The use of linear programming to integrate forest operations

Silvana NobreAtrium Forest Consulting, Brasil

silvana@atriumforest.com

The use of linear programming to integrate forest operations

1. Overview on Brazilian forest-based companies

2. First-sight Problem: Clonal seedlings allocation

3. Integration A: Plantation Activities Constraints

4. Integration B: Nursery Production process

5. Conclusions

1 . Overview on Brazilian forest-based

companies

Source: Bracelpa - 2010

179 units

from 400 to 237,000 ha

Total: 4,671,521 ha

Species: Eucalyptus,

Pinus, Teca,

Acácia, Paricá,

Araucária

Industries: Pulp & Paper

Solid & Panels

Steel Mills

Investments

1 - Overview on Brazilian forest-based companies

2 - Clonal seedling allocationExample….

...

2 - Clonal seedling allocation

TRU - 1 TRU - 2 TRU - 3......Fertilization x----• Clone 1 – 300• Clone 2 – 260• Clone 3 – 220

......Fertilization y----• Clone 2 – 320• Clone 3 – 270• Clone 1 – 230

......Fertilization z----• Clone 3 – 310• Clone 1 – 250• Clone 2 – 210

...

...

Stand TRU

1 1

2 1

3 2

4 2

5 3

792 4

793 3

794 4

Industry Forest Unit 95.254 haProductivity 50 m3/ha.yearClear Cut 6 years oldAnnual Replanting Area 15.876 haSpacing ( 2,5 x 3 ) 6 m2Trees per area 1.667 trees/haPlantation eficciency 95%Seedling Need 27.782.475 seedlings/year

Tipical Brazilian Stand 20 haNo Stands to plant in a year 794 standsTechinical Recomendation Units 4 UnitsDifferent Clones 6 clones

Tipical Pulp&Paper Forest Industry Unit

2 - Clonal seedling allocation

Exercise….• MS Excel ®

• 330 ha

• 5 stands

• 3 Technical Recommendation Units

• 3 Clones

Max Z = ∑ P ij × X ij

Maximizar Produção Potencial….

• Pij Probable Productivity of Stand i, if we plant clone j

• Xij Area of Stand i, we will plant clone j

Techinical Recomendation

Unit

StandsStand Area 60 70 80 50 70

330

Clone CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3

Productivity (m3 / ha)

300 260 220 300 260 220 230 320 270 230 320 270 250 210 310

Sum: Stand-Clone Area

60 0 0 70 0 0 0 80 0 0 20 30 0 0 70

Sum: Stand Area 60 70 80 50 70

330

Potential

Production (m3)18,000 0 0 21,000 0 0 0 25,600 0 0 6,400 8,100 0 0 21,700

100,800

CL1 130 >= 100 m3

CL2 100 >= 100CL3 100 >= 100 ha

TRU1 TRU2 TRU3

St1 St2 St3 St4 St5

2 - Clonal seedling allocation

Max Z = ∑ P ij × X ij

Subject to:∑ X ij ≤ Ij, I =1, 2, 3, 4, 5 (stands)

∑ X ij ≥ Cj, j=1, 2, 3 (clones)

Where: Ii, area of stand i that can be planted Cj, at least this area must be planted with clone j

• Pij Probable Productivity of Stand i, if we plant clone j

• Xij Area of Stand i, we will plant clone j

3 - Plantation Activities Constraints

Exercise….

• MS Excel ®

• Pre-Planned Harvesting Process

• Until 3 months after Harvesting

• Planning Horizon: 12 months

• Period: 1 month

• 1 team

Techinical Recomendation Unit

Stands Stand Area (ha)

60 70 80 50 70330

Clone Productivity (m3 / ha)CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3

300 260 220 300 260 220 230 320 270 230 320 270 250 210 310

Sum: Stand-Clone Area (ha)60 0 0 70 0 0 0 50 30 0 50 0 0 0 70

60 70 80 50 70

330

Potential Production (m3)18,000 0 0 21,000 0 0 0 16,000 8,100 0 16,000 0 0 0 21,700

100,800 Months

1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 02 30 0 0 0 0 0 0 0 0 0 0 0 0 0 03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 304 0 0 0 0 0 0 0 0 20 0 0 0 0 0 105 0 0 0 0 0 0 0 0 0 0 0 0 0 0 306 0 0 0 0 0 0 0 20 10 0 0 0 0 0 07 0 0 0 0 0 0 0 30 0 0 0 0 0 0 08 0 0 0 0 0 0 0 0 0 0 30 0 0 0 09 0 0 0 10 0 0 0 0 0 0 20 0 0 0 0

10 0 0 0 30 0 0 0 0 0 0 0 0 0 0 011 0 0 0 30 0 0 0 0 0 0 0 0 0 0 012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

St1 St2 St3 St4 St5

Plantation per Period (Decision Variables)

TRU1 TRU2 TRU3

One more Dimension: Time >>>> Schedule problem

Stands Available to be planted (Harvest constraint)From short term Harvest scheduling model

Techinical Recomendation Unit

Stands Stand Area (ha)

60 70 80 50 70330

1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 02 1 1 1 0 0 0 0 0 0 0 0 0 0 0 03 1 1 1 0 0 0 0 0 0 0 0 0 1 1 14 1 1 1 0 0 0 1 1 1 0 0 0 1 1 15 0 0 0 0 0 0 1 1 1 0 0 0 1 1 16 0 0 0 0 0 0 1 1 1 0 0 0 1 1 17 0 0 0 0 0 0 1 1 1 1 1 1 1 1 18 0 0 0 0 0 0 1 1 1 1 1 1 0 0 09 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0

10 0 0 0 1 1 1 0 0 0 1 1 1 0 0 011 0 0 0 1 1 1 0 0 0 0 0 0 0 0 012 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

St1 St2 St3 St4 St5

Stands Available to be planted (Harvest constraint)

TRU1 TRU2 TRU3

Techinical Recomendation Unit

Stands Stand Area (ha)

60 70 80 50 70330

Monthly Plt (ha)

Monthly Plt Constraint

1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 <= 302 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 <= 303 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 30 <= 304 0 0 0 0 0 0 0 0 20 0 0 0 0 0 10 30 <= 305 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 30 <= 306 0 0 0 0 0 0 0 20 10 0 0 0 0 0 0 30 <= 307 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 30 <= 308 0 0 0 0 0 0 0 0 0 0 30 0 0 0 0 30 <= 309 0 0 0 10 0 0 0 0 0 0 20 0 0 0 0 30 <= 30

10 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 30 <= 3011 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 30 <= 3012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 <= 30

330 = 330

St1 St2 St3 St4 St5

Plantation per Period (Results)

TRU1 TRU2 TRU3

Operational constraint: Plantation Team Capacity

3 - Plantation Activities Constraints

Max Z = ∑ P ij × X ijk

Subject to: ∑ X ij ≤ Ij, I =1, 2, 3, 4, 5 (stands) - Area Constraints

∑ X ijk ≥ Cj, j=1,2, 3 (clones) - Biological Constraints

∑ X ijk ≤ Tk, k= 1 to 12 (periods) - Plantation Team Capacity

∑ X ijk = 0 , for some stands i in some periods k - Harvest Constraints

Where: Ii, area of stand i that can be planted

Cj, at least this area must be planted with clone j

Tj, maximum productivity of a team plantation in each period k

• Pij Probable Productivity of Stand i, if we plant clone j

• Xijk Area of Stand i, we will plant clone j, in period k

4 - Nursery production process

1st Step :

Clonal Garden

2nd Step : Collect mini-cuttings

from mini-stumps

3rd Step : cuttings production

4th Step : Green House

5th Step : Open Space Growth

4 - Nursery production process

6th Step : Seedlings Deliver Space

• MS Excel ®

• Constraints:

Cutting production team

Green House Occupation

Clonal Garden Capacity

4 - Nursery production process

Stands

Replant Eficiency (%) 3% 3% 3% 3% 3%Spacing (trees/ha) 1667 1667 1111 1111 1515Stand Area (ha) 60 70 80 50 70 330 haClonesCL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3 CL1 CL2 CL3Months

CL1 CL2 CL316,250 13,835 21,425 0 0 0 0 0 0 0 0 0 0 0 0 16,250 13,835 21,42515,234 0 7,689 0 0 0 0 0 0 0 0 0 0 0 0 15,234 0 7,68922,544 0 0 0 0 0 0 0 0 0 0 0 0 0 26,325 22,544 0 26,325

6,043 0 0 0 0 0 0 16,935 0 0 0 0 0 0 18,228 6,043 16,935 18,2280 0 0 0 0 0 0 25,445 0 0 0 0 0 0 12,116 0 25,445 12,1160 0 0 0 0 0 0 21,843 0 0 0 0 0 0 17,028 0 21,843 17,0280 0 0 0 0 0 0 0 0 0 8,271 0 14,801 0 20,733 14,801 8,271 20,7330 0 0 0 0 0 0 26,152 1,172 0 7,006 0 0 0 0 0 33,158 1,1720 0 0 23,843 0 0 0 0 0 0 9,345 9,094 0 0 0 23,843 9,345 9,0940 0 0 16,250 0 0 0 0 0 0 16,323 7,177 0 0 0 16,250 16,323 7,1770 0 0 16,250 12,337 0 0 0 0 0 0 0 0 0 0 16,250 12,337 00 0 0 39,000 0 12,510 0 0 0 0 0 0 0 0 0 39,000 0 12,510

Seedling Requirements per cloneSeedling requirements per Period Per Clone

TRU1 TRU2 TRU3

St1 St2 St3 St4 St5

Seedlings requirements per clone per period

It determines mini-cuttings requirements ….

mCt NeedmCt Team

CL1 CL2 CL325,000 19,764 10,679 55,443 <= 60,00023,437 0 36,563 60,000 <= 60,00034,683 0 25,317 60,000 <= 60,000

9,297 24,193 16,827 50,317 <= 60,0000 36,350 23,650 60,000 <= 60,0000 31,204 28,796 60,000 <= 60,000

22,771 11,816 1,627 36,215 <= 60,0000 47,369 12,631 60,000 <= 60,000

36,682 13,350 9,968 60,000 <= 60,00025,000 23,318 0 48,318 <= 60,00025,000 17,625 17,375 60,000 <= 60,00060,000 0 0 60,000 <= 60,000

mini-Cutti ngs Requirements per clone

Mini-cuttings requirements

Subject to: Cutting production team constraint

It determines mini-stumps requirements ….

CL1 CL2 CL3 CL1 CL2 CL35,000 3,294 1,780 <= 16,000 8,000 10,0004,687 0 6,094 <= 16,000 8,000 10,0006,937 0 4,220 <= 16,000 8,000 10,0001,859 4,032 2,805 <= 16,000 8,000 10,000

0 6,058 3,942 <= 16,000 8,000 10,0000 5,201 4,799 <= 16,000 8,000 10,000

4,554 1,969 271 <= 16,000 8,000 10,0000 7,895 2,105 <= 16,000 8,000 10,000

7,336 2,225 1,661 <= 16,000 8,000 10,0005,000 3,886 0 <= 16,000 8,000 10,0005,000 2,937 2,896 <= 16,000 8,000 10,000

12,000 0 0 <= 16,000 8,000 10,000

mini-Stumps Requeriments per clone mini-Stumps in Clone Garden per clone

Mini-stumps requirements

Subject to: Mini-Stumps that already exists in the Nursery

Area Constraints (stands) ∑ X ij ≤ Ij, I =1, 2, 3, 4, 5

Biological Constraints (clones) ∑ X ijk ≥ Cj, j =1, 2, 3

Plantation Team Capacity ∑ X ijk ≤ Tk, k = 1 to 12

Harvest Constraints ∑ X ijk = 0 , for some stands i in some periods k

Cutting Product.Team Capacity ∑ d j X ijk ≤ D k k= 1 to 12

Greenhouse Max Space ∑ g j X ijk + g j X ijk-1 ≤ G k , for k = 1 to 12

Mini-Stumps in the Nursery ∑ s j X ijk ≤ S jk , for j =1,2, 3 and k= 1 to 12

• Pij Probable Productivity of Stand i, if we plant clone j

• Xijk Area of Stand i, we will plant clone j, in period k

Max Z = ∑ P ij × X ijk

Subject to:

Where:

Ii, area of stand i that can be planted

Cj, at least this area must be planted with clone j

Tj, maximum productivity of a team plantation in each period k

dj, nursery efficiency of clone j

Dk cuttings team capacity, in period k

gj, green house occupation factor of clone j

Gk green house maximum occupation in period k

sj, mini-stumps need factor of clone j

Sjk mini-stumps in the Nursery in production of clone j, in period k

5 - Conclusions

• MS Excel ® simplified exercise

• 5% of potential production

• 10% productivity from first to second clone

10% area >>> 1,588 ha

10% diff produc >>> 30 m3/ha

Production difference >>> 47,627 m3

Minimum Wood value >>> 50 R$/m3

Prouction Diff Value >>> 2,381,355 R$

Prouction Diff Value >>> 850,484 €

In one year, on 10% of the area you could have planted the 1st clone, but you planted the 2nd one

silvana@atriumforest.comwww.atriumforest.com