This thesis is structured as follows. Chapter 2provides the required background for the present study. It explains what combinatorial optimisation problems are and some methods frequently used for solving them, namely local search, Tabu Search, and Evolutionary Algorithms. Here are described multi-objective combinatorial optimisation problems, and some quality indicators that are used to evaluate opti- miser performance. Three widely-known multi-objective Evolutionary Algorithms
are presented, highlighting their key stages of processing. Lastly, a series of solution distance measures for combinatorial problems are introduced.
The Vehicle Routing Problem and two of its variants, specifically the Capacitated VRP (CVRP) and the VRP with Time Windows (VRPTW), are formulated in Chapter 3. Additional considerations and constraints are exposed for further va- riants of the problem. An overview of previous studies that are relevant to this thesis, that is studies that tackled the CVRP and VRPTW, is given, along with their reported results for commonly used benchmark sets. Finally, this chapter dis- cusses multi-objective VRPs and surveys further studies and objectives that have been considered for optimisation.
The developed preliminary approaches to solving VRPs, along with their results and analysis, is presented in Chapter 4. One of the major contributions of this thesis is presented in this chapter, which is the measure designed to quantify the similarity between two solutions to the VRP, as well as the proposed population diversity measure. The algorithm described in this chapter forms the basis of the final proposed Multi-Objective Evolutionary Algorithm for solving the CVRP and VRPTW.
Chapter 5 explains the adjustments made to the algorithm previously introduced in order to effectively solve the CVRP and VRPTW. Here are presented the expe- rimental studies and the analysis of results from the bi-objective and tri-objective optimisation of the concerned problems, and a discussion on the findings.
Finally, the evaluation of the proposed approach at its different stages of develop- ment, along with the main contributions of this thesis and potential directions for further research, are presented in Chapter6.
Chapter 2
Combinatorial optimisation
problems
This chapter provides the essential background knowledge to this thesis. Here is described, firstly, what combinatorial optimisation problems are and some common techniques for solving them, specifically local search, Tabu Search and Evolutionary Algorithms. Secondly, an introduction to multi-objective combinatorial optimisa- tion problems is given, as well as to three multi-objective performance metrics, that is coverage, convergence and hypervolume. Then, the key difference between single-objective and multi-objective Evolutionary Algorithms is explained, an three well-known and successful multi-objective Evolutionary Algorithms, namely PAES [144], SPEA2 [261], and NSGA-II [67], are described. Finally, a number of solution distance measures for combinatorial problems are presented.
2.1
What are combinatorial optimisation problems?
An optimisation problem consists in finding the best (optimum) solution to a given instance of the problem. These problems can be grouped into two categories: In
the first category one can find those problems with continuous variables, where we look for a vector of real numbers. In the second group we find those problems with discrete variables, which are called combinatorial, where one typically looks for an object within a finite set, or possibly countable infinite, which is generally a sub-set of the variables, a permutation, or a graph [186].
Formally, an instance of a combinatorial optimisation problem is a pair (X , f ), where X is an N -dimensional domain, and f is a function that maps f : X → R. Without loss of generality, we consider the problem of finding a solution x∗ ∈ X for which
f (x∗) ≤ f (x), ∀ x ∈ X . (2.1)
Such a solution x∗ is called a globally optimum solution to the given instance, or,
when no confusion arises, simply an optimum solution [186]. Furthermore, f∗ =
f (x∗) denotes the optimum cost, and X∗ = {x ∈ X | f (x) = f∗} denotes the
optimum solution set [2].
Sometimes we face problems for which not all x ∈ X are valid solutions, in the sense that they do not satisfy certain restrictions. Thus, all x ∈ X′ ⊆ X satisfying such restrictions are the feasible solutions, and subset X′ is the feasible domain. In this
case, the problem in (2.1) must satisfy
gi(x) ≤ 0 , (2.2)
hj(x) = 0 , (2.3)
where the gi(x) functions in (2.2) and the hj(x) functions in (2.3) are the inequality
and equality constraints, respectively, which actually define the feasible domain X′.
We will refer to function f as the objective function, to the domain X as the solution
space, and to the feasible domain X′ as the feasible region or search space.
For illustration purposes, Figure 2.1 presents one of the simplest combinatorial pro- blems, though not easy, which is the knapsack problem [139]. Here we have a set
Figure 2.1: Knapsack problem: which items should be packed to maximise the total value while the overall weight is kept under or equal to the backpack limit?
of items, each with a weight and a value, and a backpack which capacity is limited in weight. The problem consists in determining which combination of items (the solution space) should be packed so that the total weight is less than or equal to the backpack limit (inequality constraint) and the total value is as large as possible (the objective function). The Vehicle Routing Problem is a also combinatorial problem, however, in addition to selecting items, we have to define a sequence of them, which perhaps makes the problem even more difficult to solve.