# Comparison of Selected Mathematical Programming Models Used for Sustainable Land and Farm Management

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Linear Programming (LP)

#### 2.2. Positive Mathematical Programming (PMP)

#### 2.3. Weighted Goal Programming (WPG)

## 3. Mathematical Programming Models

#### 3.1. Linear Programming (LP) Model

_{j}that satisfy the following linear constraints:

_{j}≥ 0, j = 1, 2, …, n which give the maximum or minimum of the linear function

#### 3.2. Positive Mathematical Programming (PMP) Model

- z is the objective function,
- x is an n × 1 vector of production activities,
- r is an n × 1 vector of gross margins of production activities,
- c is an n × 1 vector of variable costs,
- A is an m × n matrix of technoeconomic coefficients,
- b is an m × 1 vector of output and policy constraints, and
- π is an m × 1 vector of the shadow values of factor constraints (e.g., marginal productivity of exhausted constraints).

#### 3.3. Weighted Goal Programming (WGP) Model

- Initially a set of objectives is determined that are considered the most important for farmers;
- Then the pay-off matrix of the above objectives is determined;
- Finally, the pay-off matrix is used to calculate a set of weights that optimally reflect farmers’ preferences.

_{1}(X)… f

_{i}(X) f

_{n}(X) that represents the farmers’ true objectives (e.g., profit maximization, minimization of fertilizer use, minimization of labor use, etc.).

_{ij}is the value of the i-th feature when the j-th objective is optimized. When the payoff table is complete, we solve the following system of q (number of targets) equations:

_{j}are the weights attached to each objective that reproduces the farmer’s real behavior, f

_{ij}are the elements of the payoff matrix, and f

_{i}is the price achieved for the i-th objective, according to the existing crop plan.

_{j}it is necessary to search for the best possible solution, minimizing the sum of deviational variables that finds the closest set of weights. For this purpose, a weighted goal programming problem with percentage variable deviations is created [62]. This solution is found from the following linear programming model:

_{i}represents the positive deviation from target i and n

_{i}the negative deviation from it.

## 4. Methodology

- For all farms the same crops were considered as the decision variables and the real situation crop plan is the same for all models;
- The same constraints were applied to all models;
- The objective functions of all models were optimized using the same goals. The goals were maximization of gross margin, minimization of fertilizers use, and minimization of labor use;
- All the optimum results for all models were compared to the real situation crop plan;
- Finally, a set of sustainability indicators (economic, social, environmental) is calculated to help the comparison procedure.

#### 4.1. Real Situation

#### 4.2. Goals

_{i}) participating in the crop plan. The goals that we assumed reflected the farmers’ decisions were the maximization of gross margin:

#### 4.3. Constraints for All Models

_{i}) must add up to 100. This constraint is introduced to express the results of the models as percentages.

_{i}) must be less than or equal to the total working hours (TWH).

_{i}) must be less than or equal to the total available capital (TC).

_{i}) must be less than or equal to the real crop plan fertilizers use (TF).

#### 4.4. Indicators

## 5. Results

#### 5.1. Results for Linear Programming (LP) Model

#### 5.2. Results for Positive Mathematical Programming (PMP) Model

#### 5.3. Results for Weighted Goal Programming (WGP) Model

#### 5.4. Comparison of the Sustainability Indicators Results

^{3}. The implementation of the LP model showed an increase in water use, while there was a decrease in the WGP and PMP models. Then, the use of nitrates in the real situation equals 6791 kg. It increases when applying the LP and PMP models, while when applying the WGP model a reduction was observed. The electric power was equal to 21.46 MWh in the real situation, when in LP it was 20.82 MWh, in WGP was 20.25 MWh and in PMP 20.52 MWh. Finally, the thermal power in the real situation was equal to 94.46 MWh and reduced in all three mathematical programming models.

## 6. Discussion

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Indicators | Units | |
---|---|---|

Economic | Gross Income | EUR |

Gross Margin | EUR | |

Social | Labor Use | hours |

Annual Work Units | AWU | |

Seasonality | hours/month | |

Environmental | Crop diversity | Number of crops |

Land Cover | % | |

Water Use | m^{3} | |

Nitrates Use | kg | |

Electric power | MWh | |

Thermal power | MWh |

Real | LP | ||
---|---|---|---|

Values | % Deviation | ||

Gross Margin (EUR) | 15,699 | 16,573 | 5.6 |

Fertilizers use (kg) | 6791 | 6867 | 1.1 |

Labor Use (hours) | 2715 | 2748 | 1.2 |

Cotton | 10.80 | 5.72 | −47.0 |

Common wheat | 5.03 | 0.00 | −100.0 |

Durum wheat | 27.79 | 33.29 | 19.8 |

Sugarbeets | 2.45 | 0.00 | −100.0 |

Barley | 5.15 | 6.70 | 30.0 |

Alfalfa | 7.29 | 8.74 | 20.0 |

Maize | 8.97 | 10.76 | 19.9 |

Olive trees | 1.84 | 1.93 | 4.7 |

Rice | 16.72 | 20.07 | 20.0 |

Sunflower | 1.46 | 1.89 | 29.9 |

Tomatoes | 0.92 | 1.10 | 19.8 |

Potatoes | 0.55 | 0.67 | 20.8 |

Cherry trees | 5.45 | 5.72 | 4.9 |

Apple trees | 0.85 | 0.90 | 5.4 |

Peach trees | 2.05 | 2.15 | 4.8 |

Kiwi | 0.34 | 0.36 | 4.9 |

Vetch | 2.33 | 0.00 | −100.0 |

Total | 100.0 | 100.0 |

Real | PMP | ||
---|---|---|---|

Values | % Deviation | ||

Gross Margin (EUR) | 15,699 | 16,398 | 4.5 |

Fertilizers use (kg) | 6791 | 6801 | 0.1 |

Labor Use (hours) | 2715 | 2613 | −3.8 |

Cotton | 10.80 | 4.00 | −62.9 |

Common wheat | 5.03 | 4.96 | −1.5 |

Durum wheat | 27.79 | 32.48 | 16.9 |

Sugarbeets | 2.45 | 0.00 | −100.0 |

Barley | 5.15 | 6.12 | 18.7 |

Alfalfa | 7.29 | 8.74 | 20.0 |

Maize | 8.97 | 10.76 | 19.9 |

Olive trees | 1.84 | 1.93 | 4.7 |

Rice | 16.72 | 16.55 | −1.0 |

Sunflower | 1.46 | 1.45 | −0.4 |

Tomatoes | 0.92 | 0.92 | −0.2 |

Potatoes | 0.55 | 0.67 | 20.8 |

Cherry trees | 5.45 | 5.72 | 4.9 |

Apple trees | 0.85 | 0.90 | 5.4 |

Peach trees | 2.05 | 2.15 | 4.8 |

Kiwi | 0.34 | 0.36 | 4.9 |

Vetch | 2.33 | 2.30 | −1.3 |

Total | 100.0 | 100.0 |

**Table 4.**Comparison of the Real situation crop plan vs. WGP model optimum crop plan. Source: Our elaboration.

Real | WGP | ||
---|---|---|---|

Values | % Deviation | ||

Gross Margin (EUR) | 15,699 | 16,511 | 5.2 |

Fertilizers use (kg) | 6791 | 6706 | −1.3 |

Labor Use (hours) | 2715 | 2642 | −2.7 |

Cotton | 10.80 | 0.00 | −100.0 |

Common wheat | 5.03 | 0.00 | −100.0 |

Durum wheat | 27.79 | 35.98 | 29.5 |

Sugarbeets | 2.45 | 0.00 | −100.0 |

Barley | 5.15 | 6.70 | 30.0 |

Alfalfa | 7.29 | 8.74 | 20.0 |

Maize | 8.97 | 10.76 | 19.9 |

Olive trees | 1.84 | 1.93 | 4.7 |

Rice | 16.72 | 20.07 | 20.0 |

Sunflower | 1.46 | 1.89 | 29.9 |

Tomatoes | 0.92 | 1.10 | 19.8 |

Potatoes | 0.55 | 0.67 | 20.8 |

Cherry trees | 5.45 | 5.72 | 4.9 |

Apple trees | 0.85 | 0.90 | 5.4 |

Peach trees | 2.05 | 2.15 | 4.8 |

Kiwi | 0.34 | 0.36 | 4.9 |

Vetch | 2.33 | 3.03 | 30.2 |

Total | 100.0 | 100.0 |

**Table 5.**Comparison of mathematical programming models for the sustainability indicators. Source: Our elaboration.

Real | PMP | LP | WGP | ||||
---|---|---|---|---|---|---|---|

Values | Values | % Deviation | Values | % Deviation | Values | % Deviation | |

Economic | |||||||

Gross Margin (EUR) | 15,699 | 16,398 | 4.45% | 16,573 | 5.57% | 16,511 | 5.17% |

Gross Income (EUR) | 29,499 | 29,475 | −0.08% | 30,375 | 2.97% | 29,694 | 0.66% |

Social | |||||||

Labor Use (hours) | 2715 | 2.613 | −3.77% | 2748 | 1.23% | 2642 | −2.69% |

Annual Work Units (AWU) | 1.55 | 1.49 | −3.77% | 1.57 | 1.23% | 1.51 | −2.69% |

Seasonality (hours/month) | 226.24 | 217.73 | −3.77% | 229.03 | 1.23% | 220.17 | −2.69% |

Environmental | |||||||

Crop Diversity (number) | 17 | 16 | −5.88% | 14 | −17.65% | 14 | −17.65% |

Land Cover (%) | 74.84 | 74.79 | −0.07% | 75.28 | 0.58% | 74.77 | −0.11% |

Water Use (m^{3}) | 38,435 | 35,491 | −7.66% | 40,016 | 4.11% | 36,870 | −4.07% |

Nitrates Use (kg) | 6791 | 6801 | 0.14% | 6867 | 1.11% | 6706 | −1.25% |

Electric power (MWh) | 21.46 | 20.52 | −4.38% | 20.82 | −2.99% | 20.25 | −5.65% |

Thermal power (MWh) | 97.04 | 92.85 | −4.31% | 94.02 | −3.11% | 91.73 | −5.47% |

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Moulogianni, C.
Comparison of Selected Mathematical Programming Models Used for Sustainable Land and Farm Management. *Land* **2022**, *11*, 1293.
https://doi.org/10.3390/land11081293

**AMA Style**

Moulogianni C.
Comparison of Selected Mathematical Programming Models Used for Sustainable Land and Farm Management. *Land*. 2022; 11(8):1293.
https://doi.org/10.3390/land11081293

**Chicago/Turabian Style**

Moulogianni, Christina.
2022. "Comparison of Selected Mathematical Programming Models Used for Sustainable Land and Farm Management" *Land* 11, no. 8: 1293.
https://doi.org/10.3390/land11081293