CAPÍTULO 3: ESTRATEGIA METODOLÓGICA
II. RESULTADOS CUALITATIVOS BASADOS EN LOS INSTRUMENTOS
As explained in Section III.A, finding the Pareto optimal set is not always achievable in practice. Therefore, the aim of multi-objective optimization is to find an approximate nondominated front. But it is difficult to compare the approximation sets obtained from one algorithm with those of another algorithm. Because generally, one nondominated set is not dominantly better than the other. So there is a need for unary indicators or metrics to measure the quality of the solution sets. There are several indicators that try to describe the goodness of an approximation in terms of criteria such as closeness to the Pareto optimal set, diversity, and coverage of the space (Zitzler et al. 2003).
Hypervolume measure, also known as the S-metric (Zitzler 1999) or Lebesgue measure (Laumanns, Rudolph, and Schwefel 1999, and Fleischer 2003), is the best known indicator that was first introduced by Zitzler and Thiele 1999 as “size of the space covered”. The hypervolume metric measures the amount of objective space that the approximation set dominates. In other words, it measures the volume of the approximate nondominated front w.r.t a reference point. Usually, the reference point
values in a dimension, cannot contribute in that dimension, as their distances from the nadir or reference point is zero. To avoid having a volume of zero, the worst point can be shifted (worsen) by an appropriate amount.
Comparing approximate nondominated frontiers, the one with a higher hyper- volume is better. Because it is a better approximation of the true nondominated fron- tier. The hypervolume is maximized if and only if the set contains all Pareto optimal points (Fleischer 2003). Bringmann and Friedrich 2008 showed that the calculation of hypervolume metric is #P-hard (#P-hard of counting problems is analogous to NP-hard of decision problems), which implies that a polynomial-time algorithm does not exist, unless P = NP. Many researches have been done in this area to make the calculation faster. To learn more, refer to the dissertation of Bradstreet 2011.
In this research, the Hypervolume by Slicing Objectives (HSO) algorithm (While et al. 2006) is used for calculating the hypervolume indicator. Figure. 9 shows the hypervolume created by 4 points (a, b, c, d) in a three-objective maximization problem. This figure also shows how HSO sweeps along each objective and creates cross-sectional slices. In this figure, the unfilled circles represent points in each slice that are dominated by a point in y and z axes. Here, the reference point is the origin. The pseudo-code of HSO algorithm reproduced from While et al. 2006 and Bradstreet 2011 is shown in Algorithm 3.
The inputs of the algorithm are the nondominated front, f, and the number of objectives, n. Function “Depth” calculates the distance between two consecutive points. Function “Insert” performs two tasks. First, it sweeps along the chosen objective and makes a slice between one point to the other. For doing so, it visits the points one at a time, skips the first objective of f, and add the points to a new list derived from n − 1 objectives. So, it maintains a list of all the points that are visited and contributed to the current slice.
FIGURE 9 – HSO algorithm (While et al. 2006)
Algorithm 3Pseudo-code of HSO function HSO(f, n)
f = sorted f in objective 1 in descending order
if n= 2 then
return the area created by the two dimensions of n
np = empty list for keeping nondominated points
for all p ∈ f do
depth = Depth(f, p)
Insert(np, p, n − 1)
hypervolume = hypervolume + depth × HSO(np, n − 1)
return hypervolume function Depth(f, p)
if p is a one-dimension point then
return distance from p to the reference point
return distance from p to the next point in f w.r.t first dimension function Insert(np, p, n)
skip the first objective and insert p to np remove any point in np that is dominated by p
ensure all the points in np are sorted in the next relevant objective
nondominated and also sorted w.r.t to the next objective. So when a new points is added to the slice, all the points that are dominated by the new point are removed and the remaining points are sorted by their first objective. This concept is illustrated
objective (along x axis). Point a is the first to be visited and added to the list. Then, the slice from point a to point b is created (the first objective is skipped and the remainder is the 2-dimension area created by the values of y and z axes). Then point
b is added and the slice now contains both a and b (they are nondominated w.r.t the
remaining two objectives along y and z axes). When point c is visited, it dominates both a and b along y and z axes. So, they are removed and the slice contains only point c. Finally, point d is visited an added to the slice. Since each slice creates a new list of points with n − 1 objectives, its hypervolume can be calculated recursively.
The HSO pseudo-code in Algorithm 3 is for maximization problem. The ref- erence point in the simplest form can be the nadir point. However, in maximization problems (with all positive values), origin can be a good candidate which makes cal- culations easier. For minimization problem, on the other hand, there are two options. Either using the reference point, or normalizing the problem first, then inverting it and working with the normalized and inverted problem similar to a maximization problem. In this research, the latter approach has been used. Because it is easier to work with similar-signed values (all positive or negative) as opposed to different signs (which is the case in this research) and normalization takes care of this matter.
To normalize a point, a reference point is needed. Here, the reference point is the nadir point shifted (worsen) by an appropriate amount. So in a minimization problem, fRef > fN, where fN is the nadir point. Each point f ∈ ˜Y
N is first
normalized and inverted as follows:
f0 = f − f I fRef− fI
¯
f = 1 − f0
where fI, f0,and ¯f are the ideal, normalized, and inverted points, respectively.
proximation, it is a point estimate. So, after having all the hypervolumes of all the solutions, a statistical hypothesis test (such as Kruskal-Wallis non-parametric test) is required for finding out which algorithm provided statistically significant better results.
Knowing the hypervolume indicator and the approximate nondominated front obtained using PAES method, one question is how they can help in generating the order picking strategies? The goal of next chapter is to answer this question.
CHAPTER IV
ORDER PICKING STRATEGIES