Problem $5$: Scrap yard $1$ has $500$ tons of scrap steel available, and scrap yard $2$ has $800$ tons of scrap

steel available. There are two recycling facilities, each of which can process up to $800$ tons of scrap steel.

The costs per ton to transport scrap steel from each scrap yard to each recycling facility are given in the

following table:

The cost to process scrap steel is $$\frac{55}{?}$ ton at facility $1$ and $$\frac{40}{?}$ ton at facility $2.$ For each ton of scrap

steel processed, $0\mathrm{.}95$ tons of recycled steel are produced. Recycled steel can be transported to one of

two customers. The costs per ton to transport recycled steel from each recycling facility to each

customer are given in the table below:

Customers $1$ and $2$ respectively demand $300$ and $700$ tons of recycled steel.

Part a: Formulate an LP to determine a plan for transporting and processing steel to minimize the cost of

satisfying customer demands.

Part b: Formulate an indexed LP $($using summation and "for every"$)$ that generalizes your LP from part a

to an arbitrary number of scrap yards, recycling facilities, and customers.