Minimum Cost Maximum Flow Algorithm with restrictions

**Minimum Cost Maximum Flow Algorithm with Restrictions** This algorithm efficiently allocates clients to servers while minimizing the total transmission cost and adhering to capacity constraints. It operates as follows: ```html

LOOP until stop
  Loop over servers
    SET closest = NULL
    SET shortest = INFINITY
    Loop over clients 
      IF client has server
        CONTINUE
      SET dist = distance from client to server
      IF dist < shortest
        SET closest = client
        SET shortest = dist
    ENDLOOP over clients
    IF closest == NULL
      STOP
    Connect closest client to server
  ENDLOOP over servers
ENDLOOP until stop

``` **Mathematical Optimization** Alternatively, a mixed-integer linear program can be formulated to model this problem: **Variables:** * x<sub>ij</sub>: 1 if client i is assigned to server j, 0 otherwise * y<sub>j</sub>: Number of clients assigned to server j **Known Constant:** * d<sub>ij</sub>: Distance between client i and server j **Formulation:** **Objective:** ```html min Σ_i,jd_ijx_ij ``` **Constraints:** * Each client i is assigned to exactly one server: ```html Σ_jx_ij = 1 for each client i ``` * The number of clients assigned to a server is equal to y<sub>j</sub>: ```html Σ_ix_ij - y_j = 0 for each server j ``` * Each server has at least f and at most f+1 clients assigned to it: ```html y_j - f >= 0 for each server j y_j - f <= 1 for each server j ``` * x<sub>ij</sub> and y<sub>j</sub> must be binary: ```html x_ij = 0 or 1 y_j variables are not required to be integer-valued, but will be forced to equal integers ```