Multi-objective Evolutionary Approach Research Articles

An integrative approach to protein sequence design through multiobjective optimization.

With recent methodological advances in the field of computational protein design, in particular those based on deep learning, there is an increasing need for frameworks that allow for coherent, direct integration of different models and objective functions into the generative design process. Here we demonstrate how evolutionary multiobjective optimization techniques can be adapted to provide such an approach. With the established Non-dominated Sorting Genetic Algorithm II (NSGA-II) as the optimization framework, we use AlphaFold2 and ProteinMPNN confidence metrics to define the objective space, and a mutation operator composed of ESM-1v and ProteinMPNN to rank and then redesign the least favorable positions. Using the two-state design problem of the foldswitching protein RfaH as an in-depth case study, and PapD and calmodulin as examples of higher-dimensional design problems, we show that the evolutionary multiobjective optimization approach leads to significant reduction in the bias and variance in RfaH native sequence recovery, compared to a direct application of ProteinMPNN. We suggest that this improvement is due to three factors: (i) the use of an informative mutation operator that accelerates the sequence space exploration, (ii) the parallel, iterative design process inherent to the genetic algorithm that improves upon the ProteinMPNN autoregressive sequence decoding scheme, and (iii) the explicit approximation of the Pareto front that leads to optimal design candidates representing diverse tradeoff conditions. We anticipate this approach to be readily adaptable to different models and broadly relevant for protein design tasks with complex specifications.

A Multi-Objective Evolutionary Approach to Discover Explainability Tradeoffs when Using Linear Regression to Effectively Model the Dynamic Thermal Behaviour of Electrical Machines

Modelling and controlling heat transfer in rotating electrical machines is very important as it enables the design of assemblies (e.g., motors) that are efficient and durable under multiple operational scenarios. To address the challenge of deriving accurate data-driven estimators of key motor temperatures, we propose a multi-objective strategy for creating Linear Regression (LR) models that integrate optimised synthetic features. The main strength of our approach is that it provides decision makers with a clear overview of the optimal tradeoffs between data collection costs, the expected modelling errors and the overall explainability of the generated thermal models. Moreover, as parsimonious models are required for both microcontroller deployment and domain expert interpretation, our modelling strategy contains a simple but effective step-wise regularisation technique that can be applied to outline domain-relevant mappings between LR variables and thermal profiling capabilities. Results indicate that our approach can generate accurate LR-based dynamic thermal models when training on data associated with a limited set of load points within the safe operating area of the electrical machine under study.

An adaptive evolutionary multi-objective clustering based on the data properties of the base partitions

Evolutionary multi-objective clustering (EMOC) is a modern clustering technique in which the general concepts of evolutionary multi-objective optimization are applied to the clustering problem. The design and definition of clustering are difficult problems, and the choice of the objective functions and parameter setting of the algorithms are still challenges. In this paper, we propose the adaptive evolutionary multi-objective clustering approach based on data properties (AEMOC). AEMOC considers a new metric to evaluate the base partitions (candidate solutions used in the initial population) generated by minimum spanning tree clustering. This metric is applied to: (1) determine properties of the data, such as separation and overlap; and (2) define an offline selection of objective functions and parameter settings of the multi-objective algorithm. The information regarding the data properties allows AEMOC to avoid unnecessary data processing in datasets in which the initialization provides optimal solutions and optimization is not required. AEMOC presented promising results considering a diverse set of artificial and real-life datasets, based on two aspects: (1) it succeeded in the determination of the data properties of the base partitions and verifying the potential clustering quality, presenting a correlation of 0.8 with a reference metric, and (2) it provided better clustering results than reference EMOC approaches, achieving a general gain in the clustering performance of 3% in real-life datasets and 7% in artificial datasets.

An Evolutionary Multi-Objective Neural Architecture Search Approach to Advancing Cognitive Diagnosis in Intelligent Education

An Evolutionary Multi-Objective Neural Architecture Search Approach to Advancing Cognitive Diagnosis in Intelligent Education

Analysis and multi-objective evolutionary optimization of Solar-Biogas hybrid system operated cascade Kalina organic Rankine cycle for sustainable cooling and green hydrogen production

A novel cascade system of Kalina and Organic Rankine cycle for cooling and green hydrogen production was proposed. The cascade system is designed to utilizes the heat effectively with heat recovery components for sustainable cooling and green hydrogen. The proposed system consists of a solar-biogas hybrid heat source, Kalina and ORC power generation system, dual evaporator vapor compression cooling system and solid oxide electrolyser. The system's performance is evaluated based on the availability of low-temperature heat sources (90 to 150 °C) and medium-grade heat sources (190–280 °C). It is assessed for its capability to provide cooling only or in combination with hydrogen production, and their efficient utilization is compared from energetic, exergetic, environmental, and economic perspectives. To optimize this innovative system, a multi-objective evolutionary approach is employed, considering operating conditions as constraints and incorporating performance metrics for all criteria as objectives. Utilizing multi-objective decision-making techniques, diverse optimal points are pinpointed within the trade-off space, enabling real-time adaptive responses to changes in weather, energy demands, and biogas availability. The net present value analysis demonstrates that, assuming a hydrogen selling price of $6 per kilogram and cooling costs of $4 per KW the payback periods are calculated to be 2 years for the ORC-VCRS cycle and 3 years for the integrated Kalina topping with bottoming ORC cycle. The multi-objective optimization results indicate that the optimum operating conditions for the cascade system involve boiler, absorber, and condenser temperatures set at 196 °C, 110 °C, and 25 °C, respectively. Under these conditions, the system achieves an energy utilization ratio of 0.76, an exergy efficiency of 21.56 %, and a total cost of $58,677.

A multi-objective evolutionary approach for the electric vehicle charging stations problem

This paper presents a framework that integrates a mobility and traffic simulator with evolutionary algorithms to solve the location problem of electric vehicle charging infrastructure. The framework presented in this research is configured with a real-life scenario that considers a maximum number of charging stations and a specific number of electric vehicles that are driven according to a movement distribution model for the urban city under consideration. The concerning scenario is formulated as a multi-objective optimization problem and is approximated by two multi-objective evolutionary algorithms based on two evolutionary approaches for multi-objective optimization, namely, NSGA-II and MOEA/D-gen. In this way, the adopted algorithms search for optimal station locations and capacities in terms of travel time and the number of charging stations. The analysis of the results obtained by the multi-objective approaches used in the proposed framework is two-fold. On the one hand, solutions confirm that both objectives considered in this study are in conflict. That is, the better the solution for one objective is, the more deteriorated it is for the other. On the other hand, it was found that NSGA-II obtained a better approximation to the optimal solutions according to two multi-objective performance indicators. These results give an idea of the type of algorithms that could best solve the formulated problem in terms of efficiency and effectiveness and understand the behavior of evolutionary algorithms in the multi-objective electric vehicle charging stations problem. Therefore, this research traces the route to study different scenarios and analyze the characteristics of the problem. Furthermore, it opens the door to exploring other meta-heuristics and their components to deal with this problem adequately.

An adaptive dynamic community detection algorithm based on multi-objective evolutionary clustering

PurposeCommunity detection of dynamic networks provides more effective information than static network community detection in the real world. The mainstream method for community detection in dynamic networks is evolutionary clustering, which uses temporal smoothness of community structures to connect snapshots of networks in adjacent time intervals. However, the error accumulation issues limit the effectiveness of evolutionary clustering. While the multi-objective evolutionary approach can solve the issue of fixed settings of the two objective function weight parameters in the evolutionary clustering framework, the traditional multi-objective evolutionary approach lacks self-adaptability.Design/methodology/approachThis paper proposes a community detection algorithm that integrates evolutionary clustering and decomposition-based multi-objective optimization methods. In this approach, a benchmark correction procedure is added to the evolutionary clustering framework to prevent the division results from drifting.FindingsExperimental results demonstrate the superior accuracy of this method compared to similar algorithms in both real and synthetic dynamic datasets.Originality/valueTo enhance the clustering results, adaptive variances and crossover probabilities are designed based on the relative change amounts of the subproblems decomposed by MOEA/D (A Multiobjective Optimization Evolutionary Algorithm based on Decomposition) to dynamically adjust the focus of different evolutionary stages.

A multi-objective robust evolutionary optimization approach applied to the multivariate helical milling process of super duplex steel

This work presents an approach for multivariate modeling and multi-objective robust evolutionary optimization for processes presenting as inputs control and noise variables and as outputs a set of correlated responses regarding process performance and product quality. Factor analysis is applied to obtain a transformed output set that is uncorrelated and retains most part of the information of the original outputs. Models that are robust regarding noise variables relating this transformed output set with the control variables are obtained through robust parameter design. Response models in the function of process and noise variables are obtained through the least squares method. Mean and variance models as a function of process variables are obtained through error propagation. Robust mean square error models are obtained to model the bias and variance of each latent variable. Finally, multi-objective evolutionary optimization is performed. Three evolutionary algorithms were applied and compared through hypervolume. The approach is applied in the helical milling of super duplex stainless steel UNS S32760. Tool overhang length, hole depth, and lubri-cooling flow rate were set as noise factors. Cutting forces were monitored, while roughness and roundness were measured to evaluate process performance and product quality. The three factors, obtained to represent the six outputs, guaranteed dimensionality reduction and minimized the redundancy in output space. The mean square error models obtained to these three factors as a function of control variables minimize both bias and variance regarding noise factors. The AGE-MOEA multi-objective evolutionary algorithm obtained the best performance considering the hypervolume. Some solutions with high trade-off were selected through pseudo-weights to aid the decision-making.

Synergetic renewable generation allocation and 5G base station placement for decarbonizing development of power distribution system: A multi-objective interval evolutionary optimization approach

The growing penetration of 5G base stations (5G BSs) is posing a severe challenge to efficient and sustainable operation of power distribution systems (PDS) due to their huge energy demand and massive quantity. To tackle this issue, this paper proposes a synergetic planning framework for renewable energy generation (REG) and 5G BS allocation to support decarbonizing development of future PDS. The problem is formulated as a multi-objective interval optimization (MOIO) model, wherein the 5G BS siting and sizing are co-optimized with the REG capacity allocation and their operating strategies to minimize the total economic cost and carbon emissions of the PDS. The potential flexibility benefits achievable from 5G BS operation (as responsive load demands to PDS) are explicitly considered in the proposed planning formulation by accounting for the effects of both transmit power control and on-site energy storage battery dispatch. Moreover, we adopt an interval approach to capture the uncertainties involved in the system (including REG output, market prices, carbon emissions factor, and power/communication load demands). In order to solve the proposed MOIO problem efficiently, a constrained interval multi-objective optimization evolutionary algorithm based on decomposition (CIMOEA/D) is developed by combining constraint handling, interval-based Tchebycheff aggregation function and interval ensemble comparison, which could achieve Pareto optimal solutions and an interval-formed Pareto front to the problem with all the uncertain information fully preserved in the decision-making process. The proposed model is demonstrated on a modified IEEE 33-bus test case, and the simulation results verified the effectiveness of the suggested approach.

Explainable interactive evolutionary multiobjective optimization

We present a new approach to Interactive Evolutionary Multiobjective Optimization (IEMO) guided by a preference elicitation procedure inspired by artificial intelligence and designed in line with decision psychology. For a conflicting nature of objectives, a solution optimizing all objectives simultaneously does not exist and thus the best compromise solution is searched for. In the interactive search, preference elicitation phases alternate with optimization phases. In the IEMO procedure proposed in this paper, during the preference elicitation phase, the Decision Maker (DM) gets a sample of solutions from the current population and is asked to indicate relatively good solutions in this sample. Using the Dominance-based Rough Set Approach (DRSA), this information is summarized by “if ..., then ... ” decision rules which represent DM’s preferences. They are used in the next optimization phases of the IEMO to influence the crossover so as to converge towards the part of the Pareto front containing the best compromise solution. Besides guiding the search process, the decision rules can be read as arguments explaining the DM’s preferences. In this way, the proposed method implements the postulate of transparency and explainability expected from interactive procedures. The DM has a chance to understand how their answers given in the preference elicitation phase are translated into guidelines for the algorithm in the optimization phase. This is a distinctive aspect of what we call eXplainable Interactive Multiobjective Evolutionary optimization Approach (XIMEA). The presented XIMEA-DRSA method implements this approach. From the viewpoint of behavioral psychology, the decision rules support the DM to construct and learn their preferences in the course of evolutionary optimization. To check the efficiency of XIMEA-DRSA, we performed several experiments on continuous as well as combinatorial test problems, assuming the existence of an artificial DM that iteratively provides its preferences according to a known value function. The results prove that XIMEA-DRSA is converging to the most interesting part of the Pareto front, similar to an evolutionary algorithm that optimizes the value function of the artificial DM and, therefore, the latter is used as a benchmark.

Inferring test models from user bug reports using multi-objective search

Bug reports are used by software testers to identify abnormal software behaviour. In this paper, we propose a multi-objective evolutionary approach to automatically generate finite state machines (FSMs) based on bug reports written in natural language, to automatically capture incorrect software behaviour. These FSMs can then be used by testers to both exercise the reported bugs and create tests that can potentially reveal new bugs. The FSM generation is guided by a Multi-Objective Evolutionary Algorithm (MOEA) that simultaneously minimises three objectives: size of the models, number of unrealistic states (over-generalisation), and number of states not covered by the models (under-generalisation). We assess the feasibility of our approach for 10 real-world software programs by exploiting three different MOEAs (NSGA-II, NSGA-III and MOEA/D) and benchmarking them with the baseline tool KLFA. Our results show that KLFA is not practical to be used with real-world software, because it generates models that over generalise software behaviour. Among the three MOEAs, NSGA-II obtained significantly better results than the other two for all 10 programs, detecting a greater number of bugs for 90% of the programs. We also studied the differences in quality and model performance when MOEAs are guided by only two objectives rather than three during the evolution. We found that the use of under-approximation (or over-approximation) and size as objectives generates infeasible solutions. On the other hand, using as objectives over-approximation and under-approximation generates feasible solutions yet still worse than those obtained using all three objectives for 100% of the cases. The size objective acts as a diversity factor. As a consequence, an algorithm guided by all three objectives avoids local optima, controls the size of the models, and makes the results more diverse and closer to the optimal Pareto set.

Urban flood resilience in Kampung Melayu: A multi-objective evolutionary approach

The superblock of Kampung Melayu in Jakarta, Indonesia, is an urban morphology amalgamated by the environmental and infrastructure challenges raised by Jakarta’s inevitable urban growth. Low-income settlements like Kampung Melayu are particularly susceptible as a result of the city’s rapid and uncontrolled urban sprawl, erratic tropical weather, increasing sea levels and unparalleled environmental stresses. The proposed research utilises a multi-objective evolutionary algorithm (MOEA) for an in-depth investigation of the many relationships within the urban fabric to address these difficulties. The experiments demonstrate an alternate urban strategy for a flood-resilient Kampung that investigates the selection techniques coupled with the use of population-based algorithms. While preserving the irregularity that has been ingrained in the history of the urban form, the results address the environmental and demographic stresses of the urban village.

Evolutionary Multi-Objective Optimization in Searching for Various Antimicrobial Peptides [Feature

Antimicrobial peptides (AMPs), which are parts of the innate immune response found among all classes of life, are promising in broad-spectrum antibiotics and drug-resistant infection treatments. Although AMPs effectively kill bacteria, numerous AMPs widely distributed in the sequence space remain unknown to humans. Therefore, the de novo design of AMPs involves the exploration of vast sequence space to identify peptides with high antimicrobial activity and good diversity among the known AMPs. Computational intelligence approaches have successfully identified some AMPs; however, most of them fail to address the diversity of the obtained AMPs. This paper reports an evolutionary multi-objective approach for AMP design to optimize both the antimicrobial activity and diversity among identified AMPs. Our approach employs a deep learning model to predict a peptide’s antimicrobial activity and a niche sharing method to estimate a peptide’s density. Then, an evolutionary multi-objective algorithm is presented to simultaneously optimize the objectives of antimicrobial activity and diversity. The algorithm takes the advantage of a decomposition-based framework to search for AMPs with good diversity. These AMPs are collected by an elite archive during the evolution process. Moreover, a local search strategy is applied to enhance the quality of the identified AMPs. The experimental results show that the proposed approach outperforms the state-of-the-art designs in searching for various AMPs. The AMPs generated by the proposed approach have high antimicrobial activities and are distinct from each other and among the AMPs in the datasets.

A multi-objective evolutionary approach towards automated online controlled experiments

Due to the complexity of the web and mobile software applications, engineers rely heavily on A/B testing (i.e., online controlled experiments) to evaluate and measure the impact of the new changes. However, creating and evaluating A/B tests is a time-consuming, error-prone, and costly manual activity. For the problem of automating A/B testing, there is no solution in literature that evaluates A/B testing results as multi-criteria instead of single criteria. The current single criteria based methods overly simplify the A/B test results, as in the web/mobile application industry, a typical A/B test in practice has multiple metrics instead of one metric. In this paper, we describe the Variant Creation and Evaluation (VCE) problem in A/B testing and propose MOVSW (Multi-Objective Variant Selection Wrapper) that utilizes Multi-Objective Evolutionary Algorithm (MOEA) to automate the process of creating and selecting variants as launch candidates in A/B testing. Given a set of parameters we would like to optimize, the control group in the A/B test uses the existing values of these parameters in the system. The MOVSW automatically creates variants of different parameter values and conducts A/B testing to evaluate these variants against the control group with existing parameter values based on multiple measurements. The outputs of the MOVSW are a list of non-dominated candidate variants (also known as Pareto optimal set) that lead to potential improvements on key measures according to the A/B test results. These candidate variants could be then reviewed by stakeholders to determine the final variant to be launched. We designed and conducted a case study to comprehensively evaluate the effectiveness of five MOEAs and three baseline methods on MOVSW for automating A/B tests, using user clicks logging of news articles from the Addressa Dataset. Our results indicate that MOEA/D is the most stable method with fast convergence compared with other MOEAs, produces more high-quality solutions with high precision than the single-objective methods, and thus is promising to be used in the VCE problem of A/B testing.

Evolutionary Multiobjective Design Approach for Robust Balancing of the Shaking Force, Shaking Moment, and Torque under Uncertainties: Application to Robotic Manipulators

In this paper, the environmental uncertainties are taken into account when designing a robotic manipulator to balance the shaking force, shaking moment, and torque. The proposed robust balancing design approach does not consider the probability distributions of the uncertainties and is addressed without dependence on specific trajectories. This is expressed as a nonlinear constrained multiobjective optimization problem in which the nominal performance in the time-independent terms of the shaking force balancing, the shaking moment balancing, and the torque delivery, as well as their three sensitivities to uncertainties, are simultaneously optimized to provide a set of link shapes that match link mass distributions in a single stage. The proposal is applied to a three-degree-of-freedom serial-parallel manipulator, and the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) is used to solve the associated problem. Comparative results with other design approaches reveal that the selected design achieves a suitable tradeoff in balancing the shaking force balancing, the shaking moment balancing, and the torque delivery and their sensitivities, leading to a reduction in their values and variations under mass changes in the manipulator end-effector with different operating conditions (tasks).

A Multiobjective Evolutionary Approach for Solving Large-Scale Network Reconstruction Problems via Logistic Principal Component Analysis.

Currently, the problem of uncovering complex network structure and dynamics from time series is prominent in many fields. Despite the recent progress in this area, reconstructing large-scale networks from limited data remains a tough problem. Existing works treat connections of nodes as continuous values, leaving a challenge of setting a proper cut-off value to distinguish whether the connections exist or not. Besides, their performances on large-scale networks are far from satisfactory. Considering the reconstruction error and sparsity as two objectives, this article proposes a subspace learning-based evolutionary multiobjective network reconstruction algorithm, called SLEMO-NR, to solve the aforementioned problems. In the evolutionary process, we assume that binary-coded individuals obey the Bernoulli distribution and can use the probability and natural parameter as alternative representations. Moreover, our approach utilizes the logistic principal component analysis (LPCA) to learn a subspace containing the features of the network structure. The offspring solutions are generated in the learned subspace and then can be mapped back to the original space via LPCA. Benefitting from the alternative representations, a preference-based local search operator (PLSO) is proposed to concentrate on finding solutions approximate to the true sparsity. The experimental results on synthetic networks and six real-world networks demonstrate that, due to the well-learned network structure subspace and the preference-based strategy, our approach is effective in reconstructing large-scale networks compared to six existing methods.

MULTIOBJECTIVE EVOLUTIONARY METAHEURISTIC APPROACH TO THE CONSTRAINED PORTFOLIO OPTIMIZATION PROBLEM

In this paper, we propose a multi-objective evolutionary metaheuristic approach based on the Pareto Ant Colony Optimization (P-ACO) metaheuristic and the non-dominated genetic sorting algorithms (NSGA II and NSGA III) to solve a bi-objective portfolio optimization problem. P-ACO is used to select the best assets composing the efficient portfolio. Then, NSGA II and NSGA III are separately used to find the proportional weights of the budget allocated to the selected portfolio. The results we obtained by these two algorithms were compared to designate the best performing algorithm. Finally, we performed another comparison between our results and those of an exact method used for the same problem. The numerical experiments performed on a set of instances from the literature revealed that the combination of the ant colony optimization metaheuristic and the NSGA III genetic algorithm that we proposed most often gave much better results than both the combination of the ant colony optimization metaheuristic and NSGA II on the one hand and the iterative approach on the other hand.

A Multiobjective Evolutionary Approach for Solving the Multi-Area Dynamic Economic Emission Dispatch Problem Considering Reliability Concerns

Economic dispatch (ED) problems, especially in multi-area power networks, have been challenging concerns for power system operators for several decades. In this paper, we introduce a novel approach for solving the multiobjective multi-area dynamic ED (MADED) problem in the presence of practical constraints such as valve-point effect (VPE), prohibited operating zone (POZ), multi-fuel operation (MFO), and ramp rate (RR) limitations. Different objective functions including energy not supplied (ENS), generation costs, and emissions are investigated. The reliability objective, which has been less studied in economic dispatch area, distinguishes the proposed study from other studies. A compromise has been made from economic and reliability points of view. The MADED problem in the power system is inherently a complex and nonlinear problem, considering the operational constraint increments and the intricacy of the problem. Hence, the modified grasshopper optimization (MGO) algorithm based on a chaos mechanism is presented to prevent being trapped in local optima. The proposed method is tested on two systems including a 10 unit, 3-zone test system and a 40-unit 3-zone test system, and then, the outcomes are compared with those of other evolutionary techniques such as gray wolf optimization (GWO) and modified honey bee mating optimization (MHBMO). The simulation results demonstrate that the suggested strategy is successful in resolving both single-objective and multiobjective MADED problems.

Solving Periodic Investment Portfolio Selection Problems by a Data-Assisted Multiobjective Evolutionary Approach

Classic portfolio selection problems mainly focus on high-risk financial markets with tradeoffs between returns and risk. However, more risk-averse investors pursue long-term portfolio planning with the objectives of maximizing final returns and maximizing flexibility. This article addresses a new type of the portfolio problem, called periodic investment portfolio selection problems (PIPSPs), in which investors periodically allocate resources to financial products with different periods. A multiobjective model for PIPSPs is first presented. With a mechanism for utilizing the data generated during the implementation of multiobjective evolutionary algorithms (MOEAs), a data-assisted MOEA (DA-MOEA) is proposed to solve PIPSPs. The main idea of a DA-MOEA is to combine a MOEA with a data-assisted process that consists of three components: 1) feature construction; 2) data fusion model development; and 3) obtained information utilization. To solve the addressed PIPSPs, two versions of DA-MOEAs with baselines of nondominated sorting and decomposition-based mechanisms are implemented, namely, the data-assisted NSGA-II (DA-NSGA-II) and data-assisted MOEA/D (DA-MOEA/D). In the developed DA-MOEAs for PIPSPs, a feature construction process and a data fusion model are well designed for mining data with different formats. To validate the algorithms, two sets of test instances are generated. The experimental results demonstrate the efficacy of the data-assisted process. Furthermore, the effects of the algorithm components, such as the data source sizes, information types, and information utilization strategies, are investigated.

Multi-objective evolutionary approach based on K-means clustering for home health care routing and scheduling problem

This paper suggests a bi-objective mathematical model, called HHC-MOVRPTW for the home health care routing and scheduling problem. HHC-MOVRPTW aims to minimize the overall service time while minimizing the total tardiness compared to the visiting time preferences. The considered problem is a NP hard, combining the Personnel Scheduling Problem and the Vehicle Routing Problem with time windows. Three solution approaches are proposed to solve it. Firstly, the HHC-MOVRPTW is solved with a non-scalar method, called Lexicographical method, in order to obtain a first solution for the problem. It is subsequently solved by two well-known multi-objective evolutionary algorithms, namely the Non-dominated Sorting Genetic Algorithm (NSGA-II) and the Strength Pareto Evolutionary Algorithm (SPEA2). Afterwards, a new hybridization approach, combined the evolutionary algorithm with K-means clustering technique is also suggested to improve the quality of the obtained Pareto sets. This is achieved by dividing the population of NSGA-II and SPEA-2 into sub-populations (clusters) and all sub-solutions have to be combined to find the final Pareto front. The computational experiments are performed using Solomon’s bench- mark instances. Thus, results prove the effectiveness of the proposed approaches and their suitability with the problem.