Compared to other surrogate modeling methods, kriging has the advantage of providing both the predicted mean and the associated mean square error (MSE) at an unknown location. The probability of improvement *PoI* at any location is given by
$$\begin{array}{}{\displaystyle PoI\left(x\right)=\mathit{\Phi}\left(\frac{{\mathit{y}}_{\mathit{t}}\left(\mathit{x}\right)\mathit{-}\hat{\mathit{y}}\left(\mathit{x}\right)}{\hat{\mathit{s}}\left(\mathit{x}\right)}\right)}\end{array}$$(6)

where *y*_{t} is the target of improvement, *y*̂ the kriging predicted mean at location *x*, *s*̂ is a square root of the mean square error at location *x* and *Φ*(⋅) is the cumulative distribution function.

The first improvement target *y*_{ext} is associated with the minimum value of each individual objective function; the subscript *ext* stands for “extreme value” and *y*_{ext} is given by
$$\begin{array}{}{\displaystyle {{y}_{ext}}^{n}={{y}_{min}}^{n}\cdot (1-p)}\end{array}$$(7)

where *y*_{min}^{n} is the known minimum value of the *n*^{th} objective function and *p* is the percentage of improvement to be defined; parameter *p* is discussed later in this section. The corresponding *PoI* is:
$$\begin{array}{}{\displaystyle {{PoI}_{ext}}^{n}\left(x\right)=\phantom{\rule{thinmathspace}{0ex}}\mathit{\Phi}\left(\frac{{{\mathit{y}}_{\mathit{e}\mathit{x}\mathit{t}}}^{\mathit{n}}\mathit{-}\phantom{\rule{thinmathspace}{0ex}}{\hat{\mathit{y}}}^{\mathit{n}}\left(\mathit{x}\right)}{{\hat{\mathit{s}}}^{\mathit{n}}\left(\mathit{x}\right)}\right)}\end{array}$$(8)

where *y*̂^{n}, *s*̂^{n}, *y*_{ext}^{n} and *PoI*_{ext}^{n} are the corresponding measures of the *n*^{th} objective function.

For the first improvement target, we find *n* values of *PoI*, which equals to the number of objectives, because the *PoI*_{ext} is calculated based on the extreme value of each objective function. We consider the maximum potential improvement for all individual objectives, hence
$$\begin{array}{}{\displaystyle {PoI}_{ext}\phantom{\rule{thinmathspace}{0ex}}\left(x\right)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}maximize\phantom{\rule{thinmathspace}{0ex}}\left\{{{PoI}_{ext}}^{n}\left(x\right)\right\}}\end{array}$$(9)

The second improvement target *y*_{int}^{n} (*x*) is associated withthe reference point that is defined based on the location of *x*. The subscript *int* stands for “intermediate” and *y*_{ref} is calculated as
$$\begin{array}{}{\displaystyle {{y}_{int}}^{n}={{y}_{ref}}^{n}\cdot (1-p)}\end{array}$$(10)

where *y*_{ref} is the calculated reference point.

To obtain the reference point *y*_{ref}, the algorithm finds the Pareto front for the existing design sites using non-dominated sorting. For each closest set of Pareto points (the number of points is equal to the number of objectives) it calculates the corresponding reference point. The coordinates for the reference point of each dimension is equal to the maximum value of the coordinates for these Pareto points in the same dimension. The coordinates for the corresponding reference point in the *n*^{th} dimension *Ref* (*x*^{n}) is given by:
$$\begin{array}{}{\displaystyle {{y}_{ref}}^{n}=\phantom{\rule{thinmathspace}{0ex}}max\phantom{\rule{thinmathspace}{0ex}}\{{Y}^{n}\}}\end{array}$$(11)

where *Y*^{n} is the collection of the *n*^{th} objective values for all of the points in that Pareto set.

Taking a bi-objective problem as an example, assuming the reference point *y*_{ref} is to be determined for Pareto points *P*_{1} and *P*_{2}, the coordinates of *P*_{1} and *P*_{2} are therefore denoted by [*P*_{1}.*x*^{1}, *P*_{1}.*x*^{2}] and [*P*_{2}.*x*^{1}, *P*_{2}.*x*^{2}], respectively. Note that *x*^{n} is the *n*^{th} objective value at the location in the search space associated with *P*. The *x*^{1} and *x*^{2} coordinates (in the objective space) of the reference point are thus described as follows

$$\begin{array}{}{\displaystyle {{y}_{ref}}^{1}=max\phantom{\rule{thinmathspace}{0ex}}\left\{{P}_{1}.{x}^{1},{P}_{2}.{x}^{1}\right\}}\end{array}$$(12)

$$\begin{array}{}{\displaystyle {{y}_{ref}}^{2}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}max\phantom{\rule{thinmathspace}{0ex}}\left\{{P}_{1}.{x}^{2},{P}_{2}.{x}^{2}\right\}}\end{array}$$(13)

and the corresponding *PoI* is given as
$$\begin{array}{}{\displaystyle {{PoI}_{ref}}^{n}\left(x\right)=\phantom{\rule{thinmathspace}{0ex}}\mathit{\Phi}\left(\frac{{{\mathit{y}}_{\mathit{i}\mathit{n}\mathit{t}}}^{\mathit{n}}\left(\mathit{x}\right)\mathit{-}\phantom{\rule{thinmathspace}{0ex}}{\hat{\mathit{y}}}^{\mathit{n}}\left(\mathit{x}\right)}{{\hat{\mathit{s}}}^{\mathit{n}}\left(\mathit{x}\right)}\right)}\end{array}$$(14)

where *y*_{ref}^{n}, *y*̂^{n}, *s*̂^{n}, *PoI*_{ref}^{n} and *PoI*_{ext}^{n} are the corresponding measures of the *n*^{th} objective function.

We have therefore obtained *n* values of *PoI* for the second improvement target. However, unlike the first improvement target, the second one uses a localized target. Therefore, we consider using the minimum potential improvement for all individual objectives and hence
$$\begin{array}{}{\displaystyle {LPoI}_{ref}\left(x\right)=minimize\phantom{\rule{thinmathspace}{0ex}}\left\{{{PoI}_{ref}}^{n}\phantom{\rule{thinmathspace}{0ex}}\left(x\right)\right\}}\end{array}$$(15)

Finally, the proposed indicator *LPoI* for any given point is the maximum of these two probability of improvement measures, given by
$$\begin{array}{}{\displaystyle LPoI\left(x\right)=\phantom{\rule{thinmathspace}{0ex}}maximize\phantom{\rule{thinmathspace}{0ex}}\left\{{LPoI}_{ref},{PoI}_{ext}\right\}}\end{array}$$(16)

where *PoI*_{ext}, as described by (9), is due to the fact that the minimum of each individual objective function is always present in the Pareto front, thus the *PoI* at each location *x*, over the optimal target of that function, is always considered. This term also contributes to the diversification of the Pareto front.

Furthermore, *LPoI*_{ref} – as described by (15) – can be treated as a maximum of the minimum potential improvements to a local target. This term helps to improve the Pareto front both towards the origin and in the direction of the objective value. It contributes to the diversification of the Pareto front, while the max-min method also contributes to the uniformity of the Pareto front.

To obtain the next infill sampling point, the algorithm finds the location *x* associated with the maximum *LPoI* measure in the objective space.

The parameter *p* – as seen in (7) and (10) – is associated with the magnitude of target improvement; it controls the convergence rate of the algorithm. A smaller amount of improvement will guide the solver towards existing Pareto points, while a larger value will encourage the exploration of the design space. It is crucial to use a proper *p*, since too small a value may lead to a false Pareto front, while a large value may result in a slow convergence rate or zero probability of improvement at all unknown sites. Thus it is advisable to dynamically adjust the value while monitoring the convergence.

We provide a simple self-adjusted method for parameter *p* in this paper. First, the initial improvement target percentage *p*_{initial} is defined and then the parameter *p* is calculated as
$$\begin{array}{}{\displaystyle p\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{p}_{initial}\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}max\phantom{\rule{thinmathspace}{0ex}}\{{LPoI}_{prev}\}}\end{array}$$(17)

where *LPoI*_{prev} is a complete set of *LPoI* at previous iteration.

The next infill point is taken at the location with a maximum *LPoI*. Therefore, the solver tends to minimise the localised probability of improvement and converges towards the Pareto front. When the design space is well explored, or *p* is especially small, the solver will converge towards existing Pareto fronts; at this stage, it is common for the *LPoI* to be equal, or come close, to 1 at multiple unknown sites (extremely likely to improve over the target point). In order to obtain a uniformly distributed Pareto front, the algorithm selects candidates which have the largest Euclidean distance to existing Pareto points compared to the next infill sampling points. For this reason, the maximum value of *LPoI* can be capped between 0.9 and 1 for faster exploitation of the existing Pareto front without degrading the overall performance.

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