Robust Pareto Optimal Solutions Research Articles

A bi-objective robust model for minimization of costs and emissions of syngas supply chain

Producing syngas from forest-based biomass could facilitate the transitions in energy and forest sectors by replacing natural gas; reducing emissions and wastes; and generating revenues. Optimizing the economic and environmental impacts of the biomass supply chains can help realizing these benefits. In this study, a bi-objective robust optimization model is developed for tactical supply chain planning of forest biomass gasification at a pulp mill. It optimizes the monthly flow, inventory, and preprocessing of biomass while minimizing annual costs and emissions. Robust optimization with an adjustable risk of constraint violation is used to model the uncertainties in supply and cost of biomass. The average cost and emissions of the robust Pareto-optimal solutions are 68% and 41% higher than those in the deterministic solutions, respectively. Although, the cost and emissions and their trade-off in the deterministic case are more favorable, the robust solutions ensure no biomass shortage while avoiding over-conservatism.

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).

Numerical simulation and optimization of fine-blanking process for copper alloy sheet

When the fine-blanking process is used, secondary grinding or processing can be omitted because the shear surface of fine-blanking parts can achieve almost zero fracture zone requirements. The primary objective of the fine-blanking process is to reduce the fracture zone depth and die roll zone width. This study used a 2.5-mm-thick central processing unit (CPU) thermal heat spreader as an example. Finite element analysis software was employed to simulate and optimize the main eight process parameters that affect the fracture zone depth and die roll zone width after fine-blanking: the V-ring shape angle, V-ring height of the blank holder, V-ring height of the cavity, V-ring position, blank holder force, counter punch force, die clearance, and blanking velocity. Simulation analysis was conducted using the L18 (21 × 37) Taguchi orthogonal array experimental combination. The simulation results of the fracture zone depth and die roll zone width were optimized and analyzed as quality objectives using Taguchi’s smaller-the-better design. The analysis results revealed that with fracture zone depth as the quality objective, 0.164 mm was the optimal value, and counter punch force made the largest contribution of 25.89%. In addition, with die roll zone width as the quality objective, the optimal value was 1.274 mm, and V-ring height of the cavity made the largest contribution of 29.45%. Subsequently, this study selected fracture zone depth and die roll zone width as multicriteria quality objectives and used the robust multicriteria optimal approach and Pareto-optimal solutions to perform multicriteria optimization analysis. The results met the industry’s fraction zone depth standard (below 12% of blank thickness) and achieved a smaller die roll zone width.

Grid-based dynamic robust multi-objective brain storm optimization algorithm

Rich works have been done on brain storm optimization algorithm solving static single- or multi-objective optimization problems, but less reports for dynamic multi-objective optimization problems. Based on this, a grid-based multi-objective brain storming algorithm with hybrid mutation operation is proposed to find the robust Pareto-optimal solution set over time. Grid-based clustering method partitions the objective space evenly along each objective and classifies the individuals located in the same grid into a cluster. Its computational complexity is less than k-means- and group-based clustering strategies. Traditional Gaussian-, Cauchy- and Chaotic-based mutation operators have different mutation steps and generate the new individuals with various diversity. In order to enhance the diversity and avoiding the premature convergence, a hybrid mutation strategy integrating above three mutation operators is presented. Experimental results for eight dynamic multi-objective benchmark functions show that the proposed algorithm can find robust Pareto-optimal solutions approximating the true Pareto front under more subsequent environments with the acceptable fitness threshold. The longer survival time also indicates that grid-based clustering method and hybrid mutation strategy are apt to better robustness.

Robust multiobjective and multidisciplinary design optimization of electrical drive systems

Design and optimization of electrical drive systems often involve simultaneous consideration of multiple objectives that usually contradict to each other and multiple disciplines that normally coupled to each other. This paper aims to present efficient system-level multiobjective optimization methods for the multidisciplinary design optimization of electrical drive systems. From the perspective of quality control, deterministic and robust approaches will be investigated for the development of the optimization models for the proposed methods. Meanwhile, two approximation methods, Kriging model and Taylor expansion are employed to decrease the computation/simulation cost. To illustrate the advantages of the proposed methods, a drive system with a permanent magnet synchronous motor driven by a field oriented control system is investigated. Deterministic and robust Pareto optimal solutions are presented and compared in terms of several steady-state and dynamic performances (like average torque and speed overshoot) of the drive system. The robust multiobjective optimization method can produce optimal Pareto solutions with high manufacturing quality for the drive system.

Robust Multi-Objective Optimization for EE-SE Tradeoff in D2D Communications Underlaying Heterogeneous Networks

In this paper, we concentrate on the robust multi-objective optimization (MOO) for the tradeoff between energy efficiency (EE) and spectral efficiency (SE) in device-to-device (D2D) communications underlaying heterogeneous networks (HetNets). Different from traditional resource optimization, we focus on finding robust Pareto optimal solutions for spectrum allocation and power coordination in D2D communications underlaying HetNets with the consideration of interference channel uncertainties. The problem is formulated as an uncertain MOO problem to maximize EE and SE of cellular users (CUs) simultaneously while guaranteeing the minimum rate requirements of both CUs and D2D pairs. With the aid of ε-constraint method and strict robustness, we propose a general framework to transform the uncertain MOO problem into a deterministic single-objective optimization problem. As exponential computational complexity is required to solve this highly non-convex problem, the power coordination and the spectrum allocation problems are solved separately, and an effective two-stage iterative algorithm is developed. Finally, simulation results validate that our proposed robust scheme converges fast and significantly outperforms the non-robust scheme in terms of the effective EE-SE tradeoff and the quality of service satisfying probability of D2D pairs.

The Evolutionary Algorithm to Find Robust Pareto-Optimal Solutions over Time

In dynamic multiobjective optimization problems, the environmental parameters change over time, which makes the true pareto fronts shifted. So far, most works of research on dynamic multiobjective optimization methods have concentrated on detecting the changed environment and triggering the population based optimization methods so as to track the moving pareto fronts over time. Yet, in many real-world applications, it is not necessary to find the optimal nondominant solutions in each dynamic environment. To solve this weakness, a novel method called robust pareto-optimal solution over time is proposed. It is in fact to replace the optimal pareto front at each time-varying moment with the series of robust pareto-optimal solutions. This means that each robust solution can fit for more than one time-varying moment. Two metrics, including the average survival time and average robust generational distance, are present to measure the robustness of the robust pareto solution set. Another contribution is to construct the algorithm framework searching for robust pareto-optimal solutions over time based on the survival time. Experimental results indicate that this definition is a more practical and time-saving method of addressing dynamic multiobjective optimization problems changing over time.

Post Pareto-optimal pruning algorithm for multiple objective optimization using specific extended angle dominance

For the last two decades, significant effort has been devoted to exploring Multi-Objective Evolutionary Algorithms (MOEAs) for solving complex practical optimization problems. MOEAs approximate a representative set of Pareto-optimal solutions and present them to the decision maker (DM). Recently, studies in this area have focused on decision-making techniques in order to help the DM to arrive at a single preferred solution. This paper presents a pruning algorithm which can be applied in the post Pareto-optimal phase to select a subset of robust Pareto-optimal solutions before presenting them to the DM. Our algorithm is called Angle based with Specific bias parameter pruning Algorithm (ASA). Our pruning method begins by calculating the angle between each pair of solutions using an arctangent function. We introduce a bias intensity parameter to calculate a threshold angle in order to identify areas with desirable solutions based on the DM׳s preference. The bias parameter can be tuned specifically for each objective. We also propose a technique to determine a region of interest using reference point to MOEA/D algorithm which leads to a modified version of MOEA/D (PR-MOEA/D). The experimental results show that our pruning algorithm provides a robust subset of Pareto-optimal solutions for our benchmark problems when evaluating solutions in terms of convergence to optimality.

Novel performance metrics for robust multi-objective optimization algorithms

Performance metrics are essential for quantifying the performance of optimization algorithms in the field of evolutionary multi-objective optimization. Such metrics allow researchers to compare different algorithms quantitatively. In the field of robust multi-objective optimization, however, there is currently no performance metric despite its significant importance. This motivates our proposal of three novel specific metrics for measuring the convergence, coverage, and success rate of robust Pareto optimal solutions obtained by robust multi-objective algorithms. The proposed metrics are employed to quantitatively evaluate and compare Robust Multi-objective Particle Swarm Optimization (RMOPSO) and Robust Non-dominated Sorting Genetic Algorithm (RNSGA-II) on seven selected benchmark problems. The results show that the proposed metrics are effective in quantifying the performance of robust multi-objective algorithms in terms of convergence, coverage, and the ratio of the robust/non-robust Pareto optimal solutions obtained.

Reliability-based robust design optimization: A general methodology using genetic algorithm

In this paper, we present an improved general methodology including four stages to design robust and reliable products under uncertainties. First, as the formulation stage, we consider reliability and robustness simultaneously to propose the new formulation of reliability-based robust design optimization (RBRDO) problems. In order to generate reliable and robust Pareto-optimal solutions, the combination of genetic algorithm with reliability assessment loop based on the performance measure approach is applied as the second stage. Next, we develop two criteria to select a solution from obtained Pareto-optimal set to achieve the best possible implementation. Finally, the result verification is performed with Monte Carlo Simulations and also the quality improvement during manufacturing process is considered by identifying and controlling the critical variables. The effectiveness and applicability of this new proposed methodology is demonstrated through a case study.

Multiobjective design of water distribution systems under uncertainty

The water distribution system (WDS) design problem is defined here as a multiobjective optimization problem under uncertainty. The two objectives are (1) minimize the total WDS design cost and (2) maximize WDS robustness. The WDS robustness is defined as the probability of simultaneously satisfying minimum pressure head constraints at all nodes in the network. Decision variables are the alternative design options for each pipe in the network. The sources of uncertainty are future water consumption and pipe roughness coefficients. Uncertain variables are modeled using probability density functions (PDFs) assigned in the problem formulation phase. The corresponding PDFs of the analyzed nodal heads are calculated using the Latin hypercube sampling technique. The optimal design problem is solved using the newly developed RNSGAII method based on the nondominated sorting genetic algorithm II (NSGAII). In RNSGAII a small number of samples are used for each fitness evaluation, leading to significant computational savings when compared to the full sampling approach. Chromosome fitness is defined here in the same way as in the NSGAII optimization methodology. The new methodology is tested on several cases, all based on the New York tunnels reinforcement problem. The results obtained demonstrate that the new methodology is capable of identifying robust Pareto optimal solutions despite significantly reduced computational effort.