Kalyan Shankar Bhattacharjee

Bridging the Gap: Many-Objective Optimization and Informed Decision-Making

The field of many-objective optimization has grown out of infancy and a number of contemporary algorithms can deliver well converged and diverse sets of solutions close to the Pareto optimal front. Concurrently, the studies in cognitive science have highlighted the pitfalls of imprecise decision-making in presence of a large number of alternatives. Thus, for effective decision-making, it is important to devise methods to identify a handful (7 ± 2) of solutions from a potentially large set of tradeoff solutions. Existing measures such as reflex/bend angle, expected marginal utility (EMU), maximum convex bulge/distance from hyperplane, hypervolume contribution, and local curvature are inadequate for the purpose as: 1) they may not create complete ordering of the solutions; 2) they cannot deal with large number of objectives and/or solutions; and 3) they typically do not provide any insight on the nature of selected solutions (internal, peripheral, and extremal). In this letter, we introduce a scheme to identify solutions of interest based on recursive use of the EMU measure. The nature of the solutions (internal or peripheral) is then characterized using reference directions generated via systematic sampling and the top

Efficient Use of Partially Converged Simulations in Evolutionary Optimization

{K}

Decomposition Based Evolutionary Algorithm with a Dual Set of reference vectors

solutions with the largest relative EMU measure are presented to the decision maker. The performance of the approach is illustrated using a number of benchmarks and engineering problems. In our opinion, the development of such methods is necessary to bridge the gap between theoretical development and real-world adoption of many-objective optimization algorithms.

Welding Heat Transfer Analysis Using Element Free Galerkin Method

For many real-world optimization problems, evaluating a solution involves running a computationally expensive simulation model. This makes it challenging to use evolutionary algorithms that usually have to evaluate thousands of solutions before converging. On the other hand, in many cases, even a prematurely stopped run of the simulation may serve as a cheaper, albeit less accurate (low fidelity), estimate of the true fitness value. For evolutionary optimization, this opens up the opportunity to decide about the simulation run length for each individual. In this paper, we propose a mechanism that is capable of learning the appropriate simulation run length for each solution. To test our approach, we propose two new benchmark problems, one simple artificial benchmark function and one benchmark based on a computational fluid dynamics (CFDs) simulation scenario to design a toy submarine. As we demonstrate, our proposed algorithm finds good solutions much more quickly than always using the full CFDs simulation and provides much better solution quality than a strategy of progressively increasing the fidelity level over the course of optimization.

Multiple surrogate assisted multiobjective optimization using improved pre-selection

Decomposition based approaches are increasingly being used to solve many-objective optimization problems (MaOPs). In such approaches, the MaOP is decomposed into several single-objective sub-problems and solved simultaneously guided by a set of predefined, uniformly distributed reference vectors. The reference vectors are constructed by joining a set of uniformly sampled points to the ideal point. Use of such reference vectors originating from the ideal point has so far performed reasonably well on common benchmarks such as DTLZs and WFGs, since the geometry of their Pareto fronts can be easily mapped using these reference vectors. However, the approach may not deliver a set of well distributed solutions for problems with Pareto fronts which are convex/concave or where the shape of the Pareto front is not best suited for such set of reference vectors (e.g. minus series of DTLZ and WFG test problems). While the notion of reference vectors originating from the nadir point has been suggested in the literature in the past, they have rarely been used in decomposition based algorithms. Such reference vectors are complementary in nature with the ones originating from the ideal point. Therefore, in this paper, we introduce a decomposition based approach which attempts to use both these two sets of reference vectors and chooses the most appropriate set at each generation based on the s-energy metric. The performance of the approach is presented and objectively compared with a number of recent algorithms. The results clearly highlight the benefits of such an approach especially when the nature of the Pareto front is not known a priori.

Selective evaluation in multiobjective optimization: A less explored avenue

Mesh-less methods belong to a new class of numerical methods in computational mechanics and offer several advantages over the conventional mesh-based methods. They enable modelling of processes involving high deformation, severe discontinuities (e.g. fracture) and multiple physical processes. These types of situations are usually encountered in arc welding, rendering its modelling suitable via mesh-less methods. In this paper, a mesh-less Element Free Galerkin (EFG) method has been developed to model the heat transfer during welding. The results predicted by the EFG method are found to be in close agreement with those obtained by the finite element method and those observed in welding experiments. This demonstrates the effectiveness and utilities of the EFG method for modelling and understanding the heat transfer processes in arc welding.

Alternative Passenger Cars for the Australian Market: A Cost–Benefit Analysis

In multiobjective engineering design, evaluation of a single design (solution) often requires running one or more computationally expensive simulation models. Surrogate assisted optimization (SAO) approaches have long been used for solving such problems, in which approximations/surrogates are used in lieu of computationally expensive simulations during the course of search. Existing SAO approaches use a variety of surrogate models and model management strategies, and the best choice is still a matter under investigation. Our current proposal is an attempt to exploit the best features of several strategies, and in particular compares two possible versions of pre-selection in multiobjective optimization. The proposed algorithm is based on the non-dominated sorting genetic algorithm (NSGA-II) but, instead of evaluating the potential offspring solutions directly, a surrogate assisted evolutionary search is conducted in the neighborhood of every offspring solution using the best local surrogate model (among Kriging, Radial basis function (RBF), Polynomial response surface method (RSM) of order 1 and 2 and Multilayer perceptrons (MLP)). Out of the combined set of candidate solutions generated using the above step, the most promising offspring solutions are pre-selected, and we examine and compare two versions of pre-selection, one ignoring the parents and one taking the parents into account. The performance of the proposed approach is studied using a number of well known numerical benchmarks and engineering design optimization problems.

Enhanced Pareto Interpolation Method to Aid Decision Making for Discontinuous Pareto Optimal Fronts

Population based stochastic algorithms have long been used for the solution of multiobjective optimization problems. In the context of computationally expensive analysis, the existing practice utilizes some form of surrogates or approximations. In this paper, we investigate the effects of selective evaluation of promising solutions and try to derive answers to the following questions: (a) should we discard the solution based on variable values only ? (b) should we evaluate one of its objective functions and then decide to select or discard it ? or (c) should we evaluate both its objective functions before selecting or discarding it ? While evaluation of solutions is crucial for learning, it comes with a computational cost that can be significant for problems involving computationally expensive analysis. Herein, we study the effects of various selective evaluation strategies using support vector machine (SVM) classifier coupled with non-dominated sorting genetic algorithm (NSGA-II). The performance of the strategies have been evaluated using five well studied unconstrained bi-objective optimization problems (DTLZ1-DTLZ5) with limited computational budget. The results clearly indicate the benefits of using certain strategies for certain class of problems and in certain stages of the search process. Furthermore, the results also suggest that some solutions can be discarded without any evaluation, while for others after evaluation of one or both objective(s). Selective evaluation is a rarely investigated field and we hope that this study would prompt design of efficient algorithms that selectively evaluate solutions on the fly i.e. based on the trade-off between need to learn/evaluate and cost to learn.

An Evolutionary Algorithm with Classifier Guided Constraint Evaluation Strategy for Computationally Expensive Optimization Problems

Petrol or diesel powered cars (henceforth referred as conventional vehicles (CV)) have long been in existence, and currently, 75% of cars in Australia belong to this category. Hybrids (HEVs), i.e. a vehicle with an electric drive system and an internal combustion engine running on either petrol or diesel have gained significant market share in recent years. While hybrids are regularly presented as “greener alternatives”, their competitive edge is largely dependent on existing market conditions (gasoline prices, electricity prices, purchase price and maintenance costs) and the usage (commuting distances). A new breed of cars, i.e. fully electric vehicles (EVs), is becoming increasingly popular for city commuters and expected to feature prominently in “Smart Cities” of the future. This study aims to evaluate the performance of electric vehicles (EVs), hybrid electric vehicles (HEVs) and conventional vehicles (CVs) based on three major considerations: average gasoline consumption per day, average GHG emissions in kg-CO2 equivalent (kg-CO2-eq) per day and equivalent annualized cost (EAC), for a typical Australian scenario. Each of the three objectives are assessed across a range of gasoline prices, electricity tariffs and commuting distances. Four vehicles have been considered in this study (The models for analysis have been developed based on available open source data with simplifications and assumptions.): two EVs (Nissan LEAF and a BMW i3), one HEV (Toyota Prius) and one CV (Hyundai i30). For a typical city commute of 50 km/day, the average gasoline consumption varies between 0 to 2.30 litre per day, average GHG emission varies between 8.25 to 12.57 kg-CO2 equivalent per day, and equivalent annualized cost (EAC) varies between 9.90 to 38.79 AUD per day. With such a variation, the choice of one vehicle over another is largely dependent on user preferences. The chapter also presents an approach to customize EVs i.e. effectively develop EV or HEVs to be attractive to a particular market segment.

Cost to Evaluate Versus Cost to Learn? Performance of Selective Evaluation Strategies in Multiobjective Optimization

Multi-criteria decision making is of interest in several domains such as engineering, finance and logistics. It aims to address the key challenges of search for optimal solutions and decision making in the presence of multiple conflicting design objectives/criteria. The decision making aspect can be particularly challenging when there are too few Pareto optimal solutions available as this severely limits the understanding of the nature of the Pareto optimal front (POF) and subsequently affects the confidence on the choice of solutions. This problem is studied in this paper, wherein a decision maker is presented with a few outcomes and the aim is to identify regions of interest for further investigation. To address the problem, the contemporary approaches attempt to generate POF approximation through linear interpolation of a given set of (a few) Pareto optimal outcomes. While the process helps in gaining an understanding of the POF, it ignores the possibility of discontinuities or voids in the POF. In this study, we investigate two measures to alleviate this difficulty. First is to make use of infeasible solutions obtained during the search, along with the Pareto outcomes while constructing the interpolations. Second is to use proximity to a set of uniform reference directions to determine potential discontinuities. Consequently, the proposed approach enables approximation of both continuous and discontinuous POF more accurately. Additionally, a set of interpolated outcomes along uniformly distributed reference directions are presented to the decision maker. The errors in the given interpolations are also estimated in order to further aid decision making by establishing confidence on predictions. We illustrate the performance of the approach using four problems spanning different types of fronts, such as mixed (convex/concave), degenerate, and disconnected.