Question $1(18$ points$)$:

In the lecture of Week $10,$ the following model was presented to formulate the Maximum Survival Location

Problem $($MSLP$).$

Subject to:

max,${\displaystyle \underset{\text{i}\text{i}\text{n}\text{I}}{\overset{?}{}}}{d}_i{\displaystyle \underset{\text{j}\text{i}\text{n}\text{J}}{\overset{?}{}}}s(t_{ji}+t^{\text{d}\text{e}\text{l}\text{a}\text{y}})y_{ij}$

$y_{ij}{x}_j,$iinI,jinJ

${\displaystyle \underset{\text{j}\text{i}\text{n}\text{J}}{\overset{?}{}}}{y}_{ij}=1,$iinI

${\displaystyle \underset{\text{j}\text{i}\text{n}\text{J}}{\overset{?}{}}}{x}_{j}q,$

$x_{j}in\{0,1\},$jinI

$y_{ij}in\{0,1\},$iinI,jinJ

Decision variables

$x_j=1$ if station $j$ is opened; $0$ otherwise.

$y_{ij}=1$ if station $j$ is the closest opened station to demand

node $i$;$0$ otherwise.

Part a$)$ How would you modify the model to imply that if station $1$ is opened then then station $4$ also must

be opened? $(3$ points$)$

Part b$)$ How would you modify the model to imply that if station $2$ is opened then at most two of the stations

$1,5,$ and $6$ can be opened? $(3$ points$)$

Part c$)$ How would you modify the model such that either both stations $3$ and $8$ are opened, or at least one

of the stations $5$ and $6$ is opened? $(3$ points$)$

Part d$)$ How would you modify the model if at least one of the stations $3$ and $8$ is opened then both stations

$8$ and $9$ cannot be opened? $(3$ points$)$

Part e$)$ How would you modify the model such that if station $4$ is closest opened station to node $5,$ then

either station $6$ or station $7$ must be the closest opened station to node $4?(3$ points$)$

Part f$)$ How would you modify the model if there is a limited budget for opening station? In this case, you

could suppose that $B$ is the parameter representing the maximum available budget and $g_i$ is parameters

denoting the cost of opening station $j.(3$ points$)$