spatial forest planning with integer programming lecture 10 (5/4/2015)
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Spatial Forest Planning with Integer Programming
Lecture 10 (5/4/2015)
Spatial Forest Planning
Spatial Forest Planning with
Roads
Balancing the Age-class Distribution
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Red OakRed MapleOther ZonesOther OaksOther Hrdwds.N. Hrdwds.ConifersA. Hrdwds.
Acr
es
Forest Age Class Distribution
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Red OakRed MapleOther ZonesOther OaksOther Hrdwds.N. Hrdwds.ConifersA. Hrdwds.
Acr
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Forest Age Class Distribution
Spatially-explicit Harvest Scheduling Models
• Set of management units• T planning periods• Decision: whether and when to harvest
management units– Modeled with 0-1 variables– xmt = 1 if unit m is harvested in period t, 0
otherwise
Spatially-explicit Harvest Scheduling Models (continued)
• Constraints– Logical: can only harvest a unit once, at most– Harvest volume, area and revenue flow control– Ending conditions
• Minimum average ending age• Extended rotations• Target ending inventories
– Maximum harvest area (green-up)– Others spatial concerns:
• Roads, mature patches, etc…
one constraint for each and for each t=hi,…,T
Integer Programming Model for Spatial Forest Planning
0 01
[ ]m
M T
m m m mt mtm t h
MaxZ A c x c x
0 1m
T
m mtt h
x x
0
ht
mt m mt tm M
v A x H
, 1 0l t t tb H H
, 1 0h t t tb H H
1jtj C
x C
0 01
[( ) ( ) ] 0m
M TT TT Tm m m mt mt
m t h
A Age Age x Age Age x
0,1mtx
C
subject to:
one constraint for each m=1,2,…,M
one constraint for each t=1,2,…,T
one constraint for each t=1,2,…,T-1
one constraint for each t=1,2,…,T-1
for each m=1,2,…,M and for each t=1,2,…,T
Notation
wherehm = the first period in which management unit m is old enough to be harvested,xmt = a binary variable whose value is 1 if management unit m is to be harvested
in period t for t = hm, ... T; when t = 0, the value of the binary variable is 1 if management unit m is not harvested at all during the planning horizon (i.e., xm0 represents the “do-nothing” alternative for management unit m),
M = the number of management units in the forest,T = the number of periods in the planning horizon, cmt = the net discounted net revenue per hectare plus the discounted expected
forest value at the end of the planning horizon if management unit m is harvested in period t,
Mht = the set of management units that are old enough to be harvested in period t,Am = the area of management unit m in hectares,vmt = the volume of sawtimber in m3/ha harvested from management unit m if it is
harvested in period t,Ht = the total volume of sawtimber in m3 harvested in period t,
andbl,t = a lower bound on decreases in the harvest level between periods t and t+1
(where, for example, bl,t = 1 would require non-declining harvests and bl,t = 0.9 would allow a decrease of up to 10%),
bh,t = an upper bound on increases in the harvest level between periods t and t+1 (where bh,t = 1 would allow no increase in the harvest level and bh,t = 1.1 would allow an increase of up to 10%),
C = the set of indexes corresponding to the management units in cover C, = the set of covers that arise from the problem,hi = the first period in which the youngest management unit in cover i is old
enough to be harvested, = the age of management unit m at the end of the planning horizon if it is
harvested in period t; and = the minimum average age of the forest at the end of the planning horizon.
Notation cont.
TmtAge
TAge
The Challenge of Solving Spatially-Explicit Forest Management Models
• Formulations involve many binary (0-1) decision variables– Feasible region is not convex, or even
continuous– In fact, it is a potentially immense set of
points in n-dimensional space• Solution times could increase more than
exponentially with problem size
1.) Unit Restriction Model (URM):adjacent units cannot be cut
simultaneously
2.) Area Restriction Model (ARM): adjacent units can be cut simultaneously as long as their combined area doesn’t exceed the maximum harvest opening size
Pair-wise Constraints for URM
Pair-wise adjacency constraint for AB is
1At B tx x
A, 14 B, 4 C, 10
F, 4
D, 8
E, 7
Adjacency list: AB
BC BD BE CD
AD AE
DF
What is a “better” formulation?
Maximal Clique Constraints for URM
Clique adjacency constraint for ABD is
1At B t Dtx x x
A, 14 B, 4 C, 10
F, 4
D, 8
E, 7
Maximal clique list:
BCD DF
ABD ABE
Cover Constraints for ARM
Cover constraint for FDC is:2C t D t F tx x x
A, 14 B, 4 C, 10
F, 4
D, 8
E, 7
Cover list: ABC
AE BCE BDEF CDF
AD BCD
1mtm C
x C
In general:
max 20A ha
McDill et al. 2002
GMU Constraints for ARM
A, 14 B, 4 C, 10
F, 4
D, 8
E, 7
GMU variable for BDF is: BDF tx
GMU list: A-F
BD BDE BDF BE
AB BC
CD CF
1 , ..., jt
nt jn K
x j J and t h T
The constraintin general form:
Where is the set of GMUs that contain at least one stand in max clique j; J is the set of max cliques; and hj is the first period in which the youngest stand in clique j is old enough to be cut.
max 20A ha
McDill et al. 2002 and Goycoolea et al. 2005
jtK
The “Bucket” Formulation of ARM
• Introduce a class of “clearcuts”: , and• Introduce variables that take the
value of 1 if management unit m is assigned to clearcut i in period t.
• Objective function:
Constantino et al. 2008
itmx
00
1
[ ]m
M Ti it
m m m mt mm i t h
MaxZ a c x c x
The “Bucket” Formulation of ARM (cont.)
• Constraints:
0,
1m
Titm
t t h i
x
for m = 1, 2, ..., M
max1
Mit
m mm
a x A
for i and t = hm, . . . T
it itm Qx w for , , Q m Q i m and t = hm, . . . T
1itQ
i
w
for Q and t = hm, . . . T
The “Bucket” Formulation of ARM (cont.)
• Constraints:
,
0ht
itmt m m t
m M i
v a x H
for t = 1, 2, . . . T
00
1
[( ) ( ) ] 0m
M TT TT i T itm m m mt m
i m t h
a Age Age x Age Age x
, 1
, 1
0
0
l t t t
h t t t
b H H
b H H
for t = 1, 2, . . . T-1
The “Bucket” Formulation of ARM (cont.)
• takes the value of one whenever a unit in maximal clique is assigned to clearcut i in period t;
• is a maximal clique in the set of maximal cliques.
itQw
Q
Q
Strengthening and Lifting Covers
13 14 43 50 3x x x x
24 38 1x x
18 31 40 2x x x
24 38 3
24 38 49
11
x x xx x x
18 31 40 6 15
18 31 40 8
22
x x x x xx x x x
13 14 43 50 32 3x x x x x
max 48A ha
Formalization
maxjj Ca A
1jj Cx C
max\ jj C la A
.l C
where C is a set of management units (nodes) that forma connected sub-graph of the underlying adjacency graph,and for which holds, but
for any
The minimal cover constraint has the general form of:
maximum harvest area
area of unit j
Strengthening the Minimal Covers
{ {0,1} : 1, C }.nii C
P x x C
( )N A
s( )( , ) max{ : and x 1}jj N s Cs C x x P
denote the set of all possible minimal covers thatarise from a certain forest planning problem (oradjacency graph).
Notation: Let
For every set of management units A, let the set of all management units adjacent, but not belonging, to A.
Define:
represent
Define:
Number of units in the forest
Proposition: Consider a minimal cover C ( ).s N C
* ( ) ( , ) 1 .N s C s C Define: *: Then, for all
1j sj C
x x C
is valid for P.
and
Strengthening the Minimal CoversCont.
s( )( , ) max{ : and x 1}jj N s Cs C x x P
Strengthening the Minimal CoversCont.
1C
\ ( ) ( )j s j j
j C j C N s j N s C
x x x x
( )
\ ( ) *jj N s C
C N s x
\ ( ) ( , ) *C N s s C
.x P 0sx
1sx
Proof: Consider If , then the inequality holds by the
If , then:
definition of minimal cover C.
* ( ) ( , ) 1N s C s C
How strong are these inequalities?
max 3A ha
332211ss
441ha1ha 1ha1ha1ha1ha
2ha2ha
1ha1ha
1 2 3 4 3x x x x
1 2 3 4 2 3sx x x x x 1 2 3 4 1 3sx x x x x
* ( ) ( , ) 1N s C s C
Sequential Lifting Algorithm
\{ }C
NO
YES
YES
Pick a cover, C from set .
Generate the set of management units, S that are adjacent to at least 2 units of C.
Pick a unit, s from set S.
* 1s ?
Determine *s using Proposition 1.
?
STOP
NO
YES
Generate the set of minimal covers, using the Path
Algorithm
For Cs Q , set * s s Solve ,A CP .
, 1?A CP C
NO
NO 0? i
Solve ,A CP .
, 1?A CP C YES NO
NO
YES
1. 12. 3. 1
i i
jt ij j
* ?C C
s ss Q s Q
YES
? Ci Q NO
YES
0? i
YES
\{ }
0
for an ?s
kk Q s
Cs Q
NO
* ?s s
YES
NO
S ?
NO
\{ }S S s
{ }C CQ Q s CQ ?
YES
1CQ
NO
YES
NO
Build E(C) YES
i=j=1
1. 2. 13. 1
j
i i
i t
j j
NO
i =i+1
? Ci Q NO
YES
YES
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[3]
[4]
[5] [7]
[6]
[8]
[9]
[23]
[10]
[11]
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CQ .
Build E(C)
Build E(C)
Build E(C)
[13] [14]
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[16] [17]
[18] [19]
[20]
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Strengthening and lifting
Compatibility testing
Open Questions:
1) Can one define a rule for multiple lifting?
2) Can one find even stronger inequalities?
3) Is there a better formulation?
A “Cutting Plane” Algorithm for the Cover Formulation
21 37 47 2X X X
8 18 40 2X X X
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