DOCSLIB.ORG

Bees Algorithm for Vehicle Routing Problems with Time Windows

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bees Algorithm for Vehicle Routing Problems with Time Windows International Journal of Machine Learning and Computing, Vol. 8, No. 3, June 2018 Bees Algorithm for Vehicle Routing Problems with Time Windows Malek Alzaqebah, Sana Jawarneh, Hafiz Mohd Sarim, and Salwani Abdullah annealing, or population-based approaches such as ant colony Abstract—This paper presents the bees algorithm for vehicle optimization and genetic algorithms [3]. routing problems within time windows (VRPTW). The VRPTW Population-based approaches are classified into two aims to determine the optimal route for a number of vehicles categories, which are evolutionary algorithms and swarm when serving a set of customers within a predefined time interval (the time window). The objective in VRPTW is to intelligence algorithms [4]. This classification is based upon minimize overall transportation cost. Various heuristic and natural phenomena being modeled by the algorithms. metaheuristic approaches have been developed in literature to Evolutionary algorithms apply the theory of evolution to produce high-quality solutions for this problem because of its create new species [3]. By contrast, swarm intelligence (SI) high complication rate and extensive implementation in real-life algorithms rely on metaheuristics that mimic collective applications. This research investigates the use of bee algorithms behavior of problem-solving processes in self-organized (BA) for VRPTW and identifying the strengths and weaknesses. systems [5]. The interactions between agents in social Index Terms—Foraging behaviour, bees algorithm, vehicle colonies with their environment leads to the collective routing problem with time windows. intelligence [6]. SI features have inspired some researchers to develop algorithms for VRPTW, including the ant colony optimization algorithms [7], bee colony optimization (BCO) I. INTRODUCTION algorithms [8] and the artificial bee colony (ABC) algorithms Vehicle routing problems (VRP) have an important role in [9], [10], to name a few examples. the domains of transportation, delivery, and logistics. Hence, With regards to swarm intelligence, researchers are numerous research works have been undertaken to study VRP concerned with developing algorithms that model the since 1959 [1]. The work presented by Dantzig and Ramser in behavior of honeybees. Honeybee algorithms are categorised 1959 proposed the problem as a generalized Travelling based on the behavior of bees [11]: queen, marriage, and Salesman Problem (TSP). Later, a huge number of studies foraging bee. Examples of foraging behavior algorithms can have been conducted on number of VRP divisions (e.g. VRP be found in ABC [12], bees algorithm (BA) [13], and bee with Time Windows (VRPTW), VRP with Pick-Up and colony optimization (BCO) [14], [15]. In this paper, Bees Delivery (VRPPD), Multiple Depot VRP (MDVRP), algorithm (BA) is investigated for VRPTW and some MDVRP with Time Windows (MD-VRPTW), and strengths and weaknesses are identified. Capacitated VRP (CVRP)). The remainder of the paper is organized as follows: in the VRPTW is an NP-hard problem, which is concerned with next section VRPTW is described and its formulation with determining the best routing of a set of limited capacity some relevant-literature review is provided. In the third vehicles between a central depot and a number of scattered section, the proposed algorithm is presented. The fourth customers, where customers must be visited within predefined section provides an extensive computational study comparing time duration (the time window). the results of BA and with the best-known approaches. Finally, Several approaches have been developed for VRPTW. the paper is concluded and some future research directions are Solomon [2] first introduced heuristics to solve this problem. listed. In recent years, metaheuristics has become increasingly popular. Metaheuristics can be classified either as single-based approaches such as tabu search and simulated II. THE VEHICLE ROUTING PROBLEMS WITH TIME WINDOWS This study focuses on the VRPTW in order to understand its complexity and maximize its advantages in real life. In recent years, the VRPTW has gained considerable research attention in network optimization [16]. Ellabib et al. [17] defined the VRPTW as a generalisation Manuscript received March 15, 2018; revised May 2, 2018. of the VRP. The VRPTW often appears in many real M. Alzaqebah is with the Department of Mathematics, Faculty of Science, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia (e-mail: applications. VRPTW can be also used to achieve the [email protected]). following objectives: optimize the number of vehicles that S. Jawarneh is the Department of Computer Science, Community College must serve a set of customers, to determine the customers that Dammam, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia (e-mail: [email protected]). each vehicle should serve, and to determine the order of cost H. M. Sarim and S. Abdullah are with Data Mining and Optimisation that should be minimised as much as possible. The VRPTW is Research Group, Centre for Artificial Intelligence Technology, Universiti subject to capacity of the vehicle and service time constraints. Kebangsaan Malaysia, Selangor, Malaysia (e-mail: [email protected], [email protected]). The problem contains vehicle tasks that minimise task cost. doi: 10.18178/ijmlc.2018.8.3.693 236 International Journal of Machine Learning and Computing, Vol. 8, No. 3, June 2018 According to the definition above, the VRPTW processes a [13]. The general pseudo-code for the BA is shown in Fig. 1. specific number of requests to be visited by a certain number of vehicles. All customers are supposed to require a limited Initialization: quantity of demands. Furthermore, each route must start and NumOfIte : maximum number of iteration; populationsize : #of solutions in the population; finish at depot, and each customer must be visited only once. ne: #of elite solutions; The main formulation of the VRPTW was proposed in nre: list of recruited bees assigned to elite solutions; 1987 by Solomon [2]. The standard benchmark of Solomon’s nb: #of best solutions; nrb: list of recruited bees assigned to best solutions; datasets contains 56 instances, and every instance has 100 stlim: stagnation iteration limit for abandonment solutions; customers. In the network, each vehicle represents one route. The vehicle starts routing from the departure point and arrives Initialise the population; Compute the initial value of the fitness, f(Sol); iteration ← 0; at specific destinations (customers), then returns to the best solution, Solbest ← Sol; departure point. In considering Solomon’s benchmark, each Improvement: route has its own cost cij and the traveling time tij. Each do while (iteration < NumOfIte) vehicle spends one-time slot (time unit) to travel one distance Rank all solutions in the population based on fitness value. nbSet = the top nb solutions of population; unit, so the speed of the vehicle is assumed to be constant. neSet = the top ne solutions of nbSet; The time window boundaries are defined by the earliest and for (i=1 : nrb) latest arrival times (the time interval); in which the vehicle for (j=1 : nb) Sol* ← neSetj; must arrive at the customer’s place before the latest arrival A random neighbourhood method is applied on Sol* ; time. The vehicle should wait in cases where it arrives before Update the population; the earliest arrival time. The service time of the customer must end for be taking into account, which represents the time that is spent end for for (j=1 : nre) to load or unload demands. The demands’ size is considered for (h=1 : nrb) unique for all customers. Route time (time window) is the Sol** ← neSetj; time taken by vehicles to visit all customers in the same route. random neighbourhood method is applied on Sol** ; end for Equation 1 represents the main function for VRPTW [2]: end for Solbest ← the best solution found so far; N N N random neighbourhood method is applied on the rest of (nb–ne) Minimise xc (1) ijkij solutions; i0j 0k 1 Update the population; iteration++; end do where, Fig. 1. Pseudo-code for the BA. 0 if there is no arc from node i to node j x i j,i, j {01 , , N} ijk 1 otherwise As can be observed by Fig. 1, the algorithm begins with N number of customers randomly initializing the population, where the number of scout bees is equal to the number of solutions in the K number of vehicles population. The scout bees randomly search for the food sources (solutions). Once they find the food sources, they cij cost on arc between two customers i to j. return to the hive and start recruiting other bees to exploit those food sources. Next, all solutions in the population are III. BEES ALGORITHM (BA) evaluated and ranked based on a fitness function. A number of nb highest ranked solutions are selected for a local search. BA was firstly proposed by Pham et al. [13] and algorithm Then, the scout bees recruit the foragers to search on the has been applied in various optimization fields [18]-[22]. neighbourhood of the selected solutions, as follows: BA consists of two groups of bees i.e., scout bees and Every scout bee returned from each nb (the best solutions) foragers. The scout bees are involved in searching for food will recruit the nest (the top nb solutions from the sources (solutions) and recruiting foragers to continue population) foragers for a local exploration. searching in the discovered food sources. The scout bees search for food sources randomly in the search space The scout bees that visited the first ne elite solutions (Exploration). During the search, they collect useful among the best nb solutions will recruit nre foragers and information and return back to the hive to share the apply a random neighbourhood structure. information with other bees in the hive. and the scouts then The scout bees that visited the rest (nb–ne) solutions will recruit other bees as foragers to continue searching in recruit nrb < nre foragers and apply a random previously discovered food sources (Exploitation).
Load more

Recommended publications
  • An Exhaustive Review of Bio-Inspired Algorithms and Its Applications for Optimization in Fuzzy Clustering
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2021 doi:10.20944/preprints202103.0282.v1 Article An Exhaustive Review of Bio-Inspired Algorithms and its Applications for Optimization in Fuzzy Clustering Fevrier Valdez1, Oscar Castillo1,* and Patricia Melin1 1 Division Graduate of Studies, Tijuana Institute of Technology, Calzada Tecnologico S/N, Tijuana, 22414, BC, Mexico. Tel.: 664 607 8400 * Correspondence: [email protected]; Tel.: 664 607 8400 1 Abstract: In recent years, new metaheuristic algorithms have been developed taking as reference 2 the inspiration on biological and natural phenomena. This nature-inspired approach for algorithm 3 development has been widely used by many researchers in solving optimization problem. These 4 algorithms have been compared with the traditional ones algorithms and have demonstrated to 5 be superior in complex problems. This paper attempts to describe the algorithms based on nature, 6 that are used in fuzzy clustering. We briefly describe the optimization methods, the most cited 7 nature-inspired algorithms published in recent years, authors, networks and relationship of the 8 works, etc. We believe the paper can serve as a basis for analysis of the new are of nature and 9 bio-inspired optimization of fuzzy clustering. 10 11 Keywords: FUZZY; CLUSTERING; OPTIMIZATION ALGORITHMS 12 1. Introduction 13 Optimization is a discipline for finding the best solutions to specific problems. 14 Every day we developed many actions, which we have tried to improve to obtain the 15 best solution; for example, the route at work can be optimized depending on several 16 factors, as traffic, distance, etc. On other hand, the design of the new cars implies an 17 optimization process with many objectives as wind resistance, reduce the use of fuel, Citation: Valdez, F.; Castillo, O.; Melin, 18 maximize the potency of motor, etc.
    [Show full text]
  • An Improved Bees Algorithm Local Search Mechanism for Numerical Dataset
    An Improved Bees Algorithm Local Search Mechanism for Numerical dataset Prepared By: Aras Ghazi Mohammed Al-dawoodi Supervisor: Dr. Massudi bin Mahmuddin DISSERTATION SUBMITTED TO THE AWANG HAD SALLEH GRADUATE SCHOOL OF COMPUTING UUM COLLEGE OF ARTS AND SCIENCES UNIVERSITI UTARA MALAYSIA IN PARITIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER IN (INFORMATION TECHNOLOGY) 2014 - 2015 Permission to Use In presenting this dissertation report in partial fulfilment of the requirements for a postgraduate degree from University Utara Malaysia, I agree that the University Library may make it freely available for inspection. I further agree that permission for the copying of this report in any manner, in whole or in part, for scholarly purpose may be granted by my supervisor(s) or, in their absence, by the Dean of Awang Had Salleh Graduate School of Arts and Sciences. It is understood that any copying or publication or use of this report or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to University Utara Malaysia for any scholarly use which may be made of any material from my report. Requests for permission to copy or to make other use of materials in this project report, in whole or in part, should be addressed to: Dean of Awang Had Salleh Graduate School of Arts and Sciences UUM College of Arts and Sciences University Utara Malaysia 06010 UUM Sintok i Abstrak Bees Algorithm (BA), satu prosedur pengoptimuman heuristik, merupakan salah satu teknik carian asas yang berdasarkan kepada aktiviti pencarian makanan lebah.
    [Show full text]
  • Title: a Memory-Integrated Artificial Bee Algorithm for Heuristic Optimisation
    Title: A memory-integrated artificial bee algorithm for heuristic optimisation Name: T. Bayraktar This is a digitised version of a dissertation submitted to the University of Bedfordshire. It is available to view only. This item is subject to copyright. A MEMORY-INTEGRATED ARTIFICIAL BEE ALGORITHM FOR HEURISTIC OPTIMISATION T. BAYRAKTAR MSc. by Research 2014 UNIVERSITY OF BEDFORDSHIRE 1 A MEMORY-INTEGRATED ARTIFICIAL BEE ALGORITHM FOR HEURISTIC OPTIMISATION by T. BAYRAKTAR A thesis submitted to the University of Bedfordshire in partial fulfilment of the requirements for the degree of Master of Science by Research February 2014 2 A MEMORY-INTEGRATED ARTIFICIAL BEE ALGORITHM FOR HEURISTIC OPTIMISATION T. BAYRAKTAR ABSTRACT According to studies about bee swarms, they use special techniques for foraging and they are always able to find notified food sources with exact coordinates. In order to succeed in food source exploration, the information about food sources is transferred between employed bees and onlooker bees via waggle dance. In this study, bee colony behaviours are imitated for further search in one of the common real world problems. Traditional solution techniques from literature may not obtain sufficient results; therefore other techniques have become essential for food source exploration. In this study, artificial bee colony (ABC) algorithm is used as a base to fulfil this purpose. When employed and onlooker bees are searching for better food sources, they just memorize the current sources and if they find better one, they erase the all information about the previous best food source. In this case, worker bees may visit same food source repeatedly and this circumstance causes a hill climbing in search.
    [Show full text]
  • An Improved Bees Algorithm for Training Deep Recurrent Networks for Sentiment Classification
    S S symmetry Article An Improved Bees Algorithm for Training Deep Recurrent Networks for Sentiment Classification Sultan Zeybek 1,* , Duc Truong Pham 2, Ebubekir Koç 3 and Aydın Seçer 4 1 Department of Computer Engineering, Fatih Sultan Mehmet Vakif University, Istanbul 34445, Turkey 2 Department of Mechanical Engineering, University of Birmingham, Birmingham B15 2TT, UK; [email protected] 3 Department of Biomedical Engineering, Fatih Sultan Mehmet Vakif University, Istanbul 34445, Turkey; [email protected] 4 Department of Mathematical Engineering, Yildiz Technical University, Istanbul 34220, Turkey; [email protected] * Correspondence: [email protected] Abstract: Recurrent neural networks (RNNs) are powerful tools for learning information from temporal sequences. Designing an optimum deep RNN is difficult due to configuration and training issues, such as vanishing and exploding gradients. In this paper, a novel metaheuristic optimisation approach is proposed for training deep RNNs for the sentiment classification task. The approach employs an enhanced Ternary Bees Algorithm (BA-3+), which operates for large dataset classification problems by considering only three individual solutions in each iteration. BA-3+ combines the collaborative search of three bees to find the optimal set of trainable parameters of the proposed deep recurrent learning architecture. Local learning with exploitative search utilises the greedy selection strategy. Stochastic gradient descent (SGD) learning with singular value decomposition (SVD) aims to Citation: Zeybek, S.; Pham, D. T.; handle vanishing and exploding gradients of the decision parameters with the stabilisation strategy Koç, E.; Seçer, A. An Improved Bees of SVD. Global learning with explorative search achieves faster convergence without getting trapped Algorithm for Training Deep at local optima to find the optimal set of trainable parameters of the proposed deep recurrent learning Recurrent Networks for Sentiment architecture.
    [Show full text]
  • University of Birmingham Bees Algorithm for Multimodal Function
    University of Birmingham Bees algorithm for multimodal function optimisation Zhou, Z.; Xie, Y.; Pham, D.; Kamsani, S.; Castellani, M. DOI: 10.1177/0954406215576063 License: None: All rights reserved Document Version Peer reviewed version Citation for published version (Harvard): Zhou, Z, Xie, Y, Pham, D, Kamsani, S & Castellani, M 2015, 'Bees algorithm for multimodal function optimisation', Institution of Mechanical Engineers. Proceedings. Part C: Journal of Mechanical Engineering Science . https://doi.org/10.1177/0954406215576063 Link to publication on Research at Birmingham portal General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. •Users may freely distribute the URL that is used to identify this publication. •Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. •User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?) •Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.
    [Show full text]
  • A Grouped Version of the Bees Algorithm
    computers Article Grouped Bees Algorithm: A Grouped Version of the Bees Algorithm Hamid Reza Nasrinpour 1, Amir Massah Bavani 2,* and Mohammad Teshnehlab 3 1 Electrical and Computer Engineering Department, University of Manitoba, Winnipeg, MB R3T 5V6, Canada; [email protected] 2 Electrical and Computer Engineering Faculty, Technical University of Kaiserslautern, 67663 Kaiserslautern, Germany 3 Electrical and Computer Engineering Faculty, K.N. Toosi University of Technology, Tehran 163171419, Iran; [email protected] * Correspondence: [email protected]; Tel.: +49-152-3893-7661 Academic Editor: Kartik Gopalan Received: 21 November 2016; Accepted: 24 January 2017; Published: 28 January 2017 Abstract: In many non-deterministic search algorithms, particularly those analogous to complex biological systems, there are a number of inherent difficulties, and the Bees Algorithm (BA) is no exception. The BA is a population-based metaheuristic search algorithm inspired by bees seeking nectar/pollen. Basic versions and variations of the BA have their own drawbacks. Some of these drawbacks are a large number of parameters to be set, lack of methodology for parameter setting and computational complexity. This paper describes a Grouped version of the Bees Algorithm (GBA) addressing these issues. Unlike its conventional version, in this algorithm bees are grouped to search different sites with different neighbourhood sizes rather than just discovering two types of sites, namely elite and selected. Following a description of the GBA, the results gained for 12 well-known benchmark functions are presented and compared with those of the basic BA, enhanced BA, standard BA and modified BA to demonstrate the efficacy of the proposed algorithm.
    [Show full text]
  • The Bees Algorithm – a Novel Tool for Complex Optimisation Problems
    Intelligent Production Machines and Systems D.T. Pham, E.E. Eldukhri and A.J. Soroka (eds) © 2006 Cardiff University, Manufacturing Engineering Centre, Cardiff, UK. Published by Elsevier Ltd. All rights reserved. The Bees Algorithm – A Novel Tool for Complex Optimisation Problems D.T. Pham, A. Ghanbarzadeh, E. Koç, S. Otri , S. Rahim , M. Zaidi Manufacturing Engineering Centre, Cardiff University, Cardiff CF24 3AA, UK Abstract A new population-based search algorithm called the Bees Algorithm (BA) is presented. The algorithm mimics the food foraging behaviour of swarms of honey bees. In its basic version, the algorithm performs a kind of neighbourhood search combined with random search and can be used for both combinatorial optimisation and functional optimisation. This paper focuses on the latter. Following a description of the algorithm, the paper gives the results obtained for a number of benchmark problems demonstrating the efficiency and robustness of the new algorithm. Keywords: Bees Algorithm, Function Optimisation, Swarm Intelligence 1. Introduction 2. Intelligent swarm-based optimisation Many complex multi-variable optimisation Swarm-based optimisation algorithms (SOAs) problems cannot be solved exactly within mimic nature’s methods to drive a search towards the polynomially bounded computation times. This optimal solution. A key difference between SOAs generates much interest in search algorithms that and direct search algorithms such as hill climbing find near-optimal solutions in reasonable running and random walk is that SOAs use a population of times. The swarm-based algorithm described in this solutions for every iteration instead of a single paper is a search algorithm capable of locating good solution. As a population of solutions is processed in solutions efficiently.
    [Show full text]
  • Swarm Intelligence in the Multi-Agent Environment A
    SWARM INTELLIGENCE IN THE MULTI-AGENT ENVIRONMENT A Master’s Thesis Presented to Department of Computer and Information Sciences SUNY Polytechnic Institute Utica, New York In Partial Fulfilment of the Requirements for the Master of Science Degree By Sravani Bollam December 2016 © Sravani Bollam 2016 ACKNOWLEDGEMENTS The completion of this thesis could not have been possible without the participation and assistance of my Professor Dr. Saumendra Sengupta. His contributions are genuinely appreciated and acknowledged. I would particularly like to thank my parents, Naveen Kumar and Madhavi Bollam who stood by me and gave me immense support by understanding my dreams. And I would also like to thank my siblings and friends for inspiring me and by sharing their support either morally or physically. ii ABSTRACT This thesis is focused on the use of naturally occurring concept of “swarm intelligence” to multi- agent systems, namely the relatively new system-theoretic framework known as Swarm Intelligence Systems (SIS). In this work, outlined is the general framework of an interacting agents behaving as a swarm group on a mission to deliver a service such as transporting a set of goods from a given starting point to a destination. As an example, one could think of a team of drones, for instance, providing a delivery service in a milieu where normal transportation mode may be prohibitively expensive. The entire logistic, in detail as a commercial project, is too involved for our thesis focus; instead, some of the structural interaction issues with a group are outlined and discussed here to articulate the behavior of multi-agent based swarm as a group necessary to function as a distributed entity somewhat different from the standard swarm models found in literature.
    [Show full text]
  • Bees and Firefly Algorithms for Noisy Non-Linear Optimisation Problems
    Bees and Firefly Algorithms for Noisy Non-Linear Optimisation Problems N. Chai-ead, P. Aungkulanon*, and P. Luangpaiboon, Member, IAENG engineering practice is much more challenging because of Abstract — Effective methods for solving the complex and the huge, complex and noisy solution space. Finding exact noisy engineering problems using a finite sequence of solutions to these problems turn out to be NP-hard. This instructions can be categorised into optimisation and meta- kind of complex problem requires an exponential amount of heuristics algorithms. The latter might be defined as an computing power and time, as the number of decision iterative search process that efficiently performs the variables increases. To overcome these problems, exploration and exploitation in the solution space aiming to researchers have proposed approximate evolutionary-based efficiently find near optimal solutions. Various natural intelligences and inspirations have been adopted into the or meta-heuristics algorithms as a means to search for near- iterative process. In this work, two types of meta-heuristics optimal solutions [1]. Generally, meta-heuristics work as follows: a population called Bees and Firefly algorithms were adapted to find optimal solutions of noisy non-linear continuous mathematical of individuals is randomly initialised where each individual models. Considering the solution space in a specified region, represents a potential solution to the problem. The quality of some models contain global optimum and multiple local each solution is then evaluated via a fitness function. A optimums. Bees algorithm is an optimisation algorithm selection process is applied during the iteration of meta- inspired by the natural foraging behaviour of honey bees.
    [Show full text]
  • New Algorithm for Evaluating the Green Supply Chain Performance in an Uncertain Environment
    sustainability Article New Algorithm for Evaluating the Green Supply Chain Performance in an Uncertain Environment Pan Liu and Shuping Yi * College of Mechanical Engineering, Chongqing University, Shazheng Road 174, Chongqing 400044, China; [email protected] * Correspondence: [email protected]; Tel.: +86-23-6510-6939 Academic Editor: Marc A. Rosen Received: 29 April 2016; Accepted: 12 September 2016; Published: 26 September 2016 Abstract: An effective green supply chain (GSC) can help an enterprise obtain more benefits and reduce costs. Therefore, developing an effective evaluation method for GSC performance evaluation is becoming increasingly important. In this study, the advantages and disadvantages of the current performance evaluations and algorithms for GSC performance evaluations were discussed and evaluated. Based on these findings, an improved five-dimensional balanced scorecard was proposed in which the green performance indicators were revised to facilitate their measurement. A model based on Rough Set theory, the Genetic Algorithm, and the Levenberg Marquardt Back Propagation (LMBP) neural network algorithm was proposed. Next, using Matlab, the Rosetta tool, and the practical data of company F, a case study was conducted. The results indicate that the proposed model has a high convergence speed and an accurate prediction ability. The credibility and effectiveness of the proposed model was validated. In comparison with the normal Back Propagation neural network algorithm and the LMBP neural network algorithm, the proposed model has greater credibility and effectiveness. In practice, this method provides a more suitable indicator system and algorithm for enterprises to be able to implement GSC performance evaluations in an uncertain environment. Academically, the proposed method addresses the lack of a theoretical basis for GSC performance evaluation, thus representing a new development in GSC performance evaluation theory.
    [Show full text]
  • Constrained Eigenvalue Minimization of Incomplete Pairwise Comparison Matrices by Nelder-Mead Algorithm
    algorithms Article Constrained Eigenvalue Minimization of Incomplete Pairwise Comparison Matrices by Nelder-Mead Algorithm Hailemariam Abebe Tekile * , Michele Fedrizzi and Matteo Brunelli Department of Industrial Engineering, University of Trento, 38123 Trento, Italy; [email protected] (M.F.); [email protected] (M.B.) * Correspondence: [email protected] Abstract: Pairwise comparison matrices play a prominent role in multiple-criteria decision-making, particularly in the analytic hierarchy process (AHP). Another form of preference modeling, called an incomplete pairwise comparison matrix, is considered when one or more elements are missing. In this paper, an algorithm is proposed for the optimal completion of an incomplete matrix. Our intention is to numerically minimize a maximum eigenvalue function, which is difficult to write explicitly in terms of variables, subject to interval constraints. Numerical simulations are carried out in order to examine the performance of the algorithm. The results of our simulations show that the proposed algorithm has the ability to solve the minimization of the constrained eigenvalue problem. We provided illustrative examples to show the simplex procedures obtained by the proposed algorithm, and how well it fills in the given incomplete matrices. Keywords: constrained eigenvalue minimization; incomplete pairwise comparison matrix; Nelder-Mead algorithm; AHP Citation: Tekile, H.A.; Fedrizzi, M.; Brunelli, M. Constrained Eigenvalue Minimization of Incomplete Pairwise Comparison Matrices by 1. Introduction Nelder-Mead Algorithm. Algorithms Increasingly complex decisions are being made everyday and especially in positions 2021, 14, 222. https://doi.org/ of high responsibility, they must be “defensible” in front of stakeholders. In this context, 10.3390/a14080222 decision analysis can be seen as a set of formal tools which can help decision makers articulate their decisions in a more transparent and justifiable way.
    [Show full text]
  • Comprehensive Taxonomies of Nature- and Bio-Inspired Optimization: Inspiration Versus Algorithmic Behavior, Critical Analysis and Recommendations
    Comprehensive Taxonomies of Nature- and Bio-inspired Optimization: Inspiration versus Algorithmic Behavior, Critical Analysis and Recommendations Daniel Molina*1, Javier Poyatos1, Javier Del Ser2,3,4, Salvador García1, Amir Hussain5, and Francisco Herrera1,6 1 Department of Computer Science, DaSCI Andalusian Institute of Data Science and Computation Intelligence, University of Granada, Spain, {[email protected], [email protected], [email protected], [email protected]} 2 TECNALIA, Spain, [email protected] 3 Dept. of Communications Engineering, University of the Basque Country (UPV/EHU), Spain 4 Basque Center for Applied Mathematics (BCAM), Spain 5 Edinburgh Napier University, United Kingdom, [email protected] 6 Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia Abstract Introduction: In recent years, a great variety of nature- and bio-inspired algorithms has been reported in the literature. This algorithmic family simulates different biological processes observed in Nature in order to efficiently address complex optimization problems. In the last years the number of bio-inspired optimization approaches in literature has grown considerably, reaching unprecedented levels that dark the future prospects of this field of research. Method: This paper addresses this problem by proposing two comprehensive, principle-based taxonomies that allow researchers to organize existing and future algorithmic developments into well-defined categories, considering two different criteria: the source of inspiration and the behavior of each algorithm. Using these taxonomies we review more than three hundred publications dealing with nature-inspired and bio-inspired algorithms, and proposals falling within each of these categories are examined, leading to a critical summary arXiv:2002.08136v3 [cs.AI] 30 Apr 2021 of design trends and similarities between them, and the identification of the most similar classical algorithm for each reviewed paper.
    [Show full text]