$($C$1,$ C$2,$ C$3,$ and C$4)$ to three waste disposal facilities $($Facility$1,$ Facility$2,$ and Facility$3).$ The objective is to minimize the total transportation costs while considering the following complex factors:

$1.$ The transportation costs between each collection point and disposal facility are known, and they vary depending on distance and road conditions.

$2.$ Each collection point generates waste at different rates. These rates may fluctuate seasonally or due to other factors.

$3.$ The city is committed to reducing its carbon footprint. It wants to minimize emissions associated with waste transportation.

$4.$ Different types of waste $($e$.$g$.,$ hazardous, non$-$hazardous$)$ are generated at each collection point. The disposal facilities must adhere to specific waste type regulations.

$5.$ Each disposal facility has a limited capacity for storing waste. Overfilling a facility can incur extra costs and potential environmental issues.

In the tableau below, each cell contains two values separated by a comma. The first value represents the transportation cost per unit $($in $$),$ and the second value represents the carbon emissions per unit $($in kg$).$ The transportation cost accounts for both cost and environmental impact $($assume that every unit kg of emissions cost $$0\mathrm{.}5).$

Facility$1$ Facility$2$ Facility$3$

C$15,107,156,12$

C$28,186,149,20$

C$36,125,108,16$

C$47,149,207,15$

Capacity:

C$1$:

o Hazardous waste: $15$ units

o Non$-$hazardous waste: $45$ units

C$2$:

o Hazardous waste: $10$ units

o Non$-$hazardous waste: $30$ units

C$3$:

o Hazardous waste: $20$ units

o Non$-$hazardous waste: $30$ units

C$4$:

o Hazardous waste: $25$ units

o Non$-$hazardous waste: $45$ units

$$ Facility$1$ can process $30$ units of hazardous waste and $40$ units of non$-$hazardous waste.

$$ Facility$2$ can process $20$ units of hazardous waste and $60$ units of non$-$hazardous waste.

$$ Facility$3$ can process $50$ units of hazardous waste and $40$ units of non$-$hazardous waste.

$1.$ How can the LP model incorporate carbon emissions as part of the objective function $($Simplex LP$)$ to minimize the environmental impact of waste transportation?

$2.$ Formulate constraints to ensure that each facility receives only the types of waste it can process without violating regulations.