FOREST HAR\EST SCHEDL-LIXG PROBLESI: STCDE-IXG WITHEMATIC-4 AXD CO'SSTR-ID-T PROGRALIMD-G

SOLUTIOX SSTRATEGES

.Junas Xdhiliary

B-Sc.. Trent Gniversity. Peterborough. OY Canada. 199-4

A THESIS SCBMITTED I'i P.4RTIAL FCLFILLMEYT

OF T H E REQCIREMEXTS FOR T H E D E G R E E O F

MASTER OF SCIEXCE

in the School

of

Computing Science

@ .Junas Xdhikary 1997

SIMOK FR-UER UYWERSITY

April 1991

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Abstract

The Forest Harvest Scheduiing Problem (FHSP) is an important part of the forest resource

management. [t is a comples multi-criteria optimization problem. Our review of the opti-

mization techniques applicable to forest harvesting problems in general suggests that long

term scheduling is difficdt because of the prohibitive size and the complexicy inherent to

the problem. In this thesis. we stiidy mathematicd and constraint propramming as both

rnadeUing and solving techniques for such problems. CVe discuss hou. these solution strate-

gies may accommodate the forest growth and treatment simulation model. The simulation

mode1 is an important part of the scheduling system since it makes the solution more real-

istic and irnplementable. We give a mived integer programrning (MLF') formulation for the

restricted FHSP problem and test it using real-life data from the Sorwegian ECOPL-\?i

project. Since Iarger instances were not solved in a reasonable time. t h e UIP modet aione

is not sufficient t o solve practical FHSP. We advocate the combined use of the Constraint

Satisfaction ProbIem (CSP) model and the iterative improvement technique a s a sohtion

strategy- The simuiation model can also be more easily integrated in tiuii strategv thaii iri

the SIIP model. The iterative improvement techniques wiU in general benefit from the high

quality initiai solutions. We show how a CSP formulation can be used t o find -good" initiai

solutions to the problem. based on simulation results from a stand g o w t h and treatment

simdator. The initiai solution generator can be used as a module in a n integrated forest

treatment scheduling system which WU advance the state-of-the-art in forest harvesting

practices.

Acknowledgment s

The author is extremely gateful for the continuing support and guidance of his senior

supervisor. Dr. CViarn S. Havens. Thanks t o his supervisor Dr. Lou Hafer for helping

in the mat hematical side of t h e research. I am extremely pratefui t o Cunnar Misund and

Dr. Geir Hasle lrom SIXTEE' Oslo for giving me an opportunity t o work closely with them

on the Xorwegian ECOPLXX pprject and for providing the ECOPLA?; pmtotvpe data.

Thanks t o Morten lrgens for initiating thiç collaboration between ISL a n d SIXTEF. Finaiiy

my special thanks goes t o a.ll t he members of t h e Intelligent Systems Lab for reading my

drafts and helping me improve s u bstantially.

To rny parents Gehendra and Sushila Adhikary.

Contents

.. .4pprod Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II

... Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LU

.4cknowledgment S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \-

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LY

1 Lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 -

1.1.1 Growth and Treatment Simulation . . . . . . . . . . . . . . . 1

1.2 Review of Optimized Forest Harvest Scheduling . . . . . . . . . . . . . 8

1.2.1 Mat hematical Programming . . . . . . . . . . . . . . . . . . 9

1 2 Heuristic Optimization . . . . . . . . . . . . . . . . . . . . . 10

L.2..3 MsceUaneous Met hodology . . . . . . . . . . . . . . . . . . . 1 3

1.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1:3

1 .3 Ov-erview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3

2 Formaikat ion of the FHSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1 Identifying Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

'2.1.1 0- 1 Xode-Packing Probleni . . . . . . . . . . . . . . . . . . . 19

2.2 Consistency Enforcement . . . . . . . . . . . . . . . . . . . . . . . . . 22

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Row Convexity 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conclusion 2-5

3 Solving the CCSP using SIIP 4Iodels . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 MIP Mode1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 MIP Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

13.3 Adjacency Const raints Revisited . . . . . . . . . . . . . . . . . . . . . 30

.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4. I initial experiments using lpso l re solver . . . . . . . . . . . . 31

3.4.2 Resuits using cpier solver . . . . . . . . . . . . . . . . . . . . 32

3.5 htegrating LP with Simuiation . . . . . . . . . . . . . . . . . . . . . . S I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion 39

4 Solving the FHSP using a CSP mode1 . . . . . . . . . . . . . . . . . . . . . . 40

4 . L Met hods Studied for Generating Initial Schedule . . . . . . . . . . . . 41

. . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Linear Programming -41

4.1.2 Wxed lnteger P rogrmrning . . . . . . . . . . . . . . . . . . 43

4.1.3 The CSP model and Constraint Programming . . . . . . . . 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.l.4 Conclusion 4.7

1.2 -1 method based on a CSP mode1 . . . . . . . . . . . . . . . . . . . . . 4.5

. . . . . . . . . . . . . . . . . . . . . . -4.2. L Minconflicts Heurist ic 46

4.3 Esperimental Resdts . . . . . . . . . . . . . . . . . . . . . . . . . . . -47

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ..3 .I Data Sets -Li-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Results 49

4.4 Furt her Improvements and Suggestions . . . . . . . . . . . . . . . . . 51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion 32

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 5.5 Appendices

A Data format for minconflicts . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B D a t a G e n e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

C Test Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

vii

List of Tables

3.1 Results obtained for random data using t he fpsofsolre hIIP solver . . . . . . . . 32

3.2 Results obtained for Set 1 using cpfer LIIP solver . . . . . . . . . . . . . . . . 3-1

3.3 Results obtained for Set 2 and -I using cpler lIIP solver . . . . . . . . . . . . 35

3.4 Results obtained for Set 3 and 5 using c p k z l I IP solver . . . . . . . . . . . . 36

List of Figures

1.1 An esample of a forest area divided into stands - . - . - . . - . - - . . - - . . 5

II A number of schedules as paths in the tree produced bj- a treatment simulator 7

1.3 Tight integration of simulation in a iorest harvest scheduling svstem . . . . . 9

2.1 (SI . S2) participating in a neighborhood constraint . . . . . . . . . . . . . . . ln

2.2 ?;odes i .j and k participating in an adjacency constraint . . . . . . . . . . . 21

4.1 Constraint graph of a forest in Figure 1.1 . . . . . . . . . . . . . . . . . . . . 44

4.2 Function used t o calculate height . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Variables left inconsistent and iterations (each s tand has .5 schedules ) . . . . . 49

4.4 VariabLes Ieft inconsistent and iterations (each s tand has 10 schedules) . . . 50

4.5 Variables left inconsistent and iterations (each s tand has L . 3 or 20 schedules) . -50

Chapter 1

Introduction

Governments and private sectors worldwide manage an immense area of forest. The C'nited

States Department of Agriculture alone manages a forest area almost twice the sizc of

Gennany. Forest harvest planning and scheduling probierns are an important part of t his

forest management. Recently. bot h povernments and the public worldwide are dernanding

sustaiaable harvesting practices for these areas. rvhich wodd take into account not only

economics but aiso the presermtion of biodiversit- and the est het ic. and recreat ional d u e

of the forest. Long term forest harvest scheduling aiiows communities. governmentai. and

non governmentai organizations t o check if sustainable forest ry is act u d y being practiced.

Simplified. the problem is t o assign forest treatment types and times t o treatment units (of-

ten c d e d stands) in a given forest area over a Iong period of tirne. We wiii c d t his problern

the Forest Harvest Schedulirig Problern ( FHSP 1. An optimization version of the problern

invoIves optimizing objective ccriteria such as economical. esthetic. and recreationai d u e s .

which are subject to multiple constraints. Traditiondiy. mat hematical programming models

and solution techniques have been estensively used to penerate such scheddes. and are stiU

the most widely used method (for exarnple. (Garcia 1990. YeIson et al. 1991. Yoshimoto et al.

1994) ). Artificial Intelligence (-41) techniques such as the rule-based expert systems. and

the local search. have also been applied t o forest management (for exampie. (Linehan k

Corcoran 1994. Eriksson 199-4. Murray k Church 1995b)). The above techniques have dif-

ficulties in modeiiïng a wide variety of spatial. temporal. and visual constraints and/or in

performing the ( u s u d y conficting) multi-criteria optimization that is inherent t o the prob-

Iem. In addition. the size of a real Life problem is often beyond the practicai capacity of

t hese met hods.

Siisund et al. have suggested the integrated use of Geographic information Technol-

ogy ( G I T ) and certain AI techniques (constraint reasoning and local search) t o rnodel and

solve such problems ( SIisrind d ai. KW.5 ). They describe their approach through a simpler

version of the problem n-here only clear-cutt ing is allowed. The problem is caUed the Clear-

Cut Scheduiïng Problern ( CCSP ). CCSP is rnodelled as a Constraint Satisfaction Problem

( C S P ) (Tsang 1993) and a solution is iteratively improved using Tabu search (GIover 1989).

Although the authors give some reasons for their choice of Tabu search as the iterative im-

provement technique. it is not aIi that clear whether the traditional method using a rnised

integer linear programrning ( I I I P ) rnodel can be effective.

The thesis of this dissertation is that the FHSP (as defined later) can in principle be

solved using a l I IP model and a state-of-the-art l I I P solver if the problem constraints are

sufficiently r e l a ~ e d and if some objective criteria are ignored. However. the size of the

problems solcable in a reasonable tirne is far below the size of any practical real-world prob-

lem. Furthermore. an MIP-based strâtegy can not be easily integrated wit h the simulation

models. which makes the solution hard t o implement. The term simulation in our thesis

refers to the growth and treatment simulation. which is often used bu foresters to find aii

possible treatnient schedules for a stand based on the current stand characteristics and

general forestry knodedge, The simulation results can be incorporated into a MIP-based

model too but the mode1 formulation alone becomes a demanding task. Therefore. we give

a CSP-based model where the simuIation can be integrated easily into a solution strategy.

Experimental results show t that a simple heuristic cailed minconflicts ( Xinton e t al. 1992 )

is able t o generate initial solutions that are almost always complete and consistent. The

idea is to take these initial solutions and improve them by applying other heuristic opti-

mization procedure Like Tabu search. T h e novel idea in this approach is the integration of

the simulation mode1 in a harvest scheduling process using a CSP model.

Our thesis does not solve the fuU FHSP problem. but rather provides a foundation

t o s tar t soiving practical problems using either mathematical o r constraint programniing

techniques. Our contribution to the FHSP can be outlined as foilows. First we conduct a

formal study on the restricted version of the problem (CCSP) as given in (Misund e t al.

199.5). We give a 4IIP model for the CCSP as described later in the problem definition

(cf. Section 1.1). We mode1 aii of the constraints and the objectives. Evaluating this

model on a reai-world da ta shows that it is difficult and time consuming t o soIve even

medium sized problems with aii t h e constraints and the objectives. Yext. we integrate

t h e growt h and t reatment simulation results into a CSP model u-hich can be used to solve

t h e FHSP. -1 method for using the simulation results as domain variables for the stands is

developed. Lastly. ive $;ive a n algorit hm t o generate a n initial solution for t h e FHSP using

the rninconj?icts heuristic repair method. Our esperimental results show t h a t this rnethod is

fast and makes use of the knowledge from the treatment simulation. Thereïore. the met hod

can be a good candidate for generating initiai solutions to be improved upon later using

the iterative techniques. In short. we show tha t it is possible to model the FHSP as a CSP

and use the simulation results so that be t ter forest treatment schedules can be found for

t h e practical problerns.

1.1 Problem Definition

Sustainable forest management requires t h e forest harvesting actions schedided for a long

period of tirne t o obey a variety of constraints. For management purposes. a forest landscape

is divided into basic treatment units. often equivalent t o stands. -4 s t and is a forested

a rea considered homogeneous with respect t o a selected set of properties. We will rest rict

our s t udy t o even-aged stands. tha t is. s tands containing trees of the same o r almost the

same age. We will c d the time period of the schedule as scheduling horizon. [t is a

common practice to divide the scheduling horizon into a riumber of time periods. each

period e q u i d e n t to. 5 - 10 years.

Definition 1 (Stand) -4 stand Si is an arra ichich is considerrd homogeneous u-ith respect

to certain chamcteristics. The set oJ n stands comprises a forest T . such thnf ,

For each stand Si .

L K(S;) : t ime of most recent hamest

ErLRLk'(S;) : minimum durntion between hamests

L-&fE(Sr) : maximum duration between hamests

a O P T ( S i ) : optimal time between hamests

.\iH(Si. 8 ) : lime r rqu i~d by &rues in the stand to gror from O to height H

a -ARE--i(Si) : a r a of the stand

GROCP-TH(Si. y ) : colume that may be hamested y yenrs a&r the Iast hamest

Def ini t ion 2 (Scheduling Horizon) -4 sch~duling horizon 'K is n contiguous set of m

pen'ods {Pl. . . .. Pm). where ~ a c h Pi is of a certain length in yeurs.

Defini t ion 3 (Trea tmen t ) =I trratment unit (stand) i.s g i r ~ n a nurnbr of mnnngement

option$ ocer the schedufing horkon. C f i ~ ~ i l l cal1 indi~idual management option a t reat ment.

-4 set 01 trratrn~nt types 'T = {To.. . . . Tt] is amilable for each danci- [ C i k t T , to be nul1

treatrnent or -let gow- where a stand is left uithout a n y trratmcnt. and Ti to b~ the c l~nr-

cut treatment.

Def ini t ion 4 (Forest Harvest Scheduling Problem (FHSP)) G i c ~ n an nrru offorest

3 dirided into n stands. {Si.. . . .Sn). Jind a schedule S = {.-LI.. . ...-ln) ocer .icheduiiny

horizon satisfying a .cet of constraints C = (Cl.. . . .Cc) . Hen. =li i.s the set O/ ac t i r i t i~ .~

repwsented by a set of tuples < Pi,. Tg, >. u : h e ~ each tupie reprrsents trratment p~riod and

treatrnent type for the stand Si. The constraints are usunlLg atIjac~ncy constmint. harw.st

intemal constmint et cetera.'

Xow we describe the constraints and the optimization criteria involved in the FHSP

bu defining the Clear-Cut Scheduling Problem (CCSP) in detail '. The CCSP is a special

case of the FHSP where (besides nul1 treatment ) onIy one other treat nient (cIear-cutting)

is aiiowed. and each stand is assigned only one management option over the scheduling

horizon. Figure 1.1 is an example of a forest area divided into stands. Two stands sharing

a common border are defined aç adjacent. or neighbors.

Def ini t ion 5 (Clear-Cut Scheduling Problern ( C C S P ) ) The Clear-Cut Schedding Prob-

lem is a restricted FHSP. Only tuo possible t ~ a t m e n t types are allowd. TO and 'Tl ("let

grou." and clear-cut). The clear-cut treatment is scheduled only once for cach stand dun'ng

the entire scheduling horizon.

'The constraints dl be described later 'This closely foiiows the definition given in (hfisund et al. 1995).

Figure 1.1: An esample of a forest area divided into stands

-1 CCSP can be visualized as assigning values to each of hi . 1 5 i < n. where hi is the

scheduled harvesting (clear-cut ) period for the stand ,Si. rest of the periods are assigned

the nuii treatment. The d u e of each hi is picked frorn its dornain Di = (1.. . . . nz} where

m is the total number of periods. such that the problern constraints (defined shortly) are

satisfied. We define the constraints and the optimization criteria for the FHSP as follows."

Constraints Constraints can be generaüy divided into two categories - hard and soft.

The hard constraints are defined as those constraints that must be satisfied whereas the soft

constraints can be reIased to satisfy the hard constraints or to adjust the objective function

value.

Hard Constraint Ci: Minimum height constraint Ail the neiglibors of the stand to

be harvested must have an average tree height of at least 'C meters. For CCSP.

where I ( i ) is index set for neighbors of Si.

Soft constra.int C2 : Harvest Interval Preferred harvest interval based on economical

and ecologicd considerations. -4 lower and upper threshold is @en for each stand. as

weii as the optimal harvest tirne relative to the Iast clear-cut. For CCSP.

V i E (1.. ... IL} : E=LRLY(Si) < h; - L H ( S i ) 5 L,LTE(Si) (12)

'These are taken £rom forest harvesting requirernents for the Xorwegian ECOPLAY project (Misund e t al. 1995) but are also generd enough ta be applicable to forest harvesting practices elsewhere.

Optimizat ion Criteria A f~asible solution is an assignmen t of pro blem variables s uch

that ail the constraints are satisfied. An ~ p t i ~ z a t i o n versian of the CCSP (and the FHSP)

arises when various optimization criteria are masirnized (or minimized ) in the solution.

CVe are interested in solving the optimization version of the probtern with the following

optimization criteria:

O [: Optimal Harvest Tirne The actual time between harvests shoiild be as dose to t he

optimd as possible. For CC'SP.

O?: Even Harvest Volume The estimated harvested volume EHI.&S) for an- period p

should be as close to the average harvested volume in a schedule S. .4HCF(S). [n otlier

words. the variance between the liarvest volumes should be minirnized.

03: Old Forest A specified minimum area of old forest (;LOF). tliat is. the total area of

the stands wit h average age above a certain threshold. shoiild be maintained over the

scheduling horizon. Let O L D ( p . S) be the area of old forest in period p.

rn

Winirnize rnnx(0. ;LOF - O L D ( p . S)) yp= l

Xote t hat the optimal would be O when the area of old forest is greater t ban or equal

to AOF.

04: Visual Impact Yinimize the visual damage of clear-cutting relative to a set of view-

points. Let C - f S ( S i . p. kW) denote visual impact from stand Si in period p relative to

viewpoint C'. Xote that the foilowing equation o d y assumes one viewpoint. whereas

in the rea! case there is usually a set of viewpoints.

The rninimirrn height constmint is enforced so t hat a large area of forest is not clear-cut

simuitaneously causing a great damage to the regeneration process and the wildlife habitats.

Scheduling horizon

Figure 1.2: A number of schedules as pat hs in the tree produced by a t reatment simulator

It is also referred to as adjacency c o n s h i n t because cutting adjacent stands at the same

time period is prohibited by t h e constra.int. This restriction is u s u d y valid for certain

specified periods such that the harvested area will have regenerated properly. These periods

are commonly caUed exclusion penods or green-up age. Recent lu. the adjacency const raint

and the exclusion period have been given special consideration in the majoritv of research

dealing with the forest harvest scheduling (according to our review in the next section).

1.1.1 Growth and Treatment Simulation

in t his t hesis. the term simulation refers to the growt h and t reatment simulation of areas of

forest as practiced and defined in the forestry Literature ( Hoen 1992. Lappi 1992). Crowth

simulation is u s u d y implemented in a simple way using growth curves for a variety of

tree species. On the ot her hand. treatment simulation is more involved and ~equires more

forestry knowledge and stand characteristics data. Various factors corne into play when

generating an dowable treatment for a given stand. Mthough the simulation assumes

that each stand is independent from the others. a state of the stand changes each time a

treatment is applied. Depending on t his state. future treatments are simulated.

Figure 1.2 is a s m d esample of the groivth and creatment simulator result for a stand.

The simulator in effect produces a tree where each node represents a period in tlie scheduiing

horizon. which is increasing in the S-&S. Different treatments are represented in the Y - a i s

at every branch of the tree. For example. the first level could be the -let grow" treatment

or do nothing. the second clear-cut. the third commercial thinning and so on. Every patli

in the tree arnounts to a complete schedule for the stand. Thus. the number of leaf nocles

in the tree is eqiiivalent to the number of aiternate schedules available for the stand.

It is evident that any harvest scheduler that integrates simulation results produces a

correct and implementable scheduie. The problem is that each tree as in Figure 1.2 can

have hundreds of leaf nodes. For efficiency. only sonie of the leaf nodes can be selected as

the possible schedules during the scheduiing phase. There lias not been any published work

( to our knowledge) that tries to integrate the simulation model into tlie scheduiing process

in such a waÿ that the probtem constraints are aiso satisfied simultaneously. Almost aii

of the scheduiing systems to date do not take the treatment simulation into account while

penerating harvest schedules. h o t her problem wit h such simulation is the independence-

of-stands assumption. -4 scheduler has to take care of all the confiicts that may arise from

the constraints between the stands such as the adjacency constraint. Thus. the scheduler

will perform better if every stand has a number of alternate schedules.

Our goal is to integrate the growth and treatment simulation into the liarvest scheduling

process. We study whether this can be done with eit her the mathematical or the constraint

programming. Ideally. we would iike to have an initial solution that uses the simulation

r~sul ts and the optimizer improves the solution with the help o l tlie simulator. This is

illustrateci in Figure 1.3. Tt is a tight integmtion model. If the simulator resiilts are compiled

beforehand and given to the optimizer. then we have a loose integmtion motlei. In oiir thesis.

we only deal with the loose integration model.

1.2 Review of Optimized Forest Harvest Scheduling

There are two major classes of forest harvest scheduling algorithms. One is based on math-

ematical programming and the other on heuristic techniques (with or without using simula-

tion in both cases). The former is a global procedure whicli finds an optimal solution to the

forest management model whereas the latter is usua.liy a locd procedure which iterativelÿ

improves the solution without any guarantee of finding the optimal one. hlatliematical

Initiai solution n

Figure 1.3: Tight integration of simulation in a forest harvest scheduling system

programming is the technique that is commonly used in practice and consequently a large

percentage of the research activity is devoted to it. There are also some algorit hms which

do not f d under these two categories. and that vie will discuss under the misceilaneous

met hodology-

1.2.1 Mathematical Programming

The general forest planning problern. where harvesting is a n important process. has been

studied for some time. Early forest management models (X'avon L%l. .Johnson S- Crim

1986) rvere developed using Linear Programrning ( LP). Two well known LP-based scheduling

packages in use are FORPLA.1' (Johnson S: Rose 1986) and h I r S K ' (Johnson S: Jones

1979). Johnson and Scheurrnan reviewed and anaiyzed many LP-based forest planning

systems (Johnson S; Scheurman L917). Ln this paper. the authors group LP models into

Mode1 I and Mode1 11. which are frequently used in forest planning. Later Garcia in his

review of LP in forest planning (Garcia !990) revises the classification. .Ill these rnodels

are geared towards LP-based solution strategies.

LP models use continuous variables. Defining spatial relationships requires the use of

integer decision variables. For example it is oot possible to capture the adjacency constra.int

just b- using continuous variables. Thus. CP-based model are aspatid models. These

models t herefore have been used at strategic planning level. But implementing the plan

ôt an operational levet, that is. designating a spatial area of forest for harvesting, is very

difficult and may be impossible. Furthemore. the solution obtained frorn such a model is

nonintegai. Consequently. LP-based solutions are difficult to interpret. For esample. how

informative would be a solution suggesting treatment 2.3 for a stand or period 1.5?

Since LP-based models are not able to espress spatial relationships. the research began to

shift towards the mixed integer programming (.LIIP) models (Kirby ct al. i.986. .Jones e t al.

199 1 ). A n integer decision variable is used to espress a particiilar harvesting decision. This

ailows the rnodel to express spatial relationships in the form of adjacency constraints. In

Chapter 3. we wiil give a lIIP model for the CCSP such that the spatial relationsliips

are capt ured in the constraints. Since the gneral integer programming algorit hms are

not efficient when a large number of integer variables exist. the SIIP-based strategies are

restricted to srnad problems. One way of making the MIP models efficient is using an im-

proved representation of adjacency const raints ( 1Ieneghin e t al. 1988. Torres-Rojo Jr Brodie

1990. Jones e t al. 199 1. lkshimoto Si Brodie 1994). Researchers have been steadily solving

larger and larger SIIP problems using the improved constraint formiilations and a variety of

heurist ic algorit hrns. Weint raub e t al. solved a MIP forest hzrvesting model wit h adjacency

const raints for multiple time periods using Lnear programming. and a sop histicated round-

ing heuristic ( LVientraub e t al. 199-&). The LP relaxation of the LIIP model is solved and

passed to a MIP solver which attempts to assign integer values using the heuristic algorithm.

The LP rnodels are sometimes used toget her \vit h a grorvt h and t reatrnent sirnulator.

The simulator is used to find ail possible treatment schedules for aU the stands. In most

cases. each stand is treated independently and the necessary information is provided as

the simulation proceeds. In LP models al1 the information has to be encoded beforehand

in the model. Some researchws have tried ta combine t hese two techniques (G,ikkk-1-LP

(Hoen 1992) and JLP (Lappi LS9-L) are good examples). In both cases. the growth and

treatrnent sirnulator is used to define aLi the ailowable treatments for each stand and the

output is optimized by a LP solver. The net present d u e of the forest is often used as the

optirnization criteria. It is calculated using various economic and forest data such as the

interest rate. the volume harvested. the species of the harvested trees e t cetera.

1.2.2 Heuristic Optimization

T here have been several st udies exploring the heurist ic op timization techniques for the

FHSP wit h (or without ) mathematical programming. One of the approaches considered

successful is the sampling heuristic c d e d Monte Cario tnteger Programming ( W I P ) (Xel-

son S- Brodie 1990). It is a biased sampling scheme that generates a number of feasible

solution alternatives. The soIution becomes better as the number of sarnpI~s increases.

Therelore. the optimal or the near optimal solution ma>* be possible only if a very large

nurnber of samples are generated. Cnfortunately. this increases significantly the time nec-

essary to find a solution. Lockwood and Moore use Simulated Annealing (SA) to generate

harvest schedules with spatial constraints ( Lockwood 8~ LIoore 1993). SA is a stochastic

optimization technique that has been successfuiiy used to solve combinatorid optimization

problems (Kirkpatrick 1983. Kirkpatrick 1984). The authors report t o have solved large

harvest scheduling problems with adjacency constraints only. Briefl. annealing is a natural

phenomenon (for example in metals) in wkch the intemal eiements of a cooling body rear-

ranges t heir order from a high t o a low energy state. From the molten ( freely moving) state

the body cools down slowly t o a stable s t a te with minimal energy. If it is cooled too fast the

body hardens into a suboptimd state. S - 1 tries to imitate this phenornenon by graduail-

rearranging the elements of a systern from disordered to ordered state. If the systern is

cooled slowly enough. S.1 guarantees ( probabilistic) convergence to a n optimal solution.

-1 recent study compares t hree heuristic approaches to solving the forest planning prob-

lems. of which harvest scheduling is a part (hlurraq- 9i Church 199.5b). The authors mode1

the problem as a MIP which d o w s the representation of the adjacency constraints. The

only objective used in the study is the mâ,uirnization of the net revenue. They deveiop a

met hod ahich improves upon the solution produced bu Monte Carlo sarnpling process using

Hill CLimbing (HC). Simulated Annealing (SA). and Tabu Search (TS). In ail the three

approaches. the initial solution produced by Slonte Carlo sampiing is locaily improved by

generating new neighboiirhood solutions. Ln HC. only an improved solution is accepted in

every step and therefore is more likely to get stuck in a locally optimal solution. When

minimizing a l o c d y optimal solution is c d e d local minimum. These local minima hinder

the heuristic from moving towards the g l o b d y optimal solution. SA and TS accept a a-orse

solution (wit h some probability ) in hope of escaping the Local minima. These met hods were

tested using the data from (Selson k Brodie 1990). It was lound that TS performed better

more often than SA or HC. However. this does not impk tha t TS will a l w a ~ s produce a

better solution than the ot her two approaches. The authors confirmed t his using Friedman

anaiysis (Conover 1980) on the solution resuits. Given any initial solution. it is e q u d y

Lkely that a high quality solution be reached by any one of the three techniques.

PukkaIa and k-angas describe another heuristic optimization technique (Pukkala S: Kan-

gas 1993). Their rnethod uses a growth sirnulator as the first ,tep t o produce several alter-

native treatment schedules for each stand. The second step is the actual heiiristic optimiza-

tion. in which the total utility of the schedule is rnasimized bu randomly picking a t reat ment

schedule. where the ritility is calculated bu adding the d u e s of the objective function. This

method was tested on a da ta of a small forest area and ivas found to be successful. Tt is

claimed to be better than the LP-based methods because of its ability to espress nonlinear

objectives. The drawback of this method. however. is that it does not take take into account

the adjacency constraints, Therefore the scheduies generated may be of very poor quality.

Constraint Satisfaction Problems (CSPs)

=\lrnost al1 of the met hods mentioned above have an underlying mat hematical programming

rnodel for solving the forest harvesting problem. One interesting studv is ( l[ isund e t al.

199.5). where the problem is modeiied as a constraint satisfaction problem (CSP) for the first

time ( t o our knodedge). The authors suggest the CSP model and an iterative improvement

technique. such as TS. to solve the FHSP. They used the restricted version of the problem

(CCSP) as a test case. ive di study this case in niore detail in Chapter 2 and Chapter 3.

There are weU-studied AI search methods that can be used to solve CSPs (Tsang 1993).

These methods can be classified into constructive and iterative search methods. Constnic-

tive search. for example standard backtracking (Bitner 8~ Reingold 197.7). starts with a

particular orderin; of the variables and instantiates them one a t a time. Thus it works

with a par t id solution and tries to extend it to a fuii solution. This search niethod suffers

from a phenomenon c d e d thmshing wherebv the same t-ariable-value pair that leads to no

solution is instantiated over and over again during the search process. These algorithms

have esponential time complexity. However. this search technique is complete. that is ail

the possible solutions can be found. and the optimal one identified.

The alternative t o the constructive search is to start with an initial solution and use

a local search heuristic (for example. Tabu search (Glover 1959) or simulated annealing

(Kirkpatrick 1984)) t o iteratively improve the solution. This method has been shown to be

much more effective for large and cornplex search problems. The advantages of using such

methods are that they are fast for some problems. that the crlrrent best solution is availabie

any time, and that fairly large problems can be attempted. The drawback is that the search

can get stuck in local opt imaor minima and consequently finding the global optimal solution

can become impossible.

In addition to a search technique. a solution strate= for the CSPs can also consist

of a consistency enforcement mechanisrn. for esampie. arc-consist~ncy (S[acktvort h 1'377).

These algorithms reduce the size of the search space by eliminating those parts that can

not contain a solution. Therefore. a typical solution s t r a t e s - for a CS P interleaves a search

mechanism wit h the consistency enforcement.

1.2.3 Miscellaneous Met hodology

There are a few ot her studies that cio not f d under the above categories. One study

uses 0-1 integer programming t o determine patterns for forest harvesting with adjacency

restrictions and forbidden regions modeiled as a grid-packing problem (Snyder 9- ReVeUe

1996). . h o t her recent st udy suggests a met hod bâjed on Bayesian stat istical concepts

(Van Deusen 1996). S t U these methods do not solve the FHSP problem as Xe defined it.

especiaily the multi-criteria opt irnization part.

Other techniques are those based on control t h e o r . and on non linear and dynamic

prograrnming (Roise 1986. Hof & .Joyce 1992). Even though man? objectives in the FHSP

can be written as nonlinear constraints. there has been very Little work done in this area.

Most of the nonlinear programrning research d e d wit h the problem of habitat scheduling

d o n g with the harvest scheduling. However. the size of the problem t hat is actually being

solved is c-ery s m d because of the additional habitat constraints and the inefficiency of

general nonlinear program solvers.

1.2.4 Summary

Most of the methods mentioned before in the review bave been tested with very s m d prob-

lems compared t o the practical ones that exist. Many studies use adj- 5-7 periods and

100 stands whereas any real-life problem is bound to have stands in the order of thousands

and approximately 20 periods (usually of Iength .5 years). The problem size is kept s m d

because MiP models tend t o consist of a large number of binary integer variables (o r inte-

grality const raints ). These constraints restrict the size bounds for problems because of the

ümitations of general integer programming solution procedures. For example. in one study

considering only adjacency relations the number of constraints is e?(nrn). where n and rn

are the number of stands and periods respectively (Murray 8i Church L995b). The number

of binary variables needed to formulate t hese constraints is nm. Furthermore. each such

variable participates in a number of constraints. -1 practical problem typically has a t least

5000 stands and 20 periods. That is. 100.000 integer variables - a large number for current

MIP solvers.

Hoivever. some studies have developed and added various heuristics to solve increasingly

larger harvest scheduling problerns. Furt hermore. in most of the esperiments the scheduling

horizon is kept ver? s m d to keep the number of binary variables (and thus the integrality

constraints) under control. But these methods do not deal with ail the constraints and

the objectives simultaneously as we have illustrated in the problem definition (Section 1- 1 ).

Therefore. a 1IIP model which tries to miaimize the number of binary integer variables

using efficient constraint formulations and/or good cut constraints is needed. These cut

constraints help produce an integrai solution iaster. ?Ve will study some of t h e cut con-

straints r e l e ~ ~ n t to our problem in the first part of Chapter 2.

l los t of t he algorit hms we reviewed implements g o w t h and yield simulation using grotvt h

and yield curves for a variety of tree species. But the treatment simulation is almost never

used. W e reviewed some research which t ried to integrate the treat ment simulation wit h a

LP solver to find harvest schedules (Hoen 1992. Lappi 1992). However. the problem with

these approaches is the LP model since not even the adjacency constraint (spatial relations)

can be represented in the model. Lntegrating the treatment simulation with a 1IIP has not

been done so far and ive will see in Chapter 3 that this may be possible if ive can utilize

column generation technique. Hovvever. a 4IIP model formulation for this is a challenging

task. Therefore. an alternative model is needed. in which the integration of the simulation

would be easier than in the MIP models. A constraint satisfaction problem (CSP) mode1 is

int roduced in ( Misund et ai. 1995). The aut hors integrate constraint reasoning techniques

wit h geographical information technology ( GIT ). GIT is helpful in visualizing the solution

t o esplore the possibilhty of furt her irnprovements. This is effective for esample regarding

the visual effect of a particular schedule. It is t herefore imperative t hat models be built such

tha t these techniques can also be esploited properly. The CSP model is attractive for this

purpose since new models can be integrated easier than in a SIIP. \Ve s u g e s t this rnodel

for integrating the simulation and the schedder for the FHSP.

1.3 Overview of the Thesis

The rest of the thesis is organized as follows. Chapter 2 describes a formal study on the

structure of the CCSP to gain a deeper understanding of the problem. Good cut constraints

(defined in the nest section) are needed to solve a SIIP model for integer solutions effectively.

L i é identif- a constraint formulation in the CCSP and explain why it is usefui during the SIIP

solution process. Studying the adjacency constraint in the CCSP reveals that the problem

is very loosely const rained. It rneans t hat preprocessing the problem using consistency

techniques f 'ilackworth 1977) is not worthwhile. Consistency enforcement is a technique

commody used wide solving the CSPs. Furthermore. it irnplies that satisfying just the

adjacency constraint in CCSP is easy. This does not necessarily Say anything about the

hardness of the rvhole problern when the objective criteria are aiso included. In Chapter 3.

we give a UIP model for the CCSP as described in Section 1.1. The adjacency constraints

are forrnulated in such a way that they behave as the cut constraints. Mé also outline some

empirical evidence showing t hat small-sized problems wit h some constraint reIasat ions can

be solved using the model. This elcercise aiso shows that solving practical problems using

currently available MIP solvers is unrealistic.

Since it is unlikely to solve the fuii FHSP using just the SIIP techniques, we give a

model based on the CSP model. Në also integrate the growth and treat nient sirriulation in

this model. Based upon the research to date. our reason for doing this is twofold. Firstly.

SfIP can give solutions but the solvable size of the problems is very smaU compared to the

practical problems that esist. Secondly. incorporating simulation results into a LIIP woufd

be difficult and the problem decomposition techniques (for esample. the column generation)

have to be figured out first. This type of problem is most likely solved iising some sort o l

iterative improvement technique together with a wide r-ariety of amilable heuristics. For

these techniques to work weii. it is believed that the initiai solution has to be reasonably

-good". LVe t herefore present a met hod of generating an initiai solution that uses the known

minconflicts heuristic (Minton et al. 1992). Our measure for goodness is the number of

consistent variables in the initial solution. Our esperimentai results show t hat t his method

can be used t o generate consistent solutions.

FinaUy. we give concluding remarks and some future research directions in Chapter 5 .

Chapter 2

Formalization of the FHSP

The Forest Harvest Scheduling Problem ( F H S P ) is a comples problem involring spatial and

non-spatial constraints and (conflicting) objective criteria (cf. Chapter 1.1). It is a n esample

of a hard combinatorial optirnization problem. T h e search space for these problerns is huge.

To iUust rate. let us take an esample of t he CCSP ( o n l - considering the adjacency const raint )

a i t h 500 stands and 20 periods. Then the total nurnber of schedules. both feasible and

infeasible. is -LO"*'. which is more than the number of atoms in the universe! Furt hermore.

the optimization criteria add complesities to the CCSP. Such cornples probIenis can be

solved effectively only if problem-specific heuristics are developed that searche only part

of the search space. h o t h e r approach is to reduce the size of the search space itseIf by

eiiminating those parts which do not contain any feasible (or optimal) solution.

The probleni solviiig techniques we are going to study are niathematical programming.

NIP in particular. and constraint programming using a CSP model. To solve a W P mode1

for an integral solution. we need an efficient constraint formulation and we have t o identify

good cut constraints (defined shortly). Efficient constraint formulations help reduce the

number of integer variables in the model and the cut constraints help quicken the hIIP

solution process by converging to the integrai solution faster. We study the CCSP structure

to identify these formulations. -4 constraint solving technique is more effective if the search

space is reduced t o a manageable size. However. the search space is huge even for the CCSP

which is already a restricted version of the FHSP. Nevertheless. it is usualiy the case that

a large area of search space can be thrown away by enforcing consistency techniques. for

example arc-conçistency. We study the problem structure to see if such techniques can be

used effectively t o solve the CCSP (and t h e FHSP).

ne first study the CCSP problem structure to identify t h e efficient constra.int and cut

constraint formulations for a SIIP model. Then ive study const raint structure in the CC'SP

to see if any of t h e consistency enforcement techniques can be applied to speed-up t h e

s o h t ion process.

2.1 Ident-g Cuts

Ln this section ive will study the re la~ed version of the FHSP problem (CCSP). main-

its structure and t h e constraints. We study the node-packing problem and show that the

CCSP consists of many such problems as sub-problems. We st udy the adjacency const raint

structure in particular. Firs t . some definitions are required.

Definition 6 (Set-packing Problem (Nemhouser & Wosley 1988)) Any probkm that

can be defined as linear system of inequalitia Ax 5 b. such that A is a ( O . 1) matrir. b i.5

an unit wctor and x k binary.

Definition 7 (Node-packing Problem) Gicen a gmph C = (1: E ) . r h c ~ L' is the set

of rertices and E is the set of edges. find l - C Le. and Il-1 2 2. such that no pair of nodes

in I ; is j o i n ~ d by an ~ d g e .

Definition 8 (Branch-and-Bound) It is the most popular i n t ~ g e r progmmming algo-

rithm. The original problem is dicided into s m a l k r and s m a l k r sub-pro6knc.s. The di-

d i n g (branching) is the process of partitioning the en t im set 01 solutions into crnalkr and

srnalier subsets. Then. the best solution in each subsrt is bounded and the sub.c~t discnrded

i j the bound indicates that it can not contain optimal solution for the original p m 6 i ~ m . The

discarding process is also called fat homing.

Definition 9 (Cutting plane (Cut)) -4 cutting plane (cut) Ior on integer program i.5 a

neu7 c o n s h i n t that eliminntes some jeasible solution of the original L P relaration whik keep-

ing ece y integral feasible solution intact. Cuts tighten the bound obtained in the bounding

step of the branch-and-bound technique and improces its performance.

Some researchers have shown that a binary integer (0-1) model for the node-packing

problem has better chance of converging into a n integer solution in practice (for esample.

(Gebotys S; E h a s r y 1993. Hoffman S: Padberg 1991)). Gebotys and Elmas- app. a

Figure 2.1: ( S 1. S2) participating in a neighborhood constraint

node-packing problem representat ion for architect ural synt hesis pro blems. ;\rchitectural

synt hesis is an important part of the VLSI design cycle. The objective is to transform the

input algorit hm into a hardware architecture t hat minimizes a cost lunction while satisfying

a set of constraints. They mode1 this problem as a 0-1 integer programming problem and

solve the previously unsolved instances. Their success cari be attributed to the constraint

formulations which act as good cuts during the branch-and-bound algorithm for integer

solutions.

LVe now niodel the CCS P (cl. definition iri Cliapter 1.1 ) as a 0- 1 integer program. We

start with a binary (0.1) variable x, for each stand Si and each period p,. where i = 1 . . . n

and j = 1 . . . m. That is. if xi, = 1. then stand Si is harvested in period j.l Sorv our

problem is t o assign integer values to d xi, such that the CCSP constraints are satisfied.

We can represent this problem by a system of inequalities Ax 5 b. where the r o m of

A correspond to the coefficients (values) [xL 1 . . .q,. 2 2 1 . . . xt, . . . c,l. . .mm]. where x is

binary and b is a unit vector,

Thus. by definition t his is a set-packing problem. Set packing problems can be transformed

into a node-packing graph G(V,E) (Nemhouser & Wosley 1988). Therefore. the CCSP can

' I t is standard practice in ïorestry iiterature (cf. Chapter 1)

CHAPTER 2. FORMALIZ,-1TIOY OF T H E FHSP

be t ransformed into a node-pacliing graph G( L E ) .

In Figure 2.1. tivo stands (Si. S1) are neighbors and it is assumed that the tirne for

trees t o grou- to the minimum height of say 2 meters. MH. is 2 periods and the lenpth of

scheduling horizon is 7 periods. Then. we irnmediately see that

That is. a stand is harvested oniy once ' in the scheduling horizon. The soüd iine ellipsoids

in Figure 2. I group the variables participating in these const raints. Anot her const raint of

this E n d that becomes apparent is the minimum heiqht constraint. For esample. in Figure

-2.1 if stand SI is harvested in period 4. then rl4 = 1. which means that stand S2 can not

have been harvested 2 periods before and can not be harvested 2 periods after the period

1. We can represent t his in general by the equation:

l i t + C ; 1. V neighbors (S i .S j ) . (t-.UH) moci m S n < _ ( t + M H ) mod m

We can g raph icdy view this equation in Figure 2. i as dotted cones. This equation is

syrnrnetric for the stands Si and S2 since rve are assuming on- one treatment per stand

in the CCÇP. it can not be the case otherwise. This is a siibgraph of the node-packing

graph. For esample. in Figure 2.1. I.- = { r i j } Vi E {l.. . .. n) and V j E {l. - . .. r n } and

E = {(Si. S,) [Si and .y, can not be sirnultaneously 1). Then ive can choose ui E Si and

u2 E S2 where u l . u;? E I V . and that m-ould be our solution. Yote that it is trivial to solve

node packing in this case. just choose r l j from Si and choose the node q k from Sz such

that Ij - CI > .\If?. Hoivever. it may not be so simple if Si or S2 is also adjacent to some

other stand.

2.1.1 0-1 Node-Packing Problern

We noted eariier that branch-and-bound is a technique for solving general integer program-

ming problems. It is an efficient technique in the sense that the aigorithm identifies and

ignores the feasible solutions which can not be the optimal one. Xevertheless. it is still re-

stricted to problems with a smail number of integer variables due t o the very large number of

the feasible solutions which may exist. Generai 0- I integer program is SP-complete (Sem-

houser & LVosley 1988). Hoivever, in sorne special cases. polyhedral theory can be applied

*1t is an assurnption in the definition of the CCSP

to solve t hem more efficientl- .ln>- bouncied system of iinear ineqiialities can be visualized

as a polytope such that the solution(s) can onlx be the vertices of the polytope. An integral

polytope is a polytope where a l l the vertices that are the solutions are also integral. It ha5

been proven that for every polytope there esists an integer polytope that can be solved as

a Linear program and always produce an integer solution (Semhouser SC Lvosley 1988). ;UI

integrd j'acets of the poiytope are necessary to be formulated as the cons t raints to identify

its corresponding integer polytope. where a lacet is a hyper-plane of dimension one less than

that of the polyhedron.

There exïst problems for which all such integal facets are characterized. for esample.

matching and total unir no du la rit^ (Semhouser Sr Wosley 1988). But for the node-packing

problem. it is only p a r t i d y characterized. Finding all integal facets for the node-packhg

problem is XP-complete (Gebotys S- EImasry 1993). It is stiiI worthwhile finding some of

t hese facets because t hey make the mode1 tight . t hat is. reduce the size of t he soliition space.

and give good bounds for the problem. This in turn is expected to make a branch-and-bound

algorit hm more effective.

In graph-theoretic terms. a set C 2 I.- is cailed a clique if every pair of nodes in C have

an edge. In other words. C' is a complete gaph. Therefore. in a node-packing. only one node

from a clique is ailowed in the solution. Ziote t hat the ellipsoids in Figure 2.1 are ciiques.

A clique is mnn'mal if it can not be contained in any ot her cliques. Each maximal clique

in a node-packing problem can be Forniulated as a clique constraint or clique inequality.

The clique const raint is a known integral facet of the node-packing problem ( Nemhouser k

CVosley 1988). Therofore. it i s in oiir interest to find atl t h e maximal cliques in the problcm.

These inequaiities aiIl not define all the integal facets. However. it is Likely t hat t hey wili

rediice the number of branch and bounds later while solving for optimal integer solution.

Lemma 1 The cliques repmsentrd by the ellipsoids in Figure 2.1 are marimal nssurning

.bf H 2 sizeo f theschedulinghorilon. Let n uindocc for xij of stand S,, . Cc. &E the set of

nodes in S, that can not be simultaneously one uith ri,. cchenerer Si and S, are neighbor.5.

The clique mpresented by the cone jor ewry cariable r , in Figure 2.1 is also maximal if

W;, is not contained in any other windoio for the caRa6L.s other than r,,.

ProoJ Xote that the cliques represented by the bigger eilipsoids are maximal. The ones

represented by the dotted cooes are maximal because ail of t hem are unique. This is because

each x, together with the r j k where t is a window. Hi',. of interval [ ( t - M H ) mod m. ( t +

Figure 2 2 : Xodes i .j and

M H ) mod ml forme a clique. [f this is not contained in any other window for variables

ot her than xi,. t hen it is unique and hence the clique is maximal. rn Sote that t his ma- not be the case when we consider the complete problem. There m a -

exist a nurnber of st mct ures similar to Figure 2.1 as in Figure 2.2. where the nodes i. j and

k are part of separate cliques. We c m c d t the grap h on the Ieft of the figure a n nd~acency

gmph. where the nodes are stands and edges represent the adjacency constraint. Let ri. c,

and ~ - r , be one of the binary variables for the nodes i .j and k in respective order. Then

ci + ( P, + YB;) 5 1: ci # Q: Q + ( c, + -\[Hi ) 5 1. Thus. what used to be a masimal clique

for any two nodes does not remain so any longer because it can be estended. For example.

the clique represented as the cone for nodes i and j can be extended wit h another node c k .

which also shares the same variables from the node j to form a clique.

To use clique constraint as integral facet. ail the maximal cliques in the original adjacency

graph has to be determined. Since the graph is planar. size of the masimal cliques is

bounded by 4. .U1 these cliques can be found by loolcing a t aii possible combination of three

and four vertices in the g a p h . Thus. the complexity is of 0(n4). where n is the nurnber

?iot to be confuseci wit h the maximai cliques of b i n q variables

CH-4 PTER 2. FOR,Cl-4LIZ-4TIO.V OF T H E FHSP

of vertices in C. hnother known integral facet for node-packing is CiEc xi 5 q. for

al1 chordless cycles C. ICI 3 5 . Finding these facets for large problems can lead to integral

solutions faster.

Lemma 2 For every pair of neighbors (Si.Sj). it is suficient to incltrde in the J I I P mode1

the clique ineqriahlies defining the ndjacency constraint frosm only one etand.

Proofs It is easy to see that the constraints are symmetric and t herefore, one set of

constraints originat ing from Si also essentially captures the set of const raints originatinp

from S,. rn rsing Lemma 1 and 2. we expect that the chances of finding integer solutions to our

mode1 with reasonable nurnber of constraints are higher. W e describe our MIP mode1 in

detaii in Chapter 3 ,

2.2 Consist ency Enforcement

The FHSP can also be represented as a constraint network (defined below ). There are vari-

ous dgorithms t hat eliminate parts of the search space for such netvorks by t hrowing away

inconsistent solutions. One such aigorit hm is arc-consistency. However. t hese algorit hms

are ~vasteful if the problem (constraint network) is loosely constrained. Therefore. we want

to find out how -loosen our problem is. To fo rmdy study this. we adopt van Beek's concept

of constraint looseness which gives the lorver bound on the inherent level of local consistency

i n a hinary constraint network (Van Beek 199.1). For g a p h coloring problems. these bounds

are tigh t .

Definit ion 10 (Binary Const raint Network (Montanari 1974)) .A binanj const raint

network consists of a set S of n cariables {x l . X*. . . . . x, ). a domain set Di oJ possible

uufues for mch cariable. and a set of binary relations between cariables. -4 binaq relation

or b i n n y constraint. Rij betiïeen cariables xi and x, is n subset of cartesian product of

their do.mains. -4 n instantiation of üa Rables in S is a n-t uple {Si. -Y2. . . . . Sn} such ihat

,Xi E Di is assigned to x i . -4 consistent instantiation of a network. afso calied solution. is

an instantiation of üariabies such that aU the constmints between cnriables are satisfied.

Definition 11 (Strong Bconsistency (Freuder 1978, Freuder 1982)) -4 network is

L-consistent ifl givert any k - 1 consistent instantiations of cariables there exists another

consistent instantiation for the kth cariable s i ~ h that al1 the P instnntiation.5 are consistent

with one nnother. iJk = 2 or k = 3. the n e t ~ o r k is cnlled arc-consistent or pat h-consistent

wspecticely. -4 ncticork is cnlled stronply k-consistent i f and only i f it is ;-consistent for d l

j 5 k.

Definition 12 ( m-looseness (Van Beek 1994)) .4 binnnj constraint i.5 m-loose i f e cery

rou: and e c e y column O/ the (O. 1)-matni that dejînes the constrnint has nt 1ea.rt m ones.

uhere O 5 m 5 (DI - 1. -4 binary constrnint neticork is rn-lms~ if d l it.5 binnnj constraints

are rn-loose. The (O. Ij-mntrix repres~ntation of the con.strninl between ran'nbles ri and x,

is gicen by

L i f n + b ~ l a - b I + (i- j ( Rij.ub =

O othemise

Theorem 1 ((Van Beek 1994)) If a binary corwtruint nettcork is m-loose and al1 do-

mains are of ai32 1 DI or hss . thrn the network is slmngiy ([&])-consistent.

The above definitions and the theorem can be understood in terms of a graph coloring

example. say b-coloring problem. Then. the problem is k-consistent. lt is d s o k - 1 loose

because every binary constraint between two variables disallows the assignment of the same k colors for both. Therefore. the probiem is --consistent. that is. k-consistent. Above

Xote that for any row. the maximum number of zeroes is 2 t M H - 1) + 1. that is. in any

row i. the columns that are ( M H - 1) to the Left and the right and the ith column itseii can

be zero. Therefore. the minimum number of ones is bounded by [ Di] - 2( .CI H - 1 ) f 1. The

matriv is diagonaily symmetric. Thus. same number is aiso the lowver bound for the columns.

Tt is usually the case that in a practicai FHSP ( D i ] is much greater tlian -UR. For example.

definitions are generalizations of thiç observation.

Representing node-packing constraint from Figure 2.1 in ( O . 1 ) - matrix form gives the following mat ris.

R i j =

O O l l l l l

0 0 0 1 1 1 1

L O ( i 0 t l L

1 I O 0 O 1 1

l l l O 0 0 1

1 1 1 1 0 0 0

1 1 1 1 1 0 0

lDil = 20 and -11 H = 5. Thus. this network is m - loose. where m = 1 Di[ - 2( Mfl - 1 ) + 1

o r 13. Then. according t o Theorem 1. t he network is strongly 3-consistent. Thus. it is

usuaiiy the case t hat sucfi networks a re aiready pat h-consistent. --lrc-consistency a n d pat h-

consistency algorithms have tirne complexïty of C3(ed2) and L3(eQ3) respectively. where F

is the number of variables and d is the size of the largest domain. Therefore. mnning

consistency enforcement algorithrns rnay be wasteful for many instances of FHSP if only

const raint satisfaction is sufficien t.

2.2.1 Row Convexity

A globally consistent network yields a backtrack-free solution. that is. a solution can be

found in n s teps where n is the number of variables in the network. -4s seen eariier. the

constraints in t he problem are fairiy loose and usuaily path-consistent. Such networks can

be identified t o be globally consistent o r minimal using the fotlowing theorem.

Definition 13 (Row Convex (Van Beek & Dechter 1995)) .4 b i n a q relation R,, rep-

rrsented as a ( O . 1)-nantrlr is row concer if and only if in ~ a c h row al1 of the ones are

consecutire: that is no t u o ones in a r0.w are separateri by a zero.

Theorem 2 ((Van Beek & Dechter 1995)) there eb.sts an ordering of the domains

Dr. - - . . D, of a path-consistent network .\- such that al1 the relutions are roc concex. then

\; is minimal or globally consistent.

Theorern 3 ((Booth & Lueker 1976)) =In m x n (0.1)-matriz specified by its f-nonzero

entriEs c m 6.e tested for ~ h d h e r a permutation of the columns exist such that the matriz is

m2ü conmx i n O(m + n f J ) steps.

Furthermore. row convesity property can be checked in linear time as Theorem 3 sug-

gests. Even though ou r network is path-consistent. t he binary relations are not row-convex.

This can be seen bu analyzing the m a t r k representation of R i j . Some rows in t h e middle

of the rnatn'r d o not have the consecutive ones property unlike some rows at the top and

a t t he bottom. Shifting the columns with non-consecutive ones so as t o put ail t h e ones

consecutively. will ahvays resuit in a t Least one o ther row containing non-consecutive ones.

2.3 Conclusion

We showed tha t a sub-problem of the CCSP can be viewed as a node-packing problem. The

advantage of rnodelling the CCSP as a node-pacliing is the use of clique constraints wliich

act as integral facets in the node-packing polytope. Finding integral facets for a partictilar

integer prograrnming pro blem has a lwa-s been a n in terest ing research issue. W e believe

that the clique constraints ailow iis t o mode! t h e FHSP as a LIIP using O- L integer variables

and provide a good reason as to why t his Formulation of the adjacency constraint is bet te r

in practice.

We also showed tha t the constraint s t ruc ture in the sub-problems of the CCSP is loose.

T hat is. rnost practical consistency e n forcement algori t hms are not use fui iv hile solving the

CCSP as a constra.int network. Problern specific heuristic or iterative improvement may be

more effective wit hout t h e consistency enforcement.

Chapter 3

Solving the CCSP using MIP Models

The Forest Harvest Scheduling Problem ( FHSP) and forest planning problems in generd are

combinatorial optimization problems. Briefi. the problem is to develop a forest manage-

ment plan for a large area of forest satis-ing multiple constraints and balancing competing

objectives. Traditionaiiy. mathematical programming has been and still is the most widely

used technique for solving such problems. For the general forest harvesting prob1em. there

esist a number of LP and hIIP rnodeis (cf. licerature review in Chapter 1). In this chapter

we study a restricted version of the the FHSP. caiied the Clear-C'ut Scheduling Problem

(CCSP) (defineci in Chapter 1.1) as a test case. CC> attempt to model and to solve the

CCSP using a MIP model. CVe wiU not model the problem as a LP since the literature

survey shows that LP-based strategies are difficult to interpret and may be impossible to

implement. largely due to their inability to express spatial relationships.

We wiii start with an example of a MIP mode1 for the CCSP From Johansen and Mis-

und's paper (dohansen k &und 1995). Their model comprises only the minimum height

constraint (CL). and the harvest intercd constraint (C2) as given in equation ( 1 ) and (21

(cf. Chapter 1.1). Their model is incomplete since it does not include any of the objective

criteria. We will briefly describe this approach and outline its drawbacks. -4s a next step

we present our h1IP mode1 formulation which deals wit h bot h constraints and objective

criteria. We test our mode1 using reai-life data from the Xorwegian ECOPLXN project

(Misund e t al. 199.5). Our intention is to test the hÿpothesis that 41IP models can only

be used effectivelj- for s m d instances of the CCSP. hny practical instance of the FHSP is

likely to be more complex and larger in orders of magnitude.

3.1 MIP Mode1 1

Johansen and Misund's report describes their attempt ro formulate the CCSP as a SHP

mode1 (Johansen k Misund 199.5). They define n (real) variables, h l . . . .. h,. where hi

corresponds to the harvesting period for stand Si. They jtistify t heir use of real variables for

harvesting periods by pointing out that if. for example. some hi is 10.3. t hen t his may give

an additional information about when in the period 10 the stand Si should be harvested.

The report describes only the minimum height and the harvest interid constraints.

The harvest interval const raint is included in the model sim ply by permit ting the values

of hi to be in the aiion-able harvest interval [E-4RLI ' (S i ) + L H ( S i ) . L-4TE(Si) + L H ( S i ) ] .

for ail i. The intervai may be relaved if needed. Yote that t his does not reff ect the definit ion

of the optimal harvesting time objective (cf. equation ( 3 ) in Chapter 1).

The minimum height constraint is rnodelled by introducing a binary integer variable for

each pair of neighboring stands (Si. S, ). The value of the binarq- variable is either O or 1

and is determined by the foilou-ing equation:

m-here m is the number of periods in the scheduling horizon. 3ow assuming that hi f h,

and .\IH be the time required for the stands CO grow up to the minimum height for the

harvest. the minimum height constraint can be represented by the foilowing equations:

The first equation is neglected if 6;; = O. assuming t h a t no stand takes more than the

length of the plan to grow up to the height of S meters. Lf' 6, = 1. then it states that

h; - .MH(S;) - hj must be greater than O. This reflects the I-meter constraint when

hi > h,. The ot her equatioa works similarl.

Johansen and 4Iisund's report (Johansen S; 4Iisund 1995) stops here as far as the MIP

model is concerned. It mentions t hat the aut hors were unabIe to model ot her constraints

and objective functions. Later in ( Misund e t al. 1995). the authors mention t hat t his model

is not suitable for the FHSP.

CKAPTER 3- SOLC'IXC: T H E CCSP I;SfXG JlIP JIODELS 28

tVe tried t o extend the above mode1 by adding the rest of t h e objective criteria. CVe

found it t o be a difficult process. Even t h e minimum height constraint formulation seems

awkward. Furt hermore. we t hink t ha t representing harvesting periods bu real variables may

make the solution hard t o implement a t t h e operational level. Therefore. ive present ano t her

MIP model which is capable of expressing the objective criteria in addition to the spat ial

relationship const raints.

3.2 MIP Mode1 II

We abandoned the real variables h,. . . . . h, and s ta r ted with a binary (0.1) variable si, for

each s tand Si. where i = 1. - .n a n d j = 1.. .m. where n is t h e number of stands and m is

the number of periods.

It may seem counter-intuitive t o replace some compact-looking constraints with a large

number of constraints with binary variables. But t he reason it works weU is t ha t t he CCSP.

in t his formulation. has node-packing as a subproblem (cf. C h a p t e r 2). Therefore. we can

espect t o benefit from t h e clique constraints t ha t a r e Likely to quicken the UIP solution

process. Furt hermore. as s t a t e d in Chap te r 2. t his formulation has proven to v+.ork weH in

practice.

Our JIIP mode1 consists of n s tands. {Si . . .Sn}. Each harvest period p for a s t and Si

is represented by a binary variable x ip- i ve assume t hat each s t a n d is harvested only once

t hroughout the scheduling horizon. Sow. ive formulate t he const raints and the objective

functions for t he CCSP as s t a t ed in t h e probIem definition (cf . Chap te r 1.1 1.

Constraints In addition t o constraints Ci and C3 we have an additional constraint such

tha t every s tand is harvested exact ly once during the scheduling horizon.

Ci: Minimum height constraint For aii t from 1.. .m.

t i t + C x j n 5 1. V neighbours (Si . S j ) . t-.CIH<n<t+JIH

C2: Harvest interval const raint : Sirnply d o not include the binary variables associated

with the undesirable periods for ail stands in the model-

C3: At least one harvest per plan 1, xim = 1. V i .

CHAPTER 3. SOLVTXG THE CCSP I'SIA-C; MIP .MO DELS

Objective Criteria -U escept t h e objective criterion U4 is formdated as foUotvs.

Oi: Optimal Harvesting Time The actual time beta-een harvests should be as close to

the optimal as possible. For ail i in 1 . . . n.

where j goes from L . . .m. ai 2 O. Here ai is penalty for not harvesting stand i in i ts

optimal harvest time. Let Po,, = Ci ai. Then. t h e optimal harvesting time objective

is achieved by minimizing P,, in the objective function.

O?: Even Harvest Volume The estimated harvested volume E H I,(S) for any period p

should be as close t o average hamested volume in a schedule S. :lHE*(S). In ot her

words. the variance between the harvest volunies s h o d d be minimized.

Let r , be estimated volume of forest obtained by cutt ing s tand Si in period j. Then.

C; = Er=, x, ch. V j is t o t d volume harvested in period j . Set constraints

Then objective is t o minimize C,,, - Cm;,. which effectively minimizes the vari-

ance between volumes harvested. making a n o v e r d schedule wit h even consumption

t hroughout the planning horizon.

Cl3: Old Forest -1 specified minimum area of old forest ( *-!OF). t hat is. stands \vit h aver-

age age above a certain threshold. should be maintained throughout the scheduling

horizon. Let O L D ( p . S) be the area of old forest in period p.

Let Oit = 1 if s tand i is not cut before it becomes old in period t. otherwise Oit = 0- That is.

Let be the area of old forest a t period t . which is xi Oit * rit. Vi. We want t o keep

Ot as close t o -\OF (desired a rea of old forest) as possible. That is. for aii t in 1 . . . m .

Then. the objective function wül rninirnize C, y t .

04:Visual Impact This is not put into the 4IIP model.

CH-APTER :3. SO L\?r\;C: T H E CCSP i31'iC -W MOD ELS

This t y e of model can be estended to handIe aiternative treat ments. For an additional t

treatments. t (nrn) - nrn variables WU be added to the model. The same order OC constraints k wiil aiso be added. of type xijkf Ck=- x;,k 5 1. which means that one treatment type inhibits

other treatments for a certain number of periods. The assurnption t hat only one treatment

is allowed per scheduling horizon can also be dropped.

3.3 Adjacency Const raints Revisited

As noted earlier in Chapter 1. a significant amount of current research in operational forest

planning is devoted to the formdation of adjacency constraints using integer variables in a

'LIIP rnodel. This constraint is the reason research shifted from LP to blIP i n forest harvest

planning and scheduiing. Larious formulations for t his const raint esist in the literat ure.

T h e most straight forward and weli-known method is known as the pairwise approach

whereby every pair of adjacent stands. (SI.&). share a constraint of type r l + 2- 5 1.

svhere si is 1 if stand Si is to be bar\-ested and O otherwise. This approach !vas conimonly

used untii Sleneghin el al. proposed a new type I adjacency constraints (Xeneghin et aL

1958). They proposed that o d y one constraint be shared between the mutuaLIy adjacent

stands. Thus. if 2 stands are m u t u d y adjacent. then they WU share a pairu-ise adjacency

constraint. But there is also a possibiiity of 3 o r 4 stands being mutually adjacent. Then

they wiLI shate a tripIet o r quatrupkt constraints. These constraints ahogether are called

the type t constraints. The authors also present a method t o identify a minimal set of these

constraints. First. all quatmplet constraints are generated. then the triplets that are not

contained in already identified quatruplet constraints are identified. Finaux. ali the pairwise

constraints that are not proper subsets of a quatruplet or a triplet constraints are identified.

There are other methods proposed in the Literature that identif- minimai set of adjacency

constraints. Esamples inchde Ordinaru Adjacency Mat rix ( OASI ) const raint set ( loshi-

moto S- Brodie 1994). and compartrn~ntal constmints ( Torres-Rojo S: Brodie 1900. Murray

Sr Church 1994) e t cétera.

Church and Slurray 95 review these formulations and give empiricai eealuations on 6

different cases (hiurray k Church 1995a). T h e authors conclude t hat the type 1 constraint

formulation is the winner over aii other formulations when solving these data sets in terrns of

computation time even though the number of constraints is higher than the others. Iiowever.

they do not give an expianation for this. We beiieve that the explanation Les in the use

CH-WTER 3. SOL1,'lNG THE CC'SP IÏS1,\I'C: Jf1P JfODELS

of the clique inequalities. The tvpe I formulation alioivs the problern to be vietved as a set

of node-packing su bproblems. R e c d the definit ion of a set-packing problem: Any problem

that can be defined a s a linear system of inequalities 4s 5 6. such that A is a (O. 1 ) matris,

b is a n unit vector and x is bina- Any set packing problem can be transiormed into a

node-packing graph ('iemhouser S- Wosley 1988). Csing the tvpe 1 formulation. -4 becornes

a (O. I )-mat rix. and b a unit vector (cf. 2). in other adjacency constraint formulations t his

is not the case. Therefore. t hese iormulations may not generate integral facets. This may

be tt-h~- the tvpe 1 constraint formulation came out as the winner in Church and Murray's

empirical evaluat ion ( Murray S- Church 1995a).

SIeneghin et aFs type 1 adjacency const raints can also be esplained simply by using the

gaph-t heoretic term clique. A U r n u t u d y adjacent stands form a clique when each stand is

treated as a node and adjacent relations as e d g s in a graph. Furthermore. such a graph

wiU be a planar graph since it is a representation of a geographical region. It is a known

resuit that a planar graph can not contain cliques of size .5 or more. Therefore. finding

the clique inequalities for masimal cliques of size 4. 3.and 2 will be the minimal set of

adjacency constraint formulation. Thus. what Meneghin el ni found Kas precisely the clique

inequali t ies.

3.4 Experiment al Results

First tve describe empiricai results for random data. with only the adjacency and harvest

intervd constraints (Cl and C2). using a public domain SIIP solver l p , ~ o l c ~ rersion 1 . 5

( I p ~ o l c e nad.). Xest. ive present the results obtWned from solving instances based on

the EC0PL.U prototype da ta (Misund e t ai. 1995). using a commercial .\.HP solver c p k r

rersion 4.0.4 (CPLEX n.d.) with a l l the constraints and the objective criteria as described

in our SIIP.

3.4.1 Initial experiments using fp-soire soiver

Forest data was randomly generated using data description as described in ( Misund e t al.

199.5). ' This d a t a is parsed and another input file is constructed for the 411P solver

Ipso1 ce. Only the adjacency and the harvest interval constraints (Ci and C2) were used

during the data generation. The objective value was set so that the optimal value is n for

'This data set can not be cornpareci with the data sets used with cpiex solver

CHAPTER 3. SOLL7XG THE CCSP C'SIXC: JlP .CIODELS

1 n 1 obi. d u e 1 remark 1 - 3

10 20 3 0 40 5 0

60 - 9.5 LOO

- 3

10 20 :3 0 40 .j O

X/A LOO

optimal optimal optimal optimal optimal optimal

mernory fad t op t irnal

Table 3.1: Resuits obtained for random data using the lpsolre JIIP solver

instances with n stands, Our intention \vas to test how hard these problems were when

considering only constraint satisfaction and if it was possible to solve the model with some

additionai objectives.

W e generated problem instances for a variable number of stands. but the planning hori-

zon was left constant at 100 periods with each period equivalent to 1 y a r . The nuniber of

stands were mried frcrn .5 to 100. The results using a Sun sparc.5 workstation is as shown

in Table 3.1. The solution time taried from 5 minutes to 2 hours. The results show that

for aimost hall the instances Ipso1 ce esits abnormaliy wit h a memory fault. The rest are

solved to optimalit- For our randomly generated data. the neighboring stands for any given

stand were low resulting in a loose adjacency constraint structure. This may esplain whu

instances with LOO stands were solc-ed to optirnaiity. h o t her reason is the domain size of

LOO periods. -1lthough some of the periods were disdowed by the harvest interval constraint

(C2). every stand has stiu between 7.5 to 100 doniain values t o pick from. In fact. when

the number of periods were reduced to 20. the solver was not very successfui. Furthermore.

memory faults caused by more t han half the instances iead us to choose a more powerful

511P solver cpler.

3.4.2 Results using cplex solver

The Xorwegian ECOPLAY project's prototype data was used t o test our 41IP model using

cpler Y IP solver. This data set is different t han the one used wit h lp ~ o l v e solver in t hat the

former has realistic adjacency relations and the number of periods for each stand. The area

of each stand was generated randordy to occur between the largest and srnaJiest area in the

actual data set since we were not able to get the data set a i t h the real area information.

.U the instances were run on a Sun sparcio workstation-

The ECOPL.45 prototype data for 494 stands was divided into sniaiier instances of n

stands where n = {2.i.50.100.200.300. -LOO. -494). Each of these instances cvas tested wit h 5 .

10. 1.5 and 20 periods ( rn ) as the scheduling horizon. Furt hermore. each of t hese instances

was used to generate .5 different sets a i t h different objective criteria. S 1.. . . . S.S. as input for

the c p k x eolver. Among t hese sets. S 1 contains data for const raint satisfaction oniy. T hat

is, oniy the adjacency constraint (Cl). the harvest i n t e n d constraint (C-). and one harvest

per scheduling horizon ( C 3 ) were present ( besides the integraiity const raints) wit hout any

of the objective criteria. The objective d u e tvas set to be n for a data set with n stands. ' AU other sets also contain these constraints but the objective criteria differ in each of

the sets. 5 2 contains objective 01 (optimal harvest time). S3 contains objective 0- (even

harvest volume). S4 contains objective O3 (old forest ). and 5-5 contains a l l three objective

criteria (01. O?. and 03)-

Table 3.2 shows the cples output for the ECOPLAX prototype data tvhen the green-up

age for adjacency constraint [vas set to 3 periods. wit h each period being equal to 5 pars.

The table show that it \vas difficult to satisfy most of t h e instances. T'here were no

integer feasible solution for di of the data sets with .i periods. Xote that if an integer

feasible solution does not e,uist for the set S 1. t hen no integer solution esists for anj- other

sets. Furthermore. if no integer feasible solution esists for a data set with n stands and m

periods. no integer feasible solution esists for any of the other data sets with m periods and

stands greater than a. Only the data sets with pairs (25. 10). (25. 20). and (50. 10) were

solved to optimality. where each pair represents ( n. m j.

The entries in the table marked with a "**- niay seem au-kward. for esample. why is

t here no integer solution for the instance wit h 25 stands and 1.5 periods when t here exists

an optimal solution for the instance ivith 25 stands and 10 periods? The reason is that

during the data generation. if the dowable harvest interval for a stand goes beond the

scheduling horizon. it is relaxed so that the stand mqv be dowed to be harvested during

the last .j periods. This wu done so that every stand is harvested once. Thus. in the case

of the instance with 10 periods. some stands got a larger harvest interval than in the case - - - - - - --

2 ~ r n o u n t s to cutting a l i stands once. 3This was the specification given to ECOPLAN project by the Xorwegian Forest Owner's Association. "Since al1 the adjacency relations are contained in the larger problem.

CH-WTER 3. SOLl;7YG T H E CCSP C-SIXC JUP JIODELS

- Set s 1 s 1 s 2 s:3 S4 SS

**S 1 S 1 S2 s:3 s-4 S5 S 1 S2 S3 s4 SS

**S 1 s 1 -

- read int sol

s -1 2.5 103

Ï'003.5 202 1 ï 7 2170.51

S A 2 -5 6

3 1.5564 181096 496684

50 -2.53

3 10699 476260 $-L3134

5 --1 x -1

time iter BB nodes rema no int

in feasi

no int in feasi

Table 3.2: Results obtained for Set L usinp r p k x MIP solver

n: number of stands. m: number of periods. read: problem tead time, C: number of constraints. V: number of variables.opt: is so lu t i~n optimal?. int sol: integer solution. time: cpu tirne ( in secs). iter: nurnber of Branch and Bound (BB) iterations. BB nodes: number of BB nodes tried

with 1.5 periods.

Careful analyses of the solver output for the unsatisfiable instances revealed that the

harvest interval constraint for some stands was too tight. that is. the stands could be

harvested in only 1 or 2 periods. This caused the sotver to conclude integer infeasibility

relatively quickly. We were interested in testing the SIIP solver for larger instances with a

nuniber of objective criteria. Therefore. ive relased the harvest intermi constraint so t hat a

stand can be harvested in any period during the scheduling horizon. This is not an unreaiistic

relaxation since the mi~iimization of the objective Cli essentiaUy has the equivalent effect

on the solution. =Uso. to increase the number of periods. we oniy generated da ta with 20

periods.

Set 2 t irne

2 II 3 7 .5:3 L30

2538 1200 7200 7200 1200 7200

nodes Set -4

t ime

Table 3.3: Results obtained for Set 2 and 4 using c p k x MIP solver

n: number of stands, C: nurnber of constraints. L': number of variables. 0,: optimal harvest time value (* opt), O=: old forest objective value (* opt) . time: cpu time (secs). CC: number of clique cuts applied. nodes: number of Branch and Bound nodes tried. iYT , inturrupted

Data instances tvith n = {15.100. L5O. 500.250.300.3.50.400.250. 4W} %vas generated for

aii 5 sets S 1 to SJ. Lvhile solving some instances it tvas observed t hat the solver \vas taking

more than 2 hours just to find an initial solution. These instances were reruu with the

solver's rounding heuristic turned on. The solver used a sophisticated rounding heuristic to

find the first integer solution. [n addition. the solver had also clique and cover cuts options

to improve performance of the branch and bound phase.

In Table 3.3 snider instances have been solved to optimaht. The objective value for

O1 denotes the total number of periods off the optimal harvesting period for a.il the stands.

For esample. if two stands were harvested 5 periods either before or after the optimal. and

the rest were schedded on the optimal harvesting time. then the value of Oi is 10. The

d u e of the objective O3 (in t housands of square kilometers) is the total area of forest not

above the old forest threshold for d the periods. The old forest threshold was set to 10% of

the total area for di the periods in our experiments. which cornes to LU0 square kilometers

(in t housands). The solver could not find optimal solutions for the set S2 starting lrom

n = 2.50. For the set S4. no integer solution was found within the time limit of 2 hours for

n starting from 150.

CHAPTER 3 . SOLVIN: THE CCSP LÏS1.YG J1P JlODELS 36

Set :3 0 3

L 14-8 36 1.1 S.522 :3 L 7.6 366

4-5-52 202.6 no int no int no int I?iT

Set 5 0 1 l 0 2

12.5 22 1 209 :].j.5 511

no int no int no int no int no int

Table 3.4: Results obtained for Set 3 and 5 using cpkr !dIP solver

n: number of stands. C: number of constraints. V: number of variables, Or: optimal harvest time value. O?: even consumption objective value. 03: old forest objective value ( * opt). CC: number o f clique cuts applied. oodes: number of Branch and Bound nodes tried. I X T interrupted

Table 3.4 shows results for the sets S3 and S.S. The value of the objective O2 (in

t housands of square kilometers) is the difference between the bigest and the srnailest harvest

volume during the scheduling horizon. ;UL the instances were stopped after 2 hours of

execution. that is. no optimal solution was Found. S o integer solution for the set S:l was

foiind for values of n starting from 300. For the set 55. this d u e rvas 200.

in a.ii the instances. the number of clique cuts generated was high. This means that

t urning t his option on during solving \vas probably a good idea since t hese cuts give tighter

bounds and help branch and bound algorithm to find an integer optimal solution faster.

After the elirnination of the harvest i n t e n d constraint. the solver was able to find the

optimal solution for all of the instances in SI. that is. the adjacency constraint was satisfied.

When the objective criteria were added. the solver required more time to find the optimal

o r an integer solution. From the results. it seems that the addition of the even harvest

volume objective makes the MIP much harder since both the sets S3 and 5.3 that contain

the objective were not solved to optimality even for s m d e r instances. It remains to be seen

h o a good these solutions are since currently ive have no means of comparing them. One

rneasure could be the optimal solution of the full problem wit h n = 454. However. running

CHAPTER 3. SO L'L'IXC: THE CCSP I'SING .NP JfOD ELS

the problem for 3 3 hours did not even produce a n integer feasible solution.

These results show that the solver is capable of finding integer solutions to the ECOPL-45

prototype data when the harvest intervai constra.int is relaved and no objective criteria is

in the model. ;Uso. an optimal solution was found for srnaller instances (usually 2.5 and -50)

even after the addition of all of the objective criteria. However. the rest of the instances

were eit her not solved to optimality or not even to an integer feasibility. This suggests that

for practicai problems. with orders of magnitude larger than those of our test problem. t his

method alone is not realistic.

3.5 Integrating LP wit h Simulation

As noted eariier in Chapter 1. the growth and treatment simulation are an important part

of the forest harvest scheduling procedure and our goal is to integrate the two processes.

in t lüs section 1.e will describe Hoen's work on integrating the simulation wit h a LP solver

( Hoen 1992 ) . The simulator used is the growt h and treat ment simdaror called C;Akkk-1 and

the integrated product is cded GAYALP.

if' Y is the total number of schedtdes and K the number of stands. Hoen formulates the

LP problem of mavimizing net present value ( X P V ) of the harvest schedule as follows:

subject to

w here

The objective function. Yet present d u e ( X P V ) is most cornmonly used.

3 x 1 (column) vector. Decision variables representing the number of hectares to be

treated by a management schedule (activity ).

1 x 'i (row) vector containing NPV value of schedule.

LI x 3 matrix produced by the simulator. It contains production coefficients. net

payment. and harvest volumes for each schedule.

51 x 1 (column) vector of resource constraints.

CH.4PTER :3. SOLC'TYG THE CCSP C*SI;t'G MIP JrODELS

A? K x ?i matris of area related coefficients. where element (y) is 1 if activity j belongs

to stand i. otherwise it is O.

bz K x 1 (column) vector consisting of the initial area of the stands.

The problematic part about the simulation is that it pr~duces a large number of treat-

ment schedules for the stands in a forest. often in the order of thousands. This resiilts in

a LP formulation with a large number of columns as compared to the rows. CVe can see

this in the above LP formulation as weii. where the number of columns in the LP rnatris is

the total number of schedules for aii of the stands and the number of rows are the resource

and the area constraints. Column generation as a problem deccmposition technique can be

used CO increase computational efficiency of this model (for detail see (Chvatal 1933)). The

solution of the LP probiem can be determined incrernentdy by looking at a subset of the

colurnn. n,. The problem with n, columns. LP,. is solved. The dual variables from t h e

solution of L Pa is used to e d u a t e an entering variable among .V - n, columns for the basis

of the prima1 LP, problem. The redoced cost for S - n, column is calculated and colunins

with positive costs are chosen as candidates for the entering variables. Then. non-basic

variables from optimd soliition of LP, are repIaced with the candidate variables and the

problem is resolved. This process continues until the optimal solution to the original LP

has been found or certain other criteria are met.

Aoen reported solving models with only a few hundred constraints using G.41-A-LP.

Thus. it has not proven to work at ail with a large number of stands and schedules as would

be the case in any practicai problem. Furthermore. the disadvantage of using this model

is that no constraints between the stands are represented. Therefore. the plans generated

by C.I\:A-LP may not be feasible spatiâlly. That is, taking x hectares with management

activity a . as suggested in the pian. ma? not be possible. Therefore. adjacency constraint

has to be satisfied in the solution. But. the addition of this constraint in the model maF

not be a trivial task. =Uso. it is not obvious liow to model the objective criteria iike even

consumption. old forest et cetera.

G-ktP-A-LP is an example of loose integration of the simulation with an optirnization

process. .U1 possible schedules are compiled in advance before running the LP solver. if

the LP solver can ask the simulator to generate a number of additional schedules as it

progresses. then we can achieve the true integration of a sirnulator and an optimizer in the

harvest scheduling process. ?io LP or W P scheduling system for forest harvesting to date

( t o Our knowledge) has achieved this level of integration.

3.6 Conclusion

From o u r esperirnental results we can conclude tha t ou r 4IIP mode1 can be used for the

CCSP and t h a t s m d instances are solved to opt imal i t - if only some objective criteria a re

tised. For larger instances the harvest intervai constraint was too tiglit and had thus to be

retaued. Then. satisfying the adjacency const raint was eas .

It was when ive started introducing some additional objectives. t hat the solving s tar ted

t o become tirne consuming. in particular. the even Bow objective seenied to be hard t o

optimize. T h e clique and cover cuts option in the solver ivas turned on while solving these

instances. A high number of clique cuts was geenerated for all the instances suggesting tighter

bounds for the branch and bound phase. However. t h e largest instance with ail the objective

criteria was not even solved for an integer feasible solution after 33 hours of esecution in a

Sun sparc 10 workstation. Furt hermore. cpf ea: is considered to be the state-of-t he-art among

the current MIP solvers. The 4IIP can be made more efficient if the problem decornposition

technique c d e d coliimn generation can be utilized. However. modeUing the coiist raints and

the objectives t o use t his technique is not a triviai task. Thus. our MIP mode1 d o n e is not

realistic for solving large practical FHS P.

Chapter 4

Solving the FHSP using a CSP

rnodel

The Forest Harvest Scheduling Problem ( FHSP ) is a cornplex combinatorial optimization

problem with multiple constraints and objectives. The most popular method to solve these

problems is based upon mat hernatical programming. However. as seen in Chapter 3. the

SIIP model for FHSP can be solved only for the srnail instances of the problem. Tsuaily.

the size of the rnodel for large problems is too big for current LISP solvers. After testing

the SIIP model. w e believe that unless problem specific heuristics are developed and used

along with the MIP solver. it is very difficult to find solutions to the FHSP that is readily

acceptable in a reasonable time. Since the r e d data is of an order of magnitude larger

t han the ones we have solved. t his method done is not sufficient. Hence. it is most likely

that t h e large FHSP wiil require some other technique. -4 suitable candidate is [terative

Improvement Technique ( [IT).

Iterat ive improvement met hods can be described in general as consisting of 3 algorit hrns

- cl. B and C (Papadimitriou et al. 1990. Johnson et al. 1988). The algorithm -4 finds

an initial solution. The algorithm B checks for the optimality of the solution and the

algorithm C finds a new feasible solution. The process is iterated on algorit hm B and C

until a terminal condition is met. The process can be interrupted any time and the most

recent - the best so Far - solution can be obtained.

For the o v e r d success of such algorithms in practice. it is believed tliat algorithm -4

has to give a -goodW initial solution. not just any feasible solution (Papadimitriou e t al.

CH-4PTER 4. SOLV13-C; THE FHSP C:SlXC; -4 C'SP JlODEL 4 1

1990). Since we are aiming at solving a large foresc harvesting problem. it is wort hwhile to

spend some time in generating a good initial solution. The measure for this goodness is the

number of consistent variables.

In this chapter. we outline a few approaches to generate a good initial solution for

the FHSP. Yext. we choose a method based upon Constraint Satisfaction Problem (CSP)

model. This initial solution can be used by an iterative improvement technique for further

optirnization. This met hod incorporates forest ry knowledge in the problem solution process

by using the growth and treatment simulation. In the process. we develop a C f + class

tibrary c d e d FHSPLib.

4.1 Met hods St udied for Generating Initial Schedule

Different approaches can be used to generate an initiai solution for the FIlSP depending

upon the mode1 used t o describe the FHSP. The rnost popiilar models are based on rnath-

ematicd progarnming where one formulates the problem as a LP or a. SIIP. Then. the

respectis-e soIvers look for the optimal solution of the model. An alternative mode1 is based

on the CSP which aliows one to use the constraint programming techniques. such as con-

structive or iterative search rnethods. to look for the solution. We give a brief description

of how these approaches m a - be used to generate an initiai solution for the FHSP. We as-

sume t hat the growt h and t reatment sirnulator ( for example. G--lk--A ( Hoen 1992) ) is to be

included in the solution process so that more realistic solutions can be obtained.

In Hoen's GAY-A-LP. a linear programming solver linked with the simulator G.A\i-.A. GAk--A

generates aU possible schedules for a l of the stands and then the LP solver selects one

eschedule for each stand. depending upon an economic objective XPV (Net Present Value)

of the forest. Hoen has integrated JLP (Lappi 1992). a specialized LP solver. with G.II-.l.

and this reportedy works better than G-IY-4-LP '. There are t hree approaches based on G.lkkkl and .JLP. We wiii say that G-Ak--A outputs

a tree for each stand. Each path in the tree is a schedule. The number of leaf nodes in the

'Sütonen's SIELA (Sütonen 1993) and Baskent and Jordan's GIS-based method (Baskent Sc Jordan 1991) are aiternatives to GAk'r\.

CHAPTER 4. SO LCTVG' T H E FHSP CTSiXG -4 CSP JI0 D E L

tree is the number of possible schedules. Figure 1.2 in Chapter L is a small example of the

t ree,

1. 4Iake singleton schedule for each stand. so t hat (J.Ak*A twil l only produce one pat h for

each stand. This could be treated as a simple initial solution. The path chosen bu the

simulation for each stand depends upon the o v e r d quality of the stand and various

ot her forest ry criteria-

2. According t o Hoen. C;.AY;\ can be rnodified such that only a few schedules are gen-

erated for each stand based upon the econornic criteria. If this is t h e case. then ive

should be able to generate any number of' schedules per stand as required. Observe

t hat this approach is not independent - it requires an additional LP solver.

3 . (3-11--4-JLP can be used to select the optimal schedule for each stand based upon the

econornic criteria-

The first approach above can be implemented as a function that takes as input an

individual stand data. The second approach seems Like a good idea. provided t hat we can

control the number of schedules generated per stand. Otherwise. there can be size problems

with C k K l . The number of pattis generated as possible schedules for each stand is quite

large. For IO time penods and J O difFerent treatments. it could be as large as LOO0 (per

stand). Only a s m d subset of the treatnient set is possible depending upon t h e current

state of a stand. of t hese t reatments finallu translate into Less t han 10 basic t reat ments

such as clear-cut. commercial thinning. and fertilization. In one case. GAY-1-JLP was used

for 6000 hectares of forest. approximately 14.000 - 15_000 stands. and 7 ten-year periods.

For t his problem. t here tvas a n average of about 25 schedules per stand generated by GAY-1-

of JLP (in the third approach) requires definition of an objective function. The

XPV seerns to be the most commonly used for this purpose. .Idditional da ta - the number

of trees per stand and the distance the trees have to be transported after being harvested

- is required to calculate the YPV. The ECOPLAX prototype data that we used for our

e'cperiments in Chapter 3 do not have t his information.

Summarizing. we need

1. GAY-\ formatted input da t a

2. Economic da ta to calculate the NPV for use with GAkX and with the LP solver part

C k e n this. a restricted (relaued) version t h e FHSP may be solved t o obtain the optimal

solution which could be used as an initiai solution for the full FHSP.

4.1.2 Mixed Integer Programming

T h e FHSP can be better modelled as a 51IP. But the problematic part of this approach

is t he size of the model. since the number of variables and constraints become prohibitive

even for a relatively small problem. Therefore. before using t his model. a decomposit ion of

t h e problem is required. This can be done by eit her partitioning the problem into smaller

subproblems o r using the UIP formulation t ha t aUows a column generation technique. To

o u r knowledge there is not any system that has integrated the simulation rnodel into a MIP

model.

If the problern is geographicaüy partitioned into subproblems. then the subproblems

can be solved independent- and the solutions combined. if it is the case t hat FHSP has

constraints tha t are fair- loose from one geographical area to another. it may be possible

t o combine the subproblem solutions into a global one without much effort. However. the

problems wit h t his approach a re t ha t i t is not obvious how the simulation results can be used

a n d tha t current 5IIP solvers can only handle s m d problem instances. in short. making

this approach work efficientIy is not a trivial task.

4.1.3 The CSP model and Constraint Programming

Let S = { x l . . . . . r,,} be the stands in a &-en forest. Di = {Pi.. . . . f i ) be the set of pa ths

obtained from a simulation for s tand xi. a n d RG C Di x LI;. Each path Pi = {a i.....n,).

where each a; is a node in the pa th consisting of a tuple < pi. 1; >. where t ; is a treatment

and pi the treatment period. We can use t h e adjacency constraint to construct C = {c,} .

where ci j is a set of allowable pairs of values for xi and x,. Then. by definition. t he triple

(X. D.C) is a CSP. where .Y is a set of variables. D is a set of domains and C is a set of

constraints. If w e draw a constraint graph for this CSP. each node in the graph represents

a s t and and each edge a n adjacency constraint between two nodes (see Figure 4.1). This

will give us a binary constraint network (cf. definition in Chapter 2) . Let us assume tha t

we want t o use the results of the simulation ' in this model.

*for exampie, GAYA

CH-4PTER 4. SOLVI...-G T H E FHSP I'SIXG -4 C5P JfODEL

adjacency constraint

(D31 /D51

Figure 4.1: Constraint graph of a forest in Figure 1.1

1. start instantiating the variables in some static ordering rl.. . . . x,. Then. r 1 - P, E

DI -

2. for i = 2 to n. x ; - d, E Di such that adjacency constraints are satisfied greedily

with .ri . -

3. if not ail consistent, use local repair heuristic. for example. mincon flicts (Srinton e t nl.

1992). o r tree search to generate a n initiai feasible solution

Tt is obviously not possible to represent a.ii of the possible paths from the simulation

as domains. A s m d e r subset of the set of paths have to be used for efficiencj-. ive can

then order the ~ar iables and start instantiating (step 1 from above). But to take care of

the adjacency constraint. each schedule s h o d d have the erclusion period information. that

is. the periods a t which the stand is below the minimum height. This can be obtained frorn

the input da ta direct1y using the age and growth information.

Instead of using a repair method in step 3. we couid use a tree search. However. we

believe that the problem has exponentiai search space and that therefore the tree search

may not be a good idea.

Sote that the ordering does not have t o be in terms of the stands but can be in the

periods. One can s tar t instantiatbn from the period I to the period m. The algorit hm moves

from period i t o period i + 1 when a local ohjecti\-e criterion. for esarnple. the area o r the

volume of the harvest- is fulfiiied. CVe can check during each period t o see if the assignments

satisf:? the adjacency constraints with the assignments from the previous periods.

. h o t her approach could be to randomly select the t reatments and the periads from the

simulation result. But the solution in most cases ma!* not be feasible since the simulation

result does not take into account any constraints between the stands including adjacency

constraints. However. the solution can be made feasible bu using an iterative repair heuristic

o r a constructive t ree search met hod.

Surnrnarizing- t o use a CSP model ive wiii need

L. a s m d number of paths generated for each stand by a simulator (item 2 from previous

section)

2. a suitabie constructive search or an iterative repair method

4-1.4 Conclusion

It seems that in order to use a SUP mode1 to generate an initial solution. larious other

prob1ems like decomposition. suitable constraint formulation et c e t ~ r n have to be considered

first. Linear programming model with the G-4\i\i\i-l simulator seems to be an easier choice.

Even GAYA alone could be used t o generate a simple initial solution if only one schedule

is picked for every stand. X CSP model is easier t o fornulate and a number of solution

strategies can be adopted from constraint prograrnming. The approaches that use simulation

results intuitively seem to be better since the growth and yield modeIs are then incorporated

in the solution process. W e give a method based on a CSP model in the nest section.

4.2 A method based on a CSP model

The FHÇP problem is modelled as a CSP and a weii known mincon flic& heuristic is used

t o i tera t ivel repair a greedily assigned initial ( tr ial) solution. We wiil describe our im-

plementation and empiricai ecëduation of this method in the following sections. -4s noted

eariier. we develop a Cf+ class library. FHSPLib. in ttiis process. The implementation uses

an es ternal Cf+ class Iibrary called LEDA 3.

Ln this section we wiii focus more on the input requirements and o n the outputs from

the FHSPLb L i b r a - -1 typical application while using t his Library would also have a d a t a

generator which prepares the d a t a for the probiem instantiation routine in the tibrary. The

data generator typically works o n the simulator results t o prepare a nuniber of schedules

for the stands.

T h e input da ta file is assumed t o have a particular format. which is the output format

of the data generator ( o r d a t a preprocessor). -1ppendis A has a detailed esplanation of the

format.

Some of the d a t a items ( for esample. t he nurnber of trees, t h e volume et cetem) are not

currently used in the initial solution generation process but they a r e t e p t there assuniing

t hat aIi the results [rom the s t a n d simulator \VU be used in the future.

4.2.1 Minconflicts Heuristic

It is often the case tha t while solving a CSP using the iterative techniques. a n initial trial

solution is generated randornly o r greedily. In niost cases this solution is infeasible as it

violates tarious const raines. O n e w a - t o rnake t his a feasible solution is iteratively repairing

it by changing the value assigned t o a variable. Which variable t o choose for the change is

a research issue and often depends upon the problem. One of the methods of choosing the

variable t o change is the minconflicts heuristic. This heuristic is very simple-niinded. uet

it has proven to be effective o n large scheduling problems. A n algori thm using it may be

descri bed as foiiows:

Algorithm 1

1. create initial trial solution

2. pick a variable. S. in conflict

:3. assign value ri E d o m n i n ( S ) t o X such tha t the conflicts created bu it is minimal

4. if no variables in confict exit

5 else goto step 2

3Library of efficient data structures and aigorithrns (http://www.mpi-sb.rnpg.de/LEDh/www/)

CH-4PTER 4. S0LVL.G THE FHSP I'SIJG -4 CSP JfODEL

Figure 4.2: Function used to calculate height

Step I of the aigorithm can be implernented either greedity o r randomly. We ha&-e chosen

a greedy nay in our irnplementation. This step d s o creates two sets of variables. LFars Done

and IearsLe/ t - with the former containing the consistent and the latter containing the

inconsistent variables. S tep 2 picks a variable - ive picli randomlv - from the set Vars Done.

Li step 3 . all the d u e s for the variable picked a re checked and the one t hat minimizes the

number of conflicts with other variables is assigned to it. Then. the two sets are updated.

Step 4 checks for the terminating condition. in our implementation the number of iterations

is aiso included as the terminating condition.

4.3 Experimental Results

We have a da ta generator that takes the foLlowing input parameters and produces stand

specific da ta suitable for the library routines. f he d a t a file used ( first argument ) was the

ECOPLAX prototype data (Misund et al. 1995). The source code for the da ta generator

and the test application are found in Appendis B and Appendix C respectivel.

For the height calculation. a simple Linear function as shown in Figure -42 is used. The

necessary points in the function are given as input parameters to the data generator. -1

more det ailed parameter informat ion is s hown below.

The executable gen-data takes 10 a r p e n t s ,

char* file = argv Ci] ; // data f i l e

int HIB-AGE = a t o i (argv C2J ) ; // for clearcnt

CH-4PTER 4 . SOLC'TXC T H E FHSP LS1h-G -4 CSP JIODEL

int MAX-AGE = ato i (argvC31); // for clearcut

int age-1 = ato i (argv[4] ) ; // for height c a ï c

int age-2 = ato i (arpvC51) ; // ... float height-1 = atof (argvC61); // ... float height-2 = atof (argvC71) ; // . . . Int nSch = atoi (argvL81) ; // nnmber of schedules (domain s ize)

int period = ato i (argv [9] ) ; // number of periods

int pLength = ato i (argvClOl) ; // period length

Example :

gen-data stands-db 50 80 20 110 18.0 22.0 1 10 5

4.3.1 Data Sets

Two basic data sets were used. both generated using the ECOPL-4S prototype data. How-

ever. these sets are not comparable to the sets used with cplex hIIP solver in Chapter 3 . The

latter donot have treatments other than the clear-cut and are restricted to one treatment

per stand. These restrictions were primarily placed due to the inefficiency of general integer

programming algorithms. The two data sets generated for this experiment with mincon-

flicts heuristic have multiple treatments and a stand may have rarious treatments during the

scheduling horizon. The difference between the parameters used to generate these two data

sets was the rdues of parameters MI.V-4GE and -if = L S - - G E . ahich determined the Iegal

intercal (in years} for the clear-cut treatment. For DataSetl . and Dataset2. the intervals

were [35.90] and [30. LOO] respectively. From each set. further test sets were generated using

the foliowing argument to the data generator.

castberg-stands-hp 35 90 55 150 10.0 22.0 $nSch$ $nPer$ 5 ,

where n P ~ r is set (10.1.5.20) and for nPer = 10. nPer = 1.5, and nPer = 20. respective

values for nSch were sets {.5.10). {5.10,1.5). and (5.10.20). Thus. a total of nine test sets

were generated from each data set.

Dataseteti was tested with 10.100. and 500 iterations. At this point. it was observed

that some of the consistent assignments were the schedules tbat had no treatment for al1

of the periods. This was because of the legd treatment intervai which prohibitcd some

C'H.4PTER 4. SOLI'LVC: THE FHSP CrSIYC -4 CSP .MODEL

Inconsistent Variables Vs. Iterations (5 scnedulesl

t , I O 500 la00 1500 2000 25ûû

Nurnber of iterations

Figure 4.9: Variables left inconsistent and iterations (each stand has 5 scheduIes)

stands from having t r e a t m e n t ~ . ~ Since this should not happen when using a simulator. the

data generator was modified such that aii the schedules generated had a t least one \-alid

t reatment. This {vas done by randomly picking one period in which to aiiow the treatment.

Dataset2 was generated in such a way and used for further esperiments. For clarity. ive

divided Dataset- into 3 sets. S e t l . Set-. and Set3 . with nPer = IO. nPer = 1-3. and

nPer = 20 with a respective nSch of sets (5.10). (5.10.1.5). and {5.10.20).

4.3.2 Results

Figure 4.3 shows the result of running our algorithm for instances in tvhich each stand has

.5 schedules in its domain. from each of the three sets. Figure 4.4 is a similar pIot but

with instances where each stand has 10 schedules in its domain. In Figure -1.5. results from

instances of Set;! and Set3 only are plotted. with each stand having 1.5 or 20 (onlx in S e t 3 )

schedules as domains.

The figures show that after some iterations the number of inconsistent variables starts

"For example. there are some stands with current age greater than 100.

Inconsistent Variables Vs. lterations (10 schedules) T 1 L 4 1 1 I I 1

O 50 100 150 200 250 300 350 400 450 500 Number of iterations

Figure 4.4: Variables left inconsist ent and iterat ions (each stand has LO schedules )

Figure 4.5: Variables left inconsistent and iterations (each stand has 15 o r 20 scheddes)

CHAPTER 4- SOLL73G THE FHSP LS1,VG -4 CSP JIODEL

to fluctuate wit hin a small interval. The position of t his interval depends upon the number

of schedules initially açailable to each stands. In Figure 4.3 and Figure 4.4. this interwl

is [:30.60] and [.5.10] respectively. In Figure 4.5. all the variables are consistent within 60

iterations. In fact. t his interial is the largest for instances wit h the least number of schedules

as domain. This is as expected - the bigger the size of the domain. greater the chances of

finding a consistent solution wit h less numbers of iterations.

Furt hermore. ail the instances in Figure 4.3 and Figure 4.4 were dowed to run t ill 10.000

iterations in hope of finding a complete consistent instantiation. However. the nurnber of

the inconsistent variables rernained wit hin a constant interval throughout the run. Eit her.

our algorithm was trapped in the local minima or there were no better solution to be found.

The results show that this method can be used to find the initiai solutions. [t also

suggests that for the ECOPLAX prototype data. if 10 or more schedules are generated

for each stand using a simulator. then the chances are that the method will converge to

a cornplete or almost complete consistent instantiation. This is not a ver- demanding

requirement since a stand treatment simulator Like GAYX ( Hoen 1992) can be modified easily

to output such schedules. However. it is still to be seen whether the scheddes generated by

such a simulator behave as the randomly generated ones in our experiment.

The repair algorithm took .5-7 minutes of processing time in a sparc- 10 SCY workstation

for each of the instances.

4.4 Furt her Improvements and Suggestions

There are many ways to improve the performance of the FHSPlib library. The minconf l ic t s

heuristic implementation requires a programmer to make certain choices while implement-

ing the trial solution generation process. the confiict set builder et cetem. The foilowing

approaches may improve the performance of the Library.

1. Right now the graph class from LEDA is minimdy used. However. it can be utilized

to order the variable set employing some other heuristic. The rational for not using

it now is t hat it creates an overhead t hat is not necessary in our test case. If the test

case is compiicated and large. then it may be beneficial to cluster the nodes of the

grap h according t O t heir degree, neighbors. topology ( for example. cliques ) . et cetera

to order the variables before starting the repair algorit hm.

2. The initial trial solution can be constructed randornly instead ol geedily if the nimber

of the solutions is Large. The greedy assignments add overhead that ma)- not be

necessary.

3. It might be tlseful t o introduce some optimization. sitch as the area harvested. the vol-

ume harvested. the net present value. et cetera. in addition to const raint satisfaction.

4-5 Conclusion

We implemented a library that provides users to represent the FHSP Like problems as a

CSP. We mostIy focussed on the initial schedule building mechanism for the FHSP. The

e'rperimental results showed that our test case can be easily solved if onIy the adjacency

constraint is considered. Other constraints are usiiaily formulated as objectives whicli can

be iteratively optimized. [t aIso showed that the number of schedutes available for each

stand directly effects the running time and the quaJitu of tlie initiai solution. The qiiality of

the initial solution also depends upon the input data provided. Thus. i f the stand simulator

is robust and can provide good varying schedules for each stand. t hen the initial solution

builder can also provide a good starting point for iteratively optimizing tlie solution.

Overail. tliis method has shown to be useful for our test da ta which is a real-life prototype

data from Xorwa?. However. more empiricd evaluations with Iarger d a t a sets and simulator-

generated schedules have to be done to fdiy evaluate the usefiilness of this approach.

Chapter 5

Conclusion

Forest harvest scheduling is a miilti-criteria optimizat ion problem t hat requires multi-disciplinar:

expertise. Most of the research to date has been in mat hematical programming based rnod-

els for solving the problem. W e studied the MIP and the CSP as both rnodeiling and

the solution strategies for this problem. Our study of the adjacency constraint structure

has shoan that some of the formulations of the constraint help the MIP solution process

by giving tight bounds to the problem. Furthermore. the adjacency constraint seemed t o

be -loose^. that is. consistency enforcement algorithms. Like arc-consistency. ma? not be

wort hwhile in t hese cases,

We gave a complete 4IIP model for a restricted version of the problem c d e d the CCSP-

We tested the rnodel with a red-life data from the ECOPLAS project in So rwa . The

results showed that the model is efficient for srnall instances of the CCSP and that the

optimal solution can be found ver- quicidy. Hoivever. we were not able to solve the full

problern to optirnaiity even after mnning the solver for 3-1 hours. This test data is very

s m d as compared to the practicai problems that esist. Furthermore. the g o w t h and

treat ment simulation results were hard to integrate into the MIP model. Therefore we gave

an alternative model based on CSP and showed how the simulation can be integrated into

it-

We implemented a met hod that provides the users with a way to represent the FHSP

Like problems in a CSP setting. in particular ive focussed on the algorithms for building

the initial solutions using the growth and treatment simulation results. Our test results

showed that the ECOPL-43 prototype problem can be easüy solved if oniy the adjacency

const rkints are considered. O t her const raints are usually formulated as objectives which can

be iteratively optimized. Tabu searcli and simulated annealing are good candidates for the

iterative improvernent technique. W e also showed that the number of schedules available

for each stand directly effects the running time and the quality of the initial solution. The

quality is aiso dependent upon the input data provided. The generated initial solution c m

be improved by fu& i n t ega t ing the stand simulator combined \vit h minconflicts heurist ics.

The FHSP problem has a ~i~gpificant spatial component. consider for example. the adja-

cency const raints. Recent advances in geograp hicd in format ion technology ( Ci IT ) sliould be

exploited to handle. analyze and visualize the data. This is particularly the case when eval-

uating the visual impact of a schedule. and in general while handling ecologicai. recreationai

and esthetic constraints/criteria. We believe t hat the CSP mode1 can be easil- integrated

with the GIT.

We did not solve the FHSP in this thesis but rather provided a good foiindation for

both m o d e h g and solving the problem using either the XIP or the CSP technique. Future

research would be to formulate the constraints and the objectives in the 5fIP mode1 so that

the column generation technique can be utilized. On the CSP area. the initial solution

builder should be extended to solve the complete problem. .Uso. both of t hese approaches

should be compared using the sarne data and penalty functions for aiI of the objectives.

CCé believe that this WU help advance the state-of-the-art in the forest harvest scheduling

pract ices.

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Appendix A

Data format for rninconflicts

//f irst l i ne

I f of stands, number of periods, period l e n e h

//rest of the f i l e

stand i d , age, volume, treeslf, t ree height, area, neigh. -1, # of schedules(n)

// schednle 1

treatment type, f U-st pexiod, -1, age, volume, # t r ees , t r ee height, area

. * . treatment type, l a s t period, -1, age, volume, # trees, t r e e height, area

// schedule 2

// schedule n

... stand i d , age, volume, trees#, t ree height, area, neigh. -1, # of schednles

Example :

The follouing is data f o r 1 stand, a i t h 1 schednle with 10 periods

of length 5 years each.

4 E L 4 . D,4T--i FORJI-4T FOR 31LVCO.VFLICTS

Appendix B

Data Generator

f l o a t height (int yr , int a i , i n t a2, f l o a t h l , f l o a t h2) ;

int dbLoad (istreamL, in t&, const in t , const i n t , const f loat , const f l o a t ) ;

// the approximate function t o calculate the height i s as follous :

// case a: y = (hi /ai ) * x

// case b: y = ((aî*hl - al*h2 + (=-hl)) / (étî-al)) * x

// case c : y = h2

// a l : agei a2: age2 , hl: heightl , h2: height2

/ /

f l o a t height ( i n t y r , int a l , int a2, float h l , f l o a t h2) //------------------------------------------------------------------

E if (yr < al) // case a

return ( (h l /a l ) * yr) ;

else if (yr <= a2) // case b

retnrn ( ((a2*hl - al*h2 + h2-hl) / (a2-al)) * yr); e l s e // case c

retnza (U);

//---------------------------------------------------------------

int dbLoad ( i s t r e a d in, intk age, const int age-1, const int age-2,

- const float height-1, const float height-2)

int id, Last Earvest h g , EarlyHarvest h g , Opt imalHarvest ing , Lat eHarvest ing ;

int rid;

float TimeForGroathToWeter, maxVol, Area;

in >> id;

cont << id << " ";

in >> LastHarvesting;

age = LastHarvesting * -1 ;

cout << age << " "; in >> TimeForGrouthToSHeter ;

in >> EarlyHarvesting;

in >> OptimalHarvestuig;

in >> LateHarvest h g ;

in >> maxVolYeax;

in >> maxvol;

cout C c O << " lu; // # o f trees

cout << maxVol << " ";

cout << height(age, age,l, age-2, height-1, height-2) << " "; in >> Area;

tout << Area << " "; in >> rid;

while (rid != -1) C tout << rid << " ";

in >> rid;

1 cout << "-1 ";

return (1);

//------------------------------------------------------------------

int main(int argc, char* argv )

//-----------------------------------------------------------------

E int nnm, i, j, k , ok, age, origAge;

ch- f i l e = argv [il ; int KIN-AGE = a t o i (argv C21) ; int MAX-AGE = a t o i ( a r g d 3 I ) ;

int age-1 = a t o i (argvC4I);

in t age-2 = a t o i (argvC51);

f loat height-1 = atof (argv CG] ) ;

float height-2 = atof (argvC7I);

int nSch = a t o i (asgvc81);

int period = a t o i (argvC91);

int pLength = a t o i (argvC101);

i f (argc C 10) {

return (O);

> i f stream in(f i l e ) ;

in >> n m ;

i f ( !in.good() ) (

re tu rn (-1);

> tout << nunt << " " << period CC "

for (i = 1; i <= nnm; i++) (

i f ( !in.good() ) < return (-1);

>

// da ta f i l e

// f o r c learcnt

// f o r c learcnt

// f o r height ca lc

// ... // -.. // . * . // number of schednles (domain s i z e )

// number of periods

// period length

" CC pLength CC endl;

ok = dbLoad(in, age, age-1, age-2, height-1. height-2);

i f (ok != 1) < return (-1) ;

1 cout << nSch << endl;

// noir the "nSch" schedules

origAge = age; int something = O ; int A C501;

for (j = 1; j <= nSch; j++) (

age = origAge;

something = 0;

f o r (k = 1; k C= period; k++) (

if ( (age > (MfB-AGE + (j-1) * plength) ) &&

Cage < (nBX-AGE + (j-1) * pLength)) ) C Ab] = 1;

something = 1;

age = 0;

> else .C

A&] = 0;

> age = age + plength;

>

if (something != 1) C // if no 1ega.l treatment fonnd A[(random() % 9) + 11 = 1;

1

age = origAge;

for (k = 1; k <= period; k++) {

if ( ACk] == 1) < coat << 1 << " " << k CC -1 ";

age = 0;

> e l s e E

cout << O << " " << k << " -1 ";

> //tout << "O O O O 0" << endl;

cout << "O O " ; coat << height(age, age-1, age-2, height-1, height-2) " "; cout << "O 0- << endl;

age = age + plmgth;

>

Appendix C

Test Application

bool read = true;

mcschednle myschedule;

string dir = "-/development/src/app/minconf";

read = myscheduîe-dbload ( d i r . maxstands);

myScheduie.solwe();

TEST TARGET (QA-3)

APPLIEO A IMAGE. lnc 1653 East Main Street - -. -, Roctieçter, NY 14609 USA -- == Phone: 71 61482-0300 -- -- - - Fm 7W288-5989

O 1993. Applied Image. [W.. All Righs R w r w d