Late Acceptance Hill Climbing Research Articles

Cancer classification using RNA sequencing gene expression data based on Game Shapley local search embedded binary social ski-driver optimization algorithms

Cancer remains a significant health concern due to its high mortality rates. Recent decades have witnessed substantial progress in cancer research, driven by advancements in high throughput sequencing technology and the application of diverse machine learning (ML) methods, particularly in the analysis of gene expression data. However, the proliferation of high-dimensional datasets, such as RNA-seq data, underscores the need for more robust ML techniques capable of efficiently handling large volumes of data to enable accurate treatment decisions. This paper introduces a novel hybrid feature selection (FS) algorithm, termed Game kernel SHapley Additive exPlanations (kSHAP), which combines with binary Social Ski Driver (bSSD), Adaptive Beta Hill Climbing (ABHC) and Late Acceptance Hill Climbing (LAHC) algorithms. The study comprehensively investigates three novel FS algorithms—kSHAP-bSSD, kSHAP-ABHC, and kSHAP-LAHC for cancer classification tasks using RNA sequencing (RNA-seq) datasets. An experiment conducted on five well-established RNA-seq cancer datasets: Lung Adenocarcinoma (LUAD), Stomach Adenocarcinoma (STAD), Breast Invasive Carcinoma (BRCA), lung squamous cell carcinoma (LUSC) and uterine corpus endometrial carcinoma (UCEC). The objective is to enhance cancer classification accuracy, robustness, and scalability using RNA-seq datasets. Additionally, the study evaluates six classifiers—Autoencoder (AE), Support Vector Machine (SVM), K-Nearest Neighbors (KNN), Naive Bayes (NB), Neural Network (NN), and Random Forest (RF) with AE consistently out performing others. Evaluation metrics include accuracy, recall, precision, box plot, F1-score, radar plot, confusion matrix, ROC and statistical analysis. Our approach is compared against recent state-of-the-art FS algorithms, showing improvements in gene selection and classification accuracy. The kSHAP-bSSD demonstrates superior performance across all datasets compared to traditional methods, achieving an accuracy rate of 99.9% in LUAD and exhibiting higher accuracy rates and robustness in STAD, BRCA, LUSC, and UCEC datasets. Assessment across multiple metrics affirms the superiority of kSHAP-bSSD and kSHAP-ABHC combinations, underscoring their effectiveness in cancer classification tasks.

Dynamic mutation late acceptance hill climbing aided red fox optimization for metabolomic biomarkers selection from lung cancer patient sera

The accurate selection of serum metabolomic biomarkers for early lung cancer screening remains a significant challenge in the clinical context. Consequently, this study introduces the Red Fox Optimization (RFO) that integrates Dynamic Mutation Late Acceptance Hill Climbing (DM-LAHC), with the aim of selecting a panel of serum metabolomic biomarkers suitable for distinguishing between benign and malignant pulmonary nodules. The key innovation is the dynamic adjustment of the mutation probability in the Late Acceptance Hill Climbing algorithm, which greatly enhances the local search capabilities. And the RFO's reproduction mechanism has been improved through the utilization of a more efficient interpolation form. The biomarker selection model employs a multi-objective fitness function that takes into account both accuracy and quantity. After this, the optimal model yielded a biomarker panel, including Inosine, Hippuric acid, Alanine, and other metabolites. This model demonstrates outstanding performance on an independent test dataset, achieving a fitness value of 0.9136, an AUC (Area Under the Curve) of 0.9926, a sensitivity of 0.9643, and a specificity of 0.9412. Furthermore, the clinical net benefit is highlighted across various risk thresholds by decision curve analysis. These results underscore the significance of DM-LAHC aided RFO in the selection of serum metabolomic biomarkers for lung cancer. The supporting source codes of this work can be found at: https://github.com/zzl2022/DM-LAHC-aided-RFO.

Matheuristics vs. metaheuristics for joint lot-sizing and dynamic pricing problem with nonlinear demands

This paper considers a joint dynamic pricing and production planning decisions problem for a profit-maximizing firm that produces and sells multiple products. The objective is to develop a coordinated decision approach for multi-product pricing and lot sizing decisions for a manufacturer considering a limited production capacity. The demand for each product is assumed to be iso-elastic and integrates the complementarity and the substitution effects between the products. First, the problem is formulated as a non-convex mixed integer nonlinear programming model (MINLP) incorporating capacity constraints, setup costs, and nonlinear demand functions. Then, since the model is nonlinear and non-convex, a set of approximate approaches based on the Genetic algorithm, Late Acceptance Hill Climbing and Simulated Annealing methods are designed to solve this problem. Based on this study, the performances of two variants of approximate methods: matheuristics and metaheuristics are discussed and analyzed. The extensive experimental study, performed on real-world inspired instances, shows that matheuristic methods with setup-variables encoding scheme outperform the rest of the methods. The research outcomes show that coordinating a decision-making process by optimizing both prices and production plans simultaneously can result in significant profit for a company. However, one must consider the joint effect of the parameters of the demand function as well as the impact of the production capacity. This comprehensive understanding enables the company to avoid excessive investments in less lucrative products with lower sales potential, thereby ensuring resource allocation aligns with profitability and market demand.

Research of a Multi-Level Organization Human Resource Network Optimization Model and an Improved Late Acceptance Hill Climbing Algorithm

Complex hierarchical structures and diverse personnel mobility pose challenges for many multi-level organizations. The difficulty of reasonable human resource planning in multi-level organizations is mainly caused by ignoring the hierarchical structure. To address the above problems, firstly, a multi-level organization human resource network optimization model is constructed by representing the turnover situation of multi-level organizations in a dimensional manner as a multi-level network. Secondly, we propose an improved late acceptance hill climbing based on tabu and retrieval strategy (TR-LAHC) and designed two intelligent optimization operators. Finally, the TR-LAHC algorithm is compared with other classical algorithms to prove that the algorithm provides the best solution and can effectively solve the personnel mobility planning problem in multi-level organizations.

Late acceptance hill climbing aided chaotic harmony search for feature selection: An empirical analysis on medical data

In today’s era of data-driven digital society, there is a huge demand for optimized solutions that essentially reduce the cost of operation, thereby aiming to increase productivity. Processing a huge amount of data, like the Microarray based gene expression data, using machine learning and data mining algorithms has certain limitations in terms of memory and time requirements. This would be more concerning, when a dataset comes with redundant and non-important information. For example, many report-based medical datasets have several non-informative attributes which mislead the classification algorithms. To this end, researchers have been developing several feature selection algorithms that try to discard the redundant information from the raw datasets before feeding them to machine learning algorithms. Metaheuristic based optimization algorithms provide an excellent option to solve feature selection problems. In this paper, we propose a music-inspired harmony search (HS) algorithm based wrapper feature selection method. At the beginning, we use a chaotic mapping to initialize the population of the HS algorithm in order to better coverage of the search space. Further to complement the inferior exploitation of the HS algorithm, we integrate it with the Late Acceptance Hill Climbing (LAHC) method. Thus the combination of these two algorithms provides a good balance between the exploration and exploitation of the HS algorithm. We evaluate the proposed feature selection method on 15 UCI datasets and the obtained results are found to be better than many state-of-the-art methods both in terms of the classification accuracy and the number of features selected. To evaluate the effectiveness of our algorithm, we utilize a combination of precision, recall, F1 score, fitness value, and execution time as performance indicators. These metrics enable us to obtain a comprehensive assessment of the algorithm’s abilities and limitations. We also apply our method on 3 microarray based gene expression datasets used for prediction of cancer to ensure the scalability and robustness as a feature selection method in real-life scenarios. In addition to this, we test our approach using the COVID-19 dataset, and it performs better than several metaheuristic based optimization techniques.

Improved Artificial Bee Colony Algorithm-based Path Planning of Unmanned Aerial Vehicle Using Late Acceptance Hill Climbing

Improved Artificial Bee Colony Algorithm-based Path Planning of Unmanned Aerial Vehicle Using Late Acceptance Hill Climbing

FastStereoNet: A Fast Neural Architecture Search for Improving the Inference of Disparity Estimation on Resource-Limited Platforms

Convolutional neural networks (CNNs) provide the best accuracy for disparity estimation. However, CNNs are computationally expensive, making them unfavorable for resource-limited devices with real-time constraints. Recent advances in neural architectures search (NAS) promise opportunities in automated optimization for disparity estimation. However, the main challenge of the NAS methods is the significant amount of computing time to explore a vast search space [e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.6\times 10^{29}$ </tex-math></inline-formula> ] and costly training candidates. To reduce the NAS computational demand, many proxy-based NAS methods have been proposed. Despite their success, most of them are designed for comparatively small-scale learning tasks. In this article, we propose a fast NAS method, called FastStereoNet, to enable resource-aware NAS within an intractably large search space. FastStereoNet automatically searches for hardware-friendly CNN architectures based on late acceptance hill climbing (LAHC), followed by simulated annealing (SA). FastStereoNet also employs a fine-tuning with a transferred weights mechanism to improve the convergence of the search process. The collection of these ideas provides competitive results in terms of search time and strikes a balance between accuracy and efficiency. Compared to the state of the art, FastStereoNet provides <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5.25\times $ </tex-math></inline-formula> reduction in search time and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$44.4\times $ </tex-math></inline-formula> reduction in model size. These benefits are attained while yielding a comparable accuracy that enables seamless deployment of disparity estimation on resource-limited devices. Finally, FastStereoNet significantly improves the perception quality of disparity estimation deployed on field-programmable gate array and Intel Neural Compute Stick 2 accelerator in a significantly less onerous manner.

Enhancing Software Reliability and Fault Detection Using Hybrid Brainstorm Optimization-Based LSTM Model

An essential attribute of software quality is software reliability. To achieve higher reliability, the testing phase with detected and corrected flaws is incorporated in the software development. The fault correction process (FCP) includes the fault detection process (FDP) to develop the software reliability growth model (SRGM). This is difficult to integrate because due to several reasons, including the effects of staffing levels and the interdependence of faults. It limits the applicability of the analytical model. Because of the adoption of data-driven methodologies such as Artificial Intelligence (AI) technology, no precise FCP and FDP assumptions are necessary. In this article, we proposed a hybrid long short-term memory (LSTM) with BrainStorm Optimization and Late Acceptance Hill Climbing (BSO-LAHC) algorithm of a stepwise prediction model for software fault detection and correction. The fault detection and correction procedure has great influence by considering the testing effort. While compared to the existing methods, the proposed hybrid with the BSO-LAHC algorithm demonstrated superior results by using Firefox and bug tracking system Bugzilla datasets. The proposed model’s effectiveness is confirmed via empirical study. Based on the Bugzilla and firefox datasets, the proposed mean square error performance is 1.92 and 21.44 respectively. Additionally, the proposed method is less expensive and takes less time to execute. In Bugzilla version 5.0.4, releases 2 and 3 had a determination coefficient of 99.2% and 98.9%, respectively. The FCP is 27% more effective than previous approaches, and the FDP is 32% more effective.

A Sequential Handwriting Recognition Model Based on a Dynamically Configurable CRNN.

Handwriting recognition refers to recognizing a handwritten input that includes character(s) or digit(s) based on an image. Because most applications of handwriting recognition in real life contain sequential text in various languages, there is a need to develop a dynamic handwriting recognition system. Inspired by the neuroevolutionary technique, this paper proposes a Dynamically Configurable Convolutional Recurrent Neural Network (DC-CRNN) for the handwriting recognition sequence modeling task. The proposed DC-CRNN is based on the Salp Swarm Optimization Algorithm (SSA), which generates the optimal structure and hyperparameters for Convolutional Recurrent Neural Networks (CRNNs). In addition, we investigate two types of encoding techniques used to translate the output of optimization to a CRNN recognizer. Finally, we proposed a novel hybridized SSA with Late Acceptance Hill-Climbing (LAHC) to improve the exploitation process. We conducted our experiments on two well-known datasets, IAM and IFN/ENIT, which include both the Arabic and English languages. The experimental results have shown that LAHC significantly improves the SSA search process. Therefore, the proposed DC-CRNN outperforms the handcrafted CRNN methods.

Hybrid Bird Mating Optimizer With Single-Based Algorithms for Combinatorial Optimization Problems

Bird mating optimizer (BMO) is a population-based metaheuristic that has been recently extended to solve combinatorial optimization problems. Even though the algorithm shows promising performance in solving combinatorial optimization problems, it suffers from slow convergence and poor efficiency which leads to poor solution quality for some problem instances. This is due to the limited capability of BMO in exploiting the search space and identifying more promising regions. Therefore, in this work we propose a hybrid BMO with five single-based metaheuristics: hill-climbing, late acceptance hill-climbing, simulated annealing, iterated greedy heuristic and variable iterated greedy heuristic. Each of these algorithms is used inside the BMO to exploit the search space, and improve the quality of solution generated from the BMO population. This work also compares which one of these five is better for hybridizing with BMO. The performance of these algorithms is tested on two combinatorial problems: travelling salesman problem and berth allocation problem. Experimental results demonstrate that the hybrid algorithm is superior to BMO when applied to both problems and it improved the BMO by 1.13% for BAP and by 4.13% for TSP. Furthermore, the hybrid algorithm is able to match the best-known results for most of the instances. In addition, the proposed hybrid approaches perform well over both tested domains and obtain competitive results when compared to the best-known results that have previously been presented in the scientific literature.

Extending and Solving the Refrigerated Routing Problem

In recent years, cold food chains have shown an impressive growth, mainly due to customers life style changes. Consequently, the transportation of refrigerated food is becoming a crucial aspect of the chain, aiming at ensuring efficiency and sustainability of the process while keeping a high level of product quality. The recently defined Refrigerated Routing Problem (RRP) consists of finding the optimal delivery tour that minimises the fuel consumption for both the traction and the refrigeration components. The total fuel consumption is related, in a complex way, to the distance travelled, the vehicle load and speed, and the outdoor temperature. All these factors depend, in turn, on the traffic and the climate conditions of the region where deliveries take place and they change during the day and the year. The original RRP has been extended to take into account also the total driving cost and to add the possibility to slow down the deliveries by allowing arbitrarily long waiting times when this is beneficial for the objective function. The new RRP is formulated and solved as both a Mixed Integer Programming and a novel Constraint Programming model. Moreover, a Local Search metaheuristic technique (namely Late Acceptance Hill Climbing), based on a combination of different neighborhood structures, is also proposed. The results obtained by the different solution methods on a set of benchmarks scenarios are compared and discussed.

Path Planning for Indoor UAV Using A* and Late Acceptance Hill Climbing Algorithms Utilizing Probabilistic Roadmap

The main objective of an unmanned aerial vehicle (UAV) path planning is to generate a flight path that links a start point to an endpoint in an indoor space avoiding obstacles. Path planning is essential for many real-life applications such as an autonomous car, surveillance mission, farming robots, unmanned aerial vehicles package delivery, space exploration, and many others. To create an optimal path, we need to adopt a specific criterion to minimize the distance the UAV must travel such as the Euclidean distance. In this paper, we provide our initial idea of creating an optimal path for indoor UAV using both A* and the Late Acceptance Hill Climbing (LAHC) algorithms. We are adopting an indoor search environment with various complexity and utilize the Probabilistic Roadmap algorithm (PRM) as a search space for both algorithms. The basic idea following PRM is to generate random sample points in the space and search these points for an optimal path. The developed results show that the LAHC algorithm outperforms the A* algorithm.

Hybrid Brain Storm Optimization algorithm and Late Acceptance Hill Climbing to solve the Flexible Job-Shop Scheduling Problem

The Brain Storming Optimization (BSO) algorithm is a novel swarm intelligent algorithm that simulates the brainstorming process of humans. This paper presents the BSO algorithm as a solution to the Flexible Job-Shop Scheduling Problem (FJSSP). In aim to improve the global search of the BSO algorithm, a new updating strategy is proposed to adaptively perform several selection methods and neighborhood structures. Furthermore, BSO algorithm has good ability in exploring the search space by clustering the solutions and searching in each cluster independently, thus leading to slow convergence speed, to enhance the local intensification capability and to overcome the slow convergence of the BSO algorithm, we introduce Late Acceptance Hill Climbing (LAHC) with three neighborhoods to the BSO algorithm. Extensive computational experiments were carried out on four well-known benchmarks for FJSSP, and the performance of the BSO algorithm was compared with that of the proposed algorithm. The results demonstrate that the proposed algorithm outperforms the BSO algorithm. Furthermore, the proposed approach overcomes the best-known algorithms in some datasets and it is comparable with these algorithms in other datasets.

Late Acceptance Hill-Climbing Matheuristic for the General Lot Sizing and Scheduling Problem with Rich Constraints

This paper considers the general lot sizing and scheduling problem with rich constraints exemplified by means of rework and lifetime constraints for defective items (GLSP-RP), which finds numerous applications in industrial settings, for example, the food processing industry and the pharmaceutical industry. To address this problem, we propose the Late Acceptance Hill-climbing Matheuristic (LAHCM) as a novel solution framework that exploits and integrates the late acceptance hill climbing algorithm and exact approaches for speeding up the solution process in comparison to solving the problem by means of a general solver. The computational results show the benefits of incorporating exact approaches within the LAHCM template leading to high-quality solutions within short computational times.

Performance Analyses of Graph Heuristics and Selected Trajectory Metaheuristics on Examination Timetable Problem

Examination timetabling problem is hard to solve due to its NP-hard nature, with a large number of constraints having to be accommodated. To deal with the problem effectually, frequently heuristics are used for constructing feasible examination timetable while meta-heuristics are applied for improving the solution quality. This paper presents the performances of graph heuristics and major trajectory metaheuristics or S-metaheuristics for addressing both capacitated and un-capacitated examination timetabling problem. For constructing the feasible solution, six graph heuristics are used. They are largest degree (LD), largest weighted degree (LWD), largest enrolment degree (LE), and three hybrid heuristic with saturation degree (SD) such as SD-LD, SD-LE, and SD-LWD. Five trajectory algorithms comprising of tabu search (TS), simulated annealing (SA), late acceptance hill climbing (LAHC), great deluge algorithm (GDA), and variable neighborhood search (VNS) are employed for improving the solution quality. Experiments have been tested on several instances of un-capacitated and capacitated benchmark datasets, which are Toronto and ITC2007 dataset respectively. Experimental results indicate that, in terms of construction of solution of datasets, hybridizing of SD produces the best initial solutions. The study also reveals that, during improvement, GDA, SA, and LAHC can produce better quality solutions compared to TS and VNS for solving both benchmark examination timetabling datasets.

Performance Analyses of Graph Heuristics and Selected Trajectory Metaheuristics on Examination Timetable Problem

Examination timetabling problem is hard to solve due to its NP-hard nature, with a large number of constraints having to be accommodated. To deal with the problem effectually, frequently heuristics are used for constructing feasible examination timetable while meta-heuristics are applied for improving the solution quality. This paper presents the performances of graph heuristics and major trajectory metaheuristics or S-metaheuristics for addressing both capacitated and un-capacitated examination timetabling problem. For constructing the feasible solution, six graph heuristics are used. They are largest degree (LD), largest weighted degree (LWD), largest enrolment degree (LE), and three hybrid heuristic with saturation degree (SD) such as SD-LD, SD-LE, and SD-LWD. Five trajectory algorithms comprising of tabu search (TS), simulated annealing (SA), late acceptance hill climbing (LAHC), great deluge algorithm (GDA), and variable neighborhood search (VNS) are employed for improving the solution quality. Experiments have been tested on several instances of un-capacitated and capacitated benchmark datasets, which are Toronto and ITC2007 dataset respectively. Experimental results indicate that, in terms of construction of solution of datasets, hybridizing of SD produces the best initial solutions. The study also reveals that, during improvement, GDA, SA, and LAHC can produce better quality solutions compared to TS and VNS for solving both benchmark examination timetabling datasets.

Optimization of a multi-constraint two-sided assembly line balancing problem using an improved imperialist competitive algorithm

PurposeAssembly is the last step in manufacturing processes. The two-sided assembly line balancing problem (TALBP) is a typical research focus in the field of combinatorial optimization. This paper aims to study a multi-constraint TALBP-I (MC-TALBP-I) that involves positional constraints, zoning constraints and synchronism constraints to make TALBP more in line with real production. For enhancing quality of assembly solution, an improved imperialist competitive algorithm (ICA) is designed for solving the problem.Design/methodology/approachA mathematical model for minimizing the weighted sum of the number of mated-stations and stations is established. An improved ICA is designed based on a priority value encoding structure for solving MC-TALBP-I.FindingsThe proposed ICA was tested by several benchmarks involving positional constraints, zoning constraints and synchronism constraints. This algorithm was compared with the late acceptance hill-climbing (LAHC) algorithm in several instances. The results demonstrated that the ICA provides much better performance than the LAHC algorithm.Practical implicationsThe best solution obtained by solving MC-TALBP-I is more feasible for determining the real assembly solution than the best solution obtained by solving based TALBP-I only.Originality/valueA novel ICA based on priority value encoding is proposed in this paper. Initial countries are generated by a heuristic method. An imperialist development strategy is designed to improve the qualities of countries. The effectiveness of the ICA is indicated through a set of benchmarks.

Late Acceptance Hill Climbing Based Social Ski Driver Algorithm for Feature Selection

Feature selection (FS) is mainly used as a pre-processing tool to reduce dimensionality by eliminating irrelevant or redundant features to be used for a machine learning or data mining algorithm. In this paper, we have introduced binary variant of a recently proposed meta-heuristic algorithm called Social Ski Driver (SSD) optimization. To the best of our knowledge, SSD has not been used yet in the domain of FS. Two binary variants of SSD are proposed using S-shaped and V-shaped transfer functions. Besides, the exploitation ability of SSD is improved by using a local search method, called Late Acceptance Hill Climbing (LAHC). The hybrid meta-heuristic is then converted to binary version by using said transfer functions. The proposed methods are applied on 18 standard UCI datasets and compared with 15 state-of-the-art FS methods. Also to check the robustness of the proposed method, we have applied it to 3 high dimensional microarray datasets and compared with 6 state-of-the-art methods. Achieved results confirm the superiority of the proposed methods compared to other meta-heuristic wrapper based FS methods considered here. Source code of this work is available at https://github.com/consigliere19/SSD-LAHC.

A Hybrid Meta-Heuristic Feature Selection Method for Identification of Indian Spoken Languages From Audio Signals

With the recent advancements in the fields of machine learning and artificial intelligence, spoken language identification-based applications have been increasing in terms of the impact they have on the day-to-day lives of common people. Western countries have been enjoying the privilege of spoken language recognition-based applications for a while now, however, they have not gained much popularity in multi-lingual countries like India owing to various complexities. In this paper, we have addressed this issue by attempting to identify different Indian languages based on various well-known features like Mel-Frequency Cepstral Coefficient (MFCC), Linear Prediction Coefficient (LPC), Discrete Wavelet Transform (DWT), Gammatone Frequency Cepstral Coefficient (GFCC) as well as a few deep learning architecture based features like i-vector and x-vector extracted from the audio signals. After comparing the initial results, it is observed that the combination of MFCC and LPC produces the best results. Then we have developed a new nature-inspired feature selection (FS) algorithm by hybridizing Binary Bat Algorithm (BBA) with Late Acceptance Hill-Climbing (LAHC) to select the optimal subset from the said feature vectors in order to reduce the model complexity and help it train faster. Using Random Forest (RF) classifier, we have achieved an accuracy of 92.35% on Indic TTS database developed by IIT-Madras, and an accuracy of 100% on the Indic Speech database developed by the Speech and Vision Laboratory (SVL) IIIT-Hyderabad. The proposed algorithm is also found to outperform many standard meta-heuristic FS algorithms. The source code of this work is available at: https://github.com/CodeChef97dotcom/Feature-Selection

A bi-objective transportation-location arc routing problem

ABSTRACT Location and routing problems belong to well-known optimization ones, whose combinations have merged as a location-routing problem (LRP). Arc routing has recently become a popular interest of researchers. Accordingly, the location-arc routing problem (LARP) is defined. This paper deals with a location-arc routing problem when there are transportation decisions from suppliers to established depots (TLARP). It is addressed by developing a bi-objective mathematical model to optimize the total costs and makespan. An augmented ε-constraint algorithm is employed to find the true Pareto solutions. Then two meta-heuristics are considered: non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective late acceptance hill-climbing (MOLAHC) algorithm. There are four heuristics regarding NSGA-II, its combination with MOLAHC, and having/not having a local search (LS). Results demonstrate that NSGA-II+LS is the best and hybrid+LS is the second best heuristics while the solving time of hybrid+LS is significantly less than the time of NSGA-II+LS.