Question $1$

$(30$ points$)$ A formulation for the Biobjective Facility Location Problem is given below. This version mod$-$

ifies the facility location problem to consider two objectives, minimizing the cost of opening up facilities

and minimizing the total distance to the sites, at the same time. $y_j$ denotes if the facility $j$ is opened or

not, $x_{ij}$ denotes the assignment of site $i$ to facility $j.$

min,$z_1={\displaystyle \underset{j=1}{\overset{m}{}}}{f}_j{y}_j$

min,$z_2={\displaystyle \underset{i=1}{\overset{n}{}}}{\displaystyle \underset{j=1}{\overset{m}{}}}{c}_{ij}{x}_{ij}$

subject $to{\displaystyle \underset{j=1}{\overset{m}{}}}{x}_{ij}=1$

${\displaystyle \underset{i=1}{\overset{n}{}}}{x}_{ij}{M}_j{y}_j$

$x_{ij}in\{0,1\}$

$y_{j}in\{0,1\}$

Consider a problem with $6$ sites and $4$ possible facility locations. Take capacity $M_j$ as $3$ for all facilities.

Define the other parameters as follows $($you can ignore the units$)$;

Apply the $lon-$constraint algorithm and find the set of all nondominated $($or weakly nondominated$)$ points.

Write and solve each model with a solver of your choice. Provide the mathematical models and write the

steps of the algorithm and your findings $($the solution of the model and its image$)$ in each iteration.

$($Note: You need to attach the printouts of your models when you submit your homework. Also, providing

only printouts is not enough. You must clearly write results of each step in a report.$)$

Question $2$

$(20$ points$)$ For each nondominated point that you have found, discuss whether it would be possible to

find that point by solving a weighted sum method. Solve the problem using W$.$S$.$ Algorithm $($Perpendicular

Search$).$ Write down nondominated set and weights for each step.