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Design optimization of engineering systems with multiple competing objectives is a painstakingly tedious process especially when the objective functions are expensiveto-evaluate computer codes with parametric uncertainties. The effectiveness of the state-of-the-art techniques is greatly diminished because they require a large number of objective evaluations, which makes them impractical for problems of the above kind. Bayesian global optimization (BGO), has managed to deal with these challenges in solving single-objective optimization problems and has recently been extended to multi-objective optimization (MOO). BGO models the objectives via probabilistic surrogates and uses the epistemic uncertainty to define an information acquisition function (IAF) that quantifies the merit of evaluating the objective at new designs. This iterative data acquisition process continues until a stopping criterion is met. The most commonly used IAF for MOO is the expected improvement over the dominated hypervolume (EIHV) which in its original form is unable to deal with parametric uncertainties or measurement noise. In this work, we provide a systematic reformulation of EIHV to deal with stochastic MOO problems. The primary contribution of this paper lies in being able to filter out the noise and reformulate the EIHV without having to observe or estimate the stochastic parameters. An addendum of the probabilistic nature of our methodology is that it enables us to characterize our confidence about the predicted Pareto front. We verify and validate the proposed methodology by applying it to synthetic test problems with known solutions. We demonstrate our approach on an industrial problem of die pass design for a steel wire drawing process.

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The steel manufacturing process is characterized by the requirement of expeditious development of high quality products at low cost through the effective use of available resources. Identifying solutions that meet the conflicting commercially imperative goals of such process chains is hard using traditional search techniques. The complexity in such a problem increases due to the presence of a large number of design variables, constraints and bounds, conflicting goals and the complex sequential relationships of the different stages of manufacturing. A classic example of such a manufacturing problem is the design of a rolling system for manufacturing a steel rod. This is a sequential process in which information flows from first rolling stage/pass to the last rolling pass and the decisions made at first pass influence the decisions that are made at the later passes. In this context, we define horizontal integration as the facilitation of information flow from one stage to another thereby establishing the integration of manufacturing stages to realize the end product. In this paper, we present an inverse design method based on well-established empirical models and response surface models developed through simulation experiments (finite-element based) along with the compromise decision support problem (cDSP) construct to support integrated information flow across different stages of a multistage hot rod rolling system. The method is goal-oriented because the design decisions are first made based on the end requirements identified for the process at the last rolling pass and these decisions are then passed to the preceding rolling passes following the sequential order in an inverse manner to design the entire rolling process chain to achieve the horizontal integration of stages. We illustrate the efficacy of the method by carrying out the design of a multistage rolling system. We formulate the cDSP for the second and fourth pass of a four pass rolling chain. The stages are designed by sequentially passing the design information obtained after exercising the cDSP for the last pass for different scenarios and identifying the best combination of design variables that satisfies the conflicting goals. The cDSP for the second pass helps in integrated information flow from fourth to first pass and in meeting specified goals imposed by the fourth and third passes. The end goals identified for this problem for the fourth pass are minimization of ovality (quality) of rod, maximization of throughput (productivity), and minimization of rolling load (performance and cost). The method can be instantiated for other multistage manufacturing processes such as the steel making process chain having several unit operations.