A Cluster-based Optimization Approach for the Multi-

Time Windows

Another way of designing a heuristic decomposition method is to split a prob- lem heuristically. There are several ways to split the problem, such as clus- tering. Dondo and Cerdá [56] proposed a three-phase heuristic to solve multi- depot heterogeneous fleet vehicle routing problem with time windows (VRPTW). The VRPTW is an extension of the VRP where vehicles to make visits must ar- rive the location within the appointment time window. In addition, the optim- isation solver cannot find the optimal solution of every instance of the multi- depot VRPTW. This three-phase heuristic could be a practical implementation to maximise the use of a mathematical solver to tackle the multi-depot VRPTW. This heuristic decomposition method by Dondo and Cerdá [56] has three hierarchical phases to generate the routing plan. The three phases are

• Phase I: cluster generation,

• Phase II: cluster assignment and sequencing, and

Phase I: Cluster Generation

Cluster generation is the key to reducing overall computation time as it clusters

locations and defines each cluster c∈ C as a node. Each cluster contains several

customer locations. Travel distances are defined between a pair of clusters. We can see that the mathematical model of cluster-based problem uses a cluster as a location node, which has smaller size than the problem defined by customer locations.

This phase processes all visit nodes T, where a node j ∈ T represents a

customer to visit, in order to generate clusters which will act as super-nodes in the mathematical model used in phase II. The process in phase I also requires the set of vehicles V, travel distances and time between nodes, service times, and loads of each node. The outcome from phase I is clusters in which their members can form feasible route such that

1. the total loads in a cluster can fit in a single vehicle,

2. there exists a route connecting nodes inside the cluster which satisfy all time window constraints,

3. vehicle waiting time should be minimum, and

4. the average travel duration per node should remain low. The process follows the following steps:

1. Sort location nodes by increasing the value of the earliest arrival time ai;

if several nodes have the same ai, sort by increasing value of the latest

arrival times bi. The sorted location list is defined as L.

2. Sort vehicles by decreasing values of the capacity- ratio qv/c fvwhere qvis

a capacity of vehicle v and c fvis a fixed cost for using vehicle v, and name

3. Select the first vehicle v of V, define as a vehicle of a new cluster cn ∈ C

then remove v from V.

4. Pick a location node j from the list L to the current cluster cn ∈ C if j

does not make the total load of the cluster exceed capacity qv of vehicle

v and distance of j to the nearest node in the cluster does not exceed the

maximum distance allowed dmax.

5. If the node j fits into the current cluster, add node j to the cluster. 6. Pick another node from the list L until reaching the end of the list.

7. Repeat step 3-6 to build the next cluster until all location nodes in L are allocated.

8. Calculate the centroid, time and distances between clusters.

Each cluster represents a super node. Therefore, the mathematical formu- lation of a cluster based problem is built based on clusters. In this case, a su- per node acts like a single location. Therefore, node components must be ad- dressed, including cluster time window and cluster service time. Cluster time windows are defined by nodes in the cluster such that the earliest start time of the cluster is the minimum earliest start time of all location nodes in the cluster,

mini∈C(ai), and the latest start time of the cluster is the maximum latest start

time of all location nodes in the cluster, maxi∈C(bi). The cluster service time is

the summation of time required by all nodes in the cluster and the travelling time between those nodes.

Phase II: Cluster-based Multi-depot Heterogeneous Fleet VRPTW

This phase assigns clusters to vehicles by using an MIP model. This process takes the set of super-nodes, the set of vehicles, super-node time windows, super-node time and distances as inputs. We can see that the data size is shrink

when replacing nodes with super-nodes. Therefore, the optimal solution to the reduced data size should be easier to find, thus, takeing less computation time. The result of this phase is assignments of vehicles to visit nodes (via super- nodes). The solution also allocates the used vehicles to the designated depots, and provides a sequence of clusters on the same tour. The solution in phase II should satisfy all constraints. However, the solution can be improved by ad- justing order of visits within a tour which will be processed in phase III.

Phase III: the Single Tour Scheduling Problem

To complete the solution, this process schedules visits inside the tour. This phase assigns vehicle arrival time to visit nodes. This phase tackles every tour provided by phase II where the tour is formulated in the same mathematical model used in phase II and then solved by the MIP solver. The problem can be considered a travelling salesman problem as it requires a vehicle to leave the depot, visit all locations in the tour and return back to the depot.

Result and Discussion

The three-phase method was used to tackle instances in Solomon datasets which considers be homogeneous VRPTW. The method combines visiting nodes into clusters, which results in node reduction in the mathematical problem by 68- 90% of the original problems. The outputs from using this approach matched the best known solution. It was shown that Phase II required the most compu- tational time.

The Solomon dataset was modified to be a heterogeneous dataset by chan- ging the capacity of individual vehicles. However, to guarantee that all loca- tions can be visited, the total fleet capacity remained the same. The result from using this approach showed the computational time increases to 2,271 seconds (from 74.15 seconds).

This approach was reported to solve the homogeneous VRPTW successfully [56]. The computational time consumed by this approach is less than the exact method. However, it provided only one sample of the heterogeneous VRPTW which was the main aim of the proposed method.

Generally, this study is an example of using decomposition in multiple de- pot problems. The decomposition used in this algorithm is not focused on par- tition a problem, but merges several visits into a single representative location. In addition, this approach also makes a decision that the visits within a cluster must be made by a single vehicle. The MIP solver is used to make decision in phase II and phase III. The process of phase II is to find inter-cluster routes and each route will become an independent MIP problem in phase III. For the HHC problem, using the phase I algorithm to cluster visits might not as simple as the VRPTW because visits in the HHC problem requires workers with satisfactory skills, while any vehicle in the VRPTW can make any visits. Therefore, determ- ining visits to be in the same cluster is almost as hard as solving the whole problem.

There are parts of the solution approach that inspired to our research. The cluster generation process will be adopted in one of our visit partition rules, location based with uniform partition (see page 124). The single tour schedul- ing (phase III) also an inspiration to the assignment conflict repair process (see Chapter 5). However, there is a slightly difference in assignment conflict repair such that some assignments may be dropped which will become unassigned visits as all assignments cannot be guaranteed to be assigned.