spatial forest planning with integer programming lecture 10 (5/4/2015)
DESCRIPTION
Spatial Forest Planning with RoadsTRANSCRIPT
![Page 1: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/1.jpg)
Spatial Forest Planning with Integer Programming
Lecture 10 (5/4/2015)
![Page 2: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/2.jpg)
Spatial Forest Planning
![Page 3: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/3.jpg)
Spatial Forest Planning with
Roads
![Page 4: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/4.jpg)
Balancing the Age-class Distribution
0K
10K
20K
30K
40K
50K
Red OakRed MapleOther ZonesOther OaksOther Hrdwds.N. Hrdwds.ConifersA. Hrdwds.
Acr
es
Forest Age Class Distribution
0K
4K
8K
12K
16K
20K
24K
28K
Red OakRed MapleOther ZonesOther OaksOther Hrdwds.N. Hrdwds.ConifersA. Hrdwds.
Acr
es
Forest Age Class Distribution
![Page 5: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/5.jpg)
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
![Page 6: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/6.jpg)
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…
![Page 7: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/7.jpg)
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
![Page 8: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/8.jpg)
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,
![Page 9: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/9.jpg)
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
![Page 10: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/10.jpg)
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
![Page 11: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/11.jpg)
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
![Page 12: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/12.jpg)
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
![Page 13: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/13.jpg)
What is a “better” formulation?
![Page 14: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/14.jpg)
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
![Page 15: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/15.jpg)
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
![Page 16: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/16.jpg)
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
![Page 17: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/17.jpg)
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
![Page 18: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/18.jpg)
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
![Page 19: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/19.jpg)
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
![Page 20: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/20.jpg)
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
![Page 21: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/21.jpg)
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
![Page 22: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/22.jpg)
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
![Page 23: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/23.jpg)
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
![Page 24: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/24.jpg)
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
![Page 25: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/25.jpg)
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
![Page 26: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/26.jpg)
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
![Page 27: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/27.jpg)
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
[1] [1] [2]
[3]
[4]
[5] [7]
[6]
[8]
[9]
[23]
[10]
[11]
[12]
CQ .
Build E(C)
Build E(C)
Build E(C)
[13] [14]
[15]
[16] [17]
[18] [19]
[20]
[21]
[22]
[24]
Strengthening and lifting
Compatibility testing
![Page 28: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/28.jpg)
Open Questions:
1) Can one define a rule for multiple lifting?
2) Can one find even stronger inequalities?
3) Is there a better formulation?
![Page 29: Spatial Forest Planning with Integer Programming Lecture 10 (5/4/2015)](https://reader035.vdocument.in/reader035/viewer/2022062906/5a4d1b6e7f8b9ab0599b45f1/html5/thumbnails/29.jpg)
A “Cutting Plane” Algorithm for the Cover Formulation
21 37 47 2X X X
8 18 40 2X X X