In last few decades, evolutionary algorithms (EAs) have been successfully used for optimization problems involving more than one conflicting objective, as these algo- rithms are capable to produce a set of solutions in a single run. This set of solutions are an approximation to the Pareto-optimal set, as described above. In 1985, the first real multi-objective genetic was proposed by Schaffer [1985]. This approach involves generating subpopulations at each generation. The number of subpopu- lations are equal to the number of objectives. Each sub-population is responsible for searching one objective. Since 1985, different evolutionary algorithms have been proposed for multiobjective optimisation using evolutionary algorithms (MOEAs). For example, [Fonseca et al., 1993; Srinivas and Deb, 1994; Fonseca and Fleming, 1996;Zitzler,1999;Zitzler and K¨unzli,2004;Zhou et al.,2011]
Multiobjective evolutionary algorithms can be classified into three groups:
Dominance-based algorithms: These are the most popular multi-objective evo- lutionary algorithms that have been proposed by many researchers. In 1999,Zitzler and Thiele[1999] proposed the Strength Pareto Evolutionary Algorithm (SPEA) for
approximating the Pareto-optimal set for multi-objective optimization problems. It combined different features from the previous multi-objective EAs in one algorithm. For example, it stores a non-dominant evaluated set of solutions in an external pop- ulation, assigns scalar fitness values to individuals based on the Pareto dominance concept and if the size of the Pareto set is larger than a predefined limit, then the Pareto set is reduced without destroying its characteristics by using clustering methods.
In 2000, Knowles and Corne proposed the Archived Evolution Strategy (PAES) that uses the dominance concept to evaluate solutions [Knowles and Corne,2000]. It is used a single-parent single-offspring EA similar to a (1+1) evolution method. They used binary strings and bitwise mutations to create offsprings. This algorithm compares the offspring with respect to the parent. If the parent is dominated by the offspring, the offspring is the next parent while if the offspring is dominated by the parent, the offspring is rejected and it finds a new one. On the other hand, if neither dominates the other, both the parent and the offspring are compared with an archive of best solutions found so far.
The Nondominated Sorting Genetic Algorithm II (NSGA-II) [Deb et al., 2002] is a commonly used MOEA. It is an elitist algorithm in which the approximation to the Pareto set (the maximal set of non-dominated solutions) is kept from generation to generation. A crowding distance strategy is used to limit the size of the Pareto set and help improve the spread of solutions across the Pareto front. The crowd- ing distance strategy is used to perform density estimation of solutions surrounding a specific solution in the population and create a Pareto rank for each individual. Nondomination individual rank and crowding distance are needed to create a pop- ulation of individuals. In addition, there are other algorithms based on dominance concepts such as the Niched Pareto Genetic Algorithm (NPGA) [Abido, 2003] and Multi-objective Differential Evolution (MODE)[Varadarajan and Swarup,2008].
gorithms is the approximation of the Pareto optimal set. Indicator based approaches use a scalar indicator such as hypervolume and generational distance to measure the quality of the Pareto front [von L¨ucken et al.,2014]. These algorithms used the indicator to direct the search. Zitzler and K¨unzli[2004] proposed the first indicator- based an evolutionary algorithm (IBEA). In this algorithm, a pair of solutions are compared using a binary indicator and it does not need any diversity preservation mechanism. In 2005, Emmerich et al. [2005] proposed S-metric selection-EMOA (SMS-EMOA) which is based on the hypervolume measure to combine the concept of a selection operator and non-dominated sorting. It is similar to NSGA-II except in selection and there is a different ranking method used for the Pareto optimal solutions. This algorithm is designed to maximise hypervolume which is the size of dominated space [Hopfe, 2009]. However, the complexity of computing the hy- pervolume indicator in high dimensions is expensive. This algorithm showed good results for two or three objective problems. In order to deal with this problemBader and Zitzler [2011] proposed the approximation of exact hypervolume values using a Monte Carlo algorithm and they presented this idea as Hypervolume Estimation Algorithm for Multi-objective Optimization (Hype).
Decomposition based algorithms: Another promising multi-objective evolution- ary algorithm for optimising multi objectives by using the scalar functions are de- composition algorithms [Jaszkiewicz,2004;Hughes,2007;Li and Zhang,2006;Zhang and Li,2007]. Two difficulties associated with solving multi-objective problems need to be determined: 1- The number of solutions to approximate the Pareto front in- creases exponentially [Ishibuchi et al.,2008]. 2- The ability of search will deteriorate. The advantage of these algorithms is computational efficiency for calculating the scalarisation function. One of the most popular decomposition methods is MultiOb- jective Evolutionary Algorithm based on Decomposition (MOEA/D which has been developed byZhang and Li [2007]. A lot of literature demonstrated that MOEA/D has a high ability of search on different test problems and it has a low computation of complexity [Chang et al.,2008;Ishibuchi et al.,2009,2010;Konstantinidis et al.,
2010; Li and Zhang, 2009; Peng et al., 2009; Zhang et al., 2010]. The main idea behind MOEA/D is to decompose the multiobjective optimisation problem into a number of scalar optimisation subproblems rather than solving a MOP as a whole. There are three approaches which have been used by Zhang and Li [2007] for the decomposition process. The simplest one is the weighted sum aggregation method. This method works well when the Pareto Front (PFs) is concave, but that disad- vantage is the nonconcave Pareto front can not be handled. The second one is the Boundary Intersection method which is used with nonconcave PFs. The last one and most popularly used is the Tchebycheff approach. This approach can be used with fronts that contain concave and convex regions. We have therefore used this method for the work presented here and the algorithm is described in more detail in Chapter 3.
In this thesis, we will use MOEA/D to optimise two conflicting objectives simultane- ously to produce a set of candidate solutions in evolutionary optimisation method; this will be discussed in chapters 3 and 4.