Obtained Pareto Front Research Articles

The multi-depot vehicle routing problem with profit fairness

One of the main obstacles in horizontal collaborative transportation is the proper distribution of gains among partners. In practice, a posteriori gain sharing mechanisms rarely guarantee that all partners feel treated fairly. In this paper, we present the multi-depot vehicle routing problem with profit fairness (MDVRP-PF), a bi-objective optimization problem that adds a fairness objective function to the classical cost minimization function. By studying the MDVRP-PF, we explore the effects of integrating fairness in the optimization process. To that end, we approximate the Pareto front of any problem instance using an adaptive large neighbourhood search algorithm, embedded within an ϵ-constraint scheme. This problem is solved for different instance types and time horizons. All the obtained Pareto fronts are characterized by six measures. Results show that the economic cost of fairness remains below 5%, even though remarkable differences exist among different instance types. We also observe that such costs tend to decrease when fairness is considered for longer time horizons.

Population-Aware Sequential Flight Path Optimization for Low-Noise and Low-Fuel Consumption Departure Trajectory

With the increase in air traffic over the past decades, the reduction of aircraft noise is one of the major challenges facing stakeholders. Flight operating conditions that decrease noise may possibly increase the fuel consumption of aircraft, which is an important factor in airline cost management. In this paper, we propose a methodology to support flight path planning with the aim of optimizing both perceived noise and fuel consumption. We decompose an aircraft flight trajectory into surface and altitude paths to model relevant air transportation constraints. The shortest path for the surface projection is found using Dubins path method and an improved algorithm, which considers guide points according to the flight destination, runway angles, spatial separation of aircraft near the airport, population distribution, and steering motion. The altitude path is optimized for low-perceived noise and low-fuel consumption, which is determined by solving the longitudinal governing equations of motion of flight using the distance computed from this surface path. A modified nondominated sorting genetic algorithm II for discrete optimization is developed to obtain Pareto fronts of the optimum altitude paths with reduced computational effort. The methodology is demonstrated by simulating flights departing from Hong Kong International Airport to two compulsory air-traffic-service reporting points. The results are then compared with Quick Access Recorder data and Standard Instrument Departure (SID) tracks. Although certain factors in air transportation that affect departure path planning, such as weather patterns and air traffic mix, are not considered in this method, the resulting surface path exhibits a close similarity with SID tracks. The resulting Pareto fronts of the altitude path exhibit reductions in fuel consumption and perceived noise levels. The tradeoffs between fuel consumption and perceived noise levels are also discussed based on the relevant flight physics for the different routes.

Reprint of: The multi-depot vehicle routing problem with profit fairness

One of the main obstacles in horizontal collaborative transportation is the proper distribution of gains among partners. In practice, a posteriori gain sharing mechanisms rarely guarantee that all partners feel treated fairly. In this paper, we present the multi-depot vehicle routing problem with profit fairness (MDVRP-PF), a bi-objective optimization problem that adds a fairness objective function to the classical cost minimization function. By studying the MDVRP-PF, we explore the effects of integrating fairness in the optimization process. To that end, we approximate the Pareto front of any problem instance using an adaptive large neighbourhood search algorithm, embedded within an ε-constraint scheme. This problem is solved for different instance types and time horizons. All the obtained Pareto fronts are characterized by six measures. Results show that the economic cost of fairness remains below 5%, even though remarkable differences exist among different instance types. We also observe that such costs tend to decrease when fairness is considered for longer time horizons.

An intelligent design for Ni-based superalloy based on machine learning and multi-objective optimization

The simultaneous consideration of multiple conflicting properties in alloy optimization design is necessary yet remains challenging. A comprehensive strategy of machine learning (ML) and multi-objective optimization based on thermodynamic simulation data was proposed to accelerate the composition design of Ni-based superalloys. The microscopic parameters were determined by Pearson correlation analysis and domain knowledge as the key affect factors of tensile strength and elongation. The Multi-objectives Evolutionary Algorithm (MOEA) was adopted to search the well-built surrogate by ML meta-heuristically for the Pareto front of three objectives and its responding Pareto optimal solution set of composition. Furthermore, nine high-performance superalloy samples selected from the obtained Pareto front were well verified by fabricating and testing in the laboratory. Specially, a new composition among the nine as-fabricated samples was the best one according to the pre-defined design preferences with the γ′ solvus temperature, γ′ volume fraction, and TCP phases content approximating to 1210 °C, 65%, and 0.01%, respectively. This intelligent cooperation strategy based on ML and MOEA extends the methodology for multi-composition and multi-property design materials, which can optimize multiple conflicting objectives simultaneously rather than do one by one.

Modeling the Dynamic Exclusive Pedestrian Phase Based on Transportation Equity and Cost Analysis.

The exclusive pedestrian phase (EPP) has proven to be an effective method of eliminating pedestrian–vehicle conflicts at signalized intersections. The existing EPP setting conditions take traffic efficiency and safety as optimization goals, which may contribute to unfair interactions between vehicles and pedestrians. This study develops a multiobjective optimization framework to determine the EPP setting criteria, with consideration for the tradeoff between transportation equity and cost. In transportation equity modeling and considering environmental conditions, the transportation equity index is proposed to quantify pedestrian–vehicle equity differences. In cost modeling, traffic safety and efficiency factors are converted into monetary values, and the pedestrian–vehicle interaction is introduced. To validate the proposed optimization framework, a video-based data collection is conducted on wet and dry environment conditions at the selected intersection. The parameters in the proposed model are calibrated based on the results of the video analysis. This study compares the performance of the multiobjective evolutionary algorithm based on decomposition (MOEA) and the nondominated sorting genetic algorithm II (NSGA-II) methods in building the sets of nondominated solutions. The optimization results show that the decrease in transportation equity will lead to an increase in cost. The obtained Pareto front approximations correspond to diverse signal timing patterns and achieve a balance between optimizing either objective to different extents. The sensitivity analysis reveals the application domains for the EPP and the traditional two-way control phase (TWC) under different vehicular/pedestrian demand, yielding rate, and environment conditions. The EPP control is more suitable at intersections with high pedestrian volumes and low yielding rates, especially in wet conditions. The results provide operational guidelines for decision-makers for properly selecting the pedestrian phase pattern at signalized intersections.

Learn to decompose multiobjective optimization models for large‐scale networks

AbstractInfrastructures can be modeled as large‐scale networks consisting of nodes and arcs, making network optimization a popular modeling option for arising problems. In specific, providing timely restoration plans for interdependent infrastructures facing disruptions has been a challenge for decision makers. In this study, we focus on geospatial (co‐location) and functional interdependencies to capture the impact of cascading failures on infrastructure systems. The dynamics of real networks are more complicated to be captured by one objective function. Therefore, we define three objective functions in three pillars of sustainability: (a) economic, (b) social, and (c) environmental. To solve the multiobjective optimization model, we develop a learn‐to‐decompose framework, consisting of a multiobjective evolutionary algorithm based on decomposition module and a Gaussian process regression (GPR) module to periodically learn from the obtained Pareto front and guide the search direction. We also included a heuristic module to address two significant challenges in restoring interdependent infrastructures: the island scenario and co‐location interdependencies. We applied the proposed framework to benchmark problems and interdependent water and transportation networks in the City of Tampa, FL. We carried out sensitivity analyses to monitor the performance of the GPR by different kernel functions. We also provided insights for decision makers by finding the trade‐off between fortification (proactive) and restoration (reactive) costs. The result demonstrates the proposed framework is feasible and applicable for large‐scale networks.

An application of adaptive normalization evolutionary optimization ANMOGA for missile fin design based on trajectory parameters

System design is a complicated and iterative process. Accordingly, a robust optimization procedure should be involved during the design process to minimize complications and time consumed for the whole process. The present research proposes a new methodology to design missile fins that can lessen the gap between the conceptual and final phases during the missile design process. The proposed methodology is based on a new dynamic adaptive multi-objective optimization algorithm accompanied by a selection scheme to choose a single optimum design amongst a Pareto set of solutions. The adaptive normalization multi-objective evolutionary algorithm can be used in any system design. This algorithm is capable of predicting Pareto fronts for multi-objective optimization problems at very good dispersion. In addition, a selection scheme is provided to help decision-makers choose a single optimum amongst Pareto front points. The proposed algorithm is tested against several multi-objective test functions as well as the application of a selection scheme to the obtained Pareto fronts. The results showed the robustness of the proposed algorithm in the prediction of convex fronts. However, it still needs more improvement to deal with non-convex ones. An application for the considered algorithm is also provided as a case study to design tail fins for a conventional missile configuration. The output configuration of the designed fins has very reasonable dimensions when compared to similar configurations.

A bi-objective humanitarian logistics model considering equity in the affected zones: application to a recent earthquake in Mexico

In this paper, we propose a bi-objective program to model a post-disaster strategical decision problem. We consider the situation after a catastrophic disaster occurred, in which temporary distribution centers must be located. The distribution centers consolidate aid to be delivered to affected people. We assume that affected people go to collect needed aid from temporary located distribution centers. Hence, a predefined mobility radius is considered, that indicates the distance that people are willing to travel to receive aid. Additionally, needed aid required by affected individuals is consolidated in an affected demand zone and equity constraints are included to balance the aid given to those affected zones. One objective of the proposed model is to minimize the time employed by demand zones to collect aid. In humanitarian logistics it is common that the decision maker is associated with either government or non-profit organizations that are in charge of relief. Usually, there is a limited budget to conduct the operations. Hence, the decision maker also aims to minimize the cost of locating temporary distribution centers. Both objectives are simultaneously considered. Hence, to obtain efficient solutions of this bi-objective problem, an exact AUGMECON method is proposed, which is an improved version of the classic ε-constraint method for multi-objective optimization. To overcome with the computational limitations shown by the exact method, a genetic algorithm is also designed and used to approximate the Pareto front. To conduct the computational experience, a case study and additional random instances are considered. The case study is based on an earthquake that recently occurred in Mexico. The results obtained by both implemented methods are compared by using different well-known metrics, such as, the number of solutions, the k-distance, the size of the space covered, and a coverage measure. It is shown that, on average, the proposed genetic algorithm outperforms the AUGMECON when comparing the quality of the obtained Pareto fronts. Results offer the possibility for the decision maker to prioritize either time or cost when locating temporary distribution centers in a catastrophic situation.

Multi-objective optimisation of split-plot designs

Modern experiments allow scientists to tackle scientific problems of increasing complexity. Often experiments are characterised by factors that have levels which are harder to set than others. A possible solution is the use of a split-plot design. Many solutions are available in the literature to find optimal designs that focus solely on optimising a single criterion. Multi-criteria approaches have been developed to overcome the limitations of the one-objective optimisation, however they mainly focus on estimating the precision of the fixed factor effects, ignoring the variance component estimation. The Multi-Stratum Two-Phase Local Search (MS-TPLS) algorithm for multi-objective optimisation of designs of experiments is extended, in order to ensure pure-error estimation of the variance components. The proposed solution is applied to two motivating problems and the final optimal Pareto front and related designs are compared with other designs from the relevant literature. Experimental results show that the designs from the obtained Pareto front represent good candidate solutions based on the different objectives.

Deep learning based optimization under uncertainty for surfactant-enhanced DNAPL remediation in highly heterogeneous aquifers

The optimization for surfactant-enhanced aquifer remediation (SEAR) of dense non-aqueous phase liquid (DNAPL)-contaminated aquifers is usually accompanied by uncertainties, which may arise from the characterization of complex aquifer heterogeneity and DNAPL source zone architecture (SZA) due to measurement sparsity. Optimization under uncertainty is computationally expensive as it involves an enormous number of model runs. The huge computational burden can be alleviated by utilizing a surrogate model for repeated model evaluations. However, most of the developed surrogates are often limited to low-dimensional optimization problems that only consider simplified aquifer heterogeneity. In this study, we developed a multi-objective simulation-optimization framework to optimize the SEAR schemes considering the characterization uncertainties from both highly heterogeneous aquifer permeability and complex SZA. A fast-to-run convolutional neural network (CNN)-based surrogate model was developed to approximate the high-dimensional and highly complex input-output mapping of the DNAPL multiphase flow simulation model. We first used the rejection sampling strategy to generate random realizations of permeability and SZA conditioning on their limited measurements and then formulated a multi-objective optimization under uncertainty problem based on these realizations. The developed 3-D CNN was trained and used as the surrogate for repeated model runs in optimization to identify the optimal SEAR schemes under uncertainty. A 3-D numerical experiment was used to test the performance of the CNN-based simulation-optimization framework. Comprehensive analysis on the obtained Pareto fronts demonstrates that the proposed framework can efficiently identify reliable Pareto-optimal solutions with a 99.8% speedup compared to the traditional optimization coupled with the forward model. Moreover, the optimization considering multiple realizations enables us to perform the risk assessment to locate the risk zone where the NAPL phase possibly exists after remediation, which provides useful information for decision-making.

Multi-objective optimization of a CSP-based dish Stirling field layout using Genetic Algorithm and TOPSIS method: Case studies in Ouarzazate and Madrid

CSP systems have experienced significant growth with great ability to supply clean and eco-friendly electrical power. In this paper, multi-objective optimization is carried out to design the most energy and cost-effective Solar Dish Stirling (SDS) power plant. Meteorological data from two different sites have been considered to simulate the technical and economical performances of 10 MW SDS power plant. Two main energy and economic conflicting objective functions were considered with the ultimate goal to maximize of the annual Capacity Factor and minimize the Total Cost of Site Improvements simultaneously. Several parameters such as the separation distance between collectors, ground slope, slot gap height and width have been parametrically assessed to find out effects on the Capacity Factor and the Total Cost of Site Improvements. First, a third-order multiple non-linear surrogate model was constructed. The model validity was demonstrated by the R-square values which were found to be 0.995 and 0.9964 for Ouarzazate and Madrid, respectively. Then, multi-objective optimization using Elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II) has been applied to address the trade-off between the two objective functions. The obtained Pareto Front was identified accordingly and final optimal solution was determined using Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) decision-making method. Final optimal solution corresponded to Capacity Factor (CF) values of 22.99% and 12.91% and a Total Cost of Site Improvements (TCSI) of 2.69 M$ and 2.60 M$, for Ouarzazate and Madrid, respectively.

Enhancing Robustness and Transmission Performance of Heterogeneous Complex Networks via Multiobjective Optimization

Transmission networks are ubiquitous in modern society and play critical roles in facilitating the delivery and movement of information, power, and people between various locations. Robustness and transmission capacity are two pivotal and universal properties of practical networks, and much research effort has been devoted to investigating these two properties in the past decade. In this article, we consider a heterogeneous transmission network consisting of hosts and routers and aim to optimize both transmission capacity and robustness of this network simultaneously. To solve this problem, we propose a multiobjective evolutionary algorithm (MOEA) for optimizing transmission capacity and robustness. Moreover, in order to achieve optimized transmission performance and robustness at reasonable and balanced computational cost, the proposed MOEA adopts a two-phase design, i.e., a sampling phase and an optimization phase. Simulation results on scale-free and realistic transmission networks demonstrate the effectiveness of the proposed algorithm. Moreover, comprehensive analysis of the solutions from different parts of the obtained Pareto fronts shows distinct characteristics and provides various choices for optimizing transmission functionality and robustness of networks.

Proposal of a new low-temperature thermodynamic cycle: 3E analysis and optimization of a solar pond integrated with fuel cell and thermoelectric generator

The present research deals with a novel hybrid system that comprises of a salinity gradient solar pond (SGSP) integrated with a proton exchange membrane fuel cell (PEMFC) and a thermoelectric generator (TEG). The key novelty of this research is the use of PEMFC waste heat to increase the performance of the low-temperature heat source (solar pond). Also, a thermoelectric generator is employed in the proposed setup for complete waste energy recovery. In this paper, the proposed system's energy, exergy, and economic (3E) performance are investigated, as well as compared with two additional systems that were modelled by excluding TEG and PEMFC to showcase the benefits of the sub-systems in the final configuration. A parametric study is also conducted to investigate the effect of significant parameters on the overall system performance. The simulation results indicated that the integrated system's net output power is 2288.8 kW at a base case operating condition, with energy and exergy efficiency of 11.26% and 13.17%, respectively, and a system cost rate of 394 $/h. In comparison to other components in the integrated system, SGSP alone accounts for 75% of total exergy degradation and has the highest investment cost rate of 66.70 $/h. Further, to achieve the best system performance, a multi-criteria optimization is performed for the suggested system with net output power, energy efficiency, exergy efficiency, and system cost rate as objective functions. Besides, the multi-criteria decision analysis (MCDA) is also carried out on the obtained Pareto front using TOPSIS decision-maker to select the best operating condition that improves the output power, energy, and exergy efficiency by 52.6%, 49.6%, and 61.8%, respectively, at the expense of 5.2% increase in the system cost rate. The current investigation demonstrates how an appropriate optimized design of an integrated energy system using SGSP, PEMFC, and TEG can improve the overall performance with reasonable cost increment.

Analyzing Behavioral Patterns of Bus Passengers Using Data Mining Methods (Case Study: Rapid Transportation Systems)

The aim of analyzing passengers' behavioral patterns is providing support for transportation management. In other words, to improve services like scheduling, evacuation policies, and marketing, it is essential to understand spatial and temporal patterns of passengers' trips. Smart Card Automated Fare Collection System (SCAFCS) makes it possible to utilize data mining tools for the purpose of passengers' behavioral pattern analysis. The specific goal of this research is to obtain functional information for passenger's behavioral pattern analysis in city express bus which is called BRT, and classification of passengers to improve performance of bus fast transportation system. Additionally, it is attempted to predict usage and traffic status in a line through predicting passenger's behavior in a bus line. In this paper, smart card data is applied to provide combinational algorithms for clustering and analysis based on data mining. To this end, we have used a combination of data mining methods and particle swarm optimisation algorithm and leveraged multivariate time series prediction to estimate behavioral patterns. Results show that price and compression ratio features are the most influencing features in the separability of transportation smart card data. According to obtained Pareto front, four features include a card identification number, bus identification number, bus line number, and charge times are influencing clustering criteria.

Data-driven economic dispatch for islanded micro-grid considering uncertainty and demand response

The variability and intermittency of renewable energy and power load bring great pressure to the dispatch of Micro-Grid, especially intra-day dispatch. In order to reduce the intra-day dispatch pressure, this paper proposes a data-driven two-stage day-ahead dispatch model for islanded Micro-Grid. The first stage dispatch model considers multiple demand responses, and applies phase space reconstruction and machine learning to predict renewable energy output and power load. Multi-objective particle swarm optimization algorithm is used to solve the first stage dispatch model. For the obtained Pareto Front, weight multiple objectives by entropy weight method to get the optimal solution. Since the deviation between the predicted value and the actual value can lead to renewable energy curtailment or load loss, the role of the second stage is to predict the renewable energy curtailment and load loss after the first stage dispatch. Firstly, extreme gradient boosting is used to predict when renewable energy curtailment and load loss occur. Secondly, extreme learning machine is used to predict the amount of renewable energy curtailment and load loss at the corresponding time points. Finally, linear programming and mixed integer linear programming are used to solve the second stage dispatch model. By comparative cases analysis, simulation results show that the enhancement of demand response to system efficiency is at the cost of increasing dispatch cost and reducing reliability. In contrast, the proposed two-stage dispatch method using regulation reserve capacity not only reduces dispatch cost, but also improves system efficiency and reliability.

Multi-objective optimization of a composite orthotropic bridge with RSM and NSGA-II algorithm

To address the problems of pavement damage and fatigue cracking of orthotropic steel deck (OSD) in bridges with traditional asphalt overlay, an innovative composite bridge deck composed of OSD with open ribs and ultra-high performance concrete (UHPC) layer was proposed. This paper firstly investigates the stress responses of fatigue-prone details in the composite bridge deck by refined two-scale finite element analysis. The results show that the rib-to-deck joint can achieve an infinite fatigue life, while the floorbeam detail of rib-to-floorbeam joint indicates finite fatigue life. Then, approximate regression models of stress ranges of fatigue details and structure weight were derived via both the central composite design and response surface method. The analysis of correlation and variance demonstrates that the response surface models are significant and sufficiently accurate to predict the objective responses. And 3-D response surface plots were developed to display the interaction effect between each pair of structural parameters on objective responses. Finally, to improve the fatigue performance for achieving an infinite fatigue life under relatively low structure weight, the multi-objective optimization was executed by an Improved Non-dominated Sorting Genetic Algorithm (NSGA-II). The obtained Pareto front shows that there is a strong competition between the stress range of fatigue-prone detail and structure weight. Under the principle of relatively low structure weight, the obtained structural parameters satisfying infinite fatigue life imply that the deck plate thickness is 10 mm, and the floorbeam spacing is 2.4 m as well as the floorbeam thickness is not less than 12 mm.

Multi-objective optimization of a small sized solar PV-T water collector using controlled elitist NSGA-II coupled with TOPSIS

Small-sized solar photovoltaic thermal (PV-T) collector can achieve better overall performance due to less aperture plane heat losses and the possibility of concentrating more radiant flux per unit aperture area compared to their large counterpart designs. Despite its potential, studies on its performance optimization have not yet been explored. This study aims to perform a multi-objective optimization of the performance of a small-sized PV-T water collector with channel type absorber design by using a two-step procedure i.e., implementation of Non dominated sorting genetic algorithm (NSGA-II) followed by a decision making method i.e., a technique for order preference by similarity to ideal solution (TOPSIS). Based on the Taguchi orthogonal array design for three control parameters i.e., mass flow rate, inlet temperature, and inclination angle, an experimental campaign is undertaken. The design space created on the basis of Taguchi design of experiment is used to conduct a total 16 number of experiments. An analysis of variance is carried out to obtain the significant design and operating parameters controlling the collector performance, which are used to fit the objective function equations for both thermal and electrical efficiency. A multi objective optimization strategy is then developed by a variant of NSGA-II to simultaneously optimize its average thermal and electrical efficiencies. From the obtained Pareto fronts, the most optimal solution is obtained by using TOPSIS method. The highest thermal and electrical efficiency of 82.55% and 10.45% is obtained under an optimal mass flow rate of 0.02 kg/s, inlet temperature 32 °C and inclination angle 38.88°. The influence of solar radiation on its optimized performance demonstrates a little less higher thermal efficiency of 81.05% and electrical efficiency of 11.43% obtained at a radiation of 700 W/m2. On experimental verification of the optimized performance, the highest thermal and electrical efficiency is obtained within an agreement of ± 3.51% and ± 6.9%, respectively. The present optimized efficiency values are higher compared to that of a typical experimental day with un-optimized input conditions and also higher than some of the recent published papers.

Crashworthiness optimization of a multicellular thin-walled tube with triangular cells

This article optimizes a multicellular thin-walled tube with N-equilateral triangles (MTTN), which was inspired from the microstructure of the cross-section of bamboo. The crashworthiness of MTTN under axial impact was analyzed, and the sensitivity analysis was conducted for the geometric parameters of MTT24, which exhibited the best crashing performance among MTTNs with N ≤ 30. It was found that the order in radial direction has a weak influence on the crashworthiness of MTT24, while the cell-wall thickness and the inner-tube diameter could significantly affect the crashworthiness of MTT24. Then, a multi-objective optimization was carried out for the MTT24 to further improve its crushing performance. The wall thickness and inner diameter of the thin-walled tube were selected as design variables, while the minimum peak crushing force (PCF) and the maximum specific energy absorption (SEA) were set as objectives. The obtained Pareto front enables the thin-walled tube to achieve the best possible crush resistance under the conditions of maximum SEA and minimum PCF. With a constraint of PCF ≤ 200 kN, the SEA and MCF of MTT24 were improved by 11.40% and 68.20%, respectively, compared with the benchmark.

Multi-objective design of aircraft maintenance using Gaussian process learning and adaptive sampling

Aircraft maintenance design aims to identify strategies that render the aircraft reliable for flight in a cost-efficient manner. These are often conflicting objectives. Moreover, existing studies on maintenance design often limit themselves to only one type of maintenance strategy, overlooking other potentially dominating designs. We propose a framework for aircraft maintenance design with explicit reliability and cost-efficiency objectives. We explore the design space of a variety of maintenance strategies ranging from traditional time-based maintenance to predictive maintenance. To explore this design space, we propose an adaptive algorithm using Gaussian process learning and a novel adaptive sampling method. Gaussian process learning models rapidly pre-evaluate new maintenance designs, while adaptive sampling selects for further exploration only those designs that are expected to improve the available Pareto front of maintenance designs. This framework is illustrated for the maintenance of multi-component aircraft systems with k-out-of-n redundancy. The results show that novel predictive maintenance designs based on Remaining-Useful-Life prognostics dominate other maintenance designs, especially in the knee region of the obtained Pareto front, where the most beneficial balance between conflicting objectives is achieved. Our proposed exploration algorithm also outperforms other state-of-the-art exploration algorithms with respect to the quality of the Pareto front obtained.

Multi-stage guided stochastic search for optimization and standardization of free-form steel double-layer grids

There has been a growing interest in the use of free-form structures with irregularly curved yet aesthetically pleasing configurations in the recent decades. Although design optimization of regular steel grids has been well addressed in the literature of structural optimization, still limited work has been devoted to optimum design of real-size free-form grid structures. On the one hand, a main obstacle when dealing with real-size free-form steel grids is the excessive computational effort associated with contemporary evolutionary optimization algorithms. On the other hand, it is generally perceived that the obtained final designs using conventional optimization algorithms may not necessarily be favored in practice if certain provisions are not stipulated by the algorithm to preclude an abundance of distinct steel section sizes in the final design. Hence, instead of offering a single optimum or near optimum design, it would be more desirable to provide the designer or decision maker with a Pareto front set of non-dominated design alternatives taking into account both the minimum weight as well as the assortment of available steel section sizes in the final design. Accordingly, in this paper, a computationally efficient multi-stage guided stochastic search algorithm is proposed for optimization and standardization of real-size free-form steel double-layer grids. A gradual design-oriented section elimination approach is followed where in the first optimization stage, a complete set of commercially available steel sections is introduced to the algorithm and in the succeeding stages, the size of section list is reduced by eliminating the redundant sizes. Two variants of the algorithm are employed to demonstrate the usefulness of the proposed technique in challenging test examples of free-form steel double-layer grids, and the obtained Pareto fronts are plotted to illustrate the trade-off between minimum weight and assortment of steel section sizes in the final design.