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According to the fitness assignment scheme of PICEA-g, it can be observed that if the goal vectors are exclusively generated in a region then candidate solutions inside this region will be more likely to survive during the search. The reason is that these candidate solutions can dominate more goal vectors and so are more likely to gain higher fitness, while candidate solutions outside this region can only dominate a few goal vectors and so are likely to have lower fitness. Therefore, candidate solutions will exhibit a tendency to approach to the specified region. This has also been demonstrated in Section 3.7.1 that by using different goal vector bounds, PICEA-g can approximate different parts of the Pareto optimal front.

(a) Aspirations (b) Weights

Figure 6.6: Illustration of the goal vectors generated in different cases.

Inspired by this thinking, goal vectors in iPICEA-g are generated only within the region defined byzR,wand θ. Specifically, goal vectors are generated in the shaded region in

Figures6.6(a)and6.6(b)when the preferences are modelled by aspirations and weights, respectively. By co-evolving candidate solutions with these specially generated goal

vectors, candidate solutions are expected to be guided towards the ROIs. Similarly, when the preferences are specified using the brushing technique, goal vectors are generated in a similar way as shown in Figures 6.6(a) and 6.6(b) for the Scheme 1 and Scheme 2, respectively.

(i) When searching with aspiration levels, goal vectors are generated in both G1 and

G2, extending both towards and away from theideal point. This is to handle the case where the supplied zR is unattainable. In this case, the Pareto-dominance

relation is applied to determine whether a solution meets (i.e., dominates) a goal vector or not.

(ii) When searching with weights, goal vectors are generated along the directionwand spanned withinθ radians, that is, goal vectors are generated in the shaded region closed by pointsgmin,gf1,gf2 andgmax.

Note that in this case, instead of using the Pareto-dominance relation, iPICEA-g employs the Pareto cone-dominance (Batista et al., 2011) to determine whether a goal vector g is met by a candidate solution s or not. Specifically, gj is said

to be Pareto cone-dominated by si, if and only if the angle between the vector

−−→

sigj and the vector −−−−→sigmin is not larger than θ. For example, in Figure 6.6(b),

g1 is Pareto cone-dominated by candidate solution s1 while g2 is not Pareto cone-dominated by s1. Mathematically, given two feasible vectors x and y we say x Pareto cone-dominates y, x _cone y : if and only if y ∈ _C, where C is

a generated cone, given by weight vector w1,w2,· · ·,wM, i.e. C = {z : z =

λ1w1+λ2w2,· · · , λMwM,Pλi = 1,∀λi > 0}. Figure 6.7 illustrates the gener-

ated cone ofxin a bi-objective space.

Figure 6.7: Illustration of a generated cone in 2-objective case.

Compared with Pareto-dominance, the use of Pareto cone-dominance can further em- phasise the solutions along the reference direction, i.e., assigning higher fitness to these

solutions. For example, in terms of Pareto-dominance,s2 can satisfy some goal vectors and therefore might be retained in the evolution. However, s2 is not in the ROI. When using Pareto cone-dominance,s2cannot satisfy any goal vector, having the lowest fitness and then would be more likely to be disregarded in the evolution.

In both cases the lower,gmin, and upper,gmax, goal vector bounds are estimated based

on the objective values of the non-dominated solutionsf(S∗) found so far and the refer- ence point, zR,

gmin = min (zRi∪fi(S∗)), i= 1,2,· · · , M

gmax = max (zRi∪fi(S∗)), i= 1,2,· · ·, M

(6.2)

In addition to the benefit that iPICEA-g can handle different types of DM preference, i.e. aspiration levels and weights, another major benefit of iPICEA-g is that multiple ROIs can be explored by simultaneously generating goal vectors for all the ROIs. It is also anticipated that iPICEA-g will perform well on many-objective problems because PICEA-g has been demonstrated to have good performance in this environment. Finally, the interaction of the analyst and the decision-maker when using iPICEA-g is as follows:

(i) The analyst asks the decision-maker to specify his preferences. The preferences can be expressed by aspirations, weights or brushed regions. Based on these pref- erences, the three parameters (zR,wand θ) are configured.

(ii) Next, the algorithm sets the population size of candidate solutions, the number of goal vectors and the stopping criterion. Subsequently, iPICEA-g is executed until the stopping criterion is met.

(iii) The obtained solutions are presented to the decision-maker. If the decision-maker is satisfied with the provided solutions then stop the search process. Otherwise, ask the decision-maker to specify new preference information. For example, if the decision-maker would like to narrow the range of the obtained solutions, then reduce the setting ofθ; if the decision-maker would like to bias one objective, e.g. fi, then increase the value of wi. Certainly, decision-maker can specify a different

zR. After updating the preferences, return to step (ii).

An additional issue

Intuitively, driving the search towards a narrow region at the beginning of an optimisa- tion process might cause a lack of population diversity as the candidate solutions may have similar phenotypes and thus result in low search efficiency and converge to a local optimum (Coello Coello et al., 2007, pp. 131-143). This issue is more likely to appear

in a priori decision-making as the decision-maker might be only interested in a small Pareto region, i.e., the search range is small. In order to avoid this problem, we suggest starting the search with a large search rangeθ0 and gradually, decreasingθ0 to a preferred search rangeθ. The decreasing process is defined:

θuse=θ0−(θ0−θ)×(

gen

maxGen)

α_{; (6.3)}

wheregen is the current generation,maxGenis the maximum generations, and α is to control the speed of decreasing. As shown in Figure6.8,θis decreased from π₄ radians to

π

9 radians with differentα. A largeαcorresponds to a slow decreasing speed in the early stages and a fast decreasing speed in the later stages of evolution. α = 2 is suggested for use after a preliminary analysis. The benefit of this strategy will be illustrated next in Section6.3.1. 0 20 40 60 80 100 20 25 30 35 40 45 50 generation α = 2 α = 1 α = 0.5 α = 4 α = 0.25 θ’ θ

Figure 6.8: Illustration of the decrease process ofθwithmaxGen= 100.

6.3

Experiments

In this section, we examine the performance of iPICEA-g on different benchmarks from the ZDT, DTLZ and WFG test suites. The selected test problems have convex, concave, connected, as well as disconnected, Pareto front geometry. In all the experiments, the simulated binary crossover (SBX, pc = 1, ηc = 15) and polynomial mutation (PM,

pm = 1/nper decision variable andηm= 20, wherenis the number of decision variables)

are applied as genetic operators. To quantitatively measure the performance of iPICEA- g, the generational distance (GD) metric is employed to evaluate the convergence of the obtained solutions.

To gain insight into the effect that the parameters zR, w and θ have in iPICEA-g, we

study them in isolation in Section 6.3.1. In Section 6.3.2, we describe the performance of iPICEA-g when searching with aspiration levels or weights for regions of interest in an a priori preference articulation setting. In Section6.3.3, we introduce what we refer to as the brushing technique (Buja et al.,1996;Hauser et al.,2002), which is useful for interactive preference articulation and decision-making.