Furthermore, to account approximately for busy periods, we want to extend the maximum covering problem in the following three ways: (i) Each facility being located can have at most H max demands...

Furthermore, to account approximately for busy periods, we want to extend the maximum covering problem in the following three ways:

(i) Each facility being located can have at most H_max
demands assigned to it.

(ii) Each demand must be fully assigned to a facility or possibly to a number of different facilities (even if the facility or facilities to which it is assigned is/are more than Dc, distance units away).

(iii) Each facility must be within DF distance units of at least one other facility. We will refer to DF as the backup facility distance. To formulate this problem, we define the following notation:

Indices and Sets

I = set of demand nodes (indexed by i)

J = set of candidate locations (indexed by j and k)

Inputs

d_ij
= distance between demand node i
 I and candidate location j
 J

D_c
= coverage distance

D_F
= backup facility distance

(Note that b_jj
= 0 for all j
 J so that we prevent a facility from serving as its own backup facility.)

h_i
= demand at node i
 I

H_max
= maximum demand that can be assigned to any facility

M = a very large number

Decision Variables

Y_ij
= fraction of demand at node i
 I that is assigned to a facility at node j
 J

X_j
= 1 if we locate at candidate location j
 J

0 if not

With this notation, formulate the weighted objective function and constraints defined below:

MAXIMIZE

The total covered demands as the primary objective

MINIMIZE

The demand-weighted distance between demand nodes and the facilities to which they are assigned for the uncovered demands as the secondary objective

Note that these should be formulated as one objective function.

SUBJECT TO:

Locate exactly P facilities

 All of the demand at a node is assigned to some facility

 Capacity of the facilities

 Demands can only be assigned to open facilities

 Each facility must have at least one other facility within DF distance units

 Nonnegativity and integrality