# Coarse Woody Debris Management with Ambiguous Chance Constrained Robust Optimization

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## Abstract

**:**

## 1. Introduction

#### 1.1. Deadwood

#### 1.2. Aspects to Consider for Deadwood Management Decision Making

## 2. Materials and Methods

#### 2.1. Deadwood

#### 2.2. Dealing with Uncertainties

- Stochastic optimization: For this purpose, the essential requirement is a randomly distributed uncertainty. Then the uncertainty can be defined in a probabilistic way and described using distribution functions. Typically, the main focus of these models is the expected value of the objective. However, there are also expansions like chance constrained programming, which deal with questions on the actual form of the solution space.
- Robust optimization: In contrast to the stochastic approach any assumptions on the distribution of uncertainty are disclaimed. Only the borders of the uncertainty region are defined. The focus lies on finding solutions that cope with all possible values of the uncertain data in an optimal way. The main interest is not on the expected value of the objective but on the feasibility of a solution that incorporates most of the possible values of the uncertain parameters [47]. Consequently, the solution is also feasible for the edge of the uncertainty space [48]. The objective is typically expressed as a min-max- or ($\mu $,$\sigma $)-condition.

#### 2.3. Data Demanding Stochastic Model

#### 2.3.1. Data Uncertainty

#### 2.3.2. Expected Return Uncertainty

#### 2.4. Robust Model for Situations without Information about the Probability Distribution of the Uncertainty

- $\parallel a-{\mu}_{m}{\parallel}_{\infty}\le \beta $: The constraint can be reformulated as ${\parallel d\parallel}_{\infty}\le 1$ with $d:={\beta}^{-1}{max}_{m}|a-{\mu}_{m}|$. Then, we transform the “greater than” constraint of the robust problem of Equation (14) into a “less than” constraint to be able to linearise it in the end:$$\begin{array}{cc}\hfill {a}^{\mathsf{\top}}x& \ge Z\phantom{\rule{1.em}{0ex}}\forall a\hfill \\ \hfill \equiv -{a}^{\mathsf{\top}}x& \le -Z\phantom{\rule{1.em}{0ex}}\forall a\hfill \end{array}$$$$\begin{array}{cc}\hfill -{a}^{\mathsf{\top}}x& \le -Z\phantom{\rule{1.em}{0ex}}\forall a\hfill \\ \hfill \equiv {\overline{a}}^{\mathsf{\top}}x-{\beta}^{-1}{d}^{\mathsf{\top}}x& \ge Z\phantom{\rule{1.em}{0ex}}\forall d\hfill \end{array}$$$$\begin{array}{cc}\hfill \equiv -{\overline{a}}^{\mathsf{\top}}x+{\beta}^{-1}{d}^{\mathsf{\top}}x& \le -Z\phantom{\rule{1.em}{0ex}}\forall d\hfill \end{array}$$$${d}^{\mathsf{\top}}x=\sum _{i}{d}_{i}{x}_{i}\le \sum _{i}|{d}_{i}\left|\right|{x}_{i}|\le \sum _{i}|{x}_{i}{|=\parallel x\parallel}_{1}$$$$-{\overline{a}}^{\mathsf{\top}}x+{\beta}^{-1}{\parallel x\parallel}_{1}\le -Z$$$$\begin{array}{cc}& \underset{x,h}{max}\phantom{\rule{4pt}{0ex}}Z\hfill \\ \hfill \mathrm{with}\phantom{\rule{1.em}{0ex}}& -{\overline{a}}^{\mathsf{\top}}x+{\beta}^{-1}\sum _{i}{h}_{i}\le -Z\hfill \\ & -{h}_{i}\le {x}_{i}\le {h}_{i}\phantom{\rule{1.em}{0ex}}\forall i\hfill \\ & x\in \Omega \hfill \end{array}$$
- $\parallel a-{\mu}_{m}{\parallel}_{2}\le \beta $: The constraint is approximated by a cone (or ellipsoid, ball), using the Euclidean norm. This is the most effective way to count for uncertainties as the “ball shape” of the uncertainty region emphasizes the more likely values of the uncertain data by cutting off the unlikely “corners”. Using the definition of d from above, the constraint can be reformulated as ${\parallel d\parallel}_{2}\le 1$. As the ball constraint must hold for all d, it holds also for the worst-case, so the term ${d}^{\mathsf{\top}}x$ in the constraint of Equation (18) can be reformulated (Cauchy-Schwarz inequality):$${d}^{\mathsf{\top}}x\le {\parallel d\parallel}_{2}{\parallel x\parallel}_{2}\le {\parallel x\parallel}_{2}$$$${\overline{a}}^{\mathsf{\top}}x-{\beta}^{-1}{\parallel x\parallel}_{2}\ge Z$$$$\begin{array}{cc}& \underset{x}{max}\phantom{\rule{4pt}{0ex}}Z\hfill \\ \hfill \mathrm{with}\phantom{\rule{1.em}{0ex}}& {\overline{a}}^{\mathsf{\top}}x-{\beta}^{-1}{\parallel x\parallel}_{2}\ge Z\hfill \\ & x\in \Omega \hfill \end{array}$$

#### 2.5. Model Implementation

#### 2.6. Model Application

## 3. Results

#### 3.1. Analysing the Two Forest Enterprises Shows the General Tendencies

- All scenarios
- (a)
- The higher the desired deadwood amount is set, the higher are the linked costs.
- (b)
- The longer the time horizon to reach the desired deadwood amount is chosen, the more cost-effective the target that can be reached.
- (c)
- The actual costs depend on the stand structure of the forest enterprise.
- (d)
- There is no clear ranking between the spruce and the beech scenarios.
- (e)
- A segregative approach leads to lower costs than the integrative patterns.

- Spruce scenarios
- (a)
- Starting from deadwood amounts of 20 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$ and more, the crown material option is more expensive than the heavy timber strategy. The total tree approach is the most cost-effective. For low deadwood targets (5 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$) the crown material option is the cheapest.
- (b)
- Within a larger forest enterprise more deadwood objectives can be reached with crown material.

- Beech scenarios
- (a)
- In most scenarios the total tree strategy has the biggest cost advantage.
- (b)
- From deadwood amounts of 40 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$ and more, the heavy timber strategy is more cost-intensive than the use of total trees. (In the case of “Eichelberg” this is already true for a deadwood target of 20 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$.)
- (c)
- The ranking between the crown material and the heavy timber strategy depends on the enterprise.

#### 3.2. Comparison of the Robust Model with the Stochastic Approach Reveals Its Stability

#### 3.3. Analysis of Deadwood Supply Derives Effective Strategy Combinations

## 4. Discussion

#### 4.1. Robust and Stochastic Approach Lead to Similar Results

#### 4.2. Some Results Are Comparable to Other Studies

#### 4.3. Other Results Reveal Additional Important Aspects of Deadwood Management

#### 4.4. Supply Analysis as A Management Approach to React on Market Prices

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Forest enterprise “Eichelberg”: Costs in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] for target deadwood amounts in [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$], to be maintained within 5 years. Blue columns show spruce (sp) and orange beech strategies (be). tt: Total tree, ht: Heavy timber, cm: Crown material.

**Figure 2.**Forest enterprise “Eichelberg”: Costs in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] for target deadwood amounts in [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$], to be maintained within 20 years. Blue columns show spruce (sp) and orange beech strategies (be). tt: Total tree, ht: Heavy timber, cm: Crown material.

**Figure 3.**Forest enterprise “Eichelberg”: Costs in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] for target deadwood amounts in [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$], to be maintained within 30 years. Blue columns show spruce (sp) and orange beech strategies (be). tt: Total tree, ht: Heavy timber, cm: Crown material.

**Figure 4.**Forest enterprise “Neureichenau”: Costs in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] for target deadwood amounts in [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$], to be maintained within 5 years. Blue columns show spruce (sp) and orange beech strategies (be). tt: Total tree, ht: Heavy timber, cm: Crown material.

**Figure 5.**Forest enterprise “Neureichenau”: Costs in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] for target deadwood amounts in [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$], to be maintained within 20 years. Blue columns show spruce (sp) and orange beech strategies (be). tt: Total tree, ht: Heavy timber, cm: Crown material.

**Figure 6.**Forest enterprise “Neureichenau”: Costs in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] for target deadwood amounts in [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$], to be maintained within 30 years. Blue columns show spruce (sp) and orange beech strategies (be). tt: Total tree, ht: Heavy timber, cm: Crown material.

**Figure 7.**Forest enterprise “Eichelberg”: Costs or necessary compensations (incentives) in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] of absolute deadwood amounts in [${\mathrm{m}}^{3}$], to be maintained within 30 years, using crown material of beech. The different colours correspond to the average deadwood volume from 5 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$ to 90 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$.

**Figure 8.**Forest enterprise “Eichelberg”: Costs or necessary compensations (incentives) in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] of absolute deadwood amounts in [${\mathrm{m}}^{3}$], to be maintained within 30 years, using total trees of beech. The different colours correspond to the average deadwood volume from 5 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$ to 90 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$.

**Figure 9.**Forest enterprise “Eichelberg”: Costs or necessary compensations (incentives) in [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}{\mathrm{a}}^{-1}$] of absolute deadwood amounts in [${\mathrm{m}}^{3}$], to be maintained within 30 years, using heavy timber of beech. The different colours correspond to the average deadwood volume from 5 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$ to 90 ${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$.

**Table 1.**Numerical example of where to provide a certain amount of deadwood. DBH is the diameter at breast height.

Stand | A | B |
---|---|---|

Species | beech | spruce/beech |

Age [years] | 100 | 80 |

Spruce trees [N] | - | 500 |

Beech trees [N] | 300 | 350 |

Volume of beech [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$] | 450 | 250 |

Avg. DBH of beech [cm] | 35 | 30 |

Timber value [$\u20ac\mathrm{h}{\mathrm{a}}^{-1}$] | 65 | 40 |

Risk of disturbance | low | medium |

**Table 2.**Main characteristics of the investigated forest enterprises located in the east of Lower Bavaria, Bavaria, Germany. Volume and increment are values without bark volume.

Enterprise | Eichelberg | Neureichenau |
---|---|---|

Ownership | private | public |

Political location | rural district of Passau | rural districts of Passau |

and Freyung-Grafenau | ||

Geographical location | river Danube | river Danube and Bavarian Forest |

Altitude [$\mathrm{m}$] | 400 | 300–1360 |

Area [$\mathrm{h}\mathrm{a}$] | 220 | 17,300 |

Main tree species | spruce, fir, beech, oak | spruce, beech, fir |

Volume [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$] | 450 | 313 |

Increment [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$] | 10.7 | 8.6 |

Avg. temperature [${}^{\circ}\mathrm{C}$] | 8.3 | 5.5–8.3 |

Precipitation [mm] | 850–950 | 806–1309 |

Initial deadwood [${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$] | 4.7 | 0 |

**Table 3.**Parameters for the construction of the deadwood scenarios. All available combinations sum up to a set of 720 possible scenarios.

Target [m^{3} ha^{−1}] | Tree Species | Time Horizon [years] | Sorting Strategy | Segregation [%] |
---|---|---|---|---|

5, 20, 40, 65, 90 | beech (be), | 5, 20, 30 | total tree (tt), | 100, 50, 25, 20, |

spruce (sp) | heavy timber (ht), | 10, 5, 2, 1 | ||

crown material (cm) |

**Table 4.**Forest enterprise “Eichelberg”: Median costs and influence of the degree of segregation. The single strategies are abbreviated as shown in Table 3. Costs 1 shows the costs as the loss of annuity within the enterprise, while Costs 2 shows the costs related to the area of the individual tree species of the scenario. The Min. column shows the result for 100% segregation, the Max. column the result for 1% segregation.

Time Horizon | Strategy | Target | Costs 1 | Costs 2 | Min. | Max. |
---|---|---|---|---|---|---|

[years] | [m${}^{3}$ha${}^{-1}$] | [€ha${}^{-1}$a${}^{-1}$] | [€/ha/a] | [€ha${}^{-1}$a${}^{-1}$] | [€ha${}^{-1}$a${}^{-1}$] | |

5 | sp/ht | 5 | 22.6 | 39.6 | 39.6 | 39.7 |

20 | 44.0 | 77.3 | 77.3 | 86.3 | ||

40 | 101.7 | 178.4 | 176.5 | 182.5 | ||

65 | 189.3 | 332.2 | 327.6 | 356.4 | ||

sp/tt | 5 | 27.4 | 48.0 | 48.0 | 48.0 | |

20 | 43.8 | 76.9 | 76.8 | 77.9 | ||

40 | 90.6 | 159.0 | 158.6 | 164.8 | ||

65 | 157.1 | 275.7 | 272.4 | 282.7 | ||

90 | 232.8 | 408.7 | 395.5 | 427.2 | ||

sp/cm | 5 | 11.8 | 20.7 | 20.6 | 27.2 | |

be/ht | 5 | 7.2 | 33.3 | 33.3 | 33.9 | |

20 | 27.7 | 127.4 | 122.8 | 135.5 | ||

40 | 81.3 | 373.7 | 359.5 | 395.1 | ||

65 | 196.4 | 902.9 | 902.9 | 902.9 | ||

be/tt | 5 | 9.9 | 45.5 | 45.5 | 45.5 | |

20 | 16.1 | 74.0 | 73.5 | 76.0 | ||

40 | 39.2 | 180.3 | 177.8 | 186.6 | ||

65 | 74.5 | 342.5 | 336.9 | 350.4 | ||

90 | 120.5 | 553.9 | 539.8 | 572.7 | ||

be/cm | 5 | 4.4 | 20.5 | 20.4 | 21.4 | |

20 | 26.1 | 120.0 | 118.3 | 125.8 | ||

40 | 76.2 | 350.5 | 344.2 | 360.5 | ||

20 | sp/ht | 5 | 22.6 | 39.6 | 39.6 | 39.7 |

20 | 34.4 | 60.3 | 60.3 | 66.3 | ||

40 | 70.6 | 123.9 | 123.4 | 157.6 | ||

65 | 124.3 | 218.2 | 212.3 | 262.4 | ||

90 | 186.3 | 327.0 | 315.0 | 350.5 | ||

sp/tt | 5 | 27.4 | 48.0 | 48.0 | 48.0 | |

20 | 37.2 | 65.4 | 65.3 | 67.7 | ||

40 | 68.2 | 119.7 | 119.7 | 125.5 | ||

65 | 112.7 | 197.9 | 196.6 | 257.8 | ||

90 | 160.9 | 282.3 | 276.9 | 319.2 | ||

sp/cm | 5 | 10.2 | 17.8 | 17.8 | 25.2 | |

be/ht | 5 | 6.5 | 29.7 | 29.7 | 29.7 | |

20 | 15.5 | 71.4 | 70.6 | 73.9 | ||

40 | 40.5 | 186.4 | 181.5 | 199.2 | ||

65 | 82.0 | 377.2 | 363.9 | 393.6 | ||

90 | 146.1 | 671.8 | 660.9 | 682.7 | ||

be/tt | 5 | 9.9 | 45.4 | 45.4 | 45.4 | |

20 | 11.9 | 54.9 | 54.8 | 59.1 | ||

40 | 25.3 | 116.4 | 115.8 | 120.8 | ||

65 | 46.6 | 214.5 | 211.4 | 258.0 | ||

90 | 71.4 | 328.5 | 320.6 | 352.2 | ||

be/cm | 5 | 3.5 | 15.9 | 15.9 | 16.1 | |

20 | 15.6 | 71.8 | 70.8 | 77.1 | ||

40 | 44.8 | 205.8 | 201.8 | 220.7 | ||

30 | sp/ht | 5 | 22.6 | 39.6 | 39.5 | 39.6 |

20 | 30.7 | 53.9 | 53.9 | 59.1 | ||

40 | 57.7 | 101.2 | 101.0 | 111.6 | ||

65 | 95.6 | 167.8 | 165.6 | 185.1 | ||

90 | 140.5 | 246.6 | 236.6 | 344.4 | ||

sp/tt | 5 | 27.4 | 48.0 | 48.0 | 48.0 | |

20 | 34.7 | 61.0 | 60.9 | 61.3 | ||

40 | 58.3 | 102.4 | 102.4 | 117.4 | ||

65 | 91.9 | 161.3 | 160.6 | 184.8 | ||

90 | 127.2 | 223.2 | 220.9 | 242.3 | ||

sp/cm | 5 | 8.6 | 15.2 | 15.1 | 23.5 | |

be/ht | 5 | 6.5 | 29.7 | 29.7 | 29.7 | |

20 | 12.2 | 55.9 | 55.6 | 65.5 | ||

40 | 28.4 | 130.5 | 129.0 | 135.2 | ||

65 | 55.5 | 255.0 | 245.5 | 306.1 | ||

90 | 89.5 | 411.7 | 394.8 | 438.9 | ||

be/tt | 5 | 9.9 | 45.4 | 45.4 | 45.4 | |

20 | 10.8 | 49.7 | 49.7 | 50.3 | ||

40 | 20.4 | 93.7 | 93.3 | 106.7 | ||

65 | 35.2 | 162.0 | 161.0 | 176.2 | ||

90 | 52.2 | 239.8 | 235.9 | 250.8 | ||

be/cm | 5 | 3.5 | 15.9 | 15.9 | 16.1 | |

20 | 11.4 | 52.5 | 51.8 | 55.3 | ||

40 | 31.4 | 144.4 | 141.5 | 153.0 | ||

65 | 66.3 | 304.6 | 294.4 | 369.4 | ||

90 | 112.6 | 517.8 | 510.9 | 524.7 |

**Table 5.**Forest enterprise “Neureichenau”: Median costs and influence of the degree of segregation. The single strategies are abbreviated as shown in Table 3. Costs 1 shows the costs as the loss of annuity within the enterprise, while Costs 2 shows the costs related to the area of the individual tree species of the scenario. The Min. column shows the result for 100 % segregation, the Max. column the result for 1 % segregation.

Time Horizon | Strategy | Target | Costs 1 | Costs 2 | Min. | Max. |
---|---|---|---|---|---|---|

[years] | [m${}^{\mathbf{3}}$ha${}^{-\mathbf{1}}$] | [€ha${}^{-\mathbf{1}}$a${}^{-\mathbf{1}}$] | [€ha${}^{-\mathbf{1}}$a${}^{-\mathbf{1}}$] | [€ha${}^{-\mathbf{1}}$a${}^{-\mathbf{1}}$] | [€ha${}^{-\mathbf{1}}$a${}^{-\mathbf{1}}$] | |

5 | sp/ht | 5 | 18.9 | 22.4 | 19.6 | 28.8 |

20 | 93.9 | 111.1 | 105.5 | 121.3 | ||

40 | 215.4 | 254.8 | 244.3 | 270.7 | ||

65 | 376.2 | 445.1 | 434.1 | 497.7 | ||

sp/tt | 5 | 13.0 | 15.4 | 14.6 | 17.3 | |

20 | 65.0 | 76.9 | 73.6 | 90.0 | ||

40 | 151.3 | 179.1 | 173.1 | 205.8 | ||

65 | 269.1 | 318.4 | 304.5 | 340.5 | ||

90 | 406.7 | 481.2 | 447.8 | 547.7 | ||

sp/cm | 5 | 9.4 | 11.2 | 8.2 | 17.9 | |

20 | 115.8 | 137.0 | 125.8 | 154.2 | ||

be/ht | 5 | 1.0 | 43.7 | 42.9 | 44.7 | |

20 | 2.4 | 101.1 | 100.6 | 101.6 | ||

40 | 6.6 | 277.7 | 276.5 | 281.0 | ||

65 | 10.7 | 448.1 | 446.3 | 451.3 | ||

90 | 15.9 | 663.2 | 660.9 | 676.0 | ||

be/tt | 5 | 1.3 | 53.9 | 53.5 | 55.8 | |

20 | 2.6 | 108.1 | 106.8 | 109.1 | ||

40 | 4.3 | 181.8 | 181.0 | 182.6 | ||

65 | 6.7 | 282.2 | 280.8 | 286.9 | ||

90 | 10.1 | 423.9 | 422.6 | 429.5 | ||

be/cm | 5 | 2.0 | 82.4 | 80.3 | 83.2 | |

20 | 2.3 | 94.2 | 92.9 | 96.2 | ||

40 | 5.1 | 212.6 | 211.7 | 218.0 | ||

65 | 9.8 | 411.7 | 410.5 | 417.1 | ||

90 | 16.0 | 670.8 | 668.9 | 686.0 | ||

20 | sp/ht | 5 | 11.0 | 13.0 | 11.8 | 14.7 |

20 | 58.9 | 69.6 | 67.5 | 99.5 | ||

40 | 137.1 | 162.3 | 156.7 | 225.8 | ||

65 | 244.9 | 289.8 | 274.7 | 355.5 | ||

90 | 356.3 | 421.5 | 400.6 | 455.5 | ||

sp/tt | 5 | 7.8 | 9.2 | 8.6 | 9.6 | |

20 | 41.1 | 48.7 | 47.6 | 55.6 | ||

40 | 96.2 | 113.8 | 110.8 | 134.8 | ||

65 | 171.2 | 202.6 | 196.7 | 227.1 | ||

90 | 253.9 | 300.4 | 286.0 | 366.8 | ||

sp/cm | 5 | 1.4 | 1.7 | −2.0 | 10.2 | |

20 | 59.4 | 70.2 | 67.1 | 90.7 | ||

40 | 188.9 | 223.5 | 202.5 | 265.2 | ||

be/ht | 5 | 0.3 | 11.5 | 10.6 | 12.4 | |

20 | 1.2 | 49.3 | 48.2 | 50.4 | ||

40 | 3.1 | 131.4 | 130.6 | 132.8 | ||

65 | 6.1 | 253.2 | 252.2 | 258.2 | ||

90 | 9.9 | 413.5 | 411.5 | 427.0 | ||

be/tt | 5 | 0.6 | 26.5 | 25.0 | 27.9 | |

20 | 1.3 | 52.7 | 51.5 | 54.4 | ||

40 | 2.4 | 100.3 | 98.9 | 100.6 | ||

65 | 4.0 | 167.5 | 166.2 | 167.9 | ||

90 | 6.0 | 251.2 | 250.7 | 252.2 | ||

be/cm | 5 | 0.6 | 25.2 | 23.7 | 26.6 | |

20 | 0.6 | 25.3 | 24.4 | 26.5 | ||

40 | 2.3 | 95.0 | 94.2 | 97.6 | ||

65 | 5.2 | 215.9 | 214.5 | 224.5 | ||

90 | 8.8 | 366.3 | 364.6 | 379.6 | ||

30 | sp/ht | 5 | 9.0 | 10.6 | 9.0 | 11.2 |

20 | 45.6 | 54.0 | 52.4 | 78.3 | ||

40 | 103.2 | 122.0 | 118.4 | 147.4 | ||

65 | 180.6 | 213.7 | 207.0 | 244.7 | ||

90 | 267.6 | 316.6 | 298.4 | 401.0 | ||

sp/tt | 5 | 6.2 | 7.3 | 6.2 | 7.8 | |

20 | 32.5 | 38.4 | 36.4 | 42.8 | ||

40 | 72.9 | 86.2 | 84.7 | 129.3 | ||

65 | 129.0 | 152.6 | 148.5 | 197.5 | ||

90 | 187.6 | 222.0 | 215.2 | 253.8 | ||

sp/cm | 5 | 0.2 | 0.2 | -2.0 | 1.7 | |

20 | 41.0 | 48.5 | 44.9 | 91.9 | ||

40 | 124.5 | 147.3 | 130.4 | 283.3 | ||

be/ht | 5 | 0.8 | 33.6 | 33.1 | 34.6 | |

20 | 1.4 | 59.5 | 58.7 | 62.1 | ||

40 | 3.3 | 138.7 | 137.9 | 140.0 | ||

65 | 4.9 | 204.8 | 204.0 | 206.9 | ||

90 | 8.0 | 332.6 | 331.6 | 335.6 | ||

be/tt | 5 | 1.1 | 46.3 | 45.8 | 47.4 | |

20 | 1.7 | 69.8 | 68.6 | 71.8 | ||

40 | 2.6 | 107.2 | 105.8 | 107.8 | ||

65 | 3.6 | 148.8 | 148.1 | 149.2 | ||

90 | 4.9 | 205.5 | 205.1 | 206.1 | ||

be/cm | 5 | 1.6 | 65.9 | 63.9 | 66.7 | |

20 | 1.4 | 58.5 | 57.8 | 58.8 | ||

40 | 2.4 | 101.5 | 100.3 | 101.9 | ||

65 | 4.9 | 205.2 | 203.8 | 207.6 | ||

90 | 7.3 | 304.7 | 303.5 | 315.5 |

**Table 6.**Mean annuities and standard deviations of the stochastic model and the robust model derived from 100 Monte Carlo simulations of the uncertain possible revenues of each stand. Random values are chosen from the part of the uncertainty space that includes negative derivations from the nominal value. For abbreviations see Table 3.

Time Horizon | Strategy | Target | Stochastic | Robust | Diff. | |||
---|---|---|---|---|---|---|---|---|

Mean | Std.dev. | Mean | Std.dev. | Mean | Std.dev. | |||

[years] | [m${}^{3}$ha${}^{-1}$] | [€ha${}^{-1}$a${}^{-1}$] | [€ha${}^{-1}$a${}^{-1}$] | [€ha${}^{-1}$a${}^{-1}$] | [€ha${}^{-1}$a${}^{-1}$] | [%] | [%] | |

baseline | 0 | 598.0 | 67.4 | 583.1 | 54.1 | −2.5 | −19.7 | |

5 | sp/tt | 20 | 561.8 | 70.6 | 547.7 | 60.5 | −2.5 | −14.4 |

65 | 420.6 | 43.4 | 413.4 | 39.1 | −1.7 | −9.8 | ||

be/tt | 20 | 579.8 | 64.8 | 568.5 | 53.3 | −1.9 | −17.7 | |

65 | 515.7 | 49.1 | 503.9 | 36.2 | −2.3 | −26.3 | ||

20 | sp/tt | 20 | 551.5 | 60.2 | 542.1 | 54.5 | −1.7 | −9.5 |

65 | 469.9 | 49.7 | 460.8 | 40.8 | −1.9 | −17.9 | ||

be/tt | 20 | 581.9 | 65.2 | 569.2 | 55.7 | −2.2 | −14.6 | |

65 | 552.0 | 53.5 | 538.0 | 38.6 | −2.5 | −27.8 | ||

30 | sp/tt | 20 | 562.7 | 64.1 | 546.8 | 52.4 | −2.8 | −18.2 |

65 | 496.1 | 62.2 | 480.2 | 45.5 | −3.2 | −26.8 | ||

be/tt | 20 | 586.3 | 59.7 | 573.7 | 53.7 | −2.2 | −10.1 | |

65 | 551.5 | 64.5 | 537.9 | 49.3 | −2.5 | −23.7 | ||

Mean | −2.3 | −18.2 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Härtl, F.; Knoke, T. Coarse Woody Debris Management with Ambiguous Chance Constrained Robust Optimization. *Forests* **2019**, *10*, 504.
https://doi.org/10.3390/f10060504

**AMA Style**

Härtl F, Knoke T. Coarse Woody Debris Management with Ambiguous Chance Constrained Robust Optimization. *Forests*. 2019; 10(6):504.
https://doi.org/10.3390/f10060504

**Chicago/Turabian Style**

Härtl, Fabian, and Thomas Knoke. 2019. "Coarse Woody Debris Management with Ambiguous Chance Constrained Robust Optimization" *Forests* 10, no. 6: 504.
https://doi.org/10.3390/f10060504