A system for solving spatial forest planning problems

A system for solving spatial A system for solving spatial forest planning problemsforest planning problems

Karl R. WaltersUgo Feunekes

Andrew CogswellEric Cox

IntroductionIntroduction

•

Ongoing relationship–

Remsoft

•

small software developer specializing in forest & fire management

–

Champion International Corp.•

multinational integrated forest products company

•

Solution to a difficult planning/scheduling problem

Presenter

Presentation Notes

Today I’m going to discuss an ongoing relationship between two very different companies. One company is a privately owned Canadian software development firm with fewer than 10 employees that has become a leader in providing decision support tools to the forestry community. The other is a multinational, integrated forest products company controlling millions of acres of forest land in North and South America. My talk will discuss how Remsoft and Champion have worked together to craft a methodology for addressing a very complex forest planning problem. First, I will briefly introduce the problem and some alternative mathematical formulations of it and their relative strengths and weaknesses. Then I will present the methodology used in the Remsoft Spatial Planning System, and apply it to a representative case study. Finally, I will discuss Champion’s implementation of the software and and how it has become a key element in their forest planning process.

Historical perspectiveHistorical perspective

•

Champion controls more than 5 million acres in US

•

traditional southern pine plantations–

large uniform plantations

–

highly concentrated age classes –

basic PNV maximization LP models

–

manual harvest blocking/scheduling

Presenter

Presentation Notes

First, some historical background. Champion International, like many other large industrial forestland owners in the US has substantial holdings in the U.S. Southeast where the climate is suitable for short rotation pine plantations. Plantations are typically several hundred to several thousand acres in size, and historically they have been harvested every 15 to 35 years, depending on site quality. In order to minimize costs of harvesting, road maintenance and moving equipment, plantations have been established in concentrated areas, resulting in large aggregate areas of similar age. Linear programming has been widely used to schedule the harvests of these plantations since the late 1960’s. Objective functions typically maximize present net value, subject to constraints on volume and product flows. Since plantations were typically liquidated in one harvest operation, manual blocking methods were sufficient.

Historical perspectiveHistorical perspective

Presenter

Presentation Notes

This slide shows the configuration of slash pine plantations in a forest tract in Georgia. Each color represents the year in which the plantation was established. The background color represents non-plantation lands (agriculture, water, scrub, etc) and the white areas represent cypress ponds and bays that are common in the area and give the plantations a swiss-cheese appearance. The hexagon in the lower right represents an area of 20 acres, so you can see that some of the plantations are indeed very large

Changing timesChanging times

•

1995: AF&PA adopts Sustainable Forestry Initiative (SFI)–

greatly reduced harvest block areas

–

buffers separate concurrent blocks–

multi-year green-up

intervals separate

adjacent blocks•

1996: SFI compliance becomes a condition of AF&PA membership

Presenter

Presentation Notes

In 1994, the member companies of the American Forest and Paper Association drafted Sustainable Forestry principles to guide forest management on industrial forest lands. In addition to demanding industry support of basic research, reforestation and other business reforms, the SFI drastically changed the way harvesting is carried out, particularly compared to traditional southern pine plantation management. SFI was adopted in 1995 by the AF&PA board of directors and in 1996, adoption of sustainable forestry principles was required for membership in AF&PA. Numerous memberships were terminated for non-compliance in the next year.

Changing timesChanging times

•

Sustainable Forestry guidelines–

no clear-cut harvest areas > 240 ac

–

clear-cut harvest areas < 120 ac unless absolutely necessary

–

contemporary clear-cut harvest blocks separated by buffers 120 -

300’

–

no clear-cut harvesting adjacent to a recent harvest until 4-5 years elapse

Presenter

Presentation Notes

As Champion was one of the principals in developing SFI, it is no surprise that the company’s Sustainability and Stewardship guidelines meet or exceed the guidelines of SFI. Although there are many details in the S&S document, the ones of prime concern to forest planners were the limits on clear-cut harvest areas. In addition to significantly reducing the size of clear-cuts, the guidelines also required significant areas of forest as buffers, forcing harvests to be dispersed over the landscape.

Southern pine plantationsSouthern pine plantations

Presenter

Presentation Notes

This slide illustrates the kind of impact the S&S guidelines had on forest operations. In the upper half of the slide, we see three different plantations that were clear-cut harvested and replanted in single harvesting operations. The lower half of the slide illustrates a possible harvest sequence for those plantations under S&S guidelines. The plantations have been subdivided by a hexagonal grid to make it easier to visualize the maximum opening size and proximity relationships. Since each side of a 20 ac hexagon is just over 300 feet, we can readily determine that the buffer distance requirement is met if concurrent harvests are separated by at least one cell. Since the maximum opening size is 120 ac, no harvest block should be larger than 6 cells. One approach would be to alternate 120 acre clear-cut areas with 1-cell wide buffers, but such leave-strips are difficult to harvest later on, if they can be harvested at all. A better alternative is to harvest compact blocks, scheduled such that the green-up interval is met. This allows the entire area to be harvested, but over a much longer period of time than was the case prior to S&S being implemented. Many different configurations are possible by different colorings of the hexagons, and in fact the problem is combinatorial in nature. The problem is, which coloring yields the best overall solution?

Unit Restriction ModelUnit Restriction ModelMaximize(1) Z = Σi

Σt αit

xit

Subject to(2) Σt

xit

<

1 ∀i(3) Σi

βit

xit

>

Lt

∀t(4) Σi

βit

xit

<

Ut

∀t(5) xit

+ xjt

<

1 ∀i, t, j ∈

Ni

(6) xit

= (0, 1) ∀i, t

⎧1 if unit

i is treated in period txit

= ⎨

⎩

0 otherwise.

i

= index of planning units,t

= index of time periods,αit

= benefit or revenue associated with treating unit i

in period t

βit

= volume contribution for treating unit i

in period tLt

= lower bound on total volume produced in period t

Ut

= upper bound on total volume produced in period t

Ni

= set of planning units adjacent to unit

i

Presenter

Presentation Notes

If you consult the forestry literature on spatial planning problems, the bulk of the papers deal with a mathematical formulation that Alan Murray calls the unit restriction model. The underlying assumption is that harvest units are smaller than the maximum opening size allowed, and that by disallowing the simultaneous harvest of any adjacent units, the maximum opening size restriction is guaranteed to be met. Each decision variable is binary, and only one of the harvest options (timing and/or treatment) for each harvest unit may be chosen. Pairwise adjacency constraints prevent simultaneous harvesting of adjacent units. The objective is to maximize overall benefit.

Area Restriction ModelArea Restriction ModelMaximize(7) Z = Σi

Σt αit

xit

Subject to(2) -

(4), (6)(8) ƒit

(vi

x) <

A ∀ i, t

⎧1 if unit

i is treated in period txit

= ⎨

⎩

0 otherwise.

A = maximum permissible contiguous area treated

vi

= area of unit

i

ƒit

(vi

x)

= recursive function summing all treated neighboring units associated with xit

(if xit

=1)

Presenter

Presentation Notes

Murray defines an alternative formulation called the area restriction model, which is the same as the unit restriction model except that opening size restrictions are modeled directly within the adjacency constraints. Equation (8) replaces equation (5) of the URM, and the structure of the objective function (7) is the same as in (1), but the decision variables are different. Instead of preventing all simultaneous harvesting of adjacent units, the constraint function allows simultaneous harvesting as long as the total opening size is not exceeded. This new adjacency constraint function is not linear like the pairwise adjacency constraints it replaces. Since the total opening size depends not only on a harvest unit and its neighbors, but also the neighbors of its its neighbors, the function is recursive.

ComparisonComparison

•

URM–

as an MIP can be solved exactly

–

limited problem sizes solved

–

requires prior block delineation

–

formulation may not represent real problem

•

ARM–

unlikely to be solved exactly

–

heuristics do not yield optimal solutions

–

block layout part of solution

–

directly models regulatory constraints

Presenter

Presentation Notes

The advantage of the URM is that it can be formulated as a mixed integer programming problem, and thus in theory can be solved exactly using commercial software. Unfortunately, integer programming suffers the curse of dimensionality and the size of problems that can be practically solved is rather limited. Moreover, if realistic harvest blocks are desired, then some sort of prior delineation of them is necessary, which can be a time-consuming chore. If natural stand boundaries are used, some stands may exceed the maximum opening size; if GIS overlay techniques such as our hexagon grid are used to subdivide the forest, the resulting solutions may be an unrealistic representation of the problem because much larger harvest blocks are allowed by regulations. The nonlinear constraint function of the ARM not only prevents formulation as an MIP, but it also means that exact methods are unlikely to ever be developed to solve it. However, heuristic methods can be applied to it, though there is no guarantee of optimality. Moreover, the method does not require delineation of harvest blocks a priori. Since opening size restrictions are represented directly in the model, the block layout is generated as part of the solution

RemsoftRemsoft’’s approachs approach

•

Develops commercial applications•

Most literature solutions unsatisfying–

specialized applications (research)

–

limited to small problem instances–

clumsy/limited user interfaces

–

poor data management features–

little or no documentation or technical support available

Presenter

Presentation Notes

Remsoft has had ongoing research and development in spatial forest planning since 1993. Much of the early work evaluated research tools developed by graduate students for specific tasks, and how to link them into a more cohesive system. Many of these research studies showed great promise but as individual tools, they were too difficult to use and lacked the flexibility necessary in commercial software. As much effort was spent on devising appropriate data linkages and structures for data integrity as on improving the efficiency of these tools themselves. Woodstock and Stanley were developed as separate products, but over time their feature sets have become complementary, and they work seamlessly together. With the next revision of Woodstock, all the software components of the spatial planning system will be fully 32-bit Windows applications.


•

Simplify the problem•

Most of the management decisions are made in strategic model

•

Tactical decisions reduced to minimizing deviations from strategic

•

Only types scheduled during tactical planning horizon are blocked

Presenter

Presentation Notes

Given the scope of the planning problem, and the limitations inherent to the mathematical formulations outlined earlier, the obvious solution was to simplify the problem, reducing it to tractable sub-problems that could be addressed with current technology. What many people seem to ignore is the fact that most of the important decisions in forest planning are made at the strategic level. This is where goals such as desirable volume and product flows, silviculture budgets and resource constraints are determined. Although spatial restrictions are important, they largely constrain implementation, not management objectives. This is why linear programming has a long tradition in timber harvest scheduling: it is an effective means of efficiently allocating scarce resources. The need for spatial detail in planning has not obviated this key advantage.Instead, the LP harvest schedule is really just a relaxation of the spatial constraints. Regardless of the length of the planning horizon, the harvests for the initial few periods must come from stands that currently exist, and one cannot feasibly substitute juvenile stands for mature ones. Therefore only a subset of the forest is truly eligible for tactical planning, and any spatial harvest schedule will have to use these stands or leave them unharvested. By recognizing this fact, a great deal of the complexity of the strategic planning phase can be shed in the tactical phase. There is no need to be concerned with stand types harvested beyond the tactical planning horizon. Individual stands within stand types are freely substitutable by definition and where deviations are necessary, reduced cost information from the LP can guide the choice of substitutes.


•

2-stage ARM (Jamnick & Walters)–

Use LP to determine an optimal schedule of stand-types to cut

–

Use heuristics to allocate harvest treatment prescriptions to stands

•

Contiguous stands assigned the same treatment in the same period defines a block

•

Harvest blocks must meet maximum size, proximity and green-up restrictions

Presenter

Presentation Notes

The hierarchical planning approach I just outlined was first developed by Jamnick and Walters in 1993. Remsoft’s Spatial Planning system is based on the Jamnick-Walters approach, but builds significantly on it by providing integrated data management tools and far more sophisticated blocking/scheduling algorithms. The system relies on the power of linear programming to develop a strategic harvest schedule. From the initial periods of the LP schedule, a list of harvest activities and their associated stand types and timing choices is identified for blocking purposes. The remaining forest types are ignored. The Stanley software handles most of the data preparation tasks for blocking and scheduling. It scans the geographic information databases to determine which polygons are eligible for blocking and to determine adjacency and proximity relationships among those polygons. Once the user has specified the spatial restriction parameters, the algorithm begins searching for polygons or groups of polygons that match the LP schedule and can be harvested simultaneously as a feasible harvest block. Using a combination of local improvement, random restart and specializing flow balancing algorithms, the Stanley software tries to minimize deviations from the LP schedule while maintaining spatial feasibility. Solutions that violate spatial restrictions or flow tolerances are discarded but a solutions that is feasible and comes closer to meeting the LP output targets than the current incumbent solution is retained as the new incumbent.

Stage OneStage One

•

Stratify forest according to developmental characteristics

•

Assign each forest stand (map polygon) to one stratum

•

Generate and solve LP harvest schedule using Woodstock

•

Identify outputs to be used to measure goal attainment in Stanley

Presenter

Presentation Notes

There are basically three stages of analysis using the Remsoft system. First of all, geographic information for the forest of interest must be obtained, and a stratification scheme must be devised. Every stand in the forest must be assigned to a specific stratum so that later on the stratum-based harvest schedule can be disaggregated. Next, the analyst develops an acceptable harvest schedule using Woodstock. Although I appear to be glossing over this step, in fact, it is the most important step in the entire process. Once the strategic harvest schedule has been developed, the analyst is ready to begin block harvest scheduling.

Stage TwoStage Two

•

Set parameters (harvest block size, proximity distance, green-up interval)

•

Set acceptable flow variations from LP targets

•

Generate spatial harvest schedules under different scenarios

•

Retain best solution found

Presenter

Presentation Notes

At this point, spatial restrictions on harvesting come into play. Because we are using a modified ARM approach, we cannot assume each individual stand or polygon in the model is a feasible harvest block as we must in the URM approach. Therefore, in addition to maximum opening sizes, we must also specify a minimum opening size. We must also recognize that because the decision variables in the spatial scheduling model are integral, it may be impossible to have flows that are exactly equal to the flows coming out of the LP relaxation. It is up to the analyst to determine how tightly the flows must follow the the LP solution by specifying flow fluctuation limits. Basically, if the percentage deviation in flow between the periods with the lowest and highest deviations is less than the specified percentage, then the solution is considered feasible. Typically, ranges of 5 to 15% yield acceptable solutions. The analyst must identify the location of the Woodstock solution files so that the Stanley software can determine yield coefficients and operability limits of individual stands. This information is used to determine which timing choices are feasible for each block. The run time necessary for Stanley to generate good solutions depends on a number of factors, including the number of planning periods, stands and activities to be blocked. Given the stochastic nature of its heuristics, the probability of finding a better solution increases with time but not uniformly.

Stage ThreeStage Three

•

Make adjustments to Stanley solution to reflect operational realities

•

Iteratively re-run Stanley until acceptable solution results

•

Generate mapped solutions•

Incorporate Stanley solution into Woodstock LP model to test long-

term sustainability

Presenter

Presentation Notes

No algorithmic approach is going to completely replace the need for professional judgement in forest planning. There are many variables in determining a feasible operational plan, including weather, human nature and market forces that cannot easily be represented in a tabular database. Stanley’s map tools allow analysts to modify harvest blocks by changing their timing choices, adding or deleting stands from the block, or by manually delineating blocks from scratch. User feedback has indicated that 60 to 70% of blocks coming from a Stanley solution make intuitive sense and require no adjustments. The remaining 30% must be manually adjusted. Once the corrections are made and stored, the algorithm can be run again, but with some blocks now fixed. Over the course of a few iterations, a feasible block harvest schedule is developed. Map products illustrating the harvest schedule across the landscape and through time can be immediately produced. To test the long-term sustainability of the spatial harvest schedule, an analyst can incorporate the spatial schedule back into Woodstock. The simulator in Woodstock carries out the activities specified in the spatial schedule, updates the forest appropriately, and then generates a new LP solution for the remaining planning periods. As long as there are no large-scale deviations in the projected harvest flows compared to the initial analysis, one can be reasonably confident that the spatial schedule is sustainable.

Quality of SolutionsQuality of Solutions

•

Woodstock/Stanley approach generates satisficing

solutions only

•

URM has optimal scheduling solution but requires block layout a priori

•

Stanley yields block layout as part of solution but schedule is not optimal

•

Use Stanley blocks in an MIP formulation to determine quality

Presenter

Presentation Notes

Because the LP relaxation does not consider spatial relationships explicitly, and the block harvest scheduling algorithms are heuristic, the solutions generated by the Remsoft software are not optimal. But determining an optimal solution to compare to is difficult. The URM can produce an optimal schedule for a specific block layout, but there is no way to determine what block layout would yield a global optimum. A reasonable compromise is to compare a Stanley solution to an MIP formulation using the same block configuration that was in the Stanley solution. Although objective function value can be readily compared, solution times cannot because much of the processing time in the Stanley algorithms is used to develop the block layout whereas the MIP formulation only schedules the blocks.

Case studyCase study

•

Forest of pine plantations, cypress ponds and bottomland hardwoods

•

87 000 acres, 13 000 map polygons•

25 year strategic, 10 year tactical planning horizons (1 year periods)

•

Maximize PNV subject to non- declining flow constraints on harvest

volume

Presenter

Presentation Notes

A representative case study was developed based on 87,000 acres of southern pine plantations in Georgia. A 25 year strategic harvest schedule was developed with Woodstock, maximizing present net value subject to non-declining flow constraints on harvest volume.


Presenter

Presentation Notes

In this slide, the dark blue area represents pine plantations and the orange areas represent cypress ponds that are interspersed throughout the forest, but are not harvested with the pine. Other colors represent hardwood and noncommercial forest types that are also not considered in this analysis.


•

Champion S&S guidelines–

10 ac minimum blocks

–

120 ac maximum blocks–

300 ft proximity distance

–

5 year green-up delay•

Stanley parameters–

allow +/-5% deviation in periodic flow

–

run time = 15 min (Pentium II-266)

Presenter

Presentation Notes

Champion’s Sustainability and Stewardship guidelines for clear-cut harvesting were applied. Although the guidelines do not specify a minimum block size, professional judgement indicated that blocks smaller than 10 ac were not generally economic to harvest. Because this forest had no riparian or other special treatment zones, the standard 120 ac maximum opening and 300 foot proximity restrictions were used throughout. At least 5 years must elapse before harvesting stands adjacent to a new clear-cut area. To give Stanley some flexibility in generating solutions, a 5% flow tolerance was specified. This means that if the best period achieves 90% of the strategic output target and the worst period achieves 85% of the strategic output target, then the solution is considered to have feasible flow. If the worst period only achieves 82% of the strategic target, then the solution is not feasible. The program was allowed to run for 15 minutes on a Pentium II processor running at 266MHz.

Case study resultsCase study resultsProgram Execution time SolutionWoodstock 44 s, matrix generationC-Whiz 20 s, LP solution 45 525 cunits/yearLP2WK conversion 3 s,Stanley 900 s 34 266 cunits/year min

76.4% of LP optimalMIP formulation(maxmin)

3893 s, stopped after4 integer solutions

35 224 cunits/year min77.4% of LP optimal

Flow variation Stanley – 4.9% MIP – 0.3%

Presenter

Presentation Notes

To generate the entire spatial harvest schedule, including the strategic and tactical planning phases took 967 seconds. Although the LP relaxation suggested an optimal harvest flow of 45,525 cunits per year, Stanley was only able to schedule about 75% of that volume under spatial constraints. Although this seems low, remember that this forest has been managed under traditional southern pine management, and many of the mature stands of timber are all in close proximity to one another. With the 120 ac maximum clear-cut size and the the 300 foot proximity buffers, a large portion of the forest is inaccessible due to spatial restrictions. Although Stanley evaluates solutions in terms of achieving a specified output target, the algorithms are actually driven by penalties for deviating from the LP solution. There is no direct analog for this in an MIP objective function, so we formulated the problem as a maxmin objective, so that we could compare the worst performing period in each solution more fairly. Obviously we could have used a goal programming formulation or a simple maximization subject to periodic flow constraints as well to achieve similar but not the same schedules. That said, the MIP was able to schedule about 1% more of the volume target than Stanley, and the flow fluctuations were very small. However, the MIP took more than 4 times as much processing time as Stanley yet yielded a 1% improvement. Thus the Stanley algorithms are performing quite well.

ChampionChampion’’s experiences experience

•

Initially drawn to Woodstock due to its flexible modeling structure–

Acquired two copies of Woodstock for testing purposes in 1995

–

Woodstock adopted company-wide in 1996 as strategic planning model

•

Stanley acquired as tactical planning model in 1996-97

Presenter

Presentation Notes

Champion first contacted Remsoft about Woodstock in 1994 because they felt its flexible modeling structure was ideal for the wide range of forest types and operations the company faced in its US operations. They also learned of Remsoft’s development plans for Stanley and asked to be kept up to date. In 1995, the company licensed two copies of Woodstock for evaluation purposes at its Jacksonville Technology Center. The software proved robust and promising and in 1996, Woodstock was adopted by all of Champion’s US regional offices. Remsoft was contracted to provide comprehensive training in the software to all of Champion’s planning and support staff. In 1996, Stanley was commercially released and over the course of the next year was acquired by virtually all of Champion’s regional offices.


•

Nearing completion of a new unified forest information system–

Woodstock/Stanley integral part of it

–

yield models link directly to Woodstock through dynamic link libraries

–

standard procedures ensure integrity of data across strategic & tactical levels

–

minimal in-house proprietary software

Presenter

Presentation Notes

Previously, forest inventory data was stored on a variety of mainframe, workstation and local databases. After several years of development, Champion is nearing completion of a new unified forest information system, with Woodstock and Stanley as key components. The new system will allow analysts to generate accurate growth and yield estimates directly from online inventory data, and these estimates will link directly into Woodstock through its support of dynamic link libraries. Because manual data preparation steps are being replaced by automated procedures, data integrity is maintained across all levels of planning. This development is unique in an industry that has historically relied on proprietary systems. Although the Remsoft planning software is available to Champion’s competitors, company staff feel no loss of competitive advantage. Instead, they feel that competitive advantage arises from knowing how to make the best use of available tools, and capitalizing on them early. The Remsoft planning software allows their analysts to make better informed decisions


•

User satisfaction high–

system based on sound theory

–

solutions that make intuitive sense–

software interface makes it easy for planners to apply professional judgment

–

holistic approach to data management ensures integrity across planning levels

–

quality software and technical support

Presenter

Presentation Notes

When faced with new systems and procedures, analysts are often skeptical. However, after using the software for several years, planning staff have expressed satisfaction with Woodstock and Stanley. The planning process incorporated into the software is based on sound theory and solutions generated by it make intuitive sense. Although the mathematical programming and algorithmic techniques employed by the software are very important, it is the software interface and automated data manipulation features that make the system truly powerful to analysts. Because they can readily impose changes on solutions without worrying about corrupting the underlying databases, planners are more confident and willing to conduct more and better analyses. The spatial planning software relies on a large amount of data from a variety of sources, and is itself rather complex. Although by and large the implementation of Remsoft’s spatial planning system at Champion has gone smoothly, there have been some difficulties. Champion has found Remsoft to provide excellent technical support and a strong willingness to work with them to resolve problems. In turn, Remsoft has incorporated new features into its software that not only address Champion’s needs, but also improve the marketability of the software. Remsoft’s spatial planning system has been implemented by several forest companies, consultants and government agencies in Canada and the United States.

ConclusionsConclusions

•

Remsoft developed general modeling tools with flexibility in mind–

good use of available OR technology

•

Champion sought software solution adaptable to wide range of conditions–

same software can be used for very different forestland/operations

–

ongoing relationship with developers

Presenter

Presentation Notes

I have introduced to you a successful implementation of

A system for solving spatial forest planning problems

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