Content placement in WMNs can be addressed by considering it as a Facility Location Problem. In operations research, optimizing the access cost of clients in different demand locations I to access a set of facilities J is an important problem. To address the problem, we need an efficient solution of a feasible cost (i.e., build and operate the facilities). This problem is formulated as a FLP such that many facilities are set up and each one is assigned with the demands of a subset of clients. There are several examples of FLP such as emergency points, education centers, public transport stations, and retail services.
We model the problem of content placement from the perspective of the FLP in the context of WMNs. In our model, a node can be either a demand node (client) or a facility node (replica server). FLP provides a mathematical formulation of optimization aspects. The formulation includes the cost to open facilities, and the distances between the clients and facilities that normally satisfy metric properties (e.g., the triangular inequality). The total cost relies importantly
on the number of facilities to open and their locations. To address the FLP, we need to specify the set of facilities to be opened, and assign the clients to the open facilities. For a typical FLP, J and I are the inputs. The output solution is to open a subset of the facilities J ⊆ I and assign clients to their closest open facility. The solution tries to minimize the total cost that consists of two parts:
1. Opening cost: The opening cost fj for a facility j depends on the targeted problem. For a solution, the opening cost is the sum of costs for all open facilities.
2. Weighted distance cost: This is the weighted distance cost from a client i ∈ I to a facility j ∈ J denoted by hidij. Where, hi is the demand at node i and dij is the shortest distance from node i to facility j. It is assumed to be symmetric and satisfies the triangle inequality. The sum of distance costs of all clients is the total cost for the solution.
A number of variants for the FLP exist by combining these costs in different ways [32, 33, 34]. The number of facilities to open can be a constant (p-median problem), limited number of served clients by a facility (capacitated FLP) or unlimited (uncapacitated FLP). Most variants of FLP are NP-complete [35]; hence, approximation algorithms that find solutions close to the optimal solution are under investigation. Following are some variants of the FLP:
2.2.1
p-center problem
This is also known as the minimax problem, as we seek to minimize the maximum distance between any demand and its nearest facility. The cost here is not weighted by hi. Then ∀j ∈ J, select up to p facilities to minimize the total cost:
C(I, J, p) =X ∀i∈I
X ∀j∈J
where m(i) ∈ J is the facility j closest to i.
2.2.2
p-median problem
When a limited budget is available for opening the facilities, and the opening cost of all the facilities are approximately the same. Then ∀j ∈ J, select up to p facilities to minimize the total cost:
C(I, J, p) =X ∀i∈I
X ∀j∈J
hidi,m(i) (2.2)
where m(i) ∈ J is the facility j closest to i.
2.2.3
Uncapacitated FLP (UFLP)
When the opening cost is considered and the number of facilities to open depends on a joint optimization for opening cost and distance cost, we have the UFLP. The solution is to open a set of facilities J to minimize the joint cost C(J, I, f ), where a facility j can serve an unlimited number of clients:
C(I, J, f ) = X ∀j∈J fj+ X ∀i∈I X ∀j∈J hidi,m(i) (2.3)
where m(i) ∈ J is the facility j closest to i.
2.2.4
Capacitated FLP (CFLP)
In CFLP, we assume that a facility can have a constraint in resources dedicated to its clients, so it is important to limit the number of clients assigned to a facility. We open a set of facilities J to minimize the joint cost C(J, I, f ), while ensuring that each facility j can only serve at most uj clients:
C(I, J, f ) = X ∀j∈J fj+ X ∀i∈I X ∀j∈J hidi,m(i) (2.4)
where m(i) ∈ J is the facility j closest to i and cj is the number of clients i attached to facility j such that cj ≤ uj.
2.2.5
Multiple commodity facility location problem
The FLP can be extended to a problem of multiple commodities served at a facility. Let L denote the set of commodities L = 1, ..., M . Each commodity x ∈ L has a subset of clients. To extend the cost function, we consider an optimization for all commodities and assume the same opening cost f for every commodity x, the joint cost can be expressed as:
C(I, J, L, f ) = X ∀j∈J X ∀x∈L fj(x)+ X ∀i∈I X ∀j∈J X ∀x∈L hidi,m(i,x) (2.5)
where m(i, x) ∈ J is the facility j holding x closest to i. If we consider the CFLP, we have the number of clients i demanding any commodity x attached to facility j such thatP
∀x∈Lcj(x) ≤ uj.