The continuous CFLP

Two capacitated continuous CFLP problems are discussed, the CCCP (capacitated centred clustering problem) and the capacitated MSWP (multisource Weber prob-lem). Unlike the CCCP, customers are not limited to single-source depot assign-ments in the capacitated MSWP. The capacitated MSWP also has a fixed number of depots to open and a constant capacity is assumed for all depots, whereas in the CCCP this is not necessarily the case.

3.1 The facility location problem (FLP)

3.1.3.1 The capacitated multi-source Weber problem (MSWP)

The capacitated MSWP is the continuous variant of the CpMP, because the locations of the depots are not limited to fixed points. Similar to the CpMP, the number of depots, p to select is known beforehand. Different from the CpMP the depot supply capacities Wj are replaced with a fixed depot capacity, W . If the transportation cost is a linear function of the distances between points, the problem can be formulated as follows (Zainuddin and Salhi, 2007):

Minimise ^X is the coordinates of customer i with ai ∈ <². The coordinates of depot j are rep-resented by the additional decision variables Xj = (X_j¹, X_j²) with Xj ∈ <². These coordinates need to be determined in such a way that the distances, d(Xj, a_i), asso-ciated with the transportation costs, are minimised. Normally Euclidean distances, calculated with Eq. (2.33) are assumed and the distance can also be written as the Euclidean norm: d(Xj, a_i) = kXj − a_ik₂ (Brimberg et al., 2000). The decision variable xij is the amount to be delivered from depot j to customer i.

Constraints (3.29) ensure the constant depot capacity is adhered to while constraints (3.30) ensure all customer demands are met. Constraints (3.31) limit the decision variables to positive values, while Constraints (3.32) limits the depot locations to the Euclidean plane.

3.1.3.2 The capacitated centred clustering problem (CCCP)

The CCCP is similar to the CCP but unlike the CCP, the CCCP does not have a list of potential cluster centres (or depots) because the problem is a continuous FLP. The centre of a cluster, is called the depot of the cluster and is calculated as the point with the minimum total distance from all points assigned to the cluster and itself. It can be located anywhere within the Euclidean space where the points

3.1 The facility location problem (FLP)

are located. According to Negreiros and Palhano (2006), the CCCP is similar to the normal clustering problem of assigning n points into p clusters, but it assumes that each cluster has its own capacity constraint, making it an unique problem.

Negreiros and Palhano (2006) define two different versions of the CCCP: the p-CCCP and the generic p-CCCP. In the p-p-CCCP the number of clusters is predefined and the problem can be formulated as follows:

Minimise ^X the Euclidean coordinates of point i and the means vector of the points in cluster j is Xj. The number of customers assigned to cluster j is nj and Wj is the capacity constraint of cluster j.

The problem is single-source and because xij is binary, constraints (3.34) ensure each point is assigned to only one centre, while constraints (3.35) restrict the amount of points in each cluster to nj. Constraints (3.36) locate the depots as the means vector X_j, while constraints (3.37) enforce capacity constraints. The depots can be placed anywhere on the Euclidean plane while the assignment variables must be binary, enforced by Constraints (3.38).

In the generic CCCP, the objective is to minimise the costs associated with the dissimilarity between points and depots and the depot opening costs, fj, given the cluster capacity, Wj per depot j. This is similar to the objective function of the fixed charge CFLP. It allows for different depot capacity constraints and depot opening costs and assumes the transportation costs are the same as the squared Euclidean distances in the objective function. The generic CCCP is formulated by Negreiros and Palhano (2006) as follows:

3.1 The facility location problem (FLP) where zj and xij are the decision variables associated with opening depot j and assigning point i to depot j respectively, I is the set of customers and J is the set of chosen depots. Similar to the p-CCCP, ai = (a¹_i, a²_i) is the Euclidean coordinates of point i and the means vector of the points in cluster j is Xj.

Constraints (3.40) ensure each point is assigned to one and only one depot, while Constraints (3.41) are used to calculate the coordinates of the depots. Constraints (3.42) enforce the depot capacity constraints and constraints (3.43) limit the decision variables.

In order to lower the travelling costs, the best solution will be to create as many depot clusters as possible. To let the model choose to open less clusters, there must be an incentive. This is achieved by restricting the capacity per cluster Wj and by associating a cost fj to the opening of cluster j in the generic CCCP.

A similar problem is called the transportation problem (TP). The TP can be de-scribed as follows: given a set of depots with given supply capacity and associated transportation costs, assign a set of customers each with a given demand to min-imise the total transportation cost. The TP can be solved optimally in polynomial time. See Winston (2004, Ch. 7) for an example of how to solve the TP optimally using the simplex method. The difference between the TP and the CCCP problem is that the single-source constraints are not enforced in the TP. Unlike the capaci-tated MSWP and CCCP, the depot locations are already known and all depots are assumed open in the TP, implying that customers can be assigned to any depot without an additional depot opening cost charge.

3.1 The facility location problem (FLP)