Robust Pareto Research Articles

A robust Pareto model for electric vehicle charging station deployment in urban areas considering psychology effects of drivers

A successful transition from gas-powered to electric vehicles (EVs) depends on identifying the most convenient locations for electric vehicle charging stations (EVCS), particularly in urban areas. While EVCS location problems have been addressed in the literature, this study considers the ambiguity of EV drivers' range anxiety and charging demand to explore the EVCS deployment in continuous and discrete solution spaces, representing roads and parking facilities in the real-world. Additionally, our paper highlights the novelty of including the negative psychology effects experienced by both electric vehicle (EV) and fuel vehicle (FV) drivers due to the ICEing problem, where fuel vehicles (FVs) block EVCS access. This paper proposes a comprehensive framework that includes a spatio-temporal Gaussian process model for predicting charging demand, a multi-objective EVCS location model for an EVCS deployment, and a Scenario-based Multi-Objective min-max Robust Pareto (SMORP) model with ambiguous charging demand and drivers' range anxiety for a robust Pareto EVCS deployment. The proposed algorithms identify the optimal and robust Pareto fronts for EVCS deployments. We validate the models using a case study of an urban area. The resulting EVCS deployment enables the selection of optimal EVCS locations among discrete parking facilities and identifies continuous coordinates for curb parking space for EV charging.

An Improved Ant Colony Algorithm with Deep Reinforcement Learning for the Robust Multiobjective AGV Routing Problem in Assembly Workshops

Vehicle routing problems (VRPs) are challenging problems. Many variants of the VRP have been proposed. However, few studies on VRP have combined robustness and just-in-time (JIT) requirements with uncertainty. To solve the problem, this paper proposes the just-in-time-based robust multiobjective vehicle routing problem with time windows (JIT-RMOVRPTW) for the assembly workshop. Based on the conflict between uncertain time and JIT requirements, a JIT strategy was proposed. To measure the robustness of the solution, a metric was designed as the objective. Afterwards, a two-stage nondominated sorting ant colony algorithm with deep reinforcement learning (NSACOWDRL) was proposed. In stage I, ACO combines with NSGA-III to obtain the Pareto frontier. Based on the model, a pheromone update strategy and a transfer probability formula were designed. DDQN was introduced as a local search algorithm which trains networks through Pareto solutions to participate in probabilistic selection and nondominated sorting. In stage II, the Pareto frontier was quantified in feasibility by Monte Carlo simulation, and tested by diversity-robust selection based on uniformly distributed weights in the solution space to select robust Pareto solutions that take diversity into account. The effectiveness of NSACOWDRL was demonstrated through comparative experiments with other algorithms on instances. The impact of JIT strategy is analyzed and the effect of networks on the NSACOWDRL is further discussed.

Bi-level game theoretic approach for robust design: A case study of path-generating four-bar

This study addresses the bi-level multi-objective optimization problems (MOP) that raise in robust design and optimization of engineering systems through establishing a state-of-the-art game theoretic scenario. A novel leader-follower decentralized decision-making scenario is proposed, leveraging the synergy of game theory, Robust Design Optimization (RDO), Monte Carlo Simulation (MCS), and Artificial Intelligence (AI). The proposed algorithm can be employed for optimum robust Pareto design of a wide range of dynamical systems. In order to achieve a robust design, both the mean and variance of each objective function are considered as players in a multi-agent game setting. In this approach, both Stackelberg and cooperative games are utilized to model the behaviors of the players. Genetic Programming (GP) meta-models are employed to capture the Stackelberg protocol between two levels specifically for constructing the follower's rational reaction set (RRS). Additionally, the Nash bargaining function is -utilize to model the cooperative behaviors among players in each level. The proposed approach is applied and demonstrated through a case study involving multi-objective robust design of planar four-bar linkages. In this manner, four objective functions are assigned to four players within the system. Each player is responsible for optimizing a specific objective criterion, namely the mean of tracking error (TE), variance of tracking error, mean of transmission angle and variance of transmission angle (TA) of the linkage. As a result, the four-objective optimization problem of mechanism is transformed into a single-objective robust synthesis problem. The comparisons of the results show a significant enhancement in the robust behavior of the linkage, while ensuring that deterministic criteria such as quality of motion and precision are preserved.

Approximating robust Pareto fronts by the MEOF-based multiobjective evolutionary algorithm with two-level surrogate models

The multiobjective optimization problems (MOPs) under uncertain environments are very challenging to be solved due to the sensitivities of some robust decision variables. To find the robust Pareto fronts (PFs) of these MOPs, the mean effective objective function (MEOF) is often used for evaluating the qualities of solutions in the existing evolutionary multiobjective optimization (EMO) algorithms. In the MEOF evaluation, the objective function values of multiple solutions in the neighborhood of a certain solution should be averaged. As a result, the MEOF-based EMO algorithms consume a large number of function evaluations to find robust PFs with high qualities. To overcome this weakness, we propose a new MEOF-based EMO framework with two-level surrogate models, denoted by EMO-MEOF/TS, which utilizes radial basis function and Gaussian process model to predict high-quality robust solutions at the levels of global search and local search. Some experiments are conducted to evaluate the performance of the proposed framework on some modified MOPs with robust decision variables. Our experimental results demonstrate that EMO-MEOF/TS is advantageous against several robust MOEAs in approximating the PFs of MOPs with robust characteristics.

Pareto Optimal Strategy Under H∞ Constraint for the Mean-Field Stochastic Systems in Infinite Horizon.

This article focuses on the mean-field linear-quadratic Pareto (MF-LQP) optimal strategy design for stochastic systems in infinite horizon, which is with the H∞ constraint when the system is disturbed by external interferences. The stochastic bounded real lemma (SBRL) with any initial state in infinite horizon is first investigated based on the stabilizing solution of the generalized algebraic Riccati equation (GARE). Then, by discussing the convexity of the cost functional, the stochastic indefinite MF-LQP control problem is defined and solved based on the MF-LQ theory and Pareto theory. When the worst case disturbance is considered in the collaborative multiplayer system, we show that the Pareto optimal strategy design with H∞ constraint [or robust Pareto optimal strategy, (RPOS)] can be given via solving two coupled GAREs. When the worst case disturbance and the Pareto efficient strategy work, all Pareto solutions are obtained by a generalized Lyapunov equation. Finally, a practical example shows that the obtained results are effective.

Multi-objective topology optimization design of truss structures based on evidence theory under limited information

Uncertainty information is often limited and has great discreteness, how to ensure the structure having good robustness under epistemic uncertainty becomes a technical problem for multi-objective topology optimization design. In this paper, the evidence theory is used to quantify the epistemic uncertainty; the plausibility measurement expressing upper limit of failure probability is applied to be a new evaluation of reliability. The total weight and the reliability of structures are taken as the optimization objectives, and a multi-objective robust topology optimization design model based on evidence theory is proposed. Parallelization technique based differential evolution for multi-objective optimization (DEMO) is preferable to search above robust Pareto front due to its merits of non-requirement of any gradient and superior mechanisms of non-dominate strategy. In order to verify the effectiveness of the proposed method, the multi-objective reliability optimization of a wood truss structure is implemented with considering elastic modulus and structural load as uncertain variables. According to the optimization results, six same truss specimens are made for the static random loading test. The failure probability of the truss structure is judged by the stress and node displacement obtained from the test, so as to verify the feasibility of the reliability optimization method based on evidence theory. The experimental results also indicate that the proposed method can avoid the deviation of the optimization results caused by the fluctuation of epistemic uncertainty, and provide a new method for designers to make the optimization results robust even when the data information is insufficient and the cognitive level is limited.

Fuzzy multi-objective truck scheduling in multi-modal rail–road Physical Internet hubs

The Physical Internet (PI) is an innovative concept that has the potential to significantly improve the efficiency, cost-effectiveness, and sustainability of the global supply chain industry, particularly in cross-docking operations. This paper addresses the truck-scheduling problem in rail–road PI-Hubs, taking into account simultaneously both uncertainty and multi-objective decision-making, which has not been fully explored in the literature, particularly for PI-structures. Our proposed approach defines a Multi-Objective Mixed-Integer Programming model (FMO-MIP) that incorporates fuzzy chance-constrained programming and ϵ-constraint to minimize both the total delay and the sum of PI-containers traveled distances, while considering the uncertainty on truck arrival times. This work takes into account the particularities of the Physical Internet and presents a novel decision-making solution to generate a robust Pareto front that aligns with decision-makers’ attitudes towards risk (optimistic/pessimistic) while balancing trade-offs between conflicting objectives.

The point-based robustness gap for uncertain multiobjective optimization

In robust single-objective optimization, the robustness gap is a measure of the distance between the robust optimal objective value and the optimal objective values of the scenarios. While robust multiobjective optimization is a growing field of study, no notion of a robustness gap has been proposed. A concept of a point-based robustness gap for uncertain multiobjective optimization problems is introduced. The gap is defined as the minimal distance between the robust Pareto set and the Pareto sets of the scenarios. It is shown that the gap is zero whenever the uncertainty is constraint-wise and objective-wise, supplementing a major result about the single-objective robustness gap. Because the distance between Pareto sets is hard to compute, lower and upper bounds on the gap are constructed for convex problems. Specific results about the zero gap and the bounds are presented for linear problems. Numerical examples are included.

Nondeterministic multiobjective optimization of 3D printed ceramic tissue scaffolds

Despite significant advances in the design optimization of bone scaffolds for enhancing their biomechanical properties, the functionality of these synthetic constructs remains suboptimal. One of the main challenges in the structural optimization of bone scaffolds is associated with the large uncertainties caused by the manufacturing process, such as variations in scaffolds’ geometric features and constitutive material properties after fabrication. Unfortunately, such non-deterministic issues have not been considered in the existing optimization frameworks, thereby limiting their reliability. To address this challenge, a novel multiobjective robust optimization approach is proposed here such that the effects of uncertainties on the optimized design can be minimized. This study first conducted computational analyses of a parameterized ceramic scaffold model to determine its effective modulus, structural strength, and permeability. Then, surrogate models were constructed to formulate explicit mathematical relationships between the geometrical parameters (design variables) and mechanical and fluidic properties. The Non-Dominated Sorting Genetic Algorithm II (NSGA-II) was adopted to generate the robust Pareto solutions for an optimal set of trade-offs between the competing objective functions while ensuring the effects of the noise parameters to be minimal. Note that the nondeterministic optimization of tissue scaffold presented here is the first of its kind in open literature, which is expected to shed some light on this significant topic of scaffold design and additive manufacturing in a more realistic way.

Supporting strategy selection in multiobjective decision problems under uncertainty and hidden requirements

Decision-makers are often faced with multi-faceted problems that require making trade-offs between multiple, conflicting objectives under various uncertainties. The task is even more difficult when considering dynamic, non-linear processes and when the decisions themselves are complex, for instance in the case of selecting trajectories for multiple decision variables. These types of problems are often solved using multiobjective optimization (MOO). A typical problem in MOO is that the number of Pareto optimal solutions can be very large, whereby the selection process of a single preferred solution is cumbersome. Moreover, preference between model-based solutions may not be determined only by their objective function values, but also in terms of how robust and implementable these solutions are. In this paper, we develop a methodological framework to support the identification of a small but diverse set of robust Pareto optimal solutions. In particular, we eliminate non-robust solutions from the Pareto front and cluster the remaining solutions based on their similarity in the decision variable space. This enables a manageable visual inspection of the remaining solutions to compare them in terms of practical implementability. We illustrate the framework and its benefits by means of an epidemic control problem that minimizes deaths and economic impacts, and a screening program for colorectal cancer that minimizes cancer prevalence and costs. These examples highlight the general applicability of the framework for disparate types of decision problems and process models.

Robust Optimization for a Bi-Objective Green Closed-Loop Supply Chain with Heterogeneous Transportation System and Presorting Consideration

In this study, we propose a robust bi-objective optimization model of the green closed-loop supply chain network considering presorting, a heterogeneous transportation system, and carbon emissions. The proposed model is an uncertain bi-objective mixed-integer linear optimization model that maximizes profit and minimizes carbon emissions by considering uncertain costs, selling price, and carbon emissions. The robust optimization approach is implemented using the combined interval and polyhedral, “Interval+ Polyhedral,” uncertainty set to develop the robust counterpart of the proposed model. Robust Pareto optimal solutions are obtained using a lexicographic weighted Tchebycheff optimization approach of the bi-objective model. Intensive computational experiments are conducted and a robust Pareto optimal front is obtained with a probability guarantee that the constraints containing uncertain parameters are not violated (constraint satisfaction).

An environment-driven hybrid evolutionary algorithm for dynamic multi-objective optimization problems

In dynamic multi-objective optimization problems, the environmental parameters may change over time, which makes the Pareto fronts shifting. To address the issue, a common idea is to track the moving Pareto front once an environmental change occurs. However, it might be hard to obtain the Pareto optimal solutions if the environment changes rapidly. Moreover, it may be costly to implement a new solution. By contrast, robust Pareto optimization over time provides a novel framework to find the robust solutions whose performance is acceptable for more than one environment, which not only saves the computational costs for tracking solutions, but also minimizes the cost for switching solutions. However, neither of the above two approaches can balance between the quality of the obtained non-dominated solutions and the computation cost. To address this issue, environment-driven hybrid dynamic multi-objective evolutionary optimization method is proposed, aiming to fully use strengths of TMO and RPOOT under various characteristics of environmental changes. Two indexes, i.e., the frequency and intensity of environmental changes, are first defined. Then, a criterion is presented based on the characteristics of dynamic environments and the switching cost of solutions, to select an appropriate optimization method in a given environment. The experimental results on a set of dynamic benchmark functions indicate that the proposed hybrid dynamic multi-objective evolutionary optimization method can choose the most rational method that meets the requirements of decision makers, and balance the convergence and robustness of the obtained non-dominated solutions.

A variable-fidelity multi-objective optimization method for aerospace structural design optimization

Variable-fidelity (VF) surrogate models have been widespreadly applied to aerospace structural design and optimization problems with multiple objectives to alleviate the optimization cost. To enhance the performance of the existing multi-objective optimization algorithms based on VF surrogate model, a variable-fidelity hypervolume expected improvement (VF-HVEI) method is proposed. Co-Kriging model is utilized to replace computational expensive objective functions in the proposed method, and it is sequentially updated with the VF-HEVI method during the optimization process. The proposed infilling criterion effectively considers the prediction uncertainty of the VF surrogate model, the contribution of sample points of different fidelity on the improvement of the current Pareto front and the computation cost of different simulation models at the same time. The test results in analytical and engineering examples indicate that the proposed method obtains more accurate and robust Pareto front under the same simulation cost.

Multi-Objective and Robust Design of a Semi-Active Suspension System

This paper presents a robust multi-objective optimal design (RMOP) of a passenger car with a semi-active suspension system. The mean-effective values of the root mean square of the passenger’s head acceleration, suspension travel, and tire deflection are considered as design objectives. The passive components of the suspension and the design details of the Linear Quadratic Regulator (LQR) algorithm are used as design parameters. During the design, global sensitivity analysis is carried out using the Fourier Amplitude Sensitivity Test (FAST) to specify the elements of the model that can highly alter the design objectives. The mass of the passenger’s head and upper body, the mass of the passenger’s lower body and cushion, passenger and cushion’s elastic properties, and the sprung mass of the vehicle are selected for the sensitivity analysis. Results show that the design criteria are very sensitive to the variations in the sprung mass of the vehicle as compared to the other parameters. As a result, the variations in this parameter and passive elements of the suspension system are considered. Constraints are applied on the objectives in compliance with the requirements of ISO 2631-1 on the design of car suspension systems. The optimization problem is solved by the NSGA-II (non-dominated sorting genetic algorithm II) and robust Pareto front and set are obtained. The Pareto set includes multiple design options from which the decision-maker can choose to implement. Responses of the passenger’s head acceleration, suspension travel, and tire deflection show that the robust multi-objective design algorithm (RMOA) is very effective and guarantees less sensitivity to the suspension passive elements.

An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design. To find robust Pareto fronts, multi-objective robust optimization (MORO) methods with inner–outer optimization structures usually have high computational complexity, which is a critical issue. Generally, in design problems, robust Pareto solutions lie somewhere closer to nominal Pareto points compared with randomly initialized points. The searching process for robust solutions could be more efficient if starting from nominal Pareto points. We propose a new method sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points where MOOPs with uncertainties are solved in two stages. The deterministic optimization problem and robustness metric optimization are solved in the first stage, where nominal Pareto solutions and the robust-most solutions are identified, respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and robust-most points. The proposed SARPF method can reduce a significant amount of computational time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values, which can reduce computational efforts further. The global solvers, NSGA-II for multi-objective cases and genetic algorithm (GA) for single-objective cases, are used in corresponding optimization processes. Three examples with the comparison with results from the previous method are presented to demonstrate the applicability and efficiency of the proposed method.

Model-based production optimization under geological and economic uncertainties using multi-objective particle swarm method

Optimization of the water-flooding process in the oilfields is inherently subject to several uncertainties arising from the imperfect reservoir subsurface model and inadequate data. On the other hand, the uncertainty of economic conditions due to oil price fluctuations puts the decision-making process at risk. It is essential to handle optimization problems under both geological and economic uncertainties. In this study, a Pareto-based Multi-Objective Particle Swarm Optimization (MOPSO) method has been utilized to maximize the short-term and long-term production goals, robust to uncertainties. Some modifications, including applying a variable in the procedure of leader determination, namely crowding distance, a corrected archive controller, and a changing boundary exploration, are performed on the MOPSO algorithm. These corrections led to a complete Pareto front with enough diversity on the investigated model, covering the entire solution space. Net Present Value (NPV) is considered the first goal that represents the long-term gains, while a highly discounted NPV (with a discount rate of 25%) has been considered short-term gains since economic uncertainty risk grows with time. The proposed optimization method has been used to optimize water flooding on the Egg benchmark model. Geological uncertainty is represented with ensembles, including 100 model realizations. The k-means clustering method is utilized to reduce the realizations to 10 to reduce the computing cost. The Pareto front is obtained from Robust Optimization (RO) by maximizing average NPV over the ensembles, as the conservative production plan. Results show that optimization over the ensemble of a reduced number of realizations by the k-means technique is consistent with all realizations’ ensembles results, comparing their cumulative density functions. Furthermore, 10 oil price functions have been considered to form the economic uncertainty space. When SNPV and LNPV are optimized, considering uncertainty in oil price scenarios, the Pareto front’s production scenarios are robust to oil price fluctuations. Using the robust Pareto front of LNPV versus SNPV in both cases, one can optimize production strategy conservatively and update it according to the current reservoir and economic conditions. This approach can help a decision-maker to handle unexpected situations in reservoir management.

A framework for enhanced decision-making in aircraft conceptual design optimisation under uncertainty

AbstractThis paper presents a framework to support decision-making in aircraft conceptual design optimisation under uncertainty. Emphasis is given to graphical visualisation methods capable of providing holistic yet intuitive relationships between design, objectives, feasibility and uncertainty spaces. Two concepts are introduced to allow interactive exploration of the effects of (1) target probability of constraint satisfaction (price of feasibility robustness) and (2) uncertainty reduction through increased state-of-knowledge (cost of uncertainty) on design and objective spaces. These processes are tailored to handle multi-objective optimisation problems and leverage visualisation techniques for dynamic inter-space mapping. An information reuse strategy is presented to enable obtaining multiple robust Pareto sets at an affordable computational cost. A case study demonstrates how the presented framework addresses some of the challenges and opportunities regarding the adoption of Uncertainty-based Multidisciplinary Design Optimisation (UMDO) in the aerospace industry, such as design margins policy, systematic and conscious definition of target robustness and uncertainty reduction experiments selection and prioritisation.

Estimation Methods for the Upper Tail of the Wealth Distribution

This paper investigates methods to estimate the upper tail of the wealth distribution. I compare data types and estimation methods using data from the Netherlands for the period 1993–2018, exploiting the unique availability of multiple types of data for this context. In addition to comparing the existing methods of OLS regression, Maximum Likelihood, and Generalized Pareto interpolation, I develop a new method to combine data from several sources. This method, called Robust Pareto Regression, combines local estimates of wealth concentration from individual data sources, and uses fixed effects methods to correct for the heterogeneity across data sources and years. Several conclusions emerge: (i) No data source on its own accurately captures the top tail, meaning that all sources need to be adjusted or combined to estimate top wealth. (ii) Combining surveys with rich lists is highly sensitive to the quality of the underlying data sources; generalized Pareto interpolation partly addresses this concern, but straight Pareto regression and Maximum Likelihood methods do not. (iii) Robust Pareto Regression is preferable to existing methods, since it more adequately adjusts for data heterogeneity, and easily shows trends over time.

Yield-aware multi-objective optimization of a MEMS accelerometer system using QMC-based methodologies

This paper proposes a novel yield-aware optimization methodology that can be used for mixed-domain synthesis of robust micro-electro-mechanical systems (MEMS). The robust Pareto front optimization of a MEMS accelerometer system, which includes a capacitive MEMS sensor and an analog read-out circuitry, is realized by co-optimization of the mixed-domain system where the sensor performances are evaluated using highly accurate analytical models and the circuit level simulations are carried out by an electrical simulator. Two different approaches for yield-aware optimization have been implemented in the synthesis loop. The Quasi Monte Carlo (QMC) technique has been used to embed the variation effects into the optimization loop. The results for both two- and three-dimensional yield-aware optimization are quite promising for robust MEMS accelerometer synthesis.

Multi-Objective Adaptive Robust Voltage/VAR Control for High-PV Penetrated Distribution Networks

In active distribution networks, high penetration of distributed photovoltaic power generation may cause voltage fluctuation and violation issues. To conquer the challenges, this paper firstly proposes a load-weighted voltage deviation index (LVDI) to quantify network voltage deviation. Secondly, this paper proposes a multi-objective adaptive voltage/VAR control (VVC) framework which coordinates multiple devices in multiple timescales to minimize voltage deviation and power loss simultaneously. Then, a multi-objective adaptive robust optimization method is proposed to obtain robust Pareto solutions under uncertainties. Accordingly, solution algorithms based on different multi-objective programming algorithms and a column-and-constraint generation algorithm are developed and systematically compared. The proposed method is verified through comprehensive tests on the IEEE 123-bus system and simulation results demonstrate high effectiveness of the LVDI, high efficiency of the solution algorithms and full operating robustness of the proposed VVC method against any uncertainty realization.