# Comparison of Three Computational Approaches for Tree Crop Irrigation Decision Support

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. System Definition

#### 2.2. Crop Yield

^{3}/m

^{3}]:

^{3}/m

^{3}] and ${\theta}_{w}$ [m

^{3}/m

^{3}] are the field capacity and wilting point of the soil, respectively, and $p$ is the fraction of potentially available soil moisture that can be depleted from the root zone or depth ${Z}_{r}$ before moisture stress occurs. Daily soil moisture $\theta \left(t\right)$ can be estimated using a simple water balance model that neglects the terms of runoff and groundwater:

#### 2.3. Strategies

#### 2.4. Simple Multicriteria Approach

_{j}, we will subjectively assign a utility value u

_{ji}provided the resulting production yield after its implementation is ${S}_{i}$ ($i$ = A,B,C,D,E). Utility values express the cost or advantage for each pair of alternative action—future state [53]. The factors defining each scenario’s utility are normalized and combined with utility indexes for each potential state:

#### 2.5. Multicriteria Approach with Posterior Information

#### 2.6. Multicriteria Fuzzy Approach

#### 2.7. Decision Tree and the ID3 Algorithm

- which is the next attribute to split,
- when splitting is terminated, and
- how to assign terminal nodes to a class.

_{2}0 is assumed as 0. If decisions for all objects are the same, then entropy will be zero. This means that it is not necessary to split the node on the corresponding decision level [60]. Every time a split is done, S is updated. If E(S) = 0 or no attribute remains to be split, then this branch is terminated. If all branches are terminated, then classification is complete. Let $Ti$ be the set of subsets created from splitting set $S$ by attribute Ai: Information gain for each attribute is computed as follows [60]:

#### 2.8. Decision Variable Importance

## 3. Case Study

#### 3.1. Study Area

^{3}[70], but the nonuniform distribution in space (a negative gradient of almost 300 mm from west to east and a strong orographic effect) and time (dry summers) makes dry season precipitation a very small but crucial portion of the total supply [71]. Figure 2 and Table 1 show dry season (March to November) weather parameters for three characteristic years—dry, normal and wet—with probabilities of occurrence equal to 20%, 60% and 20%.

#### 3.2. Argicultural Input and Water Cost

^{3}with an average (normalized per distributed m

^{3}) of 0.13 €/m

^{3}[79] or 13 mm/ha irrigated, depending on water scarcity or quality. Here, we tested three low-to-moderate price scenarios of 0.05, 0.10, and 0.18 €/m

^{3}assuming that under high demand or climate change conditions, values will increase. Additionally, we considered that each irrigation event causes an additional expense of 50 €/event.

#### 3.3. Crop Yield

#### 3.4. Alternative Conditions and Strategies

**Table 3.**Soil moisture at field capacity (${\theta}_{fc}$ ) and wilting point (${\theta}_{w}$ ) for texture classes at 2.5%w organic matter (OM), no salinity, gravel or density adjustment (values after [84]).

Soil Texture Class | ${\mathit{\theta}}_{\mathit{f}\mathit{c}}$ | ${\mathit{\theta}}_{\mathit{w}}$ |
---|---|---|

Loamy Sand (LS) | 0.15 | 0.07 |

Sandy Loam (SL) | 0.23 | 0.11 |

Clay (Cl) | 0.36 | 0.22 |

#### 3.5. Simple Multicriteria Approach

- The relative amount (in percentage) of the disposed water used during the irrigation compared to the full-scale irrigation in order to reach the field capacity (100%, 75%, 50%).
- The reduction in frequency (number of irrigation times) during the cultivation season compared to the number recommended for maximum crop yield (recommended n -1, recommended n-2).
- The profit from the farm. This is calculated by subtracting the costs of irrigation from the revenue from selling the crop.

#### 3.6. Multicriteria Approach with Posterior Information

#### 3.7. Multicriteria Fuzzy Approach

- Excellent application (Fuzzy Element 1).
- Good application (Fuzzy Element 2).
- Moderate application (Fuzzy Element 3).

#### 3.8. Shortcomings of Probabilistic Approaches

#### 3.9. Decision Tree and the ID3 Algorithm

- Soil type. Possible soil types are loamy sand, sandy loam, and clay.
- Weather during cropping season. Wet, normal, and dry.
- Management practices. They are chosen to be: M1 heavy pruning and M2 light pruning. Tree pruning may bring down the total production, but it is a wise choice during a dry year (low in precipitation).
- The irrigation amount as a percentage of the recommended amount per irrigation event.
- The reduction in irrigation events related to the recommended.

## 4. Results and Discussion

#### 4.1. Simple Multicriteria Approach

#### 4.2. Multicriteria Approach with Posterior Information

#### 4.3. Multicriteria Fuzzy Approach

#### 4.4. Decision Tree and the ID3 Algorithm

#### 4.5. Limitations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Cumulative precipitation at the station of Moires, Crete, during the dry season (March to November) for a Dry (D), Normal, and Dry (D) year.

**Figure 3.**Histogram of olive tree farm yield [t/ha] for selected municipal units of Heraklion, Crete, for the period 2000–2007 [72].

**Figure 4.**Decision tree using the ID3 algorithm for (

**a**) 75% of the recommended irrigation and (

**b**) 100% of the recommended irrigation.

**Figure 5.**Gini index for the variables of the decision tree. Boxes depict the range of the 25–75% percentile of variable Gini’s importance over all training iterations, and the bar is the median value. The whiskers depict the minimum and maximum values and the dots are outliers.

**Table 1.**Total precipitation and average temperature $T$ at the station of Moires, Crete, during the dry season (March to November) for a Dry (D), Normal, and Dry (D) year and respective total $E{T}_{0}$ values calculated using Blaney-Criddle [37].

Scenario | Total Precipitation * [mm] | Average Temperature [°C] | Total Reference Evapotranspiration [mm] |
---|---|---|---|

Wet year (W) | 251.9 | 20.7 | 1347.8 |

Normal year (N) | 149.2 | 21.1 | 1361.3 |

Dry year (D) | 93 | 21.4 | 1374.9 |

**Table 2.**Crop yield classes and likelihood of occurrence based on [72].

Yield Class | Yield Range [t/ha] | $\mathbf{Likelihood}\text{}\mathbf{p}\left({\mathbf{s}}_{\mathbf{i}}\right)$ |
---|---|---|

E | [1, 2] | 0.024 |

D | [2, 4] | 0.667 |

C | [4, 6] | 0.214 |

B | [6, 8] | 0.067 |

A | ≥8 | 0.029 |

**Table 4.**Modeled irrigation strategies relative to the optimum irrigation amount per irrigation event ${S}_{Ix\%}$ and to the optimum number of irrigation events ${S}_{IE-n}$.

Scenario | Relative Irrigation | Reduction of Irrigation Events |
---|---|---|

1 | 100% | 0 |

2 | 75% | 0 |

3 | 50% | 0 |

4 | 100% | 1 |

5 | 75% | 1 |

6 | 50% | 1 |

7 | 100% | 2 |

8 | 75% | 2 |

9 | 50% | 2 |

Yield Label | $\mathbf{Likelihood}\text{}\mathbf{p}\left({\mathbf{s}}_{\mathbf{i}}\right)$ |
---|---|

A | 0.029 |

B | 0.067 |

C | 0.214 |

D | 0.667 |

E | 0.024 |

Precipitation | $\mathbf{Likelihood}\text{}\mathbf{p}\left({\mathbf{c}}_{\mathbf{k}}\right)$ |
---|---|

Dry | 0.2 |

Normal | 0.6 |

Wet | 0.2 |

Irrigation Strategy | 50%/n-2 | 50%/n-1 | 50%/n | 75%/n-2 | 75%/n-1 | 75%/n | 100%/n-2 | 100%/n-1 | 100%/n | |
---|---|---|---|---|---|---|---|---|---|---|

Profit | ||||||||||

Low | 7 | 6 | 5 | 6 | 5 | 4 | 5 | 4 | 3 | |

Medium | 8 | 7 | 6 | 7 | 6 | 5 | 6 | 7 | 4 | |

High | 9 | 8 | 7 | 8 | 7 | 6 | 7 | 8 | 5 |

Profit Class | Total Score Range |
---|---|

Moderate | [1, 5) |

Good | [5, 7] |

Excellent | (7, 9] |

Scenario → | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

$\mathit{E}\left({\mathit{a}}_{\mathit{j}}\right)$ | 3.242 | 3.242 | 3.242 | 3.6341 | 4.8799 | 4.9742 | 4.0259 | 5.6277 | 6.5173 |

Precipitation State | DRY | NORMAL | WET |
---|---|---|---|

$\mathit{p}\left(\mathit{A}|{\mathit{c}}_{\mathit{k}}\right)$ | 0.1667 | 0.1667 | 0.1296 |

$\mathit{p}\left(\mathit{B}|{\mathit{a}}_{\mathit{k}}\right)$ | 0.8333 | 0.8333 | 0.7037 |

$\mathit{p}\left(\mathit{C}|{\mathit{c}}_{\mathit{k}}\right)$ | 0 | 0 | 0.1667 |

$\mathit{p}\left(\mathit{D}|{\mathit{c}}_{\mathit{k}}\right)$ | 0 | 0 | 0 |

$\mathit{p}\left(\mathit{E}|{\mathit{c}}_{\mathit{k}}\right)$ | 0 | 0 | 0 |

Scenario → | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

$\mathit{E}\left({\mathit{a}}_{\mathit{j}}\right)$ | 8.3334 | 8.3334 | 8.3334 | 9.3409 | 12.5435 | 12.5435 | 10.3484 | 14.4659 | 16.7527 |

Irrigation Strategy | 50%/n-2 | 50%/n-1 | 50%/n | 75%/n-2 | 75%/n-1 | 75%/n | 100%/n-2 | 100%/n-1 | 100%/n | |
---|---|---|---|---|---|---|---|---|---|---|

Profit | ||||||||||

Low | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 1/2 | μ_{mod} = 0 | μ_{mod} = 1/2 | μ_{mod} = 1 | μ_{mod} = 1/2 | μ_{mod} = 1 | μ_{mod} = 1 | |

μ_{good} = 1/2 | μ_{good} = 1 | μ_{good} = ½ | μ_{good} = 1 | μ_{good} = ½ | μ_{good} = 0 | μ_{good} = ½ | μ_{good} = 0 | μ_{good} = 0 | ||

μ_{exc} = 1/2 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 0 | ||

Medium | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = ½ | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 1 | |

μ_{good} = 0 | μ_{good} = ½ | μ_{good} = 1 | μ_{good} = ½ | μ_{good} = 1 | μ_{good} = ½ | μ_{good} = 1 | μ_{good} = ½ | μ_{good} = 0 | ||

μ_{exc} = 1 | μ_{exc} = 1/2 | μ_{exc} = 0 | μ_{exc} = 1/2 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 0 | μ_{exc} = 1/2 | μ_{exc} = 0 | ||

High | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = 0 | μ_{mod} = ½ | |

μ_{good} = 0 | μ_{good} = 0 | μ_{good} = ½ | μ_{good} = 0 | μ_{good} = ½ | μ_{good} = 1 | μ_{good} = ½ | μ_{good} = 0 | μ_{good} = ½ | ||

μ_{exc} = 1 | μ_{exc} = 1 | μ_{exc} = 1/2 | μ_{exc} = 1 | μ_{exc} = 1/2 | μ_{exc} = 0 | μ_{exc} = 1/2 | μ_{exc} = 1 | μ_{exc} = 0 |

$\mathbf{p}\left({\tilde{\mathbf{F}}}_{\mathbf{excellent}}\right)$ | $\mathbf{p}\left({\tilde{\mathbf{F}}}_{\mathbf{good}}\right)$ | $\mathbf{p}\left({\tilde{\mathbf{F}}}_{\mathbf{moderate}}\right)$ | $\sum _{\mathbf{m}=1}^{3}}\mathbf{p}\left({\tilde{\mathbf{F}}}_{\mathbf{m}}\right)$ |
---|---|---|---|

0.3149 | 0.4444 | 0.2407 | 1 |

Scenario → | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Simple multicriteria$\mathit{E}\mathbf{\left(}{\mathit{a}}_{\mathit{j}}\mathbf{\right)}$ | 3.2420 | 3.2420 | 3.2420 | 3.6341 | 4.8799 | 4.9742 | 4.0259 | 5.6277 | 6.5173 |

Fuzzy$\mathit{E}\mathbf{\left(}{\mathit{a}}_{\mathit{j}}\mathbf{\right)}$ | 3.0000 | 3.0000 | 3.0000 | 3.3627 | 4.5156 | 4.6029 | 3.7254 | 5.2077 | 6.0309 |

Management Practice | Soil Type | Climate | Relative Irrigation | Trip Reduction Times | Irrigation/Trip (mm/ha) | Water Price (€/m^{3}) | Profit (€/ha) | Profit |
---|---|---|---|---|---|---|---|---|

M1 | Cl | Normal | 100% | 1 | 134.4 | 0.05 | 1480.1 | Medium |

M1 | SL | Wet | 50% | 2 | 129.6 | 0.05 | 281.8 | Low |

M1 | SL | Normal | 50% | 2 | 72 | 0.13 | 679.5 | Low |

M1 | SL | Dry | 50% | 0 | 43.2 | 0.13 | 579.5 | Low |

© 2020 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**

Christias, P.; Daliakopoulos, I.N.; Manios, T.; Mocanu, M.
Comparison of Three Computational Approaches for Tree Crop Irrigation Decision Support. *Mathematics* **2020**, *8*, 717.
https://doi.org/10.3390/math8050717

**AMA Style**

Christias P, Daliakopoulos IN, Manios T, Mocanu M.
Comparison of Three Computational Approaches for Tree Crop Irrigation Decision Support. *Mathematics*. 2020; 8(5):717.
https://doi.org/10.3390/math8050717

**Chicago/Turabian Style**

Christias, Panagiotis, Ioannis N. Daliakopoulos, Thrassyvoulos Manios, and Mariana Mocanu.
2020. "Comparison of Three Computational Approaches for Tree Crop Irrigation Decision Support" *Mathematics* 8, no. 5: 717.
https://doi.org/10.3390/math8050717