Solution approaches for the continuous CFLP

A couple of solution approaches to solve the continuous CFLP are discussed next.

Two methods are given to solve the capacitated MSWP as well as two methods to solve the CCCP. The methods are as follows:

3.1.4.1 A perturbation-based heuristic

Zainuddin and Salhi (2007) suggest an approach called the perturbation-based heuris-tic to solve the capacitated MSWP. The method is based on Cooper’s alternating transportation-location heuristic (ATL). The ATL starts by selecting p random cus-tomer points as the cluster centres or depots of the starting solution. The TP of this solution is solved assigning customers to each depot based on the depot capacity constraint. Because the problem is multi-source, a customer’s demand can be met by more than one depot. New cluster centres are now calculated using Weiszfeld’s algorithm. For each cluster, this algorithm iteratively solves the Weiszfeld equation until the change in coordinates is significantly small. The Weiszfeld equation is given as follows:

where wi is the demand of customer i allocated to depot j. The coordinates of customer i are ai = (a¹_i, a²_i) with ai ∈ <². The coordinates of depot j at iteration t are Xjt = (Xj¹t, X_j²_t) with Xjt ∈ <². The distance between customer i and depot j at iteration t is d(ai, Xjt). Initial coordinates for the depot, Xj0 = (X_j¹₀, X_j²₀), are needed as input variables.

The ATL heuristic continues to alternate between solving the TP and using Weiszfeld’s algorithm to calculate depot centres until the improvement in cost is below a pre-determined threshold value.

Instead of using random starting points, Zainuddin and Salhi (2007) introduce the furthest distance rule (FDR) to select depots. The first depot is still selected ran-domly, from there onwards the method determines the customer point with the furthest distance from all depots in the current depot set and adds this customer to the depot set. The method continues in this manner until p depots have been selected.

Zainuddin and Salhi (2007) propose a perturbation-based method that first solves the uncapacitated problem for different K starting solutions. A subset of the K solutions

3.1 The facility location problem (FLP)

that differ a substantial enough cost amount from each other are selected to solve the capacitated problem. The ATL heuristic is used to solve the capacitated problem, but two extra steps are added to this heuristic. Here all customers are assigned to their closest depot after Weiszfeld’s algorithm has been used to determine the depot centres. The demand of the customers are therefore moved to a single depot and Weiszfeld’s algorithm is again used to determine the depot centres.

A post optimisation search is used to find better solutions by clustering "border-line" customers and assigning them to their closest depot. A customer is seen as a borderline customer if the distance from the customer to the closest and second closest depots are almost the same. In these cases the ratio of the closest to second closest distances will be very small. A predefined threshold ratio parameter can be used to identify them. These customers are excluded from the TP if their demand cannot fully be met by the nearest depot due to capacity constraints. After the TP has been solved the customers are introduced back into the method. The reason behind this removal is to move depot centres further away from each other in order to reduce the number of borderline customers.

Zainuddin and Salhi (2007) also note that solving the TP for the different sets of depots can become quite time consuming. They suggest using a subset of depots at a time. Forbidden regions are also used to prevent starting solutions from selecting previously chosen depots. The authors also mention that selecting the best un-capacitated starting solution does not necessarily result in the optimal un-capacitated solution, which is why they use multiple starting solutions.

3.1.4.2 A region-rejection based heuristic

This method was suggested by Luis et al. (2009) to solve the capacitated MSWP. The heuristic forbids new depots from being placed too close to previously selected depots by placing a forbidden radius around these depots. The radius is then adjusted dynamically. The heuristic generates initial starting solutions which are then solved using the ATL heuristic discussed above.

A similar approach is followed by the same authors in Luis et al. (2011). They make use of restricted regions when constructing a restricted candidate list in the guided reactive GRASP metaheuristic. The GRASP method is discussed in more detail in Section 3.3.6.2. The construction of the restricted candidate list (RCL) is guided with a parameter, α and the search is enhanced with a learning process to define the bounds of α. The restricted regions are also used to prevent new solutions from being generated too close to already tested ones.

3.1 The facility location problem (FLP)

3.1.4.3 The cluster search GA metaheuristic to solve the CCCP

Chaves and Lorena (2010) suggest a cluster search algorithm to solve the CCCP.

This algorithm consists of four phases: The first phase generates random CCCP solutions. The second phase groups solutions together based on similarity and the cluster centres and number of similar solutions found are updated. The third phase analyses the information gathered about the grouped solutions. If a certain grouped solution has been identified in a predefined number of solutions, this is identified as a candidate solution to explore further. In the fourth phase, a local search heuristic is used to explore the candidate solution.

Chaves and Lorena (2011) refined the cluster search heuristic to use a genetic algo-rithm (GA) to generate solutions for the CpMP. Their reasoning behind this is that it is less time consuming to evaluate the CpMP objective function than the CCCP objective function. These solutions are then used in phase one to identify possible candidate solutions. If a candidate solution is detected, it is then evaluated with the CCCP objective function in the local search algorithm.

An additional parameter, called the inefficiency rate, is introduced in the second phase. The parameter keeps track of the number of consecutive times the local search algorithm could not improve the objective function of a candidate solution.

This parameter prevents the heuristic from repeating a local search in an area that has already proven inefficient.

3.1.4.4 A two-phase heuristic method to solve the CCCP

Negreiros and Palhano (2006) suggest a two-phase heuristic method to solve the CCCP, the first phase is a constructive phase and the second is a refinement phase using a variable neighbourhood search (VNS) as follows.

Constructive phase

Two methods can be used in the constructive phase depending on the problem.

The log-polynomial geometric tree search method was used to solve the generic CCCP. The unconstrained to constrained method was used in conjunction with an unconstrained solution to solve the p–CCCP.

• The log-polynomial geometric tree search method

This method constructs clusters from subtrees of the minimum spanning tree (MST) graph. The MST is redrawn to balance points evenly throughout the graph, from a centre point. This graph is referred to as a balanced q–tree. It is unconstrained, since there are no limits on the capacities of the subtrees.

The only focus is to balance the demand equally.

3.1 The facility location problem (FLP)

Once the unconstrained balanced q–tree has been constructed, a subtree is chosen and if the demand of the customers in the subtree is less than the capacity of a cluster, the whole subtree is assigned to one cluster. The method then looks at the next subtree, to see if it can add it to the created cluster or if it is too big, assign it to a new cluster or find the closest cluster centre with capacity. This is repeated until all the points have been assigned to clusters.

The above tree search methods can be used for the generic CCCP where the number of clusters is unknown beforehand.

• The unconstrained to constrained method

For the p–CCCP, an unconstrained to constrained method was suggested. The method can be described as follows:

1) An initial unconstrained solution for p clusters is created with either the h–means or j–means methods.

2) The points in the clusters are then ordered by decreasing demand and infeasible points are moved to feasible clusters. The ordering can also be based on distance or the demand / distance ratio as discussed in the CCP solution approach above.

3) If no feasible cluster could be found, the point is moved to a new cluster and p = p + 1. (No mention is made about the merging of clusters to bring p back to its original value.)

4) Once a capacitated feasible initial solution has been created, either the h–

means or the j–means methods can be called to better the solution. Here the methods are restricted and only feasible exchanges to other clusters are allowed.

Refinement phase

The refinement phase is based on a Reduced Variable Neighbourhood Search (RVNS), (Burke and Kendall, 2005). The refinement phase is an iterative process with the following steps:

1. A fixed number of points in a local neighbourhood are selected randomly to exchange between clusters.

2. The points can either be exchanged with their closest centres found in the constructive phase or a new set of random clusters are selected to exchange the points with. The randomly chosen clusters do not have to be distinct.

3. Once the exchanges have been made, the objective function is evaluated.

4. If the objective function has improved, the solution is updated to reflect the new assignments to clusters. If not, the solution defaults back to the original solution.