Pareto Optimal Frontier Research Articles

Mathematical model based on nonlinear differential equations and its control algorithm

Abstract A practical examination of the traditional robotic arm (RA) in operation revealed a significant limitation in its ability to control the position of motion. This underscores the urgent need to enhance the current RA’s position control capabilities. Therefore, this study proposes the use of nonlinear differential equations (NDEs) to establish a mathematical model and the design of NDE-based RA motion control algorithms in conjunction with a central pattern generator neural network. A comparison of the control effects showed that the proposed method was highly fitted to the target trajectory. The joint node (JN) motion tracking trajectories of the three RAs were similar, up to 90–85% to the target trajectories of the JNs. In addition, the control of the motion position was similar up to 95–98% to the target motion position trajectories. The motion control algorithm based on NDEs was effective in improving the average execution time of the Pareto optimal frontier of the RA by 58.29%. The joint velocity and angle changes of the three types of RAs under the NDE control algorithm exhibited a high degree of similarity to the fluctuations observed in the expected and predicted curves. These observations contribute to an understanding of the effectiveness of the system observer in observing the joint angle changes. This indicates that the motion control based on NDEs can effectively enhance the tracking effectiveness of the JN positions of the RA, improve the control ability of the RA motion, and increase the joint stability of the RA.

Multi-objective preventive maintenance strategy and optimization considering unavailability and cost: A case study on VOBC

The Vehicle On-Board Controller (VOBC) is a core component of urban rail transit trains, with high maintenance costs and serious consequences in case of failure. Ensuring long-term VOBC operation while minimizing maintenance costs is of paramount importance. This paper proposes a multi-objective preventive maintenance (PM) strategy and optimization considering system unavailability and maintenance costs. Firstly, we provide a statement of the problem and analyze the impact of different maintenance actions on VOBC. Next, the decreasing process parameter is introduced to adjust the PM interval dynamically. The mathematical models of system unavailability and maintenance costs are derived to construct a multi-objective PM strategy. Then, we develop a hybrid heuristic algorithm including multi-objective particle swarm optimization and genetic algorithm (MOPSO-GA), which is used to obtain the optimal Pareto frontier for maintenance strategies. In the study, we select maintenance strategies by introducing an aversion factor to fulfill the preferences of different decision-makers. A comprehensive sensitivity analysis is performed to illustrate the impact of several critical parameters on the multi-objective PM strategy. The case study on Beijing subway’s VOBC is also executed. The outcomes indicate that the proposed method can dynamically adjust to match the preferences of diverse decision-makers and the MOPSO-GA algorithm has excellent robustness on most parameters. This approach allows for creating field maintenance strategies catering to the varied maintenance requirements of urban rail transit VOBC.

Data-driven surrogate modeling and optimization of supercritical jet into supersonic crossflow

For the design and optimization of advanced aero-engines, the prohibitively computational resources required for numerical simulations pose a significant challenge, due to the extensive exploration of design parameters across a vast design space. Surrogate modeling techniques offer a viable alternative for efficiently emulating numerical results within a notably compressed timeframe. This study introduces parametric Reduced-Order Models (ROMs) based on Convolutional AutoEncoders (CAE), Fully Connected AutoEncoders (FCAE), and Proper Orthogonal Decomposition (POD) to fast emulate spatial distributions of physical variables for a supercritical jet into a supersonic crossflow under different operating conditions. To further accelerate the decision-making process, an optimization model is developed to enhance fuel-oxidizer mixing efficiency while minimizing total pressure loss. Results indicate that CAE-based ROMs exhibit superior prediction accuracy while FCAE-based ROMs show inferior predictive accuracy but minimal uncertainty. The latter may be ascribed to the markedly greater number of hyperparameters. POD-based ROMs underperform in regions of strong nonlinear flow dynamics, coupled with higher overall prediction uncertainties. Both AE- and POD-based ROMs achieve online predictions approximately 9 orders of magnitude faster than conventional simulations. The established optimization model enables the attainment of Pareto-optimal frontiers for spatial mixing deficiencies and total pressure recovery coefficient.

Heat charge performance prediction and optimization of locally refined bionic fin heat exchanger with PCM and nanoparticles based on NSGA-II

Phase change energy storage plays an important role in the popularization of renewable energy sources (wind, solar, geothermal, etc.). However, previous studies have paid little attention to the correlation and optimal solution of phase transformation thermal performance prediction. This research combines Response Surface Methodology with the Non-dominated Sorting Genetic Algorithm II to forecast and enhance the performance of tree-shaped biomimetic latent heat storage units, targeting optimal functionality under operational conditions. Initially, the study delves into a mechanistic analysis of the heat transfer in latent heat storage units, focusing on the coupling of conductive and natural convective heat transfer during the melting phase. The synergistic enhancement of heat conduction and convection is realized through locally refined and punched fins. Subsequently, Response Surface Methodology was applied to analyze the interactions between the design variables and the objective functions, leading to the attainment of Pareto-optimal frontiers using the Non-dominated Sorting Genetic Algorithm II. The results demonstrate that the Pareto-optimal solutions substantially improved the Nusselt number of novel biomimetic latent heat storage units by 42.58%–45.58 %, relative to tree-shaped fin latent heat storage units. Concurrently, the Fourier number was reduced by 17.54%–19.22 %. Moreover, the effect of adding nanoparticles (aluminum, copper oxide, and copper) to the phase change material was studied. The results show that the proposed locally refined optimal fin structure is superior to the addition of nanoparticles.

Constructal design for H-shaped compound heat transfer path in a rectangular heat generation body

An H-shaped compound heat transfer path (HSCHTP) model in a rectangular heat generation body (RHGB) is built in present article. With specified total volumes of HSCHTP and RHGB, constructal design for HSCHTP model is conducted by varying path length ratios and radius. The minimum complex functions composed of thermal and flow performances are acquired, and the multi-objective optimizations are performed furtherly. Three decision-making approaches are introduced to obtain the optimal design scheme (ODS) from Pareto-optimal frontiers. It indicates that complex function after triple-optimization is reduced by 14.7 % compared with that of initial design. Compared with performance of RHGB with no heat conduction path, complex function after optimization is reduced by 77.5 %. Thus, performance of the HSCHTP is apparently superior to that with no heat conduction path in this case. The complex function minimization is a special case of the multi-objective optimization (MOO). The fifth order path radius, adjacent order and internal order length ratios acquired by MOO are distributed within the ranges of 1.50mm-1.60 mm, 1.39–1.47 and 0.066–0.90, respectively. By chasing for the smallest deviation-index, optimal structure of HSCHTP acquired by LINMAP decision-making method is taken for the ODS when multiple requirements are considered in the heat dissipation designs.

Multi-Objective Optimization Design of 40 MW High-Temperature Superconducting Motor

Abstract To reduce the length of the superconducting wire of the superconducting motor, the no-load back EMF E and the magnetic field component perpendicular to the wire B⊥ were selected as the optimization goals. Therefore, a multi-objective optimization method based on the Response Surface Method (RSM) and Genetic Algorithm (GA) was proposed by taking a high-temperature superconducting motor (HTSM) with a rated power of 40 MW as the research object. Firstly, the overall structure, mathematical model, and output characteristics of HTSM were introduced. Secondly, the sample points required for constructing the RSM equation were obtained by Finite Element Analysis (FEA), and the multi-objective optimization of the RSM equations was performed by applying GA to obtain the Pareto optimal frontier. Finally, the calculated optimal design was simulated and verified. Compared with the initial design, the E of the optimized HTSM was increased by 4.47%, and the B⊥ was reduced by 4.83%. Based on the motor parameters obtained by multi-objective optimization, the final optimized design of HTSM was obtained. The length of the superconducting wire used in the motor has been reduced by 10.2%, which greatly reduces the cost.

MOBCA: Multi-Objective Besiege and Conquer Algorithm.

The besiege and conquer algorithm has shown excellent performance in single-objective optimization problems. However, there is no literature on the research of the BCA algorithm on multi-objective optimization problems. Therefore, this paper proposes a new multi-objective besiege and conquer algorithm to solve multi-objective optimization problems. The grid mechanism, archiving mechanism, and leader selection mechanism are integrated into the BCA to estimate the Pareto optimal solution and approach the Pareto optimal frontier. The proposed algorithm is tested with MOPSO, MOEA/D, and NSGAIII on the benchmark function IMOP and ZDT. The experiment results show that the proposed algorithm can obtain competitive results in terms of the accuracy of the Pareto optimal solution.

A two-stage direction-guided evolutionary algorithm for large-scale multiobjective optimization

Large-scale multiobjective optimization problems (LSMOPs) have exponential growth in the search space as the decision variables increase, and the vast search space poses a challenge to the performance of multiobjective evolutionary algorithms (MOEAs). Many current large-scale MOEAs need to consume a large amount of computational resources to get good performance. This paper proposes a two-stage direction-guided evolutionary algorithm for large-scale multiobjective optimization (LMOEA-S2D) to balance the performance and computational resource overhead. The algorithm exploits the Pareto-optimality property of domination and the diversity-preserving property of decomposition to optimize the performance in the two stages, respectively, and designs a corresponding direction-guided mechanism to improve search efficiency. LMOEA-S2D designs global direction search and local direction search in the domination-based stage for efficient exploitation to accelerate population convergence. To promote greater population diversity, a hybrid direction search was devised to aid diversity exploration in the decomposition-based stage, and this facilitates even distribution of candidate solutions across the Pareto optimal frontier. LMOEA-S2D is compared with five state-of-the-art large-scale MOEAs on some large-scale multiobjective test suites with 100 to 5,000 decision variables. The experimental results show that LMOEA-S2D significantly outperformed all compared algorithms under limited computational resources.

Multi-Objective Dynamic Economic and Emission Generation Scheduling of Coordinated Power System Considering Renewable Energy Sources and Pumped Storage Hydro Plants via Fuzzy Surrogate-Assisted Coronavirus Herd Immunity Optimization

Globally, the power generation from renewable energy sources (RES) has received considerable attention to reduce pollutant emissions and the total operating costs of power generation. In this context, this study aims to extend the multi-objective dynamic economic and emission dispatch (MODEED) problem by incorporating RES and pumped storage hydro plants. The complex constrained MODEED problem is solved by a fuzzy surrogate-assisted coronavirus herd immunity optimization (FSACHIO) algorithm in which a self-adaptive speed factor and Lévy flight mechanism are utilized to ensure populace diversity, prevent premature convergence and achieve an ideal stability among the exploration and exploitation of the algorithm. Surrogate worth tradeoff methodology is used to discover a solution that provides the optimal balance between the objectives such as operating expenses and emission pollutants. The performance of FSACHIO algorithm is validated on a small-scale and a large-scale test systems, involving 10-unit, and 40-unit test systems, and compared with that of the coronavirus herd immunity optimization, red fox optimizer, Remora optimization algorithm, and other erstwhile approaches. Besides, a fuzzy constraint handling method based on a dominance relationship is integrated to fulfill the restrictions of the MODEED problem. The simulation results reveal that: (i) the operating costs and emissions are reduced by 173785.54 $/day and 2992.1848 lb/day in the small-scale test system and by 6383.9903 $/day and 216631.4877 lb/day in the large-scale test system, respectively for the MODEED problem with the presence of RES and pumped storage hydro plants; (ii) the suggested approach may offer a well-distributed Pareto-optimal frontier and superior tradeoffs among the operating expenses and pollution objectives in comparison to the other approaches.

Multi-objective optimization of methanol reforming reactor performance based on response surface methodology and multi-objective particle swarm optimization coupling algorithm for on-line hydrogen production

The utilization of high-temperature exhaust to drive methanol steam reforming for on-line hydrogen production in engines is an effective approach to solving hydrogen storage and transportation challenges. In this work, the response surface methodology (RSM) is combined with the multi-objective particle swarm optimization (MOPSO) algorithm, and is first utilized for multi-objective optimization of the performance of a methanol reforming reactor. Regression models for optimization objectives are developed based on RSM, and the reliability and significance of regression models are determined through analysis of variance. The influence mechanism of the interaction between design variables on optimization objectives is elucidated by visual analysis of the response surface. A multi-objective optimization of reactor performance is performed utilizing the MOPSO algorithm, and the Pareto optimal solution is obtained from the Pareto optimal frontier based on the technique for order preference by similarity to ideal solution method. Optimization results demonstrate that the Pareto optimal solution is obtained at an exhaust temperature of 585.21 K, an exhaust flow rate of 0.0097 kg·s−1, a steam to methanol ratio of 1.48, and a weight hourly space velocity of 1.2. The hydrogen yield, methanol conversion and carbon monoxide selectivity corresponding to the Pareto optimal solution are 68.53 mol·h−1, 0.83 and 0.011, respectively. Under optimal operating parameters, the methanol reforming reactor achieves 21.92 % exhaust energy recovery.

A comprehensive parametric study and multi-objective optimization of turbulent jet array impingement for uniform cooling of gas turbine blades with minimized compression power

In the present paper, a comprehensive parametric study and multi-objective optimizations on jet array impingement cooling are conducted for mid-chord sections of gas turbine blades to maximize the heat transfer uniformity on the target plate and minimize the air compression power consumption at different desired Nusselt numbers. The validated numerical method based on RANS equations is utilized to determine the effects of Reynolds number (2500≤Re≤35000), jet spacings (3≤Px,Py≤8), and the jet-to-target distance (0.75≤Pz≤3) on air compression power (Wc), average Nusselt number (Nu‾), and heat transfer uniformity index (UI). According to the parametric study, the increase of Re improves Nu‾ and UI, while intensively increasing Wc. The effect of jet-to-target distance is a function of jet spacings; with the increase of Pz at low jet spacings, Wc,Nu‾, and UI are reduced. Although, at large spacings, Nu‾ and UI increase with Pz, and Wc is independent of Pz. Additionally, the increase of Px decreases Wc, Nu‾, and UI. Increasing Py reduces Nu‾ and Wc. But at small Px and Pz, the UI is descending; while at large Px and Pz, the UI tends to ascend. Three high-accuracy surrogate models are developed using backpropagation artificial neural networks (ANN) for estimating Wc, Nu‾, and UI for input design variables. Sobol global sensitivity analysis is also performed based on the developed models for quantifying the influence of design variables and their interactions on objective functions. As expected, the results indicate that Wc and Nu‾ are the most sensitive to Re, and UI is mainly affected by Px, whereas Py is more affecting the uniformity index rather than compression power. Finally, to find out the best design and flow conditions, optimizations are conducted by the NSGA-II algorithm. The optimal Pareto frontier and final decided solutions by TOPSIS and LINMAP methods are then demonstrated for the desired Nusselt number (NuD) of 70. The TOPSIS method indicates Px=4.70, Py=3.25, Pz=1.40, and Re = 13800 as the best compromise for optimization. The analysis of Pareto solutions in the range of NuD from 35 to 130 suggests a variety of optimal flow and geometrical arrangements for a trade-off between objective functions; therefore, at most, 50% less compression power or 5% more uniformity is approachable corresponding to the design requirements.

Pedestrian evacuation planning under dam-break flood disaster considering road risk and road pedestrian demand

When facing disasters, a well-planned evacuation of people can significantly reduce casualties and disaster losses. Hurricane Katrina struck New Orleans in 2005, causing multiple dam failures and resulting in Lake Pontchartrain flooding back into the city. Taking this disaster as an example, this paper evaluates the risk of pedestrian evacuation on roads during dam-break flood disasters. The evaluation considers three aspects: road evacuation difficulty, disaster risk and road pedestrian demand. The fuzzy VIKOR model was used to evaluate 8 indicators for 50 candidate shelters. A dual-objective planning model was proposed to consider the time-varying and diverse emergency needs of disaster victims. By assessing the risk of pedestrian evacuation on the road and delineating restricted areas, the optimal path distance is calculated. Pareto optimality is then employed to achieve a trade-off between the convenience, cost, and applicability of the evacuation plan. In a complex post-disaster environment, the model can adapt flexibly to emergency rescue situations at various stages by adjusting the parameters in the objective and constraint equation. The Pareto optimal frontier can assist in determining the most suitable solution for the current scenario. In the case of New Orleans, 22 and 18 shelters were respectively selected from 34 candidate shelters for evacuation based on the simulated scenarios.

Optimizing warehouse operations for environmental sustainability: A simulation study for reducing carbon emissions and maximizing space utilization

Loading and unloading operations for Stock Keeping Units (SKUs) in warehouses are critical to logistics management systems. However, these operations also have a significant impact on the environment, particularly in terms of carbon dioxide (CO2) emissions. As such, warehouse management must consider not only operational efficiency but also environmental impact. The reduction of CO2 emissions in warehouses is becoming increasingly important, both for legal compliance and to meet sustainability targets. In this article, we will emphasize the environmental impact of warehouse operations, particularly on CO2 emissions, and explore ways to minimize them while still maximizing warehouse performance. We will review various optimization models proposed to address this issue and highlight the importance of considering environmental objectives when designing warehouse operations. We will also describe a simulation study conducted to determine the Pareto optimal frontier for a warehouse design, considering transportation, space utilization, and CO2 emissions. The outcomes of implementing this simulation's results include reduced CO2 emissions and increased space utilization, which demonstrate the potential benefits of considering environmental objectives in warehouse design and management.

Multi-Objective Optimization of Injection Molding Process Parameters for Moderately Thick Plane Lens Based on PSO-BPNN, OMOPSO, and TOPSIS

Injection molding (IM) is an ideal technique for the low-cost mass production of moderately thick plane lenses (MTPLs). However, the optical performance of injection molded MTPL is seriously degraded by the warpage and sink marks induced during the molding process with complex historical thermal field changes. Thus, it is essential that the processing parameters utilized in the molding process are properly assigned. And the challenges are further compounded when considering the MTPL molding energy consumption. This paper presents a set of procedures for the optimization of injection molding process parameters, with warpage, sink marks reflecting the optical performance, and clamping force reflecting the molding energy consumption as the optimization objectives. First, the orthogonal experiment was carried out with the Taguchi method, and the S/N response shows that these three objectives cannot reach the optimal values simultaneously. Second, considering the experimental data scale, the back propagation neural network updated by the particle swarm optimization method (PSO-BPNN) was applied to establish the complex nonlinear mapping relationship between the process parameters and these three trade-off objectives respectively. Then, the Pareto optimal frontier of the multi-objective optimization problem was attained by multi-objective particle swarm optimization using a mutation operator and dominance coefficient algorithm (OMOPSO). And the competitive relationship between these objectives was further confirmed by the corresponding pairwise Pareto frontiers. Additionally, the TOPSIS method with equal weights was employed to achieve the best optimal solution from the Pareto optimal frontier. The simulation results yielded that the maximum values of warpage, sink marks, and clamping force could be reduced by 7.44%, 40.56%, and 5.56%, respectively, after optimization. Finally, MTPL products were successfully fabricated.

A distributed multi-objective optimization method for scheduling of integrated electricity and hydrogen systems

The growing diversity in energy demand has led to an increasingly intertwined relationship between the electric power system (EPS) and hydrogen energy system (HES). However, these systems are presently managed by entities with distinct interests, resulting in competition and privacy concerns during the scheduling of integrated electricity and hydrogen systems (IEHSs). To address this issue, this paper proposes a multi-objective IEHS scheduling model with the aim of minimizing costs for electricity and hydrogen suppliers, taking into account bi-directional energy transactions between EPS and HES. Furthermore, we present an innovative approach based on alternating direction multiplier method (ADMM) for distributed ξ-constraint optimization. This approach efficiently captures the optimal Pareto frontier while maintaining the privacy of EPS and HES. The proposed methodology is validated on an IEHS composed of an IEEE 33-bus EPS and a 20-node HES. The results demonstrate that no matter how many solution sets, a more uniform pareto front can be obtained than the traditional method. Furthermore, it simplifies the multi-objective optimization problem and effectively protects the privacy of participants at the expense of acceptable solution time.

Renewable energy utilizing and fluctuation stabilizing using optimal dynamic grid connection factor strategy and artificial intelligence-based solution method

As a crucial aspect of pollution emissions, the transformation and upgrading of the power industry urgently require attention. Utilizing renewable energy generation techniques is an effective means to promote energy conservation and emission reduction within the power industry. However, the utilization rates of renewable energy generation are hindered by significant randomness and uncertainty associated with renewable energy power. In light of these challenges, this study constructed a hybrid economic emission dispatch (HDEED) model to explore the establishment of effective regulation mechanism under renewable energy grid connection. First, taking into account dynamic constraints, the wind-solar-thermal-storage HDEED model was constructed with operation costs and pollution emissions as objective functions. Second, an optimal dynamic grid connection factor strategy and the GWOH-based solution method were proposed. Finally, the proposed strategy and algorithm were verified using an improved IEEE-39 bus test system. The results revealed that under the same test conditions, the Pareto optimal frontiers obtained by the GWOH algorithm demonstrated the most competitive performance. Furthermore, the system cost achieved by GWOH was 4.52% lower than that without renewable energy access, while emissions were reduced by 11.84% under the optimal dynamic factor grid connection strategy. This study significantly contributes to the improvement of energy management efficiency and the enhancement of renewable energy utilization rates in the power industry.

Design and motion planning of a 7-DOF assembly robot with heavy load in spacecraft module

Currently, there is great demand for automated assembly of equipment inside spacecraft module, which would require an assembly robot with small size, large workspace, high load capacity, and high flexibility. To meet this demand, a redundant seven-degree-of-freedom (7-DOF) assembly robot with three prismatic joints and four rotating joints is proposed in this study. First, a kinematic model of the assembly robot is established, and the positional forward kinematics are derived. Then, the positional inverse kinematic equations are derived by using the fixed joint method and the coordinate inverse transformation method for the characteristics of the robot with redundant degrees of freedom. The inverse solution optimization model of the assembly robot is designed based on genetic algorithm, and B-spline interpolation algorithm is used for joint trajectory planning. Next, NSGA-2 algorithm is used to construct the trajectory optimization model, and the time-pulsation multi-objective Pareto optimal frontier is obtained. Finally, the joint simulation model of mechatronics and the prototype experiment are established to verify the effectiveness of the optimization results. Analysis shows that the proposed assembly robot has better load-bearing capacity and dynamic characteristics than similar existing assembly robots. Furthermore, the segmented trajectory optimization strategy for the assembly robot using inverse solution optimization, interpolation algorithm, and NSGA-2 based multi-objective optimization gets each motion joint to realize the trajectory tracking function quickly and stably, and the trajectory tracking error is small enough to meet the requirements of end positioning accuracy.

Optimization design of space radiation cooler based on response surface method and genetic algorithm

The cooling capacity of a radiator under complex environmental conditions is improved by controlling the detector temperature within a range of the thermal control index. An optimal design method for the radiation cooler based on the response surface method (RSM) and genetic algorithm is proposed, and the area and location of the radiation cooler of the space ultraviolet imager detector are optimized. First, the physical model and thermal simulation model of the space ultraviolet imager are established. Then, the detector temperature is used as the objective function, and the length, height, and inclination angle of the radiator in space are used as the design variables. The mathematical model of the objective function is established by the RSM, and the rationality of the mathematical model is verified by analysis of variance (ANOVA). Finally, the Pareto optimal frontier is obtained using the genetic algorithm. The optimization results confirm that the optimal design parameters of the radiator are XW = 165.58 mm, XH = 160.29 mm, θ1 = 7.28°, θ2 = 5.87°. Compared with before optimization, the area of the radiator is reduced by 18.18%, and the weight is reduced by 48.78 g. Based on verifying the optimization results, the optimized radiation cooler can control the detector within the target temperature of −20 to −10 °C. Therefore, the optimal design of the radiation cooler meets the requirement of lightweight while improving the cooling capacity of the radiator.

Multi-objective Capacity Planning for Expressway Microgrid Considering Photovoltaic Panel Dust Removal Maintenance

The increasing passenger volume and the rapid development of the electric vehicle industry make the electricity consumption of the expressway service area grow day by day. It is of great significance to study the planning and design method of the microgrid in the expressway service area. This paper established a wind-photovoltaic-storage capacity planning model for the microgrid in expressway service areas, which considered the dust removal maintenance of photovoltaic panels. The multi-objective optimization problem was solved by the improved fast non-dominated sorted genetic algorithm (NSGA II) with the aim of minimizing the total system investment, and operating cost and maximizing the self-consistent rate of electricity consumption of the microgrid. The Pareto optimal solution frontier with uniform distribution and wide range is obtained, and the capacities of PV, wind power, energy storage, and contact line are obtained under different self-consistent rates of electricity consumption. This method can not only make full use of distributed PV resources that can be developed on the expressway to improve the self-consistent level of energy but also comprehensively evaluate the economy of the microgrid system and meet the requirements of power supply reliability.

Using an exact bi-objective decoder in a memetic algorithm for arc-routing (and other decoder-expressible) problems

We address the bi-objective Capacitated Arc Routing Problem (CARP) by considering two levels of solution interpretation: implicit and explicit solutions. An algorithm that translates implicit solutions into explicit solutions is called a decoder. In this work, the decoder takes as input a permutation of the required edges and generates a Pareto frontier of CARP solutions. While bi-objective CARP was our main focus and starting point, we could also use the proposed framework to solve a bi-objective version of the traveling salesman problem by plugging-in a different decoder. Recall that bi-objective CARP asks to service (the demands of) a set of required edges using a fleet of vehicles of limited capacity so as to minimize: (i) the total travelled distance and (ii) the length of the longest route. Any permutation s of the required edges constitutes an implicit CARP solution. The decoder constructs all non-dominated explicit solutions that service the edges in the order indicated by s, i.e., the decoder is an exact algorithm that returns the optimal Pareto frontier subject to the service order s. To achieve competitive CARP results it is also important to reinforce the decoder using a local search operator that acts on explicit routes (and that may change the service order s). For nine instances, the resulting algorithm was even able to find a new total-cost upper bound, improving upon the best solutions reported in the (considerably larger) mono-objective CARP literature. This shows that (some of) the proposed ideas can also be useful in single objective optimization: the second objective can be seen as a guide for the mono-objective search process.