typically, a relatively small number of the most recent members are maintained on these "tabu" lists, so that decision variables are not excluded for the remainder of the process, but long

" . . . . t

�9 Generate initial feasible solution. Begin ~ Zero out TABUshort and TABU_long.

~' , Count = O.

Lc .. . . . ~ I'

Accept improvement solution. ] [ Jt Zero out F_list�9

Add zeroed out variables to ~ Ideutify variables set equal to zero. TABU short. ~

Yes

For a zero variable, set vadaNe equal to one and zero out any vadable that now violates adjacency or temporal constraints.

Make roading adjustments.

No

on TABU_s

Add vadable to list of feasible alternatives, F_list.

Is there a solution on @ @ F - list that , . improves, No

Yes

~q 8top

Fig. 1. Flowchart of the TABU process

Accept best feasible solution vadaNe on F list that is NOT on TABU_long.

Add accepted variable to TABU_long.

Add zeroed out variables to TABU_short.

Yes

@ ~ TABU long?

enough to serve the purpose of preventing the process from repeating previous solution transitions. The use of the arrays and the counter will become more apparent as the process is more fully explained. After the Count is incremented, non-harvested variables in all time periods are identified and F_ list is initialized to zero. F_ list keeps track of the neighborhood solutions of all time periods. That is, those solutions that maintain feasibility after solution transition. This is done by evaluating each non- harvested variable in all time periods in turn. This variable is set in to be harvested and the solution is adjusted to eliminate any adjacency or temporal violations. If the variable is not on TABU_short, the roading decisions are adjusted. Next the volume and revenue constraints are tested for feasibility. If the solution is feasible, the objective function is evaluated and the unit decision variable and time period is placed on F_list. After all non-harvested units have been examined, all feasible solutions, F_ list, are evaluated for the best solution. If this solution is better than the previous best solution, the new solution is accepted, zeroed out variables are placed on TABU_short, and the evaluation process begins again (recall that zeroed out variables are those that violated adjacency or temporal restrictions). If there is no local improvement, then Count is tested against its limiting value, MaxT. If Count has reached its limit or there are no items on F_list, then the process terminates. Otherwise, if there is a F_list solution not on TABU_long, then the best solution is accepted and the new variable is placed on TABU_long,

A.T. Murray, R.L. Church: Heuristic solution approaches to operational forest planning problems 199

+ i ii :i:i:i

iiiii ...... . . . . .

Fig. 2. MacMillan Bloedel tree farm Vancouver, British Columbia (Nelson and Brodie 1990)

along with placing zeroed out variables on TABU_short. Next, Count is incremented and the process begins again. If there are no feasible solutions that are not on TABU_long, the process also terminates.

It should be apparent that Tabu searches the solution space in a more systematic manner than does simulated annealing. Also, input parameters allow for significant control over how the search is carried out, which is true for simulated annealing as well. A unique feature of Tabu is that the parameters may be adjusted so that the Tabu process reduces to the Interchange process. As a final point, the diversification strategy of repeated initial starts is utilized, as suggested for the previous two heuristic processes.

The complexity of each heuristic process has increased in the order in which they have been described, and in turn there is a parallel increase in needed computing resources. Upon analysis of the solution results, it will be demonstrated that the increased complexity and computing time per solution is well worth the additional effort.

Results

All three heuristic processes have been coded in For t ran and results will be given for applications of these solution techniques to two forests located in British Columbia, Canada. Figure 2 outlines the MacMillan Bloedel tree farm in Vanvouver, British Columbia (Nelson and Brodie 1990), which is the first data set to be analyzed. The MacMillan data has 45 compartments and 52 road linkages, which span 3 time periods. The MIP formulation

Table 1. Parameters used for the MacMillan Bloedel data set

Simulated annealing TABU search

Initial temperature, T = 35 Short-term memory length, 10 Cooling rate,/~ = 0.40 Long-term memory length, 15 Maximum iteration, N = 20 Maximum iterations, MaxT = 50 Tolerance value, tol = 0.10

contains 300 decision variables (291 integer variables) and 740 constraints (Nelson and Brodie 1990). Nelson and Brodie (1990) report a solution time of approximately 60 h on a 386/33 personal computer using LINDO/PC. The amount of time needed to generate a single solution is unacceptable in an interactive decision making environment, where alternative solutions must also be generated. For this reason, operational forest planning problems are ideal candidates for developing heuristic solution approaches.

For each heuristic process applied to the MacMillan data set, 1300 solutions were generated, each process be- ginning with the same initial random solution configuration. We chose 1300 for our sample size to conform with earlier work involving MCIP by Nelson and Brodie (1990). Table 1 provides all of the parameters used by Simulated Annealing and Tabu for the MacMillan data set. The optimal solution to this maximization problem is 5953.20. This is a different optimal solution from that reported in Nelson and Brodie (1990) due to a roading constraint that was excluded from their MCIP process.


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For comparison purposes, we excluded the same constraint from the original MIP formulation and re-solved for the optimal solution using L I N D � 9 (Schrage 1989). This allows for a direct comparison to the MCIP results of Nelson and Brodie (1990). The best solutions identified by each process were: Interchange, 5883.74; Simulated Annealing, 5897.12; and Tabu, 5932.55. These compare very well to the optimal solution, being within 2% of optimal for the Interchange process and within 10/0

for the Simulated Annealing and Tabu processes. Also, these solutions surpass the best solution identified by MCIP of 5748.38. Figure 3a depicts how the 1300 solutions are distributed. The Monte Carlo distribution represents the initial randomly generated feasible solutions. It is clear that all of the processes significantly improved upon the Monte Carlo process, that is, improvements are consistently made to the initial Monte Carlo solution. Interestingly, there was no correlation between the quali-

A, T. Murray, R.L. Church: Heuristic solution approaches to operational forest planning problems 201

ty of the initial solution and the quality of the improvement solutions, so that any initial solution is equally like- e- ly to produce a high quality final solution by any one of = o the solution processes. This is confirmed by conducting a _.z

o

Friedman analysis (Conover, 1980) on the Interchange, Simulated Annealing and Tabu final solution results. ~ Specifically what is tested is the following hypothesis:

Ho: Each solution approach is equally likely to produce ,o a superior final solution for a given initial Monte ~ Carlo solution.

Rejection of this hypothesis would indicate that one approach consistently outperforms any of the other approaches. We do not reject the hypothesis at a level of a significance of ~ = 0.25. This indicates that no one approach can identify superior solutions on a consistent basis for any initial Monte Carlo solution. This does not, however, conclude anything about overall performance c- in general. ._.9.

To gain a better perspective about overall performance "5 of the solution processes, we focus more on the results -6 given in Fig. 3 a. Figure 3 b provides a focused view of the

m 8 frequency distributions of the three heuristic processes. < This figure eliminates the Monte Carlo distribution of b- Figure 3 a. There is a clear shift between each of the distributions, where the means are the following: Interchange, 5374.03; Simulated Annealing, 5437.02; and Tabu, 5538.04. All of these means are within 10% of the optimal solution, but there is a decided difference in the mean b values, with Tabu having a mean solution value closest to the optimal solution.

The next question to examine is does one process con- tinually dominate another process, in the sense of taking an initial solution and consistently finding a superior final ~" o solution. Figure 4a presents a plot of each final Inter- ~ change solution versus the corresponding Simulated An- healing final solution. The diagonal line represents the line y = x, where any point lying on this line would repre- m < sent an initial Monte Carlo solution that had the same b- final solution for each heuristic. Points on the Inter- change side of the diagonal represent initial solutions for which the Interchange process outperformed the Simulat- ed Annealing process in terms of a final improvement objective value. Figure 4b, c make similar comparisons e for Interchange vs. Tabu and Simulated Annealing vs. Tabu. Figure 4b, c show that all in all, Tabu tends to perform the best, which was also demonstrated in Fig. 3 a, b. An interesting finding is displayed in Fig. 4 b. One might expect that Tabu would do at least as well as Interchange and while this is generally the case, Fig. 4 b shows that it is not always the case. This is due to the use of the short-term memory list TABU_short. TABU_short prevents cyclic moves encountered in the Interchange process, thus, the Tabu solution will not necessarily take the identical solution path that Interchange does.

To this point, solution time has not been discussed. Nelson and Brodie (1990) generated 1300 solutions by MCIP in 8 h of computer time on a 386/33 personal computer, which they translate to 25 s per solution. We report the following approximate solutions times, also on a 386/ 33 personal computer: Interchange, 3 h, Simulated An-

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Fig. 4. Solution plot of corresponding initial solutions a Simulated annealing vs. interchange b Interchange vs. TABU e Simulated annealing vs. TABU

nealing, 11 h; and Tabu, 30 h. The average performance of these processes is 8.60, 30.86, and 82.60 s per solution, respectively. It should be clear, however, that a reduction in the sample sizes of these processes will not dramatically effect the quality of the final solutions produced, but will substantially reduce computation time. This is especially true since the mean Tabu solution is within 7% of the optimal solution. In fact, Table 2 provides a breakdown of the number of solutions obtained that are greater than a specified percentage of the optimal solution. It is re- markable that Interchange, Simulated Annealing, and

202

Table 2. Number of solutions within specified percentage of optimal. (Optimal = 5953.20)

A.T. Murray, R.L. Church: Heuristic solution approaches to operational forest planning problems

MCIP" Interchange Simulated TABU annealing

90% (5357.8797) 175 646 840 1121 95% (5655.5397) 12 95 161 349 97% (5774.6037) 1 20 37 133 98% (5834.1357) 0 5 8 53 99% (5893.6677) 0 0 1 20

a Nelson and Brodie (1990) findings using an optimal solution of 5909.07 which is less than the corresponding solution to their MCIP results

Table 3. Parameters used for the Hardwick Island data set

Simulated annealing TABU search

Short-term memory length, 10 Long-term memory length, 15 Maximum iterations, MaxT = 50

Initial temperature, T = 10 Cooling rate, # = 0.60 Maximum iteration, N = 20 Tolerance value, tol = 0.10

Tabu have approximately 7%, 12%, and 27% of their total solutions within 5 % of the optimal solution, respectively, as compared to less than 1% of the MCIP solutions falling into this range.

The second data set that these processes were tested on is the Hardwick Island forest on the west coast of Vancou- ver Island, British Columbia. This forest has 431 compartments where no roading decisions were imposed. Therefore, the majority of the constraints were adjacency conditions. The optimal solution for this problem is 2212.0 with a solution time exceeding 24 h on a 486/50 personal computer (Murray and Church 1994). The heuristics for this data set were run on a 486/50 personal computer using a solution sample size of 200 for the parameters given in Table 3. The best solutions found by each process were: Interchange, 2108.0; Simulated An- nealing, 2092.0; and Tabu, 2176.0. All of these solutions are within 6% of the optimal solution and Tabu is within 2%. The mean solution values were 2019.16, 1949.93, and 2114.15 and the average performance solution times were 39.45, 7.44, and 96.71 seconds per solution, respectively. Even though Tabu had a significantly higher time per solution, the mean solution for Tabu was greater than the best solution identified by either of the other two processes. Friedman analysis on these results also does not reject the null hypothesis, confirming that no process consistently produces a better final solution for all Monte Carlo generated initial solutions.

Discussion and conclusion

Interchange, Simulated Annealing and Tabu search are all viable solution approaches to the operational forest planning problem and the analysis of the final solution results is very encouraging. An important fact that per-

tains to all of these approaches is that extensions and additional constraints are not difficult to incorporate. The added complexity presented by more constraints should not be a factor in the performance of any of these methods.

Simulated Annealing can be dependent on parameters and thus, performance is very sensitive to the settings of these parameters, as seen in the results of the data set applications. It is worth noting that the Simulated An- nealing applications of Lockwood and Moore (1993) and Dahlin and Sallnas (1993) are dramatically different from the approach described in this paper. Both of these approaches begin their processes with an infeasible solution and proceed towards feasibility and improvement. Thus, they may terminate with an infeasible solution. Addition- ally, much of the solution time is spent in search of a feasible solution, which detracts from improvement search.

One of the most important issues is robustness across data sets. It is obvious that solution time may be a factor, but by reducing sample size Tabu still may be the pre- ferred approach. Sample size is typically sacrificed in practice (see Dahlin and Sallnas 1993), since time constraints tend to be a major factor in analysis of plans. Tabu does extremely well and confidence can be placed in a much smaller sample size, as compared to MCIP and for that matter, Interchange and Simulated Annealing as well. The best MCIP solution identified for the MacMil- lan Bloedel problem was 5748.38 with the next best being 5576.66. However, generating just 5 consecutive solutions using Tabu gives the following final solutions: 5821.69, 5265.33, 5525.61, 5292.12, and 5765.63. Two of these solutions are superior to the best found by MCIP in 8 h as compared to 7 min using Tabu. This matches expecta- tions based upon the summary presented in Table 2 (which indicates that approximately every 1 in 10 solutions is within 3% of the optimal solution).

This paper has developed three heuristic approaches for solving the operational forest planning problem: In- terchange, Simulated Annealing and Tabu search. Each of these approaches represents both new and unique solution applications to the operational forest planning problem. These processes were applied to two forests and the results show that all three approaches produce high quality solutions in a relatively short amount of time. How- ever, the Tabu approach consistently identified close to optimal solutions and is less likely to be affected by re- duced sample sizes.

Acknowledgements. The authors wish to acknowledge Phil Aune and the Pacific Southwest Forest Research Station and U.S. Forest Service personnel from Regions 5, 6 and the Washington Office Land Management Planning for research support in this coopera- tive research effort. We want to particularly note the help of Klaus Barber (Region 5) and Richard Dyrland (Region 6). Also, we wish to thank J. D. Nelson for supplying data sets used in this paper.

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