Multi-objective Robust Optimization Research Articles

Robust Multi-Objective Supply Chain Optimization of Surgical Supplies Considering Costs and Satisfaction of Surgeon, and ranking Suppliers Using ARAS Method

Robust Multi-Objective Supply Chain Optimization of Surgical Supplies Considering Costs and Satisfaction of Surgeon, and ranking Suppliers Using ARAS Method

Necessary Optimality Conditions for Robust Nonsmooth Multiobjective Optimization Problems

Abstract This paper deals with a robust multiobjective optimization problem involving nonsmooth/nonconvex real-valued functions. Under an appropriate constraint qualification, we establish necessary optimality conditions for weakly robust efficient solutions of the considered problem. These optimality conditions are presented in terms of Karush-Kuhn-Tucker multipliers and convexificators of the related functions. Examples illustrating our findings are also given.

The structure design of integrated urban drainage systems: A view of robust optimization

Rapid urbanization puts a lot of pressure on urban water pollution from point and non-point sources, calling for the practical, specific, and integrated management of urban drainage systems (UDS). The structural design of an integrated UDS is essential for highly complex and uncertain urban water management. In this study, we developed a multi-objective robust optimization model to explore the optimal structures of UDS considering system uncertainty. We applied this model to City B, northern China, to illustrate its effectiveness. The results show that the model can produce optimal designs with a more robust performance in terms of structural uncertainty. When the uncertainty degree ranges from 5% to 20%, a considerable extra cost (increased by 1.10–2.68 times) is required to improve the robustness of UDS. With the increase in structural uncertainty, the fraction of the cost invested in the stormwater subsystem increased from 10.2% to 27.2%. The findings showed that stormwater management is efficient in coping with system uncertainty. The research results promote an understanding of robust urban drainage systems.

Multi-Objective Robust Optimization for the Sustainable Location-Inventory-Routing Problem of Auto Parts Supply Logistics

A great loss of transportation capacity has been caused in auto parts supply logistics due to the independent transportation from auto parts suppliers (APSs) to the automobile production line (APL). It is believed that establishing distribution centers (DCs) for centralized collection and unified distribution is one effective way to address this problem. This paper proposes a unified framework simultaneously considering the location-inventory-routing problem (LIRP) in auto parts supply logistics. Integrating the idea of sustainable development, a multi-objective MIP model is developed to determine the location and inventory capacity of DCs and routing decisions to minimize the total system cost and carbon emissions while concerning multi-period production demand. In addition, a robust optimization model is developed further in the context of uncertain demand. Numerical experiments and sensitivity analyses are conducted to verify the effectiveness of our proposed deterministic and robust models. The results show that synergistically optimizing the location and capacity of DCs and routing decisions are beneficial in reducing total system cost and carbon emissions. The analysis can provide guidelines to decision-makers for the effective management of auto parts supply logistics.

Exploring trade-offs in agro-ecological landscapes: Using a multi-objective land-use allocation model to support agroforestry research

Finding the optimal land allocation for providing ecosystem services, conserving biodiversity and maintaining rural livelihoods is a key challenge of agricultural management and land-use planning. Agroforestry has been widely discussed as a sustainable land-use solution and as one strategy to improve the provision of multiple ecological and economic functions in agricultural landscapes. In this study, we use the backdrop of agroforestry research to evaluate a method from the multi-criteria decision analysis toolbox: robust multi-objective optimization. The key feature of this modelling approach is its capacity to integrate uncertain ecological and socio-economic data. We illustrate the optimization model with a case study from eastern Panama, showing how the model can bring together scientific and practical knowledge to provide potentially desirable landscape compositions from the perspective of farmers, a public perspective, and a compromise solution. Example results of our case study show how to assess whether agroforestry is a desirable component in a landscape composition to satisfy multiple objectives of different interest groups. Furthermore, we use the model to demonstrate how different objectives influence the optimal area share and type of agroforestry. Due to its parsimonious nature, the model could be used as a starting point of an interactive co-learning process with decision-makers, researchers and other stakeholders. The model, however, is not yet suitable for an exact prediction of future land-use dynamics, for questions of spatially explicit land-use configuration, studies going beyond the regional scale or for socio-economic interactions of agents. Therefore, we outline future research needs and recommendations for other types of models or hybrid approaches.

Approximate solutions for robust multiobjective optimization programming in Asplund spaces

In this paper, we study a nonsmooth/nonconvex multiobjective optimization problem with uncertain constraints in arbitrary Asplund spaces. We first provide necessary optimality condition in a fuzzy form for approximate weakly robust efficient solutions and then establish necessary optimality theorem for approximate weakly robust quasi-efficient solutions of the problem in the sense of the limiting subdifferential by exploiting a fuzzy optimality condition in terms of the Fréchet subdifferential. Sufficient conditions for approximate (weakly) robust quasi-efficient solutions to such a problem are also driven under the new concept of generalized pseudo convex functions. Finally, we address an approximate Mond-Weir-type dual robust problem to the reference problem and explore weak, strong, and converse duality properties under assumptions of pseudo convexity.

Robust optimization of sustainable food supply chain network considering food waste valorization and supply uncertainty

Increasing pressure from customer expectations to government regulation for sustainable food products warrants research in the sustainable food supply chain (FSC). One-third of the food produced is wasted, despite the potential to valorize food wastes. Additionally, the uncertainty associated with the supply of food products has a significant impact on the FSC decisions, steering the need for robust and sustainable models. Lastly, the issue of food perishability is missing in most optimization models. In line with the gaps identified, this study presents an integrated robust multi-objective optimization model for designing the FSC network considering all three dimensions of sustainability (economic, social and environment), whilst including FSC perishability, food waste valorization and supply uncertainty simultaneously. The proposed model is applied to a real-time case of the Indian mango pulp supply chain and yields several insights for transforming the FSC towards sustainability. This study shows the impact of focusing only on one dimension of sustainability on the other dimensions and the trade-offs among sustainability dimensions which would help the decision-maker select an appropriate non-dominated solution that aligns with the organization's sustainability goals.

Robust Multi-Objective Design Optimization of Water Distribution System under Uncertainty

The multi-objective design optimization of water distribution systems (WDS) is to find the Pareto front of optimal designs of WDS for two or more conflicting design objectives. The most popular conflicting objectives considered for the design of WDS are minimization of cost and maximization of resilience index which are considered for the current study. Robust multi-objective optimization is to find the optimal set of the Pareto front considering demand is uncertain. The robustness is controlled by a single parameter that defines the size of the uncertainty set it can vary. The study explores ellipsoidal uncertainty set with different sizes and co-variance matrices. A combined simulation–optimization framework with a combination of self-adaptive multi-objective cuckoo search (SAMOCSA) and the fmincon optimization algorithm is proposed to solve the robust multi-objective design problem. The proposed algorithm is applied to medium and large WDS. The main contribution of this paper is to study the effect of demand uncertainty and the correlation on the WDS designs in a multi-objective framework. The study shows that the inclusion of correlation into the multi-objective design framework can significantly affect the optimal designs.

A novel machine learning-based multiobjective robust optimisation strategy for quality improvement of multivariate manufacturing processes

The primary objective of this study was to develop a novel data-driven machine learning-based multiobjective robust optimisation strategy to improve the overall quality of multivariate manufacturing processes. The new strategy was conceptualised considering a manufacturing environment with unreplicated non-normal data observations and limited opportunity for off-line sequential design of experiments. At a macro level, the new strategy adopts suitable artificial intelligence-based process models and a fine-tuned non-dominated sorting genetic algorithm-II (NSGA-II) to derive robust efficient process setting conditions. These robust solutions are iteratively derived considering process model predictive uncertainties, process setting sensitivities, and variance-covariance structure of uncontrollable multivariate non-normal inputs (or covariates). These solutions are also ranked based on multicriteria decision-making (MCDM) techniques to facilitate implementation. In this study, the quality of the best-ranked solutions was compared (w.r.t. closeness to specified multiple targets and predicted multivariate output variabilities) with those of the solutions obtained from parametric and commercial software-based approaches using three different real-life manufacturing cases.

Equitable post-disaster relief distribution: a robust multi-objective multi-stage optimization approach

PurposeThis paper aims to address a location-distribution-routing problem for distributing relief commodities during a disaster under uncertainty by creating a multi-stage model that can consider information updates during the disaster. This model aims to create a relief network that chooses distribution centers with the highest value while maximizing equity and minimizing response time.Design/methodology/approachA hybrid algorithm of adaptive large neighborhood search (ALNS) and multi-dimensional local search (MDLS) is introduced to solve the problem. Its results are compared to ALNS and an augmented epsilon constraint (AUGMECON) method.FindingsThe results show that the hybrid algorithm can obtain high-quality solutions within reasonable computation time compared to the exact solution. However, while it yields better solutions compared to ALNS, the solution is obtained in a little longer amount of time.Research limitations/implicationsIn this paper, the uncertain nature of some key features of the relief operations problem is not discussed. Moreover, some assumptions assumed to simplify the proposed model should be verified in future studies.Practical implicationsIn order to verify the effectiveness of the designed model, a case study of the Sarpol Zahab earthquake in 2017 is illustrated and based on the results and the sensitivity analyses, some managerial insights are listed to help disaster managers make better decisions during disasters.Originality/valueA novel robust multi-stage linear programming model is designed to address the location-distribution-routing problem during a disaster and to solve this model an efficient hybrid meta-heuristic model is developed.

Robust optimization of a bi-objective tactical resource allocation problem with uncertain qualification costs

In the presence of uncertainties in the parameters of a mathematical model, optimal solutions using nominal or expected parameter values can be misleading. In practice, robust solutions to an optimization problem are desired. Although robustness is a key research topic within single-objective optimization, little attention is received within multi-objective optimization, i.e. robust multi-objective optimization.This work builds on recent work within robust multi-objective optimization and presents a new robust efficiency concept for bi-objective optimization problems with one uncertain objective. Our proposed concept and algorithmic contribution are tested on a real-world multi-item capacitated resource planning problem, appearing at a large aerospace company manufacturing high precision engine parts. Our algorithm finds all the robust efficient solutions required by the decision-makers in significantly less time than the approach of Kuhn et al. (Eur J Oper Res 252(2):418–431, 2016) on 28 of the 30 industrial instances.

An interval-based multi-objective robust design optimization for vehicle dynamics

This study presents an Interval-based Multi-objective Robust Design Optimization (IB-MORDO) algorithm applied to a vehicle dynamic problem. The proposed algorithm optimizes a full 15 degrees-of-freedom (15-DOF) vehicle model, subjected to a double-lane change (DLC) maneuver under random road profiles, to attain driver comfort and safety. This study does not make assumptions about uncertain parameter statistics; instead, the uncertainties are quantified using a non-probabilistic α-cut level interval analysis. These uncertainties are applied to the system parameters and design variables to ensure robust results. After the optimization process, the root mean square (RMS) vertical acceleration at the driver’s seat resulted in a robust solution of 1.041 m/s2 and a parameter interval radius (IR) equals to 0.631 m/s2, whereas the RMS lateral acceleration at the driver’s seat resulted in a solution of 1.908 m/s2 with an interval radius of 0.168 m/s2. Unlike the Robust Optimization, the algorithm proposed herein considers uncertainties at system parameters and design variables without assuming any statistical data. HIGHLIGHTS An Interval-based Robust Multi-objective Optimization procedure is proposed and tested on a 15-DOF vehicle model. Αn α-cut level methodology is used to deal with the uncertainty propagation. Resulted optimal suspension parameters minimize center and interval radius of driver’s vertical and lateral accelerations.

Robust multi-objective optimization of gear microgeometry design

Gear microgeometry is important as it can change the tooth flank contact pattern effectively. In gear design, it is important to minimize the peak-peak transmission error (PPTE) for lower Noise, Vibration and Harshness (NVH), and to minimize the generated stresses for higher durability that can help to avoid tooth failure. In the current article, a structured approach is developed that combines the advantages of applying meta-models and robust optimization. For the Design of Experiments (DOE), the LDP software tool was used to obtain the objective functions of PPTE, contact stress, and root stresses at all design points. Probability distribution and worst-case scenario measures were applied to weigh the objective functions and make them robust against torque. The generated data were used in MATLAB to build meta-models for each objective function using squared exponential Gaussian regression. The generated meta-models were then used in a multi-objective optimization algorithm to obtain a Pareto set of solutions. These solutions were examined and ranked from the highest to the lowest, based on the weights of the PPTE and stress safety factors. Using Rank 1 design, the gains of -25, -7, +3 and -5% for the PPTE, contact stress, gear 1 root stress and gear 2 root stress respectively were obtained. However, these gains can be different in other design cases because these gains were calculated using Rank 1 design as compared to the manually selected benchmarked design which mainly depends on the engineer's experience. The developed approach can save time and help to obtain a unique optimal design solution in a structured format.

Robust Multi-Objective Optimization of a 3-Pole Active Magnetic Bearing Based on Combined Curves With Climbing Algorithm

With the extensive application of active magnetic bearings (AMBs), the robust multiobjective optimization of the structure seems to be a priority. However, it is a challenge due to the high dimension and huge computational cost of finite-element analysis. In this article, a robust multi-objective optimization method is proposed to pursue good performance for a three-pole AMB. To increase the efficiency of the optimization process, the Kendall correlation coefficient is applied to assist in determining the sensitivity. The parameters are divided into three layers, and a three-level multiobjective optimization structure is established. Meanwhile, Kriging model is employed to improve the optimization efficiency. The selection of the final solution in Pareto curves is always an issue. The proposed optimization structure can only ensure the performance of the AMB, rather than robustness. Thus, a robust solution selection method is proposed based on the climbing algorithm. The robustness can be easily shown in the Pareto curve obtained through the optimization structure. The final solution is selected with good robustness in terms of suspension force and force ripple. The experimental results based on a prototype are provided to verify the effectiveness of the proposed optimization method.

Crashworthiness analysis and multiobjective robust optimization of two-stage variable thickness expansion tube under impact loading

Crashworthiness analysis and multiobjective robust optimization of two-stage variable thickness expansion tube under impact loading

The influence of discounting ecosystem services in robust multi-objective optimization – An application to a forestry-avocado land-use portfolio

Discounting is standard in economics to consider time preferences of people and account for future market changes. However, so far discounting has mainly been applied to monetary flows and ignored for many ecosystem services. In multi-objective optimization, selectively disregarding time preference for some non-monetary services create bias. Here we study how discounting a range of ecosystem service indicators influences a public planner's optimal land allocation. We used a robust multi-objective optimization approach to model a mixed forestry-avocado farm portfolio in South Africa. The objectives for optimization were the provisioning of various ecosystem services and disservices represented by four indicators: net present value, payback period, carbon sequestration, and fertilizer use. To account for time preferences concerning indicator flows, we applied specific discount rates to each ecosystem service indicator, depending on its character (non-monetary or monetary indicators). We demonstrate that discounting reduces the standard deviations of the discounted sum of the indicators, which leads to less diversified land-use portfolios. To account for discount rate uncertainty, we introduced three indicator sets simultaneously, each using a different discount rate, which was off setting the effect of decreasing diversification.

Reliability-Based Robust Multi-Objective Optimization (RBRMOO) of Chemical Process Systems: A Case Study of TEG Dehydration Plant

Natural gas (NG) production has significantly increased in the past decade as new unconventional oil and gas wells are being discovered. NG as obtained from the wellhead requires processing before being considered as pipeline grade. The process consists of the removal of acidic gases followed by dehydration. NG processing is associated with toxic emission having substantial environmental and health impact. Difficulty in NG processing arises from varied flow rate and uncertain feed composition that provides a challenge in efficient design as well as finding the optimal operating condition. The present work used a stochastic approach to characterize natural gas composition and its importance on the product and waste emission is studied. Under the uncertain feed composition, optimal operating condition of the controllable variables was attained by a reliability-based robust multi-objective optimization (RBRMOO) technique that mitigates BTEX emission while fulfilling NG pipeline specification. Chemical process simulator is used to find the impact of the control process settings and variation of uncertain feed condition on NG dehydration and BTEX emission. The best prediction models were developed using machine learning algorithm, chosen from a family of metamodels. RBRMOO is performed using metaheuristic algorithm to determine the optimal process condition of the control variables. The impact of uncertain feed composition in process modeling and subsequent optimization demonstrates optimal process condition where the rate of emission is lower by ~83 ton/yr when compared to that from the deterministic model where median value of uncertain feed composition is used for analysis, portraying the limitations of traditional sustainability assessment methods that do not account for uncertainty.

Scenario‐Based Multiobjective Robust Optimization and Decision‐Making Framework for Optimal Operation of a Cascade Hydropower System Under Multiple Uncertainties

AbstractForecast errors of multiple information sources of a cascade hydropower system cause risks of water and energy supply in real‐time operation. Mechanisms minimizing multiple risks with budgeted cost under oscillations of multiple forecast uncertainties through robust operation are not yet well‐investigated. This study proposed a multiobjective robust optimization (RO) and decision‐making framework comprising series of models for risk analysis, robust control, and decision making. The risk analysis model identifies and analyzes dependent risks that stem from forecast errors of supply and demand‐side information by Copula functions. A RO model was established for minimizing risk probabilities, vulnerabilities of ecological protection, water and energy supply, as well as revenue loss from energy. Thereafter, a multiattribute decision‐making model was incorporated for determining the compromise solution from competing non‐dominated solutions. The proposed framework was applied to the cascade hydropower system of the Xiluodu, Xiangjiaba, Three Gorges project and Gezhouba reservoirs on the Yangtze River, China. The principal findings were as follows. (a) Addressing the singular forecast uncertainty of streamflow is inadequate to obtain robust solutions. (b) In a normal year, RO can reduce the average conditional value‐at‐risk of ecological water shortfall, consumptive water shortfall, and energy shortfall by 25.1%, 35.3%, and 16.5% than that of chance‐constrained programming. (c) Multiple risks are dispersed through risk avoidance and hedging that reduces outflow variance and homogenizes mean outflow, lowering down efficiencies of energy production and water usage as a tradeoff. The proposed model framework could be applied to inform operation of cascade hydropower systems.

On approximate optimality conditions for robust multi-objective convex optimization problems

In this paper, we are interested in the study of approximate optimality conditions for weakly ϵ-efficient solutions to robust multi-objective optimization problems ((RMOP) for short) in view of its associated minimax optimization problem (MMOP). To this end, we first establish the relationship between a weakly ϵ-efficient solution to the problem (RMOP) and an α-solution to the problem (MMOP), where and . Then, we explore the representation of the so-called β-normal set (where is a given parameter) to a closed convex set at some reference point by two methods. At last, by employing the α-subdifferential of the max-function and the obtained representation of the β-normal set, we establish an approximate necessary optimality condition for the problem (RMOP). Moreover, we also give an example to illustrate our results.

Robust optimal attitude determination using interval analysis

The attitude determination of the satellite leads to solving the Wahba problem. The Wahba problem utilizes a set of at least two independent sensor measurements and reference vectors for attitude determination. These input vectors are not accurate due to sensor noises, misalignment, and low-order approximations. But these uncertainties do not consider in the classic Wahba problem directly. This paper presents a robust multi-objective optimization method for solving the static attitude determination with bounded uncertainty. The bounded uncertainty due to modeling error, measurement noise, and biases have modeled using interval analysis. Then, the Wahba problem is reformulated to a robust multi-objective optimization problem that consists of two performance indexes. The effectiveness of the proposed approach is evaluating by solving several examples.