# A Bilevel Linear Programming Model for Developing a Subsidy Policy to Minimize the Environmental Impact of the Agricultural Sector

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}(x, y∗)

_{1}(x, y∗) ≤ 0

_{1}(x, y∗) = 0

_{2}(x, y)

_{2}(x, y) ≤ 0

_{2}(x, y) = 0

_{1}, f

_{2}, g

_{1}, g

_{2}, h

_{1}, h

_{2}are linear functions, then the model in Equations (1)–(6) is a BLP model, which makes its solution considerably easier [17].

#### 2.1. The Lower Level Model

**Sets**

**Parameters**

_{i}: Cost of crop i per unit of product

_{i}: Yiel d of units of product of crop i per unit of farmland

_{i}: Subsidy per unit of farmland with crop i

^{k}: Available units of farmland of farmer k

_{i}: Maximum subsidized units of farmland per crop i

**Variables**

_{i}is considered a parameter for the LLM but is derived by the ULM, where it is a decision variable, whereas x

_{i}

^{k}and X

_{i}

^{k}are decision variables for the LLM but parameters for the ULM.

#### 2.2. The Upper Level Model

**Sets**

**Parameters**

**Variables**

_{i}denotes the environmental impact of crops and can be replaced with the metric to be minimized in each case, such as water usage, energy usage, greenhouse gas emissions, or pesticide use. Of course, due to the Water-Energy-Food-Climate Nexus, minimizing one resource’s usage in the agricultural sector might positively influence all other interlinked resources. For example, minimizing the water usage for irrigation would also reduce the overall energy usage, since a majority of the energy used in agriculture is used for pumping out groundwater. Of course, more conditional constraints could be added to the ULM depending on each case’s needs, as discussed later.

#### 2.3. Formulation of Single Level Model

**Sets**

**Parameters**

**Variables**

#### 2.4. Case Study

## 3. Results

**Sets**

**Parameters**

**Variables**

_{i}is treated as a parameter and not a variable. To achieve this, the subsidy value is set as the break-even point where the subsidized crop is more profitable than the most profitable non-subsidized crop. This relationship between subsidy value s

_{i}and most profitable non-subsidized crop is shown in Equations (28) and (29).

## 4. Discussion

_{mini}is the minimum production of crop i.

_{k}, which denotes the number of individual farmers that belong in each category k. Finally, parameter e

_{i}of Equation (18) is the energy expenditure of crop i.

_{i}can be substituted for any metric required without affecting the model’s useability.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Crops | Total Farmland (m^{2}) | Yearly Production (kg) | Total Product Value (€) | Crop Yield per Farmland (kg/m^{2}) | Value per Kilo of Crop (€/kg) | Crop Value per Farmland (€/m^{2}) | Water Use (m^{3}/m^{2}) | Energy Use (kJ/m^{2}) |
---|---|---|---|---|---|---|---|---|

Rice | 1,823,146 | 1,511,072 | 453,321.55 | 0.829 | 0.3 | 0.25 | 1.134 | 2866.7 |

Corn | 224,804,489 | 264,718,258 | 47,649,286 | 1.178 | 0.18 | 0.212 | 0.547 | 1851 |

Other Cereals (I) | 42,558,786 | 335,204,399 | 2,007,327 | 0.635 | 0.2 | 0.127 | 0.5 | 1769.6 |

Vegetables (I) | 70,909,286 | 248,468,992 | 268,163,519 | 3.227 | 0.8 | 2.58 | 0.658 | 2042.9 |

Fruit (I) | 112,216,852 | 717,649 | 223,622,093 | 2.214 | 0.9 | 1.99 | 0.7 | 2115.82 |

Citrus (I) | 755,589 | 25,201,356 | 215,294 | 1.45 | 0.3 | 0.435 | 0.755 | 2211.84 |

Olives (I) | 129,807,752 | 22,334,289 | 58,467,144 | 0.194 | 2.32 | 0.45 | 0.409 | 1611.81 |

Potatoes | 10,940,576 | 2,816,550 | 7,817,001 | 2.04 | 0.35 | 0.71 | 0.4 | 1596.43 |

Lentils | 24,299,555 | 49,097,293 | 4,506,479 | 0.116 | 1.6 | 0.18 | 0.31 | 1440.06 |

Sugar beets | 7,115,935 | 247,865,051 | 1,472,918 | 6.9 | 0.03 | 0.2 | 0.551 | 1857.56 |

Cotton (I) | 915,527,563 | 3,755,551 | 148,719,030 | 0.27 | 0.6 | 0.16 | 0.613 | 1965.63 |

Tobacco | 10,628,109 | 425,891,847 | 13,144,430 | 0.353 | 3.5 | 1.24 | 0 | 903 |

Wheat | 1,358,529,260 | 67,321,529 | 51,107,021 | 0.313 | 0.12 | 0.03 | 0 | 903 |

Other Cereals (NI) | 88,896,717 | 10,036,637 | 13,464,305 | 0.312 | 0.2 | 0.06 | 0 | 903 |

Vegetables (NI) | 215,531,784.3 | 2,874,543 | 2,299,634 | 1.13 | 0.8 | 0.9 | 0 | 903 |

Fruit (NI) | 2,552,328.26 | 148,621,518 | 133,759,366 | 0.67 | 0.9 | 0.6 | 0 | 903 |

Citrus (NI) | 278,591.23 | 264,602 | 7,938,073,442 | 0.95 | 0.3 | 0.28 | 0 | 903 |

Olives (NI) | 218,642,706.2 | 42,448,100 | 98,479,594 | 0.194 | 2.32 | 0.45 | 0 | 903 |

Nuts | 39,950,885.66 | 13,588,696 | 23,100,784 | 0.34 | 1.7 | 0.58 | 0.51 | 1786.32 |

Cotton (NI) | 1,605,045.55 | 327,213 | 196,328 | 0.2 | 0.6 | 0.12 | 0 | 903 |

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**MDPI and ACS Style**

Ziliaskopoulos, K.; Papalamprou, K.
A Bilevel Linear Programming Model for Developing a Subsidy Policy to Minimize the Environmental Impact of the Agricultural Sector. *Sustainability* **2022**, *14*, 7651.
https://doi.org/10.3390/su14137651

**AMA Style**

Ziliaskopoulos K, Papalamprou K.
A Bilevel Linear Programming Model for Developing a Subsidy Policy to Minimize the Environmental Impact of the Agricultural Sector. *Sustainability*. 2022; 14(13):7651.
https://doi.org/10.3390/su14137651

**Chicago/Turabian Style**

Ziliaskopoulos, Konstantinos, and Konstantinos Papalamprou.
2022. "A Bilevel Linear Programming Model for Developing a Subsidy Policy to Minimize the Environmental Impact of the Agricultural Sector" *Sustainability* 14, no. 13: 7651.
https://doi.org/10.3390/su14137651