Bonus problem $2$: $(6$ points$)$ Give an LP formulation of this process model. Explain the variables with clear definitions and a diagram. Make the objective function and constraints clear.

Consider the following production process modeling, which is similar to what are discussed in the textbook and homework. $($Note that the problem is not exactly the same$).$ Write the constraints of the LP formulation of the problem. $($The objective function is not needed.$)$ Non$-$negativity constraints are not required. Use the variables given below.

Chemco produces three products: $1,2,$ and $3.$ Each pound of its raw material costs $$45.$ It undergoes a process and yields $2$ oz of product $1$ or $1$ oz of product $2.$ Or it can produce any combination of product $1$ or product $2$ following the ratio. For example, one pound of the raw material can produce $1$ oz of product $1$ and $\frac{1}{2}oz$ of product $2.$ It is also possible to produce $\frac{1}{2}$ oz of product $1$ and $0\mathrm{.}75$ oz of product $2$ using one pound of the raw material. It costs $$1$ and $2$ tons of water to process each pound of raw material. Each ounce of product $1$ can be used in one of these three ways.

It can be sold for $$\frac{12}{o}z.$

It can be processed into $1$ oz of product $2.$ This requires $3$ tons of water and costs $$1.$

It can be processed into $2$ oz of product $3.$ This requires $2$ tons of water and costs $$2.$

Product $3$ is sold for $$\frac{50}{o}z.$

Each ounce of product $2$ can be used in one of these three ways.

It can be sold for $$\frac{20}{o}z.$

It can be processed into $1$ oz of product $1.$ This requires $2$ tons of water and costs $$1.$

It can be processed into $1$ oz of product $3.$ This requires $1$ ton of water and costs $$6.$

A total of $1,000$ tons of water is available.

x $0$ : number of pounds of raw material used

x $1$ : number of ounces product $1$ produced directly from the raw material

x $2$ : number of ounces product $2$ produced directly from the raw material

x $11$ : number of ounces product $1$ sold without product conversion from product $1.$

x $12$ : number of ounces product $2$ converted from product $1.$

X $21$ : number of ounces product $1$ converted from product $2.$

x $13$ : number of ounces product $3$ sold converted from product $1.$

x $22$ : number of ounces product $2$ sold without product conversion from product $2.$

$23$ : number of ounces product $3$ sold converted from product $2.$

$($If you need any additional variables, add them.$)$