Eighth Method

The Eighth Method Company makes several lines of environmentally friendly

cleaning materials. They produce them at $3$ different locations in North America. If

they choose to produce cleaners at a particular location then they will incur a fixed

cost for tooling and equipping that location for production of cleaners; if they

choose not to produce cleaners at a production location, then they will not incur the

fixed cost there. Each production location has a maximum capacity that can be

installed for cleaner production.

Finished products will be sent to warehouses in the United States where they will

then be sent on to $4$ different customers. They have three possible warehouse

locations identified in the United States. Costs for shipping a pallet of cleaner from

production locations to warehouse locations are given in the data in the table below.

Note that due to packing and shipping restrictions, there is a maximum capacity that

can be shipped from a specific production facility to a specific warehouse while

there is also a minimum capacity for any production$-$warehouse shipping pair that is

used; that is$,$ Eighth Method can choose to not use an arc, but if they do choose to use

any given arc then it must have a minimum number of units shipped on that arc.

The specific numbers are given in the data. As with the production facilities, if a

warehouse is used it incurs a fixed cost.

Each of the $4$ customers has a stated demand for cleaners to sell to their customers.

The cost of shipping from$/$to each warehouse$/$customer pair is given in the data.

Note that the same type of shipping restrictions exist for each warehouse$-$customer

pair with a min $($if used$)$ and max.

Eighth Method would like to find a way to satisfy customer demand in the least cost

manner.