Pareto scaling

Dividing each variable by the square root of the standard deviation is called Pareto scaling. 

Pareto scaling [60] (uses the inverse of the square root of the standard deviation, has an effect, that is, somehow intermediate between using raw variables and scaling, in the sense that large variance variables are less down-weighted) ... 

The SMB process is a dynamic process operating on periodic cycles which makes it a challenging optimization problem. The IPOPT optimizer (Wachter and Biegler, 2006) was used within the IND-NIMBUS software (as an underlying optimizer) to produce new Pareto optimal solutions. The IPOPT optimizer was chosen because it has been developed for solving large scale optimization problems. 

In Figure 4-15 the average total stock in tons per period is plotted against the number of shipments ordered per period. Each Pareto-optimal solution is represented by a grey-scaled point whereby the shade indicates the realized fi-service level of this solution. Bright solutions show a fi-service level of about 40% whereas dark-coloured refer to a fi-service level close to 100%. 

Figure 4.15 Scatterplot of Pareto front (/ -service level in grey scale)...

In Figure 4- Pareto front is evaluated over the area spanned by the sample depicted in Figure The grey scale, representing the jS-service level, is the same as before. 

Figure 4.16 Grey-scaled levelplot of estimated Pareto front...

Let time-to-failure y,- be distributed according to the Pareto distribution P X, a) with known scale parameter X and unknown shape parameter a Notice that transformation transforms the shape parameter (so called tail index) of Pareto to the scale parameter of Erlang distribution. At the other hand, we consider the scale parameter X of Pareto to he known. This is, however, in the risk community a common assumption. The assumption of X being known is cpiite typical, because, as for example (Philbrick (1985)) states, "although there may be situations where this value must be estimated, in virtually all insmance applications this value will be selected in advance . 

It could be argued that the apparent lack of trade-off between the two objectives could be a consequence of the ISE not being a good criterion. However, this impression might be also a question of scaling (i.e. check figure 9d). Moreover, replacing the ISE metric by others, like the ITSE, led to similar results for the Pareto front. In any case, we would like to stress that these are a posteriori conclusions which can only be taken if the multi-objective problems are properly solved with robust methods, the main objective of our chapter (otherwise, the results can be artifacts due to the nonconvexity of the NLPs, as we discussed). 

In modern physics and economics, phase tfansitions and nonlinear dynamics are related to power laws, scaling and unpredictable stochastic and deterministic time series. Historically, the first mathematical application of power-law distributions took place in the social sciences and not in physics. We remember that the concept of random walk was also mathematically described in economics by Bachelier before it was applied in physics by Einstein. The Italian social economist Vilfredo Pareto (1848-1923), one of the founder of the Lausanne school of economics, investigated the statistical character of the wealth of individuals in a stable economy by modeling them with the distribution y c , where y is the number of people with income x or greater than x and v is an exponent that Pareto estimated to be 1.5 [29]. He noticed that his result could be generalized to different countries. Therefore, Pareto s law of income was sometimes interpreted as a universal social rule rooting to Darwin s natural law of selection. 

Multi-Objective Evolution Strategies (MOES) [25]. This is a multi-objective variant of classic Evolution Strategies (ES), an evolutionary algorithm based on mutation only. Mutation simply adds to each component of the solution a random number drawn from an adaptive distribution. Solutions are then ranked, based on their fitness values, to obtain the Pareto front. We set minimum step size /Xmin = 0.01, initial step size innu = 0.2, life span LS = 30, scaling factor a = 0.2, and learning rate t = 1. 

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