Question $1$:

There are $93$ districts which have to receive service. You are given the distances between

these districts, the population of each district, and the risk coefficient of each link that connects

the districts $($if the risk coefficient is high then the probability that the link is out of use is also

high.$)$

The municipality plans to open distribution centers for basic aid. Since the roads may be

damaged, people may walk to those distribution centers from districts.

a$)$ Assume the municipality will open $5$ distribution centers. Give a mathematical model

that determines the locations of DC$\text{'}$s while minimizing the longest distance $($R$)$ that a

person needs to travel to reach a DC$.$ Remember that each district should be covered by

at least one DC$.$ Give your findings $($results$).$

b$)$ Now assume the municipality decides that each district should be covered by at least

two DCs within a range$($R$)$ with at most $p$ DCs$.$ Use the $R$ value that you have found in

part a$).$ How do you model this situation? Give your findings $($results$).$

c$)$ Conduct a sensitivity analysis on the number of distribution centers $($p$)$ for part b$)$ and

give your results $($corresponding $R$ values$).$

d$)$ Consider part a$)$ again. But this time assume the municipality also wants to consider the

risk of destruction of the roads. $(1=$ remains perfectly, $0=$ totally demolished$)$ So$,$ they

want each district can access the DC that they are assigned to$,$ using a link with a risk

coefficient greater than $0,7.$ Again, minimize the longest distance $($R$)$ that a person

needs to travel to reach a DC$.$