## 4 Modelling
### Biophysical Constraints  
The determination of a crop rotation plan for a set of plots can be limited biophysically by factors related to the biological characteristics of the crops and the physical characteristics of the soil. As biophysical constraints, we have:  
* Two crops cannot be planted in the same plot [24]. In fact, a plot is entirely affected to a crop for the duration of its growing process and its harvest. \[\sum_{i=1}^{N}\sum_{r=0}^{t_{i}-1}x_{i(j-r)k}\ \leq 1\] (1) With \(j=1\ldots M,\ k=1\ldots L\).
* Succession of crops from the same botanical family on the same plot is not allowed [24]. This problem is mainly due to the fact that crops of the same family have similar nutrient requirements and defects (risk of contracting the same diseases or hosting the same weeds). This compromises the agronomic sustainability of the cropping system. \[\sum_{i\ \epsilon\ F(p)}\sum_{r=0}^{t_{i}}x_{i(j-r)k}\ \leq 1\] (2) With \(p=1\ldots NF,\ j=1\ldots M,\ k=1\ldots L\). Constraint (2) limits the sum of all crops \(i\) of botanical family \(p\) over their production period to a maximum of one crop.
* Two crops of the same botanical family should not be planted at the same time on two adjacent plots [24; 33]. This problem is also due to the fact that crops of the same family share common attributes. Thus, planting them on two adjacent plots at the same time is equivalent to planting the same crop on both plots, which does not optimize the optimal distribution of crops on the different plots.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter/Variable** & **Meaning** \\ \hline \(M\) & Number of periods (in months) \\ \(P_{i}\) & Price of crop \(i\) (in XAF—currency in Central Africa) \\ \(L\) & Number of plots \\ \(C\) & Set of main crops (cash crops) \\ \(N\) & Cardinality of \(C\) \\ \(NF\) & Number of botanical families \\ \(F(p)\) & Set of crops of the botanical family \(p\) \\ \(t_{i}\) & Production time of the crop \(i\) \\ \(r\) & Variables for indexing the production time of crop \(i\) \\ \(Cost_{i}\) & Production cost associated with the use of the resource \(l\) (in XAF) \\ \(A[k]\) & Set of parcels adjacent to parcel \(k\) \\ \(l\) & Number of independent plots \\ \(Z_{N*k}\) & Amount of nitrogen applied to plot \(k\) over a time interval \(\alpha\) (in Kg/ha) \\ \(S_{Ni}\) & Minimum amount of nitrogen required to start the crop \(i\) (Kg/ha) \\ \(B\) & Budget allocated for a production cycle (XAF) \\ \(q_{ijl}\) & Quantity of resource \(l\) needed for crop \(i\) over period \(j\) \\ \(R_{i}\) & Crop yield \(i\) (in ton) \\ \(x_{ijk}\) & A decision variable that tells whether crop \(i\) begins in period \(j\) on the plot \(k\) \\ \(D_{ij}\) & Demand for crop \(i\) over period \(j\) (in ton) \\ \(Q_{l}\) & Quantity of resource \(l\) \\ \(\alpha\) & Length of the interval \(\omega\) (in month) \\ \(O\) & The set of resources \\ \(m\) & The cardinality of \(O\) \\ \(\omega\) & Number of intervals of \(M\) (for amendments) \\ \hline \hline \end{tabular}
\end{table}
Table 3: List of parameters and variables.  
\[\sum_{i\;\epsilon\;F(p)}\sum_{r=0}^{t_{i}}\sum_{u\;\epsilon\;A[k]}x_{i(j-r)u}<=I \left(1-\sum_{i\;\epsilon\;F(p)}\sum_{r=0}^{t_{i}}x_{i(j-r)k}\right) \tag{3}\]  
With \(k=1\ldots L\;,\;j=1\ldots M\).  
Constraint (3) states that the number of crops from a botanical family \(p\) during their periods of production on the set of plots \(A[k]\) adjacent to a plot \(k\) must be equal to 0 if a crop \(i\) of the same botanical family \(p\) is already sown on plot \(k\). Otherwise, the number of crops is at most equal to the number of independent plots.