The discrete CFLP

Drezner and Hamacher (2001, Ch. 3) identified eight different discrete facility lo-cation models: set covering, maximal covering, p–center, p–dispersion, p–median, fixed charge, hub and maxisum. The problem formulations of the p–median and fixed charge FLP are relevant to the CLRP and discussed in further detail. The ca-pacitated clustering problem (CCP) by Mulvey and Beck (1984) is another variant

3.1 The facility location problem (FLP)

of the CFLP discussed below.

3.1.1.1 The capacitated p–median problem (CpMP)

In a p-median problem, the goal is to find the median(s) and minimise the sum of the within-clustered Euclidean distances between customers and the median(s), (Drezner and Hamacher, 2001). It is a discrete problem where the list of potential depots is the same as the list of customers. Any customer can serve as a potential depot and the selected depots are called the medians or concentrators of each cluster.

If only a limited capacity can be assigned to each of the p centres, then the problem is referred to as the CpMP. This can also be referred to as a capacitated p–concentrator location problem, the CCLP (Ceselli et al., 2009).

The number of depots to select is restricted to p. The objective is to minimise the distance from the chosen p depots to the customers assigned to them. Each customer must be assigned to a depot and can only be assigned to one depot to enforce the single-source constraints.

Problem formulation

The problem can be represented by a graph G(V,A), where V is the set of all points in the graph and A is the set of directed edges between the points, used to represent customer assignments. The CpMP can now be formulated as the following integer linear programming (ILP) model:

where V is the set of n customers to be clustered. The decision variables (xij∀i ∈ V, ∀j ∈ V) are binary with xij = 1 if customer i is assigned to depot j. If customer j is a depot, then xjj = 1. The number of depots allowed is p. The distance used in the objective function, dij, is the Euclidean distance between point i and point j

3.1 The facility location problem (FLP)

as given in Eq. (2.32). The demand of the customer i is wi and W is the maximum capacity of any depot.

3.1.1.2 The capacitated clustering problem (CCP)

The capacitated clustering problem (CCP) is a discrete clustering problem with a set of candidate centres or depots to select from, where exactly p capacitated clusters needs to be created. Similar to the CpMP it assumes that each customer has a demand and each facility/depot has a limited supply. According to Negreiros and Palhano (2006), the CpMP is a specific case of the CCP where, the capacity constraints for all the depots are homogeneous and the coefficients of the objective function are distances.

In general, for p–median problems, the list of potential cluster centres is the same as the list of points, whereas for the CCP and discrete CFLP the list can differ from the list of points. According to Mulvey and Beck (1984), the CCP is NP–hard.

Problem formulation

Given a set I of n customer points with an associated wi weight (or demand) per point, group the points into p clusters in order to minimise the total dissimilarity between the points in a cluster and the cluster centre, given a maximum capacity constraint per cluster, Wj. The set J represents the list of m potential cluster centres and the set of all points in the graph is then V = I ∪ J. The formulation for the CCP given by Mulvey and Beck (1984) as follows:

Minimise ^X where wi is the demand of the customer i, Wj the maximum capacity of cluster j, p is the specified number of clusters to select so that m ≥ p. xij = 1 if point xi is assigned to cluster j and the binary variable yj = 1 if point j is chosen as a cluster

3.1 The facility location problem (FLP)

centre. The dissimilarity dij is calculated between a point and the median.

Constraints (3.8) ensure that each point is assigned to only one cluster centre, while constraints (3.9) limit the number of clusters to exactly p. Constraints (3.10) allow points to only be assigned to the selected open cluster centres and constraints (3.11) enforce the capacity constraint per cluster. The last constraint ensures all variables are binary.

3.1.1.3 The fixed charge capacitated facility location problem (CFLP) In the fixed charge variation of the FLP, the goal moves from minimising distances to minimising costs that also include fixed depot costs. The constraint on the number of depots is replaced with a fixed cost per depot. The number of depots then becomes a trade-off between the fixed costs and distribution costs.

In the literature, reference to the FLP assumes the fixed charge variant. See, for example, Erlenkotter (1978), Ceselli et al. (2009), Klose and Görtz (2007). In this study when referring to the FLP, the fixed charge variant is meant, except where explicitly stated otherwise.

Figure 3.1 illustrates an example of the costs versus the number of depots used. It is similar to the generalised cost graphs for the FLP as illustrated by Ballou (2004, p. 574) and Rand (1976). It shows how the distribution costs decrease exponentially as more depots are added.

Figure 3.1: Typical costs associated with the FLP problem.

3.1 The facility location problem (FLP)

In practical situations the fixed costs per depot will typically not include the com-plete once off set-up costs for opening the depots in the objective function. The reason for this is that they are too high and impractical. Rather, the depot costs are divided by the expected pay-off term to calculate a periodic instalment cost.

This cost together with the expected operating cost over the same time period is then used to create a fixed cost per depot.

In practice, if the fixed depot costs are not available and the number of depots to open is unknown, another popular approach is to determine the number of depots for which the decrease in distribution costs is still significant. The acceptable level of significance will then be determined by the decision maker.

In the FLP, the set of candidate depots does not have to be the same as the list of customers and a completely different set of possible depot locations can be used.

An example of the discrete FLP is illustrated in Figure 3.2. The points denote the customers to be visited, the squares denote the candidate depots and the filled squares denote the chosen depots.

Figure 3.2: An illustration of the discrete FLP problem.

If there are no capacity constraints, customers should be assigned to the closest open depot in order to minimise distribution costs. If the depots are restricted by a supply capacity, the problem is called the capacitated FLP (CFLP) and if customers can only be assigned to one depot, the problem is called the single-source capacitated facility location problem (SS-CFLP). Most researchers refer to the single-source

3.1 The facility location problem (FLP)

variant when referring to the CFLP, therefore in this work the single-source variant is assumed. In cases where the set of potential depots is equal to the list of customers, the problem is referred to as the capacitated concentrator location problem, the CCLP (Simchi-Levi and Bramel, 1997, Ch. 12 and Ceselli et al., 2009).

Problem formulation

The mathematical formulation for the discrete CFLP is as follows:

Minimise ^X where I is the set of all n customers, J is the set of all m candidate depots and fj

is the fixed cost for opening depot j. The distribution cost to deliver to customer i from depot j is cij. If depot j is open, yj = 1 and if customer i is assigned to depot j, xij = 1.

The capacity can differ per depot. This is referred to as Wj while wi is the demand of the customer i. The total demand of customers assigned to a depot j should be equal or less than Wj as shown in constraints (3.16).

The problem formulation differs from the p–median problem in the following ways:

A new variable yj, is introduced to determine whether depot j is open or not and the constraint on the number of open depots equal to p has been removed.