Agricultural land allocation $-$ LP relaxation

The Collective of Agricultural Communities $($CAC$)$ refers to a cluster of three agricultural communities $($A$,$ B$,$ and C$),$ which are communal farming communities. The Central Planning Authority $($CPA$)$ oversees the collective planning for this group. Presently, the CPA is mapping out the agricultural production for the upcoming year.

Table $1$ presents information on the agricultural yield of each agricultural community, which is restricted by the amount of irrigable Land available and the quantity of irrigation water allotted by the Department of Irrigation and Drainage $($DID$),$ a government official at the national level.

The region$$s crop options, namely Rice, Corn, and Buckwheat, are under consideration for the upcoming season. These crops vary regarding their net return per hectare and water consumption. Moreover, this cluster is subject to a maximum quota for the total Land allocated to each crop, as presented in Table $2,$ by the CPA.

Table $1$: Data on resources

Agricultural communities $$ Available Land $($hectares$)$ Water allocation $($cubic hectometre$)$

A $479525$

B $592694$

C $415403$

Table $2$: Crop data

Crop plants $$ Maximum quota $($hectares$)$ Water consumption $($cubic hectometre$)$ Net return $($USD$/$hectare$)$

Rice $6553980$

Corn $4802775$

Buckwheat $3751530$

Due to the limited water availability for irrigation, the CAC cannot use its entire irrigable Land for crop cultivation in the upcoming season. To ensure fairness among the three agricultural communities, they have agreed to plant the same proportion of their respective available irrigable lands. For instance, if agricultural community A plants $239\mathrm{.}50$ hectares out of $479$ available hectares, then agricultural community B will plant $296\mathrm{.}00$ hectares out of $592$ available hectares, while agricultural community C will plant $207\mathrm{.}50$ out of $415$ available hectares. The Central Planning Authority needs to determine the number of hectares each crop will be allotted to the communities while adhering to the specified restrictions. The objective is to maximize the total net return of the Collective of Agricultural Communities as a whole. The communities may grow any combination of crops.