Bi-objective Ranking and Selection Using Stochastic Kriging

TL;DR

This work tackles the problem of identifying the true Pareto-optimal designs in bi-objective problems when objective values are observed with noise. It introduces SK-MORS, a sequential ranking-and-selection framework that uses stochastic kriging to build predictive distributions and guide resampling. Two acquisition criteria, the expected hypervolume difference (EHVD) and posterior distance (PD), are combined to balance front improvement with predictive accuracy, and two screening procedures reduce computational load. Empirical results on artificial test problems and an industrial supply-chain case show that SK-MORS outperforms static allocation (EQUAL) and the state-of-the-art MOCBA, while benefiting other methods through the use of SK information. The approach offers practical improvements for reliable multiobjective decision making under uncertainty and points to extensions to higher objective counts and parallelized resampling.

Abstract

We consider bi-objective ranking and selection problems, where the goal is to correctly identify the Pareto optimal solutions among a finite set of candidates for which the two objective outcomes have been observed with uncertainty (e.g., after running a multiobjective stochastic simulation optimization procedure). When identifying these solutions, the noise perturbing the observed performance may lead to two types of errors: solutions that are truly Pareto-optimal can be wrongly considered dominated, and solutions that are truly dominated can be wrongly considered Pareto-optimal. We propose a novel Bayesian bi-objective ranking and selection method that sequentially allocates extra samples to competitive solutions, in view of reducing the misclassification errors when identifying the solutions with the best expected performance. The approach uses stochastic kriging to build reliable predictive distributions of the objective outcomes, and exploits this information to decide how to resample. Experimental results show that the proposed method outperforms the standard allocation method, as well as a well-known the state-of-the-art algorithm. Moreover, we show that the other competing algorithms also benefit from the use of stochastic kriging information; yet, the proposed method remains superior.

Figures (14)

• Figure 1: Bi-objective (left) and tri-objective (right) Pareto fronts with different geometries.
• Figure 2: Left panel: True performance of a given set of solutions, for a bi-objective minimization problem. Center panel: Observed performance based on the sample means after $r$ replications per solution. Right panel: Observed performance after $r+1$ replications per solution.
• Figure 3: Left panel: Hypervolume (shaded area) dominated by a given non-dominated set (filled points) with respect to a reference point $\mathbf{r}$. Right panel: the EHVC (shaded area) of a set of non-dominated points with respect to a reference point $\mathbf{r}$. The filled circles denote the current observed front (set P), and the filled squares the expected new performance after new samples are allocated (set Q). Empty circles represent the dominated points.
• Figure 4: The five cases to be considered when calculating the EHVD.
• Figure 5: Left panel: Sample means (black dots), posterior means (red line) and posterior uncertainty (gray area) for a single objective. The distance between the prediction and the sample mean is denoted with a blue line. Right panel: The posterior distance between the sample means (circles) and predicted means (squares). The observed and predicted fronts are depicted with filled circles and squares respectively.

Theorems & Definitions (1)

• Definition 1