団地化での森林資源管理
吉本 敦
統計数理研究所

集約化：団地化
どの林分群を集約（団地化）するのか？

Base Unit時空間的最適化モデル
集約化された場所での施業時期を考慮
解法：集約候補の生成により対応

Base Unit傘伐管理
3 cut system

Windthrow

5つのストリップの連続群の配置

Windstorm Prevention Problem

Strip Shelterwood Management
2 cut system

Strip Shelterwood Management
3 cut system

Strip Shelterwood Management Scheme
Windthrowprogress from windward to leewardHyper Unit and Neighbors
NBi(0) : i-th unit itself, 0 degree adjacency
NBi(1) : neghbors of the 1st degree adjacency for the i-th unit
NBi(2) : neghbors of the 2nd degree adjacency for the i-th unit
NBi(n ) : neghbors of the n-th degree adjacency for the i-th unit
j- 1NBi( j ) : those units adjacent to {U NBi(k ) }M

k =1

HU i = U NB
k =1

(k )i

Strip Shelterwood Management Scheme
Windthrow

Removal Cut, Establishment Cut, Preparatory Cut

Developing Hyper Unit
• Adjacency for Units from windward to leeward
12

⎛ 0
⎜⎜
⎜⎜ 0
⎜⎜
A = ⎜⎜ 0⎜⎜⎜⎜ 0⎜⎝ 0310000010004001000 ⎞⎟⎟⎟0 ⎟⎟⎟0 ⎟⎟⎟⎟1 ⎟⎟⎟⎟0 ⎟⎠5Prevention of overlapping by adjacency
HU for 3 cut system
HUHUHUExample for Schelterwood Scheme
P e r iod

Cu t

1SRF

2345Str ip
6 7 8

112S
3R S
4F R S
25F R S
6F R S
7F R
38F
9S – seeding cut, R – removal cut, F – final cut

P e riod
1Cu t

1SF

2

12S
23F
4356
S – seeding cut, F – final cut

3SF

4SF

5SFSRF

9SRF

1 0 11 1 2 1 3SRFS
RFStrip
6 7 8SF SF SF

9SRFSRF1 0 11 1 2 1 3SFSFSRFSFSFSF

Elements of Hyper Unit
2 Cut System

5HU i = U NB(k )ik =13 Cut System

7HU i = U NB
k =1(k )i

Optimization Problem• Objective: max # of strips to be treated• Spatial Constraints– Avoid overlapping of HUs– Avoid the same treatment for adjacent units atthe same time (normal adjacency)

Windthrow

Forest Stands to Strips

Windthrow

5 Strips Connection
2 Cut System
4 strips uncut

Windthrow6 Strips Connection
5 strips uncut

Windthrow

7 Strips Connection
3 Cut System
11 strips uncut

Aggregation Problem with Spatial Adjacent
Constraints
We assume that one clique is created based on one unit
with certain buffer connection

Base Unit

Stepsa) Create candidate cliques (Hyper Units) for aggregation based onunits, and define decision variable matrix:

Original Unit Set: ui = {i}Hyper Unit Set : HUi = {indices combined for clique based on ui }b) Develop extended land accounting constraints for overlappingc) Develop adjacency constraints for HUs and original unitsd) Formulate spatially constrained problem with area restriction
under multiple harvests

a) Create candidate clique
for aggregation based on each unit
Hyper Units:HUi = {indices combined for clique based on ui }

Introduce decision variables for HUsXY⎧⎪
1⎪(m ×n) : x i, j = ⎨⎪
⎪⎩0⎧
⎪1
⎪
(m ×n) : yi, j = ⎨⎪⎪⎩0

if the j -th treatment is implemented for ui
otherwise
if the j -th treatment is implemented for HU iotherwise

Decision Variable Matrix for HU and u⎛X ⎞⎟⎜W = ⎜⎜ ⎟⎟⎜⎝Y ⎟⎠(2m ×n)b) Develop extended land accounting constraints
for overlapping
Ordinal land accounting = Overlapping of treatments

X1n ≤ 1m
Different treatments are overlapping for one unit

Different Treatments for HU and u are overlapping
over each unit

HU jHUi

Introduce Matrix for Overlapping
For overlapping between ui and u j
AOu (m ×m)⎧⎪⎪1 if ui ∩ u j ≠ ∅Ou: ai, j = ⎨⎪0 if ui ∩ u j = ∅⎪⎪⎩For overlapping between ui and HU jOhA(m ×m):aOhi, j⎧⎪⎪1 if ui ∩ HU j ≠ ∅=⎨⎪0 if ui ∩ HU j = ∅⎪⎪⎩Overlapping Matrix for HU and uOOuOhA = (A ,A ) : (m × 2m)

Extended Land Accounting Constraints
AO = (AOu ,AOh ) : (m × 2m)(AO ⊗1n′ )ivec(W ′)≤1m⎛ x⎞⎟⎜⎜ 1,1 ⎟ui
HUi ⎜⎜⎜  ⎟⎟⎟⎟⎟⎟⎜⎜⎟⎟ ⎛ ⎞⎜⎜ x⎛ 11 …
⎞⎟⎟⎜⎜ m,1 ⎟⎟⎟ ⎜⎜ 1 ⎟⎟0⎜⎜Oh⎟⎟⎜⎟⎟⎟⎜⎜ ⎜′A⊗1≤⎟n⎟⎜⎜⎟⎟⎜⎜⎜  ⎟⎟⎟ ⎜⎜⎜  ⎟⎟⎟⎟⎠⎜ y⎟⎜⎝ 0 11⎜⎜ 1,1 ⎟⎟⎟ ⎜⎝ 1 ⎠⎟⎟⎜⎜⎟⎟⎜⎜ ⎟⎟⎜⎜ y⎜⎝ m,n ⎟⎟⎠Spatial Adjacency for HU and uc) Develop Adjacency Constraints
for HU and uHyper Units:HUi = {indices combined for clique based on ui }+NBi = {index set of units adjacent to the i -th unit}

Spatial Adjacency for HU and u

⎛
⎞⎟⎜⎜
⎟⎟
⎜⎜
⎟
u to HU ⎟⎟⎜⎜ u to u⎟⎟S⎜⎟⎟
A = ⎜⎜⎟⎟⎜⎜⎜⎜ HU to u HU to HU ⎟⎟⎟⎟⎟
⎜⎜⎟⎟⎠
⎜⎝

(2m × 2m)

adjacency constraints for HU and u
Spatial Adjacency for HU and u⎛ A⎜⎜Su-u
A = ⎜⎜⎜⎝ AHU-u⎧⎪1⎪⎪u-HU
Au-HU : ai,j=⎨⎪⎪⎪⎩0⎧⎪1⎪⎪HU-HU
AHU-HU : ai,j=⎨⎪⎪⎪⎩0Au-HU ⎞⎟⎟⎟⎟ (2m ×2m)
AHU-HU ⎟⎟⎠′ )Au-HU (=AHU-u

if ui ∈  NBk and ui ∩ HU j = ∅k∈HU jotherwise
if HUi ∩  NBk ≠ ∅ and HUi ∩ HU j = ∅k∈HU jotherwiseAdjacency Constraints for HU & uSVA = A ⊗A+⎛X ⎞⎟W = ⎜⎜⎜ ⎟⎟⎜⎝Y ⎟⎠(2mn × 2mn)

(2m × n)⇓ + diag(A ⋅ 1 )] ⋅ vec(W ′) ≤ A ⋅1[A2mn2mn

Summarize:
Definition of Decision variable Matrix
Decision variable matrix⎛X ⎞⎟W = ⎜⎜⎜ ⎟⎟⎜⎝Y ⎟⎠X for a set of {ui }, Y for a set of {HUi }⎧⎪1 if the j -th treatment is implemented for ui⎪
x i, j = ⎨⎪⎪⎩0 otherwise⎧⎪
1 if the j -th treatment is implemented for HUi⎪yi, j = ⎨⎪⎪⎩0 otherwise

Definition of Coefficient Matrix
Coefficient matrix⎛C ⎞⎟C = ⎜⎜⎜ ⎟⎟⎜⎝C ⎟⎠(2m ×n)

ci, j : coefficient of x i, j , ci, j : coefficient of yi, jci, j = (1 + α) ∑ ck, j , α : reward ratek ∈HUi

Definition of Volume Matrix
Volume flow matrix at period p⎛ p⎜⎜ v1,1⎜⎜⎜⎜ ⎜ p⎛V ⎞⎟ ⎜⎜ v1,m
⎜ p
Vp = ⎜⎜ ⎟⎟⎟ = ⎜⎜⎜⎜⎜⎝V ⎟⎠ ⎜ v pp⎜⎜ 1,1⎜⎜⎜⎜ 
⎜⎜ v p⎜⎝ 1,m⎞⎟p… v1,n⎟⎟⎟  ⎟⎟⎟⎟⎟p vm,n ⎟⎟⎟⎟⎟⎟p⎟⎟… v1,n⎟⎟  ⎟⎟⎟⎟⎟p vm,n ⎟⎟⎠(2m ×n)

vi,jp : volume flow from x i.j , vi,jp : volume flow from yi.jv = ∑v
pi,jk∈HUipk,jd) Formulation with area restriction undermultiple harvests

Max PNVmn

Z = max tr(C ′W ) = ∑ ∑ (ci,j ⋅ x i,j + ci,j ⋅yi,j )X

i=1 j =1st.
(AO ⊗ 1n′ ) i vec(W ′) ≤ 1m Extended Land Accounting Constraints′ ) ≤ (1 + α)v , p = 1, 2,,T(1 − α)v ≤ tr(V W0p0

Harvest Flow Constraints

 + diag(A ⋅1 )]⋅ vec(W ′) ≤ A ⋅1[Amnmn = AS ⊗ AVAAdjacency Constraints