Scatter Search is another population based approach proposed by Glover et al. [31]. It presents a more deterministic approach to population based search methods. The population of solutions, the so-called reference set, is initialized to be diverse explicitly by selecting a diverse subset of solutions from a construction method. To create new solutions, asubset generationmethod creates subsets of solutions which are later combined by asubset combinationmethod, improved and then, based on theupdatestrategy, included into the population.
In their ongoing work they also presented techniques for the subset generation and combi-nation [30, 40]. For the subset generation an iterative approach was suggested. The first set of subsetsU1(type 1) consists of all pairs of the solutions
U1=[
(σi, σj), wherei= 1, . . . ,|RefSet| −1, j > i. (3.2) The following type 2 setU2 holds all subsets by creating a subset for each type 1 subset and adding the best solution not already in the subset to it.
U2=[
(σi, σj, σk), where(σi, σj)∈ U1, k=min{k|i6=k6=j, σk∈RefSet}. (3.3) The same procedure is applied for type 3 subsetsU3 usingU2. As this might create duplicate subsets (i.e. subsets containing the same solutions) only one of the duplicates is generated. If the number of solutions in the reference set is larger than4then subsets of sizei= 5, ...,|RefSet|
containing theibest solutions are generated, where|RefSet|is the size of the reference set, i.e., U =U1∪ U2∪ U3∪(σ1, σ2, σ3, σ4, σ5)∪. . .∪(σ1, σ2, . . . , σ|RefSet|). (3.4) As combination method they presented path relinking. Path relinking generates solutions using the neighbourhood space. Given two solutions, the initiating solutionand the guiding solution, the method identifies the intermediate solutions by calculating a pathπof moves from the initiating solution and the guiding solution. Each entry inπ is a neighbour of the previous solution, starting with a neighbour of the initiating solution and ending with the last solution before reaching the guiding solution.
After creating new solutions these are improved using for example local search heuristics and afterwards some of these are put into the reference set. Glover et.al. discussed this process and introduced atier-based approach. In this approach, the reference set is split into 2 or 3 tiers.
In tier 1 the best solutions are stored, starting with the best solutions to the worst. This part will be updated by replacing worse solutions with the best new solutions created. Tier 2 holds a diverse set of solutions where solutions are stored and replaced based on their contribution to the overall diversity of the set. Tier 3 was proposed by Laguna and Marti [37] where so-called best generatorsare stored. These generators are solutions contributing best when used in the combination method.
Related work
Based on their research of the NRP for hospitals in Belgium, Burke et al. [13] used a scatter search to tackle the problem more efficiently. They used a constructive solution combination technique (see also section 5.3) and tested a hill-climbing and a VND as local search procedure.
Compared with the MA approach in [13] the scatter search underperforms when using hill-climbing and produces better results when using VND. However, when VND was used, the computation time increased significantly.
Russel et al. [50] presented a scatter search for the VRPTW using an arc subset generation approach and a global route combination approach where a set partitioning algorithm is used to create new solutions. As local search procedure a previously proposed reactive tabu search [23]
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was used. Their approach was able to find 6 best known solutions out of 52 instances [51].
Other research using scatter search was done only for related problems, like the VRPTW with split deliveries by Belfiore [5]
Motivation
As a more deterministic approach, the scatter search seemed promising to find reasonable good solutions using more thorough search techniques then the other selected metaheuristics. The possible high runtimes needed to produce competitive solutions as for related problems was also considered, but the possible gain in terms of solution quality favoured the use of this approach.
Additionally the scatter searches’ pool might also be used to refine multi day solutions, although the small size might not help as much as the population created by the MA.
CHAPTER 4
Existing Approach
The metaheuristics implemented for this thesis are built into the already existing solving ar-chitecture created at the AIT. This framework includes several vital procedures like loading problem instances, creation of the travel time matrix and logging as well as the representation and evaluation of the solution.
This existing architecture also contains a constraint programming (CP) approach to gener-ate initial solutions in addition to a random generation method and a VND-VNS improvement heuristic with several neighbourhood structures. As the metaheuristics implemented in this the-sis use this framework and will be compared to the existing approach, this chapter will provide details of parts of the framework.
4.1 Solution Representation
In the existing approach a solution σ = (R,S) is created by assigning jobsj ∈ J to routes R ∈ R. Each of the jobj ∈ J is assigned to at most one tour Rnand each job j ∈ Rn is assigned to a specific position in tourRn, i.e., for each job j ∈ J assigned to tour Rn has a unique positioniin the tour represented byRni, wherei= 1, . . . ,|Rn|. An example is shown in Fig. 4.1:
The set of starting timesS is then constructed using a 2-step greedy algorithm illustrated in Fig. 4.2: In a first step, the earliest possible start times for each jobRni for nursen∈ N, where i = 1, . . . ,|Rn|, are computed under consideration of the travel time from the location of the previous job loc(Rni−1) (or the the home location loc(n), ifi = 1) to the location job of Rni as well as all time windows, i.e., starting time window of jobRni and the nurses’ working time windowsT Wn. While the travel times are never violated, the time windows are considered as soft constraints (see Sec. 2.3), thus violations of these windows are possible and allowed.
In a second step, all jobsRni ∈ Rn except the last are moved backwards as far as possi-ble. After each of the two steps, either version can be better due to particular soft constraints, like meeting the desired start times of the customers. Therefore, the assignment is selected
R1 j3 j1 j7 j2
Figure 4.1:Illustration of the representation (a) and actual tours (b) using an exemplary assignment
Rn1
Figure 4.2: An example for the 2-step greedy heuristic for determining job starting times for nurse n∈ N: first step (a), starting with the first job, all jobs are set to the earliest possible start time considering the job durations as well as the travel times between the jobs (dashed lines); second step (b), starting from the second to last job, the jobs are moved as far as possible to the end (with respect to their durations and travel times between them).
that results in a better objective value which will be described next. This improvement step is performed whenever a solution is constructed or has been modified.