Shortest Path Problem: Write down the mathematical optimization model of Shortest Path Problem. Explain your decision variables and parameters. No partial credits will be given in this question.

B$)(10$ points$)$ Suppose we want to open up $"p"$ number of stations among $"n"$ possible customer locations. To promptly serve the customers, the decision maker wants no customer to be further than $"$K$"$ kilometers away from the station he$/$she has assigned. Each customer must be assigned to one station and each customer $"i"$ has a demand of $"D_i".$ Facilities have serving capacity of $"C"$ amount of customer demand and each location has fixed setup cost of opening $"F_i".$ Cost can be thought as weighed transportation $($i$.$e$.,$ distance $x$ demand$)$ and fixed setup costs of opening up facilities. Distance between point $"$ i $"$ and point $"$ j $"$ is denoted by $d[I,j].$ Propose the mathematical optimization model that selects the locations of the facilities with the minimum cost. Explain everything that is used in your formulation.

Add the required constraints for the following items:

B$1)(3\mathrm{.}75$ points$)$ Suppose customer $"k"$ and customer $"m"$ cannot assigned to the same station.

B$2)(3\mathrm{.}75$ points$)$ Suppose customer $"k"$ and customer $"m"$ cannot assigned to the different stations.

A$)$ Propose a mathematical optimization model for the k$-$center problem where the primary goal is to minimize the maximum distance of customers, and the secondary $($relatively less important$)$ objective is to minimize the total fixed cost of locating facilities. Moreover, we can open at most $k$ facilities. While the distances are denoted by $D(i,j)($ i denotes the index of a customer and j is the index for the facility$)$ and fixed cost of opening a facility at location j is denoted by $F_j.$ Also define decision variables before you propose the model.

B$)$ Suppose we change the objective function of the k$-$center problem to minimizing the total cost of opening facilities such that each customer is assigned to a center that is at most $20$ km further from their location. Use the same decision variables and parameters defined in part $A.$

C$)$ Suppose we adopt the iterative mathematical modeling algorithm to solve TSP and the resulting solution after the first iteration of the algorithm is given below. Please introduce the subtour elimination constraint to avoid the related subtour. Use Dantzig, Fulkerson and Johnson's subtour elimination constraint. D denotes the depot.