In this chapter we introduce CODEA, a COoperative DEcentralised Architecture for the development of cooperative agent systems. We also discuss four other feature-rich frameworks for multi-objective optimisation: JMetal, ECJ, HeuristicLab and ParadisEO. None of these frameworks seem to provide explicit mechanisms to create cooperative systems of agents to tackle multi-objective optimisation problems. We present CODEA as a multi-objective optimisation framework to fill this gap. We provide a high-level overview of the main features of the last two versions of CODEA: CODEA v2 and CODEA v3. CODEA v2 is an evolution of the first version that provides a number of features to deal with multi-objective optimisation problems. CODEA v3 is the result of the hybridisation of CODEA v2 and ParadisEO-MOEO. Instead of implementing more multi-objective algorithms on CODEA, we built CODEA v2 on ParadisEO. This exten- sion enables ParadisEO to work with cooperative agent systems, and allows CODEA
v3 to use ParadisEO’s objects at all levels. The product of this hybridisation brings up a number of new ready-to-use features which are not present in state-of-the-art multi- objective optimisation frameworks, such as cooperation and organic groups.
This chapter also presents four research works related to CODEA v2. These works con- cern four different combinatorial optimisation problems: 1) Vehicle Routing Problem with Time Windows (VRPTW), 2) Travelling Salesman Problem (TSP), 3) Steiner Tree Problem (STP) and Multicast Routing Problem, and 4) University Course Time Tabling Problem (UCTTP). These problems have been tackled with a MODPSO-based algo- rithm. Our MODSPO has been adapted to each problem domain by specifying three problem-dependent components: 1) a set of operators, 2) evaluation function(s) and 3) a solution data structure. The only work that has not been implemented in CODEA is the one concerning the Steiner Tree Problem and Multicast Routing Problem (Section 3.4.2). The other three research projects have been directly benefited from CODEA. This has shown the potential of CODEA for applying agent-based solving methodologies to other problem domains.
The following chapters discuss a number of studies in which CODEA has been in- volved. The next chapter presents the Dynamic Lexicographic Approach (DLA). DLA is a multi-objective ranking approach to discriminate solutions using random lexico- graphic orderings based on certain probability distributions.
Currently, we are focusing our efforts on the development of new algorithms and the generation of documentation of the API. Further work should include the incorpora- tion a number of tutorials explaining basic uses of CODEA. Once the documentation is ready, we plan to submit CODEA to ParadisEO’s project webpage.
Dynamic Lexicographic Approach
for Multi-objective Optimisation
Summary
There is a variety of methods for ranking objectives in multi-objective optimisation and some are difficult to define because they require information a priori (e.g. es- tablishing weights in a weighted approach or setting the ordering in a lexicographic approach) [52]. In many-objective optimisation problems, those methods may exhibit poor diversification and intensification performance. We propose the Dynamic Lex- icographic Approach (DLA). In this ranking method, the priorities are not fixed, but they change throughout the search process. As a result, the search process is less liable to get stuck in local optima and therefore, DLA offers wider exploration ability in the objective space.
DLA is one of the novel features introduced in CODEA v2. This ranking scheme was implemented in order to study alternative ranking methods in multi-objective optimi- sation. This chapter compares the performance of DLA to that of Pareto dominance and lexicographic ordering according to the hypervolume [271]. These methods are tested on a Multi-Objective Discrete Particle Swarm Optimisation (MODPSO) algorithm tack- ling the Vehicle Routing Problem with Time Windows.
4.1
Introduction and motivation
Multi-objective Optimisation Problems (MOPs) have a number of objectives that are usually in conflict, so improving one objective leads to worsen another. In particular, many-objective optimisation problemsinvolve the optimisation of four or more objectives, presenting a considerable challenge for some solution methods. Most research in this area has focused on the study of MOPs with two or three objectives. This is due to the assumption that the scalability of multi-objective methods to many objectives would be straightforward. Recently, several papers have investigated this issue. Khare et al. [152] investigated the scalability of a number of Multi-Objective Evolutionary Algo- rithms (MOEA) with respect to 2 to 8 objectives. Results were assessed according to three criteria: 1) ability to converge to Pareto front, 2) diversity in the obtained non- dominated solution set, and 3) running time. The authors concluded that the results obtained with 2 or 3 objectives cannot be generalised to a larger number of objectives. Hughes [138] compared the performance of NSGA-II [65] to that of multiple single ob- jective optimisers in both MOPs and many objectives optimisation problems. Results indicated that NSGA-II looses efficiency as the number of objectives increases. Similar results were found by Wagner et al. [253]. That study concluded that a Pareto-based approach cannot succeed in dealing with many-objective problem instances.
Pareto dominance (Section 2.3.1) uses a strict ranking scheme that sometimes fails to discriminate between solutions, as it only accepts improvements in all objectives at the same time. Other methods like the lexicographic approach (Section 2.3.3) impose a static behaviour, as objectives are ranked according to a fixed relative importance. Two main alternatives have been proposed to deal with the scalability problem in many objectives optimisation problems [48]. One is the relaxation of the form of Pareto opti- mality, so that it is possible to accept solutions which worsen some objectives in certain quantity, if others witness an improvement [224]. Another one considers the reduc- tion of objectives of the original MOP [23]. However, due to the difficulty involved in reducing the dimensionality of the MOP, the first alternative is more popular in the literature [171].
Dynamic Lexicographic Approach (DLA) is an alternative ranking approach for many- objective optimisation. DLA presents a relaxation, not of the form of Pareto dominance, but of the form of the lexicographic approach. DLA offers an intuitive approach to establish a dynamic ranking among objectives. Rather than establishing a fixed priority among the objectives, the decision maker establishes a preference. This preference is then used with a probability mass function (pmf ) to generate a vector of priorities that changes dynamically throughout the search process.
This chapter is organised as follows. The algorithm for the Dynamic Lexicographic Approach is detailed and exemplified in Section 4.2. We describe our experiments in Section 4.3 and discuss results in Section 4.4. Finally, our contribution and proposed further research are summarised in Section 6.9.