Problem $3.$ Single & Multiple, Discrete Location Problems

The Seven Guys fast food restaurant chain needs a distribution center to prepare the food ingredients it

provides to its individual restaurants. The company is attempting to determine the location of a

distribution center that will service five restaurants, and it has identified $4$ possible locations in which a

new distribution center can be built.

The table below shows the distances between the $5$ restaurants and $4$ candidate locations $($L$1,$ L$2$ L$3,$ and

L$4)$ in miles. It also shows the annual number of $40-$foot trailer trucks transported to each restaurant in the

last column. Consider each trip of a truck from a new distribution center to a restaurant as a two$-$way trip

$($i$.$e$.,$ trucks have to come back to the distribution center after the delivery$).$

$3-1)$ Select a location that minimizes the maximum travel distance per trip.

$3-2)$ Select a location that minimizes the total travel distance per year.

$3-3)$ Suppose that the Seven Guys restaurant chain is facing the same problem described in Problem

$2$ again. This time, the Seven Guys restaurant chain wants to know whether having multiple

distribution centers can save more cost than having only one. The unit travel cost is $$10$ per mile

and the annual fixed cost to run each distribution center is $$60,000.$ How many distribution centers

should be built in order to minimize the annual total cost $($total cost $=$ travel cost $+$ fixed cost$)?$

Which location$($s$)$ should be selected?

$3-4)$ Do you need to explore cases with more than $3$ distribution centers to solve $2-3)?$ Justify your

answer.