Two small communities, Alpha and Beta,

each have a small landfill. They are considering

building a joint landfill to serve

both communities. $($See map, Figure $3-26.)$

a$.$ Write an objective function that can

be used to minimize the total cost of

disposal where the towns would have the option of taking the refuse to their

own landfill or to the common facility.

They will not be able, however, to

take the refuse to each other$$s landfill.

Use the following variables, and state

all the constraints:

Fi fixed$-$cost landfill i $($$$100,000$

per year, regardless of size$)$

Cki transportation cost from town

k to landfill i $($$$2\mathrm{.}00$ per km per

ton$)$

xk refuse generated in town k

$(2000$ metric tons per year for

Alpha, and $3500$ metrictons per

year for Beta.$)$ yi $1$ if the landfill is operated, y

$0$ if it is not

ci operating cost of landfill i

$($$$10\mathrm{.}00$ per metric ton$)$

b$.$ Solve the objective function for two

conditions: $(1)$ when the towns use

their own landfills exclusively and

$(2)$ when only the central shared

facility is used by both towns.

c$.$ Use an optimization software package

to find the optimum $($least$-$cost$)$ solution.