# A Progressive Hedging Approach to Solve Harvest Scheduling Problem under Climate Change

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Problem Description

#### 2.2. Scenario Generation

^{3}ha

^{−1}] harvested on each management unit. To address this problem several climate change scenarios were used, which were then transformed into growth and yield scenarios using a growth and yield model that take into account climatic data as predictor variables (i.e., a process-based model). These kind of models simulate detailed physiological processes that describe system behavior, while the empirical approaches rely on correlative relationships, which do not explicitly describe system behaviors and interactions [37].

#### 2.3. The Model

**Sets:**

^{+}: minimally infeasible clusters, $\mathrm{C}\in {{\mathsf{\Lambda}}^{+}}^{+}$

**Parameters:**

^{3}in stand i in period t per ha., under scenario S;

_{i}: area of stand i (ha.);

**Variables:**

^{3});

**Objective function:**

**Constraints:**

- Stand harvest 0–1$${{\displaystyle \sum}}_{t\text{}\u03f5\text{}T}{x}_{i,s}^{t}=1\forall i\in I,\text{}\forall s\in S;\text{}$$
- Volume harvested in period t$${{\displaystyle \sum}}_{i\text{}\in \text{}I}{a}_{i,s}^{t}{A}_{i}{x}_{i,s}^{t}={V}_{s,t}\text{},\text{}\forall t\in T,\text{}\forall s\in S;$$
- Even-flow of harvest$${V}_{s,t}-\left(1+\alpha \right){V}_{s,t-1}\text{}\le 0,\text{}\forall t\text{}\u03f5\text{}\left\{2,\text{}\dots T\right\},\text{}\forall s\in S\text{}$$$$\left(1-\alpha \right){V}_{s,t-1}-\text{}{V}_{s,t}-\le 0,\text{}\forall t\text{}\u03f5\text{}\left\{2,\text{}\dots T\right\},\text{}\forall s\in S;$$
- Demand constraints$${D}_{t}\text{}\le {V}_{s,t},\text{}\forall t\in T,\text{}\forall s\in S;$$
- Adjacency constraints$${{\displaystyle \sum}}_{i\text{}\u03f5\text{}C}{x}_{i,s}^{t}\le |C|-1,\text{}\forall C\in {\Lambda}^{+},\text{}\forall t\in T,\text{}\forall s\in S;$$
- Non-anticipativity rule$${x}_{i,s}^{t}=\text{}{x}_{i,\text{}s\prime \text{}}^{t}\forall s,\text{}{s}^{\prime}\in \text{}{N}_{t},\text{}\forall t\text{}\in \text{}T,\forall i\in I,\text{}s\text{}\ne s\prime ;$$
- Binary requirements$${x}_{i,s}^{t}\in \left\{0,1\right\},\text{}\forall i\text{}\in I,\text{}\forall t\in T,\text{}\forall s\in S.\text{}$$

#### 2.4. Progressive Hedging Approach

#### 2.4.1. Methodology

Algorithm 1. Progressive Hedging |

begin |

STEP 0 |

$k\text{}:=0$ |

${w}^{0,s}\text{}(\forall s\text{}\in \text{}S);$ |

$\overline{x}=0$ |

STEP 1 |

Solve each scenario by max: |

$ma{x}_{{\overrightarrow{x}}_{s}\text{}\in \text{}{Q}_{s}}\text{}{\displaystyle {\displaystyle \sum}_{t\text{}\u03f5\text{}T}}{\displaystyle {\displaystyle \sum}_{h\text{}\u03f5\text{}H}}{r}_{i,s}^{t}{A}_{i}{x}_{i,s}^{t};$ |

STEP 2 |

Compute the average solution in each node: |

$\overline{x}={\displaystyle {\displaystyle \sum}_{{}_{s\text{}\in S}}}\mathrm{Pr}\left(\text{s}\right)\overrightarrow{{x}_{s}}$ |

STEP 3 |

If the solutions are equal according to the criterion: |

$||\overrightarrow{x}-\overline{x}||<\epsilon ;$ |

then terminate. |

STEP 4 |

Update $\rho $ |

Update the penalty factor: |

${w}_{k,s}=\rho \left(\overrightarrow{x}-\overline{x}\right)+{w}_{k-1,s}\text{\hspace{1em}}\left(\forall s\text{}\in \text{}S\right);$ |

STEP 5 |

Solve each scenario with the penalty term: |

$ma{x}_{{\overrightarrow{x}}_{s}\text{}\in \text{}{Q}_{s}}\text{}{\displaystyle {\displaystyle \sum}_{t\text{}\u03f5\text{}T}}{\displaystyle {\displaystyle \sum}_{h\text{}\u03f5\text{}H}}{r}_{i,s}^{t}{A}_{i}{x}_{i,s}^{t}+{w}_{k,s}{\overrightarrow{x}}_{s}+\frac{\rho}{2}||\overrightarrow{x}-\overline{x}{||}^{2}$ |

STEP 6 |

Update iteration number |

k = k+1 |

Return to STEP 2 |

End |

#### 2.4.2. PH Heuristic

**Fixing Variables and Compact Model**

**Penalty Factor**

**Hot Starts**

**Dynamic Gap**

#### 2.5. Instances of the Original Problem

#### 2.6. Hardware and Software

## 3. Results

#### 3.1. Configurations

#### 3.2. Performance Comparison

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A scenario tree consisting of four scenarios and four stages (time periods) and 10 nodes (

**1**

**a**) is shown. The scenarios are also shown in a disaggregated form (

**1**

**b**); the non-anticipativity constraints (information sets Nt) are reflected by the balloons grouping the nodes. For example, in period two (t = 2) node two shares scenarios one and two and therefore the decisions for this node (harvest or not harvest the stand) for scenarios one and two must be the same.

**Table 1.**Comparison of performance of PH algorithm (PH) versus the extended formulation with CPLEX (EF) for all the instances including adjacency restriction and demand levels D. PH gap is computed using the best bound given by CPLEX when solving EF. The results shown in this table are the averages found for each of the 15 forests generated for each instance. NO SOL means that no solution was found in more than 50% of the 15 forest repetitions.

EF | PH | Statistics | |||
---|---|---|---|---|---|

Instance | Gap [%] | Time [s] | Gap [%] | Time [s] | Saved Time [%] |

100SD0% | NO SOL | NO SOL | 1.2 | 354.5 | - |

100SD25% | 23.2 | 36,000 | 7.5 | 2458.1 | 91.7 |

100SD50% | 5.2 | 36,000 | 4.8 | 920.2 | 98.3 |

100SD75% | 4.3 | 36,000 | 3.9 | 469.3 | 98.6 |

100SD90% | 3.6 | 36,000 | 3.5 | 765.2 | 98.7 |

100SD95% | NO SOL | NO SOL | 0.9 | 226.00 | - |

100SD100% | NO SOL | NO SOL | 1.9 | 5974 | - |

200SD0% | NO SOL | NO SOL | 3.8 | 608.4 * | - |

200SD25% | 49.4 | 36,000 | 5.4 | 3963.5 | 88.9 |

200SD50% | 27.9 | 36,000 | 6.2 | 827.1 | 97.7 |

200SD75% | 6.4 | 36,000 | 6.1 | 261.7 | 98.5 |

200SD90% | 2.9 | 36,000 | 2.7 | 224.6 | 98.7 |

200SD95% | NO SOL | NO SOL | 1.6 | 5410.6 | - |

200SD100% | NO SOL | NO SOL | 1.3 | 5853.8 | - |

400SD0% | NO SOL | NO SOL | 3.4 | 3006.7 | - |

400SD25% | 6.1 | 36,000 | 5.8 | 3996.8 | 69.6 |

400SD50% | 5.2 | 36,000 | 4.9 | 1082.2 | 95 |

400SD75% | 7.2 | 36,000 | 6.8 | 514.4 | 97.1 |

400SD90% | 5.0 | 36,000 | 4.7 | 429.0 | 97.5 |

400SD95% | NO SOL | NO SOL | 2.6 | 6887.6 * | - |

400SD100% | NO SOL | NO SOL | 1.7 | 6068.4 | - |

800SD0% | NO SOL | NO SOL | 2.3 | 3109.4 | - |

800SD25% | 1.0 | 11,092 | 1.1 | 3268.6 | 66.6 |

800SD50% | 1.5 | 9029 | 1.7 | 2275.1 | 76.3 |

800SD75% | 7.8 | 36,000 | 7.5 | 1406.8 | 92.2 |

800SD90% | 8.8 | 36,000 | 8.1 | 1390.9 | 93.8 |

800SD95% | NO SOL | NO SOL | 4.7 | 9519.8 * | - |

800SD100% | NO SOL | NO SOL | 2.7 | 11,603.2 * | - |

1000SD0% | NO SOL | NO SOL | 1.8 | 4844 | - |

1000SD25% | 0.9 | 6061 | 1.0 | 1574 | 91.4 |

1000SD50% | 1.0 | 8279 | 1.3 | 5283 | 91.6 |

1000SD75% | 7.9 | 36,000 | 7.5 | 2404 | 92.9 |

1000SD90% | 8.8 | 36,000 | 8.6 | 1787 | 93.8 |

1000SD95% | NO SOL | NO SOL | 13.7 | 36,000 | - |

1000SD100% | NO SOL | NO SOL | 13.1 | 36,000 | - |

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## Share and Cite

**MDPI and ACS Style**

Garcia-Gonzalo, J.; Pais, C.; Bachmatiuk, J.; Barreiro, S.; Weintraub, A. A Progressive Hedging Approach to Solve Harvest Scheduling Problem under Climate Change. *Forests* **2020**, *11*, 224.
https://doi.org/10.3390/f11020224

**AMA Style**

Garcia-Gonzalo J, Pais C, Bachmatiuk J, Barreiro S, Weintraub A. A Progressive Hedging Approach to Solve Harvest Scheduling Problem under Climate Change. *Forests*. 2020; 11(2):224.
https://doi.org/10.3390/f11020224

**Chicago/Turabian Style**

Garcia-Gonzalo, Jordi, Cristóbal Pais, Joanna Bachmatiuk, Susana Barreiro, and Andres Weintraub. 2020. "A Progressive Hedging Approach to Solve Harvest Scheduling Problem under Climate Change" *Forests* 11, no. 2: 224.
https://doi.org/10.3390/f11020224