POLÍTICA Y ESTRATEGIA

A way of guaranteeing that an optimal solution has been found is to analyse all possi- ble solutions. Such process is called enumeration. For small problems enumeration is a feasible approach. Enumerating the solutions often helps to understand the struc- ture of the problem. When enumerating it is common to use a tree based structure that represents all possible solutions. The tree could potentially hold all different possibilities of variable configurations. The root node of the tree holds branches for every variable depending on their finite number of values. The leaf nodes, i.e. those without children, in the tree represent final values that cannot be branched further. The strategy to search the tree could yield different results. The strategy needs to be defined at the beginning of the optimisation process. Common strategies include a depth-first approach which explores an specific area until a terminal node, i.e. leaf, is found and subsequently it backtracks to the nearest junction. Another strategy, known as breadth-first, explores the same level of the tree and whilst doing so, it is able to prune sections of the tree which given their configuration could not lead into feasible solutions. A depth-first strategy tends to find feasible solutions quickly but it neglects regions of the tree which might have better ones. A breadth-first strategy consumes a lot of memory resources but can compare across the tree and facilitates the removal of dominated subsolutions. (Lawler and Wood, 1966; Mitten, 1970; Hillier and Lieberman, 2010)

As the size of the problem increases, the size of the tree that contains all possible solution grows explosively. Branch and Bound aim to reduce the number of nodes to be analysed in the tree whilst still maintaining optimality. In the case of large problems, the algorithm is better if performing branching only in selected regions of the tree. The regions that are bounded, hence the name, for two values: an upper bound and an estimated lower bound. The branch and bound helps to prove that some partial solutions represented in the tree structure will not lead to optimal solutions hence discarding them from the search this process is called pruning.

Branch and Price refers to the combination of branch and bound and column gen- eration methods. It consists on decomposing the original combinatorial optimisation problem into two types of sub-problems. A master problem and a pricing problem. It is a method commonly used to solve large inter programming models and mixed integer ones. (Feillet, 2010; Danna and Le Pape, 2005)

Branch and Cut uses branch and bound in combination with cutting planes techniques to gradually reduce the search space of the problem. Cutting planes iteratively refine a feasible set by adding linear constraints that satisfy all feasible integer points but

violate the current fractional value within the tree structure. (Mitchell, 2002; Martin, 2001)

A methodology that has been very useful to tackle WSRP is branch and price. Branch and price refers to using a branch and bound approach with column generation (Barn- hart et al., 1998; Feillet, 2010). The advantage of using column generation is that the problem can be relaxed and solved with a reduced set of columns, which might not be an exhaustive enumeration of all possible routes for every employee, but at any time provides a solution if it exits. In the literature, the personnel scheduling constraints side of the problem is commonly solved by heuristics to generate columns. On the other hand, the routing component can be tackled via branching. Kallehauge et al. (2005) showed that the problem formulation can be decomposed into a master prob- lem and a pricing problem. The master problem is a set partitioning problem and the subproblem a series of shortest path problems with resources constraints (Irnich and Desaulniers, 2005; Feillet et al., 2004).

Models applied to VRPTW have also been aplied to WSRP, in particular multi- commodity network flow models with time windows and capacity constraints. When using branch and price, many authors have modelled the master problem as either a set partitioning problem or as a set covering problem. There is not much difference between these two. In the first one, each customer is in one route only, whereas in the second one, more than one route could visit the same customer location.

3.1.2

Heuristics Algorithms

In this section a description of metaheuristics methods used in the tackling of work- force scheduling and routing is presented. For each metaheuristic a brief overview is performed and then reference to relevant work in the literature is provided.

Metaheuristics are high-level search methods which guide and influence other heuris- tics to increase their chances of finding good valid solutions in the search space. They offer a framework structure that is applicable to any domain which makes them non-problem specific. Metaheuristics use domain specific knowledge in their imple- mentation(Osman and Laporte, 1996; Glover and Laguna, 1999; Voßet al., 1999). Metaheuristics can be classified according to more than one criteria. Among the most common ones are the following: origin or inspiration of the algorithm therefore there are nature-inspired and non-nature inspired metaheuristics. Number of solutions si- multaneously, single point search or population based. Single point search act over

only one solution trying to improve it with every iteration. Population based have many solutions which evolve by combining characteristics of the solutions to pass them to the next generation. Classification based on the objective function nature which could be static or dynamic. Number of neighbourhoods, most metaheuristics use one neighbourhood but the possibility of using more than one in order to change the topology of the space search is a way of differentiating metaheuristics. The final classification is whether memory structures are used or not. Memory-less metaheuris- tics perform iterations based only on their current state without remembering good solutions or regions with potential to explore (Blum and Roli, 2003)