You are the transportation manager and are responsible for scheduling the daily delivery of the blended oil$($standard oil, premium oil and green oil$)$ to $5$ stations with the minimum total shipping cost.

Table $3$ summarizes the shipping cost per litre of the blended oil. Be aware that each shipment contains one blended oil only.

Table $3$: Shipping cost per litre of the standard oil, premium oil and green oil

SUPPLY$/$DEMAND STATION $1$ STATION $2$ STATION $3$ STATION $4$ STATION $5$

STANDARD OIL $$2$ $$1\mathrm{.}5$ $$3$ $$1\mathrm{.}5$ $$1\mathrm{.}25$

PREMIUM OIL $0$ $$2\mathrm{.}5$ $$2\mathrm{.}5$ $$1\mathrm{.}50$

GREEN OIL $$2\mathrm{.}5$ $$30$ $$2$ $$2$

$($Note: If a cost is missing in the table, the corresponding route is not available$)$

Assume today is $2022-12-31$ and you need to schedule the delivery. The total supply and demand are listed below.

Table $4$: Total Supply of Engine Oil on $2022-12-31$

Product Standard Oil Premium Oil Green Oil

Total Supply $($liters$)83001250011283$

Table $5$: Total demand for Engine Oil on $2022-12-31$

Stations Station $1$ Station $2$ Station $3$ Station $4$ Station $5$

Total Demand $($liters$)85007500708555003550$

However, when scheduling the delivery, you noticed that the total supply of $2022-12-31$ was not enough, so you contacted an engine oil retailer to provide extra Green Oil and asked the retailer to deliver the difference to each station directly. The retailer charges $$5$ per liter for shipping in general. $($Here$,$ you are the transportation manager, and you only consider the shipping cost, including those form the retailer to each station$)$

There are several special requirements that you $,$must satisfy:

$$ Station $3$ only needs standard oil and premium oil and the amount of premium should be at least twice the standard oil

$$ Station $4$ requires at least $1000$ liters of standard oil and $1000$ liters of premium

Questions:

$1.$ Design the LP network for this transportation schedule. Set up an LP model for this and use Solver to solve the problem

$2.$ What is the total shipping cost?

$3.$ How many units do you need to ask the retailer to deliver and deliver to which station?