# Multistage Sample Average Approximation for Harvest Scheduling under Climate Uncertainty

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## Abstract

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## 1. Introduction

- Introduce a method to handle climate change uncertainty in forest harvest scheduling that allows to make sound decisions in early stages of planning. Poor decisions in early stages are significantly more detrimental to many businesses compared to decisions in later stages. Hence, later stage decisions can be considered as secondary.
- Propose a method to generate the set of scenarios to reduce the sample size and keep the optimization model tractable. If we generate possible scenarios of forest growth in i.i.d (independent identical distributed) fashion, then we might have a very large sample size before SAA solution converges to optimality. This large sample size can make the problem computationally intractable. We propose a sampling scheme that focuses at having higher number of replications for distant stages.

## 2. Material and Methods

#### 2.1. Problem Description and Formulation

#### 2.2. Climate Change Data

`Lower`and

`Upper`in the table correspond to the lowest and highest possible growth change, respectively, at each stage. Formally speaking, [

`Lower, Upper`] at stage t, represents ${\Xi}_{t}$ of the random parameter ${\xi}_{t}$.

#### 2.3. Multistage Sampling

#### 2.4. Solution Method

#### 2.5. Importance of SAA

#### 2.6. Experiment

## 3. Results

#### 3.1. Effect of Sample Size on the Optimality Gap

#### 3.2. Effect of Sampling Scheme on the Optimality Gap

`1555`and

`5551`lead to the same number of scenarios (125 scenarios). As we can see from Figure 3, the optimality gap and its variability when ${N}_{1}$ = 1 is larger than when ${N}_{1}>1$. The lowest optimality gap is obtained with ${N}_{1}=3$. In general having higher values of ${N}_{t}$ for higher t seems to be favorable.

#### 3.3. Advantage of SAA in Stochastic Harvest Scheduling over the Deterministic Approach

## 4. Discussion and Conclusions

`2356`and

`3344`yielding sample sizes of 180 and 144, respectively, have smaller optimality gaps and variances compared to a sample size of 625 which stemmed from a sampling scheme of

`5555`. In conclusion, when we adopt the adequate sampling strategy that allows to sufficiently explore the first stage and generate large samples for future stages, we can limit the number of scenarios necessary for the SAA solution to converge to the true optimal solution. This computational challenge is common to stochastic programs in forestry [29].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Deterministic Harvest Scheduling Model

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**Figure 2.**Performance measurement of SAA for harvest scheduling. Error bars indicate 95% confidence interval.

**Figure 3.**Change of optimality gap values due to the sample size and the configuration of the sampling scheme using severe climate change data.

Sets | |
---|---|

$\mathcal{S}$ | set of stands or forest management units ($s\in \mathcal{S}$) |

$\mathcal{T}$ | set of time period in which the decision are implemented or stage at which the decisions are taken ($t\in \mathcal{T}$) |

$\Omega $ | set of scenarios (${\xi}^{i}\in \Omega $) |

Variables | |

${x}_{s}$ | binary variable: 1 if management unit s is scheduled to be harvested in the first period (now); and 0 otherwise |

${w}_{s}^{{\xi}^{i}}$ | binary variable: 1 if management unit s should not be harvested during the whole planning horizon given the scenario ${\xi}^{i}$; and 0 otherwise |

${y}_{st}^{{\xi}^{i}}$ | binary variable: 1 if the stand s should be harvested in t given the scenario ${\xi}^{i}$; 0 otherwise |

${H}_{t}^{i}$ | volume harvested in year t under scenario ${\xi}^{i}$ (${m}^{3}$). We omit the superscript i for $t=0$ |

Parameters | |

${r}_{s}$ | discounted net harvest revenue from stand s if the stand is harvested now ($) |

${r}_{st}^{{\xi}^{i}}$ | net harvest revenue from stand s if the stand is harvested in year t according to scenario ${\xi}^{i}$ ($) |

${r}_{s0}^{{\xi}^{i}}$ | discounted value of the stand at the end of the planning horizon if it is not harvested during the whole planning horizon under scenario ${\xi}^{i}$ ($) |

${v}_{s}$ | merchantable yield of stand s in the first period (m^{3}/ha) |

${v}_{st}^{{\xi}^{i}}$ | projected merchantable yield of stand s if harvested in year t according to scenario ${\xi}^{i}$ (m^{3}/ha) |

${a}_{s}$ | area of stand s (ha) |

$ag{e}_{st}$ | age of stand s at the end of the planning horizon if harvested in year t (yr) |

$ag{e}_{s.}$ | current age of the stand s (yr) |

$ag{e}_{s0}$ | age of stand s at the end of the planning if not harvested during the planning horizon (yr) |

$\alpha $ | acceptable lower bound on the fluctuation of volume of wood from one period to the next |

$\beta $ | acceptable upper bound on the fluctuation of volume of wood from one period to the next |

$\gamma $ | acceptable lower bound fluctuation of volume of wood between two non consecutive periods |

$\lambda $ | acceptable upper bound fluctuation of volume of wood between two non consecutive periods |

Stage | Lower | Upper |
---|---|---|

1 | −1.2 | 11.1 |

2 | −2.4 | 22.2 |

3 | −3.6 | 33.3 |

4 | −4.8 | 44.4 |

$|\Omega |$ | $\mathit{\u03f5}$ | $\mathit{z}\left(\overline{\mathit{x}}\right)$ | $\mathit{z}\left({\mathit{x}}^{*}\right)$ | Infeasible Scen. | VSS | VSS (bp) |
---|---|---|---|---|---|---|

256 | 1 | 7,775,108 | 7,782,679 | 0 | 7571 | 9.96 |

200 | 20 | 7,160,950 | 7,256,069 | 3 | 95,119 | 132.83 |

200 | 40 | 4,940,204 | 6,737,368 | 60 | 1,797,163 | 3637.83 |

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**MDPI and ACS Style**

Bagaram, M.B.; Tóth, S.F. Multistage Sample Average Approximation for Harvest Scheduling under Climate Uncertainty. *Forests* **2020**, *11*, 1230.
https://doi.org/10.3390/f11111230

**AMA Style**

Bagaram MB, Tóth SF. Multistage Sample Average Approximation for Harvest Scheduling under Climate Uncertainty. *Forests*. 2020; 11(11):1230.
https://doi.org/10.3390/f11111230

**Chicago/Turabian Style**

Bagaram, Martin B., and Sándor F. Tóth. 2020. "Multistage Sample Average Approximation for Harvest Scheduling under Climate Uncertainty" *Forests* 11, no. 11: 1230.
https://doi.org/10.3390/f11111230